ENOC-2005, Eindhoven, Netherlands, 7-12 August 2005
ENOC-2005, Eindhoven, Netherlands, 7-12 August 2005
ID of contribution 17-088
NUMERICAL AND EXPERIMENTAL STUDY OF REGULAR AND CHAOTIC BEHAVIOR OF TRIPLE PHYSICAL PENDULUM Jan Awrejcewicz
Bogdan Supeł*
*Department of Automatics and Biomechanics Technical University of Ł´od´z Poland
[email protected]
[email protected]
Grzegorz Kudra*
Grzegorz Wasilewski*
Paweł Olejnik*
[email protected]
[email protected]
[email protected]
Abstract The experimental and numerical analysis of triple physical pendulum is performed. The experimental setup of the triple pendulum with the first body externally excited by the square function, including the measurement equipment, is described. The mathematical model of the system is then presented. The parameters of the model are identified by the minimization of the sum of squares of deviations between the signal from the simulation and the signal obtained from the experiment. A good agreement between results from the experiment and from simulation is shown in few examples. Key words triple pendulum, experimental investigations, model identification 1 Introduction A pendulum occupies a very important place in the history of the mechanics since that simple system can be used as an illustrative example of many interesting nonlinear phenomena (Yagasaki, 1994; Bishop and Clifford, 1996; Bishop and Sudor, 1998). It is well known, that simple pendulum harmonically excited can exhibit regular or irregular behavior, the classical bifurcations (like saddle-node, symmetry breaking and period-doubling bifurcations), coexisting attractors, etc. But a single degree-of-freedom models are only the first step to understand a real behavior of either natural or engineering systems. Many physical objects are modeled by a few degrees of freedom. In mechanics, but even in physics, an attempt to investigate coupled pendulums is recently observed (Skeledon and Mullin, 1992; Skeledon, 1994). It can be noticed also an interest focused on exper-
imental investigation of either simple or coupled mechanical and electronic setup of pendulums. As the single or double pendulum in their different forms are quite often a subject of experiment (Blackburn et al., 1987; Heng et al., 1987; Bishop and Sudor, 1998), the triple physical pendulum is rather rarely presented in literature as experimental model. For example in the work (Zhu and Ishitobi, 1999) the triple pendulum excited by horizontal harmonic motion of the pendulum frame is presented with few examples of chaotic attractors. The present paper is a part of larger project of investigations (Kudra, 2003; Awrejcewicz et al., 2003), where the triple physical pendulum with rigid limiters of motion was analyzed numerically. This work bases on the earlier investigations and its aim is to provide an experimental confirmation of those results. In the present stage, the real physical pendulum without impacts is built, and some examples of regular and irregular attractors exhibited by both the experiment and the numerical simulations are shown. 2 Experimental Setup The experimental rig (see figure 1) of the triple physical pendulum consists of the following subsystems: pendulum, driving movement subsystem of the pendulum, damping subsystem, and the measurement subsystem. The pendulum consists of a stand that is a symmetrical steel welded construction. The stand consists of a massive frame and two brackets of • • long. Each bracket is connected with two blocks that make pendulum affixed to the stand by means of the bearings (the first joint of the pendulum) as well as the driving system and the measuring system. One used axial bearings (AXK 1730) and radial bearings (NKI 17/20). The bearings are of high resistance to large loads along with their small dimensions and they provide the elimination
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of backlash. The joints are made of small aluminum blocks. One of them, that belongs to the first link, functions as a pin of the bearing and the second one serves for the fixing a shaft, where the bearing is placed. The blocks also serve for fixing disks, where a part of the measuring system and a viscous damper are located. These elements will be discussed in the sequent part of the chapter. The aluminum blocks play an important role, namely they are guides for aluminum rods connecting the joints. The rods and the blocks make links of the pendulum. These four rods are bored transversely at distances of 10mm. They are able to move in the bores of these aluminum blocks. Due to this fact, one can adjust spacing of the joints i.e. changing the length of the links. The rods are affixed to the blocks by means of steel threaded pins ended with nuts. The blocks and joining rods are made of aluminum alloy in order to decrease mass of a movable piece of the pendulum. Steel
as change the number of additional weights. The triple physical pendulum is designed as a system of two symmetrically joined pendulums to eliminate stress caused by torques or forces that do not act in the plane of motion. The pendulum has a module structure and one can easily disassemble it or change its configuration. Moreover, one can also change parameters such as the mass of the links mi , the position of mass centre of the link ei , the moment of inertia Bi , and the length of the link li . The pendulum driving subsystem con-
Figure 2. The engines: 1. the commutator; 2. the second engine without the commutator; 3. the engine stator; 4. the engine rotor.
Figure 1. The experimental rig.
shafts, that are elements of the bearing, are hollowed axially in order to decrease mass of the pendulum as well. The shafts bores also enable sending a measuring signal in optical way. That will be discussed in a sequent part of this chapter. There are aluminum blocks, placed between the joints and the rods, that serve for attaching steel weights. The blocks can be displaced along the rods and one can set their position as well
sists of two engines of slow alternating currents and optoelectronic commutation. The engines with optical system are placed in a mechanical part of the pendulum (see figure 2). The engine stator consists of sixteen aircore coils (the stator coils) and eight optoisolators. The coils are affixed to two round plates of iron ”ARMCO” (low magnetic hysteresis) and are connected in logic circuit, eight sections of four coils (one for each engine). The plates, with the coils on it, are affixed to the earlier described steel blocks and together with the system of eight optoisolators make the engine stator. The part of the optoelectronic system, including eight optoisolators (located on a tape of elastic ”Rezotex”) is screwed on the circumference of one of the plates. The stator disk possesses eight symmetry axes (22 5 ) and in order to provide the desired torque, the coils are on a circle of radius 120mm long. The engine stator has been designed in such a way that the current intensity in the windings is linearly dependent on the engine torque. On this purpose, one dropped out the ferromagnetic cores because the signal controlling the engine
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torque varies according to sine function and in some instants it takes on values close to zero. Taking heed of this fact, the interaction between the stator cores and the rotor magnets would be very high. It would disturb operation of the engine and make its operating impossible. Engines designed in this manner operates after applying to them nominal voltage.
3 Measurement Analog signals incoming from measuring devices (angle sensors) are processed in LabView measurement software. The modular LabView measurement package is presently widely used in industry as complete programmable set of test instruments which are, in particular, in cooperation with the well developed blockdiagram building software. Dynamic data acquisition is made with the use of the following test instruments: chassis PXI-1011, SCXI-1125 module installed in the chassis, the terminal block SCXI-1313 (high-voltage attenuator), which are in cooperation with the PCI6052E PC computer’s card. The PXI-1011 chassis integrates a high-performance 4-slot PXI subsystem with an 8-slot SCXI-subsystem (SCXI-1125 module) to offer a complete solution for signal conditioning and switching applications. The PXI section of the chassis accepts a MXI-3 interface, and a wide variety of modules–such as multifunction I/O (MIO), digital I/O (DIO), and computer-based instrument modules. The eight SCXI slots integrate signal conditioning modules into the PXI system. The modules provide analog and digital input conditioning, isolation, and other functions. The SCXI-1125 module is the eight-channel analog input conditioning module with programmable gain and filter settings on each channel. Each channel has 12 programmable gain settings from 1 to 2000 and two programmable filter settings of either 4Hz or 10kHz. The National Instruments SCXI-1125 provide 250 Vrms to 300 Vrms of working isolation and lowpass filtering for each analog input channel. This architecture is ideal for amplification and isolation of milivolt sources, volt sources, from 0 to 20 mA, 4 to 20 mA, and thermocouples. It can multiplex these eight channels into a single channel for the DAQ device (SCXI6052E PC computer ’s card), and some other modules can be add to increase channel count. The SCXI-1313 terminal block extends the input range of the previously described module to 300 Vr ms or 300 VDC, on a per-channel basis programmatically through software commands. The SCXI-1313 also includes an onboard temperature sensor for thermocouples cold-junction compensation. The LabView software environment offers the complete library of numerical and mathematical tools which allow processing of experimental data. Blocks are connected by lines of various colors and pattern in the environment and represent some pre-defined application procedures (reading and writing to channel in-
Figure 3. The main block diagram in the scheme in LabView.
puts and outputs, numerical analysis). The series of measured data are possible to be stored in text files and then showed on any waveforms graphs. The main diagram of the measurement-calculating blocks is presented in figure 3. This diagram has been used to measure all signals incoming from angle sensors installed directly on the three links presented in figure 1. The signals are then stored as data in the floating point representation in computer’s memory. In figure 3 in the main loop running until the break of measurements appears the following general blocks are observed (starting from the left hand side): Wave -for measurement of data from hardware channels; Array Element -for simple extraction of the measured data appearing in measurement channels as a waveform generated with a sample period; Constant, Substract, Divide -mathematical transformations for signals scaling in degrees; Derivative, IR Filter -numerical transformations of series of the stored data for the velocity of the three pendulum’s links calculation and the filtering
Figure 4. Time histories (a) and velocity dependencies (b) of the pendulum’s links.
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a way and on the basis of various combinations it possible to determine other location compositions of links of the investigated pendulum. Some of experimental
Figure 5. Location characteristics of the pendulum’s links.
process of them, respectively; For -conditional loop for data preparing to draw; Bitmap Graph -graphical interfaces with the optional parameters of graphs. The previously described process of data acquisition has been used in measurements on the pendulum’s real laboratory rig. Graphs which were made thanks to graphical interface of the diagram presented in figure 3 are presented in figures 4-6. It visible in figure 4a that the pendulum’s links motion is 3-, 4-, and 5-periodic as well. The amplitude of the first link is about 20 , while the amplitudes of the others is at the same value about 30 . changes of angular velocity of the pendulum’s links are visible in figure 4b. It can be observed that the first link rotates with the velocity above three times less then the next two. A comparison of the time changes of both angle and angular velocity of the pendulum’s links presented in figure 4 brings an natural conclusion, the same angles of rotation of links 2 and 3 have another values of angular velocities. On the basis of location characteristics presented in figure 5 the sets of relative location of the appropriate links are possible to find. For example, if the first pendulum’s link was observed in the zero angle position, the third of them could be observed only in positions of +5 and 15 (see figure 5b). In such
Figure 6. Phase space projections on the planes
2 ( 2 ),
1 ( 1 ),
3 ( 3 ).
phase space projections on the phase planes of pendulum’s links have been shown in figure 6. According to the earlier conclusions, pendulum’s links rotate in a periodical manner what is exactly proved by the loops visible in the figure.
4 Mathematical Model As a physical model of the real system presented in previous section, we use the system of three rotationally coupled rigid bodies moving in the vacuum, in the gravitational field of acceleration g, shown in figure 7. The pendulum moves in the plane and its position is determined by three angles i (i = 1 2 3). In the joints Oi the viscous damping with the coefficient ci (i = 1 2 3) is present. It is assumed that the mass centers of the links lie on the lines including the joints and one of the principal central inertia axes (zci ) of each link is perpendicular to the pendulum movement
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plane. The masses of the pendulums are mi and the moments of inertia with respect to the axes zci are Jzi (i = 1 2 3), respectively.
where M( ) =
B1 N12 cos( 1 N13 cos( 1
N12 cos( 1 B2 3 ) N23 cos( 2
N13 cos( 1 N23 cos( 2 B3 3) 2)
2)
3)
3)
N( ) =
0 N12 sin( N13 sin(
N12 sin( 0 N23 sin( 3)
1
1
1
N13 sin( N23 sin( 0 3)
2)
2)
2
1
2
3)
3)
c1 +c2 c2 0 c2 c2 +c3 c3 0 c 3 c3
C=
p( ) =
fe (t) =
f (t) 0 0
M1 sin M2 sin M3 sin
=
1
2
3
=
1
2
3
1
2
3
1
=
2
3
2
2 1 2 2 2 3
=
(2)
The parameter vector of the pendulum is = [B1 B2 B3 N12 N13 N23 M1 M2 M3
(3)
c1 c2 c3 ]
Figure 7.
Model of the triple physical pendulum.
where B1 = Jz1 + e21 m1 + l12 (m2 + m3 )
B2 = Jz2 +
e22 m2 e23 m3
+
l22 m3
B3 = Jz3 + N12 = m2 e2 l1 + m3 l1 l2
The first pendulum is externally excited by the squareshape moment of the force f (t) with the amplitude q, angular velocity and with the initial phase 0 , as shown in figure 8.
N13 = m3 e3 l1 N23 = m3 e3 l2
(4)
M1 = m1 ge1 + (m2 + m3 )gl1 M2 = m2 ge2 + m3 gl2
M3 = m3 ge3 Note that the parameters of the external forcing (q, , ) are treated separately. For more details on the triple pendulum equations and their derivation see works (Kudra, 2003; Awrejcewicz et al., 2003).
Figure 8. The external forcing function.
5 Parameter Identification In order to identify the model parameters , firstly we lead to the experimental stand the external forcing (input signal) of the known parameters: q = 2 Nm, fr = 0 4 Hz and 0 = 0 163 rad (where fr = 2 ). Then the response (output signal) i (t) (i = 1 2 3) of the real system starting from zero initial conditions ( i (0) = 0, i (0) = 0, i = 1 2 3) is measured on time interval t [0 te ] as the following time series
The governing equations of the system seen in the figure 7 are
+ N( )
2
+C
+ p( ) = fe (t)
M( )
(1)
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(0) i
(1) i
(2) i
(N ) i
(i = 1 2 3)
(5)
where
(k) i
( i (0) = 0 i (0) = 0 i = 1 2 3), for the excitation amplitude q = 2 Nm and the initial phase 0 = 0 163 rad. The experimental results are obtained for the same input signal and zero initial conditions. The figures 9-12 show periodic solutions obtained numerically (black line) compared with the experimental data (grey line) for the excitation frequency fr equal to 0.4, 0.6, 0.8 and 1 rad correspondingly. In order to eliminate the transient motion, the solutions are restricted to the time interval t [200 300 s]. As it is seen, the results from the model match the results from the object very well. Moreover, the numerical simulation shows existence of a chaotic region near the frequency fr = 0 7 rad as shown in the figure 13, where we present an exemplary solution with the first link of the pendulum performing full rotations from time to time. The figure 14 shows the corresponding experimental results that confirm the numerical prediction very well.
=
i (k
t)
k=0 1 2
N
te t= N
and where te = 100 s and N = 104 ( t = 0 01 s) were taken. We want to find such a vector , for which the model matches the real system most of all in the sense of the least deviation between the output signals from the model and the real pendulum. Therefore we define the following criterion-function of matching the output signals
N
3
2
F( ) =
(k) i
(k) i
(6)
i=1 k=0
where 4
(k) i
=
t)
i (k
3
2
is the output signal of the model for the same input signal and initial conditions, obtained from numerical simulation. The global minimum of F ( ) is now the solution of the problem. The MATLAB environment for evaluation and minimum searching of the function F ( ) is used. For minimum finding the simplex search method is used that does not use numerical or analytic gradients, and finds a local minimum of a scalar function, starting at an initial estimate. First we roughly estimate the parameter vector and use it as a starting point 0 in the procedure to find opt with the following elements
1
y1
0
-1
-2
-3
-4 -0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
y1 Figure 9. The experimental (grey line) and numerical (black line)
results for the fr = 0 4 Hz.
B1 = 0 2705 kgm2
M1 = 11 020 kgm2 s2
2
B2 = 0 1547 kgm
2
M2 = 5 6508 kgm s
2
2
B3 = 0 0480 kgm
2
M3 = 3 7383 kgm s
N12 = 0 1176 kgm2
2
(7)
c1 = 0 1203 kgm2 s
N13 = 0 0776 kgm2
c2 = 0 0025 kgm2 s
2
2.5
2
1.5 1
0.5
N23 = 0 0452 kgm
2
c3 = 0 0028 kgm s
y1
0
-0.5
which minimize the function F ( ) at least locally and, as shown in the next section, gives a good agreement between the model and the real system.
-1 -1.5 -2 -2.5 -0.5
6 Numerical Results Here some examples of numerical simulations of the triple pendulum model together with corresponding experimental results from the real object are presented. The model simulations are performed for the optimal parameter vector opt (7), zero initial conditions
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
y1 Figure 10. The experimental (grey line) and numerical (black line) results for the fr = 0 6 Hz.
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2.5
p
2 1.5 1 0.5
y1
0
-0.5 -1 -1.5 -2 -2.5 -0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
p
y1
y1 Figure 11. The experimental (grey line) and numerical (black line) results for the fr = 0 8 Hz.
p
1.5
1
p
0.5
y1
0
y1
-0.5
The numerical results for the fr = 0 7 Hz.
Figure 13.
-1
-1.5 -0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
y1
p
Figure 12. The experimental (grey line) and numerical (black line) results for the fr = 1 Hz.
p
7 Concluding Remarks In the paper an experimental stand of the triple physical pendulum and the corresponding mathematical model have been presented. The pendulum is a rich source of nonlinear dynamics phenomena. Furthermore, it can be used for modeling of a number of real objects being observed in nature and engineering. One can mention for example about arms of robots and cranes. A subtle reconstruction of that pendulum’s experimental rig can be made by adding a few obstacles bounding the rotational motion of its last link. Such an operation will provide a possibility for modeling of the piston -connecting rod -crankshaft system moving by the way in a cylinder with backlash. The model parameters identification have been performed by the minimization of the scalar criterionfunction in the form of the sum of squares of deviations between the numerical and experimental series. It should be noted, that the parameter vector opt , which minimizes the function (6), is optimal in the sense of best matching of output signals of the model and the real object. Therefore, it unnecessarily matches best the real physical parameters of the object, that can be computed from (4). As shown in section 6, the numerical simulation of the model for parameters opt , can be successfully used for the prediction of behavior of the
p
y1
p
p
p
y1 Figure 14.
The experimental results for the fr = 0 7 Hz.
real triple physical pendulum. 8 Acknowledgment This work has been supported by the Polish Scientific Research Committee (KBN) under the grant no. 5 T07A 019 23.
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References Awrejcewicz, J., G. Kudra and C.-H. Lamarque (2003). Dynamics investigation of three coupled rods with a horizontal barrier. Special Issue of Meccanica 38(6), 687–698. Bishop, S.R. and D.J. Sudor (1998). The ”not quite” inverted pendulum. Int. J. Bifurcation and Chaos 9(1), 273–285. Bishop, S.R. and M.J. Clifford (1996). Zones of chaotic behaviour in the parametrically excited pendulum. J. Sound and Vib. 189(1), 142–147. Blackburn, J.A., Y. Zhou-Jing, S. Vik, H.J.T. Smith and M.A.H. Nerenberg (1987). Experimental study of chaos in a driven pendulum. Physica D 26(1-3), 385– 395. Heng, H., R. Doerner, B. Hubinger and W. Martienssen (1987). Approaching nonlinear dynamics by studying the motion of a pendulum. i. observing trajectories in state space. Int. J. Bifurcation and Chaos 4(4), 751– 760. Kudra, G. (2003). Analysis of bifurcations and chaos in triple physical pendulum with impacts. PhD thesis. Technical University of Ł´od´z. Skeledon, A.C. (1994). Dynamics of a parametrically excited double pendulum. Physica D 75(3), 541–558. Skeledon, A.C. and T. Mullin (1992). Mode interaction in a double pendulum. Phys. Lett. A 166, 224–229. Yagasaki, K. (1994). Chaos in a pendulum with feedback control. Nonlin. Dyn. 6, 125–142. Zhu, Q. and M. Ishitobi (1999). Experimental study of chaos in a driven triple pendulum. J. Sound and Vib. 227(1), 230–238.
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