Journal of Vibration Testing and System Dynamics

Volume 2 Issue 2 June 2018

ISSN 2475‐4811 (print) ISSN 2475‐482X (online)

Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics Editors Jan Awrejcewicz Department of Automation, Biomechanics and Mechatronics The Lodz University of Technology 1/15 Stefanowskiego Str., BLDG A22, 90-924 Lodz, Poland Email: [email protected]

C. Steve Suh Department of Mechanical Engineering Texas A&M University College Station, TX 77843-3123, USA Email: [email protected]

Xian-Guo Tuo School of Automation & Information Engineering Sichuan University of Science and Engineering Zigong, Sichuan, 643000, China Email: [email protected]

Jiazhong Zhang School of Energy and Power Engineering Xi’an Jiaotong University Xi’an, 710048, China Email: [email protected]

Associate Editors Jinde Cao School of Mathematics Southeast University Sipailou 2# Nanjing, 210096, China Email: [email protected]

Yoshihiro Deguchi Department of Mechanical Engineering Tokushima University 2-1 Minamijyousanjima-cho Tokushima 770-8506, Japan Email: [email protected]

Yu Guo McCoy School of Engineering Midwestern University 3410 Taft Boulevard Wichita Falls, TX 76310, USA Email: [email protected]

Hamid R. Hamidzadeh Department of Mechanical and Manufacturing Engineering Tennessee State University Nashville, TN 37209-1561, USA Email: [email protected]

Jianzhe Huang Department of Power and Energy Engineering Harbin Engineering University Harbin, 150001,China Email: [email protected]

Meng-Kun (Jason) Liu Department of Mechanical Engineering National Taiwan University of Science and Technology Taipei, Taiwan Email: [email protected]

Zhi-Ke Peng School of Mechanical Engineering Shanghai Jiao Tong University Shanghai, P. R. China 200240 Email: [email protected]

Alexander P. Seyranian Institute of Mechanics Moscow State Lomonosov University, Michurinsky pr. 1, 119192 Moscow, Russia Email: [email protected]

Dimitry Volchenkov Department of Mathematics & Statistics Texas Tech University 1108 Memorial Circle Lubbock, TX 79409, USA Email: [email protected]

Baozhong Yang Schlumberger Smith Bits 1310 Rankin Rd Houston, TX 77073, USA Email: [email protected]

Guirong (Grace) Yang Department of Civil, Architectural and Environmental Engineering Missouri University of Science and Technology Rolla, MO 65409, USA Email: [email protected]

Shudong Yu Department of Mechanical and Industrial Engineering Ryerson University Toronto, Ontario M5B 2K3 Canada Email: [email protected]

Nyesunthi Apiwattanalunggarn Department of Mechanical Engineering Kasetsart University JatuJak Bangkok 10900 Thailand Email: [email protected]

Junqiang Bai School of Aeronautics Northwestern Polytechnical University Xi’an, P. R. China Email: [email protected]

Editorial Board Farbod Alijani Department of Precision and Microsystems Engineering Delft University of Technology The Netherlands Email: [email protected]

Continued on back materials

Journal of Vibration Testing and System Dynamics Volume 2, Issue 2, June 2018

Editors Jan Awrejcewicz C. Steve Suh Xian-Guo Tuo Jiazhong Zhang

L&H Scientific Publishing, LLC, USA

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Journal of Vibration Testing and System Dynamics 2(2) (2018) 91-107

Journal of Vibration Testing and System Dynamics Journal homepage: https://lhscientificpublishing.com/Journals/JVTSD-Default.aspx

A Fuzzy Logic PI Trajectory Following Control in a Chaotically Loaded Real Mechatronic Dynamical System with Stick-Slip Friction Wojciech Kunikowski, Pawel Olejnik†, Jan Awrejcewicz Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Faculty of Mechanical Engineering, 1/15 Stefanowski Street, 90-924 Lodz, Poland Submission Info Communicated by S. C. Suh Received 10 January 2018 Accepted 3 March 2018 Available online 1 July 2018 Keywords Discontinuous dynamical system Stick-slip phenomenon Fuzzy logic control Time-frequency control

Abstract Fuzzy logic control algorithms are regarded to as a relatively new concept in modern control theory. This paper presents a comparative analysis of two qualitatively different approaches used for angular velocity control of a DC motor subject to chaotic disturbances coming from a gear with a transmission belt carrying a vibrating load. The purpose is to achieve accurate control of speed of the DC motor (a plant), especially, when the motor parameters and some external loading conditions are partially unknown. First, the classical approach based on the PID control is considered, and then a fuzzy logic based alternative is proposed. Two different controllers are developed, i.e. the classical PID controller and a Mamdani type fuzzy logic PI controller. Both control algorithms are implemented on an 8-bit AVR ATmega644PA microcontroller. Based on step responses of the plant, an analysis as well as an interesting comparison of the controllers’ performance has been presented. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Regulation of speed of a DC motor is a common technical issue. The DC motor can be treated as single input, single output (SISO) object. Voltage applied to the winding terminals is the input value and the output is the rotational speed of the rotor. The most commonly used technique for speed regulation of DC motors is the use of the PID controller. It allows to maintain a constant engine speed relative to the variable load, but it requires careful selection of the PID coefficients. Classical methods of regulator tuning are based on accurate knowledge of the mathematical model of the object under control as well as loading applied to it. Often, such a model is very difficult to build and identify, which complicates the correct synthesis of the PID controller. Fuzzy logic is a mathematical concept striving to imitate human perception. Instead of numerical values some linguistic descriptions are used to characterize input and output variables. The control strategy is derived from expert knowledge and stored in a base of fuzzy rules. This enables the design engineer to describe the behaviour of the object under control with the use of words (linguistic † Corresponding

author. Email address: [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2018.06.001

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Wojciech Kunikowski et al. / Journal of Vibration Testing and System Dynamics 2(2) (2018) 91–107

variables), rather than with complex mathematical expressions. Another advantage of the intelligent approach in control of uncertain dynamical systems is the ability of fuzzy logic controllers to handle the control by means of incomplete portion of information. Fuzzy logic has been used in a wide variety of applications like process control, electrical engineering, information technology, image recognition, telecommunications, banking (Zadeh L. A., 1989, 2008; Bai Y. and Wang D., 2006; Jager R., 1995). Interesting fuzzy logic controller implemented in an industrial controller was presented in (Arrofiq M. and Saad N. 2008). The plant consists of a DC motor subjected to varying load parameters. The Takagi-Sugeno fuzzy logic PI controller was implemented on a PLC controller. A self-tuning fuzzy algorithm, which calculates a gain coefficient for the main controller, was also implemented. The regarded system’s performance was tested for three types of conditions: varying load parameters, varying setpoint velocity and varying setpoint velocity with changing load parameters. In (Velagic J. and Galijasevic A., 2009) a robust fuzzy controller’s application has been described for the permanent magnet DC motor. The system parameters concerning the load and the DC motor’s constants were unknown. The algorithm was implemented on dSPACE rapid prototyping controller board and connected to Matlab/Simulink programming environment. The fuzzy logic controller (FLC) has two inputs, a voltage error and its derivative. The control action generated by the controller is the actual voltage supplying the DC motor. It was concluded from the experimental data that the elaborated FLC has achieved shorter response times on pulse input signals and smaller oscillations about the setpoint than the adequate PID controller. A hybrid solution of a fuzzy logic and a PI controller was presented in work (Teeter J.T., Chow M. and Brickley J.J., 1996). The object under control consisted of a DC motor with a metal disc mounted on its shaft. The load was generated by applying a magnetic field to the disc. The fuzzy logic component was responsible for calculating a gain coefficient for the PI controller. The fuzzy part has three inputs: the reference speed, measured speed, and the control action of the whole controller from the previous time period. The base of fuzzy rules consists of only one rule, and its aim is to reduce the output of the PI controller in low setpoint speeds. This is due to the nonlinear behaviour caused by frictional effects in the mentioned region. This friction compensation method yields faster response of the system and smaller settling time. An example of a self-tuning FLC was presented in (Jee S. and Korem Y., 2004). A Takagi-Sugeno fuzzy logic PD controller was applied on a 3-axis milling machine for contour milling. Based on the position error, the change of the error, the velocity feedback, and the fuzzy control action from the previous time period, the output is calculated and sent to the amplifier which drives the motor. The performance of the controller affects the adaptive algorithm that expands or contracts the input fuzzy sets or even shifts the position of the output sets in the numeric domain. In the mentioned paper the described FLC was compared to a well-tuned PID controller. The comparison showed that for cutting straight lines the adaptive FLC achieves two times lower root of mean square contour error and three times lower maximum contour error than the PID controller. In (Neto P., Mendes N., Pires J.N., Norberto J. and Moreira A.P., 2010), a PI fuzzy force controller was applied to regulate movement of an industrial robot’s end-effector. Without any obstruction during its operation, the robot moves along its pre-programmed path. However, when contact with the “foreign” object along the way of motion appears, the force control system adjusts the end-effector’s position. The force control ensures that the contact forces and moments converge to a desired value. Comparing to the classical PI algorithm, a smaller overshoot and average constant force were achieved. Kalavathi and Reddy (Kalavathi M.S. and Reddy C.S.R., 2012) show a comparative performance analysis of fuzzy logic and classical PID controllers. Both solutions are used as a brushless DC motor speed regulators. Numerical analysis shows improved performance in terms of disturbance rejection and resistance to load parameter variation in favour of fuzzy logic controllers. An application of FLC in electrical engineering was presented in (Baˇsi? M., Vukadinovi?D. and


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Poli?M., 2013). The controlled object was a self-excited induction generator. The control system was given a task to maintain the steady level of output voltage. Mamdani and Takagi-Sugeno type fuzzy logic PI controllers were developed in the discussed work. Performance of proposed solutions was tested against a classical PI controller. Both controllers were programmed in Matlab/Simulink and used to control a model of the self-excited induction generator. It has been concluded from the simulation and experimental data that the fuzzy logic controllers offer significantly better performance compared to the optimally tuned PI type controller in terms of the response time, settling time, and robustness. The downside in the described application is reflected in an increase of the computational performance requirements, especially for the Mamdani type FLC. A kind of nonlinear system with two degrees of freedom in the form of a plunge displacement and the pitch angle stands for another exemplary system which produces limit cycles oscillation (Li G., Cao J., Alsaedi A. and Ahmad B., 2017). A very interesting approach to an adaptive fractional-order fuzzy controller is presented to suppress flutter in the alaeroelastic system. In work by (Jang J.S.R., 1991) a self-learning Takagi-Sugeno controller was used for identification purposes. Mentioned controller has two input variables with three bell shaped membership functions each. The output of every rule is a linear function of input variables. To implement a gradient-descent learning algorithm, the controller was designed in a form of generalized neural network. The task of the algorithm was to modify the weight coefficients of the neural network to ensure convergence of the output from the fuzzy-neural network to three sets of training data. One of them was obtained from a real object and the other from different mathematical functions. After 200 iterations of the algorithm’s execution for each set of data, an average percentage error obtained for the first set of training data was reduced to 1.57%. For the remaining sets the error was reduced to 0.47% and 0.014%, respectively. Although the number of fuzzy rules and input membership functions are preliminarily specified, the algorithm achieved satisfactory results regarding the error and tuning time. In work by (Lin F., Fung R. and Wai R., 1998) A comparative study of sliding-mode control and fuzzy neural network control was presented. The controlled object was a toggle mechanism, consisting of rigid bodies. The mechanical system was driven by a permanent-magnet synchronous servo motor. Basing on the mechanical system’s equation of motion a sliding-mode controller was developed. Next a fuzzy controller in form of a 4 layer neural network was presented. For tuning the fuzzy neural network an on-line learning algorithm using the gradient descent method was applied. Both controllers were used to recreate a periodic step and sinusoidal trajectories of a mechanical slider (part of the toggle mechanism). The research showed that both algorithms provided a robust control performance. Additionally authors pointed that with a fuzzy neural controller no complex modelling of the mechanical system was needed. Moreover it did not demonstrate the chattering phenomena during operation. Therefore, the fuzzy neural network controller was pointed as more suitable for further use. Recognizing the benefits of fuzzy logic approach in control theory, a fuzzy logic PI controller is proposed as an alternative for the classical PID equivalent. Efficiency of both solutions in controlling a two degrees of freedom (2-DOF) discontinuous dynamical system with friction subjected to an irregular external excitation is investigated. In second part, of the paper the experimental station and its main components are discussed. Derivation of the mathematical model used to estimate the impact of irregular loading has been shown in the third section alongside the selected results of numerical solution. Detailed description and a comparative study of efficiency of both tested numerical algorithms has been presented in parts four and five respectively. Main difficulties of the study were devoted to (i) exact measuremement of the process variable, and then, a sufficiently fast implementation the theoretical formulas on the microcontroller; (ii ) application of the FLC PI regulator states a bigger challenge than application of the classical or even modified PID controller with integer order of the derivative part; (iii ) the amount of the tuned variables (see Tab. 3) and the fuzzy rules in the Rule Base prove to be a great downside of the fuzzy logic approach.

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Fig. 1 A schematic view of the mechatronic system (ω bp – process variable, cg – centre of gravity of a body M).

Fig. 2 Speed variations of the uncontrolled system.

Because the variables of the tuned system have clear physical interpretations, the tuning process have to be rather intuitive. In reality, it is very time consuming and demands also a broad knowledge about the controlled object. This work shows that all parameters of the investigated system should be tuned.

2 Experimental station A model of the dynamical system, being a source of instability of the belt driven by the controlled DC motor with a gear (compare with element 1 in Fig. 4.), has been shown in Fig. 1. The friction-induced vibration of mass m on the moving belt is mainly responsible for the dynamically changing loading of the DC motor’s shaft that drives the transmission belt. The noncontrolled velocity of the belt pulley is very irregular as depicted in Fig. 2. To inspect the behaviour of the conveyer belt shown in Fig. 2, the belt pulley’s angular velocity was measured by an incremental encoder. Fourier transform of the system’s step response depicted in Fig. 3 confirms coexistence of many frequencies of oscillations. In that trial, the system was controlled by a P controller with the proportional gain K p = 2.25. The presented spectrum of amplitudes shown in Fig. 3 ends at the Nyquist frequency fc = 1/2h ≈ 16.66 [Hz] for the assumed sampling time h = 0.03. As can be observed, the system’s dynamics is highly irregular. Many variables need to be taken into consideration while designing any optimal controller. To measure the speed of the transmission belt on which the oscillating mass vibrates, an incremental encoder of 5000 imp/rev was used. The 4x encoding was applied for the angular velocity measurement.


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Fig. 3 Fourier transform of the system’s step response.

Fig. 4 Picture of the experimental stand: 1 – the DC motor PZTK 62-42J with a 15:1 worm gear, 2 – RN12 DC motor driver, 3 – the physical object under control, 4 – IVO GI333 incremental encoder, 5 – Atnel Testing Board 1.03 with ATmega644PA microcontroller.

This means that in, a constant time period, both rising and falling edges of two shifted measurement lines A and B of the encoder are counted. This method allows one to virtually increase the base resolution of the sensor (Petrella R., Tursini M., Peretti L. and Zigliotto M., 2007). Using the 4x encoding and setting the acquisition time to 30 ms yields a measurement error of 0.1 rpm. This means that each subsequent pulse counted with mentioned sampling time increases the output of the velocity recognition algorithm by 0.1 rpm. The encoder is not perfect, thus the length of the high state tends to vary due to the manufacturing tolerances. The used method of encoding could magnify that phenomena, hence the speed measurement error may be bigger than the assumed 0.1 rpm. The contribution of that error to the shape depicted in Fig. 2 is unknown. No filtering technique was applied to the measurement to capture the dynamics of analyzed system as accurate as possible. Figure 4 presents the experimental stand on which the experiments have been carried out. Regulation of the angular velocity of the PZTK 62-42J DC motor (1) is carried out by a control system composed of the Atnel Testing Board (ATB) 1.03 with ATmega644PA microcontroller (5) and the RN12 driver (2). The Physical system under control (3) influences the speed of the transmission belt measured by the incremental encoder (4). The L293D is powered by 15 V power source. The DC motor is powered by the MATRIX 60 V (RN12 power circuit) and 30 V (RN12 logical circuit) – regulated direct current sources.

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For the purpose of noise cancelation in the measuring circuit, a M74HCT14N Schmitt inverter was used as a buffer between the encoder and microprocessor. It damps voltage peaks causing incorrect number of pulses to be counted by the microcontroller. To enable compatibility between the microcontroller and the RN12 motor driver, a L293D integrated circuit was implemented. The RN12 driver accepts a pulse width modulated (PWM) signal with amplitude ranging from 12 to 15 V, but the microprocessor is equipped only with 5 V TTL outputs. L293D served as an amplifier for the PWM signal. The known parameters of the DC motor are as follows: motor electrical constant ke = 0.104 V/rad/s, motor mechanical constant km = 0.39 N·m/A, armature resistance Rw = 1.1 Ω, armature inductance Lw = 0.001 H. 3 The dynamics of external loading and resistance of motion Electromechanical model of the DC motor follows: Lw i˙w = UV − ke ω (t) − iw (t)Rw ,

(1)

JDC ω˙ (t) = km iw (t) − Mload (t) − bω (t) − TC (ω (t)),

(2)

where: dot denotes the first derivative with respect to time, Lw – winding inductance, Rw – winding resistance, ke and km – electromotive force and motor torque constants, respectively, iw – winding current, UV – input voltage, ω – angular velocity of the rotor, JDC – mass moment of inertia of the rotor, b – viscous friction coefficient, Mload – time varying load, TC – Coulomb friction force given by: Tst for ω (t) = 0, (3) TC (ω (t)) = sign(ω (t))Tkn for ω (t) = 0, where: Tst – static friction, Tkn – kinetic friction. The equation of motion of the belt pulley is presented in the form: Jbp ω˙ bp (t) = M2 − TS (t)rbp ,

(4)

where: Jbp – mass moment of inertia of the pulley, ωbp – pulley’s angular velocity, rbp – radius of the pulley, M2 – output torque of the worm gear, TS – chaotically varying friction force caused by unpredictable behaviour of the mass m. Equations (5)-(7) constitute the simplified model of the worm gear: M1 = M2 ηρ ,

(5)

ω (t) = ωbp (t)ρ , ω˙ (t) = ω˙ bp (t)ρ ,

(6) (7)

where: η – efficiency of the angular velocity transmission system, ρ – a transmission ratio, M1 and M2 – input and output torques of worm gear. Substituting Eq. (4) to (5) with regard to (6) yields (8), which describes the total loading torque applied by the transmission system to the DC motor: Mload = M1 =

Jbp 1 (ω˙ (t) + TS (t)rbp ). ηρ ρ

(8)

Combining Eq. (3) and (8) we get Eq. (9), which is an equation of motion of the transmission system reduced to the DC motor’s shaft: (JDC +

Jbp rbp )ω˙ (t) = km iw (t) − TS (t) − bω (t) − TC (ω (t)). ηρ 2 ηρ

(9)


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The controlled object consists of a conveyer belt with a block of mass m oscillating on it along the x direction and a bracket of mass M and the mass moment of inertia J, rotating around point S about ϕ angle. The bracket is attached to the block by means of two linear springs (see Fig. 1). Depending on the linear displacement of the block m and the angular displacement of the bracket J, the friction force in a contact surface between the block and the belt changes rapidly. Moreover, the friction force characteristics switches itself between its kinetic and static form. The experimentally observed irregular changes in frictional force in the block-on-belt model are responsible for the significantly varying load transferred on the DC motor’s shaft driving the mechanical system shown in Fig. 1. Constantly changing amplitude of load affects the speed of the gear’s conveyer belt. A more detailed description of the object of control can be found in works of the authors (Awrejcewicz J. and Olejnik P., 2003, 2005; Kunikowski W., Awrejcewicz J. and Olejnik P., 2013). Two characteristic phases of movement can be distinguished for the examined system. The “stick” part of movement occurs when the block moves with constant velocity equal to Vb – velocity of the conveyer belt. When the static friction reaches its maximum value and does no longer compensate for the resultant force of springs, then the contact between the mass and the moving belt is lost, hence, the “slip” phase of the motion starts. The block of mass m moves then with accelerated motion in the opposite direction to the belt’s motion until the kinetic friction forces compensate the resultant force of springs and inertia of the sliding body. Expressions (10)-(14) describe the behaviour of the oscillating object and the friction intensification mechanism realized by the single pendulum (bracket, see in Olejnik P., 2013). The detailed derivation of aforementioned model can be found in (Pilipchuk V., Olejnik P. and Awrejcewicz J., 2015). Block of mass m is connected to a fixed wall by linear spring k1 and to a bracket body of mass M (the pendulum) by means of springs k2 and k3 . The virtual dashpots c1 and c2 represent some effects of system damping in the bearings and air. The investigated 2-DOF mechanical system is described by the two second-order ordinary differential equations: k3 x1 x2 (y1 + 1 ) = −TS , (10) x¨1 m + z˙1 c1 + (k1 + k2 )x1 + k2 y1 + r 2r x2 Q J (11) x¨1 2 + z˙1 c1 + y˙1 c2 + k2 z1 + k3 (y1 + 1 )Mgλr = − , r 2r r where: g – gravitational constant, Q – resistance torque in the pivot point S. √ y3 y1 y2 2l (1 + − 12 − 13 ), λr = (12) 2r r 2r 6r where: l is the distance between the centre of rotation and the centre of gravity of rotating bracket of mass M. Parameter z1 and its first derivative represent an internal state variables expressed as follows: z1 = x1 + y1 , z˙1 = x˙1 + y˙1 . The friction force in the frictional contact of the block-on-belt model is given by Eq. (13) (Pilipchuk V., Olejnik P. and Awrejcewicz J., 2015). It consists of normal force component and relative velocity dependent friction coefficient. Fluctuations of the friction force obtained from the numerical model are shown in Fig. 5. c2 dy1 M k3 x2 (y1 + 1 ) − ). (13) TS = μ (Vrel )mg(1 + λr − m mg 2r mg dt The experimentally verified friction coefficient dependent on the relative velocity is proposed:

μ (Vrel ) =

μ0 β (1 + ) tanh αVrel , 1 + γ |Vrel | cosh αVrel Vrel = x˙1 +Vb ,

(14) (15)

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Fig. 5 Time history of the friction force.

Fig. 6 Diagram of the closed-loop control system (ωsp – reference angular velocity of the rotor, ωbp – process variable).

where: α , β , and γ are the friction law parameters controlling the shape of curve given by Eq. (14), and Vb = ωbp · rbp is the velocity of the conveyer belt. Many physical phenomena, such as wear of the worm gear, viscous friction in all of the bearings of the mechanical system, stiffness of the transmission belt, radial run-out of belt pulley shafts, and unevenly distributed value of the dry friction coefficient on the surface of the transmission belt, have a direct impact on stabilization of the belt’s linear velocity of movement. Time series presented on Fig. 2 and 5 distinctly show the impact of the oscillating object on the DC motor loading. In order to mitigate the fluctuations of the conveyer belt’s speed, an advanced control algorithms described in Sec. 4 were used.

4 Control algorithms High difficulty of accurate modelling of all physical phenomena impacting the load parameters increases the complexity of the control task. The black-box approach to the control problem was used while any further mathematical modelling of the controlled object was omitted. Tuning of the controller was done manually by observing the real time plot of the encoded angular velocity measurement. The goal here was to achieve the lowest possible oscillations of velocity of the belt pulley after reaching the desired setpoint. Both presented algorithms were implemented in C programming language on the ATmega644PA microcontroller (Kunikowski W., Olejnik P. and Awrejcewicz J., 2015). In Fig. 6, the assumed closed-loop control system has been presented. Basing on the setpoint speed (reference angular velocity ωsp ) and the counted number of pulses w from the encoder, the proposed control algorithm calculates the adequate duty cycle uPW M of the PWM signal. It is then sent to the RN12 driver which generates appropriate voltage uV for the DC motor. A 10-bit PWM signal with


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9.7 kHz frequency was used. Therefore, the variable describing its duty cycle can be changed from 0 to 1023. The control algorithm corrects the PWM duty cycle every 30 ms. The feedback loop time regime is restricted by the accuracy and resolution of the speed measurement. 4.1

The classical PID controller

The classical discrete PID controller’s formula in n-th iteration follows: n

uPW M (n) = KP e (n) + KI ∑ e (k) − KD (w (n) − w (n − 1)) ,

(16)

k=0

where: uPW M – output of the regulator, e – error of regulation, K p – proportional gain, KI – integral gain, KD – derivative gain, w – measured angular velocity. To eliminate rapid responses of the derivative part when the setpoint speed changes (the so called “derivative kick”), the regulator’s variable was changed from the error of regulation to the difference between two successive measurements of speed. The PID controller’s output was limited to the range [0,1023] and treated as the new value of the PWM duty cycle. Tuning of PID parameters was done manually. Observing how changes of all of regarded attributes impact the step response of system, the best set was chosen. 4.2

Fuzzy logic PI controller

The second regulator taken into consideration is the Mamdani type fuzzy logic PI controller. It takes two inputs, i.e. error of regulation e and the difference δ e between the current value of error and the value of error from the last cycle. The output value is ΔuPW M (n) – the increase in the PWM duty cycle. A classical alternative that would match the regulator considered earlier is given below: ˆ (17) uPW M (t) = KI e (t) dt + KPe (t) , where: e – error of regulation, K p – proportional gain, KI – integral gain, uPW M – regulator output. If one differentiates both sides of the previous equation with respect to time, and subsequently, introduces approximated transition to the discretized time, then the equation takes the form: ΔuPW M (n) = KI e (n) + KP (e (n) − e (n − 1)) ,

(18)

where: Δu PW M ≈ du PW M /dt – output of the regulator, e– error of regulation, e(n) − e(n − 1) ≈ de(t)/dt, K p – proportional gain, KI – integral gain. The numerical range of each input variable was divided into five triangular, two piecewise linear fuzzy sets. The output variable was divided into seven one-element sets (so-called “singletons”). Graphical interpretation of this classification is presented in Fig. 7-9. Formulas for calculating the value of membership function of each type (see Fig. 10) of fuzzy set are given in Eq. (19)-(21). In Fig. 7-9 the thresholds of fuzzy sets are shown. Linguistic variables assigned to the sets are as follows: NB – negative big, NM – negative medium, NS – negative small, Z – zero, PS – positive small, PM – positive medium, PB – positive big. ⎧ ⎪ ⎨

1 b−x μL (x, a, b) = ⎪ ⎩ b−a 0 ⎧ ⎪ ⎨ x − a0 μg (x, a, b) = ⎪ ⎩ b−a 1

for x ≤ a, for a < x ≤ b,

(19)

for x > b, for x ≤ a, for a < x ≤ b, for x > b,

(20)

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Fig. 7 Fuzzy sets of the first input variable e – error of regulation in k-th measurement sample.

Fig. 8 Fuzzy sets of the second input variable δ e – increase of the error.

Fig. 9 Fuzzy sets of output variable Δu PW M .

Fig. 10 Membership functions of the assumed fuzzy sets.

⎧ ⎪ 0 ⎪ ⎪ ⎪ x−a ⎪ ⎨ a μT (x, a, b) = bc − −x ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c − b0

for x ≤ a, for a < x ≤ b, for b < x ≤ c,

(21)

for x > c.

Rules Ri bind together both input variables and output variables in cause-effect relations. Table 1 contains all possible control rules. This arrangement is the so-called MacVicar-Whelan RB and is very common for fuzzy logic PI regulators (MacVicar-Whelan P.J., 1977; Cheong F. and Lai R., 2007). It was preliminary assumed in this work that it is optimal. Both MacVicar-Whelan RB and its modification done by the authors were tested and compared.


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Table 1 Fuzzy logic PI controller Rule Base. e/δ e

NB

NM

NS

Z

PS

PM

PB

NB

NB(0)

NB(1)

NB(2)

NB(3)

NM(4)

NS(5)

Z(6)

NM

NB(7)

NM(8)

NM(9)

NM(10)

NS(11)

Z(12)

PS(13)

NS

NB(14)

NM(15)

NS(16)

NS(17)

Z(18)

PS(19)

PM(20)

Z

NB(21)

NM(22)

NS(23)

Z(24)

PS(25)

PM(26)

PB(27)

PS

NM(28)

NS(29)

Z(30)

PS(31)

PS(32)

PM(33)

PB(34)

PM

NS(35)

Z(36)

PS(37)

PM(38)

PM(39)

PM(40)

PB(41)

PB

Z(42)

PS(43)

PM(44)

PB(45)

PB(46)

PB(47)

PB(48)

Fig. 11 The inference scheme.

Figure 11 presents the inference scheme of our controller for two exemplary rules. In the first step, the inputs of the error and increase in the error are fuzzified. The algorithm determines to which fuzzy sets they belong and calculates the value of membership function for each regarded set. Next, the firing levels τ i are calculated for each rule using the t-norm min(.). To obtain in Eq. (22) the crisp output Δu PW M (n) from the regulator, the weighted average defuzzification method has been used: ΔuPW M (n) =

Σi1 τi · Δui , Σi1 τi

(22)

where: τ i – firing level of the fuzzy rule, Δu i – output of the fuzzy rule, i – number of the fuzzy rule. Similarly to the PID controller, the variable containing the PWM duty cycle was limited to the range [0, 1023]. Tuning of the FLC parameters was done manually. The acceptable extremes of the error of regulation and an increase of the error were assumed. Then, the fuzzy sets were formed symmetrically between those two values. The output singletons were chosen with regard to the controller’s feedback loop timing so that the PWM duty cycle would not increase in a rapid manner. Later parameters were corrected by observing the step responses of the system. 4.3

Modification of the rule base

To increase the performance of the fuzzy logic PI algorithm, a modification to the Rule Base has been made (Table 2). The modified RB is now nonsymmetric without “Z” on its main diagonal. This change has allowed for separation of the region of high absolute values of error from the region where oscillations around the setpoint value occur. Such a change enables the usage of higher gain (more aggressive output from the regulator) when the DC motor gains velocity.

102


Table 2 The modified Rule Base. e/δ e

NB

NM

NS

NB

NB(0)

NB(1)

NB(2)

NM

NB(7)

NB(8)

NB(9)

Z

PS

PM

PB

NB(3)

NB(4)

NM(5)

NM(6)

NM(10)

NM(11)

NS(12)

NS(13)

NS

NB(14)

NB(15)

NM(16)

NM(17)

NS(18)

Z(19)

Z(20)

Z

NM(21)

NM(22)

Z(23)

Z(24)

Z(25)

PS(26)

PM(27)

PS

Z(28)

Z(29)

PS(30)

PS(31)

PM(32)

PM(33)

PB(34)

PM

PM(35)

PM(36)

PB(37)

PB(38)

PB(39)

PB(40)

PB(41)

PB

PB(42)

PB(43)

PB(44)

PB(45)

PB(46)

PB(47)

PB(48)

Fig. 12 Time history of the angular velocity controlled by the tuned PID algorithm (KP = 1.2, KI = 2.9, KD = 1.35) and setpoint speed ω sp = 15 rpm.

Due to the absence of symmetry in the RB, the controller responds differently while increasing and decreasing the velocity of the DC motor. Comparing to the MacVicar-Whelan RB, it can be observed that very small gains were programmed in the case when the speed goes down to the setpoint after the overshoot. More decisive output values are used while the velocity drops below the setpoint value. The modifications to the RB were done manually on the basis of observation of the time series of the output speed and the authors experience and knowledge regarding the experimental stand.

5 Test results The tests were carried out for the setpoint speed of the belt pulley at the values of 5, 10 and 15 rpm. Figures 12-14 show the obtained step responses for both types of regulators and the version of the fuzzy logic PI regulator with modified RB. Figures 15 and 16 present the time series of trajectory tracking trial. In Table 3, a short comparison of their performance regarding the settling time and a sum of absolute values of error over time is provided. Some features like the execution time of the algorithm and the number of tuned variables were compared. As it can be observed, only marginal gain in control quality of the regulated system was achieved. Both regulators fail to respond properly to chaotic changes in load parameters. Modification of the Rule Base had a great impact on the settling time of the system. The FLC with tuned RB almost instantly generates adequate output to reach the setpoint speed while in the case of the PID regulator,


103

Fig. 13 Time history of the angular velocity controlled by the tuned fuzzy logic PI algorithm and setpoint speed ω sp =15 rpm.

Fig. 14 Time history of the angular velocity controlled by the tuned fuzzy logic PI algorithm with modified Rule Base and setpoint speed ω sp = 15 rpm.

a decreasing build-up closer to the setpoint can be observed. On the other hand, the fuzzy logic controllers present lower stability of angular velocity after the transition period. The sum of absolute values of the control error over time was used as a measurement on stability. It was calculated for the first 5 seconds of motion. For the FLC with tuned RB the results are similar to classical PID controller. After taking into consideration the fast build-up of angular velocity for the FLC with modified RB, it can be concluded that it provides lower stability of speed after transition period. In Fig. 14 and 15 a trajectory tracking trial was presented. An interesting thing to notice is the observable phase displacement in the output of the PID algorithm. The fuzzy logic controller does not demonstrate such behaviour and achieves the desired trajectory in much shorter time. From engineering point of view, the FLC PI regulator proposes a bigger challenge than any classical PID controller. While nowadays, the computation time and the volume of code is in most of cases insignificant, the amount of the tuned variables proves to be a great downside of the fuzzy logic

104


Fig. 15 Trajectory following by the classical PID algorithm.

Fig. 16 Trajectory following by the fuzzy logic PI algorithm with modified Rule Base.

approach. In Table 3, the amount of tuned variables is given by a sum of two values. Number 45 reflects the amount of parameters describing input and output fuzzy sets, and the 49 is the amount of fuzzy rules in the Rule Base. Initially, the first set of variables was tuned, and the RB was assumed to be optimal. In the second approach, both the parameters regarding fuzzy sets and the RB were tuned. Because the variables of the tuned system have clear physical interpretations, the tuning process seems to be very intuitive. Unfortunately, in reality, it is very time consuming and demands also a broad knowledge about the controlled object. This work shows also that all parameters of the investigated system should be tuned. In this case, the widely used MacVicar-Whelan RB did not provide any satisfactory results. Analysis of sensitivity of the fixed-point control algorithm of the angular velocity ω bp on the change in the parameters of fuzzy sets of the output variable Δu PW M can be performed on the basis of 17 test measurements carried out on the laboratory station. As a conclusion, bigger extremes of fuzzy sets generate larger deviations of the fixed-point control object response.


105

Table 3 Performance comparison of considered algorithms. Property

Lines of code required

PID 0.93 0.96 0.99 101.9 153.3 197.2 105

Fuzzy logic PI 0.84 1.12 1.65 143.0 277.1 516.2 280

Fuzzy logic PI mod. RB 0.21 0.24 0.33 100.8 157.6 194.1 280

Number of tuned variables

33

45

45+49=94

Execution time of algorithms [μ s]

73

112

112

Settling time [s] Sum of absolute values of error over time

rpm 5 10 15 5 10 15

Fig. 17 Sensitivity of the FL PI control algorithm with modified RB on the change in the parameters of τ i .

6 Conclusions The fuzzy logic controller described in this paper provides similar quality of control comparing to the classical solution. Mostly, the settling time has benefited from the usage of the fuzzy approach. Despite the interesting concept standing behind it, no substantial improvement in smoothing DC motor’s angular velocity in the setpoint region was achieved. Additionally, the very complex tuning process extends the time in which promising results were obtained. To reduce the difficulty of tuning, the fuzzy logic approach would require its own learning algorithm which would pick the right parameters and adopt them actively to the experiment. Applying a fuzzy PID regulator could also help with the system stability. The derivative action of the controller should enhance the control quality near the setpoint, but the exponentially growing number of variables to tune could be problematic. Another idea would be to develop a hybrid controller. For large values of error of regulation, it could act as a FLC, and for lower values near the setpoint, regular PID controller would take the control on the system.

106


Table 4 Standard deviation of the fixed-point response of the angular velocity of the belt pulley subject to chaotic loading about ω bp = 15 rpm for the parameters NB . . . PB of the fuzzy sets τ i of the output variable Δu PW M . σ

NB

NM

NS

Z

PS

PM

PB

test number

0,576535

-6

-4

-2

0

2

4

6

1

0,584219

-10

-3

-1

0

1

3

10

2

0,598901

-3

-2

-1

0

1

2

3

3

0,603286

-8

-4

-2

0

3

5

10

4

0,608808

-16

-8

-5

0

4

8

22

5

0,61113

-15

-4

-2

0

3

6

12

6

0,613337

-9

-6

-3

0

3

6

9

7

0,622198

-16

-8

-4

0

4

8

16

8

0,630073

-15

-4

-2

0

2

4

15

9

0,658156

-15

-10

-5

0

5

10

15

10

0,68503

-8

-4

-2

0

4

6

10

11

0,689061

-8

-4

-2

0

3

6

12

12

0,700815

-8

-4

-2

0

2

4

8

13

0,705592

-8

-4

-2

0

3

8

15

14

0,724522

-8

-4

-2

0

3

5

10

15

0,851666

-30

-20

-10

0

10

20

30

16

1,442073

-60

-30

-15

0

15

30

60

17

column graph


107

References [1] Arrofiq, M. and Saad, N. (2008), A PLC-based self-tuning PI-fuzzy controller for linear and non-linear drives control, IEEE International Conference on Power and Energy, Johor Bahru, 701-706. [2] Awrejcewicz, J. and Olejnik, P. (2005), Friction pair modelling by 2-DOF system: numerical and experimental investigations, International Journal of Bifurcation and Chaos, World Scientific, 15(6), 1931-1944. [3] Awrejcewicz, J. and Olejnik, P. (2007), Occurrence of stick-slip phenomenon, Journal of Theoretical and Applied Mechanics, 45(1), 33-40. [4] Bai, Y. and Wang, D. (2006), Fundamentals of fuzzy logic control - fuzzy sets, fuzzy rules and defuzzifications, In Y. Bai, H. Zhuang, D. Wang (ed.), Advanced Fuzzy Logic Technologies in Industrial Applications, 17-36, Springer London, London. [5] Baˇsić, M., Vukadinović, D., and Polić, M. (2013), Fuzzy logic vs. classical PI voltage controller for a selfexcited induction generator, In Natasa Trisovic, Deolinda Rasteiro (ed.), Math. Applications in Science and Mechanics, 189-194, WSEAS Press, Dubrovnik. [6] Cheong, F. and Lai, R. (2007), Simplifying the automatic design of a fuzzy logic controller using evolutionary programming, Soft Computing, 11(9), 839-846. [7] Jager, R. (1995), Fuzzy logic in control, Ph.D. thesis, T.U. Delft. [8] Jang, J.S.R. (1991), Fuzzy modelling using generalized neural networks and Kalman filter algorithm, AAAI’91: Proceedings of the Ninth National Conference on Artificial Intelligence, 2, 762-767, Anaheim. [9] Jee, S. and Korem, Y. (2004), Adaptive fuzzy logic controller for feed drives of a CNC machine tool, Mechatronics, 14, 299-326. [10] Kunikowski, W., Awrejcewicz, J., and Olejnik, P. (2013), Efficiency of a PLC-based PI controller in stabilization of a rotational motion affected by the chaotic disturbances, In J. Awrejcewicz, M. Ka´zierczak, P. Olejnik, J. Mrozowski (ed.), Dynamical Systems - Applications, 173-184, Publishing House of Lodz University of Technology, Lodz. [11] Kunikowski, W., Olejnik, P., and Awrejcewicz, J. (2015), An Overview of ATmega AVR Microcontrollers Used in Scientific Research and Industrial Applications, Pomiary Automatyka Robotyka, 19(1), 15-20. [12] Li, G., Cao, J., Alsaedi, A., and Ahmad, B. (2017), Limit cycle oscillation in aeroelastic systems and its adaptive fractional-order fuzzy control, International Journal of Machine Learning and Cybernetics, March, doi: 10.1007/s13042-017-0644-1. [13] Lin, F., Fung, R., and Wai, R. (1998), Comparison of sliding-mode and fuzzy neural network control for motor-toggle servomechanism, IEEE/ASME TRANSACTIONS ON MECHATRONICS, 3(4), 302-318. [14] MacVicar-Whelan, P.J. (1977), Fuzzy sets for man-machine interactions, Int. Journal of Man-Machine Studies, 8(6), 687-697. [15] Neto, P., Mendes, N., Pires, J.N., Norberto, J., and Moreira, A.P. (2010), CAD-based robot programming: the role of fuzzy-PI force control in unstructured environments, IEEE Conference on Automation Science and Engineering, 362-367, Toronto. [16] Olejnik, P. (2013), Numerical Methods of Solution, Analysis and Control of Discontinuous Dynamical Systems, Scientific Books of Lodz University of Technology, (1151), Lodz 2013, 117. [17] Petrella, R., Tursini, M., Peretti, L., and Zigliotto, M. (2007), Speed measurement algorithms for lowresolution incremental encoder equipped drives: a comparative analysis, ACEMP ’07. International Aegean Conference on Electrical Machines and Power Electronics, 780-787, Bodrum. [18] Pilipchuk, V., Olejnik, P., and Awrejcewicz, J. (2015), Transient friction-induced vibrations in a 2-DOF model of brakes, Journal of Sound and Vibration, Elsevier, (344), 297-312. [19] Teeter, J.T., Chow, M., and Brickley, J.J. (1996), A novel fuzzy friction compensation approach to improve the performance of a DC motor control system, IEEE Transactions on Industrial Electronics, 43(1), 113-120. [20] Velagic, J. and Galijasevic, A. (2009), Design of fuzzy logic control of permanent magnet DC motor under real constraints and disturbances, IEEE Multi-Conference on Systems and Control, 461- 466, Saint Petersburg. [21] Zadeh, L.A. (2008), Is there a need for fuzzy logic?, Information Sciences, 178, 2751-2779. [22] Zadeh, L.A. (1989), Knowledge representation in fuzzy logic, IEEE Transactions on Knowledge and Data Engineering, 1(1), 89-101.



Extreming Curves and the Parameter Space of a Generalized Logistic Mapping Diogo Ricardo da Costa1†, Matheus Hansen2 , Edson D. Leonel1 , Rene O. Medrano-T3 1

Departamento de F´ısica, UNESP - Univ Estadual Paulista, Av. 24A, 1515, Bela Vista, 13506-900, Rio Claro, SP - Brazil, 2 Instituto de F´ ısica da Universidade de São Paulo, Rua do Matão, Travessa R 187, Cidade Universit´ aria, 05314-970 São Paulo, SP - Brazil, 3 Departamento de F´ ısica, UNIFESP - Universidade Federal de São Paulo, Rua São Nicolau, 210, Centro, 09913-030, Diadema, SP, Brazil Submission Info Communicated by J.Z. Zhang Received 8 January 2018 Accepted 3 March 2018 Available online 1 July 2018 Keywords Parametric perturbation Extreming curves Parameter space Generalized logistic mapping

Abstract A logistic map with parametric perturbation is studied. We confirm the model exhibits self-similar structures in the parameter space known as shrimps. The organization of such structures may be describe through extreme curves, giving exactly the position of each one of them in a complicated behavior in the parameter space. Boundary crisis are also discussed. They lead to an abrupt destruction of the chaotic attractor and how such destruction affects the dynamics of the system is discussed, particularly affecting the time before the orbit leaves to infinite. By a specific choice of the parameters we show the existence of strange non-chaotic attractors in the parameter space. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Dissipative dynamical systems have been study for many years and they are subject of study in several areas, for example, in physics, biology, mathematics, and many others [1–13]. These systems may be modeled by nonlinear differential equations and discrete nonlinear mappings [14–18]. In particular, the one-dimensional logistic map is among the most investigated dynamical systems. Motivated as a model of population growth [19–21], the second polinomial degree logistic mapping was intensively studied presenting complex and very fundamental features in nonlinear dynamics despite its simplicity. It presents cascades of period-doubling bifurcations leading to chaos, the Feigenbaum scenario [22], tangent bifurcations giving rise to periodic windows, and intermittent behavior and crisis events (boundary, interior and merging chaotic bands crisis) [23, 24] between others. In dissipative systems, the knowledge of attractors and their properties is crucial to build applications. In this sense, it has been considered different versions based on the logistic map where behaviors not presented in the original ones develop. For instance, the so-called B-exponential map can be applied † Corresponding


110

Diogo Ricardo da Costa, et al. / Journal of Vibration Testing and System Dynamics 2(2) (2018) 109–118

Fig. 1 (a) Bifurcation diagram for the second degree logistic map (ε = 0 and α = 1). Item (b) shows an enlargement near the boundary crisis at R = 4, highlighted as the arrow. The initial condition used was X0 = X ∗ .

to generate pseudo-random numbers due to the presence of robust chaos in its parameter space [25], while forced logistic map presents configurations of periodicity domains according to its winding number [26] and Lyapunov exponents [27] similarly observed in forced flows [28]. Therefore, how the attractors change when a control parameter is varied is a question that has been awakening great interested in the scientific community for long. In our work we discuss the properties of the generalized logistic map in the one and two dimensional parameter spaces. Firstly, we focus in the extinction of attractors under a boundary crises. This phenomenon is known to occur when an unstable periodic orbit touches the chaotic attractor which is suddenly destroyed [24]. Here, we explain this phenomenon from the homoclinic bifurcation point of view. It is shown that our explanation covers also the breaking of chaotic transient. In the sequence, planes of parameters are investigated calculating the Lyapunov exponent [29]. Different complex periodic windows are found embedded in chaotic regions. We bring to light how they are organized by applying the concept of extreme curves: periodic windows are along this curves [14]. This paper is organized as follow. In Sec. 2 we describe the equations of the generalized logistic map and some properties of this mapping are shown. In Sec. 3 the boundary crises is discussed and its properties analyzed. Sec. 4 shows planes in the parameter space of the generalized logistic map. Our final remarks are drawn in Sec. 6.


___

R=4.00

R=4.16

Xn+1

1

111

0 0

(a)

1

0.5

0

0.4

(b)

Xn

0.6

1

Xn

Fig. 2 (a) Homoclinic bifurcation. (b) Homoclinic orbit that passes through x = 0.4.

2 The generalized logistic map A generalization of the second degree Logistic mapping we investigate in this paper is written as Xn+1 = Rn Xnα (1 − Xnβ ).

(1)

Here X is the dynamical variable of the system, and α and β are two control parameters that can be chosen to reproduce not just the traditional second degree logistic map [1, 30, 31], when α = β = 1 and Rn = R (constant value), but the cubic [1, 32], and quartic maps as well. In our study we explore the case α = β . Following Refs. [15, 16], we consider that Rn has a parametric perturbation given by Rn (ε , ω ) = R[1 + ε cos(ω nπ )].

(2)

Here R is a control parameter and ε gives the amplitude of the parametric perturbation. Typically, as the control parameter Rn varies, attractors emerge [15] or change stability. The parameter ω affects the periodicity of the perturbation if it is a rational number. When it is irrational, strange non-chaotic attractors arises as discussed in Ref. [33]. In order to consider periodic perturbation it is convenient to express ω = 2/q, with q = 0. The argument of Eq. (2) becomes n2π /q where q gives the period of Rn . For example, ω = 2 (q = 1) leads to R0 = R1 = R2 = · · · = RM = R(1 + ε ),

(3)

for any M. For ω = 1 (q = 2), Rn is a period 2 function since R2 = R0 , where R0 = R(1 + ε ), R1 = R(1 − ε ), R2 = R(1 + ε ).

(4)

112


Fig. 3 (a) Bifurcation diagram for the second degree logistic map. Item (b) shows an enlargement of (a). The initial condition used was X0 = 0.4.

A period 3, R3 = R0 , is obtained considering ω = 2/3 (q = 3) with R0 = R (1 + ε ), R1 = R (1 − ε /2) , R2 = R (1 − ε /2) , R3 = R (1 + ε ).

(5)

The procedure can be followed for any period q. We notice that R0 assumes the expression R(1 + ε ) for any ω . 3 Boundary crises To understand the boundary crises we notice that, in general, trajectories are mainly concentrated around the local maximum of the maps. Observe that the generalized logistic map is unimodal in the range 0 < X < 1, i.e., there is just a maximum in this region at (X ∗ , Xmax ), with X ∗ = ( α α+β )1/β is


12

α=0.5 α=1 α=1.5 α=2

10

R'

113

8 6 4 0.2

0.4

X0

0.6

0.8

Fig. 4 R as function of X0 for different values of α and considering ε = 0. The cyan curve is the lowest possible value of R .

obtained equaling the derivative of the right side of Eq. (1) to zero and Xmax = Rn (X ∗ )α [1 − (X ∗ )β ]. For simplicity, let us consider the second degree logistic map (α = β = 1 and ε = 0), where X ∗ = 1/2 and Xmax = R/4. Taking this in account, we call f (X ) = RX (1 − X ) and study f (i) (X ∗ ), the ith iteration from X0 = X ∗ with respect to the parameter R. In Fig. 1, f (i) is presented colored for i = 1, 2, · · · , 7 and in black for i > 7. It is clear that the crossings between these curves show unstable periodic orbits. Figure 1(b) shows an enlargement of Fig. 1(a). Note that all colored curves, except f (1) , are intercepting each other at the same point (X , R) = (0, 4) where, as pointed by an arrow, the boundary crisis happens. This indicates that the fixed point, at X = 0, touched the attractor annihilating it. After that the curves have a predictable behavior and all orbits tend towards −∞ very quickly. Another interpretation for the destruction of the attractive region is the appearance of a bi-asymptotic trajectory to the fixed point x = 0, for R = 4, called homoclinic orbit [see Fig. 2(a)]. This orbit achieves the fixed point at the origin at n → ±∞. Now, since this system is not invertible, there are several trajectories embedded in the chaotic region that can achieve and be leaded by the homoclinic orbit to the fixed point at the origin [34]. Therefore, the homoclinic bifurcation is the boundary of the attractive behavior: for R < 4 trajectories are always between 0 and 1 because Xmax < 1. In the other hand, since for R > 4 Xmax is greater then 1, an orbit close to Xmax is mapped to a point X < 0, evolving quickly to −∞. This is what happens when a trajectory achieves the region X ∈ (0.4, 0.6) for R = 25/6 ≡ 4.16¯ [see Fig. 2(b)]. When considering other values of X0 , which are different from X ∗ , some interesting phenomena occur. For example, in Fig. 3 it is shown a bifurcation diagram obtained just considering X0 = 0.4. Even for R > 4, where the region X ∈ [0, 1] no longer confines the dynamics, the orbits behave chaotically for a while before diverging for some special values of R. The curves f (i) are now according the new initial condition. Thus, the crossing point of curves f (5) , f (6) , and f (7) at (X = 0, R 4) indicates that a homoclinic orbit was achieved from X0 = 0.4. For R slightly larger, all trajectories from the 5th iteration are in the diverging region (X < 0) and a window poorly populated is formed in between 4 < R < 4.025. Similar mechanism justify the others windows when R is increasing.


114

λ=-2.5

-2

-1.5

-1

-0.5

0

0.5

Period = 4

6

8

10

12 14

1.2

α

1.1

1

0.9

0.8 3.35 (a)

3.4

3.45 R

3.5

3.35 (b)

3.4

3.45

3.5

3.55

R

Fig. 5 For ω = 1 (q = 2) we have the parameter space α vs R for ε = 0.0991414. In (a) the color represents the Lyapunov exponents, while in (b) the color represents the period, where the extreming orbits are highlighted.

Figure 3(b) shows an enlargement of Fig. 3(a). It is observed several windows due the presence of homoclinic orbits and a phenomenon similar to the boundary crisis at R = 4. After the curves f (i) , i ≥ 2, touching each other, at R > 4.16¯ the orbits tend to −∞ very quick and no longer chaotic transient is observed. Although the phenomenon is not exactly a boundary crisis, the mechanism behind it is the same: a homoclinic orbit was formed as showed in Fig. 3(b). The position in which the colored curves touch each other can be found solving the following expression f (2) = f (3) = f (4) = · · · = 0,

(6)

f (1) = 1.

(7)

or, in a simpler way, the solution for Solving this in Eq. (1), one can find that R =

α

(X0 )

1 . 1 − (X0 )α

(8)

For X0 = X ∗ , one can show that the real boundary crisis occurs at R = 4. For different values of X = X ∗ , the colored lines will touch each other following Eq. (8). For X0 → 0 or X0 → 1 the value of R in which we have the destruction of the intermittent behavior tends to infinite, as shown in Fig. 4, which presents R as function of the initial condition X0 , for different values of α . For the traditional second degree logistic mapping (α = 1) the position of the first boundary crisis occurs at R = 4, and in Fig. 4 it occurs in the lowest possible value of R . 4 Parameter space On the previous section we observed how the dynamics of the system is changed when a control parameter is varied. In this situation, it is possible to observe that an initial condition may be chaotic


115

Fig. 6 Plot showing the ten first extreming orbits and superstable orbits for different values of α . As one can see there exist a complex behaviour of these curves. The item (b) is an magnification of the red rectangle shown in (a).

for a certain value of control parameter, but when considering a small perturbation in the control parameter, this behavior changes. This information yields another question: what happens with the system if two control parameters are changed simultaneously? One way to answer this question is to consider a parameter space. To help us to describe regular and chaotic regions from the parameter space, we use Lyapunov exponent [19, 35]. For one-dimensional systems, the Lyapunov exponents are given as 1 ∞ λ = lim ∑ |F (Xi )|, (9) n→∞ n i=1 where F (xi ) is the first derivate of the function F evaluated in Xi . The Lyapunov exponent may be classified as follows: for λ < 0 the orbit is either periodic or quasi-periodic, however, for λ > 0 the orbit is chaotic. Let us check now the parameter space when two control parameters are varied, say R vs α for Fig. 5(a). We computed the Lyapunov exponent after a long transient time and the color is linked to the value of λ found. It is possible to see in Fig. 5(a) there are many periodic, seemingly self-similar structures embedded in chaotic regions, called shrimps.

116


0.5

0.4

1.176

0.1032

0.2

-0.5

0

1.092

0.0946

-0.2 α

-1 -1.5

ε

-0.4

1.008

0

0.086

-2

-0.6 0.924

-2.5

-0.8

0.0774

-3

-1 0.84 (a)

3.492

3.51

0.25

3.528 3.546 3.564 R

-3.5

-1.2

0.0688

-1.4

(b)

3.4272 3.4578 3.4884 3.519 3.5496 R 0.3999

0.05 0

0.23

-0.05 0.23994

ε

ε

-1

-0.1

0.21

-0.15 0.07998

3.04

3.06 R

3.08

3.1

-0.2

2

0

0.15996

0.22

3.02

-4.5

1

0.31992

0.24

0.2 (c) 3

-4

0 (d)

-2 -3

3.08

3.19

3.3 R

3.41

3.52

-4

Fig. 7 Considering different values of ω some parameter spaces are shown for: (a) α vs R with ω = 2/4 and √ ε = 0.105; (b) ε vs R with ω = 2/3 and α = 1; (c) ε vs R with ω = π and α = 1; (d) ε vs R with ω = (1 + 5)/2 and α = 1.

Another way to characterize the parameter space is to replace the Lyapunov exponent for periods, which is shown in Fig. 5(b), where each color represents the period of the periodic structures for each parameter combination. The lines shown in Figs. 5(b) are called as extreming curves, which are shown and detailed in Ref. [14]. We have used the following rule to name the extreming curves: for example, the extreming curve with period three was named as (3)R0 . The intercept between two extreming curves gives us the position of a shrimp. So, the extreming curves play an important role, given us the position of the shrimps and also showing how is the organization of periodic structures in the parameter space. As an example, we show in Fig. 6(a) the first ten extreming curves and superstable orbits for α = 1 and ω = 1 when varying ε vs R. As one can see, the curves have a complicated behavior, and it is not so easy to describe its behavior. Fig. 6(b) shows a enlargement in the red rectangle shown in fig. 6(a). There exist an accumulation of lines, which occurs at R = 4 (boundary crisis). Fig. 6(c) shows the first ten extreming and superstable curves for α = 0.5. Finally, Fig. 6(d) shows the curves for α = 2.


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5 Parameter space for different values of ω Until now, we have found the results for q = 2, in other words, for ω = 1. The dynamics of the system is strongly changed when considering different values of q. For example, in Fig. 7(a) we show the parameter space α vs R considering ω = 2/4 and ε = 0.105, where we highlight a chain of shrimps. For ω = 2/3 we also have similar structures in the parameter space, but when considering irrational values of ω , for example, when considering ω = √ π [see Fig. 7(c)], the shrimps apparently disappear. The same also occurs when considering ω = (1 + 5)/2, as shown in Fig. 7(d). Probably it occurs because some strange non-chaotic attractors are created, as shown in Ref. [33].

6 Conclusions We studied a generalization of the Logistic mapping, where a parametric perturbation was introduced in the system. Some properties of the bifurcation diagram were studied, where we highlighted the existence of an apparent second bifurcation diagram in this system, and presenting a intermittent behavior before the orbit tends to infinite. We also show the parameter space and highlighted the extreming orbits. We observed a complicated behavior of the extreming curves and superstable curves, leading to a accumulation near the boundary crisis. In the last section we studied how the parameter space changes when considering irrational values of ω . Acknowledgements DRC acknowledges to PNPD/CAPES. MH thanks to CAPES for the financial support. EDL thanks to CNPq (303707/2015-1), FUNDUNESP and FAPESP (2017/14414-2), Brazilian agencies. RMT thanks to FAPESP (2015/50122-0). This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).

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A Series of Symmetric Period-1 Motions to Chaos in a Two-degree-of-freedom van der Pol-Duffing Oscillator Yeyin Xu, Albert C.J. Luo† Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA Submission Info Communicated by S. C. Suh Received 22 February 2018 Accepted 23 March 2018 Available online 1 July 2018 Keywords 2-DOF van der Pol-Duffing oscillator Period-1 motions to chaos Periodic motion series Implicit mapping Mapping structures Nonlinear frequency-amplitudes

Abstract In this paper, independent periodic motions in the two-degree-offreedom (2-DOF) van der Pol-Duffing oscillator are investigated. From the semi-analytical method, the 2-DOF van der Pol-Duffing oscillator is discretized to obtain implicit discrete mappings. From the implicit mapping structures, periodic motions varying with excitation frequency are obtained semi-analytically, and the corresponding stability and bifurcation are obtained by eigenvalue analysis. The frequency-amplitude characteristics of periodic motions are also presented. Thus, from the analytical prediction, numerical simulations of periodic motions are performed for comparison of numerical and analytical results. The harmonic amplitude spectrums of periodic motions are also presented for harmonic effects on the periodic motions. Through this study, the order of symmetric period-1 to chaotic motions (i.e., 1(S)⊳1(A)⊳3(S)⊳2(A)⊳· · ·⊳m(A)⊳(2m+1)(S)⊳· · · ) (m → ∞) is discovered. Chaotic motions or catastrophe jumping phenomena between the two independent periodic motions exist. The independent periodic motions can be used for specific applications in phase locking, and such results can be useful to develop series of the van der Pol-Duffing circuits for applications. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Periodic motions in nonlinear dynamic systems have been of great interest for over 300 years. Since the 18th century, the perturbation method was developed for approximate solutions of periodic motions in nonlinear dynamical systems. In 1788, Langrage [1] developed the method of averaging for periodic motions of a three body problem from a perturbation of two body problem. The method of averaging was based on the solutions of the corresponding linear systems and with coefficient variation method. In the end of the 19th century, Poincare [2] extended the idea of averaging and developed the perturbation method for the approximate analytical solutions of periodic motions of the celestial bodies. In 1920, van der Pol [3] used the method of averaging to determine periodic oscillation in the † Corresponding

author. Email address: [email protected]

ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2018.06.003

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Yeyin Xu, Albert C.J. Luo / Journal of Vibration Testing and System Dynamics 2(2) (2018) 119–153

circuits. In 1928, Fatou [4] gave, for the first time, the proof of the asymptotic validity of the method of averaging through the solution existence theorem of differential equations. In 1935, Krylov and Bogoliubov [5] further developed the method of averaging for nonlinear oscillations in nonlinear vibration systems. Since then, the perturbation method became a unique way for periodic solutions in nonlinear dynamical systems. In 1964, Hayashi [6] used the perturbation and simple harmonic balance method for periodic motions in nonlinear systems. The multiscale perturbation method was extensively used for nonlinear vibration (e.g., Nayfeh [7]; Nayfeh and Mook [8]). Recently, one used the perturbation method for periodic motions and chaos in nonlinear systems. In 2016, Maaita [9] investigated the bifurcation of the slow invariant manifold of completed oscillators. In 2017, Yamgoue et al [10] studied the approximate analytical solutions of a constrained nonlinear mechanical system. Shayak and Vyas [11] used the Kyylov-Bogoliubov method to the Mathieu equation. Rajamani and Rajasekar [12] discussed the response amplitude of the parametric Duffing oscillator. Although chaotic motions in nonlinear systems were investigated through the perturbation analysis, such a way for chaotic motions in nonlinear dynamical systems is inadequate. The perturbation methods required the corresponding linear solutions to determine the approximate periodic solutions of the original nonlinear systems, and the perturbation expansion with small parameters was adopted. In 2012, Luo [13] developed the generalized harmonic balance method for analytically determining periodic motions in nonlinear dynamical systems. Luo and Huang [14] employed such a method to determine approximate analytical solutions of periodic motions in the Duffing oscillator, and Luo and Huang [15] gave the analytical bifurcation trees of period-m motions to chaos in the Duffing oscillator. For the twin-well Duffing oscillator, bifurcation trees of periodic motions to chaos are presented in Luo and Huang [16,17]). In 2013, Luo and Lakeh [18] obtained the analytical solutions of period-m motions of the van der Pol-Duffing oscillator. From the generalized harmonic balance method, Wang and Liu [19] developed a numerical scheme to compute coefficients in the finite Fourier series expression of periodic motions in nonlinear dynamical systems. Luo and Wang [20] used the generalized harmonic balance method for approximate analytical solutions of periodic motions in the rotor dynamical systems. The generalized harmonic balance methods and applications can be found in Luo [21, 22]. In the earlier stage, one used the perturbation method to determine the approximate solutions of periodic motions in weakly nonlinear systems. However, the approximate solutions cannot describe the periodic motions in the van der Pol oscillator. In 1972, Greenspan [23] used two numerical methods to investigate the periodic solutions of the van der Pol equation. In 1996, Sanjuan [24] investigated a van der Pol -Duffing oscillator based on the 4th order Runge-Kutta method. For the small parameters, different chaos was found, and for the large parameter of the cubic term, a period-adding sequence of saddle-node bifurcations was observed. One used the existing numerical methods to do numerical simulations. The numerical errors cause the numerical results not to be accurate, especially for chaotic motions. From the aforementioned reason, in 2015, Luo [25] developed a semi-analytical method for the prediction of periodic motions in nonlinear dynamic systems. This method can control computational errors of periodic motions very well, and stable and unstable periodic motions can be determined, and bifurcation trees of periodic motions to chaos can be obtained. Luo and Guo [26] used such a semianalytical method for the bifurcation trees of periodic motions in the hardening Duffing oscillator. The rich dynamic characteristics of periodic motions to chaos in a twin-well Duffing oscillator were also observed in Guo and Luo [27]. In addition, Luo and Guo [28] used such a semi-analytical method to investigate bifurcation trees of periodic motions to chaos in the periodically excited pendulum (also see Guo and Luo [29]). Guo and Luo [30] presented the bifurcation trees of periodic motions in the parametrically driven pendulum. The discrete mapping method can provide the analytical predictions of periodic motions with high accuracy, and such a method can predict the stability and bifurcation of periodic motions accurately. Compared to the generalized harmonic balance method, this semi-analytical method can be used to any nonlinear dynamical systems with certain computational


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accuracy. The generalized harmonic balance method can just be used to the polynomial nonlinear dynamical systems, and the mathematical derivations for periodic motions are very complicated. Such a method was systematically presented in Luo [31]. In 2015, Luo and Xing [32,33] applied such a semianalytical method to investigate the time-delayed Duffing oscillator. Luo and Xing [34] also discussed the time-delay effects on periodic motions in the time-delay Duffing oscillator. Xing and Luo [35] found the time-delayed, twin-well Duffing oscillator possessed infinite bifurcation trees of period-1 motions to chaos. In 2016, Luo [36] systematically discussed memorized dynamical systems, and time-delay, and the semi-analytical methods for time-delayed nonlinear systems were presented. In this paper, periodic motions of the periodically forced, two-degree-of-freedom van der Pol-Duffing oscillator will be investigated through the aforementioned semi-analytical method. The corresponding differential equations will be discretized first to obtain the implicit mappings. Using such implicit mappings, series of periodic motions in the 2-DOF von der Pol-Duffing oscillator will be predicted, and the corresponding stability and bifurcation will be determined via the eigenvalue analysis. Based on the discrete Fourier Series, the frequency-amplitude characteristics of periodic motions will be discussed. Numerical simulations of sampled periodic motions will be provided. 2 Discretization and mapping structures Consider a periodically forced, 2-DOF van der Pol-Duffing oscillator as x¨1 + (−α11 + α12 x21 )x˙1 + β11 x1 + β12 x31 + γ11 (x1 − x2 ) = Q0 cos Ωt, x¨2 + (−α21 + α22 x22 )x˙2 + β21 x2 + β22 x32 + γ21 (x2 − x1 ) = 0,

(1)

where {α11 , α21 } and {α12 , α22 } are linear and nonlinear damping coefficients; {β11 , β21 } and{β12 , β22 } are the linear and nonlinear stiffness coefficients; γ11 and γ21 are coupling coefficients; Ω and Q0 are excitation frequency and amplitude, respectively. Equation (1) becomes in form of x˙1 = y1 , y˙1 = −(−α11 + α12 x21 )y1 − β11 x1 − β12 x31 − γ11 (x1 − x2 ) + Q0 cos Ωt, x˙2 = y2 ,

(2)

y˙2 = −(−α21 + α22 x22 )y2 − β21 x2 − β22 x32 − γ21 (x2 − x1 ). From Luo [25, 31], for a small time interval of t ∈ [tk−1 ,tk ], equation (2) is discretized by a midpoint scheme to form a mapping Pk (k = 1, 2, 3, · · · ) as Pk : (x1,k−1 , y1,k−1 , x2,k−1 , y2,k−1 ) → (x1,k , y1,k , x2,k , y2,k ), (x1,k , y1,k , x2,k , y2,k ) = Pk (x1,k−1 , y1,k−1 , x2,k−1 , y2,k−1 ). The discrete implicit map is given by h x1,k =x1,k−1 + (y1,k + y1,k−1 ), 2 1 1 1 y1,k =y1,k−1 − h{ (y1,k + y1,k−1 )[−α11 + α12 (x1,k + x1,k−1 )2 ] + β11 (x1,k + x1,k−1 ) 2 4 2 1 1 h 3 + β12 (x1,k + x1,k−1 ) + γ11 [(x1,k + x1,k−1 ) − (x2,k + x2,k−1 )] − Q0 cos Ω(tk−1 + )}, 8 2 2 h x2,k =x2,k−1 + (y2,k + y2,k−1 ), 2 1 1 1 y2,k =y2,k−1 − h{ (y2,k + y2,k−1 )[−α21 + α22 (x2,k + x2,k−1 )2 ] + β21 (x2,k + x2,k−1 ) 2 4 2 1 1 + β22 (x2,k + x2,k−1 )3 + γ21 [(x2,k + x2,k−1 ) − (x1,k + x1,k−1 )]}, 8 2

(3)

(4)

122


where h = tk − tk−1 is the discrete time step. For period-1 motion of the 2-DOF van der Pol-Duffing oscillators, the mapping structure is P = PN ◦ PN−1 ◦ · · · ◦ P1 , | {z } N−actions

P : (x1,0 , y1,0 , x2,0 , y2,0 ) → (x1,N , y1,N , x2,N , y2,N ), (x1,N , y1,N , x2,N , y2,N ) = P(x1,0 , y1,0 , x2,0 , y2,0 ).

(5)

Within each map step, the mapping relations are: P1 : (x1,0 , y1,0 , x2,0 , y2,0 ) → (x1,1 , y1,1 , x2,1 , y2,1 ), ⇒ (x1,1 , y1,1 , x2,1 , y2,1 ) = P1 (x1,0 , y1,0 , x2,0 , y2,0 ); P2 : (x1,1 , y1,1 , x2,1 , y2,1 ) → (x1,2 , y1,2 , x2,2 , y2,2 ), ⇒ (x1,2 , y1,2 , x2,2 , y2,2 ) = P2 (x1,1 , y1,1 , x2,1 , y2,1 ); .. . Pk : (x1,k−1 , y1,k−1 , x2,k−1 , y2,k−1 ) → (x1,k , y1,k , x2,k , y2,k ), ⇒ (x1,k , y1,k , x2,k , y2,k ) = Pk (x1,k−1 , y1,k−1 , x2,k−1 , y2,k−1 ); .. . PN : (x1,N−1 , y1,N−1 , x2,N−1 , y2,N−1 ) → (x1,N , y1,N , x2,N , y2,N ), ⇒ (x1,N , y1,N , x2,N , y2,N ) = PN (x1,N−1 , y1,N−1 , x2,N−1 , y2,N−1 ). The algebraic equations for the mapping structure are:  h   x1,1 =x1,0 + (y1,1 + y1,0 ),   2    1 1 1  2   y1,1 =y1,0 − h{ (y1,1 + y1,0 )[−α11 + α12 (x1,1 + x1,0 ) ] + β11 (x1,1 + x1,0 )   2 4 2    1 1 h  3  + β12 (x1,1 + x1,0 ) + γ11 [(x1,1 + x1,0 ) − (x2,1 + x2,0 )] − Q0 cos Ω(tk−1 + )}, 8 2 2 for P1 ; h    x2,1 =x2,0 + (y2,1 + y2,0 ),   2    1 1 1  2   y2,1 =y2,0 − h{ (y2,1 + y2,0 )[−α21 + α22 (x2,1 + x2,0 ) ] + β21 (x2,1 + x2,0 )   2 4 2    1 1  3  + β22 (x2,1 + x2,0 ) + γ21 [(x2,1 + x2,0 ) − (x1,1 + x1,0 )]} 8 2 .. .  h   x1,k =x1,k−1 + (y1,k + y1,k−1 ),   2    1 1 1  2   y1,k =y1,k−1 − h{ (y1,k + y1,k−1 )[−α11 + α12 (x1,k + x1,k−1 ) ] + β11 (x1,k + x1,k−1 )   2 4 2    1 1 h  3  + β12 (x1,k + x1,k−1 ) + γ11 [(x1,k + x1,k−1 ) − (x2,k + x2,k−1 )] − Q0 cos Ω(tk−1 + )}, 8 2 2 for Pk ; h    x2,k =x2,k−1 + (y2,k + y2,k−1 ),   2    1 1 1  2   y2,k =y2,k−1 − h{ (y2,k + y2,k−1 )[−α21 + α22 (x2,k + x2,k−1 ) ] + β21 (x2,k + x2,k−1 )   2 4 2    1 1  3  + β22 (x2,k + x2,k−1 ) + γ21 [(x2,k + x2,k−1 ) − (x1,k + x1,k−1 )]} 8 2

(6)


123

.. .

 h   x1,N =x1,N−1 + (y1,N + y1,N−1 ),   2    1 1 1  2   y1,N =y1,N−1 − h{ (y1,N + y1,N−1 )[−α11 + α12 (x1,N + x1,N−1 ) ] + β11 (x1,N + x1,N−1 )   2 4 2    1 h 1  3  + β12 (x1,N + x1,N−1 ) + γ11 [(x1,N + x1,N−1 ) − (x2,N + x2,N−1 )] − Q0 cos Ω(tk−1 + )}, (7) 8 2 2 for PN h    x2,N =x2,N−1 + (y2,N + y2,N−1 ),   2    1 1 1  2   y2,N =y2,N−1 − h{ (y2,N + y2,N−1 )[−α21 + α22 (x2,N + x2,N−1 ) ] + β21 (x2,N + x2,N−1 )   2 4 2    1 1  3  + β22 (x2,N + x2,N−1 ) + γ21 [(x2,N + x2,N−1 ) − (x1,N + x1,N−1 )]} 8 2 and the periodicity conditions for period-1 motion is (x1,N , y1,N , x2,N , y2,N ) = (x1,0 , y1,0 , x2,0 , y2,0 ).

(8)

Once the analytical prediction of nodes x∗k (x∗k = (x1,k , y1,k , x2,k , y2,k )T , k = 0, 1, 2, · · · , N) of periodic motions are achieved, the stability and bifurcation can be analyzed through eigenvalue analysis corresponding. For the mapping in Eq.(3), consider a small perturbation ∆xk−1 in the vicinity of x∗k−1 (i.e., xk−1 = x∗k−1 + ∆xk−1 , k = 1, 2, · · · , N), the perturbation ∆xN is approximated by ∆xN = DP∆x0 = DPN · DPN−1 · . . . · DP2 · DP1 ∆x0 , {z } |

(9)

N−multiplication

with ∆xk = DPk ∆xk−1 , 

∂ x1,k  ∂ x1,k−1   ∂y 1,k    ∂ x 1,k−1 ∂ xk DPk = [ ] ∗ ∗ =  ∂x 2,k ∂ xk−1 (xk ,xk−1 )    ∂ x1,k−1   ∂y 2,k  ∂ x1,k−1

∂ x1,k ∂ y1,k−1 ∂ y1,k ∂ y1,k−1 ∂ x2,k ∂ y1,k−1 ∂ y2,k ∂ y1,k−1

∂ x1,k ∂ x2,k−1 ∂ y1,k ∂ x2,k−1 ∂ x2,k ∂ x2,k−1 ∂ y2,k ∂ x2,k−1

 ∂ x1,k ∂ y2,k−1   ∂ y1,k    ∂ y2,k−1   ∂ x2,k   ∂ y2,k−1   ∂ y2,k   ∂ y2,k−1

for k = 1, 2, · · · , N

(10)

(x∗k−1 ,x∗k )

where

∂ x1,k ∂ x1,k−1 ∂ x1,k ∂ x2,k−1 ∂ x2,k ∂ x1,k−1 ∂ x2,k ∂ y2,k−1

∂ x1,k h ∂ y1,k h ∂ y1,k , = ( + 1); 2 ∂ x1,k−1 ∂ y1,k−1 2 ∂ y1,k−1 ∂ x1,k h ∂ y1,k h ∂ y1,k = , = ; 2 ∂ x2,k−1 ∂ y2,k−1 2 ∂ y2,k−1 ∂ x2,k h ∂ y2,k h ∂ y2,k = , = ; 2 ∂ x1,k−1 ∂ y1,k−1 2 ∂ y1,k−1 ∂ x2,k h ∂ y2,k h ∂ y2,k = ( + 1), = 1+ ; 2 ∂ y2,k−1 ∂ x2,k−1 2 ∂ x2,k−1 = 1+

(11)

124


∂ y1,k ∂ x1,k−1 ∂ y1,k ∂ y1,k−1 ∂ y2,k ∂ x1,k−1 ∂ y2,k ∂ y1,k−1 ∂ y1,k ∂ x2,k−1 ∂ y2,k ∂ x2,k−1 ∂ y1,k ∂ y2,k−1 ∂ y2,k ∂ y2,k−1

1 [(γ21 h − α21 y2km ) − ∆1 ∆101 ∆203 ], ∆ 1 = s[(∆2 − α21 y2km ) − ∆1 ∆102 ∆203 ], ∆ ∂ y1,k = ∆1 ∆103 + ∆1 ∆101 , ∂ x1,k−1 ∂ y1,k = ∆1 ∆103 + ∆1 ∆102 , ∂ y1,k−1

(12)

1 [∆1 (γ11 h − α11 y1km )∆203 − ∆201 ], ∆ ∂ y1,k = ∆1 ∆103 − ∆1 (γ11 h − α11 y1km ), ∂ x2,k−1 1 2 = [(1 + )∆203 − ∆202 ], ∆ h ∂ y1,k 2 = ∆1 ∆103 − ∆1 (1 − ). ∂ y2,k−1 h

(13)

=

=

with ∆101 = (∆11 − β11 h − 3β12 x21,km h − γ11 h), h2 3h2 h2 β11 + β12 x21,km + γ11 , 4 4 4 2 3h2 h h2 β12 x21,km + γ11 , ∆103 = ∆14 + ∆13 + β11 + 4 4 4 ∆201 = ∆21 + β21 h + 3hβ22 x22,km + γ21 h,

∆102 = ∆12 + ∆13 +

h2 3h2 h2 β21 + β22 x22,km + γ21 , 2 4 4 2 2 h2 h 3h ∆203 = ∆24 + ∆23 + β21 + β22 x22,km + γ21 , 4 4 4 h h x1,km = (x1,k + x1,k−1 ), y1,km = (y1,k + y1,k−1 ); 2 2 h h x2,km = (x2,k + x2,k−1 ), y2,km = (y2,k + y2,k−1 ); 2 2

∆202 = ∆22 + ∆23 +

∆ = ∆1 ∆103 ∆203 − ∆2 , ∆1 =

(14)

h2 4 ; γ , ∆ = 21 2 γ11 h2 4

∆11 = y1,km [α11 + 2α12 x1,km h], ∆12 = y1,km [α11 +

h2 α12 x1,km ]; 2

h2 h ∆13 = [α11 + α12 x21,km ], ∆14 = 1 + α12 x1,km y1,km ; 2 2 h2 ∆21 = y2,km [α21 + 2α22 x2,km h], ∆22 = y2,km [α21 + α22 x2,km ]; 2 2 h h ∆23 = [α21 + α22 x22,km ], ∆24 = 1 + α22 x2,km y2,km . 2 2

(15)


125

The total Jacobian matrix is

∂ xN ∗ ] ∗ ∗ ∂ x0 (xN ,xN−1 ,··· ,x0 ,) = DPN · DPN−1 · . . . · DP2 · DP1

DP = [

∂ xk

1

∏ [ ∂ xk−1 ](x ,x

=

∗ k

k=N

∗ ) k−1

(16)

.

For stability and bifurcation of period-1 motion, the eigenvalues of DP are computed by |DP − λ I4×4 | = 0.

(17)

From Luo [31, 36], the stability of period-1 motion can be determined as follows. i. If the magnitudes of all eigenvalues of DP are less than one (i.e. |λi | < 1, i = {1, 2, · · · , 4}), the periodic solution is stable. ii. If at least the magnitude of one eigenvalue of DP is greater than one (i.e. |λi | > 1, i ∈ {1, 2, · · · , 4}), the periodic solution is unstable. iii. The boundaries between stable and unstable periodic flow with higher order singularity can generate bifurcation and stability conditions with higher order singularity. The bifurcation conditions are as follows. iv. If λi = 1 with |λ j | < 1 (i, j ∈ {1, 2, · · · , 4}, i 6= j), the saddle-node bifurcation (SN) occurs. iiv. If λi = −1 with |λ j | < 1 (i, j ∈ {1, 2, · · · , 4}, i 6= j), the period-doubling bifurcation (PD) occurs. iiiv. If |λi, j | = 1 with λl | < 1 (i, j ∈ {1, 2, · · · , 4}, λi = λ¯ j , l 6= i, j), Neimark bifurcation (NB) occurs. From period-m motion of the 2-DOF van der Pol-Duffing oscillator, consider a mapping structure as (m)

(m)

(m)

P(m) = PmN ◦ PmN−1 ◦ · · · ◦ P2 | {z

(m)

◦ P1 , }

mN−actions

P

(m)

(18)

: (x1,0 , y1,0 , x2,0 , y2,0 ) → (x1,mN , y1,mN , x2,mN , y2,mN ), ⇒ (x1,mN , y1,mN , x2,mN , y2,mN ) = P(m) (x1,0 , y1,0 , x2,0 , y2,0 ).

with (m)

Pk

(m)

(m)

(m)

(m)

(m)

(m)

(m)

(m)

: (x1,k−1 , y1,k−1 , x2,k−1 , y2,k−1 ) → (x1,k , y1,k , x2,k , y2,k ), (m)

(m)

(m)

(m)

(m)

(m)

(m)

(m)

(m)

⇒ (x1,k , y1,k , x2,k , y2,k ) = Pk (x1,k−1 , y1,k−1 , x2,k−1 , y2,k−1 ). (k = 1, 2, · · · mN)

(19)

126


As similar to Eq.(4), the corresponding algebraic equations for each map of period-m motion are h (m) (m) (m) (m) x1,k =x1,k−1 + (y1,k + y1,k−1 ), 2 1 1 1 (m) (m) (m) (m) (m) (m) (m) (m) y1,k =y1,k−1 − h{ (y1,k + y1,k−1 )[−α11 + α12 (x1,k + x1,k−1 )2 ] + β11 (x1,k + x1,k−1 ) 2 4 2 h 1 1 (m) (m) (m) (m) (m) (m) + β12 (x1,k + x1,k−1 )3 + γ11 [(x1,k + x1,k−1 ) − (x2,k + x2,k−1 )] − Q0 cos Ω(tk−1 + )}, 8 2 2 h (m) (m) (m) (m) x2,k =x2,k−1 + (y2,k + y2,k−1 ), 2 1 1 1 (m) (m) (m) (m) (m) (m) (m) (m) y2,k =y2,k−1 − h{ (y2,k + y2,k−1 )[α21 + α22 (x2,k + x2,k−1 )2 ] + β21 (x2,k + x2,k−1 ) 2 4 2 1 1 (m) (m) (m) (m) (m) (m) + β22 (x2,k + x2,k−1 )3 + γ21 [(x2,k + x2,k−1 ) − (x1,k + x1,k−1 )]}. 8 2 (k = 1, 2, · · · , mN)

(20)

and for period-m motion, the periodicity condition is (m)

(m)

(m)

(m)

(m)

(m)

(m)

(m)

(x1,mN , y1,mN , x2,mN , y2,mN ) = (x1,0 , y1,0 , x2,0 , y2,0 ).

(21)

Due to Eqs.(18) and (21), the discretized nodes of a period-m motion in the two-degree-of-freedom van der Pol-Duffing oscillator can be obtained by solving 4(mN + 1) equations. Thus, the node points (m) (m) (m)∗ (m) (m)∗ (m)∗ (xk = (x1,k , y1,k ∗, x2,k ∗, y2,k ∗)T , k = 0, 1, 2, · · · , mN) can be used for representing the periodxk m motion, and the corresponding stability and bifurcation of period-m motion can be determined (m) (m)∗ (m) (m)∗ through eigenvalue analysis. For a small perturbation in the neighborhood of xk , xk = xk + ∆xk (k = 0, 1, 2, · · · , mN), the linearized equation of the period-m motion is (m)

(m)

(m)

(m)

(m)

∆xmN = DP(m) ∆x0 = DPmN · DPmN−1 · . . . · DP2 {z |

(m)

(m)

· DP1 ∆x0 , }

(22)

mN−multiplication (m)

where ∆xk

(m)

(m)

= DPk ∆xk−1 , 

(m)

∂ x1,k

 (m)  ∂x  1,k−1   ∂ y(m) 1,k   (m) (m)  ∂ x1,k−1 ∂ xk (m) DPk = [ (m) ](x(m)∗ ,x(m)∗ ) =   ∂ x(m)  ∂ xk−1 k k−1 2,k   ∂ x(m)  1,k−1   ∂ y(m) 2,k  (m) ∂ x1,k−1

(m)

∂ x1,k

(m)

∂ x1,k

(m)

∂ x1,k



 (m) (m) (m) ∂ y1,k−1 ∂ x2,k−1 ∂ y2,k−1   (m) (m) (m)  ∂ y1,k ∂ y1,k ∂ y1,k   (m) (m) (m)  ∂ y1,k−1 ∂ x2,k−1 ∂ y1,k−1   (m) (m) (m) ∂ x2,k ∂ x2,k ∂ x2,k    (m) (m) (m)  ∂ y1,k−1 ∂ x2,k−1 ∂ y2,k−1  (m) (m) (m)  ∂ y2,k ∂ y2,k ∂ y2,k   (m) (m) (m) ∂ y1,k−1 ∂ x2,k−1 ∂ y2,k−1 (x(m)∗ ,x(m)∗ ) k−1

(m)

(23)

k

for k = 1, 2, · · · , mN. The components of the matrix DPk in Eq.(23) is the same as in Eq.(10), and they can be determined by Eq.(11)-(15). The stability and bifurcation of period-m motion are determined by the eigenvalues, i.e., (m) (24) DP − λ I4×4 = 0,


127

where (m)

DP(m) = [

∂ xmN

]

(m)∗ (m)∗ (m) (x(m)∗ ) mN ,xmN−1 ,··· ,x0

∂ x0

(m)

(m)

(m)

= DPmN · DPmN−1 · . . . · DP2

∏

k=mN

(25)

(m)

1

=

(m)

· DP1

[

∂ xk

]

(m)∗

(m) (xk

∂ xk−1

(m)∗

,xk−1 )

.

Similarly, the criteria of the stability and bifurcation conditions for period-m motions are the same as for the period-1 motion. 3 Analytical predictions of periodic motions Define a set of discrete nodes for period-1 motion with N nodes per period as tk = t0 + kT N, t0 = 0, ∑ = (x1,k , y1,k , x1,k , y1,k ) T = 2π Ω, k = 0, 1, 2, · · · , N .

(26)

To avoid collecting all node points of periodic motions, the node points relative to the initial point and points at each period are collected in the Poincare mapping section for the period-m motion (m = 1, 2, 3, · · · ). That is, mod (k, N) = 0 (m) (m) (m) (m) tk = t0 + kT N, (27) ∑ = (x1,k , y1,k , x2,k , y2,k ) t0 = 0, T = 2π Ω, k = 0, 1, 2, · · · mN . m Consider a set of parameters as:

α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, Q0 = 1000, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7. 3.1

(28)

Periodic motions

The periodic motions of the 2-DOF van der Pol-Duffing oscillator is presented through the node points (m) (m) (m) (m) (x1,k , y1,k , x2,k , y2,k ) with mod (k, N) = 0. The periodic motion varying with excitation frequency are presented in Figs.1-2. The solid and dashed curves represent the stable and unstable solutions of periodic motions, respectively. The acronyms “SN” and “NB” are used for saddle-node and Neimark bifurcations, respectively. In Fig.1, a global view of periodic motions series varying with excitation frequency is presented. In (m) (m) Fig.1(i) and (ii), the global views of displacement (x1,k ) and velocity (y1,k ) nodes of periodic solutions of the first oscillator are presented, respectively. The three zoomed windows for displacement and velocity are presented. Because excitation strength is too big compared to the responses of the first oscillator with excitation, the solution branches of period-m motions for the first oscillator are in the small ranges. Thus, the stable and unstable motions for period-2, period-3 and period-9 motions are (m) (m) zoomed. In Fig.1(iii) and (v), the global views of the displacement (x2,k ) and velocity (y2,k ) for the second oscillator are presented, and the further zoomed views are presented in Fig.1(iv) and (vi). From the zoomed view, the appearance order of periodic motion series are as follows 1(S) ⊳ 1(A) ⊳ 3(S) ⊳ 2(A) ⊳ 5(S) ⊳ · · · ⊳ m(A) ⊳ (2m + 1)(S) ⊳ · · · .

(29)

128


In Eq.(29), the symmetric period-1 motion will appear first, which are represented by 1(S). Once the asymmetric period-1 motion (i.e., 1(A)) exists, the symmetric period-3 motion (i.e., 3(S)) will be found. Because asymmetric period-1 motion has two branches of solutions, with one branch adding, the asymmetric period-1 motion can become the symmetric period-3 motions with three branches. After symmetric period-3 motion, with one branch adding, the asymmetric period-2 motions (i.e., 2(A)) will appear. The asymmetric period-2 motion has four branches. Once the asymmetric period-2 motions vanish, with one branch adding, the symmetric period-5 motion (i.e., 5(S)) will appear. After the symmetric period-5 motion, with a branch adding, the asymmetric period-3 motion (i.e., 3(A)) will be observed. With one branch adding, after the asymmetric period-3 motion, the symmetric period-7 motion (i.e., 7(A)) will appear. Once the symmetric period-7 motion vanishes, with one branch adding, the asymmetric period-4 motion (i.e., 4(A)) will appear, continuously, the symmetric period-9 motion (i.e., 9(S)) will appear. Until the Neimark bifurcation (Ω ≈ 8.93) of the symmetric period-1 motion are approached from Ω ∈ (8.93, ∞), such periodic motion series will continue to the periodic motion with very large period number or infinity. In Fig.1, the periodic motion series are presented for Ω ∈ (0, 30). Seven independent periodic motions in the periodic motions series are presented in such a frequency range. In period-1 motion, two saddle-node bifurcations occur at Ω ≈ 2.59 and 2.60. Other two saddle-node bifurcations are close to Ω ≈ 8.80 and 9.58. Only one Neimark bifurcation happens at Ω ≈ 8.93 and, the stable period-1 motion become unstable for Ω ∈ (8.93, ∞) and quasiperiodic motions will be observed. On the periodic motion series, two symmetric period-3 motions happen for Ω ∈ (19.72, 21.47), which possesses two saddle-node bifurcations at Ω ≈ 19.72 and 21.47. The asymmetric period-2 motion exists for Ω ∈ (13.55, 14.09), and two saddle-node bifurcation occur at both ends of the frequency range. The symmetric period-5 motion happens for (11.43, 11.75), and the asymmetric period-3 motion exists for Ω ∈ (10.67, 10.43). With decreasing excitation frequency, the symmetric period-7 motion lies in the range of Ω ∈ (9.90, 10.07). The asymmetric period-4 motion is in Ω ∈ (9.57, 9.69), and the symmetric period-9 motion exists in the range of Ω ∈ (9.31, 9.41). The illustration of detailed period-1 motion is for Ω ∈ (0, 3.0), (2.587, 2.605) and (8.7, 9.7), as shown in Fig.2. The bifurcation points and the stable and unstable solutions of period-1motions are clearly presented. In Fig.2(i)-(iv), the discrete node displacement and velocity of the period-1 motion versus excitation frequency is presented for the first oscillator. For Ω ∈ (0, 2.591), there is only one stable periodic motion. As the frequency increases, for Ω ∈ (2.591, 2.601), there exists multiple solutions of period-1 motion, and two saddle-node bifurcations occur at Ω ≈ 2.591 and 2.601, respectively. The zoomed view of the discrete node displacement and velocity of the period-1 motion is presented in Fig.2(v)-(viii). For Ω ∈ (8.804, 9.582), multiple solutions of period-1 motion occur again, and the saddle-node bifurcation occurs at Ω ≈ 8.804, and the stable period-1 motion becomes unstable. One branch of period-1 motion is stable for Ω ∈ (2.601, 9.582), and one branch of periodic motion is unstable in the entire range of Ω ∈ (8.804, 9.582). The branch of period-1 motion is stable for Ω ∈ (8.804, 8.930). At Ω = 8.930, the Neimark bifurcation occurs, and the stable periodic motion turns to be unstable. For Ω ∈ (8.930, ∞), the asymmetric period-1 motion is unstable. The detailed bifurcation points and frequency range of period-m (P-m) motions are shown in Table 1. 3.2

Nonlinear frequency-amplitude characteristics (m)

(m)

(m)

(m)

From the discrete nodes of period-m motions x(m) = (x1,k , y1,k , x2,k , y2,k ) (k = 1, 2, · · · , mN), the period-m motions are approximately expressed by the Fourier series, i.e.,

M j j (m) x(m) (t) ≈ a0 + ∑ [b j/m cos( Ωt) + c j/m sin( Ωt)]. m m j=1

(30)


129

Table 1 Frequency ranges and saddle-node bifurcations with jump phenomenon of period-m motions, (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000, Ω ∈ (0, 30)). Frequency range

SN(L)

SN(R)

Stability

P-3

(10.43, 10.67)

10.43

10.67

3 branches

Symmetric

P-2

(13.55, 14.09)

13.55

14.09

4 branches

Asymmetric

P-5

(11.43, 11.75)

11.43

11.75

5 branches

Symmetric

P-3

(19.72, 21.47)

19.72

21.47

6 branches

Asymmetric

P-7

(9.90, 10.07)

9.90

10.07

7 branches

Symmetric

P-4

(9.57, 9.69)

9.57

9.69

8 branches

Asymmetric

P-9

(9.34, 9.41)

9.34

9.41

9 branches

Symmetric Symmetric

P-1

(0, 2.591)

–

2.591

Stable

(2.591, 2.601)

2.591

2.601

Unstable

(2.601, 9.582)

2.601

9.582

Stable

(8.804, 9.582)

8.804

9.582

Unstable

(8.804, 8.930)

8.804

8.930 (NB)

Stable

(8.930, ∞)

8.930 (NB)

—

Unstable

Symmetry

(m)

The 4(2M + 1) unknown coefficients of a0 , b j/m and c j/m can be determined from the analytically predicted nodes x(m) (k = 1, 2, · · · , mN) with 4(mN + 1). For M = mN 2, the node points x(m) for the period-m motion can be expressed for tk ∈ [0, mT ]

(m)

x(m) (tk ) = xk

mN /2

(m)

≈ a0 +

∑

j j [b j/m cos( Ωtk ) + c j/m sin( Ωtk )] m m

mN /2

2π k j 2π k j [b j/m cos( ) + c j/m sin( )]. mN mN

j=1

≈

(m) a0 +

∑

j=1

(31)

where ∆t = T /N = 2π /ΩN,tk = t0 + k∆t = 2π k/ΩN for t0 = 0 and

(m)

a0

=

b j/m =

1 mN (m) ∑ xk mN k=1 j 2π k 2 mN (m) xk cos( ), ∑ mN k=1 m N

(32)

2 j 2π k (m) xk sin( ) ∑ mN k=1 m N ( j = 1, 2, · · · , mN 2) mN

c j /m =

and (m)

a0

(m)

(m)

(m)

(m)

= (a1(1),0 , a1(2),0 , a2(1),0 , a2(2),0 )T ,

b j/m = (b1(1), j/m , b1(2), j/m , b2(1), j/m , b2(2), j2/m )T , c j/m = (b1(1), j/m , b1(2), j/m , b2(1), j/m , b2(2), j2/m )T .

(33)

Yeyin Xu, Albert C.J. Luo / Journal of Vibration Testing and System Dynamics 2(2) (2018) 119–153 SN 10 NB SNX2 SN

SN SN

P-1 P-9 P-4 P-7

P-3 P-5

1.2 P-2

P-3 -2.4 0.50

3.05

3.680

P-2 0.06 13.55

-6.0

P-3

P-9 3.625 10.67 9.342

2.77 14.09 10.43

0

10

9.413

20

P-2

P-3

0.5

-0.6

-1.7 0

10

20

SNSN

P-3 P-2 P-5

-3.0

-11.0 0

10

20

([FLWDWLRQ)UHTXHQF\:

(v)

(vi)

30

SN 7

SNSN

SN SN

SN

SN

P-5 0.4

P-4

P-7

P-3

P-9

P-1

-0.3

-1.0

-1.7 8.5

10.4

12.3

14.2


SNNB

SN 7

SN SN

SN SN

SN

SN

P-2

P-5 P-3 P-4

P-7

-1.0 P-9

P-1

-6.0

-11.0 8.5

30

9.39

20

P-2

SN SN

P-1

30.5 10.58 9.36

10

SNNB

1.1

4.0

5.0

P-9

P-3 61.0 13.84 10.52

(iv)

3HURGLF1RGH9HORFLW\y,kPRGkN

3HURGLF1RGH9HORFLW\y,kPRGkN

SNNBSN

32.5

P-2 70.3 13.80

;

;

SNSN

13.0

64.5


30


(iii)

72.5

(ii)

P-1 P-5

P-1

P-3

P-7 P-4 P-9

0

SN SN

P-2

-2

-48

3HURGLF1RGH'LVSODFHPHQWx,kPRGkN


SNNBSN 9 SNX2

SN SN

44

;

;

SNSN

1.6

SN 10 NB SNX2 SN

P-5 P-3

30


(i)

SNSN

90

3HURGLF1RGH9HORFLW\ykPRGkN

SNSN

4.8

;

;

3HURGLF1RGH'LVSODFHPHQWxk PRGkN

130

10.4

12.3

14.2


(m)

Fig. 1 Period-m motion varying with excitation frequency Ω: (i) Periodic node displacement x1,k , (ii) Periodic (m)

(m)

(m)

node velocity y1,k , (iii) periodic node displacement x2,k , (iv) zoomed view of displacement x2,k , (v) periodic node (m)

(m)

velocity y2,k , (vi) zoomed view of velocity y2,k . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)


SN SN

0.8

3HURGLF1RGH9HORFLW\y,k PRGkN


SN SN 3.62

3.61

P-1 3.60

3.59

3.58 0.0

1.0

2.0


(i)

0.5

P-1 0.2

-0.1 0.0

3.0

1.0

2.0

P-1

0.1

-0.4 0.0

1.0

SN

0.60

3.594

2.596


-3.5

1.0

2.602

(vi)

3.0

SN

SN

P-1

0.56

0.52

0.48 2.587

2.605

2.0


SN

3.596

2.590

2.0

(iv)

P-1

3.592 2.587

P-1

-9.0 0.0

3.0



(v)

2.0


(iii) 3.598

SN SN

7.5



SN SN 1.6

0.6

3.0


(ii)

1.1

131

2.590

2.596

2.602

2.605


Fig. 2 Zoomed view of period-1 motion varying with excitation frequency for (x1,k , y1,k , x2,k , y2,k ): (i)-(vi) Ω ∈ (0, 3), (v)-(viii) Ω ∈ (2.587, 2.605), (ix)-(xii) Ω ∈ (8.7, 9.7). (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

Yeyin Xu, Albert C.J. Luo / Journal of Vibration Testing and System Dynamics 2(2) (2018) 119–153 SN

1.50

1.33

SN

P-1

1.16

0.99 2.587

2.590

2.596

2.602

SN

42

P-1 3.7

3.6

9.20

9.45

SN

-1.2

-1.8 8.70

8.95


(xi) Fig. 2

9.20

9.45

22

8.95

(xii)

9.20

9.45

9.70

SN

NB

SN

P-1 6

-2

-10 8.70

9.70

SN

P-1

14

P-1

2.605

32

SN

-0.6

2.602

NB

(x)

NB

2.596




0.0

SN

12 8.70

9.70


(ix)

2.590


SN

NB

8.95

-3.0

(viii)

3.8

3.5 8.70

1.0



3.9

SN

P-1

-7.0 2.587

2.605


(vii)

SN

5.0



132

8.95

9.20

9.45

9.70


Continued.

The harmonic amplitudes and phases for a period-m motion can be expressed as q c1(1), j/m , A1(1), j/m = (b1(1), j/m )2 + (c1(1), j/m )2 , ϕ1(1), j/m = arctan b1(1), j/m q c1(2), j/m A1(2), j/m = (b1(2), j/m )2 + (c1(2), j/m )2 , ϕ1(2), j/m = arctan , b1(2), j/m q c2(1), j/m A2(1), j/m = (b2(1), j/m )2 + (c2(1), j/m )2 , ϕ2(1), j/m = arctan , b2(1), j/m q c2(2), j/m . A2(2), j/m = (b2(2), j/m )2 + (c2(2), j/m )2 , ϕ2(2), j/m = arctan b2(2), j/m

(34)


133

The approximation of the period-m motion for the 2DOF van der Pol-Duffing oscillator in Eq.(30) can be rewritten as  (m)   (m)      a1(1),0      x1  c1(1), j/m  b1(1), j/m                              y(m)    a(m)   mN /2  b  c j j 1(2), j/m 1(2),0 1(2), j/m 1 Ωt) + (35) sin( Ωt) ≈ + cos( ∑ b (m)  (m)       m m c         2(1), j/m x a 2(1), j/m j=1         2     2(1),0            (m)      c b   (m) 2(2), j/m 2(2), j/m y2 a2(2),0 or

  j     (m)   (m)  Ωt − ) ϕ A cos( 1(1), j/m  1(1), j/m   a     m       x1   1(1),0           j       (m)      (m)  mN/2 A1(2), j/m cos( Ωt − ϕ1(2), j/m ) y  a  1(2),0 1 m ≈ + . ∑ (m) (m) j          a2(1),0  x2  j=1        ) ϕ A cos( Ωt − 2(1), j/m  2(1), j/m         m    (m)    (m)          y2 j a2(2),0   A  2(2), j/m cos( Ωt − ϕ2(2), j/m ) m

(36)

(m)

To avoid abundant illustrations, only frequency-amplitude curves of displacement x1 period-m motions are presented. The displacements for the period-m motion are: ( ) ( (m) )  (m)  ( ) a  mN /2 b c1, j/m x1 j j 1, j/m 1,0 sin( Ωt). ≈ + ∑ cos( Ωt) + (m) (m) a  j=1 b2, j/m m m c2, j/m x 2

Thus,

(m)

and x2

for

(37)

2,0

 j  Ωt − ϕ1, j/m ) m j    A 2, j/m cos( Ωt − ϕ2, j/m ) m

(38)

c1, j/m , b1, j/m q c2, j/m . A2, j/m = (b2, j/m )2 + (c2, j/m )2 , ϕ2, j/m = arctan b2, j/m

(39)

( (m) ) x1 (m) x2

≈

where

  a(m)  1,0

a(m)  2,0

q

A1, j/m =



A1, j/m cos( mN /2  +

∑

j=1

(b1, j/m )2 + (c1, j/m )2 , ϕ1, j/m = arctan

Based on the discrete Fourier series, the frequency-amplitude characteristics of period-m motion can be presented. In all following plots, acronyms “SN” and “NB” are the saddle node and Neimark bifurcation, respectively. The stable and unstable solutions of period-m motions are also represented by the solid and dashed curves, respectively. In Figs.3 and 4, the harmonic amplitudes of the symmetric period-1 motions in the 2-DOF van der Pol-Duffing oscillator are presented for Ω ∈ (0, 30). For the symmetric period-1 motions, ai,0 = 0 and Ai,2 j = 0 but Ai,2 j−1 6= 0 (i = 1, 2; j = 1, 2, · · · ). Thus, the harmonic amplitude A1,1 varying with excitation frequency is presented in Fig.3(i)-(iv). However, the global view of the harmonic amplitude A1,1 is placed in Fig.3(i) and the zoomed views of the harmonic amplitude A1,1 are presented in Fig.3(ii)(iv). The saddle-node bifurcations are observed clearly and the corresponding quantity level is about 100 . The harmonic amplitude A1,3 varying with excitation frequency is presented in Fig.3(v)-(viii). The global view of the harmonic amplitude A1,3 is shown in Fig.3(v) and the zoomed views of the harmonic amplitude A1,3 are presented in Fig.3(vi)-(viii). The saddle-node bifurcations are also observed


SN SN

4.6

SN NB SN

3.4

2.2

1.0 0

10

20

30


SN

4.1805

SN

4.1795

4.1785

4.1775 2.587

2.590

2.596

2.602

SN NB SN

0.75

0.50

0.25

0.00 0

10

20


4.17 0.3

4.185

1.3

30

(vi)

2.3

3.3

SN

NB

8.8

8.9

SN

4.180

4.175

4.170

4.165 8.7

9.2

9.5

9.6

9.7


(iv) +DUPRQLF$PSOLWXGHRI)LUVW'LVSDOFHPHQWA

+DUPRQLF$PSOLWXGHRI)LUVW'LVSDOFHPHQWA

SN SN

1.00

4.18


2.605


(iii)

SN SN

4.19

(ii) +DUPRQLF$PSOLWXGHRI)LUVW'LVSDOFHPHQWA


(i)

(v)



134

SN SN 0.86

0.84

0.82

0.80 0.3

1.3

2.3

3.3


Fig. 3 Frequency-amplitude analysis of the first displacement in period-1 motion: (i)-(iv) A2,1 , (v)-(viii) A2,3 , (ix)-(xii) A2,35 . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

SN

SN

0.842

0.841

0.840

0.839 2.587

2.590

2.596

2.602

2.605


SN SN

SN NB SN

1.0e-4

1.0e-8

10

20

30



(ix) 7.25e-4

7.15e-4

7.05e-4

6.95e-4 2.587

2.590

2.596


(xi) Continued.

NB

8.8

8.9

2.602

2.605

(xii)

SN

0.73

0.71

0.69 8.7

9.2

9.5

9.6

9.7

SN SN 4.0e-2

3.0e-2

2.0e-2

1.0e-2

0.0 0.3

1.3

2.3

3.3


(x) SN

SN

SN

(viii)

1.0e-1

1.0e-12 0

0.75

135




(vii)

Fig. 3


0.843




3.2e-7

SN

NB

8.80

8.93

SN

2.3e-7

1.4e-7

5.0e-8 8.70

9.20


9.45

9.58

9.70

SN SN

1.8

SN NB SN

1.2

0.6

0.0 10

20

30


SN

0.48

0.44

0.40 2.587

2.590

2.596

2.602

2.605


+DUPRQLF$PSOLWXGHRI6HFRQG'LVSDOFHPHQWA

(iii) SN SN

1.2

SN NB SN

0.4

0.0 10

20


0.4 0.3

1.9

1.3

30

(vi)

2.3

3.3

SN

NB

8.80

8.93

SN

1.5

1.1

0.7

0.3 8.70

9.20

9.45

9.58

9.70


(iv)

0.8

0

0.6




SN

0.52

SN SN

0.8

(ii) +DUPRQLF$PSOLWXGHRI6HFRQG'LVSDOFHPHQWA

0

(i)

(v)




136

SN SN

1.2

0.8

0.4

0.0 0.3

1.3

2.3

3.3


Fig. 4 Frequency-amplitude analysis of the second displacement in period-1 motion: (i)-(iv) A2,1 , (v)-(viii) A2,3 , (ix)-(xii) A2,35 . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

SN

SN

0.95

0.85

0.75

2.590

2.596

2.602

2.605



SN NB SN

SN SN 1e-1

1e-6

1e-11

1e-16 0

6

12

18



(ix) 2e-6

1e-6

8e-7

2e-7 2.587

2.590

2.596


(xi) Continued.

SN

NB

8.8

8.9

2.602

2.605

(xii)

137

SN

0.06

0.03

0.00 8.7

9.2

9.5

9.6

9.7


SN SN

1e-1

1e-3

1e-5

1e-7 0.3

1.3

2.3

3.3


(x) SN

SN

0.09

(viii) +DUPRQLF$PSOLWXGHRI6HFRQG'LVSDOFHPHQWA

0.65 2.587

(vii)

Fig. 4


1.05




6.2e-11

SN

NB

8.8

8.9

SN

4.9e-11

3.6e-11

2.3e-11

1.0e-11 8.7

9.2


9.5

9.6

9.7


SN

SN

&RQVWDQWWHUPRI)LUVW'LVSODFPHQWa

1.6e-3


138

8.0e-4

0.0

-8.0e-4

-1.6e-3 13.45 13.55

14.09

14.20


4.10

4.08

4.07

4.06 13.45 13.55

13.82

14.09

13.82 ([FLWDWLRQ)UHTXHQF\:

14.09

(vi)

SN

13.82

14.09

14.20

SN

SN

5.4e-6

3.2e-6

1.0e-6 13.45 13.55

14.20

14.20


7.6e-6

4.0e-4

-2.0e-4 13.45 13.55

SN

(iv)

1.0e-3

14.09

9.80e-3

SN

SN

13.82

1.04e-2

+DUPRQLF$PSOLWXGHA

+DUPRQLF$PSOLWXGHA

1.6e-3

1.6e-2 13.45 13.55

9.20e-3 13.45 13.55

14.20


(iii)

1.7e-2

1.10e-2

4.09

SN

1.8e-2

(ii) SN

SN

SN


+DUPRQLF$PSOLWXGHA


(i)

(v)

13.82

2.0e-2

13.82

14.09

14.20


(2)

Fig. 5 Frequency-amplitude analysis of the first displacement in period-2 motion: (i)-(viii) a2,0 , A2,1/2 , A2,1 , A2,3/2 , A2,2 , · · · , A2,12 , A2,25/2 , A2,13 . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)


SN

3.81e-5

3.58e-5

3.35e-5 13.45 13.55


(vii) Fig. 5

6.4e-4

+DUPRQLF$PSOLWXGHA

+DUPRQLF$PSOLWXGHA

4.04e-5

14.09

(viii)

SN

SN

5.7e-4

5.0e-4

4.3e-4 13.45 13.55

14.20

139

13.82

14.09

14.20


Continued.

clearly. The quantity level is about 10−1 . To avoid abundant illustrations, the global view of the harmonic amplitude A1,35 is shown in Fig.3(ix) and the zoomed views of the harmonic amplitude A1,35 are presented in Fig.3(x)-(xii). For the low frequency, the quantity level is still high, but for the high frequency, the quantity level is very low, which is below 10−12 . Similarly, the global view of the harmonic amplitude A2,1 is placed in Fig.4(i) and the zoomed views of the harmonic amplitude A2,1 are presented in Fig.4(ii)-(iv). The saddle-node bifurcations are observed clearly. The global view of the harmonic amplitude A2,3 is shown in Fig.4(v) and the zoomed views of the harmonic amplitude A2,3 are presented in Fig.4(vi)-(viii). The saddle-node bifurcations are observed clearly. The quantity level is about 100 ∼ 10−2 . To avoid abundant illustration, the global view of the harmonic amplitude A2,35 is placed in Fig.4(ix) and the zoomed views of the harmonic amplitude A2,35 are shown in Fig.4(x)-(xii). For the low frequency, the quantity level is still high, but for the high frequency, the quantity level is below 10−16 . In Figs.5 and 6, the harmonic amplitudes of the asymmetric period-2 motions in the 2-DOF van der Pol-Duffing oscillator are presented for Ω ∈ (13.55, 14.09). For the asymmetric period-2 motions, (2) (2) ai,0 6= 0 and Ai,k/2 6= 0 (i = 1, 2; k = 1, 2, · · · ). Thus, the constant a1,0 varying with excitation frequency is (2)L

(2)R

presented in Fig.5(i). a1,0 = −a1,0 . The harmonic amplitude A1,1/2 varying with excitation frequency is presented in Fig.5(ii) with the quantity level of 10−2 . In Fig.5(iii), the harmonic amplitudes A1,1 of the stable and unstable period-2 motions with the quantity level of 100 are quite close, which are almost same. Such a harmonic term plays an important role in such a period-2 motion. To compare with the harmonic amplitude A1,1/2 , the harmonic amplitude A1,3/2 is presented in Fig.5(vi), which is totally different from A1,1/2 . The harmonic amplitudes A1,2 of the period-2 motions possesses the quantity level of 10−2 , as shown in Fig.5(v). The harmonic amplitudes of A1,12 , A1,25/2 , A1,13 are presented in Fig.5(vi)-(viii), and the corresponding quantity levels are 10−6 , 10−5 and 10−4 , respectively. The harmonic amplitudes A1,13 of the stable and unstable period-2 motion are also very close. In Fig.6(i), (2) constant a2,0 varying with excitation frequency is presented. The off-center of the second oscillator is much bigger than the first oscillator. In Fig.6(ii), the harmonic amplitude A2,1/2 is presented, which is much bigger than the first oscillator. This harmonic amplitude has significant effects on the period-2 motion. In other words, the asymmetry of period-2 motions is mainly determined by such a harmonic term. In Fig.6(iii), the harmonic amplitude A2,1 is presented, which is much smaller than A1,1 . Similarly, to make a comparison to A2,1/2 , the harmonic amplitude of A2,3/2 reduces to the quantity level of 10−2 , as shown in Fig.6(iv). In Fig.6(v), the harmonic amplitude A2,2 has the quantity level of 10−3 . In Fig.6(vi)-(viii), the harmonic amplitudes of A2,12 , A2,25/2 , A2,13 have the quantity levels of 10−9 , 10−9 and 10−7 , respectively. This is because the first oscillator has a strong external excitation. For the


8.0e-2

0.97

4.0e-2

0.0

-4.0e-2

-8.0e-2 13.45 13.55

13.82

14.09

0.212

13.82

14.09

9.3e-3

9.0e-3 13.45 13.55


14.09

(vi)

13.82

14.09

14.20

SN

SN

4.8e-9

3.6e-9

2.4e-9 13.45 13.55

14.20

SN

1.8e-2

6.0e-9

9.6e-3

14.20

2.1e-2

(iv) SN

SN

14.09


+DUPRQLF$PSOLWXGHA

+DUPRQLF$PSOLWXGHA

9.9e-3

13.82

SN

1.5e-2 13.45 13.55

14.20


(iii)

0.89

2.4e-2

0.224

0.200 13.45 13.55

0.93

(ii) SN

SN

SN


+DUPRQLF$PSOLWXGHA

+DUPRQLF$PSOLWXGHA

0.236

SN

0.85 13.45 13.55

14.20


(i)

(v)

SN

SN

+DUPRQLF$PSOLWXGHA

&RQVWDQWWHUPRI6HFRQG'LVSODFPHQWa

140

13.82

14.09

14.20


(2)

Fig. 6 Frequency-amplitude analysis of the second displacement in period-2 motion: (i)-(viii) a2,0 , A2,1/2 , A2,1 , A2,3/2 , A2,2 , · · · , A2,12 , A2,25/2 , A2,13 . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)


SN

7.4e-9

6.8e-9

6.2e-9 13.45 13.55


(vii) Fig. 6

1.4e-7

+DUPRQLF$PSOLWXGHA

+DUPRQLF$PSOLWXGHA

8.0e-9

14.09

(viii)

SN

SN

1.2e-7

1.1e-7

9.2e-8 13.45 13.55

14.20

141

13.82

14.09

14.20


Continued.

second oscillator, no external excitation is added. Thus the harmonic terms of A2,(2 j−1)/2 have more effects on the asymmetric period-2 motion. For the high order periodic motions, too many branches of the periodic points should be presented. Herein, the harmonic-amplitude curves of symmetric period-9 motion is presented in Fig.7 and 8 as an (9) example. For symmetric period-9 motions, ai,0 = 0 and Ai,2 j/9 = 0 but Ai,(2 j−1)/9 6= 0 (i = 1, 2; j = 1, 2, · · · ). However, the harmonic amplitude A1,1/9 is placed in Fig.7(i) with the corresponding quantity level of 4.5 × 10−4 . The harmonic amplitudes of A1,1/3 , A1.5/9 and A1,7/9 are presented in Fig.7(ii)-(iv) with the quantity levels of 9.5 × 10−4 , 2.9 × 10−3 and 8.7 × 10−3 , respectively. However, in Fig.7(v), the harmonic amplitude of A1,1 for the period-9 motion is with the quantity level of 100 , which strongly effects the periodic motion. The harmonic amplitude of A1,3 with the quantity level of 10−1 is the second important harmonic amplitude, as shown in Fig.7(vi). To avoid abundant illustrations, the harmonic amplitude of A1,9 is presented in Fig.7 (vii) with the quantity level of 10−2 . The decay rate of the harmonic amplitude with increasing harmonic orders is very slow. In Fig.7(viii)-(xi), the harmonic amplitudes of A1,91/9 , A1,31/3 , A1,95/9 and A1,97/9 are with the quantity levels of 1.5 × 10−5 , 2.9 × 10−5 , 7.8 × 10−5 and 2.0 × 10−4 , respectively. In Fig.7(xii), the harmonic amplitudes of A1,11 is presented with the quantity level of 9.9 × 10−3 . Similarly, the higher-order harmonic amplitude can be analyzed. For the second oscillator, in Fig.8(i)-(iv), the harmonic amplitudes of A2,1/9 , A2,1/3 , A2,5/9 and A1,7/9 are presented with the quantity levels of 2.1 × 10−2 , 4.6 × 10−2 , 1.4 × 10−1 and 4.5 × 10−1 , respectively. The harmonic amplitude of A2,1 ∼ 7.35 × 10−1 is presented in Fig.8(v). The order of main contributions on the period-9 motions is A2,1 , A2,7/9 , A5/9 accordingly. In Fig.8(vi), the harmonic amplitude of A2,3 with the quantity level of 1.3 × 10−2 is presented. However, the harmonic amplitude of A2,9 is with the quantity level of 2.8 × 10−5 , as is shown in Fig.8(vii). In Fig.8 (viii)-(xi), the harmonic amplitudes of A2,91/9 , A2,31/3 , A2,95/9 and A2,97/9 are with the quantity levels of 2.7 × 10−8 , 5.4 × 10−8 , 4.0 × 10−8 and 7.2 × 10−8 , respectively. In Fig.8(xii), the harmonic amplitude of A2,11 is presented with the quantity level of 6.7 × 10−6 . From the frequency-amplitude curves, the decay rate of the harmonic amplitudes of the second oscillator with the harmonic order is faster than the first oscillator. This is because the strong external excitation is added in the first oscillator, but the second oscillator does not have such external excitation except for the coupling with the first oscillator. 4 Numerical simulations To verify the analytical prediction of the periodic motions of the 2-DOF van der Pol-Duffing oscillator, numerical simulations are carried out through the midpoint integration method. Initial conditions

4.5e-4

SN

SN

3.4e-4

2.3e-4

9.375

9.413 9.424



2.86e-3

SN

SN

2.54e-3

2.38e-3 9.342

9.375

9.413

9.424


4.17835

4.17815

4.17795

4.17775 9.330

9.342

9.375


7.7e-4

6.8e-4 9.330 9.342

8.70e-3

9.413

9.424

(vi)

9.375

9.413 9.424

SN

SN

8.45e-3

8.20e-3

7.95e-3

7.70e-3 9.330 9.342

9.375

9.413 9.424


(iv) SN

SN

8.6e-4



(iii)

SN

SN

(ii)

2.70e-3

9.330

9.5e-4



1.2e-4 9.330 9.342

(i)

(v)




142

0.714

SN

SN

0.713

0.712

0.710 9.330

9.342

9.375

9.413 9.424


Fig. 7 Frequency-amplitude analysis of the first displacement in period-9 motion: (i)-(xii) A1,1/9 , A1,1/3, · · · , A1,1 , A1,3 , A1,9 , · · · , A1,91/9, A1,31/3, · · · , A1,11 . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

SN

SN

0.02644

0.02622

9.342

9.375

9.413

9.424


(vii) 2.9e-5

SN

SN

2.5e-5

2.3e-5

9.342

9.375

9.413

9.424



(ix) 2.04e-4

1.99e-4

1.94e-4

1.89e-4

1.84e-4 9.330

9.342

9.375


(xi) Continued.

9.413

9.424

(xii)

143

SN

1.10e-5

7.50e-6

4.00e-6 9.330

9.342

9.375

9.413

9.424


7.8e-5

SN

SN

7.4e-5

7.0e-5

6.6e-5 9.330

9.342

9.375

9.413

9.424


(x) SN

SN

SN

(viii)

2.7e-5

2.1e-5 9.330

1.45e-5


0.02600 9.330


Fig. 7


0.02666




9.91e-3

SN

SN

9.79e-3

9.67e-3

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Fig. 8 Frequency-amplitude analysis of the second displacement in period-9 motion: (i)-(xii) A1,1/9 , A1,1/3 , · · · , A1,1 , A1,3 , A1,9 , · · · , A1,91/9, A1,31/3 , · · · , A1,11 . (α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

SN

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(xi) Continued.

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Fig. 8


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30

45

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+DUPRQLF2UGHUk

Fig. 9 Period-1 motion of the 2-DOF van der Pol-Duffing oscillator (Ω = 2.60). (i) Displacement x1 , (ii) Displacement x2 , (iii) Trajectory of the first oscillator, (iv) Trajectory of the second oscillator, (v) Harmonic amplitude A1,k , (vi) Harmonic amplitude A2,k . (IC: x0 ≈ (3.5967, 0.5102, 1.1985, −5.5601) parameters: α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)


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Fig. 10 The period-3 motion (Ω = 19.729, x0 ≈ (−2.2969, 36.7506, −0.5773, 2.8545)): (i) the first oscillator, (ii) the second oscillator. The period-5 motion (Ω = 10.432, x0 ≈ (1.9628, 79.1908, 0.5537, 0.8670)): (iii) the first oscillator, (iv) the second oscillator. The period-7 motion (Ω = 10, x0 ≈ (3.2921, 49.4305, −1.0080, −3.0758)): (v) the first oscillator, (vi) the second oscillator. The period-9 motion (Ω = 9.324, x0 ≈ (3.6636, 31.0214,− 0.3135,− 1.7195)): (vii) the first oscillator, (viii) the second oscillator. (parameters: α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

148

Yeyin Xu, Albert C.J. Luo / Journal of Vibration Testing and System Dynamics 2(2) (2018) 119–153 120.0

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(vii) Fig. 10

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for the numerical simulation are extracted from the analytical prediction of the periodic motions. Displacement, trajectory and frequency spectrum will be presented. The harmonic phases of ϕk ∈ [0, 2π ) will also be presented. In all plots, circular symbols and solid curves represent analytical and numerical solutions, respectively. The acronym “I.C.” stands for the initial condition. The initial conditions and the periodic points are denoted as large circular symbols. 4.1

Symmetric periodic motions

A stable period-1 motion of the 2-DOF van der Pol-Duffing oscillator is simulated for Ω = 2.6, as is shown in Fig.9. The same parameters in Eq.(28) are used. The initial condition is x0 ≈ (3.5967, 0.5102, 1.1985, −5.5601) from analytical prediction at t0 = 0. The time histories of first and second displacements are presented in Fig.9(i) and (ii). In Fig.9(iii) and (iv), the trajectories of the first and second oscillators are presented. Numerical solutions match with the analytical prediction. The first oscillator possesses more van der Pol-oscillator behaviors, and the second oscillator have more cubic nonlinearity behavior, which can be observed from the frequency-amplitude in Fig.4. For a better understanding of effects of the harmonic terms on the period-1 motion, the corresponding harmonic amplitudes are plotted in Fig.9(v) and (vi). The harmonic amplitudes are from the analytical prediction. For the first oscillator, a1,0 = 0. The main harmonic terms for the first displacement are A1,1 ≈ 4.1780, A1,3 ≈ 0.8405, A1,5 ≈ 0.4296, A1,7 ≈ 0.2628, A1,9 ≈ 0.1654, A1,11 ≈ 0.1039, A1,13 ≈ 0.0653, A1,15 ≈ 0.0414, A1,17 ≈ 0.0267, A1,19 ≈ 0.0175, A1,21 ≈ 0.0115. The other harmonic terms are A1,k ∈ (10−2 , 10−9 ) (k = 23, 25, · · · , 99) with A1,99 ≈ 4.06 × 10−9 . For the second oscillator, a2,0 = 0. The main harmonic terms for the second displacement are A2,1 ≈ 0.4131, A2,3 ≈ 0.9768, A2,5 ≈ 0.1522, A2,7 ≈ 0.0376, A2,9 ≈ 0.0280. The other harmonic terms are A2,k ∈ (10−2 , 10−9 ) (k = 11, 13, · · · , 59) with A2,59 ≈ 2.09 × 10−9 . For the analytical solutions of periodic motions for the symmetric period-1 motions at Ω = 2.6, 100 harmonic terms are needed to keep the computational accuracy of 10−9 . On the periodic solutions series of the 2-DOF van der Pol -Duffing oscillator, trajectories of the first and second oscillators for the symmetric period-3, period-5, period-7 and period-9 motions are presented in Fig.10(i)-(viii). The initial conditions for the period-3, period-5, period-7 and period-9 motions are x ≈ (−2.2969, 36.7506, −0.5773, 2.8545) for Ω = 19.729, (1.9628, 79.1908, 0.5537, 0.8670) for Ω = 10.432, (3.2921, 49.4305, −1.0080, −3.0758) for Ω = 10, (3.6636, 31.0214, −0.3135, −1.7195) for Ω = 9.324. The trajectories of the first oscillator are very close and the periodic points are quite close. However, for the second oscillator, the trajectories become more complicated from period-3 to period-9 motions. The Duffing nonlinear effects plays an important role on such symmetric periodic motions.


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Fig. 11 The period-2 motion of the 2-DOF van der Pol-Duffing oscillator (Ω = 13.56). The first oscillator: (i) trajectory (left), (ii) trajectory (right), (iii) harmonic amplitudes, (iv) harmonic phases. The second oscillator: (v) trajectory (left), (vi) trajectory (right), (vii) harmonic amplitudes, (viii) harmonic phases. (Initial conditions: x0 ≈ (0.4389, 73.0093, −0.3893, 2.3119)(left) and x0 ≈ (0.4932, 73.9975, 0.8360, − 1.1507) (right)). (Parameters:α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)


+DUPRQLF$PSOLWXGHAk

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Fig. 12 Phase trajectories of the 2-DOF van der Pol-Duffing oscillator. The period-3 motion (Ω = 10.432, x0 ≈ (3.1110,57.1182,0.1386,− 7.1450)(left), (3.0310, 59.6194, 0.5004,− 2.4440) (right)): (i) and (iii) the first oscillator, (ii) and (iv) the second oscillator. The period-4 motion (Ω = 9.6, x0 ≈ (3.5329, 38.3496, −0.4288, −0.8595) (left), x0 ≈ (3.5553, 37.4718, 0.2032, −2.3442)(right)): (v) and (vii) the first oscillator, (iv) and (viii) the second oscillator. (parameters: α11 = 1, α12 = 4, β11 = 10, β12 = 20, γ11 = 7, α21 = 1, α22 = 4, β21 = 20, β22 = 30, γ21 = 7, Q0 = 1000.)

Yeyin Xu, Albert C.J. Luo / Journal of Vibration Testing and System Dynamics 2(2) (2018) 119–153 12.0

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Asymmetric periodic motions

On the periodic motions series, the asymmetric periodic motions exist between the asymmetric periodic motions. The asymmetric period-2 motion is between the symmetric period-3 and period-5 motions. In Fig.11, an asymmetric period-2 motion of the 2-DOF van der Pol-Duffing oscillator is presented for Ω = 13.56 with the other parameters in Eq.(28). The initial conditions for the paired period-2 motion are x0 ≈ (0.4389, 73.0093, −0.3893, 2.3119) and (0.4932, 73.9975, 0.8360, −1.1507) at t0 = 0. The two trajectories of the paired asymmetric period-2 motions for the first oscillator are in Fig.11(i) and (ii). The two initial conditions for the first oscillator are very close and the trajectories look very similar. The corresponding harmonic amplitudes and phases are presented in Fig.11(iii) and (iv). The constant (2)L (2)R terms for the first oscillator are a1,0 = −a1,0 = A1,0 = 9.5882 × 10−4 . The main harmonic terms for the first displacement are A1,1/2 ≈ 0.0169, A1,1 ≈ 4.0944, A1,3/2 ≈ 9.5047 × 10−3 , A1,2 ≈ 1.0525 × 10−3 , A1,5/2 ≈ 7.4328 × 10−3 , A1,3 ≈ 0.5274, A1,7/2 ≈ 4.0348 × 10−3 , A1,4 ≈ 4.1563 × 10−4 , A1,9/2 ≈ 2.8740 × 10−3 , A1,5 ≈ 1.186 ×10−5 , A1,11/2 ≈ 1.4920 ×10−3 , A1,6 ≈ 1.5204 ×10−4 , A1,13/2 ≈ 1.0283 ×10−3 , A1,7 ≈ 0.0299, A1,15/2 ≈ 5.2280 × 10−4 , A1,8 ≈ 5.3149 × 10−5 , A1,17/2 ≈ 3.5348 × 10−4 . The other harmonic terms are A1,k/2 ∈ (10−10 ,10−3 ) (k = 9, 10, · · · , 60) with A1,30 ≈ 2.13 × 10−10 . The two trajectories of the paired asymmetric period-2 motions for the second oscillator are in Fig.11(v) and (vi). The two initial conditions for the second oscillator are different, and the asymmetry of two paired asymmetric trajectories are presented. The harmonic amplitudes and phases of the second displacement are presented in Fig.11(vii) and (viii). (2)L (2)R The constant terms for the second oscillator are a2,0 = −a2,0 = A2,0 = 0.05. The main harmonic terms

152


for the second displacement are A2,1/2 ≈ 0.8752, A2,1 ≈ 0.2299, A2,3/2 ≈ 0.0179, A2,2 ≈ 9.3555 × 10−3 , A2,5/2 ≈ 1.6086 × 10−3 , A2,3 ≈ 2.5648 × 10−3 , A2,7/2 ≈ 1.5494 × 10−4 , A2,4 ≈ 6.3017 × 10−5 , A2,9/2 ≈ 1.7246 × 10−5 , A2,5 ≈ 1.8642 × 10−4 . The other harmonic terms are A2,k/2 ∈ (10−14 ,10−3 ) (k = 9, 10, · · · , 60) with A1,30 ≈ 5.10 × 10−14 . For the paired asymmetric periodic motions, the harmonic amplitudes are same, L R and the harmonic phases are ϕi,k/2 = mod (ϕi,k/2 + (1 + k 2)π , 2π ) with i = 1, 2. From the periodic solutions series of the 2-DOF van der Pol-Duffing oscillator, there are asymmetric period-2, period-3 and period-4 motions. Herein, the two paired trajectories of the first and second oscillators for the asymmetric period-3 and period-4 motions are presented in Fig.12(i)-(viii). The asymmetric period-3 motion at Ω = 10.432 have the initial conditions of x0 ≈ (3.1110,57.1182,0.1386, − 7.1450)(left) and (3.0310, 59.6194,0.5004, − 2.4440) (right). The initial conditions of the paired asymmetric period-4 motion at Ω = 9.6 are x0 ≈ (3.5329, 38.3496, −0.4288, −0.8595) (left), and (3.5553, 37.4718, 0.2032, −2.3442)(right). The trajectories of the first oscillator are still very close for the asymmetric periodic motions and the periodic points are quite close. However, for the second oscillator, the two paired trajectories exhibit the asymmetry of the two asymmetric periodic motions. The Duffing nonlinear effects plays an important role on such asymmetric periodic motions for the second oscillators. This is because the external excitation exerted in the first oscillator is too strong. For the further study, the weak excitation should be considered in sequel, and the richer dynamical behaviors should be determined. 5 Conclusions In this paper, the series of periodic motions in the periodically forced, coupled van der pol-duffing oscillator were analytically predicted, and the chaotic motions and catastrophes between the two periodic motions exist. From the semi-analytical method, the 2-DOF van der Pol-Duffing oscillator was discretized. The series of periodic motions varying with excitation frequency were presented, and the corresponding stability and bifurcation were determined. The frequency-amplitude characteristics of sampled periodic motions were also discussed. Numerical simulations of periodic motions were carried out. The order of symmetric period-1 to period-m motions (i.e., 1(S) ⊳ 1(A) ⊳ 3(S) ⊳ 2(A) ⊳ · · · ⊳ m(A) ⊳ (2m + 1)(S) ⊳ · · · ) (m → ∞) was discovered. References [1] Lagrange, J.L. (1788), Mecanique Analytique (2 vol.) (edition Albert Balnchard: Paris, 1965). [2] Poincare, H. (1899), Methodes Nouvelles de la Mecanique Celeste, Vol.3, Gauthier-Villars: Paris. [3] Van der Pol, B. (1920) A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, pp.701-710, pp.754-762. [4] Fatou, P. (1928) Sur le mouvement d’un systeme soumis ‘a des forces a courte periode, Bull. Soc. Math. 56, pp.98-139. [5] Krylov, N.M. and Bogolyubov, N.N. (1935), Methodes approchees de la mecanique non-lineaire dans leurs application a l’Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s’y rapportant. Kiev: Academie des Sciences d’Ukraine; 1935 (in French). [6] Hayashi, C. (1964), Nonlinear oscillations in Physical Systems, McGraw-Hill Book Company: New York. [7] Nayfeh, A.H. (1973), Perturbation Methods, John Wiley: New York. [8] Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillation, John Wiley: New York. [9] Maaita, J.O. (2016), A theorem on the bifurcations of the slow invariant manifold of a system of two linear oscillators coupled to a k-order nonlinear oscillator, Journal of Applied Nonlinear Dynamics, 5(2),193-197. [10] Yamgoué, S.B. and Nana, B. Pelap, F.B. (2017), Approximate analytical solutions of a nonlinear oscillator equation modeling a constrained mechanical system, Journal of Applied Nonlinear Dynamics, 6(1),17-26. [11] Shayak, B., Vyas, P. (2017), Krylov Bogoliubov type analysis of variants of the Mathieu equation, Journal of Applied Nonlinear Dynamics, 6(1): 57-77.


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[12] Rajamani, S, Rajasekar, S.(2017), Variation of response amplitude in parametrically driven single Duffing oscillator and unidirectionally coupled Duffing oscillators, Journal of Applied Nonlinear Dynamics, 6(1): 121-129. [13] Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L&H Scientific: Beijing/Glen Carbon. [14] Luo, A.C.J. and Huang, J.Z. (2012), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance, Journal of Vibration and Control, 18, 1661-1871. [15] Luo, A.C.J. and Huang, J.Z. (2012), Analytical dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No. 1250093 (29 pages). [16] Luo, A.C.J. and Huang, J.Z. (2012), Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential, Journal of Applied Nonlinear Dynamics, 1, 73-108. [17] Luo, A.C.J. and Huang, J.Z. (2012), Unstable and stable period-m motions in a twin-well potential Duffing oscillator, Discontinuity, Nonlinearity and Complexity, 1, 113-145. [18] Luo, A.C.J. and Lakeh, A.B. (2013), Analytical solutions for period-m motions in a periodically forced van der Pol oscillator, International Journal of Dynamics and Control, 1, pp.99-115. [19] Wang, Y.F. and Liu, Z.W. (2015), A matrix-based computational scheme of generalized harmonic balance method for periodic solutions of nonlinear vibratory systems, Journal of Applied Nonlinear Dynamics, 4(4), 379-389. [20] Luo, H. and Wang, Y. (2016), Nonlinear dynamics analysis of a continuum rotor through generalized harmonic balance method, Journal of Applied Nonlinear Dynamics, 5(1), 1-31. [21] Luo, A.C.J. (2014), Toward Analytical Chaos in Nonlinear Systems, Wiley: New York. [22] Luo, A.C.J. (2014), Analytical Routes to Chaos in Nonlinear Engineering, Wiley: New York. [23] Greenspan, D. (1972), Numerical approximation of periodic solutions of van der Pol’s equation, Journal of Mathematical Analysis and Applications, 39, 574-579. [24] Sanjuan, M.A.F. (1996), Symmetry-restoring crises, period-adding and chaotic transitions in the cubic van der pol oscillator, Journal of Sound and Vibration, 193, 863-875. [25] Luo, A.C.J. (2015), Periodic flows in nonlinear dynamical systems based on discrete implicit maps, International Journal of Bifurcation and Chaos, 25(3), Article No.:1550044. [26] Luo, A.C.J. and Guo, Y. (2015), A semi-analytical prediction of periodic motions in Duffing oscillator through mappings structures, Discontinuity, Nonlinearity, and Complexity, 4, 121-150. [27] Guo, Y. and Luo, A.C.J. (2017), Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings, International Journal of Dynamics and Control, 5(2), 223-238. [28] Guo, Y. and Luo, A.C.J. (2016), Periodic Motions to Chaos in Pendulum, International Journal of Bifurcation and Chaos, 26, 1650159 (64 pages). [29] Guo, Y. and Luo, A.C.J. (2017) Routes of periodic motions to chaos in a periodically forced pendulum, International Journal of Dynamics and Control, 5(3), 551-569. [30] Guo, Y., Luo, A.C.J. (2017), Complete bifurcation trees of a parametrically driven pendulum, Journal of Vibration Testing and System Dynamics,1, 93-134. [31] Luo, A.C.J. (2016), Discretization and Implicit Mapping Dynamics, Springer/Higher Education Press: Heidelberg/Beijing. [32] Luo, A.C.J. and Xing, S.Y. (2016), Symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator. Nonlinear Dynamics, 85, 1141-1166. [33] Luo, A.C.J. and Xing, S.Y. (2016), Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator, Chaos, Solitons and Fractals, 89, 405-434. [34] Luo, A.C.J. and Xing, S.Y. (2017), Time-delay effects on periodic motions in a periodically forced, timedelayed, hardening duffing oscillator, Journal of Vibration Testing and System Dynamics, 1, 73-91. [35] Xing, S. and Luo, A.C.J. (2017), Towards infinite bifurcation trees of period-1 motions to chaos in a timedelayed, twin-well Duffing oscillator, Journal of Vibration Testing and System Dynamics, 1, 353-392. [36] Luo, A.C.J. (2017), Memorized Discrete Systems and Time-delay, Springer: New York.



Lift Enhancement of Airfoil Using Local Flexible Structure and the Influences of Structure Parameters Pengfei Lei1†, Jiazhong Zhang2, Daxiong Liao1 1 2

Facility Design and Instrumentation Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan, P. R. of China School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, P. R. of China Submission Info Communicated by S. C. Suh Received 12 February 2018 Accepted 19 February 2018 Available online 1 July 2018 Keywords Local flexible structure Structure parameter Self-induced oscillation Dual time stepping Lift enhancement

Abstract The local flexible structure (LFS) is used on the airfoil to enhance the lift, and the influences of structure parameters of LFS on the lift enhancement are studied numerically. Coupling with the structure solver, the unsteady flows over airfoil with LFS are simulated by using the CBS-ALE scheme and dual time stepping. The results show that the amplitude oscillation is the key factor to enhance the lift as the self-induced oscillation is used. The large oscillation amplitude of LFS can only be induced in certain frequency range with suitable structure parameters, such as smaller elastic stiffness and damping. The oscillation of LFS can lead to the formation of individual separation bubble and vortex, which are beneficial to the lift enhancement of airfoil. With large oscillation amplitude, vortices with low pressure can be generated early and stay on upper surface of airfoil for a long time, resulting in the significant lift enhancement. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Flexible structures (e.g. shell, plate, shallow arch, membrane) have been widely used in unmanned air vehicles (UAVs) and micro-air vehicles (MAVs), and the lift enhancement, drag reduction and suppression of separation have been observed in many experiments and numerical studies [1–3]. During the flight, unsteady flows and the structures are coupling strongly. The unsteady separated flow caused by flow separation is rather unstable, small perturbation can change the flow structure and airfoil performance dramatically, especially for the flow at low Reynolds number. Therefore, under the unsteady aerodynamic forces, the structures may undergo vibration and deformation. Then, the flow over the structure will be changed by the oscillation of structure and alter the oscillation of the structure in a feedback loop. The coupling between fluid and structure can be affected by many factors, such as Reynolds number, angle of attack, structure parameters, etc., arising a challenge of multi-disciplinary computational techniques and fundamental understanding of unsteady aerodynamics and fluid–structure interaction. † Corresponding

author. Email address: [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2018.06.004

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Pengfei Lei et al. / Journal of Vibration Testing and System Dynamics 2(2) (2018) 155–165

Earlier studies of flexible structure were focused on the modeling and instability of the coupling system, by using potential theory and thin-layer flow models [4]. Later, models and numerical methods of highly coupled fluid-structure systems are investigated inspired by biological flights. Persson et al. [5] proposed a high-order discontinuous Galerkin method to study fluid–membrane interaction on the flapping flight. Chimakurthi et al. [6] presented a computational aeroelasticity framework for the flapping wing micro air vehicles, and investigated both rigid and flexible wing. Gordnier [7] studied the aeroelastic model of a two-dimensional flexible membrane airfoil. He found that the coupling between unsteady leading edge vortex shedding and dynamic response of structure has a positive impact on the aerodynamic performance of airfoil. Rojratsirikul et al. [8] studied unsteady aerodynamics for a two-dimensional membrane airfoil at low Reynolds numbers by experiments, and found the flexible structure can delay the stall. Lee et al. [9] studied the two-dimension insect flapping wing, and found that structural flexibility has a significant impact on aerodynamic performance. Also, some key physical phenomena, such as vortex pairing and vortex staying, were observed. Kang et al. [10, 11] introduced the local flexible structure to control the flow, and improved the aerodynamic performance significantly. Compared with the fully flexible wing, local flexible structure is easier to implement active control, and can be used in normal or large size airfoil. However, the interactions between airfoil and flexible structure are complicated and affected by many flow and structure parameters. Therefore, in this paper, for the flow over airfoil with local flexible structure (LFS), the influence of structure parameters, including elastic stiffness, density and damping, on the aerodynamic performance of airfoil are studied numerically. The analyses is focusing on the influences of structure parameters on the interactions between LFS and the unsteady flow, in order to find the key factors in flow control by using self-induced oscillation of LFS.

2 Aeroelastic model and numerical methods 2.1

Numerical methods for the unsteady flow

Let the chord length of airfoil L, the free-stream velocity U and density of fluid to be the characteristic quantities, the dimensionless variables could be defined as x y Ut ∗ p ui , p = , u∗i = , x∗ = , y∗ = , t ∗ = 2 L L L ρfU U

(1)

where (x, y) is the Cartesian coordinate, t the time, ui the velocity components, p the pressure, ρ f density of fluid, and (*) is dropped hereafter for simplicity. The governing equations for two-dimensional unsteady incompressible flow are Navier-Stokes (NS) equations. To deal with the moving boundary, arbitrary Lagrangian-Eulerian [12] coordinate and mesh moving technique [13] are used. For the unsteady flow induced by LFS, dual time stepping [14] is applied to obtain the accurate solution of unsteady flow at each instantaneous time. Then, the dimensionless N-S equations with dual time stepping in the ALE configuration can be written as

∂ ui = 0, ∂ xi 1 ∂ 2 ui ∂ ui ∂ ui ∂ ui ∂p + (u j − uˆ j ) + =− + , ∂τ ∂t ∂xj ∂ xi Re ∂ x j ∂ x j

(2)

ρ UL

where t is real time, τ pseudo time, uî the velocity of the grid, Reynolds number Re = fμ , μ the viscosity coefficient. At each real time step, the governing equations are solved in pseudo time, by using finite element method with characteristic based split (CBS) scheme. Following the CBS scheme, the new coordinates


157

Fig. 1 Schematics of the shallow arch.

along the characteristic line are applied to the system, then the convective terms can be removed, and the resulting equations become the simple diffusion equations in normal form, which can be solved efficiently by the standard finite element method. The CBS scheme includes three steps: Step 1: Obtain the intermediate velocities u∗i , u∗i − uni = Δt[−c j

1 ∂ 2 ui Δt 2 1 ∂ 2 ui ∂ ∂ ∂ (ui )n + ]+ ( (c j ui )n + ), ck ∂xj Re ∂ x j ∂ x j 2 ∂ xk ∂ x j Re ∂ x j ∂ x j

(3)

where c j = u j − uˆ j . Step 2: Solve the continuity equation,

∂ 2 pn+1 1 ∂ u∗i ∂ 2 pn+θ ∂ 2 pn =θ + (1 − θ ) = , ∂ xi ∂ xi ∂ xi ∂ xi ∂ xi ∂ xi Δt ∂ xi

(4)

where θ ∈ [0, 1], the scheme is implicit as θ = 1, and explicit as θ = 0. Step 3: Correct the velocities with the obtained pressure, − u∗i = −Δt un+1 i

∂ pn+θ . ∂ xi

(5)

Equations (3), (4) and (5) can be approached easily by the standard finite element method (FEM). As the solution in pseudo time line is converged, the steady solution in pseudo time will approach the unsteady solution in real time. The details of CBS can refer to [11, 15]. 2.2

Numerical methods for the local flexible structure

For two-dimensional airfoil, the flexible structure on the surface can be modeled as a shallow arch [16,17], with simply supported boundary conditions, as shown in Fig. 1. For such structure, its self-excited oscillation can be induced by the unsteady aerodynamic forces. In order to couple with fluid solver, the dimensionless structure parameters based on the characteristic quantities are used, and defined as, l h w E F ρs , F∗ = , l ∗ = , h∗ = , w∗ = , ρs∗ = , E ∗ = 2 L L L ρf ρfU ρfU2

(6)

where ρs the density of arch, h the thickness, d the damped coefficient of the structure, E the stiffness of elasticity, l the chord length of arch, F the aerodynamic load. The (*) is dropped for the sake of simplicity. Then, the governing equation of shallow arch can be written as

∂ 2w ∂ w Eh3 + w +V (y0 − w ) = F, + d (7) ∂ t2 ∂t 12 ´l 2 2 where ( ) is partial differential of x, V = 1l 0 [ Eh 2 y − y0 )]dx, y0 the initial position of arch, y the position of arch during oscillation, w = y − y0 . With the simply supported boundary conditions and initial conditions ρs h

w |x=0 = w |x=l = 0,

(8)


158

(a)

(b)

Fig. 2 Verification of numerical methods: (a) Description of the problem and (b) transient response of the flexible bottom at middle point.

∂w = 0. (9) ∂t The linear operator in Eq. (7) has a complete set of eigenfunctions sin nlπ x, n = 1, 2, · · · , +∞ , which spans an orthogonal basis of the space onto which the solution of the governing equation can be projected [17]. So the solution of Eq. (7) can be written in the following form, t = 0 : w = 0,

∞

w(x,t) =

∑ wn(t) sin

n=1

Let G (x, y, w) = ρs h

nπ x. l

∂ 2w ∂ w Eh3 + w +V (y0 − w ) − F, + d ∂ t2 ∂t 12

and following the Galerkin procedure, ˆ l mπ x = 0, m = 1, 2, 3, · · · , ∞ G(x, y, w) sin l 0

(10)

(11)

(12)

the evolution equation for each mode can be obtained as ˆ l ld mπ ρ hl 2 2 w¨ m + w˙ m + g w1 , · · · , wn , w1 , · · · , wn − x)dx = 0, m = 1, 2, 3, · · · + ∞, (13) F sin( 2 2 l 0 where g w1 , · · · , wn , w21 , · · · , w2n indicates the oscillation of LFS is nonlinear. Then, Eq. (13) can be solved by using the fourth-order Runge-Kutta algorithm. The details of the Galerkin method can refer to [17]. In this paper, the first 10 modes are considered. 2.3

Verification of numerical methods

To verify the algorithms mentioned above, the driven open-cavity with flexible bottom, which is described in Fig. 2(a), is simulated. The Reynolds number is 100, and structure parameters are shown in Figure 2(a). The comparison shown in Fig. 2(b) shows excellent agreement between present results and the Bathe and Zhang’s results [18].

3 Numerical setup The NACA0012 is chosen as the model of airfoil. The LFS is mounted on the upper surface of airfoil near the leading edge, and its length is about 0.1L, as shown in Fig. 3. The Reynolds number is set to


159

Fig. 3 Schematics of airfoil with LFS. Table 1 Structure parameters for different cases. Cases

h

ρs

d

E

α

1

0.001

2000∼8000

0.5

8×104

6◦

2

0.001

5850

0.5

1×104 ∼1×106

6◦

3

0.001

5850

0∼7

8×104

6◦

Table 2 Lift and drag coefficients of rigid airfoil obtained by different meshes for α = 6◦ . Structural segments on the airfoil

Lift coefficient

Drag coefficient

Mesh1

400

0.28821

0.0775

Mesh2

600

0.28916

0.07818

Mesh3

800

0.28965

0.07823

Fig. 4 Computational domain and mesh of flow around airfoil.

be 5000, and the angle of attack is 6◦ . The structure parameters of various cases are shown in Table 1. The computational domain is meshed by triangular elements. In order to test the independency of the results on mesh quality, numerical simulations of rigid airfoil are conducted with 400, 600 and 800 structural segments on the rigid airfoil at α = 6◦ . By comparison of the lift and drag coefficients shown in Table 2, it can be seen that the mesh with 600 segments on airfoil, as shown in Fig. 4, is suitable for the simulations.

4 Results and analysis 4.1

Effects of structure parameters on the performance of airfoil

The natural frequency of the structure is strongly affected by the structure parameters. Under the unsteady aerodynamic load, the behaviors of LFS with different structure parameters are differ. Therefore, by studying the influence of structure parameters on the dynamics of fluid and structure, the nature of interaction between fluid and structure can be revealed. Three structure parameters, namely, elasticity stiffness, density and damping of structure, are considered in this section.

160


(a)

(b)

(c)

Fig. 5 Comparison of (a) lift coefficient, (b) oscillation amplitude of first three modes and (c) main frequency of fluid with different elasticity stiffness.

4.1.1

Elasticity stiffness

Figure 5 shows the variation of lift coefficient, oscillation amplitude and main frequency of flow for different elasticity stiffness. The oscillation amplitude of ith mode Ai is the difference of maximum and minimum values of wi , Ai = wi,max − wi,min , and the frequency of flow is obtained by spectral analysis of time history of lift coefficient. Clearly, oscillation of LFS with smaller elasticity stiffness has larger amplitude, and results in higher lift coefficient correspondingly. For cases with E ≤ 1 × 105 , the lift enhancement is remarkable, and the oscillation of LFS has large amplitude A1 > 0.002, and the corresponding frequency of fluid ( f < 1.4) is smaller than the one of rigid airfoil. As the elasticity stiffness decreases further, the amplitude of oscillation increases sharply. For the case with E = 1 × 104 , the amplitude of oscillation is ten times larger than the one with E = 5 × 104 , while the lift coefficient only increases by 6%. Considering the strength of the structure, the elasticity stiffness should be limited in certain range. As elastic stiffness is increased from E = 1 × 105 to E = 2 × 105 , the lift coefficient drops due to insufficient perturbation. However, lift coefficient increases again as elastic stiffness is increased from E = 2 × 105 to E = 6 × 105 . The frequencies of flow for these cases are near 1.8, larger than the one of rigid airfoil. Compared with the cases with E = 1 × 105 , the oscillation of LFS with E = 4 × 105 has smaller amplitude of oscillation and higher lift. When the elasticity stiffness is increased further, the amplitude of oscillation will tend to zero, and lift coefficient of airfoil will approach the same as the one of rigid airfoil. 4.1.2

Density

Figure 6 shows the variation of lift coefficient, oscillation amplitude and frequency of fluid for different structure density. There are two peaks on the curve of lift coefficient, similar with the one for different elasticity stiffness shown in Fig. 5. For the case with ρs = 6000, the oscillations of LFS have large amplitude, and the lift coefficient of airfoil increases more than 50%, the corresponding frequency of flow is about 1.2∼1.4. For the case with ρs = 3000, the oscillation frequency is near 1.8 and the oscillation of the LFS with smaller amplitude can gain better lift enhancement, compared with the case with ρs = 5500.


(a)

161

(b)

(c)

Fig. 6 Comparison of (a) lift coefficient, (b) oscillation amplitude of first three modes and (c) main frequency of fluid with different density.

(a)

(b)

(c)

Fig. 7 Comparison of (a) lift coefficient, (b) oscillation amplitude and (c) frequency of fluid with different damping.

4.1.3

Damping

Figure 7 shows the variation of lift coefficient, oscillation amplitude and frequency of fluid for different damping. With smaller damping, the oscillation of LFS is amplified, resulting in large amplitude of oscillation and high lift coefficient. The frequencies of flows with high lift coefficient are also near the range 1.2∼1.4. In conclusion, the key to enhance the lift by self-induce oscillation of LFS are the large amplitude of the oscillation of LFS. However, the oscillation amplitude of LFS is strongly affected by the struc-

162


(a)

(b)

(c)

(d)

Fig. 8 Spectral analysis of lift coefficient and amplitudes of first three modes of LFS for various cases. (a) E = 1 × 104, (b) E = 5 × 104, (c) E = 1.5 × 105, (d) E = 4 × 105.

ture parameters, and the oscillation with large amplitude can only be induced in certain frequency ranges. Therefore, only with suitable structure parameters, the oscillation with large amplitude and enhancement of lift can be obtained. Two types of oscillation that can enhance the lift are observed. The one with large amplitude (A1 > 0.002) occurs as the frequency of unsteady flow ( f ≈ 1.2 ∼ 1.4) is lower than the frequency of rigid airfoil, such as the case with E = 5 × 104 . The other one with small amplitude (A1 < 0.002) occurs as the frequency of unsteady flow ( f ≈ 1.8) is larger than the frequency of rigid airfoil, such as the cases with E = 4 × 105 . 4.2

Effects of structure parameters on the performance of airfoil

In order to understand the nature of interaction of fluid and structure, the cases with E = 5 × 104 and E = 4 × 105 , corresponding to the two types of oscillations, are studied in details, and compared with other cases in this section. 4.2.1

Oscillation of LFS

Figure 8 shows the spectral analyses of lift coefficient and amplitude of first three modes of LFS for cases with different elastic stiffness. It can be seen that the frequencies of LFS and fluid are closely correlated for all the cases with different flow frequencies, regardless the amplitudes of oscillation and effects of lift enhancement. The increasing of oscillation amplitude for some cases indicates that the


(a)

163

(b)

Fig. 9 Mean flow structures (a) and pressure distributions (b) for cases with various elastic stiffness.

(a)

(b)

Fig. 10 Instantaneous flow structures (a) and pressure distributions (b) at maximum lift coefficient for cases with various elastic stiffness.

synchronization occurs between the unsteady aerodynamic load and natural frequency of LFS. For case with E = 1.5 × 105 , the frequency of the unsteady aerodynamic force cannot synchronized with the natural frequency of LFS, and the resulting oscillation with small amplitude is insufficient to change the flow. Therefore, only with suitable structure parameters, the strong interaction between the fluid and structure can be induced to enhance the lift. However, both LFS and fluid are complicated nonlinear systems, the interaction between them still needs to be studied further. 4.2.2

Unsteady flow

The time-averaged flow structures and pressure distributions for cases with various elastic stiffness are shown in Fig. 9. It can be seen that smaller separated area indicates higher lift coefficient of airfoil. There are two contributions of LFS on the lift coefficient. First, the deformation of LFS induces camber effect at the leading edge of airfoil, resulting in the low pressure near the LFS. Second, near the separation bubble, the flow structure induced by the oscillation of LFS reduces the pressure on the upper surface of airfoil further. Figure 10 shows the corresponding instantaneous flow structures and pressure distributions as the lift coefficient reaches the maximum value. The flow structures of airfoil with LFS are different with the one of rigid airfoil. Individual separation bubbles with smaller scale appear on the upper surface of airfoil, and vortices, which have concentration of vorticity and low pressure, are generated near the separation bubbles. At the leading edge, the camber effect creates strong adverse pressure gradient, and

164


the flow separation can be induced earlier. Then, by the oscillation of LFS, pressure gradient fluctuates periodically to form individual separation bubbles and vortices. As a result, with larger amplitude of oscillation, the vortices induced by the oscillation of LFS can stay longer on the upper surface of airfoil, resulting in the decreasing of pressure on the upper surface of airfoil and enhancement of lift.

5 Conclusions The influences of structure parameters are studied numerically. The results show that the key factor for flow control by using self-induced oscillation of LFS is the amplitude of oscillation. During the self-induced oscillation, the frequencies of LFS and the fluid are correlated regardless the variations of structure parameters. However, the amplitude of oscillation can only be amplified in certain frequencies, which strongly affected by the structure parameters. Conclusively, flexible structure with suitable structure parameters can enhance the lift of airfoil greatly. The oscillation of LFS can induce the formation of individual separation bubble and vortex with concentration of vorticity, which are beneficial to lift enhancement. With larger oscillation amplitude, flow separation occurs earlier, and vortices with low pressure can stay longer on upper surface of airfoil, resulting in the significant lift enhancement.

Acknowledgement The research is supported by the National Fundamental Research Program of China (No. 2012CB026002) and the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (No. 2013BAF01B02).

References [1] Shyy, W., Aono, H., Chimakurthi, S.K., Trizila, P., Kang, C.K., Cesnik, C.E., and Liu, H. (2010), Recent progress in flapping wing aerodynamics and aeroelasticity, Progress in Aerospace Sciences, 46, 284-327. [2] Yu, M., Wang, Z., and Hu, H. (2013), High fidelity numerical simulation of airfoil thickness and kinematics effects on flapping airfoil propulsion, Journal of Fluids and Structures, 42, 166-186. [3] Chakravarty, U.K. and Albertani, R. (2012), Experimental and finite element modal analysis of a pliant elastic membrane for micro air vehicles applications, Journal of Applied Mechanics, 79, 021004. [4] Dowell, E.H. and Hall, K.C. (2001), Modeling of fluid-structure interaction, Annual Review of Fluid Mechanics, 33, 445-490. [5] Persson, P.O., Peraire, J., and Bonet, J. (2007), A high order discontinuous galerkin method for fluid-structure interaction, 18th AIAA Computational Fluid Dynamics Conference, Reston, Virigina, USA. [6] Chimakurthi, S.K., Tang, J., Palacios, R., Cesnik, C.E.S., and Shyy, W. (2009), Computational aeroelasticity framework for analyzing flapping wing micro air vehicles, AIAA Journal, 47, 1865-1878. [7] Gordnier, R.E. (2009), High fidelity computational simulation of a membrane wing airfoil, Journal of Fluids and Structures, 25, 897-917. [8] Rojratsirikul, P., Wang, Z., and Gursul, I. (2009), Unsteady fluid–structure interactions of membrane airfoils at low reynolds numbers, Experiments in Fluids, 46, 859-872. [9] Lee, K.B., Kinm, J.H., and Kim, C. (2011), Aerodynamic effects of structural flexibility in two-dimensional insect flapping flight, Journal of Aircraft, 48, 894-909. [10] Kang, W., Zhang, J.Z., and Liu, Y. (2010), Numerical simulation and aeroelastic analysis of a local flexible airfoil at low reynolds number, The 8th Asian Computational Fluid Dynamics Conference, HongKong. [11] Kang, W., Zhang, J.Z., Lei, P.F., and Xu, M. (2014), Computation of unsteady viscous flow around a locally flexible airfoil at low reynolds number, Journal of Fluids and Structures, 46, 42-58. [12] Nithiarasu, P. (2005), An arbitrary lagrangian eulerian (ale) formulation for free surface flows using the characteristic-based split (cbs) scheme, International Journal for Numerical Methods in Fluids, 48, 14151428.


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[13] Batina, J.T. (1990), Unsteady euler airfoil solutions using unstructured dynamic mesh, AIAA Journal, 28, 1381-1388. [14] Gaitonde, A.L. (1998), A dual-time method for two-dimensional unsteady incompressible flow calculations, International Journal for Numerical Methods in Engineering, 41, 1153-1166. [15] Wang, Y.T. and Zhang, J.Z. (2011), An improved ale and cbs-based finite element algorithm for analyzing flows around forced oscillating bodies, Finite Elements in Analysis and Design, 47, 1058-1065. [16] Zhang, J.Z. and van Campen, D.H. (2003), Stability and bifurcation of doubly curved shallow panels under quasi-static uniform load, International Journal of Non-linear Mechanics, 38, 457-466. [17] Zhang, J.Z., Liu, Y., Lei, P.F., and Sun, X. (2007), Dynamic snap-through buckling analysis of shallow arches under impact load based on approximate inertial manifolds, Dynamics of Continuous Discrete and Impulsive Systems-Series B-Applications & Algorithms, 14, 287-291. [18] Bathe, K.J. and Zhang, H. (2009), A mesh adaptivity procedure for cfd and fluid-structure interactions, Computers & Structures, 87, 604-617.



Continuous Stabilizing Control for a Class of Non-Holonomic Systems: Brockett System Example Fazal Ur Rehman1 , A Baseer Satti1,2†, A. Ahmed Saleem1 1 2

Department of Electronic Engineering, Capital University of Science and technology, Pakistan Holmes Institute Melbourne, Australia Submission Info Communicated by S. Lenci Received 11 September 2017 Accepted 28 February 2018 Available online 1 July 2018 Keywords Lyapunov stability Brockett system non-holonomic system Adaptive back-stepping control

Abstract A continuous adaptive controller is designed for a famous example of Brockett system which is an example of a class of non-holonomic systems. The controllability Lie Algebra of the Brockett’s system contains Lie brackets of depth one. It is shown that the closed loop system is globally uniformly asymptotically stable. The advantage of this method is that it does not require the conversion of the system model into a “chained form,” and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The feedback control strategy that has been proposed in this paper applies to systems of the type: m

z˙ = ∑ gi (z) ui , i=1

z ∈ ℜn ,

(1)

where gi , i = 1, ..., m, m < n are linearly independent vector fields on ℜn , ui are piece-wise continuous and locally bounded in t, control functions defined on the interval [0, ∞). These systems are called nonholonomic systems. The stabilization of these systems has been a hot topic for the researchers over the last decade. Such systems happen repeatedly in practice and normally represent models of mechanical systems that have non-integrable velocity constraints. The non-holonomic systems are either designed with fewer actuators than the degrees of freedom or must be able to function in the presence of actuator failures. The synthesis of control laws for these systems is a considerable challenge because as point out by Brockett [1] in his famous paper that these systems cannot be stabilized by smooth or continuous static state-feedback laws and that the dependence of the stabilizing control on time is essential [2]. † Corresponding


Fazal Ur Rehman et al. / Journal of Vibration Testing and System Dynamics 2(2) (2018) 167–172

168

It was a considerable challenge to prevail over the limitations that were imposed by Brockett in his paper [1]. A lot of work has been done in this regard to stabilize these non-holonomic systems to equilibrium points. A detailed survey can be found in [3]. The proposed methods can be classified into smooth time varying controllers [4–6], discontinuous or piecewise smooth control laws [7, 8] and hybrid controllers [9, 10]. Our interest in this paper is to propose a simple method for the construction of stabilizing control for three different non-holonomic drift free systems. The proposed method depends on adaptive back stepping technique [11–13] with the objective of steering the system from any arbitrary initial state to any desired state. This approach does not necessitate conversion of the system model into a “chained form”, and thus does not rely on any special transformation techniques. The simulations show the effectiveness of the designed controller.

2 The control problem • (SP): Given a desired set point xdes ∈ ℜn , construct a feedback strategy in terms of the controls ui : ℜn → ℜ, i = 1, 2, ...., m such that the desired set point xdes is an attractive set for Eq.(1), so that there exists an ε > 0, such that x(t; 0, x0 ) → xdes , as t → ∞ for any initial condition x0 ∈ B(xdes ; ε ). Without the loss of generality, it is assumed that xdes = 0, which can be achieved by a suitable translation of the coordinate system.

3 Brockett’s system 3.1

The Kinematics model

This is a non-holonomic drift free system (three dimensional) having a control deficit of order one and its controllability lie algebra contains lie bracket of depth one. The kinematics model of the Brockett’s system is given as [14]: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 1 x˙1 ⎣ x˙2 ⎦ = ⎣ 0 ⎦ u1 + ⎣ 1 ⎦ u2 , x˙3 −x1 x2 x˙ = g1 (x)u1 + g2 (x)u2 , x ∈ ℜ3 , where

(2)

⎡

⎡ ⎤ ⎤ 1 0 g1 (x) = ⎣ 0 ⎦ & g2 (x) = ⎣ 1 ⎦ . x2 −x1

The kinematics model (2) has the following important properties: • (P1) The vector fields g1 (x) & g2 (x) are linearly independent. • (P2) System (3) satisfies the LARC (Lie algebra rank condition) for accessibility, namely that L(g1 , g2 , g3 ), the Lie algebra, L(g1 , g2 , g3 )(x) spans ℜ3 at each point x ∈ ℜ3 . To verify property P2, it is sufficient to calculate the following Lie brackets of g1 (x) and g2 (x): ⎡ ⎤ 0 de f g3 (x) = [g1 , g2 ](x) = ⎣ 0 ⎦ , −2 which satisfy the LARC condition: span{g1 , g2 , g3 }(x) = ℜ3 , ∀x ∈ ℜ3 .


3.2

169

Controller design

The Brockett’s system given in (2) can be rewritten as: x˙1 = u1

(a)

x˙2 = u2

(b)

(3)

x˙3 = x2 u1 − x1 u2 (c) Consider the equation (3a) and by defining u1 = x2 the equation (3a) becomes: x˙1 = x2 .

(4)

Now by considering x2 as the virtual control, α1 as the stabilizing function and z1 = x2 − α1 be the error variable, equation (4) can be rewritten as: x˙1 = z1 + α1 .

(5)

To work out α1 , consider the Lyapunov function: V0 = 12 x21 for (5). Then, V˙0 = x1 x˙1 = x1 (z1 + α1 ). By choosing α1 = −x1 , the above equation becomes: V˙0 = −x21 + x1 z1 . Equation (5) becomes, x˙1 = z1 − x1 .

(6)

Consider the equation (3b), and choose u2 = x3 + θ (t) where θ (t) = θˆ (t) − θss + θ˜ (t). θˆ (t) and θ˜ (t) are some-time varying functions which will be determined independently, while θss is the steady state value of θˆ (t). Then equation (3b) becomes: (7) x˙2 = x3 + θ (t). Now by considering x3 as the virtual control, α2 as the stabilizing function and z2 = x3 − α2 be the error variable, equation (7) can be rewritten as: x˙2 = z2 + α2 + θˆ (t) − θss + θ˜ (t). Since z1 = x2 − α1 = x2 + x1 its dynamics can be written as: z˙1 = x˙2 + x˙1 = z2 + α2 + θˆ (t) − θss + θ˜ (t) + z1 − x1 .

(8)

To work out α2 , consider the Lyapunov function: V1 = V0 + 12 z21 for (5) & (8). Then, V˙1 = −x21 + z1 (z2 + α2 + θˆ (t) − θss + z1 ) + z1 θ˜ (t). By choosing α2 = −2z1 − θˆ (t) + θss V˙1 = −x21 − z21 + z1 z2 + z1 θ˜ (t). Equation (8) becomes:

z˙1 = z2 − z1 − x1 + θ˜ (t).

Consider the equation (3c): x˙3 = x2 u1 − x1 u2 x˙3 = x22 − x1 (z2 − 2z1 + θ˜ (t)).

(9)

170


Since z2 = x3 − α2 = x3 + 2z1 + θˆ (t) − θss its dynamics can be written as: z˙2 = x˙3 + 2˙z1 + θ˙ˆ (t) = x22 − x1 z2 − 2x1 z1 − x1 θ˜ (t) + 2z2 − 2z1 − 2x1 + 2θ˜ (t) + θ˙ˆ (t)

(10)

Consider the Lyapunov function: V2 = V1 + 12 z22 + 12 θ˜ 2 (t) for (5), (8) & (10). Then, V˙2 = −x21 − z21 + z2 (x22 − x1 z2 − 2x1 z1 + 2z2 − z1 − 2x1 + θ˙ˆ (t)) + θ˜ (t)(z1 + 2z2 − z2 x1 + θ˙˜ (t)). By choosing

θ˙ˆ (t) = −x22 + x1 z2 + 2x1 z1 − 3z2 + z1 + 2x1 , θ˙˜ (t) = −z − 2z + z x − θ˜ (t), 1

2

2 1

V˙2 = −x21 − z21 − z22 − θ˜ 2 (t). Equation (10) becomes,

z˙2 = −z2 − z1 + c1 θ˜ (t)

where c1 = 2 − x1 .

(11)

The closed loop system becomes: x˙1 = z1 − x1 ,

z˙1 = z2 − z1 − x1 + θ˜ (t), z˙2 = −z2 − z1 + c1 θ˜ (t).

(12)

Since x1 , z1 , z2 → 0 & θˆ (t) → θss x2 = z1 − x1 → 0, x3 = z2 − 2z1 − θˆ (t) + θss → 0. It is evident from the results that a virtual control state is firstly defined and then it is forced to become a stabilizing function, which a corresponding error variable is generated. Consequently, by appropriate designing the related control input by virtue of Lyapunov stability theory, the error variable is stabilized. For convenience, we firstly defined the model with the parameter and disturbance uncertainties, the final time derivative of the Lyapunov function does not include any positive definite term; which implies that there is no limitation on choices of the feedback gains. Consequently, it is clearly deduced from (12) that the asymptotic stability of the overall control system does not depend on that the feedback gains satisfy any inequality or condition, but, it should be noted the feedback gains are responsible for the tracking performance while the adaptation gains specify adaptation performance.

4 Simulations In this section the graphical analysis of the transformed closed loop system (12) for the Brockett system example can be observed using the adaptive back stepping technique. The simulations of the models are given below. It can be seen that all the states of the system are going to zero. The aim was to steer them to a desired value which was assumed to be zero. The controller designed above guarantee that in the presence of uncertain bounded nonlinearities the closed loop systems (12) remains bounded. Simulation results demonstrates that in our proposed method the uncertainties are more specific. They consist of unknown constant parameters which appear linearly in the system equations (3). In the presence of such parametric uncertainties we have achieved both boundedness of the closed loop states and convergence of the tracking error to zero.


(a)

171

(b)

Fig. 1 The asymptotic stability and the boundedness of the 3 states of the Brockett system starting with initial condition 0.5 with error function approaching to zero in time: (a) phase portrait, (b) error responses.

Fig. 2 Representing the stability of states of the Brockett system over longest interval of time from 0 to 500.

5 Conclusion In this paper, a systematic method for the construction of steering control for the Brockett system is introduced without transforming into “chain form” using adaptive back stepping technique. The main objective was to steer the system from any initial condition to a desired state. The proposed controller has achieved the desired purpose which is evident from simulation results.

172


References [1] Astolfi, A. (1998), Discontinuous control of the Brockett integrator, European Journal of Control, 4(1), 49-63. [2] Behal, A., Dawson, D., Dixon, W., and Fang. Y. (2002), Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics, IEEE Transactions on Automatic Control, 47(3), 495-500. [3] Branicky, M.S. (1998), Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43(4), 475-482. [4] Brockett, R.W. (1983), Asymptotic stability and feedback stabilization, Differential Geometric Control Theory (Birkhauser, Boston, USA) (R. W. Brockett, R. S. Millman, and H. J. Sussman, eds.), 181-191. [5] Campion, G., Bastin, G., and d’Andrea-Novel, B. (1996), Structural properties and classification of kinematics and dynamic models of wheeled mobile robots, IEEE Trans. on Robotics and Automation, 12(1), 47-62. [6] Coron, J.M. (1991), Global asymptotic stabilization for controllable systems without drift, Mathematics of Control, Signals, and Systems, 5(3), 295-312. [7] Duleba, I. and Sasiaek, J. (2001), Calibration of controls in steering nonholonomic systems, Control Engineering Practice, 9, 217-225. [8] Guldner, J. and Utkin, V.I. (1994), Stabilization of nonholonomic mobile robots using Lyapunov functions for navigation and sliding mode control (Orlando, Florida, USA), 2967-2972. [9] Kolmanovsky, I. and McClamroch, N.H. (1995), Developments in nonholonomic control problems, IEEE Control Systems Magazine, 15, 20-36.



Remaining Useful Lifetime Prediction of Gas Turbine Bearings Based on Experiment Vibration Signals Data Boulanouar Saadat1 , Ahmed Hafaifa1† , Ali Bennani1,2 , Nadji Hadroug1 , Abdellah Kouzou1 , Mohamed Haddar2 1 2

Applied Automation and Industrial Diagnostics Laboratory, Faculty of Science and Technology, University of Djelfa 17000 DZ, Algeria L2MP Laboratory, National School of Engineers of Sfax, Tunisia. Submission Info Communicated by S.C Suh Received 14 March 2018 Accepted 1 April 2018 Available online 1 July 2018 Keywords Prognosis Failure prediction Residual life estimation Vibration data signals Gas turbine Bearing vibration

Abstract Gas turbines are widely found in industries, especially in gas compressor stations, where they are used to ensure the transportation of gas in high pressure pipelines over a long distance. Nevertheless, these rotating machines are subject to vibrations due to the influence of several causes, such as mass imbalance and shaft misalignment. These vibrations can affect directly the bearings, where their lifespan can be shorten remarkably. Furthermore they can be damaged causing catastrophic failure in the gas turbine and its installation. The main purpose of this paper is to develop an approach that can provide the estimated remaining life of the gas turbine bearings based on the vibration signals obtained via installed sensors. This approach allows us to estimate the optimal expected life of the studied gas turbine bearings by the prediction of the expected failures times. The obtained result under the proposed approach is satisfactory and shows that the use of the proposed approach can avoid the costly damage caused by the mentioned failures in the gas turbine under the imposed constraints. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Currently, the diagnostic approaches provide a set of robust tools to optimize the monitoring operations of heavy industrial equipment. However, maintenance of industrial processes is based on the active renewing of the components of such equipment using the surveillance techniques to obtain their states on real time. Indeed, these used techniques provide valuable information on the health status of the equipment, the current performance indexes and can furthermore help in predicting the expected future operation index of the equipment. In this context, this paper proposes a prognosis model based on the estimated Remaining Useful Lifetime (RUL) of the studied gas turbine bearings using the obtained vibration signals from the real operational status monitoring data model obtained onsite for a sufficient † Corresponding


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period of lifetime, where the main aim is to predict the failure of this equipment and its residual operating time. In fact, these machines are still subject to vibration caused by the misbalancing, the misalignment defects, the imperfections of the bearings and the effects of the gas flow pulses. These vibrations can shorten the useful lifetime of the gas turbine components, that can lead moreover to catastrophic failures and damages in the whole gas turbine system. Faced to this problem, this work proposes the development of a prognostic approach to measure the degradation and the damage indicators in real time of this type of rotating machine based on a series of measurements and observations of their vibration behavior. This permits to achieve the modeling of the deterioration causes based on the remaining useful lifetime estimation using the vibration signals. Recently, diagnostic systems and predictions have been widely adapted to various industrial gas turbine applications in order to find predictive solutions for conditional maintenance problems for these industrial processes; Do Won Kang and Tong Seop Kim in [1] have proposed a new model using diagnostics performance of heavy-duty gas turbines using compressor map adaptation; Xiang Li et al. in [2] have used neural networks for the estimation of the remaining useful life to elaborate a new prognostics approach. Elias Tsoutsanis et al. in [3] have studied a dynamic prognosis system for smooth operation of gas turbines. Martha A. et al. have used the prediction of gas turbine engines using bayesian hierarchical models type with a variational approach [4, 5]. Poursaeidi E. et al. in [6] have studied the emergency effects and the transient thermal fatigue lifetime of a gas turbine. Datsyshyn O.P. et al. in [7] have proposed a model of the residual value for the estimated lifetime of the elements of a tribojoint by training criteria typical of contact fatigue damage. Other works have been conducted on the gas turbine diagnosis using the approaches of artificial intelligence; Abdelhafid Benyounes et al. [8,9] have used the fuzzy logic for modeling the gas turbine vibrations, Mohamed Ben Rahmoune et al. [10,11] have proposed a diagnostic system for the gas turbine based on neural networks for vibrations faults detection. On the other hand, many applications and works on industrial gas turbines were developed. Sandy Rahme and Nader Meskin in [12] have presented an adaptive system for observing the operation mode of a gas turbine based on a sensor fault diagnosis system. Diyin Tang et al. in [13], have proposed an optimal maintenance policy with the estimation of the remaining useful lifetime for a slowly degrading system. Martha A. et al. in [14] have carried out a prediction system for a gas turbine with an integrated approach. In 2014, Vincenzo Cuffaro et al. proposed an advanced method for lifetime estimation of a gas turbine for the assessment of their components [15]. In 2012 Xiong Zhao-fei et al. presented a diagnosis system for gas turbines defects using the concept of fuzzy logic [16]. In 2011, Sikorska J.Z. et al. presented a prognosis modeling approach to estimate the remaining useful lifetime by the industry [17]. In 2006 Afgan N. H. et al. proposed an expert system for the diagnosis and monitoring of the state of the combustion chambers in gas turbines [18]. In 2009 Li Y.G. and Nilkitsaranont P. proposed a prognosis system for improving the performance of the gas turbine and their condition monitoring [19]. Taking into account the impacts of maintenance based on the lifetime cycle assessment of industrial turbines, early decisions of the acquisition cycle profoundly affect the life cycle cost of the whole installation. Indeed, the proposed prognosis system in this work takes into account all the phases of the life cycle processes and determines the remaining useful lifetime of the studied gas turbine bearings based on the vibration signals. This approach allows to estimate the optimal expected lifetime of the studied gas turbine bearings, to predict their failures and to propose the techniques that give the best modeling of the degradation of the gas turbine system. where the main aim is to increase their safety and to reduce the future decisions affecting the state of operation of this industrial equipment. This prognostic approach has the advantage of predicting the evolution of equipment degradation and following the gas turbine performance dynamic monitoring. The obtained results show the effectiveness of this approach in order to avoid the deterioration of the gas turbine performance which is proved based on the robustness test and the vibration indicators analysis of the studied gas turbine system


175

2 Residual life time modeling The prognostic is an engineering technique which is involved in the maintenance processes in many industrial applications [13,20–22]. It is used to predict the future health state or behavior of an equipment or system. Furthermore, it is used precisely for the estimation of the time to failure of an operating industrial system, where an analysis and a prediction of the Remaining Useful Life (RUL) are needed. Hence, the development of models for uncertainty representation, quantification and management of failures for the industrial system are primordially required. The primary objective of the present work is to present a prognosis model based on the monitoring data obtained from an in-operation gas turbine to predict its failure and the remaining service life. Indeed, the analysis of damage in industrial equipment is based on the series of measurements and the observations of anomalous phenomena which allows the modeling of the damage causes. This is the first step of the monitoring which is the stage of the diagnosis based on the assessment of damage indicators obtained from practical measurements. After this stage, the prognosis phase comes, which is the stage of evaluation of the remaining useful life (RUL) based on the degradation laws and the degradation indicators that have been determined in the first step of the monitoring system, as shown in Figure 1. Practically, it is important to understand that the future states and other life predictions are simply depending on various uncertainties characterized by the previous step. The objective is to estimate the remaining useful life (RUL) of a system which is by definition the remaining operating time of the system before its failure [2, 7, 12, 23, 24]. In the present paper a model of the remaining useful time estimation of a gas turbine is developed based on vibrations measurement, then a validation test under bearing damage of the gas turbine is performed. In this area, several techniques have been developed so far for the determination of the dynamic degradation model of an industrial system based on mathematical functions. These techniques have been used the approach of the remaining useful life estimation [2,7,25,26]. In this case, the degradation phenomena is depending on the reliability of the studied machine and it is expresses as follows: RΘ (t|X1 , ..., Xn ) = PΘ (X (t) ≺ L|X1 , ..., Xn ),

(1)

where (X1 , ...Xn ) are presenting the state of the system, Θ is presenting the usage conditions of the studied machine. The failure is based on a threshold L representing the level of damage which is depending also on the reliability of the studied machine. In the case of a degraded mode of operation caused by a fault, an accidental vibration peak due to any constraints and any other kind of equipment degradation at the same time over several observation time TS , the reliability of this machine can be presented as follows: RΘ (t|X1 , ..., Xn ) = PΘ ((X (t) ≺ L) ∩ (TS t)|X1 , ..., Xn ).

(2)

This equation represents the reliability law of the studied machine, in this equation the Ts is the sampling time of the observation. Practically the studied machine can operate up to the stability limits, to ensure a good operating efficiency. For this purpose, this equation is useful for the operators for the evaluation of the machine reliability and the eventual maintenance schedules. The degradation process depends on the number of the accidental vibration peaks applied to the equipment [27–30]. Therefore, the evaluation of the RUL requires the precise knowledge of the number of the accidental vibration peaks n occurring before the current time t0 . Depending on the operation mode of the machine the remaining useful time (RUL) can be estimated. For the development of analytical models of the remaining useful lifetime of industrial equipment, a reliability model is needed to be developed. Although, all the steps of the analytical modeling of the remaining useful time have to be achieved. It can be said that the prognosis is the ability to provide reliable estimation of future health of industrial equipment or a component of the equipment based on

176


Fig. 1 Degradation stages measured by residual lifetime (RUL).

Fig. 2 Gas turbine MS 3002 rotor with symmetric isotropic flexible bearings.


177

the diagnosis of the current status, the history of failures and on the maintenance operations, to give a precise and nearly perfect estimation of the remaining useful lifetime of the system. This estimation task of the remaining useful lifetime is an indispensable challenge to ensure the safety and the sustainability of the industrial system normal operation. In the following section, an estimation of the remaining useful lifetime of a gas turbine under a degraded mode is investigated.

3 Gas turbine remaining useful lifetime using vibratory signals The dynamic behavior of a gas turbine can be studied through the vibration signal measurement obtained from the vibration sensor placed on the bearing No. 01 of the studied gas turbine MS 5002, as shown in Figures 2. In order to increase the monitoring capabilities of the vibration behavior of the studied gas turbine. This work proposes to integrate a prognostic approach to predict the degradation evolution and to follow the gas turbine behavior. Indeed, the complexity and the dynamic behavior of gas turbine systems increases the difficulty in obtaining a degradation model of this type of machine. The obtained results are presented in the last section of this work, to show the effectiveness of this approach, in order to avoid the gas turbine performance deterioration. The methods of extrapolation and interpolation that are widely used in signal processing techniques, can be used to estimate the value of a discrete-time signal x[t] at the actual time (t) and for future times based on past events values. Considering (x) as the actual reached lifetime of the studied machine, and note (Tx ) as its remaining useful lifetime from the actual time. Thus, the machine is expected to fall into failure at the time (x + Tx ). The remaining useful lifetime (Tx ) is a random variable, where the probability distribution of (Tx ) is characterized by a survival function which is expressed as follows: ptx = P [Tx t] .

(3)

t/t

The term px presents the probability of failure between the instants (t) and (t + t ) of the studied gas turbine, taking into account the lifetime (x) and ptx . The probability of failure can be expressed as a function of the probability of the survival function, as follows: t/t px = P t ≺ Tx ≺ t + t = P [t ≺ Tx ] − P t + t ≺ Tx . So,

t/t

t/t

px = ptx − px .

(4)

(5)

Assuming that the gas turbine operate 24h/24h (the requirement of the oil industry) over a period of 730 days (two years), this time is corresponding to the nominal lifetime or the warranty period declared by the manufacturer, this duration is equivalent to 17520 hours. The random variable (Tx ) which represents the lifetime in days of the high pressure (HP) wheel of the gas turbine follows the 1 . The probability that the gas turbine operates without exponential law of the parameter λ = 730 breakdown for a duration of 20 days after the nominal lifetime is expresses as follows: 20

−λ .20 = e− 730 = 0.973. p750 x (Tx ≥ 750) = e

This period is chosen following the RUL which is explained furthermore in the third phase in section (4.3). In this study, the degradation due to vibration starts after the nominal lifetime (730 days), where the limit of the remaining expected lifetime of gas turbine is estimated to be 63.33 days. In this study, the Lagrange polynomial interpolation method is used for the prediction of the vibration behavior during the presented phases. The Lagrange interpolating polynomial is the polynomial L(x) of degree (≤ n − 1) which passes through the n points (xi , yi for i = 1, 2, ..., n), in this work these

178


points are presenting the real data obtained onsite from the vibration sensors. It is expressed generally as follows [31]: n

L(x) = ∑ yi li (x).

(6)

i=1

Where l j (x) is the Lagrange polynomial which is presented as follows: x − xj . j=1 xi − x j n

li (x) = ∏

(7)

j=i

This formula was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 [31]. Its degree is (≤ n − 1), it has to satisfy the following condition: li (xi ) = 1, li (x j ) = 0.

(8)

The objective of being using the Lagrange polynomial interpolation in the present work, is to find a mathematical model of the vibration behavior along a defined duration of time. This interpolation is obtained based on the set of pairs (xi , yi ) obtained from experimental data, which is presenting the results of signals measurement through the used vibration sensors. Indeed, the interpolation polynomial consists on approximating the behavior of vibration by a function which is passing through all the n points and it is presented by a unique polynomial of degree (n − 1). In the same way the Lagrange polynomial is used for the extrapolation during the different phases of this study due to its simplicity and its flexibility towards the polynomial degree. The time remaining before a failure (RUL) is a random variable at a point (x p ) that exceeds a danger threshold that the level of shutdown of the turbine. It can be written by the expression of the remaining useful lifetime (RUL), as follows: RU L = sup {t : Ai (t) ≤ (1 − a)} , n+1

RU L =

∑ αi xi ,

(9)

i=1

where A, a the amplitude of vibration and their variation respectively and ∑n+1 i=1 αi = 1 with αi ≥ 0, n the degree of the polynomial and α = x1 ≺ x2 ≺ ... ≺ xn = β in the interval [α , β ], and Ai (t) ≤ (1 − a) the vibration amplitude of this turbine in the case of normal operation to less than 1 (industrial standard). The amplitude (An−1 ) as a critical point of our system and the failure of amplitude (An ) associated with all values (n + 1) related to the interpolation node, the corresponding polynomial of degree (n) is determined by : ∏n0 n ∞ , (10) An = Ln ∞ = Max y∈Rn+1 y ∞ where (Ln ) of (ℜ) in the plan (Pn ) The vibration prediction of the studied rotating machine is based on extrapolation, it can help to solve problems of determining the unknown terms of the vibration signals as a reliable result. In the next section presents the obtained results of the proposed remaining useful lifetime estimation approach applied to a gas turbine. This prognostic approach has the advantage of not necessarily using the analytical models of degradations to predict the evolution of the degradation of these components and to follow the behavior of gas turbines, but this requires the use of the exploitation data of this machine.


179

Fig. 3 Prognosis for estimating the residual useful life (RUL).

4 Application results Figure 3. shows the different sequential steps that are used to achieve the residual useful life (RUL) estimation of the studied gas turbine. The main purpose of this application is to ensure the estimation of the remaining useful lifetime of the HP wheel which is exposed to the influence of different vibration phenomena that are produced at the bearing of the studied gas turbine MS 3002. For this, the whole cycle life of the studied gas turbine MS 3002 is divided into three phases: • The first phase, is from 0 hour to 350 hours, • The second phase, is from 350 hours to 700 hours, • The third phase, is from 700 hours to the end of life threshold (danger threshold 1550 hours). 4.1

First phase

The obtained results for the first phase are shown in Figure 4, the original data is presented in blue, whereas, the obtained predicted data based on the polynomial extrapolation is presented in red. It is obvious that the predicted values are presenting a curve which presents nearly the mean value of the real data. The error between the predicted and the real data values are presented in the same figure. It is clear that this error is minimal, which indicates that the prediction is very effective. It is important to note that during the normal operation there are always some small vibrations that are generally acceptable. These vibrations are due to some normal ordinary constraints, such as at the times t = 45 hours, t = 170hours and t = 190hours where the vibration behavior of the turbine has small changes as shown in Figure 4. Contrary, at the times t = 120 hours and t = 240 hours it is noted that the vibration signal presents accidental vibration peak values, which indicates an alarm of vibration phenomenon is occurred.

180

Boulanouar Saadat et al. / Journal of Vibration Testing and System Dynamics 2(2) (2018) 173–185 1.9

Amplitude (mm/s)

1.85

Predit data Original data

1.8 1.75 1.7 1.65 1.6 1.55 1.5

0

50

100

150

200

250

300

350

200

250

300

350

Time (h) 1

Error

0.5

0

-0.5

-1

0

50

100

150

Time (h)

Fig. 4 The vibration signals of the first phase with vibration prediction. 2.8 2.7008

2.6817

2.6322 2.6078

2.5753

2.6

2.499

2.4941

2.4497 2.3814

Amplitude (mm/s)

2.4

2.6078

2.4129

2.3639

2.3193

2.2

2

1.9167

1.8 1.6482

Prediction time period

1.6

1.4

0

100

200

300

400

500

600

700

Measurment period (h)

Fig. 5 Second phase prediction.

4.2

Second phase

The values of the second phase are predicted based on the measured values during the first phase of the gas turbine lifetime. Figure 5, shows the predicted vibration magnitudes during the second phase. The predicted values are obtained as discrete values following a selected prediction measurements times. On the other side, Figure 6 represents the predicted values and the actual real vibration signal obtained from the collected real data on site. It can be said that the predicted values are within the envelope of the measured values. The residual values between the predicted signal and the real signal are presented in Figure 7, the change in the vibration dynamic behavior of the machine is obviously remarked by the presence of some accidental peaks within the second phase. In this case the turbine is under the influence of a fault vibration caused by the bearing under study.


181

2.8 2.7008

2.6817

2.6322 2.6078

2.5753

2.6

2.499

2.4941

2.4497 2.3814

Amplitude (mm/s)

2.4

2.6078

2.4129

2.3639

2.3193

2.2

2

1.9167

1.8 1.6482 1.6

1.4

0

100

200

300

400

500

600

700

First life cycle (h)

Fig. 6 Prediction of the second phase with the actual values of vibration. 2.8

Amplitude (mm/s)

2.6 2.4 2.2 2


1.8 1.6 1.4

0

100

200

300

400

500

600

700

400

500

600

700

Time (h) 1

Peak

Error

0.5 0 -0.5 -1 0

100

200

300

Time (h)

Fig. 7 The vibration signal of the two lifetime cycles and the prediction curve with their residual.

4.3

Third phase

Once more, the third phase vibration magnitudes predicted values are presented in Figure 8, these predicted values are obtained based on the vibration measurement real data collected during the past phases. In Figure 9, It is clear that the collected real data values from the sensed vibration signal are encompassing nearly the predicted values. What is really important is the prediction of the points that are possessing peak values. where, the main aim to ensure a reliable prognostics for the detection of the expected failures in time and to prevent any kind of breakdown of the machine during full operation, on the other side to schedule the maintenance periods adequately. The whole results of the third phase of the present application are shown in Figure 10, these results represent the prediction of the whole lifetime of the studied gas turbine during its three operation phases [0 hours − 1049 hours]. It is noted that there are two peaks as shown in Figure 11 within the

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Boulanouar Saadat et al. / Journal of Vibration Testing and System Dynamics 2(2) (2018) 173–185 4 3.7892

3.7324 3.6009 3.4885 3.4482

3.5

3.5496 3.5956

3.5366 3.4728 3.3823

3.3824

3.3324 3.242

3.2441

Amplitude (mm/s)

3.2254 3

2.535 2.5

2


1.5

0

200

400

600

800

1000

Measurment period (h)

Fig. 8 Prediction of the third lifetime cycle. 4 3.7892

3.7324 3.6009 3.4482

3.5

3.5956

3.4885 3.5496

3.5366 3.4728

3.3824

3.3324 3.242

3.3823

3.2441

Amplitude (mm/s)

3.2254 3

2.535 2.5

2


1.5

0

200

400

600

800

1000

Second life cycle (h)

Fig. 9 Total lifetime cycle with the predicted values.

error curve. It is clear that the second peak is higher than the first peak, this can be explained by the transition between phases, which inevitably leads to a change in the vibration behavior of the gas turbine to a higher vibration level. 4.4

The three phase analysis

Figure 11 represents the tendency lines. Indeed, three prediction thresholds can be defined for the characterization of the degree of failure, the three prediction thresholds are defined from the operating data of the studied gas turbine. However, the degradation state is not measured but it is possible to estimate it by identification using the proposed approach. The three prediction thresholds are: Tolerance limit [0 hours − 1049 hours]: Here the turbine is in normal operation and in the permitted area where there is no risk. When the vibration level exceeds this threshold, some caution has to be taken into account.


183

4

Amplitude (mm/s)

3.5

3


2.5

2

1.5

1

0

200

400

600

1000

1200

800

1000

1200

X: 702 Y: 0.5811

X: 356 Y: 0.5065

Peak 1

0.5

Error

800

Time (h)

1

Peak 2

0

-0.5

-1

0

200

400

600

Time (h)

Fig. 10 Vibration signals of the third phase with vibration prediction and there residual. 16

The level of danger

14.7

15.3507

14 14.0682 11.7941

12

A m p l i tu d e ( m m /s )

9.9101

The alarm threshold

10 9.3

8.9284 8.3687 7.7671

8

7.489

7.5394

7.1252

6 4.4925

The tolerance level

4.5 4


2

0

0

200

400

600

800 Third life cycle (h)

1000

1200

1400

1600

Fig. 11 Trend curve of the studied gas turbine.

Alarm [1049 hours − 1400 hours]: it warns that the machine condition is started to be deteriorated and the action of maintenance is required to be done. However, the maintenance time schedule needs the shutdown of the machine which will penalize the efficiency of the industrial plant in terms of

184


production. Danger threshold [1400 hours − 1600 hours]: it warns an imminent failure where the act of prevention has to be done quickly. It is important to clarify that there is the possibility to identify intermediate levels or phases within this phase to ensure an accurate analysis for the detection and prediction of the remaining useful lifetime of the studied machine. The prediction of the remaining useful lifetime is presented from t = 1049 hours until the time for which the vibration amplitude level reaches the danger threshold 14.70 at t = 1520 hours which indicates that an incident is happening at the gas turbine, in this case. This point can be located between the two predicted points (14.07, 1500 hours) and (15.53, 1550 hours) respectively. In this case the remaining useful lifetime is estimated (RU L = 471 hours), which is equivalent to (19.63 days), it means that this remaining time is sufficient to perform all maintenance work, here it can be said that our prognosis system has succeeded.

5 Conclusion The rotating machines occupy a large place in the industrial sector, especially within the oil and gas industries. Nevertheless, the continuous exploitation of these machines require a continuous maintenance strategy, because a breakdown of these machines during operation state due to any kind of failure is very costly. The present paper deals mainly with the development of an approach to ensure the prediction of the estimated remaining useful lifetime of the gas turbine bearings based on vibration signals data acquisition, where the main aim is to improve the performance of this kind of machines that are expected to operate under a long time without stop. The presented approach is used to estimate the optimal lifetime of the considered gas turbine bearings in order to predict their expected future failures. The obtained results prove clearly the effectiveness of the proposed approach which may be a very promising solution for other industrial applications that are containing very power and very sensitive equipments.

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Aims and Scope Vibration Testing and System Dynamics is an interdisciplinary journal serving as the forum for promoting dialogues among engineering practitioners and research scholars. As the platform for facilitating the synergy of system dynamics, testing, design, modeling, and education, the journal publishes high-quality, original articles in the theory and applications of dynamical system testing. The aim of the journal is to stimulate more research interest in and attention for the interaction of theory, design, and application in dynamic testing. Manuscripts reporting novel methodology design for modelling and testing complex dynamical systems with nonlinearity are solicited. Papers on applying modern theory of dynamics to real-world issues in all areas of physical science and description of numerical investigation are equally encouraged. Progress made in the following topics are of interest, but not limited, to the journal: • • • • • • • • • • • • • • • •

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Journal of Vibration Testing and System Dynamics Volume 2, Issue 2

June 2018

Contents A Fuzzy Logic PI Trajectory Following Control in a Chaotically Loaded Real Mechatronic Dynamical System with Stick-Slip Friction Wojciech Kunikowski, Pawel Olejnik, Jan Awrejcewicz…………………….........

91－107

Extreming Curves and the Parameter Space of a Generalized Logistic Mapping Diogo Ricardo da Costa, Matheus Hansen, Edson D. Leonel1, Rene O. Medrano-T……..………………….……………………………………....………

109－118

A Series of Symmetric Period-1 Motions to Chaos in a Two-degree-of-freedom van der Pol-Duffing Oscillator Yeyin Xu, Albert C.J. Luo…………........................................................................

119－153

Lift Enhancement of Airfoil Using Local Flexible Structure and the Influences of Structure Parameters Pengfei Lei, Jiazhong Zhang, Daxiong Liao…………………..……..…..….……

155－165

Continuous Stabilizing Control for a Class of Non-Holonomic Systems: Brockett System Example Fazal Ur Rehman, A Baseer Satti, A. Ahmed Saleem...………….....……........…

167－172

Remaining Useful Lifetime Prediction of Gas Turbine Bearings Based on Experiment Vibration Signals Data Boulanouar Saadat, Ahmed Hafaifa , Ali Bennani, Nadji Hadroug, Abdellah Kouzou, Mohamed Haddar………….................................................................…

173－185

Available online at https://lhscientificpublishing.com/Journals/JVTSD-Download.aspx

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Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics

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