Left: drill-string with rigid rod for a drill-pipe to ... 3.9 (a) An incremental rotary quadrature encoder with 500 pulses per revolution. ..... amplitude p. The PDF of R of the un-controlled wave-excited ...... ergy Reviews, 6(5):405â431, 2002. [29] M.J. ...
Dynamics and Control of Nonlinear Engineering Systems A thesis presented for the degree of Doctor of Philosophy at the University of Aberdeen
Seyed Vahid Vaziri Hamaneh MSc in Complex Systems Engineering (Henri Poincar´e University)
2015
I hereby declare that this thesis has been composed by myself, that it has not been submitted in any previous application for a degree and the work described in this thesis has been performed by myself.
Abstract This thesis is focused on the dynamics and control of nonlinear engineering systems. A developed approach is applied to three specific problems: suppression of torsional vibrations occurring in a drill-string, lateral vibrations on an unbalanced rotor and vibrational energy extraction from rotating pendulum systems. The first problem deals with drill-string torsional vibrations while drilling, which is conducted in the experimental drilling rig developed at University of Aberdeen. A realistic model of the experimental setup is then constructed, taking into account the dynamics of the drill-string and top motor. Physical parameters of the experimental drilling rig are estimated in order to calibrate the model to ensure the correspondence of the research results to the experimental conditions. Consequently, a control method is introduced to suppress torsional and stick-slip oscillations exhibited in the experimental drilling rig. The experimental and numerical results considering delay of the actuator are shown to be in close agreement, including the success of the controller in significantly reducing the vibrations. In the second problem a soft impact oscillator approach is used to study the dynamics of the asymmetric Jeffcott rotor. A realistic model of the experimental setup is developed, taking into account an asymmetric physical configuration in rotor part as well as snubber rig. Several experimental bifurcation diagrams are conducted with different conditions in range around the grazing point. Experimental and numerical results based on the proposed model are compared and shown to be in close agreement. The last problem relates to initiating and maintaining the rotational motion of a parametric pendulum as an energy harvesting system. Several possible control methods to initiate and maintain the rotational motion of a harmonically-excited pendulum are proposed and then verified experimentally. The time-delayed feedback method is shown to maintain quite well the rotational motion of a sinusoidally excited parametric pendulum, even in the presence of noise. A control method for the wave-excited pendulum system is then suggested and tested in order to increase the probability of its rotational motion. This proposed control method succeeds in significantly raising the probability of rotational motion of the wave-excited pendulum.
Acknowledgments
I would like to thank my supervisor Professor Marian Wiercigroch for giving me the opportunity to work on this project, for his support, encouragement, advice, feedback and constructive criticism throughout my PhD. I also want to express my gratitude to Dr Anna Najdecka, Dr Joseph P´aez Ch´avez and Dr Yang Liu for their fruitful collaborations in different parts of this project. Throughout my studies I have received useful advice and support from many of my colleagues and friends (Dr Nandakumar Krishnan, Dr Richard Morrison, Professor Ekaterina Pavlovskaia, Dr Richard Neilson, Dr Marcos Silveira and Dr Aline Souza De Paula), whose contributions are highly appreciated. I am particularly grateful to Marcin Kapitaniak for his invaluable help and collaboration. I would also like to thank the technicians team from the Engineering Department, in particular Alan Styles and Alistair Robertson for their help with the experimental studies. Also, I wish to thank David Jason for helping me to keep the right words in the right places in this thesis. I would also like to acknowledge the financial support from the RED project, BG group and University of Aberdeen. Lastly, of course, I would like to express gratitude to my supportive parents, Seyed Kazem and Fatemeh, and my family, for encouraging and motivating me throughout the years, and to my numerous friends, who have been there for me through thick and thin.
Contents 1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Aim of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3
1 1 5
Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Literature review 2.1 Drill-string vibrations . . . . . . . . . . . . . . . . . . . . . . . . .
8 8
2.1.1 2.1.2 2.1.3
Torsional vibrations . . . . . . . . . . . . . . . . . . . . . . Coupled vibrations . . . . . . . . . . . . . . . . . . . . . . Drill-Bit and rock models . . . . . . . . . . . . . . . . . .
10 12 13
2.1.4 Experimental drilling rigs . . . . . . . . . . . . . . . . . . 2.1.5 Suppression of torsional vibration . . . . . . . . . . . . . . Lateral vibrations on unbalanced rotor . . . . . . . . . . . . . . .
15 16 19
2.2.1 2.2.2
Recent background . . . . . . . . . . . . . . . . . . . . . . Vibration suppression methods . . . . . . . . . . . . . . .
21 21
2.2.3 Rotor with snubber ring . . . . . . . . . . . . . . . . . . . 2.2.4 Background of this work in CADR . . . . . . . . . . . . . Rotational motion of parametric pendulum . . . . . . . . . . . . .
21 22 23
2.3.1 2.3.2
Renewable and wave energy harvesting systems . . . . . . Parametric pendulum as energy harvesting system . . . . .
23 25
2.3.3 2.3.4 2.3.5
Dynamics of parametric pendulum . . . . . . . . . . . . . Rotational response of parametric pendulum . . . . . . . . Rotational control of parametric pendulum . . . . . . . . .
25 26 27
2.3.6
Wave-excited pendulum . . . . . . . . . . . . . . . . . . .
27
3 Torsional vibration of drill-string 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29 30
2.2
2.3
3.2
Experimental rig . . . . . . . . . . . . . . . . . . . . . . . . . . . v
30
Contents
3.3
3.4
3.5
3.2.1
General overview of the drilling rig . . . . . . . . . . . . .
31
3.2.2 3.2.3
Drilling machine . . . . . . . . . . . . . . . . . . . . . . . Drill-string . . . . . . . . . . . . . . . . . . . . . . . . . .
33 34
3.2.4 3.2.5 3.2.6
Static force (Weight-On-Bit) . . . . . . . . . . . . . . . . . Rock samples, fixtures and cutting fluid system . . . . . . Instrumentations and control . . . . . . . . . . . . . . . .
36 37 38
Physical parameters identification . . . . . . . . . . . . . . . . . . 3.3.1 Flexible shaft parameters . . . . . . . . . . . . . . . . . . .
42 42
3.3.2
Experimental identification of bit-rock interaction . . . . . 3.3.2.1 Experimental torque modelling . . . . . . . . . . Experimental results . . . . . . . . . . . . . . . . . . . . . . . . .
44 49 51
3.4.1 3.4.2
Torsional vibrations . . . . . . . . . . . . . . . . . . . . . . Stick-slip oscillations . . . . . . . . . . . . . . . . . . . . .
51 53
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
4 Suppression of torsional vibration 4.1
4.2
4.3
4.4
4.5 4.6 4.7
57
Harmonically excited torsional model . . . . . . . . . . . . . . . . 4.1.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . .
58 58
4.1.1.1 Torque on bit . . . . . . . . . . . . . . . . . . . . 4.1.2 Numerical results and experimental verification . . . . . . Modelling of motor and gearing system . . . . . . . . . . . . . . .
60 62 65
4.2.1 4.2.2
Torque control of motor . . . . . . . . . . . . . . . . . . . Moment of inertia and damping coefficient of motor . . . .
66 69
Two disks torsional model . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . 4.3.1.1 Torque on bit . . . . . . . . . . . . . . . . . . . .
71 72 73
4.3.2 Numerical results and experimental verification . . . . . . Suppressing the torsional vibrations . . . . . . . . . . . . . . . . .
74 77
4.4.1
Sliding-mode-control . . . . . . . . . . . . . . . . . . . . . 4.4.1.1 Controller structure and sliding surface . . . . . . 4.4.1.2 Parameter uncertainties . . . . . . . . . . . . . .
78 80 81
4.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . Experimental verification of the control method . . . . . . . . . .
83 85
Modelling delay in the actuator . . . . . . . . . . . . . . . . . . . 87 4.6.1 Sensitivities in parameter estimations in experimental study 92 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 vi
Contents 5 Lateral vibration of unbalanced rotor 5.1 5.2
5.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental rotor system . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Rig description . . . . . . . . . . . . . . . . . . . . . . . .
96 97 99 99
5.2.2 Instrumentation and data acquisition . . . . . . . . . . . . 101 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4
Physical model and equations of motion . . . . . . . . . . . . . . 107 5.4.1 Snubber ring motion and contact force . . . . . . . . . . . 110 5.4.2 Numerical implementation . . . . . . . . . . . . . . . . . . 113
5.5 5.6 5.7
Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . 116 Comparison between experimental result and the model . . . . . . 118 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6 Rotational motion of a parametric pendulum
123
6.1 6.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Experimental rig . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 6.4
Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . 125 Dynamics of the harmonically excited parametric pendulum . . . 128 6.4.1 Choice of control parameter . . . . . . . . . . . . . . . . . 129
6.5
Initiating the rotational motion . . . . . . . . . . . . . . . . . . . 130 6.5.1 Bang-Bang method . . . . . . . . . . . . . . . . . . . . . . 130
6.6
6.5.2 Velocity comparison control . . . . . . . . . . . . . . . . . 131 6.5.3 Time-delayed feedback method . . . . . . . . . . . . . . . 134 Maintaining rotational motion by TDF method . . . . . . . . . . 136
6.7
6.6.1 Robustness of TDF towards noise . . . . . . . . . . . . . . 137 Wave displacement function . . . . . . . . . . . . . . . . . . . . . 138
6.8 Dynamic of wave-excited pendulum . . . . . . . . . . . . . . . . . 140 6.9 Defining measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.10 Rotational control for wave-excited pendulum . . . . . . . . . . . 146 6.10.1 Error signal; predictive control . . . . . . . . . . . . . . . . 147 6.10.2 Peak detection . . . . . . . . . . . . . . . . . . . . . . . . 148 6.10.3 Control signal; wait-and-act predictive control . . . . . . . 149 6.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7 Conclusions and recommendations for future work 157 7.1 Torsional vibration of drill-string . . . . . . . . . . . . . . . . . . 158
vii
Contents 7.1.1 7.2 7.3
Recommendations . . . . . . . . . . . . . . . . . . . . . . . 159
Lateral vibration of unbalanced rotor . . . . . . . . . . . . . . . . 160 7.2.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . 162 Rotational motion of parametric pendulum . . . . . . . . . . . . . 162 7.3.1
Recommendations . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography
164
Appendices
186
A Experimental torque modelling
187
B Existence and uniqueness of the solution of Eq. (5.3)
189
viii
List of Figures 2.1
2.2
A schematic of a typical rotary drilling rig, listing essential parts. A drill-string might be a few kilometers long, including the BottomHole-Assembly (BHA) with a maximum length of a few hundred meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photographs of the rig of [56]. The drill-string is stationary and the rock is given rotary motion. Torsional flexibility of the system is simulated using a gear-pulley-spring system. . . . . . . . . . . .
2.3
9
16
2.4
Anti Stick-slip Tool (AST). A sudden increase in torque (M2) will cause a contraction (S) to reduce the WOB (F2).(adopted from [10]). 19 A photograph of fan blades of a jet engine damaged due to an
2.5
excessive vibration (adopted from [1]). . . . . . . . . . . . . . . . Some of wave energy harvesting system: (a) Pendulor system (b)
2.6
Wave Dragon System Principle, (c) Salter’s Duck system (d) Pelamis Wave Energy Converter (adopted from [135]). . . . . . . . . . . . Working principle of parametrically-excited pendulum harvesting wave energy. Oscillations of the sea surface results in the rotation of the pendulum (adopted from [110]). . . . . . . . . . . . . . . .
3.1
24
25
General view of the drilling machine used in this study including (1) computer for data acquisition and control system, (2) breakout box for DAQ system, (3) charge amplifiers for sensors, (4) top AC motor, (5) pulley system and (6) frequency convertor for driving the motor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
20
31
(right) Photograph of the experimental rig and (left) schematic diagram of the experimental setup. The main components of the system are: sensors (top and bottom encoders, LVDT and 4component load cell), electric motor, flexible shaft, disks, BHA, drill-bit and rock sample. . . . . . . . . . . . . . . . . . . . . . . . ix
32
List of Figures 3.3
(a) Top motor, the gearing and pulley system and (b) frequency
3.4
convertor which has been used to control the speed or torque of the top motor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drill-string configurations. Common BHA, but varying drill-pipe
33
designs for different purposes. Left: drill-string with rigid rod for a drill-pipe to conduct characterization tests (bit-rock interactions: Section 3.3.2). Right: A Bowden cable or flexible shaft for a drill-pipe to conduct drill-string dynamics experiments (observing torsional vibrations: 3.4). . . . . . . . . . . . . . . . . . . . . . . . 3.5
3.6
3.7 3.8 3.9
34
Different types of drill-pipes and rocks; (a) rigid shaft, (b) flexible drill-pipe - a slender aluminium tube, (c) bowden cable (flexible shaft), (d) rock sample - limestone, (e) rock sample - sandstonequartz, (f) rock sample - granite, (g) rock sample - sandstone. . . Additional movable disks attached to the BHA in order to increase
35
the WOB. Four first disks weight 6.52 kg each and each of rest weights about 10.63 kg. This setup can be used in both configurations; with rigid and flexible shafts. . . . . . . . . . . . . . . . . .
36
Different types of drill-bits used in the experimental studies (from left to right: 2 34 ” PDC, 3 78 ” roller cone and 3 87 ” PDC drill-bits). .
37
Cooling and cleaning fluid system using water with anti-corrosion additive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) An incremental rotary quadrature encoder with 500 pulses per revolution. (b) Position transducer (P1010-60-NJC-DS-TS) used for axial position measurements. . . . . . . . . . . . . . . . . . . .
38
38
3.10 A close-up view of the two eddy current probes with a range of 5 mm, placed circumferentially 90 degrees apart on the periphery of the brass bushing for the loose bearing. The probes monitor lateral positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Kistler 9272 load-cell. (a) Four component dynamometer placed
39
beneath the rock-holder plate; (b) ICAM5073A charge amplifier. . 3.12 A photograph of the breakout box as an interface to NI PCIe 6321 DAQ card used for data acquisition and control purposes. . . . . .
40 40
3.13 A graphical interface of the Labview program for acquiring data from sensors and controlling the top motor. . . . . . . . . . . . .
41
x
List of Figures 3.14 (a) Internal structure of a flexible shaft used in the experiments, (b) setup for identification of the torsional stiffness and damping of a flexible shaft. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Identification of the torsional stiffness and damping coefficient of
43
a flexible shaft of φ10 mm in diameter as a function of the attached mass. (a) torsional stiffness in clockwise (blue square) and anti-clockwise (red circle) directions, (b) damping coefficient in clockwise (blue square) and anti-clockwise (red circle) directions. Average values of clockwise and anti-clockwise directions are denoted by ’+’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Time histories of axial displacement, TOB, WOB and angular velocity of drilling by 2 34 ” PDC bit for sandstone, using water as a
44
cleaning and cooling fluid. The average of TOB, measured in the window 202 s to 435 s is considered to be the ’expected TOB’ when drilling in the same bit speed and conditions (Fig. 3.17). . . . . . 3.17 Large scale view of the time histories of the steady state drilling starting while the drill-bit is already 20mm inside the rock. The
46
average T OB = 15.6 Nm, W OB = 3.81 kN and RP M = 68.2 rpm. 48 3.18 Experimental identification of TOB curves and their fitted curves for different values of WOB (colour map). These curves are conducted for interaction of 3 78 ” PDC drill-bit and sandstone. . . . . 3.19 Equivalent friction coefficient as a function of bit rotational speed
48
for different values of WOB (colour map). These curves are calculated based on Equation (A.1) interaction of 3 87 ” PDC drill-bit and sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 (left) Angular velocities of top motor (blue) and drill-bit (green) and (right) phase portraits of drilling experiment with (a) 20 disks
50
and straight flexible shaft, (b) 12 disks and 1.5” pre-buckled flexible shaft and (c) 6 disks and straight flexible shaft. A variety type of torsional vibrations are observed. . . . . . . . . . . . . . . . . . . 3.21 Drill-bit angular velocity measured during a typical stick-slip in a real field by the in-bit sensor and chimerical MWD vibration
52
monitor (adopted from [90]). . . . . . . . . . . . . . . . . . . . . .
53
xi
List of Figures 3.22 (left) Angular velocities of top motor (blue) and drill-bit (green) and (right) phase portraits of drilling experiment with (a) 16 disks and 1.5 inch pre-buckled flexible shaft, (b) 16 disks and straight flexible shaft, (c) 14 disks and straight flexible shaft and (d) 12 disks and straight flexible shaft. A variety of stick-slip vibrations are observed in these cases. Case (d) is typical whereas the rest are more complex. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23 (left) Angular velocities of top motor (blue) and drill-bit (green) and (right) phase portraits of drilling experiment with (a) 10 disks
54
and and straight flexible shaft, (b) 10 disks and 1.5 inch pre-buckled flexible shaft, (c) 8 disks and 1.5 inch pre-buckled flexible shaft and (d) 6 disks 1.5 inch pre-buckled flexible shaft. A variety of stick-slip vibrations are observed in these cases. Case (b) is typical whereas the rest are more complex. . . . . . . . . . . . . . . . . . . . . . . 4.1
55
Physical model of a 1-DOF lump mass torsional model with external excitation θt . The visco-elasto properties of the pipe are given by the damping-stiffness pair c, k. The reactive torque acting on
4.2
the system during drilling is represented by Tb . . . . . . . . . . . . The model has two phases: stick phase which includes 1 mode of
4.3
operation (mode A) and slip phase which has 2 modes (mode B & C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example of stick-slip oscillations occurring in the experimental
59
61
rig for Wb = 2.19 kN and the straight flexible shaft. The time histories of the angular velocities at the bottom, θ˙b , and the top, θ˙t , phase portraits from (a) experimental studies, (b) low-dimensional model and (c) FFT of the angular velocity of the top (left) and the drill-bit (right) for experimental (red curve) and model (green curve), where |θ˙t (f )| and |θ˙b (f )| denote amplitude of FFTs as a
4.4
4.5
function of frequency f . . . . . . . . . . . . . . . . . . . . . . . .
63
A physical model of the motor and gearing system as a disk (with moment of inertia Jt ) which is subject to a driving torque Tt and a viscous damping with coefficient of ct . . . . . . . . . . . . . . .
66
A schematic of DAQ and open loop torque control system for top motor. Tc , Tˆ are requested torque and estimated torque in [%] respectively, and Tt is the torque generated by motor in [Nm]. . . xii
67
List of Figures 4.6
4.7
A physical model of experiment employed to estimate the relationship between requested torque Tc , estimated torque Tˆ and generated torque Tt by the motor. . . . . . . . . . . . . . . . . . . . . . A physical model of a 2-DOF lump mass torsional system. The vis-
68
cous damping property of the motor and gearing system and the visco-elasto properties of the pipe are given by Ct , c and k, respec-
4.8
tively. The reactive torque acting on the system during drilling is represented by Tb . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example of stick-slip oscillations occurring in the experimental
72
rig for Wb = 1.79 kN and 1.5 inch pre-buckled flexible shaft. The time histories of the angular velocities at the bottom, θ˙b , and the top, θ˙t , phase portraits from (a) experimental studies, (b) lowdimensional model and (c) FFT of the angular velocity of the top (left) and the drill-bit (right) for experimental (red curve) and
4.9
model (green curve), where |θt (f )| and |θb (f )| denote amplitude of FFTs as a function of frequency f . . . . . . . . . . . . . . . . . . A zoomed-in view of Fig. 4.8 (a) experiential studies, (b) 2-DOF model together with TOB recorded in the experiment and modelled by Eq. (4.5) (blue curves). . . . . . . . . . . . . . . . . . . . . . .
75
76
4.10 The structure of the suggested sliding-mode controller for the model. In this method the controller changes the control parameter Tt in order to keep the top and bit speed θ˙t and θ˙b close to the desired rotational speed ωd . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Time histories of top angular velocity (black curve), drill-bit an-
80
gular velocity (green curve) and control signal (blue curve), and phase portrait of the simulation using sliding-mode controller with ωd = 3.1 rad/s and λ = 0.8. The controllers are switched on at t = 30 s. The stick-slip trajectory and the trajectory to the desired fixed point are shown in green and blue respectively in phase portrait. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
83
List of Figures 4.12 Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve) and control signal (blue curve), and phase portrait of the simulation using sliding-mode controller with ωd = 5 rad/s and λ = 1. The controllers are switched on at t = 30 s. The stick-slip trajectory and the trajectory to the desired fixed point are shown in green and blue respectively in phase portrait. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 The structure of the suggested sliding-mode controller for the experimental rig. In this method the controller changes the control parameter Tt in order to keep the top and bit speed θ˙t and θ˙b close
84
to the desired rotational speed ωd . . . . . . . . . . . . . . . . . . . 4.14 Time histories of top angular velocity (black curve), drill-bit angu-
86
lar velocity (red curve) and control signal (blue curve), and phase portrait of the drilling experiment using sliding-mode controller with ωd = 3.1 rad/s and λ = 0.8. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in red and blue respectively in phase portrait. . . . . . . . . . . .
86
4.15 Time histories of top angular velocity (black curve), drill-bit angular velocity (red curve) and control signal (blue curve), and phase portrait of the drilling experiment using sliding-mode controller with ωd = 5 rad/s and λ = 1. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in red and blue respectively in phase portrait. . . . . . . . . . . . . . 4.16 Time histories of the actuator input value Tc (blue curve) and the torque generated by motor Tˆ (red curve). . . . . . . . . . . . . . . 4.17 The structure of the suggested sliding-mode controller for the experimental rig with delay and dead-zone. A 0.4 s delay and a
87
minimum 22.62 Nm torque are observed in the motor. . . . . . . . 4.18 Time histories of top angular velocity (black curve), drill-bit an-
88
88
gular velocity (green curve) and control signal (blue curve), and phase portrait of a simulation considering a 0.4 s delay and minimum of 22.62 Nm torque in motor using sliding-mode controller with ωd = 3.1 rad/s and λ = 0.8. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in green and in blue in phase portrait. This result is very close to the experiment presented in Fig. 4.14. . . . . . . . . . . . . . . . . xiv
89
List of Figures 4.19 Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve) and control signal (blue curve), and phase portrait of a simulation considering a 0.4 s delay and minimum of 22.62 Nm torque in motor using sliding-mode controller with ωd = 5 rad/s and λ = 1. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in green and in blue in phase portrait. This result is very close to the experiment presented in Fig. 4.15. . . . . . . . . . . . . . . . . 4.20 Time histories of top angular velocity (black curve), drill-bit an-
90
gular velocity (red curve), and control signal (blue curve) of the drilling experiment activating the controller in two time windows [60.6, 110.46]s and [150.4, 210.2]s. The controller achieves elimination of the stick-slip oscillations in the drill-bit (red curve). All parameters used for this experiment are the same as the ones in Fig. 4.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve), and control signal (blue curve) of the
91
simulation activating the controller in two time windows [60.6, 110.46]s and [150.4, 210.2]s. The controller achieves elimination of the stick-slip oscillations in the drill-bit (green curve). All parameters used for this simulation are the same as the ones in Fig. 4.18. An excellent match observed between the simulation and the experiment presented in Fig. 4.20. . . . . . . . . . . . . . . . . . . . 4.22 Phase portraits of the drilling experiments using sliding-mode con-
92
troller. The uncontrolled stick-slip trajectories and the controlled cycle limits are shown in red and blue respectively. All estimated parameters, boundaries and controller parameters used in these experiments are presented in Table 4.3. The controller achieves (a) 47.86% (b) 59.26% (c) 51.52% (d) 57.58% (e) 66.72% (f) 64.72% reduction in vibrations. . . . . . . . . . . . . . . . . . . . . . . . .
xv
93
List of Figures 5.1
(a) A photograph of the rotor rig and (b) a schematic diagram of the experimental setup. The main components of the system are numbered as follows: 1, motor; 2, rotor with out-of-balance; 3, rotor housing; 4, snubber ring; 5, snubber ring frame; 6, flexural rods; 7, support block; 8, speed monitoring disc; 9, dashpot dampers; 10, clearance. . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2
A front page of the Labview program prepared to observe and demonstrate all measured signals, as well as phase diagrams and Poincar´e maps of the response in realtime. . . . . . . . . . . . . . 101
5.3
Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 20.7 g on the point B of Fig. 5.12(b) (517.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz. . . . . . . . . . . . 103
5.4
Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 20.7 g on the point A of Fig. 5.12(b) (517.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz. . . . . . . . . . . . 103
5.5
Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 22.3 g on the point A of Fig. 5.12(b) (557.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz. . . . . . . . . . . . 104
5.6
Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 22.3 g on the point B of Fig. 5.12(b) (557.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz. . . . . . . . . . . . 104
5.7
Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 19.1 g on the point B of Fig. 5.12(b) (477.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz. . . . . . . . . . . . 105
xvi
List of Figures 5.8
Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 19.1 g on the point A of Fig. 5.12(b) 477.5×10−6 kgm. In (a) rotor speed increased from 9 to 16.1 Hz,
5.9
in (b) rotor speed decreased from 16.1 to 9 Hz. . . . . . . . . . . . 105 Experimental bifurcation diagrams with a phase portrait and Poincar´e map showing the change of the system dynamics as rotor speed is varied, consists of 72 experiments with rotor speed increased from 9 to 16.1 Hz and with additional mass of 20.7 g on the point B of Fig. 5.12(b) (517.5×10−6 kgm). Highlighted responses: Period one (f = 11.7 Hz), Period three (f = 11.9 Hz), Period two (f = 12.2 Hz), Quasi-periodic (f = 13.7 Hz) and period two em-
bedded in a Quasi-periodic (f = 15.3 Hz). . . . . . . . . . . . . . 106 5.10 (a) A physical model of the Jeffcott rotor, (b) co-ordinate system used to derive the equations of motion. . . . . . . . . . . . . . . . 109 5.11 A geometrical configuration of the system during the contact mode. Normal and tangential unit vectors at the surface of contact are b and Tb respectively. The force exerted on the snubber denoted by N ring by its viscoelastic support is denoted by Fs , whose normal
component is transmitted to the rotor at the surface of contact. . 111 5.12 (a) An experimental rig showing the connection between the motor and the rotor, (b) an enlarged view of a rotor face showing the possible locations of additional masses, (c) the snubber ring supported by four compression springs (labeled 1–4 in the picture) attached to the snubber ring frame, (d) the frame with the attached mass to the snubber ring. . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.13 A natural response of the mass (see Fig. 5.12(d)) and the exponential fitted curve, which allows us to calculate the stiffness and damping coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.14 A comparison of the bifurcation diagrams, phase plots and time histories obtained from the mathematical model (5.1)–(5.2) (lefthand side panels) and experimental rig (right-hand side panels). The most significant dynamical responses along with their corresponding frequency intervals are summarized in table 5.3. . . . . . 119
xvii
List of Figures 6.1
(a) An experimental rig of vertically excited pendulum, (b) a schematic representation of vertically excited pendulum adopted from [111], (c) a schematic diagram illustrating the pendulum setup including measuring and controlling equipment, 1-pendulum, 2-belt-gear
6.2
assembly connected to the motor, 3-encoder. . . . . . . . . . . . . 126 (a) A front end of Labview program for data acquisition and con-
6.3
trol of the pendulum, (b) A front end of Labview program for controlling the electromagnetic shaker. . . . . . . . . . . . . . . . 127 Basins of attraction for (a) p = 0.005, ω = 2 and γθ = 0.01 (b) p = 0.02, ω = 2 and γθ = 0.01, showing co-existence of rotational and oscillatory attractors within the main resonance zone. △ marks period-two oscillation attractors (yellow) or fixed point
at the hanging down position (green). ◦ marks period-one pure rotation attractors [110]. . . . . . . . . . . . . . . . . . . . . . . 129 6.4
Experimental results showing the initiation of the rotation using the bang-bang method with k = 1 showing the pendulum displacement (red) and control signal (blue), (left) Unsuccessful initialization for sinusoidal excitation with amplitude of A = 0.300 V and f = 1.6 Hz and (right) successful initialization for sinusoidal
6.5
excitation with A = 0.400 V and f = 2 Hz. . . . . . . . . . . . . . 130 A schematic of the velocity comparison control method, where θ˙(θ=0) is the previous velocity of the pendulum at the zero posiˆ tion and θ˙(θ=π) is the previous velocity at π, θ˙(θ=0) is the velocity of pendulum expected at the zero position in period one rotational orbit, θˆ˙(θ=π) is the velocity at π. . . . . . . . . . . . . . . . . . . . 132
6.6
A rotational velocity of pendulum at θ=0 and π for a stable period one rotation against the frequency of excitation (1.1 Hz to 2.7 Hz) determined experimentally. . . . . . . . . . . . . . . . . . . . . . . 132
6.7
Experimental results for initiating the rotation of a pendulum using velocity comparison control method (with multi-switching). For both sets of forcing parameters (left) (A = 0.400 V and f = 2 Hz) and (right) (A = 0.300 V and f = 1.6 Hz) the initialization is successful. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
xviii
List of Figures 6.8
Experimental results of initiating the rotation of a pendulum using the TDF with multi-switching method. For all forcing parameters (left) (A = 0.400 V and f = 2.5 Hz), and (right) (A = 0.400 V and f = 2 Hz); the initialization was successful. . . . . . . . . . . 135
6.9
Experimental results demonstrating initiation and maintenance of the rotation of a pendulum using the TDF with multi-switching method. The time histories of the base acceleration, angular displacement and control signal are shown in green, red and blue respectively. Frequency of excitation is changed continuously from
2.5 Hz to 1.4 Hz and back to 2.5 Hz, while the controller maintains the rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.10 (left) Experimental results demonstrating maintenance of rotations using the TDF control method (with multi-switching) in a noisy system, showing: the time histories of the base acceleration (green), pendulum angular displacement (red) and control signal (blue). The amplitude of excitation is 0.420 V. The frequency of excitation is changed continuously from 2.2 Hz to 1.4 Hz. The amplitude of the noise is 10% of the maximum power of signal control. (right) Experimental examples of TDF control method (with multi-switching) application for maintaining rotational response while the forcing frequency is varied and noise added to the harmonic excitation, showing: the time histories of the base acceleration (green), pendulum angular displacement (red) and control signal (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.11 A PSD of the wave as function of ω for Hs = 10−4 , Hs = 10−3 , Hs = 10−2 and Hs = 10−1 with maximum of ωpeak equal to 125.2, 39.6, 12.5 and 3.96 rad/s respectively. . . . . . . . . . . . . . . . . 139 6.12 Five examples of simulated wave with Hs = 0.95 m, ωr = 5 rad/s, r = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.13 A power spectrum density of a wave with significant height Hs = 0.95 m. ◦ are selected frequencies for wave model based on equal spectral content within frequency range of 0 to 5 rad/s. ∗ is the
peak frequency (ωpeak = 1.1937 rad/s). . . . . . . . . . . . . . . . 142 6.14 Power spectrum density of a scaled wave with significant height Hs = 0.95 m. ◦ are selected frequencies. ∗ is the peak frequency (ωpeak = 12.0625 rad/s). . . . . . . . . . . . . . . . . . . . . . . . 143 xix
List of Figures 6.15 Examples of scaled waves with Hs = 0.95 m, ωr = 5 rad/s and r = 5. The wave characteristics are scaled to match the dimensions of the experimental setup; l = 0.271 m. The peak frequency of the wave ωpeak is scaled to twice of natural frequency of the studied pendulum, ωn (ωpeak = 12.06) and the height of the wave, y is scaled accordingly. . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.16 An example of rotational displacement of parametric pendulum (red) excited by simulated wave (green) with Hs = 0.47 m, ωr = 10 rad/s and r = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.17 Some examples of change of rotation number of the wave-excited parametric pendulum excited by simulated wave with Hs = 0.47 m, ωr = 10 rad/s and r = 5. . . . . . . . . . . . . . . . . . . . . . 146 6.18 A schematic representation of ideal position of the pendulum in peaks and valleys of the wave. . . . . . . . . . . . . . . . . . . . . 147 6.19 Simulated and smoothed wave (green and black respectively) and its major peaks (* in red) and valleys (* in blue). . . . . . . . . . 148 6.20 A schematic representation of timing of possible control signal. In the proposed method (6.18) u is applied between peaks and valleys of the wave as necessary. . . . . . . . . . . . . . . . . . . . . . . . 149 6.21 Simulated wave signals (green) with p = 0.1774, the angular positions of wave-excited pendulum (red) (a) with zero control signal (b) with control signal (blue). In this example, pendulum has (a) oscillatory response without control and (b) pure rotary response with control signal based on the proposed control method with k = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.22 Simulated wave signals (green) with p = 0.1984, the angular positions of wave-excited pendulum (red) (a) with zero control signal (b) with control signal (blue). In this example, pendulum has (a) both oscillatory and rotary response without control and (b) pure rotary response with control signal based on the proposed control method with k = 10. . . . . . . . . . . . . . . . . . . . . . . . . . 153
xx
List of Figures 6.23 PDFs of rotational numbers, R for six different values of the forcing amplitude p. The PDF of R of the un-controlled wave-excited pendulum is in blue and the controlled one is in red. Each of these graphs consists of 1000 simulations for 100 seconds with time step of 0.001 seconds. The other parameters of the simulations are presented in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . 154 6.24 PDFs of the average of the control effort for each case presented in Table 6.1. When p increases, the average of the control effort decreases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 A.1 Bit-rock interface modelled as two sliding surfaces with an equivalent friction coefficient µ. . . . . . . . . . . . . . . . . . . . . . . 188
xxi
List of Tables 3.1
3.2 3.3 3.4
4.1
Results of testing of 2 34 ” PDC drill-bit under different combinations of bit angular velocity and WOB. As an example, the forth row are the results which were presented in details on Figs. 3.16 and 3.17. Results of testing of 3 87 ” roller cone drill-bit under different combinations of bit angular velocity and WOB. . . . . . . . . . . . . .
49
Parameters of the equivalent friction coefficients for interaction of 3 78 ” PDC drill-bit and sandstone. . . . . . . . . . . . . . . . . . .
51
Parameters of the equivalent friction coefficients describing for interaction of 3 87 ” PDC drill-bit and sandstone. . . . . . . . . . . .
53
˙ Tb and Tˆ of 1 minute drilling with different The averages values of θ, WOB and Tc . The results were recorded after 20 s when the drillbit and drill-pipe settled at a constant speed; θ˙ = cte. . . . . . . .
4.2 4.3
47
69
Results of experiments including estimation for ct , Jt and error based on Eqs (4.13 - 4.15). . . . . . . . . . . . . . . . . . . . . . . The estimated parameters used in experiments (Fig. 4.22) and the
71
corresponding results. . . . . . . . . . . . . . . . . . . . . . . . . .
94
5.1
Parameter values of the experiments presenting in Figs 5.3 - 5.7. . 107
5.2
Estimated parameter values of the rotor rig according to the model shown in Fig. 5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Summary of the most significant dynamical responses of the system
5.3
(presented in Fig. 5.14) along with their corresponding frequency intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.1
List of parameters used in simulation results presented in Figs. 6.23 and 6.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
xxii
Abbreviations Designation Explanation BHA
Bottom-Hole-Assembly
DAQ
Data Acquisition
DOF
Degree-Of-Freedom
ETDF
Extended Time-Delayed Feedback
LVDT
Linear Variable Differential Transducer
LWD
Logging While Drilling
MWD
Measurement While Drilling
ODE
Ordinary Differential Equation
PDC
Polycrystalline Diamond Compact Drill-Bits
PDF
Probability Density Function
PSD
Power Spectral Density
ROP
Rate of Penetration
TDF
Time-Delayed Feedback
TOB
Torque-On-Bit
UPO
Unstable Periodic Orbit
VRF
Vibration Reduction Factor
WOB
Weight-On-Bit
xxiii
Chapter 1 Introduction 1.1
Motivation
The real world is inherently nonlinear, and particularly in mechanical systems this nonlinearity arises from one or more of the following reasons: geometry of the system, materials applied, interactions between parts of the system and nonlinear elements such as nonlinear stiffness, damping, and friction. Such nonlinearities frequently cause undesirable behaviour in engineering structures, for example instabilities, limit cycles or even chaos [142, 162]. For simplicity and due to computational limitation, these systems have traditionally been modeled with or reduced to linear equations. However, in recent decades nonlinear dynamics has vastly developed a great potential, enabling a deeper understanding and analysis of complex dynamical systems. For example, graphical tools such as phase portraits, Poincar´e maps, bifurcation diagrams, parameter space diagrams and basins of attraction are very well known tools for analysing and predicting the behaviour of nonlinear systems (as an example see Chapter 5). Ultimately, after knowing the system’s response we want to provide the ability to lead it to the desired behaviour or to avoid unwanted ones. This is convention-
1
1.1. Motivation ally classified as being part of control theory. In fact this theory also deals with improving stability, refining performance, and optimizing effectiveness. Control theory for nonlinear problems has traditionally been based on the local linearisation of the system dynamics in conjunction with linear control techniques. Therefore, in this type of method several controllers have to be designed and scheduled with respect to the operating conditions, which makes the controller very sensitive to the working conditions. However, many control problems involve uncertainties in the system parameters and the dynamical system state. This may be due to a slow time variation of the system parameters (e.g. drilling homogeneous formations), or to a rapid change (e.g. passing through an interface of formations with radically different mechanical properties). A linear controller based on inaccurate values of the model parameters may exhibit a poor performance or even instability. In addition, these controllers are also inefficient in some strongly nonlinear systems. In a mathematical sense, a strongly nonlinear system contains at least one nonlinear term with a large parameter [16]. In other words, the topological type of its phase portrait in the neighbourhood of the critical point is not completely determined by its linear terms [83]. Their effects ought to be fully understood and accurately compensated for. Linear methods cannot handle these effects so a system’s response must be predicted by nonlinear analysis techniques taking into account of these inherent nonlinearities. Nonlinear control allied with nonlinear dynamics has had a significant impact on control theory and made some topics such as robust control, adaptive control, and bifurcation and chaos control become a centre of attention for many researchers in recent decades. The main advantage of nonlinear controllers is their simplicity; achieving this goal may be simpler and more intuitive than for their linear counterparts. This apparently paradoxical statement comes from the fact that nonlinear controller designs are often deeply rooted in the physics of the 2
1.1. Motivation controlled systems [142]. Nonlinear control methods will take advantage of the given nonlinear system dynamics to generate high-performance designs. As an example, the Time-Delayed feedback control method which has been used for the parametric pendulum system (Section 6.5.3) is very simple. As it matches the nature of the periodical behaviour of the system, it is very effective and robust. The suggested method for the wave-excited pendulum is very simple yet efficient in terms of its purpose (Section 6.10). In terms of applications, many useful nonlinear control systems have been established, ranging from modern planes “without the pilot on board” to “driveby-wire” vehicles, to advanced robotic and space systems. Despite several wellestablished nonlinear control methods introduced theoretically in the last few decades, to achieve further progress, the topic of control and dynamics of nonlinear systems still needs to be studied with particular attention to experimental calibration. In this regard, three experimental projects have been the focus of study in this thesis. The motivation behind each of these projects is explained in the following sections.
Torsional vibration of drill-string The drill-string is an important component in the oil well drilling rig used for hydrocarbon exploration and in other types of exploration drilling. A highly complex dynamical behaviour can be observed in a drill-string. These complex dynamics include uncontrolled vibrations which are harmful to the drilling process. They may cause premature wear and damage of the drilling equipment, which eventually results in expensive failures. Vibrations in the drill-string include different types of vibration such as torsional vibration and its extreme case stick-slip. There have been a few attempts in the past to overcome this problem which is reviewed extensively in Section 2.1. However there is a lack of academic 3
1.1. Motivation experimental studies with real drilling processes to verify and calibrate models and control methods. This motivates me to pursue this experimental project. The motivation and the specific aims of this project are discussed further in Sections 3.1 and 1.2 respectively.
Lateral vibration of unbalanced rotor In rotating machine such as power generation, large-scale manufacturing and automobile engines, one of the most common concerns of designers is the longterm exposure of the rotating machines to vibration, which can eventually lead to catastrophic failures or accidents. This can be the case when resonance happens or when dangerous vibration-induced intermittent impacts between rotating and stationary components occur. Such unwanted vibrations are often caused by mass imbalance, which from a practical point of view can be produced by such factors as blade-loss conditions, looseness of parts and misalignment. This makes the presence of lateral vibrations unavoidable. The literature on this topic is vast and a detailed review is presented in Section 2.2. Most of the mathematical models used in these investigations are based on the Jeffcott rotor [72], which consists of a large unbalanced disk mounted midway between the bearing supports on a flexible shaft of negligible mass. In most models, the Jeffcott rotor is assumed with a symmetric support configuration. However, as a consequence of rotating machines operating in asymmetrical conditions the asymmetry often appears in many applications. For example when lateral loads act on the rotor or when gravity effects become significant. This motivates us to derive and experimentally validate a new asymmetric Jeffcott model. The motivation and the specific aims of this project are further discussed in Sections 5.1 and 1.2 respectively.
4
1.2. Aim of this study Rotational motion of parametric pendulum The concept of the application of the parametric pendulum for wave energy extraction has been proposed by Wiercigroch [161], where the vertical motion of sea waves excites a parametric pendulum fixed on a rotary platform and the vertical motion of the base is transformed into rotation of a pendulum. A parametric pendulum has been of great interest to the scientific community for many years because of its rich dynamics. As presented in Section 2.3, the dynamics of the pendulum regarding this application have been extensively studied in the last decade. Despite these conducted studies, there is still an urgent need to develop robust control algorithms for starting up the rotation under various conditions. Moreover, control methods need to be developed to increase the probability of rotational responses of the wave-excited pendulum. Therefore, this need motivates this experimental project. The motivation and the specific aims of this project are further discussed in Sections 6.1 and 1.2 respectively.
1.2
Aim of this study
This project is aimed at understanding the nonlinear dynamic behaviour in engineering systems. Moreover, the possibilities of controlling those systems simultaneously in order to make the best use of nonlinearities have been investigated. This study involves three different problems. Each of the three problems is treated separately in the following chapters of this thesis. Each chapter is self-contained with an introduction, detailed analysis of the problem, experimental and numerical results, and conclusions. The specific objectives of these problems include: • Suppression of drill-string and BHA torsional vibrations while drilling by using a closed loop controlling system: 1. Observing torsional vibrations of the drill-string in the experimental drilling rig. 5
1.2. Aim of this study 2. Constructing a realistic model of the experimental set-up, taking into account the dynamics of the drill-string and top motor. 3. Physical parameters estimation of the experimental drilling rig in order to calibrate the model, that ensures the correspondence of the theoretical predictions to the experimental conditions. 4. Development and implementation of a control method to suppress torsional vibration and stick-slip phenomenon exhibited in the experimental drilling rig. 5. Comparison of the experimental and numerical results considering delay of the actuator. • Analyzing soft impact in the asymmetric Jeffcott rotor in order to investigate vibrations of rotor dynamics subjected to mass imperfections:
6. Constructing a realistic model of the experimental set-up, taking into account an asymmetric physical configuration in rotor part as well as snubber rig. 7. Conducting experimental bifurcation diagrams with different conditions in a range around the grazing point. 8. Comparison of the experimental and numerical results based on the proposed model. • Initiating and maintaining the rotational motion of harmonicallyexcited and wave-excited pendulum as an energy harvesting system: 9. Proposing and comparing possible control methods for the harmonicallyexcited pendulum in order to initiate and maintain its rotational motion. 10. Development and implementation of control method for the waveexcited pendulum system in order to increase probability of its rotational motion. The thesis attempts to address those aspects of the three applications which have not been covered by previous studies, but which are crucial to the understanding of the dynamics of these systems. 6
1.3. Scope of the thesis
1.3
Scope of the thesis
Chapter 1 gives an introduction to the topic of this thesis, explaining the main motivation of the study as well as stating its objectives. Chapter 2 provides an overview of the literature dealing with related subjects, including dynamics of drill-string, asymmetric Jeffcott rotor and parametric pendulum under various types of excitation. Chapter 3 introduces the drilling experimental set-up, the data acquisition and control procedures as the first example of the nonlinear engineering system. In addition, the preliminary experimental stick-slip and torsional vibrations results will be presented. Chapter 4 develops a mathematical model and a control method in order to model drill-string and top motor, and to suppress torsional vibrations of the drill-string. This model and control method will be verified experimentally. Chapter 5 deals with a new model for the asymmetric Jeffcott rotor as the second example of the nonlinear engineering system. This model will be verified experimentally. Chapter 6 contains possible control methods for rotary motion of the harmonically and wave-excited pendulum as the third example of the nonlinear engineering system. The success of the rotational control of wave-excited pendulum will be measured statistically. Chapter 7 summarizes the main conclusions from this study and gives recommendations for future work.
7
Chapter 2 Literature review This chapter has three sections. Section 2.1 gives an overview of the literature dealing with dynamics of drill-string, modelling of the bit-rock interaction, experimental drilling rigs and suppression of torsional vibration. In Section 2.2 the literature relating to asymmetric Jeffcott rotor and its research background are presented. Finally, Section 2.3 provides an overview on literature dealing with renewable and wave energy harvesting systems, the dynamics of parametric pendulum under various types of excitation and their rotational control methods.
2.1
Drill-string vibrations
The drill-string is an important component in the oil well drilling rig, used for hydrocarbon exploration and in other types of exploration drilling. Owing to their slenderness, drill-strings have much similarity with one-dimensional continua such as beams, bars, rods and others. Accordingly, the dominant dynamics of drill-strings can be classified as those along the length of the drill-string and dynamics in transverse directions normal to the length of the drill-string. The dynamics involving the length-wise directions manifest as stretch and twisting of the
8
30- 80 m
2.1. Drill-string vibrations
Drill-mud cycling system
Drill-pipe 1 - 8 Km BHA 100-300 m
Drill-string
Rotary system
Drill-collar Stabilizer Drill-bit
Figure 2.1: A schematic of a typical rotary drilling rig, listing essential parts. A drill-string might be a few kilometers long, including the Bottom-Hole-Assembly (BHA) with a maximum length of a few hundred meters.
drill-string, respectively referred to as axial vibrations and torsional vibrations. The dynamics in the transverse directions manifest as bending of the drill-string, commonly referred to as lateral vibrations. These various modes of vibrations seldom occur in isolation and become coupled due to the geometrical nonlinearities caused by the slenderness, as well as due to interactions of the drill-string with the borehole. A very informative introduction to these various modes of drill-string vibrations is found in [70, 17]. Another review article covering axial, lateral and torsional vibrations of drill-strings is presented in [144]. A schematic of a typical drilling rig with a list of its various components is shown in Fig. 2.1. The above-mentioned highly complex dynamical behaviour was first discovered when Measurement While Drilling (MWD) tools were introduced in 1992 9
2.1. Drill-string vibrations to ensure better drilling efficiency [105]. These complex dynamics include uncontrolled vibrations which are harmful to the drilling process. They may cause premature wear and damage of the drilling equipment, which eventually results in expensive failures. Also vibrations often induce well-bore instabilities that can worsen the condition of the well and reduce the directional control [144]. Besides, controlling drill-string vibrations requires a good understanding of the BHA dynamics and its interactions with the drill-string. The drill-string vibrations can be self-excited and/or induced by nonlinear dynamic interactions between the drill-bit and drilled formation, and/or between the drill-string and borehole which are often characterized by erratic patterns and behaviour. Uncontrolled vibrations can decrease the rate of penetration (ROP), and consequently increase the well cost. Furthermore, excessive oscillations can interfere with MWD tools or even cause their damage. They can also cause a significant waste of drilling energy. Vibrations often induce well-bore instabilities that can worsen the condition of the well and reduce the directional control.
2.1.1
Torsional vibrations
The most dangerous manifestation of torsional vibrations of drill-strings occurs in the form of stick-slip motions. A considerable amount of literature has been devoted to the theoretical and experimental studies of this phenomenon. Torsional vibration and its extreme case stick-slip have been observed in about 50% of drilling time [20]. Stick-slip motion of drill-strings, as the name suggests, refers to a severe form of oscillations in the rotational motion of the drill-string. In this extreme form the bit occasionally comes to a complete standstill while the rest of the drill-string continues to rotate. This results in twisting of the drill-string which ultimately leads to torque buildup in the drill-string and in turn to an accelerated bit mo10
2.1. Drill-string vibrations tion. Stick-slip motion is common for PDC bits, and for inclined and deeper wells. The main cause of stick-slip motion is attributed to the speed dependent nature of the friction torque acting on the drill-string. In particular, the negative slope of the friction for higher rotational speeds creates self-excited vibrations of growing amplitudes. Analyses of stick-slip motion have been predominated by such speed-dependent friction models. However, there has also been alternative models explaining stick-slip motions due to modulations in the normal force caused by coupling between axial and torsional motions. Such models have been proposed in [149, 120, 106, 132, 49, 18]. Popp and Stelter [129] introduce the general ideas behind the role of frictional forces in generating stick-slip motions. Two discrete and two continuous systems (one of which is experimental) involving friction induced self-excitation and external excitation have been studied using various tools such as bifurcation diagrams and Poincar´e maps. In [45] a system is studied consisting of a block on a moving belt, exhibiting stick-slip motions. The discontinuous nature of the friction forces is shown to lead to an atypical fold bifurcation. Other studies related to stick-slip in higher degrees-of-freedom systems are reported in [46]. In addition to the theoretical difficulties associated with the discontinuous nature of friction forces, numerical simulations of stick-slip motions are also challenging. Numerical simulation strategies for stick-slip simulations are discussed in [92]. Stick-slip vibrations in drill-strings have been observed and reported since the 1960s. A comprehensive literature survey on torsional vibrations in drillstrings is given in [91]. In this review the literature on various aspects of torsional vibrations such as causes, modelling and control of stick-slip vibrations have been reviewed. The review concludes that much of the published literature lacks realtime downhole data, and also lacks realistic friction characteristics for various bits drilling in different formations. Theoretical analysis of stick-slip motions using a 11
2.1. Drill-string vibrations torsional pendulum model with added viscous damping and a negative sloop of friction characteristic for the torque have dominated most of the literature [84, 68]. Most of these models demonstrate the well observed role of decreased rotary speeds and increased weight-on-bit (WOB) in promoting stick-slip motions. The existence of a critical drill-string length below which stick-slip motions are absent (keeping all other parameters fixed) has been demonstrated in [98]. Some authors [20, 24, 125] utilize experimentally determined friction curves in their simulations to demonstrate stick-slip motions. Multi degree-of-freedom (DOF) models (several coupled torsion discs), but still with the friction curve at bit, have also been studied [115].
2.1.2
Coupled vibrations
Apart from the above mentioned works which are focused in torsional vibrations in drill-string, some other works have been carried out in coupling between different modes of vibrations. Coupling between axial and torsional motions occurs predominantly at the drill-bit and rock interface. The fluctuating reactions (associated with axial vibrations) at the drill-bit and rock interface lead to fluctuations in the TorqueOn-Bit (TOB) which in turn influences the torsional motions. In an early study conducted to explain axial-torsional coupling [3], the three-lobed formations created by a tricone drill-bit is utilized to explain fluctuations in rock reaction forces and moments, and the resulting fluctuations in the rotational drill-bit velocity. The experimentally observed fluctuations occur at three times the operating rotational speed, and this paper attempts to explain the same using the three-lobed formation. Other studies on axial-torsional coupling focus on an interplay between the longitudinal vibrations of drill-string and the stick-slip vibration. The earliest studies in this regard are presented in [173], focusing on the drag drill-bits 12
2.1. Drill-string vibrations with fixed cutters. The self-excited coupling of axial and torsional vibrations due to fluctuating forces at the drill-bit and rock interface has been further pursued in recent years by [132, 49]. Authors of [5] consider a coupling between torsional motion of the bit-string system and the lateral motion of the bit. The bit is considered to be in continuous contact with the wall, and the wall friction forces representing side cutting action of the bits provides the coupling between the torsional and lateral motions. A stability analysis is conducted of the coupled model. Another study of coupled torsional-lateral vibrations is presented in [170]. In this model the lateral and torsional motions of an eccentric drill collar are studied; the coupling is brought about through mass eccentricity in the collars and wall friction. A lumped parameter model with coupling between axial, lateral and torsional motions has been studied in [26], where the model has six DOF. These include two lateral, one axial, two torsional and one electrical DOF for modelling a rotary drive system at the top. Fluctuating drill-bit and rock interactions, which are due to surface irregularities, excite the axial and torsional motions, which in turn get coupled with lateral motions through eccentricities in the BHA as well as impacts with the borehole wall. Strongly coupled motions exhibiting bit-bounce, stick-slip and transient lateral vibrations are exhibited for representative parameter values.
2.1.3
Drill-Bit and rock models
The modelling of drill-bit and rock interface in all of the above mentioned papers proceeds by either developing fitted forms for the WOB and TOB, or by summing up the interaction of individual cutters and the formation. The approach of developing fitted-forms for forces, torques and penetration rates is based on extensive laboratory testing of bits under different combinations of rotary speeds, applied WOB and hydraulic parameters. Such an approach is 13
2.1. Drill-string vibrations reported in early studies (see for example [156]). Therein, empirical relations for the dependance of penetration rates are derived, based on laboratory tests on drill-bits. This initial model assumed perfect cleaning of the drilled rock. A refined model under imperfect cleaning conditions is presented in [158]. These empirical relations relate the penetration rates to the applied WOB as well as the bit radius, the rock strength and the hole cleaning parameters. There is also a need to relate the TOB with the WOB and the penetration rates. Such a relation is derived in [157] based on the simple force-balance concept in a roller-cone. This relationship has been further pursued and utilized in simulation of torsional vibrations in [145] and [26]. The approach of obtaining drill-bit rock interaction forces through summation of individual cutter forces is difficult for a roller-cone bit due to the complicated motion of the cutters in relation to the main body of the drill-bit. Such analysis for roller-cone bits can be found in the works of [104] and [138]. For Polycrystalline Diamond Compact (PDC) drill-bits, the summation of forces on individual cutters is easier due to the fixed nature of the cutters relative to the body of the drill-bit. Authors of [159] developed models to compute forces and torques required to remove rock at a given rate, starting from single-cutter results. The model predictions are compared with laboratory tests on four different bit designs drilling four different rocks. In [33, 35] relations for WOB and TOB were determined based on individual cutter force relationships. However, unlike the study of [159] the detailed cutter locations are ignored in this work and the relative cutter locations are accounted for through an empirical factor. In [89] a four DOF model for a PDC bit fitted to a rotary assembly is presented. The dynamical system is simulated, starting from individual cutter-force relations and a mesh of initial rock profile. At each time step of integration, the rock removed is accounted for and the rock profile is updated. This model is used to 14
2.1. Drill-string vibrations determine the response of three PDC bits. The backward whirling and torsional vibration tendencies of the drill-bits are compared. A similar study is reported in [55]. A recent work [132] utilized the relations developed in [33, 35] and studied the phenomena of stick-slip. This model was discussed in greater detail in [112].
2.1.4
Experimental drilling rigs
Experimental rigs for drill-string dynamics research have been developed for many years. The design and capabilities of these rigs vary according to their purpose. Early investigations on laboratory rigs to test the stability of PDC drill-bits for their whirl tendencies were reported in [55]. In the context of development Rate of Penetration (ROP) models, the authors of [156] reported on a large-scale laboratory rig capable of drilling with a range of drill-bit diameters under WOB, with rotary speed and hydraulic conditions similar to that experienced in field. In [53] torsional vibration experiments in a nearly vertical 1000 metres deep, full scale research drilling rig were reported. Apart from these large scale rigs owned by companies, a number of laboratory experiments in academic institutions are available to study drill-string dynamics. Most of these rigs consist of a slender drill-string, which is usually a 1 to 2 mm steel string driven at the top by a motor through a rotary table. The drill-bit and BHA are usually represented in the rig using discs. The presence of rock and the cutting action is usually simulated using shakers and brakes. Standard axial excitations and torque profiles are applied onto the discs through these shakers and brakes in order to study the resulting dynamics of the system. Detailed studies such as numerical simulations, generation of bifurcation diagrams etc. are undertaken for the rig system. More details of these rigs are available in [107, 79, 97]. In contrast to these rigs, the rig in [56] is designed to drill actual rocks. However, in that setup the rock is given rotary motion, while the bit and 15
2.1. Drill-string vibrations
Figure 2.2: Photographs of the rig of [56]. The drill-string is stationary and the rock is given rotary motion. Torsional flexibility of the system is simulated using a gear-pulley-spring system.
BHA are stationary. Also, the torsional flexibility of the drill-pipes is simulated through a special gear-pulley-spring system. Although drilling real rocks, this rig neglects the lateral dynamics. Photographs of the rig of [56] are shown in Fig. 2.2.
2.1.5
Suppression of torsional vibration
Several attempts have been made to investigate drill-string vibration and to overcome the difficulties encountered by field engineers. For example, Jansen and Van den Steen [69] applied an active damping technique. In their study, the
16
2.1. Drill-string vibrations electrical variables current and voltage were used to realize the required feedback control. A few years later a classical controller (PID) was applied by Abbassian and Dunayevsky [4]. At the same time a two DOF mathematical model of a drillstring was used by Serrarens et al. [137]; it only captured the torsional dynamics and applied H∞ technique to minimize the torsional vibration. Later, Gabler et al. [43] tried to improve drilling efficiency by imposing dynamic loading at the bitrock interface. Tucker and Wang [151] explored a method of controlling torsional relaxation oscillations of an active drilling assembly to reduce torsional vibrations; in their further work a coupled continuum system was considered [152]. Christoforou and Yigit [26] developed and studied a non-linear mathematical model of drill-strings in a series of papers. For example a linear quadratic regulator (LQR) based on a linearized model was designed. An inverse dynamics was used to design a nonlinear controller to track a desired bit speed by Al-Hiddabi et al. [11]. In this method the control law is based on input-output linearization. It was shown that this controller eliminates the torsional vibrations and reduces the lateral vibrations. A control strategy based on optimal state feedback is proposed in [26]. Other studies have focused on implementing control strategies for curing stick-slip vibrations [54, 85, 71]. Another control strategy [22], which the authors describe as D-OSKIL, utilizes the WOB as an additional control variable and proposes to kill stick-slip oscillations. Experimental implementation of this scheme has been reported in [103]. However, the authors used a masonry bit and a wood block to simulate bit-rock interaction. The most popular control strategy is that of modifying the impedance at the rotary table to ensure absorption of torsional waves at the surface [54]. More recently, the main focus for suppressing the torsional vibration has been in sliding-mode control. Abdulgalil and Siguerdidjane then developed a PID controller based on Sliding Mode Control [7] and a backstepping controller [6] 17
2.1. Drill-string vibrations to reduce the effect of the torsional vibrations. However, the pure sliding-mode control has been used in many practical control problems by others (see NavarroLopez and Cortes [117] for example). Recently, Puebla and Alvarez-Ramirez [130] developed a robust feedback control approach for the suppression of stick-slip oscillations at the bottom hole assembly in drill-strings. The idea behind this control approach is to lump all terms with uncertain parameters into a single term, which is then estimated and compensated for. Very recently Li et al. [96] presented an approach of combining adaptive control with time-varying sliding mode control. They designed a two-layer sliding mode adaptive controller for a rotary drilling system. Although continuous switching functions have been used in many sliding-mode controllers, very few of them have been applied to underactuated systems and no theoretical proof can be found, except for the work of Yang in [102]. Besides these scientific works in academia, some companies claimed that they have their own solutions for stick-slip vibration in drilling rigs. A good review on 50 years of modelling and control of the drilling-string vibrations can be found in [140]. The Soft Torque Rotary System (STRS) is owned and installed by Royal Dutch Shell in almost 70% of its drilling rigs [37]. Classic Soft Torque conceptually models a drillstring as a second order inertia-stiffness system. By using the motor current, they compute the applied torque and use it in the feedback loop. Although it is commercialized, there is no access to the details of this method. National Oilwell Varco has developed their own stick-slip prevention system called Soft Speed [86]. This system is basically a PI-controller with an acceleration feedback controlling the speed of the drive, which is tuned so that it dampens torsional oscillations in the drill string, and by this cure stick-slip. Although there are some follow-up papers (e.g. [119]) showing success in suppressing the stick-slip vibrations, there are no details of this method available to 18
2.2. Lateral vibrations on unbalanced rotor the public either. Moreover, there are some mechanical tools which seem successful in reducing the stick-slip oscillations in drilling rigs. The most famous of them is Anti Stick-slip Tool (AST) owned by Tomax [150]. It is basically a spring which is compressed in stick phase and thereby decrease the WOB what helps the drill-bit to overcome sticking into the surface (Fig. 2.3).
Figure 2.3: Anti Stick-slip Tool (AST). A sudden increase in torque (M2) will cause a contraction (S) to reduce the WOB (F2).(adopted from [10]).
2.2
Lateral vibrations on unbalanced rotor
The dynamics of rotating machinery has been extensively studied in the past by many researchers, mainly due to the numerous applications in industry, such as power generation, large-scale manufacturing, automobile engines, aerospace propulsion (see Fig. 2.4) and home appliances, among others. In all these applications, one of the most common concerns of designers and troubleshooters is the
19
2.2. Lateral vibrations on unbalanced rotor long-term exposure of the rotating machines to vibration, which eventually can lead to catastrophic failures or accidents. This can be the case when for example the undesired vibration becomes close to one of the natural frequencies of the machine structure (resonance), or due to dangerous vibration-induced impacts between rotating and stationary components. Such unwanted vibrations are often caused by mass imbalance, which occurs when there is a mismatch between the principal axis of the moment of inertia of the rotating element and its axis of rotation. From a practical point of view this phenomenon can be produced by blade-loss conditions, looseness of parts, misalignment, thermal deformation, factory residual imbalance etc., which makes the presence of lateral vibrations unavoidable. This undesired effect can have quite negative consequences in terms of durability, reliability and safe operation of rotating machines.
Figure 2.4: A photograph of fan blades of a jet engine damaged due to an excessive vibration (adopted from [1]).
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2.2. Lateral vibrations on unbalanced rotor
2.2.1
Recent background
The literature in this area is vast and dates back to the end of the nineteenth century with the rapid development of locomotives and steam turbines. Recent contributions in this field are reported in Lahriri et al. [88], where both theoretical and experimental considerations are presented regarding the reduction of rub contact of a rotor via backup bearing support. Ishida and co-workers [64, 65, 169, 62] have dedicated a significant amount of effort to the study of the nonlinear characteristics of the resonances caused by gravity effects and restoring forces. Their research includes theoretical investigations of the localized and non-localized normal modes both in horizontal and vertical directions, as well as experimental observations of the predicted phenomena.
2.2.2
Vibration suppression methods
Another area that has received considerable attention in the past is vibration suppression methods. One of the most common approaches is the so-called Automatic Ball Balancing (ABB), which consists of a system of balls that are free to travel along a race mounted to a rotor at a fixed radial distance from the centre of rotation. Under certain operating conditions, the balls find an equilibrium position that compensates the mass imbalance of the rotor, thus reducing the undesired vibrations. Recent development in this direction can be found in Champneys et al. [133, 134], Green et al. [52] Ishida et al. [100, 63] and van de Wouw et al. [153].
2.2.3
Rotor with snubber ring
As mentioned earlier, the present work concerns itself with the study of vibrationinduced impacts in a Jeffcott rotor operating within a snubber ring. Such con-
21
2.2. Lateral vibrations on unbalanced rotor figuration is very common in engineering applications, where the main purpose is to achieve an improved suppression and attenuation of excessive vibrations in out-of-balance conditions. See for example [61] for a recent application in washing machines. The rotor dynamics observed in the presence of an outer snubber ring can be compared to the case of a rotor with squeeze film damper bearings, whose associated nonlinear phenomena have been largely studied in the past [39, 38, 80, 27]. A comprehensive survey of different dangerous phenomena affecting the normal operation of rotating machines, together with their main causes and consequences, is presented in Ahmad [9], Jacquet-Richardet et al. [67] and Adams [8].
2.2.4
Background of this work in CADR
In the Centre for Applied Dynamics Research (CADR) of Aberdeen University, particular attention has been paid to the study of vibration-induced impacts in rotor systems. An earlier version of the experimental rig used in the present paper was employed in Gonsalves et al. [51] to study the dynamical response of an unbalanced rotor placed eccentrically within an outer snubber ring. The rig was later modified so as to be used for a research project in collaboration with the automobile industry [163]. The main purpose was to investigate the potential use of shape memory alloy (SMA) wires for control of the critical stiffness and damping characteristics for a new engine design. To this end, several mathematical models were developed and experimentally studied in order to construct a solid basis for the intended application [78, 76, 77]. The aim of the present work is to extend and complement the previous investigations undertaken at the CADR with the purpose of deriving and validating a new model to study vibration-induced impacts in unbalanced planar rotors. The new model takes into account viscoelastic characteristics of the snubber ring 22
2.3. Rotational motion of parametric pendulum support as well as anisotropy in both the snubber ring and rotor supports. The asymmetry considered in the model often appears in many applications as a consequence of rotating machines operating in asymmetrical conditions, for example when lateral loads act on the rotor or when gravity effects become significant. Moreover, the generally stabilizing property of damping elements has been applied in the past to mitigate and control undesired vibrations due to mass imbalance [66, 21]. Such approaches involve the application of active damping devices in horizontal and vertical directions, thus giving rise to anisotropic configurations. Similarly, nonlinear stiffness control can be applied in an asymmetric way by means of SMA-based actuators [141, 73, 74].
2.3 2.3.1
Rotational motion of parametric pendulum Renewable and wave energy harvesting systems
Based on Energy Roadmap 2050 scenarios [41], renewables increase their share significantly under adopted policies and would substantially rise in all decarbonisation scenarios to reach at least 22% of primary energy consumption by 2030 and 41% by 2050. One of the most abundant renewable energy sources is ambient vibrations, because of the widespread availability of the vibrations in the environment [110]. These vibrations may be produced by machinery. Most systems for harvesting energy from machinery vibration are mainly based on piezoelectric (e.g. [136, 44]). However they provide low power which might be just enough to power systems such as microelectromechanical systems (MEMS) or wireless sensors. Human motion is another source of the energy of vibrations. Wrist watches, shoes with built-in generators [87] or power generation within knee replacement implants [128] are some examples of attempts for energy harvesting from human motion. 23
2.3. Rotational motion of parametric pendulum a)
b)
c)
d)
Figure 2.5: Some of wave energy harvesting system: (a) Pendulor system (b) Wave Dragon System Principle, (c) Salter’s Duck system (d) Pelamis Wave Energy Converter (adopted from [135]). The largest source of the energy of oscillations are ocean waters. The world’s wave energy potential is estimated to be 2.5-3 TW [23]. The technologies used to harvest energy from the ocean waves can be categorised as follows: Shoreline installations (Oscillating water column (OWC) [36] and Pendulor system [28]), Near-shore installations (Wave dragon system [160]), Offshore installations (Salter’s Duck [42] and Pelamis Wave Energy Converter [126]) and undersea installations (Wave Roller System [2] and Archimedes Wave Swing [135]). Photographs of some of these systems are presented in Fig. 2.5. Many of these systems are based on initiating pendulum-type oscillations of the structure (e.g. Wave Roller, Pendular System, Salter’s Duck). Understanding the variety of the pendulum responses
24
2.3. Rotational motion of parametric pendulum to the wave excitation as well as exploring possible control methods to initiate and maintain the desired response, might open room for discussion of potentially more effective approaches.
2.3.2
Parametric pendulum as energy harvesting system
The concept of the application of the parametric pendulum for wave energy extraction was proposed by Wiercigroch [161], where the vertical motion of sea waves excites a parametric pendulum fixed on a rotary platform and the vertical motion of the base is transformed into rotation of a pendulum. Schematically the considered system consists of a pendulum mounted on a pontoon subjected to external excitation (Fig. 2.6). The dynamics of the pendulum regarding this application has been extensively studied by the Centre for Applied Dynamics Research at Aberdeen (see for example [166], [57], [110], [123] and [113]).
Pendulor Oscillating base
Figure 2.6: Working principle of parametrically-excited pendulum harvesting wave energy. Oscillations of the sea surface results in the rotation of the pendulum (adopted from [110]).
2.3.3
Dynamics of parametric pendulum
In general a parametric pendulum has been of great interest to the scientific community for many years, because of its rich dynamical behaviour. See for example numerical study of Leven & Koch [94], Koch et al. [82], Bishop & 25
2.3. Rotational motion of parametric pendulum Clifford [19], Clifford & Bishop [29, 31]. Only a few earlier studies were focused on rotational responses of parametric pendulum: Koch & Leven [81], analytically and numerically found different solutions including rotational motion. Leven et al. [95] then investigated it experimentally. Szemplinska-Stupnicka et al. [147] and Szemplinska-Stupnicka & Tyrkiel [146] studied different aspects of dynamics of parametric pendulum including its rotational responses. Clifford & Bishop [30] numerically identified and classified different types of rotational solutions of the system and Garira & Bishop [47] completed previous work by distinguishing four broad categories: pure rotations, oscillating rotations, straddling-rotations and large amplitude rotations.
2.3.4
Rotational response of parametric pendulum
Furthermore Xu et al. [168], Xu & Wiercigroch [167], Lenci et al. [93], Horton et al. [60, 58], Litak et al. [99], Horton et al. [59], Pavlovskaia et al. [123] and Wiercigroch et al. [164] have extensively explored rotation of parametric pendulum by analytical, numerical and experimental study. They showed that the rotational responses can be observed as a steady state stable solution of the parametric pendulum. Tatiana et al. [12] carried out an experimental investigation of the response of a tri-pendulum to parametric excitation. They claimed that this configuration will overcome the restriction of the narrow range of parameters for the rotational response in low frequency of ocean waves. Teh et al. [148] also presented an experimental rig including parametric pendulum excited via vertical, non-linear electromechanical excitation generated using a RLC-circuit-powered solenoid.
26
2.3. Rotational motion of parametric pendulum
2.3.5
Rotational control of parametric pendulum
As a swinging up pendulum system has been a classical control problem, a variety of methods have been developed for different conditions (see Astrom et al. [15]), which were originally aimed at stabilizing the pendulum in upright position. More recently a few attempts have been made to maintain the rotational motion of the parametric pendulum based on Time-Delayed Feedback control method (TDF) introduced by Pyragas [131] and Extended Time-Delayed Feedback control method (ETDF) proposed by Socolar et al. [143]. De Paula et al. [32] employed the ETDF method numerically to maintain stable rotational solutions of the system avoiding period doubling bifurcation and bifurcation to chaos, whereas Yokoi & Hikihara [171] experimentally investigated the tolerance of TDF method to the delay mistune.
2.3.6
Wave-excited pendulum
Almost all the works in the field considered harmonic parametric excitation which is not a good model for the wave when one considers ocean wave excitation on the pendulum system for wave energy harvesting. The wave has to be treated as a random process. Very little work has been done in this topic. Yurchenko et al. [172] identified the regions with highest and lowest probability of rotation in the forcing parameter space, for the parametric pendulum under noisy excitation, where the noise has been modelled as a narrow-band process. Alevras et al. [13] studied numerically synchronization of two pendulums excited by a random phase sinusoidal force, thus leading to stochastic parametric excitation of the pendulums. Andreeva et al. [14] modeled the wave profile using a non-harmonic periodic function with a sharp or bent crest. The authors studied the dependency of the pendulum’s rotational potential on the wave profile. They showed that the sharp
27
2.3. Rotational motion of parametric pendulum wave profile significantly deteriorates the rotational potential of the parametric pendulum. In one of their recent work, Najdecka et al. [111] theoretically and experimentally studied rotary motion of a pendulum subjected to a parametric and planar excitation of its pivot, mimicking the random nature of sea waves. The authors modeled the vertical motion of the sea wave, based on the Shinozuka approach [139] and using the spectral representation of the sea state proposed by the Pierson-Moskowitz model [127]. In this method, the time histories of the ocean waves are generated as a sum of sine waves with different frequencies and random phase angles. In this work, a method is proposed using less terms in the sine series which reduces the time of simulation, at the same time preserving the accuracy of simulation. To do that, variable frequency increments are used based on equal spectral content within adjacent frequency ranges. Following the aforementioned studies, the aim of the present work is to design and implement possible control methods for the harmonically-excited and wave-excited pendulum in order to initiate and maintain its rotational motion.
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Chapter 3 Torsional vibration of drill-string The aim of this chapter is to fully explain the details of the experimental rig, to carefully estimate its physical parameters, and to present torsional vibrations of the drill-string and the BHA that are observed in the drilling rig. This chapter is subdivided into six sections. An introduction is given in the next section. In Section 3.2, the design of the rig is then discussed, including the drilling machine, drill-string, rock, drill-bit, fixtures and fluid system, sensors, instrumentation and data acquisition system. In Section 3.3, the parameters of the experiment are determined. Detailed experiments in different drilling conditions are also described. The results presented in this section are derived from extensive experiments and curve fitting procedures which were carried out in order to develop empirical models for drill-bit and rock interaction. The experimental results presented in section 3.4 include torsional vibrations and stick-slip observed in the drilling experimental rig under different conditions. Finally, the main conclusions of the chapter are presented. It is worth noting that some of the results of this chapter have been published recently [75].
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3.1. Introduction
3.1
Introduction
The drill-string is an important component in the oil well drilling rig used for hydrocarbon exploration and in other types of exploration drilling (more details are given in Section 2.1). A highly complex dynamical behaviour can be observed in a drill-string. It was first discovered when MWD (Measurement While Drilling) tools were introduced to ensure better drilling efficiency in 1992 [105]. These complex dynamics include uncontrolled vibrations which are harmful to the drilling process. They may cause premature wear and damage of the drilling equipment, which eventually results in expensive failures. Also vibrations often induce well-bore instabilities that can worsen the condition of the well and reduce the directional control. As explained in Section 2.1, the dynamics of drilling system has been recently studied numerically and experimentally. However, the mechanism of most experimental rigs which are reported in the literature are not based on real drilling. For example, Mihajlovic et al. [107] and Khulief et al. [79], use brake systems instead.
3.2
Experimental rig
The design of the experimental rig is inspired by [56]. However, in the rig the rock is held stationary and the BHA is given rotary motion. The rig has been designed and constructed to provide a fuller understanding of stick-slip oscillations and ultimately a means to its suppression. The primary objectives of the rig is to conduct systematic drilling tests with a rigid drill-pipe in order to obtain experimental data to model drill-bit and rock interaction. A further objective is to demonstrate the various drill-string vibration phenomena, so as to verify predictions from the mathematical models describing these phenomena. This will ultimately lead to the implementation and 30
3.2. Experimental rig verification of proposed control methods to suppress these vibrations.
3.2.1
General overview of the drilling rig 4
5
6
3 1 2
Figure 3.1: General view of the drilling machine used in this study including (1) computer for data acquisition and control system, (2) breakout box for DAQ system, (3) charge amplifiers for sensors, (4) top AC motor, (5) pulley system and (6) frequency convertor for driving the motor.
31
3.2. Experimental rig As shown in Fig. 3.1 and 3.2, a motor is connected to the drill pipe and the rotary force is transmitted to the bit through a drill-pipe, BHA and a bit-holder. The angular velocity of the motor is adjustable and is measured by an encoder on the top of the drill-pipe. The angular velocity of the drill-bit is measured by another encoder connected to the BHA. Horizontal and vertical forces and torque coming from the drill-bit to the rock are detected by a load-cell placed under the rock. The rate of penetration of the bit into the rock is measured by a linear variable differential transducer (LVDT) attached to the BHA. Velocity of drilling, type of drill-pipe and drill-bit, WOB and rock are changeable will therefore in addition motor
top encoder
flexible shaft
disk BHA eddy current probes LVDT bottom encoder drill-bit rock sample load cell
Figure 3.2: (right) Photograph of the experimental rig and (left) schematic diagram of the experimental setup. The main components of the system are: sensors (top and bottom encoders, LVDT and 4-component load cell), electric motor, flexible shaft, disks, BHA, drill-bit and rock sample.
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3.2. Experimental rig allow the identification of minor failures, dynamic characteristics of drill-pipes and BHA, rock models and drilling conditions. In the next section, the main components of the rig are explained.
3.2.2
Drilling machine
An Ibarmia pillar drilling machine provides rotary and axial motions. This machine weights 1524 kg and has a 3 kw 3-phase AC motor. Figure 3.3 (a) shows the motor and the gearing system connected to it which can provide a maximum rotary speed of 1032 rpm. The lowest attainable rotary speeds in the drilling machine without any external control system is only 54 rpm. It is desirable to reduce this lower limit to as low as 1 rpm. To achieve this and also to be able to control motor torque, a frequency convertor (speed/torque-controller; ACS55001-06A9-4 R1) is used (Fig. 3.3 (b)). (a)
(b)
Figure 3.3: (a) Top motor, the gearing and pulley system and (b) frequency convertor which has been used to control the speed or torque of the top motor.
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3.2. Experimental rig
3.2.3
Drill-string
In an industrial drilling rig, the drill-string essentially comprises a series of relatively thinner drill-pipes (typically 10 metres long), moderately thicker heavyweight drill-pipes, thicker drill-collars, and the drill-bit. Apart from these major components, a drill-string may comprise of several additional components such as stabilizers, cross-over subs, jars, Measurement While Drilling (MWD) and Logging While Drilling (LWD) tools, and directional-drilling tools. These components provide specific functionality. The drill-collars along with the stabilizers, drill-bit, and other equipment below the drill-pipes are sometimes collectively called a BHA. Typical lengths of BHAs can range between 200 to 300 metres. A typical drill-string may extend downhole and be several kilometers long, while its cross section diameters are typically around 0.3 metres. On average a BHA con-
flexible shaft with diameter of 5 to 20 mm
1000 mm
Figure 3.4: Drill-string configurations. Common BHA, but varying drill-pipe designs for different purposes. Left: drill-string with rigid rod for a drill-pipe to conduct characterization tests (bit-rock interactions: Section 3.3.2). Right: A Bowden cable or flexible shaft for a drill-pipe to conduct drill-string dynamics experiments (observing torsional vibrations: 3.4).
34
3.2. Experimental rig stitutes roughly 10% of the total length of the drill-string. However, the overall height of the machine is around 3.2 metres and the effective of drill-string that could be accommodated for this height is only up to 1.2 metres. a
c
b
e
f
d
g
Figure 3.5: Different types of drill-pipes and rocks; (a) rigid shaft, (b) flexible drill-pipe - a slender aluminium tube, (c) bowden cable (flexible shaft), (d) rock sample - limestone, (e) rock sample - sandstone-quartz, (f) rock sample - granite, (g) rock sample - sandstone. In practice, two configurations are required: characterisation of the drill-bit and rock interactions in terms of forces generated for steady drilling, and capability to simulate dangerous phenomena such as stick-slip, bit bounce and whirling. A rigid drill-pipe is required for the set of experiments concerned with characterizing drill-bits and rocks, whereas the set of experiments demonstrating drill-string dynamic phenomena requires a flexible drill-pipe. The BHA element is common to both sets of experiments. The final conceptual designs of the drill-string for these two sets of experiments are shown in Fig. 3.4, left and right respectively. A photograph of the thicker rod used for the experiments on drill-bit and rock characterization is shown in Fig. 3.5(a). This rod is made of steel. A Bowden 35
3.2. Experimental rig
Figure 3.6: Additional movable disks attached to the BHA in order to increase the WOB. Four first disks weight 6.52 kg each and each of rest weights about 10.63 kg. This setup can be used in both configurations; with rigid and flexible shafts.
cable (flexible shaft) shown in Fig. 3.5(c) was chosen. The torsional stiffness of these shafts is identified in the next section.
3.2.4
Static force (Weight-On-Bit)
Although the main function of the drill-string is to transmit the rotary motion imparted at the surface by the motor to the bit downhole, the necessary force for drilling (WOB) is supplied by the self-weight of the drill-collars, while the drill-pipes are normally held in tension to prevent them from buckling. In the experimental rig the necessary WOB is provided by a set of aluminum disks, BHA and drill-bit. Each disk has a slot along its radius in order to let the drill-pipe pass through and therefore the disks are used in pairs to balance the mass. The first four disks weight 6.52 kg. A maximum of 18 more disks can be used, each weighting 10.63 kg. The drill-bit, drill-bit holder, BHA and a cap for accommodating the disks together weight 68.35 kg. In total, a WOB of 2.804 kN can be provided for this system. Fig. 3.6 shows the additional movable disks. It worth noting that in the configuration which is for characterisation of the drill-bit and rock interactions (rigid drill pipe: Fig. 3.5(a)), a WOB can be applied 36
3.2. Experimental rig through the pulley system as it is shown in Fig. 3.3(a). In this configuration a maximum WOB of 5.1 kN was tested in the rig.
3.2.5
Rock samples, fixtures and cutting fluid system
To develop a useful model, the first step is to characterize bit-rock interaction. In the rig rock types sandstone and granite are considered. To minimize the waste, it was decided to use a cuboid rock shape of dimensions 130 mm×130 mm×130 mm. The last element in the drill-string (or BHA) is a drill-bit and throughout the experimental studies the following three types of drill-bits shown in Fig. 3.7 were used: 3 78 ” PDC, 2 34 ” PDC and 3 78 ” roller cone drill-bits.
Figure 3.7: Different types of drill-bits used in the experimental studies (from left to right: 2 34 ” PDC, 3 87 ” roller cone and 3 78 ” PDC drill-bits). Two types of holders for these drill-bits were designed and manufactured. An API regular joint (2 34 ”) was utilized for the larger PDC and the roller cone drillbits. An AMMT joint (1 12 ”) was used for the smaller PDC drill-bit. A simple clamp arrangement was fabricated for holding the rock firmly on a base plate, which in turn was bolted on top of the load-cell. In the industrial drilling rig, the drill-pipes also serve the purpose of transmitting the drilling mud which lubricates the drill-bit and also removes the cuttings. In this regard, in the experiments, water with anti-corrosion additive is used as 37
3.2. Experimental rig
Figure 3.8: Cooling and cleaning fluid system using water with anti-corrosion additive.
a cooling and cleaning fluid. Fig. 3.8 shows a simple fluid system consisting of a tank and a pump. The debris gravitates to the bottom of the tank and is removed after each few experiments.
3.2.6
Instrumentations and control (a)
(b)
Figure 3.9: (a) An incremental rotary quadrature encoder with 500 pulses per revolution. (b) Position transducer (P1010-60-NJC-DS-TS) used for axial position measurements.
In order to study the axial, torsional and lateral dynamics of the drill-string, the twist and axial position of the drill-string, as well as the lateral position of
38
3.2. Experimental rig
Figure 3.10: A close-up view of the two eddy current probes with a range of 5 mm, placed circumferentially 90 degrees apart on the periphery of the brass bushing for the loose bearing. The probes monitor lateral positions.
the BHA need to be monitored. In addition, to develop characterisation of the rocks the forces and torque developed during drilling need to be measured. The twist of the drill-pipe is measured by monitoring the cross-section rotations of the drill-pipe at the top end nearest the motor as well as at the end below the BHA. Since the BHA is quite rigid it develops little twist. Hence the measurements from the encoder near the drill-bit-holder adequately characterise the rotation of the drill-bit as well. Two encoders are chosen for the purpose of making the angular measurements with a resolution of 500 pulses per revolution. In addition, the encoder is of quadrature type so that the sense of rotation can be monitored. The overall dimensions of the encoder are shown in Fig. 3.9(a). A 5V DC power supply is required for the utilization of the encoder. The axial position of the drill-bit is monitored using a position transducer arrangement(P1010-60-NJC-DS-TS). A resistance-based potentiometer gives a voltage output proportional to the change in axial position of the bit. The range of measurement of the potentiometer is 1500 mm. A photograph of the potentiometer is shown in Fig. 3.9(b). The lateral position of the BHA is monitored through two eddy current probes which are located 90 degrees apart circumferentially on the loose bearing located
39
3.2. Experimental rig (a)
(b)
Figure 3.11: Kistler 9272 load-cell. (a) Four component dynamometer placed beneath the rock-holder plate; (b) ICAM5073A charge amplifier.
in the drilling machine table. The range of the probes is chosen to be 5 mm. The sensitivity of the probes according the supplier is 2 V/mm. The calibration tests reveal that the sensitivity is around 1.71 V/mm. A picture of the eddy current probes is shown in Fig. 3.10. It is worth mentioning, the loose bearing can be changed and different size of gap can be applied between BHA and the bearing (representing the bore-hole) in order to mimic the lateral vibrations in the drilling process.
Figure 3.12: A photograph of the breakout box as an interface to NI PCIe 6321 DAQ card used for data acquisition and control purposes.
The cutting forces and torques are measured using a load-cell beneath the rock-holder plate. After considering various alternative options, the Kistler 9272 four component dynamometer was chosen. The load-cell measures the lateral forces in the plane perpendicular to the axis of the drill-bit, as well as the thrust 40
3.2. Experimental rig force along the axis of the drill-bit and the torque about the axis of the drill-bit. The load-cell is based on the principle of piezoelectricity and generates electric charge proportional to the measured forces and torques. A charge amplifier is required to convert the generated charges to corresponding voltage levels, before being digitalised and stored. A Kistler 5073A type charge amplifier was chosen for this purpose. A picture of the load-cell and the charge amplifier are shown in Fig. 3.11. The maximum torque and WOB that can be measured by the load-cell are 200 Nm and 20 kN respectively. The charge amplifier requires a 24 V DC power supply.
Figure 3.13: A graphical interface of the Labview program for acquiring data from sensors and controlling the top motor. The sensors mentioned above generate electrical signals which need to be acquired and stored for further processing. The NI PCIe 6321 card is used to acquire the data. The card has 4 counters, a maximum of 16 analog channels and 8 digital channels for data acquisition, and 2 analogue outputs for control. A photograph of the breakout box is shown in Fig. 3.12. 41
3.3. Physical parameters identification There are two control variables: ’velocity or torque of top of drill pipe provided by motor’ and ’WOB’. The first one is controlled via a Labview system. The second one is currently adjusted manually as described in Section 3.2.4. Top and bit speed, lateral and axial position of BHA, TOB, WOB and lateral forces on rock sample are measured by sensors. The data from sensors are also collected by the Labview acquisition card and program. Fig. 3.13 shows the interface of the Labview software.
3.3
Physical parameters identification
In order to obtain good agreement between the experimental observations and the mathematical models (presented in the next chapter), a careful estimation of the experimental rig’s physical parameters was carried out. Obtaining some of physical parameters such as moment of inertia or weight of the disks were straightforward. Special attention was given to the identification of the characteristics of the flexible shaft, as well as to the laws governing the bit-rock interactions. The former is done with free vibration experiments (Section 3.3.1) and the latter was quantified in terms of a TOB-bit rotational velocity response curve (Section 3.3.2).
3.3.1
Flexible shaft parameters
As described in Section 3.2.3, flexible shafts are used to mimic the mechanical properties of slender structures like a drill-string. Due to the length of a drillstring, which can be up to several kilometers, the structure has practically no transversal stiffness when compared to the axial direction. This physical configuration can be modelled in a reduced scale by a flexible shaft consisting of many layers of thin wires, as can be seen in Fig. 3.14 (a). Such shafts are used to transmit power in rotating machines as they have high torque capacity transmission 42
3.3. Physical parameters identification (a)
(b)
θb Figure 3.14: (a) Internal structure of a flexible shaft used in the experiments, (b) setup for identification of the torsional stiffness and damping of a flexible shaft. and high flexibility. Friction between wire layers plays an important role, which means that an effective damping depends on the tensile load. In the rig, flexible shafts with a diameter of 5, 7, 10, 15 and 20 mm are tested. For the purpose of this chapter, in observing stick-slip vibrations the best results are obtained using a flexible cable with a diameter of 10 mm. In order to determine the torsional stiffness and viscous damping, an initial angular displacement (θb ) is applied. The decaying free torsional oscillations are then measured when the flexible shaft is connected to the BHA and fixed at the other side to the motor, as depicted in Fig. 3.14 (b). Such a procedure is similar to the identification of the stiffness and damping of the rotor and snubber ring as described in Section 5.5. This process has been repeated for different values of tensile forces and for clockwise and anti-clockwise initial angular displacements. To ensure high accuracy of the results, each test was repeated 10 times and averaged. The estimated stiffness appears to be constant, regardless the direction of the initial angular displacement. As can be seen from Fig. 3.15 (a), the equivalent torsional stiffness does not vary with respect to the applied axial tension, contrary to the estimated damping, which shows a linear dependence for the considered 43
3.3. Physical parameters identification parameter window (see Fig. 3.15(b)). (a)
k[Nm/rad]
6
4
2
0 50
70
90
110
130
140
M[kg] (b) 0.60
c[Nms/rad]
0.55
0.45
0.35
0.25 50
70
90
110
130
140
M[kg]
Figure 3.15: Identification of the torsional stiffness and damping coefficient of a flexible shaft of φ10 mm in diameter as a function of the attached mass. (a) torsional stiffness in clockwise (blue square) and anti-clockwise (red circle) directions, (b) damping coefficient in clockwise (blue square) and anti-clockwise (red circle) directions. Average values of clockwise and anti-clockwise directions are denoted by ’+’.
3.3.2
Experimental identification of bit-rock interaction
To describe the drill-string behaviour, modelling of reaction torque coming from rock to the drill-bit is required. Mechanisms which describe the bit-rock interaction have been largely investigated in the past [92, 34, 155]. A detailed literature
44
3.3. Physical parameters identification review on studies on bit-rock interaction is given in Section 2.1.3. In the present work, the cutting process will be mathematically described in terms of frictional models (see e.g. [116, 118, 114]). In this section experimental and mathematical methods are described which were used to fit the TOB with respect to rotary speeds (RPM) and WOB. Laboratory testing of drill-bits were carried out under different combinations of bit-speed and WOB. One example of acquired data is demonstrated in Fig. 3.16. This figure shows time histories of the axial displacement, TOB, WOB and the angular velocity of the drill-bit when a 2 34 ” PDC drill-bit was used to drill on sandstone and water has been injected as a cleaning and cooling fluid. In Fig. 3.16, the result of experiment according to the following sequence are presented. At t = 0 s, data acquisition is started with the rig switched off. At t = 11 s, the drill-bit is placed on the rock. At t = 41 s, the top motor is turned on. As can be seen in the figure, a penetration of 20 mm into the rock is observed at t = 202 s. Finally, the experiment was finished at t = 435 s. Fig. 3.17 shows a large scale view of the time histories, starting at when the drill-bit has already penetrated 20mm into the rock, which is the length of the drill-bit blades. This window is considered in order to calculate the average TOB, WOB and bit angular velocity for this experiment. As it is shown in the figure, T OB = 15.6 Nm, W OB = 3.81 kN and RP M = 68.2 rpm. Extensive laboratory testing of drill-bits were carried out under different combinations of bit speed and WOB. The results for 3 87 ” roller cone and 2 34 ” PDC drill-bits are presented in Table 3.2 and 3.1, respectively. As an example, the forth row in Table 3.1 consists of the experimental data presented on Figs. 3.16 and 3.17. For the rest of this study, I focused on the 3 87 ” PDC drill-bit. Therefore more attention is given to the identification of TOB curves for this drill-bit in different 45
3.3. Physical parameters identification
Putting Drill-bit on the rock Starting the motor
End of the window
Start of the window
Penetration
220
(mm)
On the rock
20 mm penetration into the rock
200 180 160
Torque
(NM)
30 20 10 0
Vertical Force (Z-axis)
(N)
4000 2000 0
Angular Velocity of Drill-string
RPM
60 40
velocity on top velocity on bottom
20 0 -20
0
100
200
300
400
T (s)
Figure 3.16: Time histories of axial displacement, TOB, WOB and angular velocity of drilling by 2 34 ” PDC bit for sandstone, using water as a cleaning and cooling fluid. The average of TOB, measured in the window 202 s to 435 s is considered to be the ’expected TOB’ when drilling in the same bit speed and conditions (Fig. 3.17).
46
3.3. Physical parameters identification Table 3.1: Results of testing of 2 34 ” PDC drill-bit under different combinations of bit angular velocity and WOB. As an example, the forth row are the results which were presented in details on Figs. 3.16 and 3.17. TOB [Nm] 17.608 16.232 15.266 15.569 16.533 16.291 24.349 22.354 22.057 20.952 21.418 21.886
WOB [kN] 3.784 3.763 3.761 3.813 3.796 3.835 4.786 4.847 4.872 4.890 4.979 4.964
bit angular velocy [rpm] 016.834 024.608 032.190 068.223 140.066 212.262 014.187 022.735 030.523 066.989 138.984 211.089
WOB. Experimental identification of TOB curves for different values of WOB are presented in Fig. 3.18. These curves are obtained for WOB between 0.85 kN and 2.19 kN by changing the number of disks from 4 to 20 disks. For each WOB (number of disks), the TOB is captured for different rotary speeds from between 0.5 rpm and 56 rpm in 11 experiments. As can be seen in Fig. 3.18, for the higher WOB, higher TOB is observed. Also, as was expected at low speed, when the rotational speed increases the TOB decreases. However, after extending to a certain threshold, the TOB begins to increase with an increased rotary speed. Fig. 3.18 illustrates that the explained trend is not valid for WOB below 1.10 kN. Instead, the TOB is constant in the considered range of rotary speed.
47
Penetration (mm)
ROP=0.08374 mm/s
Penetration
180 170
3.3. Physical parameters identification
160
TOB=15.5699 NM
Torque
(NM)
30 20 10
WOB=38137 N
Vertical Force (Z-axis)
(N)
4500 4000 3500
RPM
RPM=68.2232
Angular Velocity of Drill-string velocity on top velocity on bottom
68.5 68 67.5 200
250
300
350
400
T (s)
Figure 3.17: Large scale view of the time histories of the steady state drilling starting while the drill-bit is already 20mm inside the rock. The average T OB = 15.6 Nm, W OB = 3.81 kN and RP M = 68.2 rpm. 10
TOB [Nm]
8
2.19 kN 1.57 kN 1.10 kN
6
2.06 kN 1.43 kN 0.94 kN 1.88 kN 1.22 kN 0.85 kN
4
2
0
0
20
40 θ˙b [rpm]
60
Figure 3.18: Experimental identification of TOB curves and their fitted curves for different values of WOB (colour map). These curves are conducted for interaction of 3 78 ” PDC drill-bit and sandstone.
48
3.3. Physical parameters identification Table 3.2: Results of testing of 3 87 ” roller cone drill-bit under different combinations of bit angular velocity and WOB. TOB [Nm] 29.689 24.745 22.447 21.639 21.514 20.700 20.338 20.224 20.036 20.867 19.927 19.442 19.217 19.141 17.965 16.545 16.438 15.762 16.109 11.790 09.822 10.149 10.246 10.505 10.468 10.313 10.642 10.774 3.3.2.1
WOB [kN] 5.428 5.382 5.398 5.372 5.350 5.279 5.359 5.255 5.241 5.206 4.164 4.150 4.122 4.130 4.172 4.102 4.106 4.070 4.072 2.844 2.829 2.825 2.838 2.845 2.838 2.800 2.783 2.781
bit angular velocity [rpm] 000.895 001.510 003.235 005.487 011.106 016.731 027.986 056.139 102.921 138.917 000.848 001.426 001.997 002.565 003.140 005.508 016.755 028.016 056.169 001.009 003.292 005.539 011.159 016.781 028.039 056.193 105.507 141.577
Experimental torque modelling
In this section, the experimental torque curves are modelled mathematically, taking into account all qualitatively different patterns which can be seen in Fig. 3.18. A simple frictional model is required that can be calibrated in terms of the experimental observations. The main idea is to use a suitably effective friction coefficient (slip-rate dependent) that captures the main phenomena observed in 49
3.3. Physical parameters identification 0.14
0.12
µ
2.19 kN 1.57 kN 1.10 kN 2.06 kN 1.43 kN 0.94 kN
0.10
1.88 kN 1.22 kN 0.85 kN
0.08 0
20
θ˙b [rpm]
40
60
Figure 3.19: Equivalent friction coefficient as a function of bit rotational speed for different values of WOB (colour map). These curves are calculated based on Equation (A.1) interaction of 3 78 ” PDC drill-bit and sandstone. the drilling tests, namely, cutting and friction between the drill-bit and the rock sample. The process of deriving the equation of this model is described in the Appendix A. The reaction torque takes the explicit form
Tb,sl (θ˙b , γf ) =
Tb,cf , T
θ˙b = 0, (3.1) ˙
b,dr (θb , γf ),
θ˙b > 0,
2 where Tb,cf = λs Wb , 3
1 2 2Wb (λs − λk ) −λd θ˙b 2 ˙2 ˙ Tb,dr (θ˙b , γf ) = λk Wb + 2 − e λ θ + 2λ θ + 2 + Wb λstr θ˙b , d b d b 3 ˙3 3 2 λd θb Wb is WOB, θ˙b is drill-bit angular velocity, λs = µs R, λk = µk R, λd = dc R, λstr = µstr R2 , R is radius of the drill-bit , dc is decay rate and µs , µk and µstr are static friction coefficient, kinetic friction coefficient and Stribeck effect coefficient, respectively. 50
3.4. Experimental results Table 3.3: Parameters of the equivalent friction coefficients for interaction of 3 78 ” PDC drill-bit and sandstone. Wb [kN] 2.19 2.06 1.88 1.57 1.43 1.22 1.10 0.94 0.85
λs 0.0059 0.0068 0.0063 0.0058 0.0063 0.0059 0.0058 0.0000 0.0000
λk 0.0051 0.0051 0.0044 0.0044 0.0043 0.0046 0.0036 0.0045 0.0044
λd λstr 0.3091 1.4890E-05 0.3678 5.9844E-06 0.1219 1.7256E-05 0.1770 2.3248E-05 0.1085 2.0700E-05 0.1298 1.8227E-05 0.0695 3.4923E-05 0.0000 0.0000E+00 0.0000 0.0000E+00
This model includes the following phases: constant friction (Tb,cf ), and statickinetic exponentially decaying friction and its combination with the Stribeck effect (Tb,dr ). Next a curve fitting of the experimental TOB (See Fig. 3.18) to Eq. (3.1) was performed, from which were derived the necessary parameters for equivalent friction coefficients (see Fig. 3.19), corresponding to torque curves for each WOB case. All the necessary parameters to construct experimental torque curves are shown in Table 3.3.
3.4
Experimental results
After identifying the physical parameters of the experimental rig, extensive drilling experiments under different conditions have been carried out in order to observe torsional vibrations and stick-slip vibrations. Some of results are presented in the following sections.
3.4.1
Torsional vibrations
One common mode of vibrations which has been observed in the oil drilling process is torsional motion. In this type of vibration, the drill-bit angular velocity is
51
3.4. Experimental results oscillating around the desired speed however it never reaches to zero. Fig. 3.20 illustrates the angular velocities of the top motor and drill-bit, as well as its phase portrait for 3 experiments under different conditions. In all three cases torsional vibrations are clearly observed. Fig. 3.20 (a) shows the results of the experiment with 20 disks when the flexible shaft is initially straight. However in the second experiment, presented in Fig. 3.20 (b), the flexible shaft is initially 1.5" prebuckled. This experiment is conducted with 12 disks. Fig. 3.20 (c) depicts the results of an experiment with 6 disks and a straight shaft. Note that the rotary speed and drill-bit speed are lower in this experiment.
1.0 0
4.3
7.0
θ˙t [rad/s]
θ˙b [rad/s]
7.0
θ˙b [rad/s]
(a)
3.5 20
10
1.0 -0.2 0.0
1.3
θt − θb [rad]
t [s]
2.0 0
4.5
7.0
θ˙b [rad/s]
θ˙b [rad/s]
7.0
θ˙t [rad/s]
(b)
4.2 20
10
2.0 -0.3
0.0
0.9
θt − θb [rad]
t [s]
2.6 0
3.4
4.2
θ˙t [rad/s]
θ˙b [rad/s]
4.2
θ˙b [rad/s]
(c)
3.1 20
10
2.6 -1.2
0.0
1.0
θt − θb [rad]
t [s]
Figure 3.20: (left) Angular velocities of top motor (blue) and drill-bit (green) and (right) phase portraits of drilling experiment with (a) 20 disks and straight flexible shaft, (b) 12 disks and 1.5” pre-buckled flexible shaft and (c) 6 disks and straight flexible shaft. A variety type of torsional vibrations are observed. 52
3.4. Experimental results
Figure 3.21: Drill-bit angular velocity measured during a typical stick-slip in a real field by the in-bit sensor and chimerical MWD vibration monitor (adopted from [90]).
3.4.2
Stick-slip oscillations
Stick-slip oscillations occur as the extreme case of torsional vibration (Fig. 3.21 which was conducted from a real field [90]). Stick-slip phenomenon has two phases: stick phase when the drill-bit stops for awhile and slip phase when the drill-bit slips fast to release the energy given by the drill-pipe. Figs. 3.22 and 3.23 present angular velocities of the top motor and drill-bit as well as its phase portrait for 8 experiments under different conditions. In all 8 cases stick-slip vibration is observed. The conditions under which each experiment is carried out is presented in Table 3.4. As can be seen in both figures, different types of stick-slip vibrations are observed in all cases. Fig. 3.22 (d) and Fig. 3.23 (b) present typical stick-slip, whereas the rest of the figures show more complex vibrations. Table 3.4: Parameters of the equivalent friction coefficients describing for interaction of 3 78 ” PDC drill-bit and sandstone. Figure number Fig. 3.22 (a) Fig. 3.22 (b) Fig. 3.22 (c) Fig. 3.22 (d) Fig. 3.23 (a) Fig. 3.23 (b) Fig. 3.23 (c) Fig. 3.23 (d)
Number of disks 16 16 14 12 10 10 8 6
53
Initial condition of the shaft 1.5" pre-buckled straight straight straight straight 1.5" pre-buckled 1.5" pre-buckled 1.5" pre-buckled
3.4. Experimental results (a)
θ˙t [rad/s]
0.0 -0.5 0
θ˙b [rad/s]
1.55 4.0
θ˙b [rad/s]
4.0
0.0 1.35 -0.5 20 -1.2
10
0.0
1.0
θt − θb [rad]
t [s]
2.5
0.8
2.5
θ˙b [rad/s]
θ˙t [rad/s]
θ˙b [rad/s]
(b)
0.0
0.0 0.5 30
-0.5 0
15
-0.5 -0.3
0.0
0.6
θt − θb [rad]
t [s]
(c)
θ˙t [rad/s]
0.0 0
3.0 30
15
8.0
θ˙b [rad/s]
3.8
θ˙b [rad/s]
8.0
0.0 -0.5 -0.5
0.0
2.0 θt − θb [rad]
t [s]
(d)
θ˙t [rad/s]
0.0 0
2.5 20
10
8.5
θ˙b [rad/s]
4.8
θ˙b [rad/s]
8.5
0.0 -0.5 -0.5 0.0
2.5 θt − θb [rad]
t [s]
Figure 3.22: (left) Angular velocities of top motor (blue) and drill-bit (green) and (right) phase portraits of drilling experiment with (a) 16 disks and 1.5 inch pre-buckled flexible shaft, (b) 16 disks and straight flexible shaft, (c) 14 disks and straight flexible shaft and (d) 12 disks and straight flexible shaft. A variety of stick-slip vibrations are observed in these cases. Case (d) is typical whereas the rest are more complex. 54
3.4. Experimental results
2.5
1.2
2.5
θ˙b [rad/s]
θ˙t [rad/s]
θ˙b [rad/s]
(a)
0.0
0.0
-0.5 0
0.8 20
10
-0.5 -0.2
0.0
0.7 θt − θb [rad]
t [s]
(b)
θ˙t [rad/s]
0.0 -0.5 0
1.5 20
10
8.0
θ˙b [rad/s]
1.8
θ˙b [rad/s]
8.0
0.0 -0.5 -1.0
0.0
2.0
θt − θb [rad]
t [s]
(c)
θ˙t [rad/s]
0.0 0
θ˙b [rad/s]
1.55 4.0
θ˙b [rad/s]
4.0
0.0 1.35 -0.5 20 -0.9
10
0.0
0.7
θt − θb [rad]
t [s]
(d)
θ˙t [rad/s]
0.0 0
1.5 20
10
4.0
θ˙b [rad/s]
1.8
θ˙b [rad/s]
4.0
0.0 -0.5 -0.4
0.0
0.8
θt − θb [rad]
t [s]
Figure 3.23: (left) Angular velocities of top motor (blue) and drill-bit (green) and (right) phase portraits of drilling experiment with (a) 10 disks and and straight flexible shaft, (b) 10 disks and 1.5 inch pre-buckled flexible shaft, (c) 8 disks and 1.5 inch pre-buckled flexible shaft and (d) 6 disks 1.5 inch pre-buckled flexible shaft. A variety of stick-slip vibrations are observed in these cases. Case (b) is typical whereas the rest are more complex. 55
3.5. Conclusions
3.5
Conclusions
In this chapter, the experimental rig including top motor, gearing and pulley system, drill-pipes, WOB system, drill-bits, rock samples, fixtures and fluid system, as well as sensors, instrumentation and data acquisition and control system were fully described and two different configurations were explained. The parameters of the experiment were then determined and the detailed experiments for different drilling conditions described. In the first configuration, the rigid shaft was used in order to develop empirical models for drill-bit and rock interaction. In this regard, extensive experimental results were conducted in this configuration. Curve fitting procedures were then carried out and the model was developed. This model includes the two phases: constant friction, and static-kinetic exponentially decaying friction and its combination with the Stribeck effect. In the second configuration, flexible shafts were used to mimic the mechanical properties of slender structures like a drill-string. The experimental results presented, obtained in this configuration, include torsional vibrations and stick-slip phenomena observed in the rig in different conditions. Two types of stick-slip were observed. The typical stick-slip is significantly similar to the one obtained in real field by the in-bit sensor and chimerical MWD vibration monitor. Nontypical stick-slip vibrations are a combination of typical stick-slip and torsional vibration in each period. In these experimental results the bit-speed reached up to 4.5 times of the top-speed. In the next chapter a one DOF model will be presented to predict the drillstring behaviour. The top motor and gearing system and frequency convertor will then be modeled in order to suppress the torsional and stick-slip vibrations.
56
Chapter 4 Suppression of torsional vibration In the previous chapter the experimental drilling rig was introduced the physical parameters of the drill-string were estimated and some experimental results on torsional vibration and stick-slip vibration during drilling were obtained. The aim of the study presented in this chapter is to model and ultimately suppress torsional vibration and stick-slip vibration exhibited in the experimental drilling rig. This chapter is subdivided into eight sections. In Section 4.1 a harmonically excited torsional model is presented. This model is able to mimic the stickslip observed in the rig. A 1-DOF model for the top motor and gearing system and its calibration are explained in Section 4.2. In Section 4.3 the whole rig is considered and a 2-DOF lumped mass model introduced to predict stick-slip results. In Section 4.4 a sliding mode control method is presented and adopted by the model in order to suppress the torsional vibrations. This control method is experimentally verified and the results are presented in Section 4.5. In Section 4.6 delay observed in the actuator is modeled and added to the full model. The main conclusions of the chapter are then presented.
57
4.1. Harmonically excited torsional model
4.1
Harmonically excited torsional model
In this section, the torsional vibration observed in the experiment (as presented in Section 3.4) is modelled. Several attempts have been made to develop models to explain the dynamics of drill-strings. These models range in sophistication from simple lumped mass models to nonlinear Finite Element models. A detailed survey of the various drill-string dynamics models is presented in Section 2.1. While some of these models have been claimed to be able to match to the experiments and even real scenarios, none of them are both fast and simple enough for control purposes. Most of these are detailed models, aimed at predicting coupled dynamic phenomena. They are useful for postmortem failure analyses but are quite difficult to calibrate. As a result, attention should be given to the simple but efficient lumped mass models. To that end a 1-DOF model is introduced, as shown in Fig. 4.1. This model is based on a torsional pendulum approach (see e.g. [155, 69]).
4.1.1
Equation of motion
The state variables and parameters of the drilling system can be defined as the real vectors u = (θb , ωb )T and α = (J, c, k, ωt , Tb,cf , ωp , Ap , θp , γf ) respectively. Here, J, c, k are moment of inertia of BHA, damping coefficient and stiffness of the flexible cable respectively. ωt , ωp ,Ap and θp are also parameters of the model as described in the previous section. γf represents the vector containing the parameters used to model the bit-rock interaction including WOB (see Section 4.1.1.1). θt is described in Eq. (4.3). The equation governing the behaviour of the system is given by: J θ¨b + c(θ˙b − θ˙t ) + k(θb − θt ) = −Tb , 58
(4.1)
4.1. Harmonically excited torsional model
θt
drill-pipe
c, k
J
BHA drill-bit θb
Tb
Figure 4.1: Physical model of a 1-DOF lump mass torsional model with external excitation θt . The visco-elasto properties of the pipe are given by the dampingstiffness pair c, k. The reactive torque acting on the system during drilling is represented by Tb .
which can be written as a first order Ordinary Differential Equation (ODE) as follows:
˙ ωb θb u˙ = , = 1 (c (ωt − ωb + Ap sin(ωp t + θp )) + k (θt − θb ) − Tb (ωb , γf )) ω˙ b J
(4.2)
where a single overdot denotes differentiation with respect to time t, the function Tb gives the reaction torque (see Eq. (4.5) in the next section) and θt is as prescribed below. The model is comprised of a massless rotary table. Its motion is described 59
4.1. Harmonically excited torsional model by the angular variable θt , i.e. the motor and gearing system are not modelled. However, the top of the drill pipe is subjected to the external excitation which is observed in the experiments θ˙t = ωt + Ap sin(ωp t + θp ),
(4.3)
where, ωt represents a constant spin speed, and Ap is a small constant that controls the amplitude of the harmonic perturbation driven at the frequency ωp and phase shift θp . This is to account for the sinusoidal fluctuations observed in real applications when large torsional vibrations take place during operation (see e.g. [69, 170, 101]). It is worth noting that for this physical model the following simplifying assumptions are made: the borehole is perfectly vertical and lateral motions are restrained; the WOB acting on the system is constant during operation; and the drill-bit (considered as a part of the BHA) never rotates counterclockwise, i.e. the instantaneous angular velocity of the BHA is always non-negative. In the proposed model an elastic shaft with rotational stiffness k and damping c connects the rotary table with the BHA, which has a moment of inertia J and an angular displacement θt . The torque generated by the interaction between the drill-bit and rock is denoted by Tb . 4.1.1.1
Torque on bit
To calculate Tb different modes of the system need to be considered. The system is assumed to operate under one of two main modes at any time; stick or slip, corresponding to the cases when the drill-bit has zero or positive angular velocity respectively. The bit-rock interaction is modelled by the function Tb , which represents the reaction torque and takes the form Tb = Tb,st (resp. Tb = Tb,sl ) during
60
4.1. Harmonically excited torsional model the stick (resp. slip) phase.
Tb (θ˙b , γf ) =
Tb,st (t, u, α), θ˙b = 0 and Tb,st (t, u, α) < Tb,cf (γf ), Tb,sl (θ˙b , γf ),
(4.4)
otherwise.
Slip phase Tb,sl
S!ck phase
θ˙b > 0
Tb,st ≥ Tb,cf mode A
mode B
mode C
(Tb,st )
(Tb,cf )
(Tb,dr ) θ˙b = 0
Tb,st < Tb,cf
Figure 4.2: The model has two phases: stick phase which includes 1 mode of operation (mode A) and slip phase which has 2 modes (mode B & C).
Equation (3.1) described the TOB of slipping phase and has two conditions. Therefore, substituting Eq. (3.1) into Eq. (4.4), the full bit-rock interaction model shows all three modes of operation as follows: Tb,st (t, u, α), θ˙b = 0 and Tb,st (t, u, α) < Tb,cf (γf ), Tb (θ˙b , γf ) = Tb,cf (γf ), θ˙b = 0 and Tb,st (t, u, α) ≥ Tb,cf (γf ), Tb,dr (θ˙b , γf ), θ˙b > 0,
61
(4.5)
4.1. Harmonically excited torsional model where Tb,cf (γf ) and Tb,dr are defined as follows: 2 Tb,cf (γf ) = λs Wb (4.6) 3 2Wb (λs − λk ) 2 −λd θ˙b 2 ˙2 ˙ 2 − e λ θ + 2λ θ + 2 Tb,dr (θ˙b , γf ) = λk Wb + d b d b 3 λ3d θ˙b3 1 + Wb λstr θ˙b . (4.7) 2 During sticking phases (θ˙b = 0 and Tb,st (t, u, α) < Tb,cf (γf )), the reaction torque is computed via Newton’s third law as follows: Tb,st (t, u, α) = c θ˙t + Ap sin(θ˙p t + θp ) + k (θt − θb ) = cθ˙t + k (θt − θb ) , (4.8) which means that the reaction torque adjusts itself to enforce the equilibrium with the external torque acting on the drill-bit. Figure 4.2 shows the system’s three modes of operation. To find the current mode of the system, when the drill-bit is stuck (mode A), the function Tb,st is monitored. The stick phase (mode A) terminates when Tb,st becomes equal to Tb,cf (γf ). At this point the reaction torque Tb,st has reached the break-away torque value Tb,cf (γf ) and the system is in the slip phase (mode B). As soon as the drill-bit begins to rotate the system is in (mode C) of the slip phase. According to the modes defined above, the system can be fully described by Eqs (4.2, 4.5 - 4.8).
4.1.2
Numerical results and experimental verification
In this section an experimental verification of the 1-DOF model is carried out. It is worth reiterating that the goal of this work is to develop a robust calibrated lowdimensional model to investigate torsional vibrations and stick-slip oscillations occurring in drill-strings. 62
4.1. Harmonically excited torsional model
8
3.9
8
6
3.7
6
θ˙t [rad/s] θ˙b [rad/s]
θ˙b [rad/s]
(a)
3.5
4
4
3.3 2
2
3.1 0
0 -1
0
9
18
t [s]
27
2.9 45
36
-1 -1
2
θt − θb [rad]
(b) 8
3.9
8
6
3.7
6
θ˙t [rad/s] θ˙b [rad/s]
θ˙b [rad/s]
1
0
3.5
4
4
3.3 2
2
3.1 0
0 -1 0
9
18
t [s]
27
2.9 45
36
-1 -1
2
θt − θb [rad]
(c) 0.08
| θ˙t (f ) |
| θ˙b (f ) |
4
0.04
0.00 0
1
0
2
0 2
4
0
2
4
f (Hz)
f (Hz)
Figure 4.3: An example of stick-slip oscillations occurring in the experimental rig for Wb = 2.19 kN and the straight flexible shaft. The time histories of the angular velocities at the bottom, θ˙b , and the top, θ˙t , phase portraits from (a) experimental studies, (b) low-dimensional model and (c) FFT of the angular velocity of the top (left) and the drill-bit (right) for experimental (red curve) and model (green curve), where |θ˙t (f )| and |θ˙b (f )| denote amplitude of FFTs as a function of frequency f .
63
4.1. Harmonically excited torsional model A response of the stick-slip oscillations of a drill-bit with straight flexible shaft is considered for WOB value of Wb = 2.19 kN. Its experimental time history and phase portrait can be seen in Fig. 4.3 (a). The top velocity (black curve) is given as a sinusoidal excitation, which results in stick-slip oscillations of almost constant amplitude. Following the curve fitting, this excitation is replicated using function: θt = ωt + Ap sin(ωp t), where ωt = 3.419 rad/s, ωp = 3.415 rad/s and Ap = 0.064 m. Note that in this particular case the angular frequency ωt and ωp are very close to each other. To confirm that, a Fast Fourier Transform (FFT) of the signal recorded for the top excitation is carried out (see red curve in left panel of Fig. 4.3 (c)). Clearly one can see a major peak at frequency f = 0.545 Hz, which corresponds directly to both ωt and ωp (ωt = 3.419 rad/s= 0.5442 Hz and ωp = 3.415 rad/s= 0.5435 Hz). The minor peaks observed in the FFT are just harmonics of the major frequency (f = 0.545 Hz) and are not present in the FFT of top excitation for the model (green curve). In order to calibrate the model and fit it to the experimental data, the parameters close to identified parameters are applied: k = 19.000 Nm/rad and c = 0.005 Nms/rad. The torsional spring stiffness k is adjusted from the value determined during flexible shaft identification (see Section3.3.1). Note that the variation of stiffness and damping is justified by the fact that the identification has been performed for the flexible shaft in tension, whereas during the experiments the flexible shaft has been kept straight but not in tension. A TOB model of Eq. (4.5) shall be used for the TOB formulation, with corresponding parameters shown in Table 3.3 for the WOB value of Wb = 2.19 kN. For the model the drill-bit responds with stick-slip oscillations of constant amplitude (Fig. 4.3 (b)). It has an excellent agreement with the experimental observations, as can be seen when comparing the phase portraits shown in Figs 4.3 (a) and (b). The agreement between experiment and performed sim64
4.2. Modelling of motor and gearing system ulation is confirmed when the FFT of the bit angular velocity for these cases is analyzed. As can be seen in right panel Fig. 4.3 (c), only the major peak corresponds to the rotation at the top at f = 0.545 Hz, which is visible also in the responses from the model (green curve). The other peaks visible in the experimental and model cases represent only harmonics of the main frequency.
4.2
Modelling of motor and gearing system
In the previous section a 1-DOF model was introduced which can explain some of the drill-string dynamics of the rig from the top-end to the bottom-end, including the bit-rock interaction. However, in this model it has been assumed that the top-end of the drill-string is harmonically excited by the small sinusoidal vibrations which were originally observed in the experiment. In this section a 1-DOF model for the top motor and gearing system is considered and its parameters are identified, in order to develop a model of the whole rig. In this regard the top motor and gearing is considered as a disk (with moment of inertia; Jt ), as shown in Fig. 4.4. The only possible motion of the disk is a rotation about an axis fixed in space. This disk is subject to a driving torque Tt and to a viscous drag torque proportional to the angular velocity through coefficient ct .
65
4.2. Modelling of motor and gearing system
Tt
ct Jt
θt
Figure 4.4: A physical model of the motor and gearing system as a disk (with moment of inertia Jt ) which is subject to a driving torque Tt and a viscous damping with coefficient of ct .
4.2.1
Torque control of motor
Before identifying the parameters of the proposed model for the top motor, the DAQ and open loop torque control system for the top motor shall be explained. As shown in Fig. 4.5, the motor torque is controlled by DAQ and control system (Labview) through the frequency convertor. Note that the control parameter (Tc ) of this convertor is expressed as a percentage as this is required in the experimental system. In the other words, Tc in [%] in the Labview program is converted to a voltage (0-5 V), and then further converted with a voltage-to-current convertor to a current (4-20 mA). The frequency convertor receives this current and it drives the motor, in order to achieve the corresponding requested torque. Also, the frequency convertor produces a corresponding current (4-20 mA) which represents an estimation of the torque generated by the motor. This current is converted to a voltage (0-10 V) by a resistor and is read by DAQ system and converted to a value Tˆ in [%]. Therefore the first step is to find a way to estimate the absolute value of torque generated by motor Tt in Nm and to estimate the relationship between requested torque Tc , estimated torque Tˆ and generated torque Tt . In this regard, the rigid shaft was utilized in the setup shown in Fig. 4.6. The
66
4.2. Modelling of motor and gearing system
T c [%]
DAQ and control system
Data from sensors
^T [%]
Frequency convertor
Motor and gearing
T t [Nm]
Drill-string, BH A and drillbit
Figure 4.5: A schematic of DAQ and open loop torque control system for top motor. Tc , Tˆ are requested torque and estimated torque in [%] respectively, and Tt is the torque generated by motor in [Nm].
equation of motion of this system can be written as follows: ˙ Tt − Tb = (Jt + J)θ¨ + ct θ,
(4.9)
where θ, Jt , J, ct , Tt and Tb are angular position of the drill-pipe (and BHA and drill-bit), moment of inertia of the motor, moment of inertia of BHA (together with the drill-bit), effective damping coefficient of the motor and gearing system, top torque generated by motor, and the reaction torque from bit-rock interaction, respectively. Note that θ and Tb are observed by sensors. Note also that Tˆ is the motor torque estimated by a sensor placed inside the frequency convertor, and expressed as a percentage of full capacity of the motor. Extensive experiments were carried out with different WOB (using a pulley system which results in variation of TOB Tb and without any changes in J) and in different requested torque Tc . In each set of parameters, the results were recorded after 20 s when the drill-bit and drill-pipe are settled at a constant speed; θ˙ = cte. ˙ Tb and Tˆ for one minute drilling in different WOB The average values of Tc , θ, and Tc are presented in Table 4.1.
67
4.2. Modelling of motor and gearing system
Tt
ct Jt rigid shaft
θ
J
Tb
Figure 4.6: A physical model of experiment employed to estimate the relationship between requested torque Tc , estimated torque Tˆ and generated torque Tt by the motor.
Equation (4.9) can be applied to the results in Table 4.1, bearing in mind that the drill-bit speed in these experiments is fixed; θ¨ = 0. Consider the results on pairs: (No. 1 and 2), (No. 3 and 4), (No. 5 and 6) and (No. 7 and 8). In each pair of results, the requested torques Tc are the same and the estimated torques Tˆ are very close. Therefore it can be assumed that the torque provided by the motor Tt in each aforementioned pair of the results are equal. Consequently the torque Tt for each pair can be calculated as follows:
Tt =
θ˙2 Tb1 − θ˙1 Tb2 , θ˙2 − θ˙1
(4.10)
where subscripts 1 and 2 denote the first and second experiments in each pair respectively. Following this calculation, the torques generated by the motor Tt 68
4.2. Modelling of motor and gearing system for each pair of experiments (No. 1 and 2), (No. 3 and 4), (No. 5 and 6) and (No. 7 and 8) are 45.29, 47.52, 45.30 and 38.33 Nm respectively. These experiments have been repeated seven times and the results have been averaged. Based on theses results the relationships between Tt , Tˆ and Tc are estimated as follows: Tt = −5.453 Tˆ + 24.844, Tc = −0.229 Tt + 5.180,
(4.11)
where Tt and Tˆ are in [%] and Tc is in [Nm] (see Fig. 4.5). ˙ Tb and Tˆ of 1 minute drilling with different Table 4.1: The averages values of θ, WOB and Tc . The results were recorded after 20 s when the drill-bit and drill-pipe settled at a constant speed; θ˙ = cte. No. Tc [%] 1 -6.00 2 -6.00 3 -5.60 4 -5.60 5 -5.35 6 -5.35 7 -3.60 8 -3.60
4.2.2
WOB [kN] 2.50 3.50 3.93 1.95 3.06 1.44 2.55 0.55
θ˙ [rad/s] 3.44 2.86 2.29 3.44 2.86 3.44 2.29 2.87
Tb [Nm] 9.51 15.52 20.8 7.32 12.23 5.55 9.60 2.26
Tˆ [%] -4.39 -4.39 -4.07 -4.01 -3.73 -3.62 -1.97 -2.04
Moment of inertia and damping coefficient of motor
After having estimated the torque provided by the motor Tt , another set of experiments were designed and carried out in order to identify the effective moment of inertia of the motor and gearing system Jt and the effective damping coefficient ct . In this set of experiments the drill-pipe is disconnected from the motor and a constant signal Tc is applied to the system. The dynamic response of the system is then observed (θt ). Based on the schematic of the model shown in Fig. 4.4, the
69
4.2. Modelling of motor and gearing system equation of motion can be written as follows: Tt = Jt θ¨t + ct θ˙t .
(4.12)
It has been observed that the motor reaches a constant speed in steady state (t = tf ) which means θ˙t (tf )=cte and θ¨t (tf ) = 0. Therefore, in this state, using Eq. (4.12) the damping coefficient ct can be calculated as follows: ct = Tt /θ˙t (tf ).
(4.13)
On the other hand, the acceleration method [48] can be applied to this motor and gearing system. In this method the effect of the viscous drag torque is negligible at the beginning of the process. Therefore, the moment of inertia can be estimated as follows: Jt = Tt /θ¨t (tl ),
(4.14)
where tl is the end of the linear part of the response (θ˙t ). To increase the accuracy of estimation, the acceleration θ¨t can be calculated based on the line-fitting to the response of the system θ˙t . This estimation has an error as the viscous damping is neglected. The error can be calculated as follows [48]:
e=|
∆Jt 1 ct T |= . Jt 3 Jt
(4.15)
Some experimental results are presented in Table 4.2. For example, in the first experiment a constant value of Tc = −2.00 % is applied to the frequency convertor and the estimated torque Tˆ = −0.65 % is observed. It is worth noting that the negative sign shows the direction of the rotation. Therefore based on Eq. (4.11)
70
4.3. Two disks torsional model the torque generated by the motor, applied to the gearing, can be estimated as Tt = 28.414 Nm. In the steady state (t = tf ), the acceleration of the gearing system becomes zero (θ¨ = 0) and the system rotates with a constant speed (θ˙t (tf ) = 2.483 rad/s). Therefore based on Eq. (4.13), ct = 11.444 Nms/rad. Considering the first 0.7 s of the experiment from t = 0 s (when θ˙t = 0) to t = 0.7 s, a line is fitted to the response (θ˙t ) and Jt is calculated based on Eq. (4.13); Jt = 12.714 kg m2 . Finally, based on Eq. (4.15) a 21 %-error is estimated. Following a series of experiments, ct , Jt and the error are calculated and averaged. For the rest of this chapter the average values for the effective moment of inertia and damping coefficient of the motor and gearing system are used, Jt = 13.93 kg m2 and ct = 11.38 Nms/rad. Table 4.2: Results of experiments including estimation for ct , Jt and error based on Eqs (4.13 - 4.15). No. Tc [%] 1 -2 2 -2 3 -3 4 -3 5 -4 6 -4
4.3
˙ f ) [rad/s] cˆt [Nms/rad] Tˆ [%] T [Nm] θ(t -0.655 28.414 2.483 11.444 -0.653 28.404 2.500 11.361 -1.583 33.477 2.926 11.440 -1.598 33.556 2.928 11.461 -2.605 39.049 3.433 11.376 -2.594 38.991 3.430 11.368
Jˆt [kg m2 ] 12.714 12.800 13.985 12.914 16.532 14.629
error [%] 21.00 20.71 19.08 20.70 26.54 26.65
Two disks torsional model
In the previous section the top motor and gearing system were modelled and parameters of the model identified. In this section, using this model a 2-DOF model is introduced in order to model the whole drilling rig. Fig. 4.7 presents the proposed model. This model is a combination of the model presented in Figs. 4.1 and 4.4. The viscous damping property of the motor and gearing system and the 71
4.3. Two disks torsional model visco-elasto properties of the drill-pipe are given by ct , c and k, respectively. The reactive torque acting on the system during drilling is represented by Tb . It is worth remembering that in the first model (Fig. 4.1) an external excitation θt was introduced on the top-end of the drill-pipe. When the frequency convertor is set in the speed control mode this external excitation is observed in the experiment. In the model presented in this section, the frequency convertor is set in the torque control mode and the top speed can be calculated from the equation of motion. Tt
ct Jt
motor θt drill-pipe
c, k J
BHA drill-bit θb
Tb
Figure 4.7: A physical model of a 2-DOF lump mass torsional system. The viscous damping property of the motor and gearing system and the visco-elasto properties of the pipe are given by Ct , c and k, respectively. The reactive torque acting on the system during drilling is represented by Tb .
4.3.1
Equations of motion
The state variables and parameters of this model can be defined as the real vectors u = (ωt , θt , ωb , θb )T and α = (Jt , J, ct , c, k, Tb,cf (γf ), γf ), respectively. Here, Jt , J, ct , c and k are moment of inertia of BHA and motor, damping coefficient 72
4.3. Two disks torsional model of the flexible cable and motor and stiffness of the flexible cable respectively. γf also represents the vector containing the parameters used to model the bitrock interaction including WOB(see Section 4.3.1.1). The equation governing the behaviour of the system presented in Fig. 4.7 is given by: Jt θ¨t + (ct + c)θ˙t − cθ˙b + kθt − kθb = Tt , Jb θ¨b − cθ˙t + cθ˙b − kθt + kθb + Tb = 0,
(4.16)
which can be written as a first order ODE as follows:
ω˙ t θ˙t u˙ = = ω˙ b ˙θb
Jt−1 (−(c
+ ct )ωt + cωb − kθt + kθb + Tt ) ωt
Jb−1 (cωt
− cωb + kθt − kθb − Tb ) ωb
,
(4.17)
where a single overdot denotes differentiation with respect to time t, the function Tb gives the reaction torque (see next section) and the control input Tt is the torque generated by the motor. 4.3.1.1
Torque on bit
TOB (Tb ) can be calculated in a similar procedure as the one in Section 4.1.1.1. The two phases (three modes) presented in Fig. 4.2 and Eq. 4.5 are valid. However during sticking phases (θ˙b = 0 and Tb,st (t, u, α) < Tb,cf (γf )), the reaction torque is re-computed via the Newton’s third law as follows: Tb,st (t, u, α) = c(θ˙t − θ˙b ) + k(θt − θb ).
73
(4.18)
4.3. Two disks torsional model According to the modes defined above, the system can be fully described by Eqs (4.5-4.7, 4.17 and 4.18).
4.3.2
Numerical results and experimental verification
In this section an experimental verification of the 2-DOF model is carried out. A response is considered of the stick-slip oscillations of the drill-bit, for WOB value of Wb = 1.76 kN, and a 1.5 inch pre-buckled flexible shaft and Tt = 39.57 Nm torque generated by motor. Its experimental time history and phase portrait can be seen in Fig. 4.8 (a). As can be seen, the top velocity (black curve) has a sinusoidal vibration, and the drill-bit velocity experiences stick-slip oscillations of almost constant amplitude (red curve). Fast Fourier Transform (FFT) of the signals recorded for the top and drill-bit are carried out (see red curves in Fig. 4.8 (c)). One can clearly see a major peak at frequency f = 0.49 Hz for both top and drill-bit velocity. In order to calibrate the model and fit it to the experimental data, values close to identified parameters are applied: k = 10.00 Nm/rad, c = 0.005 Nms/rad, Jt = 13.93 kg m2 and ct = 11.38 Nms/rad. A TOB model of Eq. (4.5) is used for the TOB formulation, with corresponding parameters shown in Table 3.3 for the WOB value of Wb = 1.79 kN. For the model, the bit responds with stick-slip oscillations of constant amplitude (Fig. 4.8 (b)). There is excellent agreement with the experimental observations, as can be seen when comparing the phase portraits shown in Figs. 4.8 (a) and (b). The agreement between experimental and performed simulation is confirmed after analyzing FFT of the top and bit angular velocities for these cases. As can be seen in Fig. 4.8 (c), there is the major peak corresponding to the rotation at the top at f = 0.49 Hz, which is also visible in the responses from the model (green curves). 74
4.3. Two disks torsional model (a)
0
0
−1
−1 0
9
27
18
36
45
t[s]
0
−1
1
2
θt − θb [rad]
(b)
7
θ˙t , θ˙b [rad/s]
7
3.5
3.5
0
0
−1
−1 0
9
27
18
36
45
t[s]
0
−1
x10
3
x10
5
| θ˙t (f ) |
| θ˙b (f ) |
3.5
1.5
0.0
2
θt − θb [rad]
(c) 3.0
1
θ˙b [rad/s]
θ˙t , θ˙b [rad/s]
3.5
3.5
θ˙b [rad/s]
7
7
1.75
0
2
4
0.0 0
2
4
f (Hz)
f (Hz)
Figure 4.8: An example of stick-slip oscillations occurring in the experimental rig for Wb = 1.79 kN and 1.5 inch pre-buckled flexible shaft. The time histories of the angular velocities at the bottom, θ˙b , and the top, θ˙t , phase portraits from (a) experimental studies, (b) low-dimensional model and (c) FFT of the angular velocity of the top (left) and the drill-bit (right) for experimental (red curve) and model (green curve), where |θt (f )| and |θb (f )| denote amplitude of FFTs as a function of frequency f .
75
4.3. Two disks torsional model
(a)
θ˙t , θ˙b [rad/s]
8
4
0 5
0
10
t[s]
Tb [Nm]
8
5
2 0
5
10
t[s] (b)
θ˙t , θ˙b [rad/s]
8
4
0 0
5
10
t[s]
Tb [Nm]
8
5
2 0
5
10
t[s] Figure 4.9: A zoomed-in view of Fig. 4.8 (a) experiential studies, (b) 2-DOF model together with TOB recorded in the experiment and modelled by Eq. (4.5) (blue curves).
76
4.4. Suppressing the torsional vibrations It is worth noting that the vibration observed in the top velocity in this experiment is much higher than that observed in the previous experimental results (compare Fig. 4.3 and Fig. 4.8). Apart from differences in WOB and state of flexible shaft, the main differences between these two experiments are the different modes of control in frequency convertor. In the first experiment the convertor was set in the speed control mode, and therefore the torque was changing in order to keep the speed at an almost constant required value (in an open loop). However in the experiment presented in Fig. 4.8 the frequency convertor is set in the torque control mode. Therefore the motor torque is kept at a constant required value (in the open loop) and top speed changes according to the dynamical behaviour of the rest of the system. Therefore the amplitude of the vibrations at the top in the first experiment (speed control) is much lower than in the second (torque control), as can be seen by comparing the black curves in Fig. 4.3 (a) and Fig. 4.8 (a). To have a deeper understanding of the system and to confirm the TOB model, Fig. 4.9 shows a zoomed-in view of Fig. 4.8 together with TOB recorded in the experiment and modelled by Eq. (4.5). The top speed in the experiment and the model (black curves) clearly shows very similar behaviour. It can also be seen that the bit speed in model (green curve) is perfectly matched to the experimental data (red curve). More interestingly, the two phases of system can be observed in experiment and model TOB data (blue curves). There is a significant drop in TOB when the stick phase starts. The TOB increases in this phase until reaching the break-away value, after which the system goes to the slip phase.
4.4
Suppressing the torsional vibrations
In the previous section the whole experimental rig was modelled and calibrated. It is worth remembering that in this model the torque Tt generated by top motor
77
4.4. Suppressing the torsional vibrations is the control input of the system. The next step is to design a suitable control method and then to apply it to the model in order to decrease the torsional vibrations and eliminate stick-slip oscillations during drilling. A detailed literature review on control of torsional vibration was presented in Section 2.1. After considering different methods such as [22] which proposed to use WOB as a control variable for extinguishing stick-slip oscillations, the most suitable method for the proposed model and the experimental rig is the sliding-mode controller [102]. This controller is an extended version of the one proposed in [118]. In this section this controller will be applied to the system.
4.4.1
Sliding-mode-control
The state variables of the 2-DOF model can be redefined as the real vectors X = (ωt , θt − θb , ωb )T . Note that here the number of states has been reduced to three as for the proposed control method it is enough to know the differences between the top and bit angular positions instead of both of them. The equation of motion can be written as a first order ODEs as follows:
x˙1 X˙ = x˙2 = x˙3
Jt−1 (−(c
+ ct )x1 + cx3 − kx2 + Tt ) , x1 − x3 Jb−1 (cx1 − cx3 + kx2 − Tb )
(4.19)
where a single overdot denotes differentiation with respect to time t, the function Tb gives the reaction torque (see Eq. (4.20)) and the control input Tt is the torque generated by the motor. Substituting a new state vector X in Eqs (4.5-4.7
78
4.4. Suppressing the torsional vibrations and 4.18) gives: Tb,st (t, X, α), x3 = 0 and Tb,st (t, X, α) < Tb,cf (γf ), Tb (x3 , γf ) = Tb,cf (γf ), x3 = 0 and Tb,st (t, X, α) ≥ Tb,cf (γf ), Tb,dr (x3 , γf ), x3 > 0, Tb,st (t, X, α) = c(x1 − x3 ) + kx2 ,
(4.20)
2 Tb,cf (γf ) = λs Wb , 3 2Wb (λs − λk ) 2 −λd x3 2 2 Tb,dr (x3 , γf ) = λk Wb + 2 − e λ x + 2λ x + 2 d 3 d 3 3 λ3d x33 1 + Wb λstr x3 , 2 where all parameters are defined as before. In order to find the fixed points of the system, the equation X˙ = 0 is considered. Two fixed points can be found for two mode of the system as follows: stick phase ⇒ θ˙b = 0 ⇒ ¯ st = (¯ X xst1 , x ¯st2 , x¯st3 )T = (0, Tt /k, 0)T , steady state drilling ⇒ θ˙b > 0 ⇒
(4.21)
¯ sl = (¯ X xsl1 , x¯sl2 , x ¯sl3 )T = (ωsl , (Tt − c ωsl )/k, ωsl )T , where ωsl is a constant angular velocity which depends on the Wb and Tt . It is worth noting that if the torque generated by the motor Tt is not high enough then Tb,st (t, X, α) cannot reach the Tb,cf (γf ) (break-away) value and there would ¯ st . Moreover, in case of existence, X ¯ st is therefore be just one fixed point X ¯ sl is locally asymptotically stable [118]. asymptotically stable and X
79
4.4. Suppressing the torsional vibrations ωd
sliding-mode controller
Tt
2-DOF model for motor and drilling rig
θb , θt , θ˙b , θ˙t
Figure 4.10: The structure of the suggested sliding-mode controller for the model. In this method the controller changes the control parameter Tt in order to keep the top and bit speed θ˙t and θ˙b close to the desired rotational speed ωd . 4.4.1.1
Controller structure and sliding surface
Figure 4.10 depicts the structure of the suggested sliding-mode controller. The objective of the controller is to lead the system to Xsl where ωsl = ωd by changing the control input Tt . Therefore a sliding surface can be defined and its derivative with respect to time can be calculated as follows:
s = (x1 − ωd ) + λ s˙ =
Z
t
t0
(x1 − ωd )dτ + λ
Z
t
t0
(x1 − x3 )dτ,
1 (Tt − (c + crt )x1 + cx3 − kx2 ) + λ(x1 − ωd ) + λ(x1 − x3 ), Jt
(4.22) (4.23)
where ωd is the desired angular velocity, t0 is the starting time of the controller and λ is a positive control parameter. Let us consider what the states of the system are in the sliding surface (s = 0). So the Tid can be found from the solution of s˙ = 0 as follows:
Tid = (c + crt )x1 − cx3 + kx2 − Jt λ(x1 − ωd ) − Jt λ(x1 − x3 ).
(4.24)
Once the system is in the sliding surface and the model is ideal without any uncertainties and extra un-modelled dynamics, Tid leads the state of the system asymptotically to the desired fixed point x¯sl (ωd ). This can be proved by using a Lyapunov function such as V = 21 (Jt (x1 − x¯sl1 ) + k(x2 − x¯sl2 ) + J(x3 − x¯sl3 )). By substituting Eq. (4.24) into V , it can be seen that V˙ ≤ 0, and V˙ = 0 for 80
4.4. Suppressing the torsional vibrations ¯ sl (ωd ) [102]. X=X 4.4.1.2
Parameter uncertainties
In order to eliminate the uncertainties in the parameters’ estimation, the equivalent control, the switching control and eventually a sliding-mode controller can be defined (same as the one in [102]) as follows: ˆ 2 − Jˆt λ(x1 − ωd ) − Jˆt λ(x1 − x3 ), Teq = (ˆ c + cˆrt )x1 − cˆx3 + kx
Tsw = − − −
(4.25)
Mc |x1 − x3 |s Mcrt |x1 |s − Rt Rt |s| + δ1 exp(−δ2 t0 |x1 − x3 |dτ ) |s| + δ1 exp(−δ2 t0 |x1 |dτ )
Mk |x2 |s MJt λ|x1 − ωd |s − Rt Rt |s| + δ1 exp(−δ2 t0 |x2 |dτ ) |s| + δ1 exp(−δ2 t0 λ|x1 − ωd |dτ ) MJt λ|x1 − x3 |s − κs Rt |s| + δ1 exp(−δ2 t0 λ|x1 − x3 |dτ )
Tt = Teq + Tsw ,
(4.26)
(4.27)
where δ1 , δ2 and κ are small positive constants chosen by the designer. ” ˆ ” denotes the estimated model parameter when their upper bounds are known as follows: |ˆ c − c| ≤ Mc , |cˆrt − crt | ≤ Mcrt , |kˆ − k| ≤ Mk , |Jˆt − Jt | ≤ MJt .
(4.28)
The stability of the sliding-mode controller (Tt = Teq + Tsw ) can be proved by defining five extra states Z = [z1 z2 z3 z4 z5 ]T and a new Lyapunov function ℓ as
81
4.4. Suppressing the torsional vibrations follows:
z1 z2 z3 z4 z5
s
Z δ1 = 2Mc exp −δ2 |x˙ 2 |dτ , δ2 s Z δ1 = 2Mcrt exp −δ2 |x1 |dτ , δ2 s Z δ1 = 2Mk exp −δ2 |x2 |dτ , δ2 s Z δ1 = 2MJt exp −δ2 λ|x1 − ωd |dτ , δ2 s Z δ1 = 2MJt exp −δ2 λ|x1 − x3 |dτ , δ2 5
1 1X 2 ℓ = J t s2 + z . 2 2 i=1 i
(4.29)
As t → ∞, zi is exponentially convergent to zero, leading to ℓ → 0 when s = 0. Therefore ℓ defined in Eq. (4.29) is a legitimate Lyapunov function with state variable [s, z T ]T . The time derivative of ℓ is given by: Z Z ˙ℓ =Jt ss˙ − Mc δ1 |x˙ 2 |exp −δ2 |x˙ 2 |dτ − Mcrt δ1 |x1 |exp −δ2 |x1 |dτ Z Z − Mk δ1 |x2 |exp −δ2 |x2 |dτ − MJt δ1 λ|x1 − ωd |exp −δ2 λ|x1 − ωd |dτ Z − MJt δ1 λ|x1 − x3 |exp −δ2 λ|x1 − x3 |dτ . ˙ it can be seen ℓ˙ ≤ −κs2 ≤ 0 and ℓ˙ = 0 for Substituting Eq. (4.27) into ℓ, s = 0 [102]. Therefore, using the controller any trajectory of the drill-string will reach and stay thereafter on the manifold s = 0 asymptotically, and according to Section 4.4.1.1, the state of the system will asymptotically converge to the desired ¯ sl (ωd ). fixed point X = X
82
4.4. Suppressing the torsional vibrations
4.4.2
Numerical results
Some numerical results using the controller are presented for two cases in this section. The identified parameters of the experiment are used including TOB parameters for Wb = 1.76 kN (Table 3.3). The remaining parameters for both cases are: cˆ = 0.0051 Nms/rad,
cˆrt = 10.47 Nms/rad,
kˆ = 10 Nm/rad,
Jˆt = 13.92 Kg/m2 ,
Mc = 0.00255,
Mcrt = 3,
Mk = 5,
MJt = 2.
7
θ˙b [rad/s]
θ˙t , θ˙b [rad/s]
7
3.5
3.5
0 −1 0
0 30
60 −1
0
1
2
−1
θt − θb [rad]
t[s]
u[Nm]
60
30
0 0
30
60
t[s]
Figure 4.11: Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve) and control signal (blue curve), and phase portrait of the simulation using sliding-mode controller with ωd = 3.1 rad/s and λ = 0.8. The controllers are switched on at t = 30 s. The stick-slip trajectory and the trajectory to the desired fixed point are shown in green and blue respectively in phase portrait.
Note that estimated parameters are chosen close to the obtained parameters of the experimental rig and the boundaries are chosen to satisfy Eq. (4.28).
83
4.4. Suppressing the torsional vibrations 9
θ˙b [rad/s]
θ˙t , θ˙b [rad/s]
9
4.5
4.5
0 −1 0
0 30
60 −1
1
2
−1
θt − θb [rad]
t[s] 120
U[Nm]
0
70
30 0
30
60
t[s]
Figure 4.12: Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve) and control signal (blue curve), and phase portrait of the simulation using sliding-mode controller with ωd = 5 rad/s and λ = 1. The controllers are switched on at t = 30 s. The stick-slip trajectory and the trajectory to the desired fixed point are shown in green and blue respectively in phase portrait. Time histories of angular velocities of motor (black curves) and drill-bit (green curves), control signals (blue curves) and phase portraits of two simulations using sliding-mode controller are shown in Figs. 4.11 and 4.12. It can be seen in both figures that the controllers are switched on at t = 30 s while the drill-bit is in stick-slip oscillations. In the phase portraits in both examples, the stick-slip trajectories are shown by the green parts of the curves. The blue parts of the curves in both phase portraits show how the controller leads the system to the desired fixed points. In the first example (Fig. 4.11), the desired speed ωd is 3.1 rad/s and the control parameter λ chosen is 0.8. However, for the second example ωd is 5 rad/s and λ chosen is 1. The rest of the control parameters for both examples are chosen as follows: δ1 = 0.01, δ2 = 1.00E − 5 and κ = 1. 84
4.5. Experimental verification of the control method
4.5
Experimental verification of the control method
The numerical study should be validated by the experiment. In this regard the sliding-mode controller is implemented in the rig using a Labview program. As was described in Section 4.2.1, a frequency convertor is used to control the motor torque. Tc is the input in this system (Fig. 4.5) and Tt is the torque generated by the motor. Their relationship is estimated by Eq. (4.11). The structure of the suggested sliding-mode controller for the experimental rig is depicted in Fig. 4.13. Time histories of angular velocities of motor (black curves) and drill-bit (red curves), control signals (blue curves)and phase portraits of two experiments using sliding-mode controller are shown in Figs. 4.14, 4.15. It can be seen in both figures that the controllers are switched on at t = 30 s while the drill-bit is in stick-slip oscillations. In phase portraits in both examples, the stick-slip trajectories are shown by the red parts of the curves. The blue parts of the curves in both phase portraits show how the controller leads the system to a limit cycle. In the first example (Fig. 4.14), the desired speed ωd is 3.1 rad/s and the control parameter λ chosen is 0.8. However, for the second example ωd is 5 rad/s and λ chosen is 1. The rest of the control parameters and estimated physical parameters are the same as the simulations presented in Figs. 4.11, 4.12. It is worth noting that both these experiments are carried out with 12 disks which weigh Wb = 1.76 kN (including BHA and the drill-bit).
85
4.5. Experimental verification of the control method ωd
sliding-mode controller
Tc
motor and drilling rig
frequency convertor
θb , θt , θ˙b , θ˙t , Tˆ
Figure 4.13: The structure of the suggested sliding-mode controller for the experimental rig. In this method the controller changes the control parameter Tt in order to keep the top and bit speed θ˙t and θ˙b close to the desired rotational speed ωd .
7
θ˙b [rad/s]
θ˙t , θ˙b [rad/s]
7
3.5
3.5
0 −1 0
0 60 −1
30
1
2
−1
θt − θb [rad]
t[s] 25
Tc [%]
0
12.5
0 0
30
60
t[s] Figure 4.14: Time histories of top angular velocity (black curve), drill-bit angular velocity (red curve) and control signal (blue curve), and phase portrait of the drilling experiment using sliding-mode controller with ωd = 3.1 rad/s and λ = 0.8. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in red and blue respectively in phase portrait.
As expected, the sliding-mode controller is successful in eliminating the stickslip oscillations and reducing the drill-bit vibrations significantly. However, unlike in the simulation, the controller cannot lead the drill-bit to the constant speed in the experiment. A delay in the motor would be one of the possible reasons for 86
4.6. Modelling delay in the actuator this difference, as discussed in the next section. 9
θ˙b [rad/s]
θ˙t , θ˙b [rad/s]
9
4.5
4.5
0 −1 0
0 60 −0.5
30
2.5
−1
θt − θb [rad]
t[s] 30
Tc [%]
0
15
0 0
30
60
t[s] Figure 4.15: Time histories of top angular velocity (black curve), drill-bit angular velocity (red curve) and control signal (blue curve), and phase portrait of the drilling experiment using sliding-mode controller with ωd = 5 rad/s and λ = 1. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in red and blue respectively in phase portrait.
4.6
Modelling delay in the actuator
In order to achieve a more precise model of the actuator, time histories of control signals are depicted in Fig. 4.16. Considering the torque control system (Fig. 4.5), the controller computes the necessary value of Tt . This value (in Nm) is converted to % Tc by Eq. (4.11) and sent to the actuator. Eq. (4.11) is a linear equation; therefore there should not be any phase shift between Tc and Tt . On the other hand, Tˆ is also related to Tt by a linear equation (Eq. (4.11)). Therefore phase shift between Tˆ and Tc is not expected. However, as can be seen in Fig. 4.16, 87
4.6. Modelling delay in the actuator
Tc [%], Tˆ[%]
12
8
4
0 0
4
8
12
16
20
t[s] Figure 4.16: Time histories of the actuator input value Tc (blue curve) and the torque generated by motor Tˆ (red curve). DAQ & control ωd
sliding-mode controller
drilling rig Tc
delay and deadzone
motor and drilling rig
θb , θt , θ˙b , θ˙t ,Tˆ
Figure 4.17: The structure of the suggested sliding-mode controller for the experimental rig with delay and dead-zone. A 0.4 s delay and a minimum 22.62 Nm torque are observed in the motor. there is a delay in Tˆ compared to Tt . This delay was derived from several tests and averaged to give a value of 0.40 s. In addition it can be seen that in the current arrangement of the control system, the minimum torque Tc which can be requested from the frequency convertor is zero (according to 4 mA input of the frequency convertor). In the other words, using Eq. (4.11) it can be seen that the motor will produce a minimum Tt of 22.62 Nm. Considering the delay and dead-zone observed in the actuator, the new structure of the suggested sliding-mode controller in experimental rig is shown in Fig. 4.17.
88
4.6. Modelling delay in the actuator
θ˙t , θ˙b [rad/s]
3.5
3.5
0 −1 0
0 30
60 −1
1
θt − θb [rad]
t[s] 110
U[Nm]
0
65
20 0
30
60
t[s] Figure 4.18: Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve) and control signal (blue curve), and phase portrait of a simulation considering a 0.4 s delay and minimum of 22.62 Nm torque in motor using sliding-mode controller with ωd = 3.1 rad/s and λ = 0.8. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in green and in blue in phase portrait. This result is very close to the experiment presented in Fig. 4.14.
Time histories of angular velocities of motor (black curves) and drill-bit (green curves), control signals (blue curves), and phase portraits of two simulations using sliding-mode controller are shown in Figs. 4.18, 4.19. In both simulations, a delay (0.4 s) and a minimum torque (22.62 Nm) are considered. In the first example (Fig. 4.18) the desired speed ωd is 3.1 rad/s and the control parameter λ is chosen as 0.8. However, for the second example ωd is 5 rad/s and λ is chosen to be 1. The rest of the parameters including physical parameters, estimated parameters, boundaries and control parameters are the
89
2
−1
θ˙b [rad/s]
7
7
4.6. Modelling delay in the actuator same as the ones in Figs. 4.11, 4.12. Results presented in Figs. 4.18, 4.19 show that the system converges to a limit cycle significantly closer to the experimental ones presented in Figs. 4.14, 4.15 respectively.
θ˙t , θ˙b [rad/s]
4.5
4.5
0 −1 0
0 60 −1
30
1
θt − θb [rad]
t[s] 140
U[Nm]
0
80
20 0
30
60
t[s] Figure 4.19: Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve) and control signal (blue curve), and phase portrait of a simulation considering a 0.4 s delay and minimum of 22.62 Nm torque in motor using sliding-mode controller with ωd = 5 rad/s and λ = 1. The controllers are switched on at t = 30 s. The stick-slip trajectory and the cycle limit are shown in green and in blue in phase portrait. This result is very close to the experiment presented in Fig. 4.15.
Finally, other experimental results and simulations using the sliding-mode controller are presented in Figs. 4.20 and 4.21 respectively. The controller is on in two time windows [60.6, 110.46]s and [150.4, 210.2]s, as represented by the blue curves in Figs. 4.20 and 4.21. The controller achieves elimination of the stick-slip oscillations in the drill-bit, as can be seen in experimental results (red curve in 90
2
−1
θ˙b [rad/s]
9
9
4.6. Modelling delay in the actuator Fig. 4.20) and simulation (green curve in Fig. 4.21). All parameters used for these experimental and numerical studies are the same as the ones in Fig. 4.14 and Fig. 4.18 respectively.
θ˙t , θ˙b [rad/s]
7
3.5
0 −1 0
100
200
t[s] 30
Tc [%]
20
10
0 0
100
200
t[s] Figure 4.20: Time histories of top angular velocity (black curve), drill-bit angular velocity (red curve), and control signal (blue curve) of the drilling experiment activating the controller in two time windows [60.6, 110.46]s and [150.4, 210.2]s. The controller achieves elimination of the stick-slip oscillations in the drill-bit (red curve). All parameters used for this experiment are the same as the ones in Fig. 4.14.
91
4.6. Modelling delay in the actuator
θ˙t , θ˙b [rad/s]
7
3.5
0 −1 0
100
200
t[s]
U[Nm]
110
80
50
20 0
100
200
t[s] Figure 4.21: Time histories of top angular velocity (black curve), drill-bit angular velocity (green curve), and control signal (blue curve) of the simulation activating the controller in two time windows [60.6, 110.46]s and [150.4, 210.2]s. The controller achieves elimination of the stick-slip oscillations in the drill-bit (green curve). All parameters used for this simulation are the same as the ones in Fig. 4.18. An excellent match observed between the simulation and the experiment presented in Fig. 4.20.
4.6.1
Sensitivities in parameter estimations in experimental study
In order to evaluate the sensitivity of the parameter estimations in the experimental results, several experiments are carried out with a variety of the estimated parameters. Table 4.3 shows the parameters used in the experiments. In this table the results is also presented in terms of the Vibration Reduction Fac92
4.6. Modelling delay in the actuator tor (VRF = Ac /Aun %) where Ac and Aun are amplitude of the “uncontrolled stick-slip oscillation” and “controlled vibration’ respectively. Fig. 4.22 presents phase portraits of the drilling experiments using slidingmode controller. The uncontrolled stick-slip trajectories and the controlled limit cycles are shown in red and blue respectively. All estimated parameters, boundaries and controller parameters used in these experiments are presented in Table 4.3. The controller achieves (a) 47.86% (b) 59.26% (c) 51.52% (d) 57.58% (e) 66.72% (f) 64.72% reduction in vibrations. (a)
(b)
(c)
7
3.5
3.5
0
0
0
−1
0
2.5
0
-2.5
θt − θb [rad]
−1 0.5
2
2
−1
9
9
θ˙b [rad/s]
θ˙b [rad/s]
1
1
(f)
(e)
0
0
θt − θb [rad]
7
θt − θb [rad]
−1
θt − θb [rad]
(d)
−1
θ˙b [rad/s]
3.5
θ˙b [rad/s]
−0.5
7
θ˙b [rad/s]
θ˙b [rad/s]
7
3.5
4.5
4.5
0
0
0
−1 −0.5
0
2.5
θt − θb [rad]
−1
−0.5
0
2.5
−1
θt − θb [rad]
Figure 4.22: Phase portraits of the drilling experiments using sliding-mode controller. The uncontrolled stick-slip trajectories and the controlled cycle limits are shown in red and blue respectively. All estimated parameters, boundaries and controller parameters used in these experiments are presented in Table 4.3. The controller achieves (a) 47.86% (b) 59.26% (c) 51.52% (d) 57.58% (e) 66.72% (f) 64.72% reduction in vibrations.
93
4.7. Conclusions Table 4.3: The estimated parameters used in experiments (Fig. 4.22) and the corresponding results. Parameter (a) (b) (c) (d) (e) (f) ωd [rad/s] 5 3.1 3.1 3.1 5 5 λ 0.4 0.4 0.8 0.4 1 1 δ1 0.01 0.01 0.01 0.01 0.01 0.01 δ2 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 κ 1 1 1 1 1 1 cˆ [Nms/rad] 0.0051 0.0051 0.0051 0.0051 0.0051 0.0051 Mc 0.00255 0.00255 0.00255 0.00255 0.00255 0.00255 ˆ k [Nm/rad] 10 10 10 10 10 10 Mk 5 5 5 5 5 5 cˆrt [Nms/rad] 2.91 2.91 10.45 10.45 10.45 10.45 Mcrt 1.45 1.45 3 2 3 2 2 ˆ Jt [Kg/m ] 6.356 6.356 13.92 13.92 13.92 13.92 MJt 3.2 3.2 2 2 2 2 VRF [%] 47.86 59.26 51.52 57.58 66.72 64.72
4.7
Conclusions
In this chapter a harmonically excited torsional drill-string model is presented. In this model the top speed is modeled as harmonic excitation and the torque model derived in the previous chapter is used. As a result, the drill-bit stickslip oscillations observed in the rig were modeled perfectly. This is confirmed by comparison of FFTs of the bit-speed obtained in the experiment and numerical simulation. In both cases, one major peak is observed at the same frequency. A 1-DOF model for the top motor and gearing system is then presented. Several systematic experiments were carried out in order to identify the model parameters. In the next step the whole rig is considered and a 2-DOF lumped mass model introduced to predict stick-slip results, having the motor torque as the control input of the model. An excellent match between experiment and simulation is achieved. This is confirmed by comparison of major peak of FFTs of the bit-speed obtained in the experiment and numerical simulation.
94
4.7. Conclusions A sliding-mode control method is presented, and applied to the obtained model in order to eliminate the stick-slip vibrations observed in the drilling rig and simulation. The stability of the controller is proved mathematically. The controller is successful in suppressing the vibration and bringing the system to the desired fixed point. These numerical results are presented for a few sets of parameters. The controller is then implemented in the experiment as verification of the numerical results. The controller is successful in eliminating the stick-slip in the experiment. However, a limit cycle is observed around the desired fixed point. Investigating the difference between experimental results and simulation, a delay and dead-zone is observed in the actuator. Adding them to the 2-DOF model achieves an excellent match between experiment and simulation. In order to examine the sensitivity of the controller to the parameters, several experiments were carried out with a variety of the estimated parameters applied to the controller. Phase portraits of the responses clearly show the success and robustness of the controller. A significant reduction in vibration amplitude is observed when the controller is applied.
95
Chapter 5 Lateral vibration of unbalanced rotor In the previous chapters, as a first example of the nonlinear engineering system, torsional vibration was presented. This chapter deals with the lateral vibration in rotating machine as the second example of a nonlinear engineering system. The structure of this chapter is as follows. Section 5.2 describes the experimental setup used to study the vibration-induced impacts between an unbalanced rotor and an outer snubber ring. Several sets of experimental results are presented in Section 5.3. Based on the previous model and the current experimental configuration, a new mathematical model is derived in Section 5.4, where particular attention is given to the motion of the snubber ring when it is in contact with the rotor. Section 5.5 presents the process of estimation of the model parameters. The main results of the chapter are then shown in Section 5.6. Here both experimental and numerical observations are presented and compared so as to verify the predictive capabilities of the mathematical model. This chapter finishes with conclusions in Section 5.7. It is worth noting that some of the results presented in this chapter have been recently published [121].
96
5.1. Introduction
5.1
Introduction
The dynamics of rotating machinery has been extensively studied in the past by many researchers, due mainly to the numerous applications in industry such as power generation, large-scale manufacturing, automobile engines, aerospace propulsion and home appliances. In all these applications one of the most common concerns of designers and troubleshooters is the long-term exposure of the rotating machines to vibration, which can eventually lead to catastrophic failures or accidents. This can be the case when for example the undesired vibration becomes close to one of the natural frequencies of the machine structure (resonance). Also, in most rotating machines there is both a rotating and non-rotating part to consider, where dangerous vibration-induced intermittent impacts between rotating and stationary components may occur. Such unwanted vibrations are often caused by mass imbalance, which occurs when there is a mismatch between the principal axis of the moment of inertia of the rotating element and its axis of rotation. From a practical point of view, this phenomenon can be produced by such factors as blade-loss conditions, looseness of parts, misalignment, thermal deformation and factory residual imbalance, which makes the presence of lateral vibrations unavoidable. This undesired effect can have quite negative consequences in terms of durability, reliability and safe operation of rotating machines. The literature in this area is vast and dates back to the end of the nineteenth century with the rapid development of locomotives and steam turbines. As described in Section 2.2, recent contributions in this field are reported in Lahriri et al. [88], where both theoretical and experimental considerations are presented regarding the reduction of rub contact of a rotor via backup bearing support. Whirling in particular is a well-known phenomena in the literature, where eccentric rotors undergo periodic oscillation [122]. Many studies have at-
97
5.1. Introduction tempted to model rotor systems subjected to out-of-balance phenomena, see e.g. [50, 51, 78, 109]. Most of the mathematical models used in these investigations are based on the Jeffcott rotor [72], which consists of a large unbalanced disk mounted midway between the bearing supports on a flexible shaft of negligible mass. On the other hand whirl has been observed in the oil-drilling process for many years. It is recognized as being one of the most damaging types of failure mechanism [108] due to its lack of detectability at the rig surface. A similar model to the Jeffcott rotor has been used for lateral vibrations of the BHA. The model idealizes the inertia of the BHA as being a disc spinning at a constant speed. The flexibility of the BHA is modelled as a simple spring providing restoring forces. Such models have been extensively studied in literature [25, 40]. Having in mind the close similarity between the whirl phenomenon observed in drilling applications and that of imbalanced rotors, the aim of the present work is to extend and complement the previous investigations undertaken at the CADR with the purpose of deriving and validating a new model to study vibrationinduced impacts in unbalanced planar rotors (see Section 2.2). The new model takes into account viscoelastic characteristics of the snubber ring support as well as anisotropy in both the snubber ring and rotor supports. The asymmetry considered in the model often appears in many applications as a consequence of rotating machines operating in asymmetrical conditions, for example when lateral loads act on the rotor or when gravity effects become significant. Since the final aim could be to develop robust control methods to avoid impacts, the main focus in this chapter is on the analysis of parameter ranges corresponding to impact regimes, for which displacement of the snubber ring is small.
98
5.2. Experimental rotor system
5.2 5.2.1
Experimental rotor system Rig description
The main components of the experimental apparatus are explicitly shown in Fig. 5.1. The rotor system is driven by a direct-current, variable speed 1.5 kW motor (“Eurodrive”, model GN100LSG2), with a nominal maximum speed of 3140 rpm. The angular speed is controlled via a single-phase thyristor in open loop feedback using a tacho-generator. The rotor is made of mild steel, with holes drilled and tapped into it for the addition of mass imbalance (bolt-nut arrangement), which can be adjusted by adding washers (see Fig. 5.12(b)). The rotor runs in two angular contact bearings, fastened with inner sleeves inside a non-rotating housing. This rotor assembly is supported by four flexural rods made of high carbon steel to resist fatigue, which are in turn clamped to a support block bolted to a heavy iron bed. The rods provide a lightly damped elastic support. Hence a pair of dashpot dampers, one in each direction, are attached to the rotor to increase the damping. The rotor assembly is placed inside a snubber ring of slightly larger diameter constructed from aluminium, and supported by four compression springs fixed to a large frame clamped to the iron bed (see Fig. 5.12(c)). This arrangement allows the rotor housing to make intermittent contact with the snubber ring during operation, which in turn produces a discontinuous stiffness effect on the rotor system.
99
5.2. Experimental rotor system (a) 9
5
10
4 8
1
2
3
6 7
(b) Eddy current probes 5 6
3
8
4
2 1
7
Motor
1 ppr
ys(t)
xs(t)
Amplifier
60 ppr
Converter
Driver
yr(t)
xr(t)
Encoder
I/O Connector Block
Computer
Potentiometer
NI PCI-6251, DAQ LabVIEW program
Figure 5.1: (a) A photograph of the rotor rig and (b) a schematic diagram of the experimental setup. The main components of the system are numbered as follows: 1, motor; 2, rotor with out-of-balance; 3, rotor housing; 4, snubber ring; 5, snubber ring frame; 6, flexural rods; 7, support block; 8, speed monitoring disc; 9, dashpot dampers; 10, clearance. 100
5.2. Experimental rotor system
5.2.2
Instrumentation and data acquisition
Figure 5.2: A front page of the Labview program prepared to observe and demonstrate all measured signals, as well as phase diagrams and Poincar´e maps of the response in realtime.
A schematic diagram of the rig instrumentation is shown in Fig. 5.1(b). Four noncontacting eddy current probes are installed to observe the horizontal and vertical displacements of the rotor (xr (t), yr (t)) and the snubber ring (xs (t), ys (t)). A speed monitoring disc is mounted on the motor shaft in order to measure the angular speed and position of the rotor. The disc is equipped with 60 equally spaced grooves and a notch cut aligned with the imbalance mass, which triggers a phototransistor coupled with a photodiode to produce a 60- and 1-per-revolution pulse signal, respectively. All the voltage signals are amplified and then sent to a National Instruments data acquisition card (NI PCI-6251) through twisted-pair wired connections. The data acquisition process is governed by a LabVIEW101
5.3. Experimental results based graphical interface that allows the real-time response of the system to be monitored in real-time using phase portraits, time histories and Poincar´e diagrams (see Fig. 5.2). The acquired data is saved for further processing and analysis using MATLAB.
5.3
Experimental results
Several set of experiments have been carried out in order to observe the responses of the unbalanced rotor under different conditions. In each set of experiments the frequency of motor f is varied in order to generate bifurcation diagrams. For each frequency I wait enough time (about 300 periods of rotation) in order to observe the steady state of the system, after which data is recorded for 100 periods. Subsequently, f is increased (or decreased) by a small amount and the same process is repeated. This is done until a predefined final frequency value is reached. This procedure allows us to visualize the qualitative changes in the long-time dynamics of the rotor model. Figs. 5.3 - 5.8 show the bifurcation diagrams for different configurations. Each graph consists of 72 experiments under different conditions and is presented in Table 5.1. In each case (a) represents the rotor speed increased from 9 to 16.1 Hz and (b) represents it decreased from 16.1 to 9 Hz. In order to explain the results, the first experiment is considered. The results (Fig. 5.3(a)) are re-plotted with details in Fig. 5.9. Specifically, this graph consists of 72 experiments with different rotor speeds increased from 9 to 16.1 Hz. As shown in the figure, several periodic responses have been highlighted; Period one (f = 11.7 Hz), Period three (f = 11.9 Hz), Period two (f = 12.2 Hz), Quasi-periodic (f = 13.7 Hz) and period two embedded in a Quasi-periodic (f = 15.3 Hz).
102
a)
b)
−4
−4.2
xr -sensor(V)
xr -sensor(V)
5.3. Experimental results
−5
−6
-5.1
−6 8
11
14
17
8
11
f (Hz)
14
17
f (Hz)
Figure 5.3: Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 20.7 g on the point B of Fig. 5.12(b) (517.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz.
a)
b) -3
xr -sensor(V)
xr -sensor(V)
−3
-3.8
-4.6 8
-3.8
-4.6 11
14
17
f (Hz)
8
11
14
17
f (Hz)
Figure 5.4: Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 20.7 g on the point A of Fig. 5.12(b) (517.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz.
103
5.3. Experimental results
a)
b) -3
xr -sensor(V)
xr -sensor(V)
−3
-3.8
-4.6 8
-3.8
-4.6 11
14
17
8
11
f (Hz)
14
17
f (Hz)
Figure 5.5: Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 22.3 g on the point A of Fig. 5.12(b) (557.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz.
a)
b) −4
xr -sensor(V)
xr -sensor(V)
−4
−5
−6
-5
−6 8
11
14
17
f (Hz)
8
11
14
17
f (Hz)
Figure 5.6: Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 22.3 g on the point B of Fig. 5.12(b) (557.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz.
104
a)
b)
-3.8
-3.8
xr -sensor(V)
xr -sensor(V)
5.3. Experimental results
-4.7
-5.6 8
-4.7
-5.6 11
14
17
8
11
f (Hz)
14
17
f (Hz)
Figure 5.7: Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 19.1 g on the point B of Fig. 5.12(b) (477.5×10−6 kgm). In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz.
a)
b) -3
xr -sensor(V)
xr -sensor(V)
−3
-3.8
-4.6 8
-3.8
-4.6 11
14
17
f (Hz)
8
11
14
17
f (Hz)
Figure 5.8: Experimental bifurcation diagrams which show the change of the system dynamics as rotor speed is varied, consists of 72 experiments with additional mass of 19.1 g on the point A of Fig. 5.12(b) 477.5×10−6 kgm. In (a) rotor speed increased from 9 to 16.1 Hz, in (b) rotor speed decreased from 16.1 to 9 Hz.
105
Orbit
yr -sensor(V)
yr -sensor(V)
yr -sensor(V)
Orbit
4.6
4.8
4.5
4.2 3.8
4.4
4
-4
-3.5
-4.5
xr -sensor(V)
yr -sensor(V)
-4
-3.5
-3.5
xr -sensor(V)
f = 11.9 Hz
Orbit 5
3.5
-4.5 -4
xr -sensor(V)
f = 11.7 Hz
-4
4
3.6
3.5 -4.5
Orbit
yr -sensor(V)
5.3. Experimental results
f = 12.2 Hz
-5
-4
-3
xr -sensor(V) f = 13.7 Hz
-5
Orbit
5.5
4.5
3.5 -5
-6
8
yr -sensor(V)
yr -sensor(V)
f = 15.3 Hz
-4
5.5
Poincar´e map
4.5
3.5
-3
-5
xr -sensor(V)
-4
-3
xr -sensor(V)
11
14
f (Hz) Figure 5.9: Experimental bifurcation diagrams with a phase portrait and Poincar´e map showing the change of the system dynamics as rotor speed is varied, consists of 72 experiments with rotor speed increased from 9 to 16.1 Hz and with additional mass of 20.7 g on the point B of Fig. 5.12(b) (517.5×10−6 kgm). Highlighted responses: Period one (f = 11.7 Hz), Period three (f = 11.9 Hz), Period two (f = 12.2 Hz), Quasi-periodic (f = 13.7 Hz) and period two embedded in a Quasi-periodic (f = 15.3 Hz).
106
17
5.4. Physical model and equations of motion Table 5.1: Parameter values of the experiments presenting in Figs 5.3 - 5.7. Figure 5.3(a) 5.3(b) 5.4(a) 5.4(b) 5.5(a) 5.5(b) 5.6(a) 5.6(b) 5.7(a) 5.7(b) 5.8(a) 5.8(b)
Frequency 9 to 16.1 Hz 16.1 to 9 Hz 9 to 16.1 Hz 16.1 to 9 Hz 9 to 16.1 Hz 16.1 to 9 Hz 9 to 16.1 Hz 16.1 to 9 Hz 9 to 16.1 Hz 16.1 to 9 Hz 9 to 16.1 Hz 16.1 to 9 Hz
Additional mass 20.7 g on the point 20.7 g on the point 20.7 g on the point 20.7 g on the point 22.3 g on the point 22.3 g on the point 22.3 g on the point 22.3 g on the point 19.1 g on the point 19.1 g on the point 19.1 g on the point 19.1 g on the point
B B A A A A B B B B A A
Imbalance mass 517.5×10−6 kgm 517.5×10−6 kgm 517.5×10−6 kgm 517.5×10−6 kgm 557.5×10−6 kgm 557.5×10−6 kgm 557.5×10−6 kgm 557.5×10−6 kgm 477.5×10−6 kgm 477.5×10−6 kgm 477.5×10−6 kgm 477.5×10−6 kgm
It is worth nothing that points A and B are an equal distance from the centre of the rotor (0.025 m). Therefore, the bifurcation diagrams are expected to be similar in each pair of figures (Figs. 5.3 and 5.4), (Figs. 5.5 and 5.6) and (Figs. 5.7 and 5.8). It can be seen in each pair that the branch of period-1 responses appears in a same frequency range of the motor. However, the type of dynamical responses do not agree for the rest of the experiments in each pair. This can be due to one of the two following reasons. Firstly, when the results are qualitatively similar, this can be the result of using a different Poincar´e section to obtain bifurcation diagrams in each experiment. Secondly, where there is no qualitative agreement, the co-exciting responses might exist in the dynamics of this system.
5.4
Physical model and equations of motion
In order to capture the qualitative behaviour of the rotor rig and validate the experimental observations, a suitable mathematical model of the experimental setup is derived below. Consider a rigid rotor M with viscoelastic support, excited by an out-of-balance mass m located at a fixed distance ρ from the geometrical
107
5.4. Physical model and equations of motion centre of the rotor (Fig. 5.10(a)). When the displacement of the rotor is large enough during operation, an intermittent contact between the rotor and the outer snubber ring is produced. The primary (rotor support) stiffness kr is assumed to be the same in both vertical and horizontal directions. Apart from this coefficient, the vertical parameters of the rotor and the snubber ring supports differ from the horizontal ones, which results in an asymmetric physical configuration. The mathematical modelling of the system is made under the following assumptions: the mass of the rotor is much larger than that of the snubber ring, and hence its mass is negligible; there is no dry friction between the rotor and the snubber ring; no angular motion of the axis of rotation occurs, thus gyroscopic forces are not considered; transients in the rotational motion are not taken into account i.e. the rotor spins at a constant angular speed ω and the gravity effects are negligible compared to the dynamic forces acting in the system. The equations of motion are derived using the coordinate system shown in Fig. 5.10(b). The origin is chosen to coincide with the equilibrium position of the snubber ring centre Os0 . The eccentricity vector (εx , εy ) denotes the displacement of the static position of the rotor centre Or0 from the origin Os0 . Due to the p geometry of the rotor system, it can be seen that ε2x + ε2y ≤ γ, where γ stands
for the radial clearance between the rotor and the snubber ring. During operation, the points (xr , yr ) and (xs , ys ) represent the current position of the centers of the rotor Or and the snubber ring Os , respectively. The distance between these points is denoted by R, which is constantly monitored in order to determine whether the rotor is in contact with the snubber ring. The system motion is characterized by two modes of operation: no contact and contact between the rotor and the snubber ring. The mathematical description of these modes is given below
108
5.4. Physical model and equations of motion (a)
cry
kr
csx
m
γ
y
kr
ρ x
ksx
crx
M
Snubber Ring
Rotor
csy
ksy
(b)
hola
y
Or
yr R
ys
Os O ro
εy
Oso
εx
xs
xr
x
Figure 5.10: (a) A physical model of the Jeffcott rotor, (b) co-ordinate system used to derive the equations of motion. 109
5.4. Physical model and equations of motion No contact: M x¨r + crx x˙ r + kr (xr − εx ) = mρω 2 cos(ωt + ϕ0 ), M y¨r + cry y˙ r + kr (yr − εy ) = mρω 2 sin(ωt + ϕ0 ),
(5.1)
ksx xs + csx x˙ s = 0, ksy ys + csy y˙ s = 0.
Contact: M x¨r + crx x˙ r + kr (xr − εx ) + FN x = mρω 2 cos(ωt + ϕ0 ), M y¨r + cry y˙ r + kr (yr − εy ) + FN y = mρω 2 sin(ωt + ϕ0 ),
(5.2)
x˙ s = χt (xr , yr , xs , ys , x˙ r , y˙ r , αk ), y˙ s = Υt (xr , yr , xs , ys , x˙ r , y˙ r , αk ), ksx d , a single overdot means and so on. In the last equation, ksy dt (FN x , FN y )T represents the normal force acting on the rotor when it is in contact where αk =
with the snubber ring. The functions χt and Υt describe the velocity of the center of the snubber ring Os during the contact mode. Explicit expressions for χt , Υt and the normal force will be derived in Section 5.4.1.
5.4.1
Snubber ring motion and contact force
In this section a detailed description is given of the motion of the snubber ring and the normal force exerted on the rotor when it is in contact with the snubber ring. During the no contact mode there is obviously no interaction between the rotor and the snubber ring. Hence the dynamics of the system is described by a simple set of linear differential equations (5.1). On the other hand, when the rotor hits the snubber ring an additional force (FN x , FN y )T is included in the equations of motion of the rotor (5.2). This additional term corresponds to the 110
5.4. Physical model and equations of motion
y Fs Tb b N
yr ys
Or ψ
Os R=γ
Rotor
Snubber Ring
Oso
xs
xr
x
Figure 5.11: A geometrical configuration of the system during the contact mode. b Normal and tangential unit vectors at the surface of contact are denoted by N b and T respectively. The force exerted on the snubber ring by its viscoelastic support is denoted by Fs , whose normal component is transmitted to the rotor at the surface of contact. normal component of the snubber ring force Fs at the surface of contact between the rotor and the snubber ring, see Fig. 5.11. It depends on both the position and velocity of the snubber ring, which are described by the third and fourth equations of (5.2). Consequently, suitable functions χt and Υt need to be found that can satisfactorily characterize the snubber ring dynamics during the contact regime. In the contact mode of operation the distance between the centers of the rotor and the snubber ring equals the radial clearance i.e. R = γ. However, for a given fixed position of the rotor, the condition R = γ does not uniquely define the location of the snubber ring; hence further conditions need to be imposed. Therefore, in order to model the motion of the snubber ring the following assumptions are made: for any fixed location of the rotor during the contact mode, the snubber ring instantaneously assumes the position that minimizes the total elastic energy stored in the springs ksx , ksy (see Fig. 5.10(a)), satisfying 111
5.4. Physical model and equations of motion R = γ. In other words, it is supposed that the motion of the snubber ring is governed by the principle of minimum total potential energy, thus neglecting any transient effect on the snubber ring for each change in the position of the rotor during the contact regime. Therefore the determination of the snubber ring location can be addressed in terms of a minimization problem as follows. Given a fixed rotor position (xr , yr ), find the snubber ring location (x∗s , ys∗) that solves the problem: minimize U(xs , ys ) =
1 2
(ksx x2s + ksy ys2) under the constraint
(xs − xr )2 + (ys − yr )2 = γ 2 , where U is the elastic energy function. By using the method of Lagrange multipliers, the solution (x∗s , ys∗) can be computed as a root of the system:
αk xs (ys − yr ) − ys (xs − xr ) L(xs , ys , xr , yr , αk , γ) = = 0. (xs − xr )2 + (ys − yr )2 − γ 2
(5.3)
The explicit solution of this nonlinear equation involves long and laborious algebraic manipulations, requiring a numerical scheme to be employed. Note that for a numerical approach to be reliable, the existence and uniqueness of the solution of (5.3) needs to be investigated. These problems, as well as how it varies with respect to (xr , yr , αk , γ) are examined in Appendix B. Once a detailed description of the motion of the snubber ring during the contact regime is given, the next step is to find explicit expressions for the normal force (FN x , FN y )T acting on the rotor during this mode (see system (5.2)). For this purpose, first the force Fs acting on the snubber ring from its viscoelastic support needs to be computed, which is given by:
ksx xs + csx x˙ s Fs = . ksy ys + csy y˙ s
112
5.4. Physical model and equations of motion Hence the normal force (FN x , FN y )T exerted on the rotor when it is in contact with the snubber ring can be calculated as (see Fig. 5.11): b cos(ψ), FN x = hFs , Ni
b i sin(ψ), FN y = hFs , N
where b = (cos(ψ), sin(ψ))T , N
cos(ψ) =
xr − xs , γ
sin(ψ) =
yr − ys , γ
and h·, ·i stands for the Euclidean inner product.
5.4.2
Numerical implementation
In the previous section a detailed construction of the functions describing the motion of the snubber ring during the contact mode, as well as of the normal force (FN x , FN y )T acting on the rotor in this regime are presented. This allows us to have an explicit form for the equations of motion (5.1) and (5.2), which should be solved in order to study the dynamics of the rotor system. Thus, in this section the numerical process is explained for integrating the equations of motion in order to generate the trajectories describing the behaviour of the system. Let us assume that the integration is started with an initial point corresponding to the no contact regime (R < γ) at some t = tini . The solution (xr (t), yr (t), xs (t), ys (t)) of the system (no contact mode): M x¨r + crx x˙ r + kr (xr − εx ) = mρω 2 cos(ωt + ϕ0 ), M y¨r + cry y˙ r + kr (yr − εy ) = mρω 2 sin(ωt + ϕ0 ), ksx xs + csx x˙ s = 0, ksy ys + csy y˙ s = 0,
113
5.4. Physical model and equations of motion is then approximated numerically. In order to detect an impact between the rotor and the snubber ring, the distance function is monitored: D(t) = (xs (t) − xr (t))2 + (ys (t) − yr (t))2 − γ 2 during the integration. Suppose that D(timp ) = 0 i.e. R = γ for some t = timp . Then if the transversality condition: dD(t) 0< dt t=timp
(5.4)
holds, an impact between the rotor and the snubber ring has occurred. In this case the solution (xr (t), yr (t), xs (t), ys (t)) is extended by integrating the system (contact mode): M x¨r + crx x˙ r + kr (xr − εx ) + FN x = mρω 2 cos(ωt + ϕ0 ), M y¨r + cry y˙ r + kr (yr − εy ) + FN y = mρω 2 sin(ωt + ϕ0 ), x˙ s = χt (xr , yr , xs , ys , x˙ r , y˙ r , αk ), y˙ s = Υt (xr , yr , xs , ys , x˙ r , y˙ r , αk ),
with the initial point: xr (t+ ) = xr (t− ), imp imp yr (t+ ) = yr (t− ), imp imp − − xs (t+ imp ) = χ xr (timp ), yr (timp ), αk , γ , − − ys (t+ imp ) = Υ xr (timp ), yr (timp ), αk , γ . The system operates under this mode as long as the rotor and the snubber ring b ≥ 0 (see Fig. 5.11). Suppose that at some t = tlost , are in contact i.e. hFs , Ni 114
5.5. Parameter estimation b becomes zero. Then if the transversality condition: hFs , Ni d b 0> hFs , N i dt t=tlost
(5.5)
holds, the contact between the rotor and the snubber ring is lost, and the system + + goes back to the no contact regime with the initial point (xr (t+ lost ), yr (tlost ), xs (tlost ), − − − − ys (t+ lost )) = (xr (tlost ), yr (tlost ), xs (tlost ), ys (tlost )). This process is repeated until some
final value t = tfin is reached. In this chapter the numerical integration is implemented by means of the standard MATLAB solvers together with their built-in event location routines. This permits an accurate detection of the transition points between the contact and no contact operation modes as well as a straightforward verification of the transversality conditions (5.4) and (5.5). Note that the numerical scheme described above requires the evaluation of the functions χ, Υ studied in Section 5.4.1, whose explicit forms are unknown. Consequently, the values xs = χ(xr , yr , αk , γ), ys = Υ(xr , yr , αk , γ) will be approximated numerically by the Newton iterations:
(n) xs (n)
ys
=
(n−1) xs (n−1)
ys
−1 (n−1) , ys(n−1) , xr , yr , αk , γ L x(n−1) , ys(n−1) , xr , yr , αk , γ , −∂q L xs s
n ≥ 1, where the operator ∂q denotes partial differentiation with respect to the variable q = (xs , ys ). For this scheme, the initial point given is chosen by (B.1), which guarantees the convergence of the iterations provided αk is sufficiently close to 1.
115
5.5. Parameter estimation (a)
(b)
A
B (c)
(d)
1
4
2
3
Figure 5.12: (a) An experimental rig showing the connection between the motor and the rotor, (b) an enlarged view of a rotor face showing the possible locations of additional masses, (c) the snubber ring supported by four compression springs (labeled 1–4 in the picture) attached to the snubber ring frame, (d) the frame with the attached mass to the snubber ring.
5.5
Parameter estimation
In order to perform a numerical validation of the experimental results (see Section 5.6), a careful estimation of the physical parameter values of the rig has been carried out. Typically this task poses several challenges, such as the fact that some parameters cannot be measured directly (e.g. viscous damping coefficients), as well as noise, sensitivity and other uncertainties. The masses of the rotor and the imbalance were measured with a high116
5.5. Parameter estimation
Displacement sensor [V]
0.61
0.00
-0.61 0.00
0.15
0.30
Time [s] Figure 5.13: A natural response of the mass (see Fig. 5.12(d)) and the exponential fitted curve, which allows us to calculate the stiffness and damping coefficients. precision digital scale, while distances (e.g. radial clearance) were determined with a micrometer and a vernier calliper. Special attention was given to the estimation of the parameters of the snubber ring support. As can be seen in Fig. 5.12(c), the rotor assembly was dismantled so as to perform separate tests, which consisted of attaching a cylindrical mass (10.17 kg) inside the snubber ring, in such a way that they can be considered as being one solid body (see Fig. 5.12(d)). Then the free vibrations of this element in both horizontal and vertical directions were recorded (see Fig. 5.13), and the damping coefficient and stiffness of the snubber ring support were calculated from the thus obtained time histories. A similar procedure was employed to determine the viscous damping and stiffness of the rotor support. Table 5.2 contains the estimated values of the physical parameters of the rotor rig.
117
5.6. Comparison between experimental result and the model Table 5.2: Estimated parameter values of the rotor rig according to the model shown in Fig. 5.10. Parameter Symbol Rotor stiffness kr Horizontal rotor viscous damping crx Vertical rotor viscous damping cry Horizontal snubber ring stiffness ksx Vertical snubber ring stiffness ksy Horizontal snubber ring viscous damping csx Vertical snubber ring viscous damping csy Rotor mass M Imbalance mass m Distance from imbalance mass to rotor centre ρ Radial clearance between rotor and snubber ring γ Horizontal rotor eccentricity εx Vertical rotor eccentricity εy
5.6
Value 9.11 × 104 N/m 97.09 Ns/m 83.47 Ns/m 2.85 × 106 N/m 3.89 × 106 N/m 70.18 Ns/m 392.26 Ns/m 10.24 kg 0.038 kg 0.025 m 5 × 10−4 m 1.7 × 10−4 m −1.9 × 10−4 m
Comparison between experimental result and the model
This section is devoted to the comparison between the results of the experimental investigation and the numerical simulations generated from the equations of motion (5.1) and (5.2), using the parameter values given in Table 5.2. For this purpose, the last experiment is considered (Fig. 5.8(a)). This experiment is repeated with smaller increments in frequency of motor (angular speed) around impact regimes (near grazing trajectories), for which soft impacts take place (11 Hz to 14 Hz). Note that so-called grazing trajectories are ones with zero velocity impacts. For the numerical investigation, initially a low frequency value is set where no impacts between the rotor and snubber ring occur. The equations of motion are then integrated over 300 periods to allow for the decay of transients. Next the numerical solution is then extended for another interval of 100 periods and plot samples of the solution at times t =
2iπ , ω
i = 1, 2, . . . , 100. Subsequently,
ω is increased by a small amount and the same process is repeated, where now 118
5.6. Comparison between experimental result and the model the final sample of the previous step is used as initial value. This is done until a predefined final frequency value is reached. This procedure will allow us to visualize the qualitative changes in the long-time dynamics of the rotor model. Following a process analogous to the one just described, the steady states of the experimental rig will be investigated. Experimental Observations
Numerical Simulations -4
x 10
-3.40
m
-3.65
4.05
3.80 79.5
81.0
82.5
ω[
rad
-3.90 72
84.0
76
rad
ω = 85.10 [rad/s]
ω = 75.51 [rad/s]
-4
ω = 75.51 [rad/s]
ω = 82.28 [rad/s]
x 10 -0.2
yr [
-2
-5 -2.0
-1.9
-3.6 1.5
xr [
0.00
5.0
]
m
-4
x 10
1.75
xr [
-4
x 10
-4.2
-4.8 -4.05
xr -sensor [
-3.60
-5
]
V
ω = 82.13 [rad/s] -4
-3.6
V
-4.15
-4.60 -4.50
3.50
]
m
yr -sensor [
]
yr -sensor [
V
]
-3.70
m
] m
yr [
83
/s]
-4
x 10 1
-4
xr -sensor [
-3
]
V
ω = 78.59 [rad/s]
-4
x 10
x 10 1.00
yr [
-1.85
yr -sensor [
yr -sensor [
] m
] m
-2
V
V
]
]
-3.5
0
yr [
79
ω[
/s]
]
xr [ ]
V
xr -sensor [ ]
4.30
-4.1
-3.7
-4.1
-4.5 -4 -4.70 0.0
0.4
t[ ]
0.8
-1.0
1.2
s
4.0
1.5
xr [
-4.7 -4.7
]
m
-4
x 10
-4.0
xr -sensor [
-3.4
ω = 83.73 [rad/s] x 10
-4
0.8
1.2
s
ω = 81.93 [rad/s]
-4
-3
-3.4 V
]
yr [
2
-1.5
xr -sensor [
V
yr -sensor [
m
]
] m
t[ ]
x 10 1.0
xr [
0.4
]
5
0.0
]
V
-4.1
-4
-4.0 -1 0.0
-4.8 0.3
t[ ] s
0.6
0.9
-1
xr [
2
-5 -5
5
]
m
-4
x 10
-4
xr -sensor [
-3
]
V
0.0
0.3
t[ ]
0.6
0.9
s
Figure 5.14: A comparison of the bifurcation diagrams, phase plots and time histories obtained from the mathematical model (5.1)–(5.2) (left-hand side panels) and experimental rig (right-hand side panels). The most significant dynamical responses along with their corresponding frequency intervals are summarized in table 5.3.
119
5.6. Comparison between experimental result and the model Table 5.3: Summary of the most significant dynamical responses of the system (presented in Fig. 5.14) along with their corresponding frequency intervals. System response
Numerical range [rad/s]
No impact Period-3 impacting motion Period-2 impacting motion Period-1 impacting motion
00.00 < ω 81.03 < ω 82.49 < ω 83.79 < ω
≤ 81.03 ≤ 82.13 ≤ 83.79 ≤ 85.10
Experimental range [rad/s] 00.00 < ω 77.82 < ω 80.73 < ω 82.11 < ω
≤ 77.82 ≤ 79.81 ≤ 82.11 ≤ 82.96
The comparison between the numerical simulations and the experimental observations is shown in Fig. 5.14. The picture includes bifurcation diagrams, phase plots and time histories of the qualitatively relevant system response. As expected, for low frequency values the amplitude of the rotor vibrations is small, and therefore no impacts between the rotor and snubber ring occur. As the frequency is increased, the amplitude of the rotor oscillations also increases until a grazing contact between the rotor and snubber ring is found. At this point, the period-1 motion bifurcates into a period-3 orbit, and this system behaviour persists for slightly higher frequency values. As the frequency is further increased, the period-3 response suddenly disappears, and a small interval of quasiperiodic motion is detected. Thereafter, a larger window of period-2 response is found, which is then followed by an interval of impacting period-1 orbits. The frequency ranges for the most significant dynamical scenarios are displayed in table 5.3. As can be seen in Fig. 5.14, there is a good agreement between the numerical results obtained from the mathematical model (5.1)–(5.2) and the experimental observations, for the considered frequency range. Specifically, the numerical and experimental results show the same qualitative dynamics. Nevertheless, some discrepancies were also observed. In first place, there is a small difference between the theoretical and experimental results related to the length and location of the frequency windows of the observed dynamical scenarios (see Fig. 5.14). Moreover,
120
5.7. Conclusions for frequency values larger than those of the considered intervals, the dynamics of the mathematical model and the experimental rig differs significantly. This mismatch at high frequencies can be caused by the following reasons: nonlinear effects acting on the experimental rig and neglected in the mathematical model, such as friction and the presence of bearings; unwanted vibrations of the experimental rig during operation (mechanical noise); fluctuations in the rotor speed; and shaft misalignment. Other sources of uncertainty are the unavoidable inaccuracies in the parameter estimation, especially of small distances (e.g. εx , εy ), and the assumption that the four springs used for the snubber ring support work in their linear region and always in compression (i.e. in contact with the snubber ring). This last assumption may fail for the current experimental setup at high frequencies, as the amplitude of the impact force can be so high that the contact between the snubber ring and the support springs can be lost.
5.7
Conclusions
This chapter presented a new mathematical model and experimental verification of the dynamical behaviour of a Jeffcott rotor within a snubber ring with anisotropic (asymmetric) support, as shown in Fig. 5.10(a). The main components and features of the experimental apparatus (Fig. 5.1) were described in Section 5.2. Several experiments were carried out and the results are presented in Section 5.3. In order to model these experiment, a physical and mathematical modelling of the experimental rig were derived in Section 5.4, which resulted in a new set of equations (5.1)–(5.2) describing the motion of the system, whose numerical implementation was described and mathematically justified in detail. A separate test was carried out in order to study the physical properties of the snubber ring support, which revealed an anisotropic nature as well as the
121
5.7. Conclusions presence of damping effects (see Section 5.5). After estimating parameters, the experimental observations and the numerical results generated from the model were compared (see Section 5.6). A frequency sweep from low to high values in the experimental rig revealed the following sequence of dynamical scenarios: non-impacting period-1 response, grazing contact between the rotor and the snubber ring, impacting period-3 orbits, quasiperiodic motion, impacting period-2 response and impacting period-1 orbits. As shown in Fig. 5.14, the mathematical model proved to be capable of predicting the sequence of dynamical scenarios just described, as well as determining with a certain degree of accuracy the frequency windows where the different qualitative phenomena occur. Nevertheless, for higher frequency values the responses of the experimental rig and the mathematical model differ, mainly due to unavoidable nonlinear effects not considered in the model and uncertainties in the parameter estimation.
122
Chapter 6 Rotational motion of a parametric pendulum In the previous chapters two examples of the nonlinear engineering systems were introduced. The aim of this chapter is to present new control methods based on the system dynamics for harmonically excited and wave-excited pendulums in order to initiate and maintain their rotational responses. This chapter is organized as follows. In Section 6.2 the experimental setup and results are explained. Next, the numerical modelling of the system is presented in Section 6.3, and in Section 6.4 the dynamics of a harmonically-excited pendulum are discussed. Different methods for initiating and maintaining the rotating solution are examined in Sections 6.5 and 6.6 respectively. Wave displacement functions are presented mathematically in Section 6.7. Section 6.8 explains the dynamics of a waveexcited pendulum system, which leads us the need for a statistical measure as is presented in Section 6.9. Section 6.10 provides a detailed description of a new control method and its results for a wave-excited pendulum. Concluding remarks are given in the final section. It is worth noting that the first part of this chapter has been recently published in [154].
123
6.1. Introduction
6.1
Introduction
The concept of the application of parametric pendulum for wave energy extraction was proposed by Wiercigroch [161], where the vertical motion of sea waves excites a parametric pendulum fixed on a rotary platform and the vertical motion of the base is transformed into rotation of a pendulum. As has been presented in Section 2.3, the rotation of parametric pendulum by analytical, numerical and experimental study has been extensively explored, showing that the rotational responses can be observed as a steady state stable solution of the harmonicallyexcited pendulum. Also, the dynamics of the pendulum regarding the wave energy harvesting application have been extensively studied in the last decade. Although all aforementioned studies confirmed the existence of rotational solutions, these responses appear within limited parameter ranges and for specific initial conditions. Therefore, for this particular application, a system for energy extraction needs to include a controller ensuring firstly that the pendulum starts rotating, and then that it will maintain rotational motion irrespective of changing forcing conditions. In addition, the control method should be easy to implement, be robust and have a low power consumption. On the other hand, from a practical point of view, it is necessary to describe the excitation in terms of the sea wave. However, because of the random nature of a wave (modeled by the composition of different frequencies, with random phase shifts), the response of a wave-excited pendulum cannot be easily obtained. In summary, despite the conducted studies, there is still an urgent need to develop robust control algorithms for starting up the rotation under various conditions. Also, control methods need to be developed to increase the probability of rotational responses of the wave-excited pendulum.
124
6.2. Experimental rig
6.2
Experimental rig
In the experimental studies the excitation of the system has been provided by an electromagnetic shaker. The setup consisting of a pendulum fixed on the shaker is shown in Fig. 6.1(a). Here the measurement and control are performed by a low inertia DC servo-motor with encoder unit attached to the pendulum. The signal from the encoder is observed in real-time using the data acquisition software Labview. A NI PCI-6251 has been used for data acquisition and real-time control. Figs. 6.2 (a and b) present front views of two Labview programs for data acquisition and control of the pendulum, and for control of the electromagnetic shaker respectively.
6.3
Mathematical modelling
A parametric pendulum forced on the plane is schematically represented in Fig. 6.1(b). The generalized equation of motion of the system can be derived by using Lagrange’s method as follows: mlθ¨ + m(¨ y + g) sin θ + clθ˙ = 0,
(6.1)
where θ defines the angular displacement of the pendulum from the stable hanging zero position, c and m are damping coefficient of the pendulum shaft and mass of the pendulum bob, l is pendulum length, y can be arbitrary function and t is time. For the classical parametric pendulum, y is given by a harmonic function such as a sinusoidal function as below y(t) = A cos(Ωt),
(6.2)
125
6.3. Mathematical modelling
(a)
(b)
encoder motor
y
m θ, c
l
shaker m (c) 3
2
motor driver
PCI-6251
1
computer electromagnetic shaker
Figure 6.1: (a) An experimental rig of vertically excited pendulum, (b) a schematic representation of vertically excited pendulum adopted from [111], (c) a schematic diagram illustrating the pendulum setup including measuring and controlling equipment, 1-pendulum, 2-belt-gear assembly connected to the motor, 3-encoder.
126
6.3. Mathematical modelling
(a)
(b)
Figure 6.2: (a) A front end of Labview program for data acquisition and control of the pendulum, (b) A front end of Labview program for controlling the electromagnetic shaker.
127
6.4. Dynamics of the harmonically excited parametric pendulum where Ω is the angular frequency, A is amplitude of excitation and t is time. Alternatively, y can be a function of the wave displacement and can be derived from the wave displacement function y(t) containing reconstructed time history of the wave, applied at the pendulum support. This is discussed in Section 6.7.
6.4
Dynamics of the harmonically excited parametric pendulum
The equation of motion for a harmonically excited parametric pendulum can be obtained by substituting Eq. 6.2 into Eq. 6.1 and non-dimensionalising as below θ′′ + γθ′ + (1 + p cos(ωτ )) sin θ = 0,
(6.3)
where ′ and ′′ denote derivatives with respect to the non-dimensional time τ , γ, p and ω are non-dimensional damping coefficient, excitation amplitude and forcing p frequency, which are non-dimensionalised with respect to τ = ωn t; ωn = g/l is
the natural frequency of the pendulum; ω = Ω/ωn where Ω is a forcing frequency; γ = c/(mωn ) where p = Aω 2 /l.
Several studies focused on the dynamical responses of the harmonically excited parametric pendulum (see for example [166], [57] and [110]). One of the most interesting properties of the parametric pendulum is the co-existence of attractors depending on its initial conditions. Fig. 6.3 shows two basins of attraction for sinusoidal excitation with (a) (p = 0.005, ω = 2), (b) (p = 0.02, ω = 2) and the damping on the shaft γθ = 0.01 for both cases. Depending on its initial conditions, the pendulum experiences clockwise and anticlockwise rotations or oscillations. As is shown in Fig. 6.3, the stable rotational motion can be achieved in a relatively narrow range of initial conditions for given forcing parameters. For
128
6.4. Dynamics of the harmonically excited parametric pendulum (b)
θ˙
θ˙
(a)
θ
θ
Figure 6.3: Basins of attraction for (a) p = 0.005, ω = 2 and γθ = 0.01 (b) p = 0.02, ω = 2 and γθ = 0.01, showing co-existence of rotational and oscillatory attractors within the main resonance zone. △ marks period-two oscillation attractors (yellow) or fixed point at the hanging down position (green). ◦ marks period-one pure rotation attractors [110]. the application energy harvesting, stable rotation is required in a wide range of the forcing parameters. Thus, application of control ensuring a desired response needs to be considered.
6.4.1
Choice of control parameter
For initiating and maintaining the rotational motion of the pendulum, a low inertia bidirectional servomotor has been used (Fig. 6.1(a)). The additional torque component u coming from the servomotor was chosen as a control parameter and can be included in non-dimensional equations as follows: θ′′ + (1 + y ′′ ) sin θ + γθ θ′ − u = 0.
129
(6.4)
6.5. Initiating the rotational motion
θ [rad]
θ [rad]
losing the rotation 2 0 -2 7
t [s]
0
14
2
u [V]
u [V]
0
2 0 -2
0
-2
7
14
7
14
t [s]
2 0
-2
7
0
switched on
14
t [s] switched off
0
switched on
t [s]
switched off
Figure 6.4: Experimental results showing the initiation of the rotation using the bang-bang method with k = 1 showing the pendulum displacement (red) and control signal (blue), (left) Unsuccessful initialization for sinusoidal excitation with amplitude of A = 0.300 V and f = 1.6 Hz and (right) successful initialization for sinusoidal excitation with A = 0.400 V and f = 2 Hz.
6.5
Initiating the rotational motion
In order to initiate the rotational motion of the harmonically excited pendulum three methods are employed: Bang-Bang method, Velocity comparison control, and Time-delayed feedback method.
6.5.1
Bang-Bang method
The main idea behind this method is to gradually increase the amplitude of the pendulum oscillations until the rotational motion is achieved. In order to do so, the bidirectional motor is employed. In each direction of the oscillation, clockwise and anticlockwise, the motor adds a constant torque in the direction of the oscillation. If the motor can provide enough torque, the pendulum will rotate in the first swing. Otherwise the amplitude of oscillation will increase gradually and the rotational motion of pendulum will be eventually achieved. The control signal u is given by: ˙ u = k sgn(θ),
(6.5)
130
6.5. Initiating the rotational motion ˙ indicates where k is gain of controller and sgn is sign function. Note that sgn(θ) the direction of the rotation. The critical point for this method is the switch off time for the motor. Immediate disconnection of the motor once the first rotation is achieved does not guarantee stabilization of the rotational motion. The position and the velocity of the pendulum at the switch off time needs to be suitable for maintaining a stable rotation. In order to find a suitable moment for switching off the motor, periodic rotational orbits from numerical analysis illustrated on Fig. 6.3 should be considered. From this figure it can be assumed that for the rotational period one solution, pendulum in horizontal position (θ = π/2 or −π/2) should have the non-dimensional velocity around 2.5 while the phase of excitation is 2nπ and n is a natural number. Therefore, if the motor is switched off at this point rotation will be stable. In a more general view, knowing the structure of the basins of attraction for particular parameters (Fig. 6.3), the required velocity for each point can be determined. Combining this information for all points and extracting the phase from the excitation makes the method more difficult to implement. Also this method is highly sensitive to parameters. Fig. 6.4 illustrates two examples of experimental results of the bang-bang control applied to initiate the rotation of a pendulum. Control signal and angular displacement are plotted against time for two types of excitation: (right) for sinusoidal excitation with amplitude 0.400 V and frequency 2 Hz. As shown in the figure the initialization is successful. For A = 0.300 V and f = 1.6 Hz (case (left)), the initiation is not successful.
6.5.2
Velocity comparison control
In the second method, in order to achieve the rotational motion the velocities of the two critical points are compared with the known velocities of the corresponding points on the desired orbit. For example, the experiment shown in 131
6.5. Initiating the rotational motion
Pendulum in π position
ˆ u = k (θ˙(θ=0) − θ˙(θ=0) )
ˆ u = k (θ˙(θ=π) − θ˙(θ=π) )
Pendulum in zero position
Figure 6.5: A schematic of the velocity comparison control method, where θ˙(θ=0) is the previous velocity of the pendulum at the zero position and θ˙(θ=π) is the previous velocity at π, θˆ˙(θ=0) is the velocity of pendulum expected at the zero position in period one rotational orbit, θˆ˙ is the velocity at π.
θ˙ [rad/s]
(θ=π)
f [Hz] Figure 6.6: A rotational velocity of pendulum at θ=0 and π for a stable period one rotation against the frequency of excitation (1.1 Hz to 2.7 Hz) determined experimentally.
132
6.5. Initiating the rotational motion Fig. 6.4(left) can be considered. The velocity of the pendulum excited by a harmonic wave with A = 0.400 V and f = 2 Hz when it is in zero position is around ±16.12 rad/s and the velocity of the pendulum when θ = ±π is around ±10.3 rad/s. In the proposed method, for an angular displacement of the pendulum between −π and 0, the difference between the previous velocity at zero position and the velocity expected there for period one orbit (16.12 rad/s) will be applied as a control signal. For angular displacement of the pendulum between 0 and π the difference between the previous velocity of the pendulum at θ = π and the velocity expected there (10.3 rad/s) will be applied as a control signal (Fig. 6.5). Then the control signal is given as
u=k
(θ˙(θ=0) − θˆ˙(θ=0) ) if −π ≤ θ < 0, (θ˙(θ=π) − θˆ˙(θ=π) ) if 0 ≤ θ < π,
(6.6)
where θ˙(θ=0) is the previous velocity of the pendulum at the zero position and, ˆ θ˙(θ=π) is the previous velocity at π, θ˙(θ=0) is the velocity of pendulum that is ˆ expected at the zero position in period one rotational orbit, θ˙(θ=π) is the velocity at π, and k is the gain. Similarly to the bang-bang method, the magnitude of the control signal which can be applied to the pendulum is limited by the maximum power of the motor used in experiments. Therefore, again a multi-switching approach has been used and the control signal modified to: (θ˙(θ=0) − θˆ˙(θ=0) ) if −π ≤ θ < 0, ˙ u = k sgn(θ) (θ˙(θ=π) − θˆ˙(θ=π) ) if 0 ≤ θ < π.
(6.7)
As visible from the Fig. 6.7, this method is very efficient at initiating the rotational motion for the presented cases. Although this method is robust and not sensitive to small changes of parameters, it requires the knowledge of the velocity 133
θ [rad]
θ [rad]
6.5. Initiating the rotational motion 2 0 -2 4
t [s]
0
8
u [V]
u [V]
0
2 0 -2
2 0
-2
7
14
7
14
t [s]
2 0
-2
4
0
switched on
8
t [s]
0
switched on
t [s]
Figure 6.7: Experimental results for initiating the rotation of a pendulum using velocity comparison control method (with multi-switching). For both sets of forcing parameters (left) (A = 0.400 V and f = 2 Hz) and (right) (A = 0.300 V and f = 1.6 Hz) the initialization is successful. of two reference points on the period one orbit for all frequencies considered. The velocity of the pendulum for stable period one rotation when the pendulum is in upper position (θ = π) and when it is downward (θ = 0) have been determined experimentally (Fig. 6.6). This method is not sensitive to amplitude variations, but it is not suitable for low frequencies. Another problem of this method is its sensitivity to the errors in velocity calculation. In the experiment, the position of the pendulum has been accurately determined with an encoder. However the velocity, which is numerically computed from the angular displacement measurement, has a lower accuracy and some discrepancies between the computed and actual values have been observed.
6.5.3
Time-delayed feedback method
A continuous method, so-called TDF control (Time-Delayed Feedback), has been proposed by Pyragas [131] in order to control chaos and stabilize the desired UPO (unstable periodic orbit). In this method the system can be stabilized by a feedback perturbation proportional to the difference between the present and a delayed state of the system. For example, the difference between velocities or
134
θ [rad]
θ [rad]
6.5. Initiating the rotational motion 2 0 -2 7
t [s]
0
14
u [V]
u [V]
0
2 0 -2
2 0
7
14
7
14
t [s]
2 0
-2
-2 7
0
switched on
14
0
t [s]
switched on
t [s]
Figure 6.8: Experimental results of initiating the rotation of a pendulum using the TDF with multi-switching method. For all forcing parameters (left) (A = 0.400 V and f = 2.5 Hz), and (right) (A = 0.400 V and f = 2 Hz); the initialization was successful. angular displacements as in the formula below u = k(θ(τ − Γ) − θ(τ )),
(6.8)
where Γ is the period of the desired UPO, τ is the non-dimensional time, θ is the angular displacement of the pendulum, and k is the gain of the controller. In order to apply this method to the parametric pendulum, the delay time (τ ) needs to be defined. It is well known from the dynamics of the system that the period of stable period n-rotation is a n-multiple of the period of excitation, where n is a natural number, so that for the period one rotation n = 1 and for the period two rotation n = 2. The main interest of this study is the period one rotation and n = 1 is considered. Therefore, the delay time has been set equal to the period of excitation. As mentioned before the pendulum displacement measurements are more accurate than the calculated velocity value. Therefore, for the system studied the TDF method can be constructed from a continuous-time perturbation with the difference between the present angular displacement and the delayed one. Also the angular displacement increases in each rotation by 2π, so the control signal
135
6.6. Maintaining rotational motion by TDF method is given by: u = k(θ(τ − Γ) − θ(τ ) + 2π).
(6.9)
In this case when the system stabilizes on a period one rotational orbit the control signal is zero and the motor switches off automatically. This control method is capable of initiating period one rotation from all initial conditions, if a period one rotational orbit for the parameters of actual excitation exists. As for the methods considered before, the velocity of the motor used in the experimental study was not sufficient and application of the multi-switching was necessary. The modified control signal is given by: ˙ u = k sgn(θ)(θ(τ − Γ) − θ(τ ) + 2π).
(6.10)
As shown by the experimental results illustrated in Fig. 6.8, this method is successful for a range of system parameters.
6.6
Maintaining rotational motion by TDF method
The TDF control method has been applied in order to sustain the stable rotational motion of parametric pendulum excited by sinusoidal wave in the stable period one orbit while the parameters of wave (frequency and amplitude) change. To implement the method, the delay time, τ , should be defined. As in the previous section, the delay time will be set equal to the period of excitation (τ = T ). In order to detect the period automatically, an accelerometer has been used in the rig. TDF has been tested with different forcing parameters. The method is shown to be robust and automatically adjusts to the changes in amplitude or frequency of the excitation.
136
6.6. Maintaining rotational motion by TDF method changing frequency
A [V]
0.5 0
-0.5 0
30
45
60
30
45
60
45
60
t [s]
θ [rad]
15
2 0 -2 15
0
u [V]
t [s] 2 0
-2 15
0
switched on
30
t [s]
Figure 6.9: Experimental results demonstrating initiation and maintenance of the rotation of a pendulum using the TDF with multi-switching method. The time histories of the base acceleration, angular displacement and control signal are shown in green, red and blue respectively. Frequency of excitation is changed continuously from 2.5 Hz to 1.4 Hz and back to 2.5 Hz, while the controller maintains the rotations. The time histories of the base acceleration, angular displacement and control signal for the two experiments are shown in Fig. 6.9. The frequency of excitation is changed continuously from 2.5 Hz to 1.4 Hz and increased back to 2.5 Hz while the controller successfully maintains the rotation.
6.6.1
Robustness of TDF towards noise
In order to study the effect of noise on the TDF method, random noise is added to the signal control. Firstly, noise with an amplitude around 10% of the maximum signal control is considered. In Fig. 6.10(left) the noise is added while the frequency of excitation is changed continuously from 2.2 Hz to 1.4 Hz. The controller maintained the rotational motion. After that the forcing frequency is varied and noise added to the harmonic excitation. The results (Fig. 6.10(right)) demonstrate that the controller is also successful in this case. 137
6.7. Wave displacement function changing frequency
0.5
A [v]
A [v]
0.5 0 13
13
0
26
17.5
35
t [s]
u [V]
u [V]
t [s] 2 0
-2 0
35
2 0 -2
2 0 -2 0
17.5
t [s]
θ [rad]
t [s]
-0.5 0
26
θ [rad]
-0.5 0
0
2 0
-2
13
26
0
t [s]
17.5
35
t [s]
Figure 6.10: (left) Experimental results demonstrating maintenance of rotations using the TDF control method (with multi-switching) in a noisy system, showing: the time histories of the base acceleration (green), pendulum angular displacement (red) and control signal (blue). The amplitude of excitation is 0.420 V. The frequency of excitation is changed continuously from 2.2 Hz to 1.4 Hz. The amplitude of the noise is 10% of the maximum power of signal control. (right) Experimental examples of TDF control method (with multi-switching) application for maintaining rotational response while the forcing frequency is varied and noise added to the harmonic excitation, showing: the time histories of the base acceleration (green), pendulum angular displacement (red) and control signal (blue).
6.7
Wave displacement function
Prior to this section, the dynamics of a harmonically-excited pendulum were considered. In the rest of this chapter, I focus on the wave-excited pendulum. The first step is to model the sea wave. In this regards the model introduced by Najdecka et al. [111] is considered. In this and most other wave models, the Power Spectral Density (PSD) of the wave for each frequency ω is formulated based on the Pierson-Moskowitz formulation [127] as follows: " # 8.1 g 2 (g/Hs )2 S(ω) = 3 5 exp −0.032 , ω4 10 ω
138
(6.11)
6.7. Wave displacement function where Hs is the significant wave height and defined as 1/3 of the highest wave observed and g is the gravity. Fig. 6.11 depicts PSD of the wave as a function of ω for four values of Hs based on Eq. (6.11) plotted on logarithmic axes. It can be seen that when Hs increases the ωpeak decreases. 10
10
log(S(ω))
10
10
10
10
10
0
−5
−10
−15
−20
Hs Hs Hs Hs
−25
−30
10
0
10
1
10
= 10−4 = 10−3 = 10−2 = 1−1
2
log(ω) Figure 6.11: A PSD of the wave as function of ω for Hs = 10−4 , Hs = 10−3 , Hs = 10−2 and Hs = 10−1 with maximum of ωpeak equal to 125.2, 39.6, 12.5 and 3.96 rad/s respectively. Unlike most wave models, Najdecka et al. [111] used variable frequency increments, based on equal spectral content within adjacent frequency ranges. This allows less terms to be used in the sine series, which reduces the time of simulation while at the same time preserving the accuracy of simulation. In this way, the closed form formula for determination of the k -th frequency component is:
ωk =
0.032( Hgs )2 ln kr + 0.032( Hgs )2 /ωr 4
!0.25
,
139
(6.12)
6.8. Dynamic of wave-excited pendulum where r, the number of frequencies, should be chosen based on the computational limitation. The histories of the sea waves can be generated corresponding to the calculated PSD (by Eq. (6.11)) using the Shinozuka method [139] of simulating a random process as a sum of sine waves with different calculated frequencies (by Eq. (6.12)) and random phase angles φk with standard uniform distribution within 0 and 2π. Therefore, the wave displacement can be reconstructed as below
Y (t) =
r p √ X 2 S(ωk )∆ωk cos(ωk t + φk ),
(6.13)
k=1
where ∆ωk = ωk − ωk−1 , ∆ωk 6= ∆ωk−1 .
(6.14)
This set of equations (6.11-6.14) provides a powerful and systematic tool to statistically model the sea wave. The first step is to identify the significant wave height Hs based on its definition. Then depending on the computational limitation, the number of frequencies r needs to be chosen. The wave displacement can then be reconstructed as shown in the examples presented in Fig. 6.12 based on the PSD presented in Fig. 6.13.
6.8
Dynamic of wave-excited pendulum
In this section the dynamics of the wave-excited pendulum is considered. It is worth reminding that for energy harvesting purposes, the desired state of the parametric pendulum is the rotary motion as was discussed in Section 2.3. As was presented in previous sections, this response can be achieved in the harmonicallyexcited pendulum. However, because of the random nature of a wave (modeled by the composition of different frequencies, with random phase shifts), the response 140
6.8. Dynamic of wave-excited pendulum
Y [m]
1 0
-1 0
50
100
t[s]
Y [m]
1 0
-1 0
50
100
t[s]
Y [m]
1 0
-1 0
50
100
t[s]
Y [m]
1 0
-1 0
50
100
t[s]
Y [m]
1 0
-1 0
50
100
t[s] Figure 6.12: Five examples of simulated wave with Hs = 0.95 m, ωr = 5 rad/s, r = 5. of a wave-excited pendulum cannot be easily obtained. Moreover, this response should be presented and compared statistically with statistical measures. Therefore first the equation of motion of the wave-excited pendulum is presented, and then a measure is introduced to assess the success of the rotational motion of the pendulum.
Equation of motion As was mentioned in Section 6.3, the equation of motion of a vertically excited pendulum is Eq. (6.1), when Y is presented in Eq. (6.13). The equation of motion
141
6.8. Dynamic of wave-excited pendulum
S(ω)/max(S)
1
0.5
0
0
2.5 ω [rad/s]
5
Figure 6.13: A power spectrum density of a wave with significant height Hs = 0.95 m. ◦ are selected frequencies for wave model based on equal spectral content within frequency range of 0 to 5 rad/s. ∗ is the peak frequency (ωpeak = 1.1937 rad/s). is non-dimensionalised as follows: θ′′ + γθ′ + (1 + y ′′ ) sin θ = 0, where ′ and
′′
(6.15)
denote derivatives with respect to the non-dimensional time τ , γ
and y are a non-dimensional damping coefficient and the base displacement which p can be calculated from ωn = g/l, τ = ωn t, γ = c/(mωn ), y = Y /l. Scaling the wave to the experiment setup For the purpose of this study, the wave characteristics are scaled to match the dimensions of the experimental setup. The peak frequency of the wave ωpeak is scaled to twice the natural frequency of the studied pendulum ωn (See Fig. 6.14 which is the scaled PSD presented in Fig. 6.13). The wave height has been scaled accordingly. The amplitude of the pendulum excitation can be expressed in terms of a non-dimensional parameter p, defined as the ratio of the significant wave height Hs , scaled to the experimental setup, to the pendulum length, l.
142
6.8. Dynamic of wave-excited pendulum
S(ω)/max(S)
1
0.5
0
0
25 ω [rad/s]
50
Figure 6.14: Power spectrum density of a scaled wave with significant height Hs = 0.95 m. ◦ are selected frequencies. ∗ is the peak frequency (ωpeak = 12.0625 rad/s). Fig. 6.15 shows the scaled wave displacement presented in Fig. 6.12.
Rotational motion of the wave-excited pendulum As discussed in this study, the focus is on the rotational motions of the parametric pendulum for energy harvesting purposes. The dynamics of a harmonicallyexcited parametric pendulum is deterministic. Therefore, initiating and maintaining the period one rotation was the aim of the controller. However in a wave-excited pendulum, due to the random nature of the wave the rotational motion cannot be easily achieved. Fig. 6.16 depicts an example of the response of a wave-excited pendulum. In the upper panel the simulated wave is presented in green, and in the lower panel the angular displacement of the pendulum is plotted in red. As can be seen, the responses are a mixture of both rotational and oscillatory motion.
143
6.9. Defining measures
y
2 0
-2 0
300
600
τ
y
2 0
-2 0
300
600
τ
y
2 0
-2 0
300
600
τ
y
2 0
-2 0
300
600
τ
y
2 0
-2 0
300
600
τ Figure 6.15: Examples of scaled waves with Hs = 0.95 m, ωr = 5 rad/s and r = 5. The wave characteristics are scaled to match the dimensions of the experimental setup; l = 0.271 m. The peak frequency of the wave ωpeak is scaled to twice of natural frequency of the studied pendulum, ωn (ωpeak = 12.06) and the height of the wave, y is scaled accordingly.
6.9
Defining measures
A measure needs to be defined in order to evaluate the number of rotations of the wave-excited pendulum and the efficiency of any proposed controllers for the system. In the literature there are some well defined measures. For instance, rotation number, R has been used for the deterministic system in some studies (for example see [59]) which was used for the wave-excited pendulum by Najdecka et
144
6.9. Defining measures
y
2
0
−2
0
50
100
τ
θ
2 0 −2 0
50
100
τ Figure 6.16: An example of rotational displacement of parametric pendulum (red) excited by simulated wave (green) with Hs = 0.47 m, ωr = 10 rad/s and r = 5. al. [111] as: 1 R = lim τ2 →∞ ωpeak (τ2 − τ1 )
Z
τ2
˙ θdτ,
(6.16)
τ1
where τ2 − τ1 is the time span, which for higher accuracy should be maximized. In this regard some tests have been carried out and a few examples of change of the rotation number, R, in time are presented in Fig. 6.17. This rotational number can be seen as an indicator of how effectively the kinetic energy of the base is transferred to the kinetic energy of rotations of the pendulum. Therefore the aim of this study is to maximize this number.
145
6.10. Rotational control for wave-excited pendulum
R
0.3
0
0
300
600
τ
R
0.3
0
0
300
600
τ
R
0.3
0
0
300
600
τ
R
0.3
0
0
300
600
τ
R
0.3
0
0
300
600
τ Figure 6.17: Some examples of change of rotation number of the wave-excited parametric pendulum excited by simulated wave with Hs = 0.47 m, ωr = 10 rad/s and r = 5.
6.10
Rotational control for wave-excited pendulum
To increase the probability of rotational responses of the system, control methods based on a predictive control method can be applied. As will be explained later, this control signal will be applied under certain conditions. Therefore, this method can be also classified as an ’act-and-wait’ control.
146
6.10. Rotational control for wave-excited pendulum pendulum in downward position
pendulum in upward position
Figure 6.18: A schematic representation of ideal position of the pendulum in peaks and valleys of the wave.
6.10.1
Error signal; predictive control
The key point of the proposed method is to predict the pendulum behaviour in forthcoming waves and if necessary, to provide a suitable control signal which drives the motor connected to the shaft of the pendulum to help the pendulum rotate. As shown in Section 6.7, waves have a probabilistic nature which is not periodic. Therefore the wave-excited pendulum does not have periodic responses. The proposed approach to tackle this problem is to determine a main feature of the desired orbits instead of an exact orbit. Then based on calculated error, the suitable control signal for an active time is produced. The proposed observable feature in future behaviour of a wave-excited pendulum is predicted angular position on the next major valley of the forthcoming waves which will be called θv . In period-one-rotation responses of sinusoidalexcited pendulum, I expect to have the pendulum in downwards position (θ = 0) when its pivot is at the peak, and in upwards position (θ = π) when the excitation 147
6.10. Rotational control for wave-excited pendulum is in valley (see Fig. 6.18). This information from pendulum dynamics leads me to define the error (in the sense of the error signal defined in control theory) as
e = θv − π,
(6.17)
where θv is predicted angular position on the next major valley of the oncoming waves.
6.10.2
Peak detection
As was explained in Section 6.10.1, the next major peak and valley of forthcoming waves have a significant role in our proposed control method. Therefore they need to be detected carefully. In simulation, the wave can be smoothed and then the peaks and valleys can be detected (Fig. 6.19). It is worth noting that in practice the forthcoming wave can be observed with the long beam attached to the system and/or by image processing methods.
y
1.5
0
−1.5
0
25
50
τ Figure 6.19: Simulated and smoothed wave (green and black respectively) and its major peaks (* in red) and valleys (* in blue). 148
6.10. Rotational control for wave-excited pendulum
6.10.3
Control signal; wait-and-act predictive control
u
e = θv − π
Figure 6.20: A schematic representation of timing of possible control signal. In the proposed method (6.18) u is applied between peaks and valleys of the wave as necessary. The ideal control system for wave-excited pendulum should provide the minimum additional torque in the optimal time in order to help the pendulum to stay quasi-synchronous in phase with the wave as the system is to harvest energy. I therefore need to use the energy minimally. In this study, by quasisynchronization I mean that the e in equation (6.17) becomes positive. Therefore, the control system waits to act for the moment when e is negative but not too negative. In contrast, the control system does not act when e is positive, which means that the pendulum will naturally follow one of the desired orbits or when they are too negative which means it is not worth helping it. The error, e, is evaluated discontinuously at certain critical points. In this method, the control signal is defined as follows: e, emin < e ≤ 0 and tp < t < tv , k ˙ sgn(θ) u(t) = (tv − tp ) 0, otherwise,
(6.18)
where k is the control gain, tp and tv are the time of the last peak and forthcoming valley respectively, e is defined as equation (6.17), emin is a non-positive value which represents the worst case scenario that activates the controller. 149
6.10. Rotational control for wave-excited pendulum It is worth noting that the first condition, emin < e1 ≤ 0, is to activate the controller when ’it is necessary’ and ’it is efficient’. However the second condition, tp < t < tv , is to guarantee that the controller will be activated in proper timing as is shown in Fig. 6.20. In this method there is no control signal when the pendulum is going through the wave from valley to peak. Also, it is important to mention that in this method, the angular position of the pendulum on the next major valley of the forthcoming wave is predicted in each peak of the wave and then e and u are calculated consequently. Therefore, the control signal either is constant or is zero (see blue plots in panels (b) of Figs. 6.21 and 6.22). Fig. 6.21(a) depicts an example of wave-excited pendulum dynamics without any control signal. In this figure simulated wave signals with p = 0.1774 are presented in green and the angular positions of wave-excited pendulum in red. In this example, the pendulum has oscillatory responses when there is no control signal applied to the system (blue). In contrast, Fig. 6.21(b) depicts an example of a wave-excited pendulum when control signal (blue) based on the proposed control method with k = 10 is applied. In this example the controller achieves rotary responses of the pendulum. Fig. 6.22(a) depicts another example of wave-excited pendulum dynamics without any control signals. In this figure simulated wave signals with p = 0.1984 are also presented in green and the angular positions of wave-excited pendulum in red. In this example, the pendulum has both oscillatory and rotary responses when there is no control signal applied to the system (blue). In contrast, Fig. 6.22(b) depicts an example of wave-excited pendulum when control signal (blue) based on the proposed control method with k = 10 is applied. As in the previous case, in this example the controller ensures pure rotary responses of the pendulum. 150
6.10. Rotational control for wave-excited pendulum Table 6.1: List of parameters used in simulation results presented in Figs. 6.23 and 6.24. Parameter p ωr [rad/s] r k
(a) (b) (c) (d) (e) (f) 0.0180 0.0538 0.0612 0.1774 0.1984 0.5612 500 150 60 60 60 60 30 15 10 10 10 10 10 10 10 10 10 10
Due to the random nature of a wave (modeled by the composition of different frequencies, with random phase shifts), the success of any control methods should be presented and compared in a statical form with a statistical measure. Therefore the Probability Density Function (PDF) of rotational numbers, R for a few different values of the forcing amplitude p (estimated based on the significant height of the wave) for vertical excitation cases are presented in Fig. 6.23. In each of the six cases presented in this figure, the PDF of R of the uncontrolled wave-excited pendulum is in blue and the controlled one is in red. Each of these graphs consists of 1000 simulations for 100 seconds with time step of 0.001 seconds. The rest of the parameters of the simulations are presented in Table 6.1. As can be seen in Fig. 6.23, in all cases the controller successfully shifts the PDF to the right side which means that the controller can increase the probability of rotational responses of the system. However, when the p is low, which means the wave is less strong, the improvement is higher and when the p is high the improvement is lower. The last results presented in Fig. 6.23 raises a question of how much control effort would have been needed to achieve these improvements in the probability of rotational responses of the system. Fig. 6.24 shows the PDF of the average of the control effort for each case presented in Table 6.1. As can be seen, when p increases, the average of the control effort decreases.
151
6.10. Rotational control for wave-excited pendulum (a)
y
0.2 0 −0.2 0
100
200
0
100
200
τ
θ
2 0 −2
τ
u
1 0 −1
0
100
200
τ (b)
y
0.2 0 −0.2 0
100
200
0
100
200
τ
θ
2 0 −2
τ
u
0.4 0.2 0
0
100
200
τ Figure 6.21: Simulated wave signals (green) with p = 0.1774, the angular positions of wave-excited pendulum (red) (a) with zero control signal (b) with control signal (blue). In this example, pendulum has (a) oscillatory response without control and (b) pure rotary response with control signal based on the proposed control method with k = 10. 152
6.10. Rotational control for wave-excited pendulum (a)
y
0.2 0 −0.2
0
100
200
τ
θ
2 0 −2 0
200
τ
1
u
100
0 −1
0
100
200
τ (b)
y
0.2 0 −0.2
0
100
200
τ
θ
2 0 −2 0
200
τ
0.4
u
100
0.2 0
0
100
200
τ Figure 6.22: Simulated wave signals (green) with p = 0.1984, the angular positions of wave-excited pendulum (red) (a) with zero control signal (b) with control signal (blue). In this example, pendulum has (a) both oscillatory and rotary response without control and (b) pure rotary response with control signal based on the proposed control method with k = 10. 153
6.10. Rotational control for wave-excited pendulum
a)
b) 33
PDF
PDF
40
0 0.1
1
0
1.2
0.2
1
1.2
d)
c)
12
PDF
PDF
40
0 0.1 0.2
0 0.2 0.4
1 1.1
e)
1
1.7
f) 4
PDF
PDF
5
0 0.1
1
1.8
0 0.7
1
1.3
1.7
Figure 6.23: PDFs of rotational numbers, R for six different values of the forcing amplitude p. The PDF of R of the un-controlled wave-excited pendulum is in blue and the controlled one is in red. Each of these graphs consists of 1000 simulations for 100 seconds with time step of 0.001 seconds. The other parameters of the simulations are presented in Table 6.1.
154
6.11. Conclusions 50
= 0.0180 = 0.0538 = 0.0612 = 0.1774 = 0.1984 = 0.5612
PDF
P P P P P P
0 0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Average control effort
Figure 6.24: PDFs of the average of the control effort for each case presented in Table 6.1. When p increases, the average of the control effort decreases.
6.11
Conclusions
In this chapter the control for initiating and maintaining the rotational motion of a parametric pendulum with two types of excitation, harmonic and wave have been studied. Three different methods of control for initiating the rotation have been applied, including bang-bang method, velocity control method and TDF with multiswitching. Experiments have been conducted to demonstrate that each of the methods discussed is capable of initiating rotational motion. The limitations of the bang-bang method include its sensitivity to parameters of excitation. The velocity method needs the exact foreknowledge of the system and the desired rotational orbit. In the next step it has been demonstrated that the TDF control method
155
6.11. Conclusions can be used to maintain the rotational response when the excitation signal is perturbed. I studied experimentally the robustness and sensitivity of the method with respect to changing parameters of excitation and added noise. In addition, it has been demonstrated that if the parameters change rapidly the natural rotation can disappear. However if TDF is applied then rotation is maintained. The wave displacement function model is then presented. The sea wave is modeled by the composition of different frequencies, with random phase shifts. The dynamics of a wave-excited pendulum system was then explained and a statistical measure (rotational numbers) was presented in order to measure probability of the rotational motion of the pendulum. The time interval necessary for the system to reach a steady state is determined applying the measure to several numerical results. In order to increase the probability of the rotational motion of a wave-excited pendulum, a wait-and-act predictive control method was suggested. The controller aimed to help the pendulum to stay quasisynchronous in phase with the wave. This method is based on the predicted angular position of pendulum on the next major valley of the forthcoming waves. It was then demonstrated that the proposed model was successful in increasing the number of rotations. The results in form of the PDF of rotational numbers were then presented. As shown in the PDFs of the average of the control effort, when the significant wave height Hs increases, the average of the control effort needed to keep the R close to one decreases.
156
Chapter 7 Conclusions and recommendations for future work This project was aimed at understanding nonlinear dynamic behaviour in engineering applications. Moreover, the possibility of controlling those systems simultaneously in order to make the best use of nonlinearities were investigated. This study involved three different problems, as follows: • Suppression of drill-string and BHA torsional vibrations while drilling by using a closed loop controlling system.
• Analyzing soft impact in the asymmetric Jeffcott rotor in order to investigate vibrations of rotor dynamics subjected to mass imperfections.
• Initiating and maintaining the rotational motion of harmonically-excited and wave-excited pendulum as an energy harvesting system. The thesis attempts to address those aspects of the three applications which have not been covered by the previous studies, but which are crucial to the understanding of the dynamics of these systems. Each of these three problems was treated separately in the previous chapters of this thesis. The detailed conclusions as well as recommendations for the future work for each project are summarized below. 157
7.1. Torsional vibration of drill-string
7.1
Torsional vibration of drill-string
The aim of this project was to suppress drill-string torsional vibrations while drilling. Therefore the experimental drilling rig including top motor, gearing and pulley system, drill-pipes, WOB system, drill-bits, rock samples, fixtures and fluid system, as well as sensors, instrumentation and data acquisition and control system were fully described and two different configurations were explained. In the first configuration, a rigid shaft was used in order to develop empirical models for drill-bit and rock interaction. In this regard, extensive experimental results were obtained in this configuration. Curve fitting procedures were then carried out and the model was developed. This model includes the two phases: constant friction, and static-kinetic exponentially decaying friction and its combination with the Stribeck effect. In the second configuration, flexible shafts were used to mimic the mechanical properties of slender structures like a drill-string. The experimental results presented include torsional vibrations and stick-slip phenomena observed in the rig in different conditions. Two types of stick-slip were observed. The typical stick-slip is significantly similar to the one obtained in real field by the in-bit sensor and chimerical MWD vibration monitor. Non-typical stick-slip vibrations are a combination of typical stick-slip and torsional vibration in each period. In these experimental results the bit-speed reached up to 4.5 times the top-speed. A harmonically excited torsional model is presented next. In this model the top speed is modeled as harmonic excitation and the torque model derived in the previous chapter is used. As a result, the drill-bit stick-slip oscillations observed in the rig were modeled perfectly. This is confirmed by comparison of FFTs of the bit-speed obtained in the experiment and numerical simulation. In both cases, one major peak is observed at the same frequency.
158
7.1. Torsional vibration of drill-string A 1-DOF model for the top motor and gearing system is then presented. Several systematic experiments are carried out in order to identify the model parameters. In the next step, the whole rig is considered and a 2-DOF lumped mass model introduced to predict stick-slip results, having the motor torque as the control input of the model. An excellent match between experiment and simulation is achieved. This is confirmed by comparison of major peak of FFTs of the bit-speed obtained in the experiment and numerical simulation. A sliding-mode control method is presented, and applied to the obtained model in order to eliminate stick-slip vibrations observed in the drilling rig and simulation. The stability of the controller is proved mathematically. The controller is successful in suppressing the vibration and bringing the system to the desired equilibrium. These numerical results are presented for a few sets of parameters. The controller is then implemented in the experiment as verification of the numerical results. The controller is successful in eliminating the stick-slip in the experiment. However, a limit cycle is observed around the desired equilibrium. Investigating the difference between experimental results and simulation, a delay and dead-zone is obtained in the actuator. Adding them to the 2-DOF model achieves an excellent match between experiment and simulation. In order to examine the sensitivity of the controller to the parameters, several experiments are carried out with a variety of the estimated parameters applying the controller. Phase portraits of the responses clearly show the success and robustness of the controller. A significant reduction in vibration amplitude is observed when the controller is applied.
7.1.1
Recommendations
• In order to improve the 1-DOF model (excluding the top motor), the next natural step would be to match the experiment and model for a wider range 159
7.2. Lateral vibration of unbalanced rotor of parameters and then to obtain experimental and numerical bifurcation diagrams. • The experimental torque model needs to be obtained for higher range of the WOB to provide deeper understanding of the interaction between the drill-bit and rock. • The top motor and gearing system should be modelled over a wider range of parameters and include the delay of the response. Alternatively, the top motor and gearing system can be replaced with an adequate DC motor in order to minimize the extra undesired dynamics in that system. • In case there is a need to work in the current configuration of the drilling rig, one needs to design a control method to overcome the delay in the motor system to avoid the limit cycle observed in the controlled-response. • In order to improve the applicability of the sliding-mode control in the field drilling rig, design an observer to estimate the drill-bit angular position and velocity to use in the controller.
7.2
Lateral vibration of unbalanced rotor
The aim of this project was to analyze soft impact in the asymmetric Jeffcott rotor in order to investigate vibrations of rotor dynamics subjected to the mass imperfections. First, the main components and features of the experimental apparatus were described. Several experimental bifurcation diagrams with different condition in range around grazing point were conducted. Each of these graphs consists of 72 experiments. For each configuration two sets of experiments were carried out. First the rotor speed was increased from 9 to 16.1 Hz with increment of 0.1 Hz then it was decreased from 16.1 to 9 Hz with decrement of 0.1 Hz. 160
7.2. Lateral vibration of unbalanced rotor Some observed common responses are period one, period three, period two, quasiperiodic and period two embedded in a quasi-periodic response. A separate test was carried out then in order to study the physical properties of the snubber ring support, which revealed an anisotropic nature as well as the presence of damping effects. Physical and mathematical models of the experimental rig were derived. In this model the system motion is characterized by two modes of operation: no contact and contact between the rotor and the snubber ring. Moreover, in order to model the motion of the snubber ring, it is assumed that its motion is governed by the principle of minimum total potential energy, thus neglecting any transient effect on the snubber ring for each change in the position of the rotor during the contact regime. This process resulted in a new set of equations describing the motion of the system whose numerical implementation was described and mathematically justified in detail. In order to verify the suggested model, a frequency sweep from low to high values in the experimental rig with small increment (0.3 Hz) was carried out. It revealed the following sequence of dynamical scenarios: non-impacting period1 response, grazing contact between the rotor and the snubber ring, impacting period-3 orbits, quasi-periodic motion, impacting period-2 response and impacting period-1 orbits. Comparing the numerical results and experimental observation, the mathematical model proved to be capable of predicting the sequence of dynamical scenarios just described, as well as determining with a certain degree of accuracy the frequency windows where the different qualitative phenomena occur. Nevertheless, for higher frequency values the responses of the experimental rig and the mathematical model differ, mainly due to unavoidable nonlinear effects not considered in the model and uncertainties in the parameter estimation.
161
7.3. Rotational motion of parametric pendulum
7.2.1
Recommendations
• The next step of this project is to design a controller to avoid the bifurcation in order to postpone the impact between the rotor and the snubber ring. • Another aim for the controller could be switching between two co-exciting responses in order to stabilize the non-impacting response. • In order to implement any control methods in the experimental rig, an actuator needs to be employed in the system. For example the motor speed can be seen as the control input.
7.3
Rotational motion of parametric pendulum
The aim of this project was to initiate and maintain the rotational motion of parametric pendulum as an energy harvesting system. First the experimental rig and the mathematical model for the parametric pendulum were described. Three different methods of control for initiating the rotation have been applied, including bang-bang method, velocity control method and TDF with multiswitching. Experiments have been conducted to demonstrate that each of the methods discussed is capable of initiating rotational motion. The limitations of the bang-bang method include its sensitivity to parameters of excitation. The velocity method needs the exact foreknowledge of the system and the desired rotational orbit. In the next step it has been demonstrated that the TDF control method can be used to maintain the rotational response when the excitation signal is perturbed. I studied experimentally the robustness and sensitivity of the method with respect to changing parameters of excitation and added noise. In addition, it was demonstrated that if the parameters change rapidly the natural rotation can disappear. However if TDF is applied then rotation is maintained. 162
7.3. Rotational motion of parametric pendulum The wave displacement function model is then presented. The sea wave is modeled by the composition of different frequencies, with random phase shifts. The dynamics of a wave-excited pendulum system was then explained and a statistical measure (rotational numbers) was presented in order to measure probability of the rotational motion of the pendulum. In order to increase the probability of the rotational motion of a wave-excited pendulum, a control method was suggested. It was then demonstrated that the proposed model was successful in increasing the number of rotations. The results in the form of the PDF of rotational numbers were then presented.
7.3.1
Recommendations
• Both this and previous studies show that the dynamics of the response of the pendulum to harmonic excitation and wave excitation are different and the aim of the controller for these two systems would be fundamentally different. To extract the wave energy, the focus of any further work in this project should be on wave-excited pendulum. • Apart from the suggested rotational control method for the wave-excited method, the velocity of pendulum in the bottom position and the phase synchronisation can be considered as feedback signals and a new method can be designed based on them. • The suggested method can be verified in the experimental rig with a wave tank. First the excitation can be limited in the vertical direction and then it can be applied in two directions. • The suggested method can be used for synchronization of a four-pendula system in order to make the whole system self-stabilized in the x-y plane.
163
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185
Appendices
186
Appendix A Experimental torque modelling In this appendix, the experimental torque curves are modelled mathematically, taking into account all qualitatively different patterns which can be seen in Fig. 3.18. It worth noting that the material presented in this appendix is adopted from [75]. A simple frictional model is required that can be calibrated in terms of the experimental observations. The main idea is to use a suitably effective friction coefficient (slip-rate dependent) that captures the main phenomena observed in our drilling tests, namely, cutting and friction between the drill-bit and the rock sample. In this regard the bit-rock interface is considered as being two circular sliding surfaces of radius R > 0 with an equivalent friction coefficient µ, which can be velocity-dependent (e.g. exponentially decaying with respect to the relative sliding velocity). One surface which represents the drilled rock sample is stationary. The second surface which represents drill-bit rotates at an angular velocity θ˙b ≥ 0 (see Fig. A.1). In this analysis, the friction coefficient at any point of the bitrock interface will in general depend on the linear slip velocity, i.e. µ := µ(r, θ˙b ) , 0 ≤ r ≤ R. Therefore, the reaction torque during the slip mode can be computed 187
dθ dTb,sl dA dr r
R
θ
µ θ˙b
Figure A.1: Bit-rock interface modelled as two sliding surfaces with an equivalent friction coefficient µ. as follows:
T b,sl (θ˙b , γf ) =
ZR Z2π 0
r 2 Wb 2Wb µ(r, θ˙b ) dθ dr = 2 2 πR R
0
ZR
µ(r, θ˙b )r 2 dr,
(A.1)
0
where Wb > 0 represents the WOB acting on the system and γf is a parameter vector. A presence of the Stribeck effect [165] can be observed. Therefore the friction model to be used for the mathematical description of the bit-rock interaction is ˙ µ(r, θ˙b ) = µk + (µs − µk ) e−dc rθb + µstr r θ˙b ,
(A.2)
which corresponds to a classical exponentially decaying law with constants dc , µs > 0, 0 < µk < µs and µstr representing the Stribeck friction coefficient. By introducing the coefficients λs = µs R, λk = µk R, λd = dc R and λstr = µstr R2 the reaction torque (A.1) takes the explicit form as presented in Eq. (3.1).
188
Appendix B Existence and uniqueness of the solution of Eq. (5.3) This appendix provides a mathematical proof of the existence and uniqueness of the solution of (5.3) which is adopted from [121] Under certain mild assumptions and considering equation (5.3) for (xs , ys ) ksx around the origin, αk = near 1, (xr , yr ) 6= (0, 0) close to the circle x2r +yr2 = γ 2 , ksy with any γ > 0 fixed, it can be shown that (5.3) has a unique solution given by smooth functions xs = χ(xr , yr , αk , γ) and ys = Υ(xr , yr , αk , γ) satisfying:
x0s = χ(xr , yr , 1, γ) = ys0 = Υ(xr , yr , 1, γ) =
xr
yr
p
x2r
p
p
+
yr2
−γ
x2r + yr2
x2r
yr2
,
+ −γ p . x2r + yr2
(B.1)
The point (x0s , ys0) corresponds to the solution of the minimization problem aforementioned for αk = 1 (i.e. ksx = ksy ), cf. [124, Section 3.2]. Now that the solvability of the nonlinear equation (5.3) is established, let us turn to the problem of finding explicit expressions for the functions χt and Υt of
189
system (5.2). Assume that xr = xr (t), yr = yr (t), xs = xs (t) and ys = ys (t) are a smooth solution trajectory satisfying:
L(xs (t), ys (t), xr (t), yr (t), αk , γ) = 0,
(B.2)
for t in some interval (a, b). Then implicit differentiation of (B.2) gives:
x˙ s = χt (xr , yr , xs , ys , x˙ r , y˙ r , αk ),
y˙ s = Υt (xr , yr , xs , ys , x˙ r , y˙ r , αk ),
with the functions: (yr − ys ) (y˙ r (xr − xs ) + x˙ r ys ) + x˙ r (xr − xs ) ((αk − 1)xs + xr ) , (yr − ys ) (αk yr − (αk − 1)ys ) + (xr − xs ) ((αk − 1)xs + xr ) αk (xr − xs ) (x˙ r (yr − ys ) + y˙ r xs ) + y˙ r (yr − ys ) (αk yr − (αk − 1)ys ) , Υt = (yr − ys ) (αk yr − (αk − 1)ys ) + (xr − xs ) ((αk − 1)xs + xr )
χt =
(B.3) (B.4)
which are well defined provided (xs (t), ys (t), xr (t), yr (t), αk , γ) lies in the definition set of the nonlinear equation (5.3) specified above. Before moving on to the computation of the snubber ring force, let us make some remarks regarding the discussion presented above. First of all, from the physical point of view it is clear that the centre of the snubber ring will stay close to its equilibrium position (0, 0) during operation. For this reason (xs , ys ) is assumed to be close to the origin. On the other hand, one of our main concerns in this section is to describe the motion of the snubber ring during the contact regime. This mode takes place when the distance between the centers of the rotor and snubber ring equals the radial clearance γ i.e. (xr − xs )2 + (yr − ys )2 = γ 2 . Since (xs , ys ) is close to (0, 0), the impacts will occur near the circle x2r + yr2 = γ 2 . Hence values of (xr , yr ) is considered close to this circle, in such a way that the definition set chosen for the nonlinear equation (5.3) is compatible with the physical application. 190
