Journal of Vibration Testing and System Dynamics

Volume 1 Issue 4 December 2017

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

Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics Editors Stefano Lenci Dipartimento di Ingegneria Civile Edile e Architettura, Universita' Politecnica delle Marche via Brecce Bianche, 60131 ANCONA, Italy 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, P. R. China Email: [email protected]

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

Associate Editors Jinde Cao School of Mathematics Southeast University Nanjing 210096, China Email: [email protected]; [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 76308, 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]

Kalyana Babu Nakshatrala Department of Civil and Environmental Engineering University of Houston Houston, Texas 77204-4003, USA Email: [email protected]

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

Kurt Vandervort Stress Engineering Services, Inc. 42403 Old Houston Highway Waller, Texas 77484-5718, USA Email: [email protected]

Dimitry Volchenkov Department of Mathematics and 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]

Editorial Board Ichiro Ario Department of Civil and Environmental Engineering Higashi-Hiroshima, Japan Email: [email protected]

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

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

Continued on inside back cover

Journal of Vibration Testing and System Dynamics Volume 1, Issue 4, December 2017

Editors Stefano Lenci 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 1(4) (2017) 281-294

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

A Time-Frequency PID Controller Design for Improved Anti-Interference Performance of a Solenoid Valve Applicable to Hydraulic Cylinder Actuation Xiu-Heng Wu1 , Zheng-He Song1†, Yue-Feng Du1 , En-Rong Mao1 , C. Steve Suh2 1

2

Beijing Key Laboratory of Optimized Design for Modern Agricultural Equipment, China Agricultural University, Beijing 100083, China Nonlinear Engineering and Control Lab, Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Submission Info Communicated by J.Z Zhang Received 3 August 2017 Accepted 3 September 2017 Available online 1 January 2018 Keywords Time-frequency PID Electro-hydraulic hybrid system Discrete wavelet transform Nonlinear control

Abstract PID control is widely used in electro-hydraulic systems. However, enhancing PID control performance in response to system nonlinearity, fluctuations of external load, and noise inevitably renders the chattering of the system that are also telltale indications of poor efficiency and dynamic instability. On the other hand, tuning down PID parameters would alleviate chatter at the expenses of reduced performance and inefficient use of resources. To address the particular issue, a novel controller concept termed as the time-frequency PID (TFPID) is developed. Firstly, a nonlinear electro-hydraulic dynamic model to be controlled is built for numerical and physical studies. Next, the working principle of the TFPID control is elaborated where the discrete wavelet transform is employed to decompose the error signal into high frequency error and low frequency error. Two unique PID controllers incorporating proportion, differential, and integral control are designed to mitigate the two error signals. The TFPID controller and system model are developed in MATLAB/Simulink to optimize the parameters and a hardware-in-the-loop test bench is employed to establish the performance of the system subject to interferences. Physical test results show that TFPID performs significantly better in anti-interference, stability, and dynamic response. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Nonlinear control methods including sliding mode (or variable structure) control, back-stepping control, adaptive control, and fuzzy control have seen increasing applications in electro-hydraulic systems whose dynamic behaviors are highly nonlinear due to complex structures of many components, time-varying parameters, and uncertainties [1–6]. Electro-hydraulic systems found in farming equipment and machinery that are characteristically large power are often employed to handle system responses that are † Corresponding

author. Email address: [email protected]

ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2017.012.001

282

Xiu-Heng Wu, et al. / Journal of Vibration Testing and System Dynamics 1(4) (2017) 281–294

of low frequency. Given the low operation speed and thus weak nonlinearity, PID controllers are in general effective in mitigating the impact of nonlinearity on stability and performance. PIDs are commonly employed to work with other control concepts or filtering methods and implemented as digital signal processors for the real-time control of many an industrial application including electro-hydraulic systems [7–9]. Theoretically, PID parameters can be optimally tuned to improve dynamic performance, reduce steady-state error, and overcome system oscillation [10]. However, doing so will inevitably lead to chatter in daily applications due to the nonlinearities of omnipresent high frequency, external load disturbance, and noise jamming. That is, adjusting parameters however slightly can induce high frequency responses that are nonlinear and inadvertently magnify the external load fluctuation and noise disturbance, rendering low energy efficiency and risking system breakdown. On the other hand, conservative controller design aiming to weaken the control action in exchange for system stability sacrifices system performance at the expense of precious resources. This is a dilemma demanding a solution to an improved controller design that is effective in mitigating disturbance, negating high frequency nonlinear oscillation, and in the meantime enhancing system performance. The sensitivities of the proportional, differential, and integral control actions of the PID method is different from each other in different frequencies. Strengthening the differential action in low frequency part of the error signal is effective in reducing overshoots, suppressing oscillations, stabilizing the system stable, and improving system response and dynamic performance. However, doing so can negatively amplify the high frequency part of the error signal that is indicative of nonlinear high frequency chattering and noise interference [10]. Similarly, the integral control can eliminate the steady state error and improve the precision of the control system. But enhancing the integral action to the low frequency can cause the phenomenon called wind-up or integral saturation. Because the oscillation of the low frequency error signal can last a long period of time, integral saturation can force the solenoid valve spool that is already reaching the end of the valve to jerk and impact [10,11]. On the other hand, the high frequency error is insensitive to larger integral coefficient because of the rapid fluctuation involved. Therefore, it is essential that a PID controller is designed exploring the parameters for targeting different ranges of frequency response. The spectral information in the frequency domain is the signature of a dynamic system [12]. Controllers therefore need be designed not only in the time domain, but also the frequency domain. Recently the discrete wavelet transform (DWT) was introduced into the traditional PID control and named as multiresolution PID (MRPID) control for its capability in performing multiscale time-frequency analysis [13–15]. MRPID decomposes the sequence of a digital error signal into several ranges from low to high depending on the frequency content, as well as the cumulative effect of many underlying phenomena such as process dynamics, measurement noise, and effects of external disturbances, which all manifest on different scales. The proportional control is applied to each one of the components and sums together at last to generate the control signal. Once the controller is designed, it’s parameters is fixed. As the error signal is constantly changing in response to the influence variable load, nonlinearities, and disturbance, the wavelet decomposed components corresponding to the changes vary accordingly, thus the controller output is always mutative with the changing frequency of error. Evidently the controller possesses the ability to change control action in the time domain in response to the changing information in the frequency domain. This is the reason that MRPID is also considered a type of time-frequency control (TFC). Recognizing that nonlinear systems are characterized by broadband spectral response having additional frequency components emerging and diminishing, a novel wavelet-based time-frequency controller was formulated and successfully applied to the robust control of many applications [16–18]. The particular time-frequency control method incorporates filtered-x least mean square (FXLMS) algorithm with DWT and performs exceptionally well for strong nonlinear systems. In addition, the controller demonstrates better computational efficiency for the simple reason


283

that FXLMS and DWT function together just as a filter does. Moreover, being an adaptive controller demonstrating good performance in the time domain, the FXLMS component in the controller design plays the critical role of manipulating the wavelet coefficients which carry frequency information in the time domain. This particular property is explored to realize controlling chaotic systems. Because TFC allows control action to be exerted in the time and frequency domains at the same time, not only the frequency response of the systems but also the feature of the control method need be considered when the controller is being designed. An improved electro-hydraulic system controller design exploring all the prominent TF features is reported herein. This paper is arranged in the following order of presentation in three parts: 1) The development of a nonlinear dynamic system model of a hydraulic cylinder that is controlled by a solenoid proportional valve. The model is for identifying the optimal control parameters and for performing numerical experiments that consider various nonlinearities including friction, dead zone, leakage and the variable effective bulk modulus. 2) The working principle of the wavelet based TFC and the design of a TFPID controller incorporating PID control and considers the traits required of the electro-hydraulic system operating in farms. 3) The hardware-in-the-loop experiments employed based on the mathematical model. Comparisons are also made with the traditional PID controller to validate the controller design.

2 System model Consider the hydraulic cylinder controlled by a solenoid proportion valve of a universal configuration such as the one shown in Fig. 1. Comprised of a double-ended hydraulic cylinder, a 4/3-way solenoid proportion valve, and an unknown load, the system is commonly applicable to either force or position control. A comprehensive nonlinear model of the system is formulated by following largely the papers listed in Refs. [6, 19–33]. According to the Newton’s 2nd Law, the dynamic equation of the hydraulic actuator can be described as (1) M x¨ + Bex˙ + Kex = (p1 − p2 ) A − FL − FF where x is the displacement of the piston, m is the total mass of the piston along with all the linked objects, Be and Ke are equivalent damping and the spring stiffness coefficient, respectively. p1 and p2 are pressure of the left and right chamber, respectively. A is active area of the piston. FL is the unknown force of nonlinearity due to the external disturbance [6]. FF is the friction force on the piston, which is one of the primary nonlinear factor affecting the dynamics of the actuator piston. The particular friction is influenced by many factors. In general it is considered to be a function of the position and velocity of the piston [19]. Many empirical models have been established and applied to specific hydraulic actuators [20–22]. In this article, a particular friction equation of motion of the actuator piston following from the Lund–Grenoble model [23] is adopted. The model is chosen for a few reasons. It is formulated based on sound hypotheses and is rigorously derived on the basis of many kinds of friction can be considered. Most importantly, it agrees well with empirical data in most situation with adjusting parameters in the formulation properly. The friction model which is a set of coupled equations is concisely given below in Eqs. (2). With z being an intermediate variable, g(x) ˙ a function describing the steady-state friction characteristics at a constant velocity [23], vsk the Stribeck velocity defined as the most unstable velocity on the Stribeck curve [24–26], α0 the Coulomb friction, α1 the Stribeck friction, α2 the viscous friction parameter, σ0 the spring constant, and σ1 the damping coefficient. For detail derivations of the model, [24, 25] are referred to.

284


V1 p1 A

qe1

qe2

FL

x

A p2 V2

qi

q2

q1

xv

u

pr

ps

pr

Fig. 1 Schematic of the hydraulic cylinder controlled by solenoid proportion valve.

z˙ = x˙ − [(σ0 |x|)/g( ˙ x)] ˙ z ˙ sk ) g(x) ˙ = α0 + α1 e−(x/v

2

(2)

FF = σ0 z + σ1 z˙ + α2 x˙ For a double-ended hydraulic cylinder experiencing leakage, the following pressure continuity equations [27] featuring the effective bulk modulus βe for the cylinder chambers can be derived [20]:

βe (q1 − Ax˙ − qi − qe1 ) V1 + Ax βe (−q2 + Ax˙ + qi − qe2 ) p˙2 = V2 − Ax p˙1 =

(3)

where V1 and V2 are the original control volumes of the left chamber and right chamber, respectively, including the volume of the servo valve, pipeline, and cylinder chambers. q1 is the supplied flow rate to the left chamber while q2 is the return flow rate of the right chamber. qe1 is leakage form chamber 1 to the external while qe2 is leakage form chamber 2 to the external and qi is leakage form high pressure chamber to low’s. The relationship between the spool valve displacement and the load flow dictates that [28]: √ √ q1 = kq1 xv s (xv ) ps − p1 + s (−xv ) p1 − pr (4) √ √ q2 = kq2 xv s (xv ) p2 − pr + s (−xv ) ps − p2 the s(xv ) is a function defined as follows [29]: s (xv ) =

1, 0,

xv ≥ 0 xv < 0

(5)

and kq1 = Cd w1 kq2 = Cd w2

ρ /2 ρ /2

(6)

where ps is the supplied pressure, pr is the return line pressure, xv is the spool displacement of the solenoid proportion valve, Cd is the discharge coefficient, w1 and w2 are the spool valve area gradients,


285

and ρ is the fluid density. It should be noted that qi , qe1 and qe2 are determined through qi = ci (p1 − p2 ) qe1 = ce1 p1

(7)

qe2 = ce2 p2 where ci is internal leakage coefficient and ce1 and ce2 are the external leakages of the left chamber and right chamber, respectively. Note also that the effective bulk modulus βe is primarily affected by the chamber pressures involved [30, 31] as follows,

βe1 = (c1 + p1 ) [1/c2 − ln (1 + p1 /c1 )] βe2 = (c1 + p2 ) [1/c2 − ln (1 + p2 /c1 )]

(8)

where βe1 and βe2 are the input chamber and output chamber effective bulk moduli of the cylinder, respectively. c1 and c2 are constants. The displacement of the shaft of the solenoid valve xv , is related to the valve’s voltage input u, via a second-order system: (9) x¨v + 2ζ ωv x˙v + ωv2 xv = Kv ωv2 u where ζ , ωv , and Kv are the damping ratio, natural frequency, and gain of the valve dynamics [32, 33]. The dynamic equations of the electro-hydraulic system depicted in Fig. 1 can now be formulated by using a set of differential functions through recasting Eqs. (1) - (9) and the following state variables: ˙ x3 = p1 , x4 = p2 , x5 = xv , and x6 = x˙v , x1 = x, x2 = x, ⎧ x˙1 =x2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ x˙2 = [(x3 − x4 ) A − Bex2 − Kex1 − FL − FF ] ⎪ ⎪ ⎪ m ⎪

⎪ ⎪ ⎪ (c1 + x3 ) 1 c2 − ln 1 + x3 c1 √ √ ⎪ ⎪ kq1 x5 s (x5 ) ps − x3 + s (−x5 ) x3 − pr x˙3 = ⎪ ⎪ V1 + Ax1 ⎪ ⎪ ⎨ −Ax2 − ci (x3 − x4 ) − ce1 x3 } (10)

⎪ ⎪

(c √ √ + x ) 1 c − ln 1 + x c ⎪ 1 4 2 4 1 ⎪ ⎪ −kq2 x5 s (−x5 ) ps − x4 + s (x5 ) x4 − pr x˙4 = ⎪ ⎪ V1 − Ax1 ⎪ ⎪ ⎪ ⎪ ⎪ +Ax2 + ci (x3 − x4 ) − ce1 x4 } ⎪ ⎪ ⎪ ⎪ ⎪ x˙5 =x6 ⎪ ⎪ ⎪ ⎩ x˙6 =Kv ωv2 u − 2ζ ωv x6 + ωv2 x5 In real-world electro-hydraulic system, there are limits on the displacement of the piston and a dead zone in the valve spool. Such physical constraints are modelled using the state variables as follows: ⎧ x1 ≤ xmin ⎨ xmin , xmin < x1 < xmax (11) x1 = x1 , ⎩ xmax , x1 ≥ xmax ⎧ min x , ⎪ ⎪ ⎪ v ⎪ ⎨ x5 , x5 = 0, ⎪ ⎪ x , ⎪ ⎪ ⎩ 5max xv ,

x5 ≤ xmin v min xv < x5 < xdv min xdv min ≤ x5 ≤ xdv max xdv max < x5 < xmax v x5 ≥ xmax v

(12)


286

Table 1 The parameters used in the model. Parameters

Value/Unit

Parameters

Value/Unit

Parameters

Value/Unit

M

9.0 kg

kq1

2.38×10−5 m5/2 /kg1/2

ωv

534 rad/s

2.38×10−5

Be

2000N/(m/s)

kq2

ζ

0.48

Ke

10 N/m

Ps

21 MPa

σ0

5.77×106 N/m

A

645 mm2

Pr

0.1 MPa

σ1

2.28×104 N/m/s

c1

99.993 MPa

ci

1097 mm3 /(sMPa)

α0

370 N

c2

0.0733

ce1

120 mm3 /(sMPa)

α1

217 N

V1

1.29×105 mm3

ce2

mm3 /(sMPa)

α2

2318 N/m/s

V2

1.29×105 mm3

Kv

vsk

10 N/m

‫ݎ‬

120

m5/2 /kg1/2

0.5

݂Ͳ ̱݂ͳ൅

݂ͳെ̱݂ʹ൅ σ

‫ݑ‬

‫ݕ‬

െ ݂݊െͳ ̱݂݊

Fig. 2 Schematic of the time-frequency control approach.

the initial position of the piston is set at the middle of the cylinder and according to its effective length, the parameter can be determined as: xmax = −xmin = 0.2m. Similarly, assuming that the valve is symmetrical and the initial position of the spool is also at the middle of the valve bush, so the value dmax = −xdmin = 0.5 mn. In addition, other parameters involved = −xmin is obtained as: xmax v v = 3 mn, xv v that are determined by structure of system or working environment are list in Table 1 [24, 25, 33]. When a specific piston motion trajectory xo (t) or a force Fo (t) is desired, the objective of the controller is therefore to generate a series of output uo (t) to the solenoid valve through adjusting the original input signal ui (t) to achieve the tracking of x1 and the control of (x3 − x4 ) subjected to the exertions of the unknown external disturbance FL and the nonlinear friction force FF , and to maintain the displacement of the piston not to fluctuate too much.

3 TFC Design A TFC controller concept incorporating PID control to realize the TFPID approach taking full advantages of both the PID and TF method, is discussed. 3.1

Physical principles

Fig. 2 shows the schematic of the TFC approach. r represents the desire or referential output. y is the actual output of the system, which represents the outward manifestation. It also implies the internal information and reflects the essence of the system. Traditional controllers, be them feedback or feedforward, always operate r and y directly to export a series signal to the system. In contrast, the results of the synthesis operation of r and y must be analyzed and processed by the time-frequency method, thus producing several sub-signals to be computed by their corresponding controller. In other words, the TFC scheme includes several controllers at the same time, all working concurrently and constituting a controller set. As a utility tool for time-frequency analysis, DWT is widely used for its superior ability in feature


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Decomposition Process

Reconstruction Process

Fig. 3 DWT process of a signal.

Fig. 4 Diagram of the decomposing process of a signal.

extraction of short data sequence. There is a fast pyramid algorithm, developed by Mallat [34–37], that can simplify the complex process of decomposition and reconstruction into inner product of vectors or matrix multiplication, allowing the processing of a succession of r and y to be implemented in real-time. Fig. 3 illustrates a two levels DWT processing of a signal x(k) using the subband coding scheme, ˆ which includes wavelet decomposition and reconstruction where g(k) ˆ and h(k) are high-pass and lowpass decomposition filters, respectively, whereas g(k) and h(k) are high-pass and low-pass reconstruction filters, respectively. The latter pair forms a quadrature conjugate mirror filter pair with the decomposition filters. Once the convolution between x (k) and decomposition filters is done, and the down-sampling are finished, the first-level detail coefficients C g1 and trend coefficients C h1 are obtained. They contain the high frequency and low frequency information of the original signal x (k). In the same token the second-level detail coefficients Cg2 and trend coefficients Ch2 can be acquired using Ch1 . Many groups of coefficients representing the frequency information from low to high can be generated by repeating the procedures. By engaging the groups of coefficients with the filters g(k) and h(k) in the reconstruction process, one obtains the signal components x 1 (k), x 2 (k) and x 3 (k) as shown. According to the Sampling Theorem, if the sampling frequency is set to be 2 fn , then the highest frequency of the actual signal one can resolve is fn . By assuming that the highest frequency of the signal x (k) is fn , the DWT process can decompose the signal into several frequency subsets from f to fn , where f represents the lowest frequency of the signal whose general value is zero. Fig. 4 illustrates the vivid process of how the complex nonlinear signal x (k) is expressed (decomposed) as simple sub-signals for easier handling. Following the two-level decomposition in Fig. 3, the value n in Fig. 4 can be 3. Although aliasing would emerge in adjacent frequency subsets, it would not affect the result of dividing the frequency domain and designing the controller set later.

288


݁ͳ ݁ʹ

‫ݎ‬

൅ െ

݁

σ

Discrete Wavelet ݁݊െͳ Transform

݁

PID Controller1

σ

݁݊

݁

‫ݑ‬

System

‫ݕ‬

PID Controller2

Fig. 5 Schematic of the TFPID control approach.

The objective of the TFC is to design an exclusive controller for each signal component xi (k), i = 1, 2, . . ., n, so that the parameter of each controller can meet the performance requirement of the system in regards to the particular frequency response. Hence, the performance of system would be optimal in all frequency domain using the controller subsets. Most important, definitive physical interpretation can be made in the TFC design process. 3.2

TFPID controller design

The TFPID controller concept incorporates the characteristics of P, I, and D control with the TF framework. The TFPID design concept is found in Fig. 5. Traditional PID controller often operates on the error e, while TFPID applies the DWT to sequences e (k) first to resolve the high frequency error eH and the low frequency error eL . The high frequency information indicative of nonlinearity, measurement noise, and external disturbances are carried by eH , whereas the main working frequency information likes phase lag is contained in eL . The electro-hydraulic system is typically working in the low frequency 0-to-2Hz range, so the error signal needs be multi-level wavelet decomposed, and the component belonging to the lowest frequency range can be taken as the low frequency error eL , while the rest be synthesized to be the high frequency error eH . The parameters of each single controller need be designed base on the principle of PID method with each PID controller considering a specific frequency band. Considering the digital control signal u(k) at time k k

u(k) =KPL eL (k) + KIL ∑ eL (k)T + KDL 0

eL (k) − eL (k − 1) T

eH (k) − eH (k − 1) + KPH eH (k) + KIH ∑ eH (k)T + KDH T 0 k

(13)

where KPL , KPL and KPL represents the coefficients of P, I and D for the low frequency error eL , KPL , KPL and KPL represents the controller coefficients for eH , and T is the sampling period. To improve the precision in controlling step response and suppressing overshoot, a simple algorithm of separated integral method is introduced to the two PID controllers, rendering the following controller output k e(k) − e(k − 1) (14) u(k) = KP e(k) + KI ∑ eI (k)T + KD T 0 The corresponding TFPID controller output is therefore k

u(k) =KPL eL (k) + KIL ∑ eLI (k)T + KDL 0

eL (k) − eL (k − 1) T

eH (k) − eH (k − 1) + KPH eH (k) + KIH ∑ eH (k)T + KDH T 0 k

(15)


and the constraint conditions of the errors are as follows ⎧ e ≤ emin ⎨ emin , emin < e < emax eI = e, ⎩ emax , e ≥ emax ⎧ eL ≤ emin ⎨ emin , emin < eL < emax eLI = eL , ⎩ emax , eL ≥ emax

289

(16)

(17)

As the TFPID controller works with two concurrent PID controllers running, the number of parameters involved is therefore twice than the traditional PID controller, thus making TFPID controller more powerful and flexible. In theory, more parameters the controller has, more difficult the design task. But with each parameter of the PID corresponding to specific response of the system, it is not demanding to design the specific TFPID to meet the control goal. Although parameters are fixed once the controller is designed, but because the constituent of different frequency in the signal is always varying in time, so the output signal u(k) is dynamical with different and changing frequency components. Hence, the TFPID is adaptive to the variation in the frequency domain.

4 Experimental verification In order to validate the control method, MATLAB Simulink is employed to design the optimal parameters and a hardware-in-the-loop test bench is used to evaluate the TFPID design against the PID controller. 4.1

Control parameters

An extensive research is performed to identify the mother wavelet to be used in the study [13,17,34,38]. The Daubechies wavelet is chosen for its properties of orthogonality and compact support. In addition, based on the rule that a wavelet with a large number of filter coefficients can match the characteristic features in a time series with greater efficiency, db3 wavelet of six filter coefficients is applied to decompose the error. Considering the working frequency of the system and the computational capacity of the microcomputer, the size of the error signal buffer (the number of observations in the time series) N f is set to equal to 512, and to be six-level decomposed. The simulation model is built based on the model in Eqs. (10) and the control algorithm is compiled using the S-Function. Table 2 tabulates the PID parameters determined that render the minimum step response time with the condition of restraining overshoot to be less than 1%. They are afterward commissioned as the references for designing the TFPID controller. The upper and lower limits of the error are chosen to be emax = −emin = 0.0005m. Per the account on PID given before, increasing the proportional and derivative coefficients for the low frequency error signal and decreasing the integral coefficient for the high frequency at the same time would make the system more stable and responding faster. Moreover, the wind-up phenomenon can also be eliminated. Enhancing the integral action by removing the derivative action to the high frequency component of the error signal can restrain the oscillation of high frequency. Table 3 summaries the parameters of the TFPID controller design. 4.2

Hardware-in-the-loop test

The system model is compiled by VeriStand software and running in real-time on mainframe PXIe8135. The control algorithm is edited in LabVIEW and downloaded to an embedded hardware device named NI myRIO-1990 as the prototype of the controller. Signal transmission between the virtual

290


Table 2 PID parameters Coefficient

Value

KP

3000

KI

60000

KD

14

Table 3 Parameters of the TFPID controller Coefficient

Value

KPL

3500

KIL

50000

KDL

16

KPH

1500

KPH

100000

KDH

0

Fig. 6 Hardware-in-the-loop test bench.

electro-hydraulic system and the controller is via cable, and the information can be uploaded to the computer at the same time. Fig. 6 shows the test bench. Three different groups of comparison tests are performed to demonstrate the validity of the new controller design. The step responses of the two controllers are seen in Fig. 7, where Fig. 7(a) is the simulation result using Simulink and Fig. 7(b) is the experimental result using the test bench. A reference step signal from 0 to 0.03m is given to the controller at the beginning and an impulse load of 10,000N is applied at t=0.5s for 0.02s. The step response times of the two control methods are similar, thus similar dynamic performances. However, there is a steady-state error see in the simulation result after stepping due to the presence of the dead zone in the valve. In addition, when it experiencing the impact that triggers the high frequency oscillation, the system under the traditional PID control starts to oscillate, and regains stability after a while. By contrast, the system restores dynamic stability immediately under the control of TFPID, thus demonstrating the robustness of the design. The results of sinusoidal tracking are also presented as simulation and experiment in Fig. 8 and

0.032

0.032

0.031

0.031

0.030

0.030

Displacement /m

Displacement /m


0.029 0.028 Ref. PID TFPID

0.027 0.026 0.025 0.0

0.5

t/s

0.029 0.028 PID TFPID

0.027 0.026 0.025 0.0

1.0

291

(a) Simulation result

0.5

1.0

t/s (b) Experimental result

Fig. 7 System step response. 0.032 Ref. PID TFPID

0.02 0.01

0.031 Displacement /m

Displacement /m

0.03

0.025

0.00

0.020

-0.01

0.015

-0.02

0.010 0.005 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

-0.03 0.0

0.5

t/s

1.0

0.030 0.029 0.028 PID TFPID

0.027 0.026 0.025 0.0

(a) Results of sinusoidal tracking

0.5

t/s

1.0

(b) Error of sinusoidal tracking


0.031 0.030 0.029 0.028 PID TFPID

0.027 0.026 0.025 0.0

0.5

t/s

1.0

(c) High frequency error component of sinusoidal tracking

Fig. 8 Simulation result of sinusoidal tracking.

Fig. 9, respectively. Fig. 8(a) gives the actual displacements of the piston in the hydraulic cylinder under the two control methods, Fig. 8(b) shows the error between the actual displacement and the reference, and Fig. 8(c) presents the high frequency components decomposed from the error signal. It can be seen in Fig. 8(a) that the system can track the sinusoid faster under the TFPID control without the high frequency oscillations displayed in the one controlled by the PID. Such oscillations are manifested as system chatter as so prominently observed in Fig. 8(b). The feasibility of the TFPID controller design is further demonstrated in Fig. 8(c) where chatter in the high frequency component is resolved. Similar observations can also be made with the corresponding experimental results found in Fig. 9. The controller out in Fig. 9(d) attests in unambiguous terms to the same conclusion that the TFPID controller is significantly better than the PID in giving less violent output. A random interference signal of 50HZ in frequency with an amplitude that is 2% of the step applied to y is added to simulate disturbance. The corresponding results are given in Fig. 10. It is evident that the TFPID control method is able to maintain the piston close to the reference position with unremarkable transitions. This further implies that TFPID is good at anti-interference. It is more robust than the PID control to the interference of high frequency disturbance.



5.0

Ref. PID TFPID

0.02 0.01

Error /mm

292

0.00 -0.01 -0.02 -0.03 0.0

0.5

0.0

-5.0 0.0

1.0

t/s

PID TFPID

PID TFPID

4.0 2.0 0.0 -2.0 0.0

0.5

1.0

t/s

1.0

(b) Error of sinusoidal tracking Error Component /mm

Error Component /mm


0.5

t/s PID TFPID

4.0 2.0 0.0 -2.0 0.0

(c) High frequency error component of sinusoidal tracking

0.5

1.0

t/s

(d) Controller output of sinusoidal tracking

Fig. 9 Experimental result of sinusoidal tracking. 0.032

0.028

Displacement /m

Displacement /m

0.032 0.030

Ref. PID TFPID

0.026 0.0

0.5

t/s


1.0

0.030 0.028 PID TFPID

0.026 0.0

0.5

t/s

1.0

(b) High frequency error component of sinusoidal tracking

Fig. 10 Results of control with disturbance.

5 Summary P, I, and D control methods were examined to explore the feasibility of considering the frequency response of an error signal as a low and a high subbands. The principle of the TFC theory was also discussed. A TFPID controller was then developed and applied to control a specific nonlinear system that is a hydraulic cylinder controlled by a solenoid proportional valve. Optimal control parameters were found using the system dynamic model derived by considering various nonlinearities including friction, dead zone, leakage, and the variable effective bulk modulus. A hardware-in-the-loop test was subsequently implemented to evaluate the performance and quality of the novel TFPID controller design. It was shown that the TFPID control demonstrates significantly better performances in antiinterference, stability and dynamic response than that of the classic PID controller design.


293

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Impact of Tool Geometry and Tool Feed on Machining Stability Achala V. Dassanayake, C. Steve Suh† Nonlinear Engineering and Control Lab, Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

Submission Info Communicated by J. Zhang Received 2 August 2017 Accepted 4 September 2017 Available online 1 January 2018 Keywords Turning dynamics Material removal Chatter Whirling Instantaneous frequency

Abstract Tool-workpiece dynamics is characterized by aperiodic responses including period-doubling bifurcation and chaos. As a state signifying the extent of machining instability, tool chatter in longitudinal turning operation is a function of nonlinear regenerative cutting force, instantaneous depth-of-cut (DOC), and workpiece whirling. The effects of tool geometry and feed rate per revolution on cutting stability are investigated using a comprehensive model previously reported in References [1–3]. The model configuration allows the coupled toolworkpiece motion relative to the machining surface to be studied in the Cartesian space as a function of spindle speed, instantaneous DOC, rate of material removal, tool geometry, and material imbalance induced whirling. It is found that chatter can be eminent using one set of tool geometry while, at the same DOC, be sufficiently suppressed by employing tool inserts of different geometric parameters. Nonlinearity of tool structure is shown to have a dominant effect on tool vibration amplitude. High feed rate contributes to stability at high DOCs, thus indicating that feed rate is among the parameters that impact cutting stability. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

Nomenclature AC ai aii aix aiy Bi bii

chip cross sectional area acceleration of the center of mass of Rotor i distance between the fixed end and the cross section sliced through the center of gravity of the i−th rotor X-direction acceleration component of ai Y-direction acceleration component of ai middle position of section i along the spindle axis distance between the pinned end and the cross section cut through the center of gravity of the i−th rotor

† Corresponding

author. Email address: [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2017.12.002

296

Ci d1 d3 dAV E Ff Fn FT FX FY FZ Gi IAV kf ki kn kZ kZC l l0 li l˙ mi MZ O r1 r3 s t0 t ti ui vi vix viy xi x′2 yi y′2 zt zt′ α αI δi δ12 δ32 ε1 ε3 ηc

A.V. Dassanayake, C.S. Suh / Journal of Vibration Testing and System Dynamics 1(4) (2017) 295–317

geometric center of Rotor i diameter of Rotor 1 diameter of Rotor 1 average diameter of the workpiece elastic modulus of workpiece friction force normal force force due to bending stiffness in the Z-direction X-component of the cutting force Y-component of the cutting force Z-component of the cutting force gravitational center of Rotor i average area moment of inertia of the workpiece friction pressure component stiffness of Rotor i normal pressure component linear portion of the stiffness of the tool nonlinear portion of the stiffness of the tool distance from the spindle end to the tool position full length of the workpiece value of l at time = 0 constant feed rate of the tool mass of Rotor i equivalent mass of the machine tool coordinate origin placed at the spindle end of the workpiece radius of the unmachined section radius of the machined section instantaneous depth-of-cut chip width in feed per revolution time variable instantaneous chip width (instantaneous feed per revolution) vibration displacement of geometric center of Rotor i velocity of the center of mass of Rotor i X-direction velocity component of vi Y-direction velocity component of vi X-direction component of distance to geometric center of Rotor i from Bi distance in X-direction between geometric and gravitational centers of Rotor 2 Y-direction component of distance to geometric center of Rotor i from Bi distance in Y-direction between geometric and gravitational centers of Rotor 2 displacement of the tool in Z - direction displacement of the tool in Z – direction one revolution (of the workpiece) before angle the workpiece has rotated from the beginning constants with i = 1, 2, 3, and 4 static deflection of Rotor i the ratio δ1 /δ2 the ratio δ3 /δ2 eccentricity of Rotor 1 eccentricity of Rotor 3 chip flow angle


θ ρ ψ0 Ω

297

angle between the force due to stiffness and the negative X- direction density of workpiece material −−−→ angle between X-axis and C1G1 at t = 0 spindle speed

1 Introduction Tool chatter is common in manufacturing where material removal is involved. It is a detrimental state of dynamic instability that compromises machine tool integrity, tool life, and the surface quality and dimensional precision of the workpiece. Although cutting dynamics and machine tool chatter have been extensively studied [4–6], however, chatter remains an issue in manufacturing industry that eludes all the efforts [7]. As an attempt to establish the underlying mechanism that correlates cutting parameters with chatter in turning operations, a comprehensive nonlinear model considering the effects of material removal, workpiece deflection under cutting forces, and workpiece whirling on machining instability was reported in [1–3]. Machining is an operation that involves a rotary motion and a translation that is either linear or curvilinear. Being the only component that vibrates, tools in milling and drilling are the exclusive source for both types of motions. Whereas in turning, the workpiece is in rotary motion and the tool is in linear translation, thus both vibrate in material removal. However, only tool vibrations in turning operations have been given exclusive considerations [8–11]. Although workpiece vibrations impact both cutting instability and product quality including surface finishing, most models developed for investigating surface roughness [12–14] do not consider workpiece vibrations at all. Regenerative effect, nonlinearity and three-dimensionality are among the essential parameters that underlie turning instability. The significance of regeneration in cutting instability has been much discussed [15]. Considering nonlinear models for the study of machining dynamics has seen much interest [16–20]. However, many considered either single DOF [16,17,21,22] or 2-DOF models [9,11,20,23] that admit only tool vibrations. The 3D machining model reported in [10] disregards the impact of workpiece vibration on cutting dynamics. In response to the reviews above, this paper studies the effects of tool geometry and feed rate using the 3D turning model that was first documented in [1–3]. The model which is nonlinear considers regenerative effect, cutting force and structural nonlinearities, simultaneous workpiece-tool vibration, and whirling induced by the material inhomogeneity of the workpiece. The model also incorporates the decrease of the workpiece in mass and stiffness due to material removal. Since nonlinearity is prominent in cutting operation, the concept of instantaneous frequency [24] is used in the paper as the characterization tool for capturing the dynamics that underlies the various states of machining instability [25]. Lyapunov spectra [26] are also employed to provide a quantitative measure for machining instabilities that are definitive chaotic. 2 Turning Model [1, 2] Fig. 1 illustrates a longitudinal turning operation in which the workpiece is comprised of an unmachined section (B1), a being-machined section (B2) and a machined section (B3). The unmachined section is mounted to the spindle on the left-hand side and the machined section is pinned to the tailstock along the spindle axis on the right-hand side. In the figure O defines the Cartesian coordinate origin and B1, B2 and B3 are located midway of each section along the spindle axis. The distance measuring from the spindle end to tool position, ˙ is an independent variable of time defined by l, ˙ the constant feed rate of the tool, and l = li − lt, li , the value of l at time t = 0. The three sections in Fig. 1 are taken as a system consisting of 3

298


Fixed to the spindle

O

B2

B3

Spindle Axis

l0

l

X

Pin jointed at the tailstock end

2r3

2r1

B1

Z

Y

t0

Fig. 1 Workpiece configuration. rotor 1 rotor 2

rotor 3

Z

Z

O : l/2 - t 0/4 l l0/2 + l/2 + t0/4

Fig. 2 Workpiece model. dl spin Y

is e ax

Z G2

B2

C2

X

Fig. 3 Rotor 2 configuration.

rigid rotors mounted on a flexible shaft of negligible mass, as shown in Fig. 2 where Rotor 1, the unmachined section, has a full length of (l − t0 /2) measuring from the origin. The tool is aligned with the spindle along the Z-axis. Rotor 2, of t0 in thickness, is where the tool engages the workpiece. Rotor 3, the machined section, has a length measuring from (l + t0 /2) to l0 . The mass of each rotor can be determined as follows m1 = ρπ r12 (l − t0 /2)

(1)

m2 = ρ π r32 t0 + ρ π r1t0 (r1 − r3 )

(2)

ρ π r32 (l0 − l − t0 /2)

(3)

m3 =

As the configuration is not exactly a cylinder, the center of mass of Rotor 2 differs from Rotors 1 and 3. Although its section thickness and mass are small compared with those of Rotors 1 and 3, nonetheless, Rotor 2 is where the cutting force exerts and a new machined surface is being generated, as seen in Fig. 3. Thus, the response of Rotor 2 dominates the dynamics of the 3-rotor model system.


299

y1

Y

G1

Y B1

D

k

B1

C1

x1

Z O

/2 l - t0 2

X

X cross section through center of gravity of rotor 1

/2 l - t0

B1, C1 and G1 are in the same plane where B1 is on the spindle axis C1 is the geometric center G1 is the center of mass B1C1 = u1 = x12 + y12 C1G1 - H - eccentricity

2.1

Fig. 4 Rotor 1 configuration.

Acceleration and Velocity of Rotor Center of Mass

At spindle speed, Ω, the Rotor 1 illustrated in Fig. 3 would see an angle α = ψ0 + Ωt between the mass center (G1) and the geometric center (C1). It is noted that ψ0 is the angle between the X-axis and −−−→ eccentricity ε1 = C1G1 at t = 0. C1 coincides with B1 when the rotor is at rest. While in motion, C1 −−−→ has x1 and y1 components in the X − and Y −direction, respectively. When C1G1 aligns with the X-axis at t = 0, ψ0 = 0. The position vector of the center of mass of Rotor, G1, is therefore −−→ −−→ −−−→ −−−→ OG1 = OB1 + B1C1 + C1G1 = 1/2(l − t0 /2)k + (x1 + ε1 cos α )i + (y1 + ε1 sin α ) j

(4)

The corresponding velocity and acceleration of G1 are l˙ v1 = (x˙1 − ε1 Ω sin Ωt)i + (y˙1 + ε1 Ω cos Ωt) j − k 2

(5)

a1 = (x¨1 − ε1 Ω2 cos Ωt)i + (y¨1 − ε1 Ω2 sin Ωt) j

(6)

Note that at constant spindle speeds α˙ = Ω, α¨ = 0. In the same token, the velocity and acceleration of the center of mass of Rotor 3 are l˙ v3 = (x˙3 − ε3 Ω sin Ωt)i + (y˙3 + ε3 Ω cos Ωt) j − k 2

(7)

a3 = (x¨3 − ε3 Ω2 cos Ωt)i + (y¨3 − ε3 Ω2 sin Ωt) j

(8)

Considering the reduction of eccentricity due to material removal, eccentricity is assumed to be proportional to the radius of each section. Thus,

ε3 =

r3 ε1 r1

(9)

It is evident from Fig. 3 that the mass center of Rotor 2 does not coincide with its geometric center. Hence it is proper to assume that this mass center (which is determined by the shape) dominates the gravitational property of Rotor 2. As the tool moves toward the spindle end, the location of Rotor 2 also changes in time. However, the configuration and orientation of the rotor do not vary. As a result, −−−→ C2 is always below G2 and C2G2 is always parallel to the X −axis. Thus, −−−→ C2G2 = x′2 i + y′2 j

(10)

300


and

−−→ OG2 = (l − t0 /2 + z′2 )k + x′2 i + x2 i + y2 j + y′2 j

1 (r1 −r3 )(2r1 −r3 ) where x′2 = −r , y′2 = 0, and z′2 = 2π (r32 +r1 (r1 −r3 )) The acceleration of G2 is therefore

2 2 t0 (3r3 +2r1 −2r1 r3 ) 2 2 6 (r3 +r1 −r1 r3 )

(11)

. It is noted that both x′2 and z′2 are constants.

a2 = x¨2 i + y¨2 j

(12)

Because G2 is fixed with respect to C2, Rotor 2 undergoes only a curvilinear translation, not a rotation. Being a function of l, the position of Rotor 2 constantly changes in time while G2 always holds the same location with respect to C2, indicating that the rotor maintains the same shape at all time. The angular velocity of Rotor 2 is therefore zero. Also note that Rotor 2 and the tool both translate towards the ˙ and that while Rotor 2 is of a constant section thickness, end of the spindle end at a constant speed, l, t0 , at all time, the thicknesses of Rotors 1 and 3 vary with time. Since all the depth-of-cuts (DOCs) considered in the study are smaller than 6% of the diameter of the cylindrical workpiece, for the purpose of finding the static deflections at certain locations of interest, the workpiece can be assumed to be a uniform shaft with an average diameter, dAV , as dAV =

t0 d1 + d3 t0 1 [d1 (l − ) + ( )t0 + d3 (l0 − l + )] l0 2 2 2

(13)

where d1 and d3 are the diameters of Rotors 1 and 3, respectively. The average area moment of dAV . The shaft is Fig. 2 is subjected to a fixed-pinned boundary inertia of the shaft is therefore IAV = π 64 condition and a concentrated load. The static deflection of the center of gravity of each rotor can then be determined as follows [27]. Assume that δ1 , δ2 and δ3 are the deflections of Rotors 1, 2 and 3, respectively, and that vibration displacements, ui , are proportional to static deflections, δi . Thus, u3 u2 u1 δ1 = δ2 = δ3 [28]. Note that ui is also the radial displacement of the center of gravity of Rotor i and the resultant cutting force acting radially on the shaft is the concentrated load. Since gravity is negligible compared to the cutting force, one has δx11 = δx22 = δx33 and δy11 = δy22 = δy33 , or equivalently, x1 = δδ12 x2 , x3 = δδ32 x2 , y1 = δδ12 y2 and y3 = δδ32 y2 . Note that deflections are functions of t0 , l0 , and l, the concentrated load and the average area moment of inertia. As deflection ratios are functions of l, and l in turn is a function of time, all accelerations ai are also functions of x2 , y2 and time. Deflection ratios can then be expressed as

δ12 =

δ1 = δ2

and

δ32 =

2.2

δ3 = δ2

l2 [3l0 − l] [(l0 − (l−t20 /2) )3 − 3l02 (l0 − (l−t20 /2) )] 2l03 + 3l 2 (l0 − (l−t20 /2) ) − (l − (l−t20 /2) ) l2 [3l0 − l][(l0 − l)3 − 3l02 (l0 − l)] + 3l 2 (l0 − l) 2l03

l2 0 /2) 3 0 /2) [3l0 − l] [(l0 − (l0 −l−t ) − 3l02 (l0 − (l0 −l−t )] 2 2 2l03 (l0 −l−t0 /2) (l0 −l−t0 /2) 2 ) − (l − ) + 3l (l0 − 2 2 2 l [3l0 − l][(l0 − l)3 − 3l02 (l0 − l)] + 3l 2 (l0 − l) 2l03

(14)

(15)

Cutting Forces on Workpiece

The forces on Rotors 1 and 3 are due to the stiffness of the shaft. Rotor 2 sees three more forces in the X ,Y , and Z−direction that together cut to the material. The force on Rotor 1 due to shaft stiffness is −−−→ k1 u1 and its orientation is shown in Fig. 5. Here u1 = B1C1. The component form of the force on Rotor 1 is therefore (16) (−k1 u1 cos θ ) i − (k1 u1 sin θ ) j


301

k1u1 B1 X

C1 T

Y

Fig. 5 Force on Rotor 1.

k 2 u2 Z

Y G2

B2

FZ

C2

FT

FY FX X

B2 - Bearing Center C2 - Geometric Center G2 - Center of mass

Fig. 6 Cutting force components on Rotor 2.

Given the definitive geometric relations that cos θ = x1 /u1 and sin θ = y1 /u1 , the forces on Rotors 1 and 3 are therefore, respectively, (−k1 x1 ) i − (k1 y1 ) j (17) and (−k3 x3 ) i − (k3 y3 ) j

(18)

In addition to the shaft stiffness induced forces, a cutting force also acts on Rotor 2. Fx, Fy and Fz shown in Fig. 6 are the three components of the cutting force on Rotor 2 and FT is the force due to bending stiffness in the Z-direction. Since the workpiece is constrained in the Z-direction, the acceleration of Rotor 2 in the Z-direction is therefore zero, FT = Fz. Thus the force acting on Rotor 2 is (19) (−k2 u2 cos θ ) i − (k2 u2 sin θ ) j − FX i + FY j which can be further simplified as (−FX − k2 x2 )i + (FY − k2 y2 ) j

(20)

dm Applying Newton’s 2nd Law, ~ F = m dv dt + v dt , in the X − and Y − directions and making use of Eqs. (5)-(12) and (17)-(20), and the time derivatives of Eqs. (1) and (3), the following two equations are obtained: dm3 dm1 + v3x (21) −k1 x1 − k2 x2 − k3 x3 − FX = m1 a1x + m2 a2x + m3 a3x + v1x dt dt

−k1 y1 − k2 y2 − k3 y3 + FY = m1 a1y + m2 a2y + m3 a3y + v1y

dm3 dm1 + v3y dt dt

Note that Rotor 2 does not change its mass while in constant speed cutting.

(22)

302


t0 Actual cut in previous rotation P'

Y

Q'

Z Path of the tool if there is no vibration

zt

zt'

Q P X 2nd rotor

P

Q

tool moving direction

Flatted outer surface

2.3

Fig. 7 Instantaneous feed per revolution.

Instantaneous Chip Width

The equation that describes the tool motion in the spindle direction (Z-axis) can be written as MZ z¨t + kZ zt + kZC zt3 = FZ

(23)

where MZ is the equivalent mass of the machine tool, kZ and kZC are the equivalent linear and nonlinear stiffness of the tool in the Z-direction, respectively, lowercase zt is the relative displacement of the tool at time t, zt′ is the relative displacement of the tool one revolution before, and FZ is the cutting force component in the Z-direction. Note that per [15] and [22] cubic nonlinearity is an inherent property of the tool in machining. Per Fig. 7 where a PQ section of Rotor 2 actively engaging with the tool and showing a flatted outer surface on one side is seen, the instantaneous chip width, ti , is ti = t0 − zt + zt′

(24)

The tool vibrates along in the Z-axis. Note that when the tool is stationary, the corresponding chip width equals to the feed rate [29]. It is important to understand that for longitudinal turning, tool feed direction is in the direction along the workpiece. Since the tool is considered infinitely stiff in the X −Y plane and the workpiece is rigidly constrained along the Z− direction, only tool responses in the Z− direction and workpiece responses in the X − and Y − directions are considered in the followings. Thus it is fundamentally flawed that tool feed is taken to follow the inward radial direction [30]. 2.4

Instantaneous DOC

When instantaneous DOCs are only affected by vibrations in the Y − direction, the corresponding DOC = r1 − r3 − y2 . However, as seen inq Fig. 8, X -direction vibrations also contribute to DOC variations, at

which the corresponding DOC = r12 − x22 − r3 . Consider vibrations in both the X − and Y −direction, the total instantaneous DOC, s, is therefore, q s = r12 − x22 − r3 − y2 (25) Note that the side cutting edge angle (lead angle) does not play a role in the instantaneous chip width or instantaneous DOC. But it does affect the chip load and hence the cutting force. However, corner radius of the tool is not being taken into account in the study as it is much smaller than the DOCs considered herein.


303

r3

r1 C2

x2 r12 - x22

Fig. 8 DOC affected by vibrations along X-direction.

2.5

Cutting Force Components

Cutting force components, FX, , FY , and FZ , can be written in the following form [10], FX = Ff (bx1 sin(ηc ) + bx2 cos(ηc )) + bx3 Fn

(26)

FY = Ff (by1 sin(ηc ) + by2 cos(ηc )) + by3 Fn

(27)

FZ = Ff (bz1 sin(ηc ) + bz2 cos(ηc )) + bz3 Fn

(28)

with Ff being the friction force, Fn the normal force, and ηc the chip flow angle. Constants brm ’s (where r = x, y, z and m = 1, 2, 3) are functions of tool rake angle, side cutting edge angle and inclination angle. The chip flow angle can be formulated as a function of DOC valid between 0.4mm and 2.5mm, as per the experimental result reported in [10]. The chip flow angle is therefore

ηc = α1 s3 + α2 s2 + α3 s + α4

(29)

where αi ’s are fitted constants. The normal force and the friction force are as follows, Fn = kn AC

(30)

Ff = k f AC

(31)

in which kn and k f are normal and friction pressure components, respectively, and the chip cross sectional area, AC , is defined using the instantaneous chip width, ti , and the instantaneous DOC, s, as AC = ti s

(32)

By substituting Eqs. (24) and (25) into the equation above, the chip cross-sectional area becomes q (33) AC = (t0 − zt + zt′ )( r12 − x22 − r3 − y2 ) Note that kn and k f are not constants, but rather they are functions of ti as follows kn = Knmti + Knn

(34)

k f = K f mti + K f f

(35)

where Knm = −6.0 × 1012 N/m2 , Knn = 2.9 × 109 N/m3 , K f m = −8.7 × 1012 N/m2 , and K f f = 3.0 × 109 N/m3 . Using Eqs. (24), (25), and (29)-(35), cutting force components FX, , FY , and FZ can be determined.

304

2.6


Rotor Stiffness

The shaft in Fig. 2 is a circular beam being fixed to the chuck at one end and pinned to the tails stock 12EIl03 at the other. The stiffness at a particular cross section is k = a3 b2 (3l0 +b) , where E is the elastic modulus, I is the area moment of inertia, a is the length between the cross section considered and the fixed end, b is the length between the cross section considered and the pinned end, and l0 is the length of the beam [31]. Per the explanation made in the previous section, the area moment of inertia, I, can be replaced with IAV . The stiffness associated with each rotor is thus, ki =

12EIAV l03 a3ii b2ii (3l0 + bii )

(36)

where aii is the distance between the fixed end and the cross section sliced through the center of gravity of the i−th rotor, and bii is the distance between the pinned end and the cross section cut through the center of gravity of the i−th rotor. 2.7

Equations of Motion

The correlation x¨1 , x˙1 , x¨2 , x˙2 , x¨3 , x˙3 with y¨1 , y˙1 , y¨2 , y˙2 , y¨3 , y˙3 can be established by considering l = li − lt˙ along with Eqs. (14) and (15) [1]. The established relationships can then be substituted into Eqs. (5)-(8) and (12). The five resulted equations, along with Eqs. (1)-(3) and (36), and the derived equations for cutting force components, can then be substituted into Eqs. (21)-(23) to obtain three equations of motions that are functions of x¨2 , x˙2 , x2 , y¨2 , y˙2 , y2 , z¨t , zt , zt′ and time, t, as follows f1 (t)x¨2 + f2 (t)x˙2 + f3 (t)x2 = f4 (t, x2 , y2 , zt , zt′ )

(37)

f5 (t)y¨2 + f6 (t)y˙2 + f7 (t)y2 = f8 (t, x2 , y2 , zt , zt′ ) MZ z¨t + kZ zt + kZC zt3 = f9 (t, x2 , y2 , zt , zt′ )

(38) (39)

where f j (t)’s and fi (t, x2 , y2 , zt , zt′ )’s are nonlinear functions defined in the followings f1 (t) = f5 (t) = A2

(40)

f2 (t) = f6 (t) = 2A3 − A6

(41)

f3 (t) = f7 (t) = A4 + A5 − A7

(42)

A1 Ω cos(Ωt) − A8 Ω sin(Ωt) − FX

(43)

A1 Ω sin(Ωt) + A8 Ω cos(Ωt) + FY

(44)

FZ

(45)

A 1 = m 1 ε1 + m 3 ε3

(46)

A2 = 2(m1 δ12 + m2 + m3 δ32 ) A3 = m1 δ˙12 + m3 δ˙32

(47) (48)

A4 = m1 δ¨12 + m3 δ¨32

(49)

A5 = k1 δ12 + k2 + k3 δ32 ˙ 12 δ¨12 − r32 δ¨32 ) A6 = π ρ l(r ˙ 2 δ˙12 − r2 δ˙32 ) A7 = π ρ l(r

(50)

˙ 12 ε1 − r32 ε3 ) A8 = π ρ l(r

(53)

f4 (t, x2 , y2 , zt , zt′ ) = f8 (t, x2 , y2 , zt , zt′ ) = f9 (t, x2 , y2 , zt , zt′ ) =

2

2

with

1

3

(51) (52)


J

305

rake face

surface A

J- end cutting edge angle Cs - side cutting edge angle i - inclination angle (back rake angle) e - end relief angle D - top rake angle - the angle between two sufaces: surface A and rake face

Cs

Top view surface A

i

e

Side view

Fig. 9 Tool angles with standard terminology. speed, v = 2 Sr: chip thickness (feed), t 0

v chip

D- rake angle speed , :

tool insert chip

tool insert

Fig. 10 Cutting action and tool rake angle.

The system of equations above are employed in the subsequent section to determine x2 , y2 , zt . Through doing so one can study (1) the motion of the workpiece relative to the tool in the X −and Y −direction and (2) the motion of the tool relative to the machining surface in the spindle direction, all as functions of the cutting force and whirling. 3 Effect of Tool Geometry In addition to speed, feed and DOC that affect Material Removal Rate (MRR) and determines cutting force and hence power consumption, tool geometry is also one of the prominent parameters that impacts machining productivity. Surface roughness, chip formation changes, and chip flow angle are also affected by tool geometry. Even though chip flow angle is related to tool angles [32], in the model presented herein, chip flow angle is a function only of DOC. The angles used to derive cutting forces are shown in Fig. 9 with standard terminology. A better view of the rake angle, α , is given in Fig. 10 while undergoing cutting action. Tool rake angle determines the flow of the newly formed chip. Usually the angle is of a value between +5 to -5 degrees. In this study, parameter values used in [10] for tool rake angle, side cutting edge angle and inclination angle are adopted for the objective of being able

306


(a) time trace

-6

2

x 10

z(m)

z(m) 0

0

1

2

3

4

-1

5

frequency(Hz)

frequency(Hz)

1

2

3

4

5

t(s)

1000

4000

2000

3

3.5

4 t(s)

4.5

500

0

5

3

3.5

(c) Lyapunov spectrum

4 t(s)

4.5

5

1 Lyapunov exponent

1 Lyapunov exponent

0

(b) instantaneous frequency

6000

0.5 0 -0.5 -1

0 -0.5

t(s)

0

x 10

0.5

1

-1

-5

1

0

100 200 no. of data points

300

0.5 0 -0.5 -1

0


300

Fig. 11 Z-direction time responses, corresponding instantaneous frequency and Lyapunov spectra for Set #1(left) and Set #2 (right) for DOC = 0.75mm at Ω = 1250rpm. Table 1 Input data for numerical simulations of periodic motions (δ = 0.5, α = −10.0, β = 10.0, Q0 = 10.0). Set number

Side cutting edge angle, (Cs )/(degrees)

Rake angle, (α ) /(degrees)

Inclination angle, (i) /(degrees)

#1

45

3.55

3.55

to compare with experimental data. Positive rake makes the tool sharp, but it also weakens the tool compare with negative rake. Negative rake is better for rough cutting. Selection of tool geometry depends on the particular workpiece and tool materials being considered. To investigate if tool angle impacts cutting stability, two groups of tool geometry, tabulated in Table 1, are considered to define the corresponding cutting force. Both sets are taken from the tool inserts that were used in the experiments in [10]. When DOC is less than 1mm, considered as non-rough cutting, the corresponding rake angle can be taken as positive. Recall that negative rake is better for roughing. Two DOCs at 0.75mm and 0.5mm are used with a 1250rpm spindle speed and 0.0965mm feed per revolution in the numerical experiment. Except for Fig. 13, Figs. 11 and 12 give time responses, instantaneous frequency responses between 3 to 5 seconds, and the corresponding Lyapunov spectra. Tool dynamical responses in the Z-direction that correspond to DOC = 0.75mm and 0.5mm are analyzed to investigate tool motions. In the figures, plots in the right column correspond to Set #1 tool geometry conditions and the left column corresponds


(a) time trace

-6

1

x 10

2

1

0

0

-0.5 -1

0

1

2

3 t(s)

4

-1

5

1

2

3

4

5

t(s)

6000 frequency(Hz)

frequency(Hz)

0

(b) instantaneous frequency

6000

4000

2000

0

-6

x 10

z(m)

z(m)

0.5

307

3

3.5

4 t(s)

4.5

4000

2000

0

5

3

(c) Lyapunov spectrum

3.5

4 t(s)

4.5

5

Lyapunov exponent

Lyapunov exponent

1 4

2

0 0


300

0.5 0 -0.5 -1

0


300

Fig. 12 Z-directiontime responses, corresponding instantaneous frequency and Lyapunov spectra for Set #1(left) and Set #2 (right) for DOC = 0.50mm at Ω = 1250rpm.

to Set #2 tool angles. Even though force fluctuations and vibration amplitudes are both prominent, Set #2 is relatively more stability. The two DOCs considered behave dissimilarly. In all the cases, the vibration history of Set #1 has amplitudes that are of nanometer in scale. On the other hand, Set #2 vibrates with amplitudes that are a few microns for DOC = 0.75mm, and several nanometers for DOC = 0.5mm. Unlike Set #1, all Lyapunov spectra for Set #2 are evident of a stable state of dynamic response. Though having positive Lyapunov exponents, Set #2 shows instability for the DOC = 0.5mm case. Although the Lyapunov spectrum in Fig. 11 indicates a stable state of tool motion for Set #1 at DOC = 0.75mm, the corresponding instantaneous frequency suggests otherwise. The instantaneous frequency plot for Set #2 at DOC = 0.5mm also contradicts the Lyapunov spectrum. A detail review of the individual instantaneous frequency mono-components in Fig. 13 reveals that the frequency at 3240Hz has bifurcated 3 times. Thus, it is a highly bifurcated state. 4 Effect of Tool Feed Per Revolution There are three parameters that define Material Removal Rate (MRR). Constraints dictating how fast material is removed are cutting speed, feed rate and DOC. Tool feed rate, t0 , can be defined as the amount that tool moves in the axial direction per revolution of the workpiece [32, 33]. Tool feed rate is also called “undeformed chip thickness” in turning. Feed rate can be as small as 0.00125mm per

308


instantaneous frequency 5000

0

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

6000 4000 2000 3000 2000

frequency(Hz)

1000 0 3000 2000 1000 0 2000 1000 0 40 20 0

t(s)

frequency spectrum (top) and its mono components for DOC = 0.50mm at Ω = 1250rpm. Fig. 13 Instantaneous

revolution for light cuts with very thin chips or as large as 2.5mm per revolution for heavy cuts. The presented model in Section 2 was derived for a feed rate ranging between 0.05mm and 0.15mm. In practice, the parameter that can be set is the tool transverse speed, fs , in the axial direction. The relationship between feed rate and tool transverse speed is fs = t0 Ω/60

(54)

where fs and t0 are both in mm per second, and spindle speed Ω is in rpm. Because it is a major parameter in increasing MRR, this section focuses on examining the effect of t0 on cutting stability. The model is simulated with two different feeds for comparison. A constant spindle speed of 1250rpm and six different DOCs are considered with a 0.0965mm chip thickness. Numerically integrated time response data are analyzed to obtain instantaneous frequency and the largest Lyapunov exponents for stability analysis. Figs. 14-25 in this section display system dynamics for two feed rates and six different DOCs. Figs. 14, 16, 18, and 20 give the X- and Z-direction model responses corresponding to tool feed t0 = 0.125mm and DOC = 1.00mm, 1.25mm, 1.62mm and 1.75mm, respectively. Since their respective Lyapunov spectra are all zero, only time history (top) and instantaneous frequency (bottom) are shown. These results are compared with the responses associated with the feed t0 = 0.0965mm shown in Figs. 11-14 in [1]. Fig. 22 carries the responses corresponding to DOC = 2.00mm. It is compared with the responses associated with t0 = 0.0965mm in Figs. 15 in [1]. Fig. 24 presents the responses


time trace

-6

1

x 10

309

-6

10

x 10

5

0

z(m)

x(m)

0.5

-0.5

0

-1 -1.5

0

1

2

3

4

-5

5

0

1

2

t(s)

3

4

5

4.5

5

t(s) instantaneous frequency 1000

frequency(Hz)

frequency(Hz)

4000

3000

2000

1000

0

3

3.5

4

4.5

800 600 400 200 0

5

3

3.5

4

t(s)

t(s)

Fig. 14 X- (left) and Z-(right) direction model responses at DOC = 1.00mm, Ω = 1250rpm and t0 = 0.125mm. 230

86

84

220

Fz(N)

Fx(N)

225

215

80

210 205

82

0

1

2

3

4

78

5

0

1

2

t(s) 270

5

3

4

5

96

Fz(N)

Fx(N)

4

97

265

260

255

250

3

t(s)

95 94 93

0

1

2

3

t(s)

4

5

92

0

1

2

t(s)

Fig. 15 X- and Z-direction cutting forces for t0 = 0.0965mm (top) and t0 = 0.125mm (bottom) at DOC = 1.00mm and Ω = 1250rpm.

310


time trace

-6

x 10

1

-6

10

x 10

5

0

z(m)

x(m)

0.5

-0.5

0

-1 -1.5

0

1

2

3

4

-5

5

0

1

2

t(s)

3

4

5

4.5

5


frequency(Hz)

frequency(Hz)

4000

3000

2000

1000

0

3

3.5

4

4.5

800 600 400 200 0

5

3

3.5

4

t(s)

t(s)

280

126

275

124

270

122

Fz(N)

Fx(N)

Fig. 16 X- (left) and Z-(right) direction model responses at DOC = 1.25mm, Ω = 1250rpm and t0 = 0.125mm.

265 260 255

120 118

0

1

2

3

4

116

5

0

1

2

t(s) 330

5

3

4

5

139

Fz(N)

Fx(N)

4

140

325

320

315

310

3

t(s)

138 137 136

0

1

2

3

t(s)

4

5

135

0

1

2

t(s)



time trace

-6

1

x 10

311

-6

x 10

10

5

0

z(m)

x(m)

0.5

-0.5

0

-1 -1.5

0

1

2

3

4

-5

5

0

1

2

t(s) instantaneous frequency

frequency(Hz)

frequency(Hz)

4

5

4.5

5

1000

4000 3000

2000

1000

0

3

t(s)

3

3.5

4

4.5

800 600 400 200 0

5

3

3.5

4

t(s)

t(s)

355

178

350

176 174

345

Fz(N)

Fx(N)


340

170

335 330

172

168 0

1

2

3

4

166

5

0

1

2

t(s) 204

425

202

Fz(N)

420

Fx(N)

4

5

3

4

5

t(s)

430

415 410

200 198 196

405 400

3

0

1

2

3

t(s)

4

5

194

0

1

2

t(s)


312


time trace

-6

1

x 10

-6

10

x 10

5

0

z(m)

x(m)

0.5

-0.5

0

-1 -1.5

0

1

2

3

4

-5

5

0

1

2

t(s)

3

4

5

4.5

5


frequency(Hz)

frequency(Hz)

4000

3000

2000

1000

0

3

3.5

4

4.5

3000

2000

1000

0

5

3

3.5

4

t(s)

t(s)

390

200

380

195

Fz(N)

Fx(N)


370

360

350

190

185

0

1

2

3

4

180

5

0

1

2

t(s)

3

4

5

3

4

5

t(s)

470

226 224

460

Fz(N)

Fx(N)

222 450

220 218

440 216 430

0

1

2

3

t(s)

4

5

214

0

1

2

t(s)



time trace

-6

x 10

x 10

10

0 -1

-2

-6

15

z(m)

x(m)

1

313

5 0

0

1

2

3

4

-5

5

0

1

t(s)

2

3

4

5

t(s) instantaneous frequency 1000 frequency(Hz)

frequency(Hz)

4000 3000 2000 1000 0

3

3.5

4

4.5

t(s)

0

5

3.5

4

4.5

5

t(s)

1 Lyapunov exponent

Lyapunov exponent

3

Lyapunov spectrum

1 0.5 0 -0.5 -1

500

0


300

0.5 0 -0.5 -1

0


300


corresponding to DOC = 2.49mm. The corresponding X- and Z-direction cutting forces in response to DOC = 1.00mm through 2.49mm with feed t0 = 0.0965mm and 0.125mm are shown in Figs. 15, 17, 19, 21, 23, and 25. Comparing with the workpiece responses in Fig. 11 in [1], the vibration amplitudes in Fig. 14 associated with DOC = 1.00mm and t0 = 0.125mm are 10% larger. While the corresponding tool vibration amplitudes do not show discernible differences, the associated instantaneous frequency of the lower tool feed (t0 = 0.0965mm) has more frequency components. Though the time-frequency data show fewer bifurcations in the lower feed case, the one broadband component is indicative of instability, as is confirmed the same by the positive Lyapunov spectrum in Fig. 11 in [1]. The instantaneous frequency of the higher feed case has more bifurcations. Nonetheless, it is dynamically stable. Note that in Fig 15 the X- and Z- direction force magnitudes increase with increasing feed rate. This is evidently true since higher chip load would result in higher cutting force. When DOC is increased to 1.25mm in Fig 16, the X-direction vibration amplitude is again higher than that of the lower feed in Fig. 12 in [1]. While the lower feed case shows stable frequency characteristics for the X-direction response, increasing feed to 1.25mm in Fig 16 is seen to impact stability in a negative way. The tool vibratory patterns associated with the two feed rates are very different at this DOC. Higher feed is accompanied by a 25% increase in tool vibration amplitude. Moreover, whirling frequency is readily visible in the tool instantaneous frequency plot of the higher feed rate case. In fact, whirling frequency is prominent in all instantaneous frequency plots corresponding to the case of 0.125mm feed rate. The associated cutting force components display very different behaviors in Fig. 17. The X-/Z-direction cutting force of the lower feed case oscillates with 18N/7N.

314


700

300

600

250

Fz(N)

Fx(N)

500 400 300

150 100

200 100

200

0

1

2

3

4

50

5

0

1

2

t(s) 540

270

530

265

Fz(N)

Fx(N)

520 510 500

4

5

3

4

5

260 255 250

490 480

3

t(s)

0

1

2

3

t(s)

4

5

245

0

1

2

t(s)


In contrast, the case of higher feed rate oscillates with a mere 0.2N/0.1N. Note that the magnitude of the X-/Z-direction cutting force increases by 20%/15% for the higher feed rate case. Increase DOC further to 1.62mm, the workpiece vibration patterns of the two feed rate cases in Fig. 18 and Fig. 13 in [1] are seen to be similar. However, the vibration amplitude of the higher feed rate case is 12% higher. The lower feed rate case demonstrates a broadband frequency characteristic. Tool vibratory characteristics of the two cases are visibly distinct. The case of lower feed rate has tool vibration amplitude about 10 times higher than that of the higher feed rate. This is the case when tool structural nonlinearity becomes dominant. Workpiece vibration amplitude of the higher feed rate case, on the other hand, is always larger than the lower feed rate case. Tool instantaneous frequency plots in Fig. 18 illustrate that higher feed is dynamical more stable than the lower feed. Oscillation of cutting force seen in Fig. 19 is also smaller for the higher feed, with the magnitude showing a similar percentage increase as in previous cases. Fig. 20 presents X- and Z-direction responses for DOC = 1.75mm and t0 = 0.125mm. Compare with the responses in Fig. 14 in [1], it can be seen that workpiece vibration responses are similar, though the higher feed case has slightly larger vibration amplitude. Despite the fact that both responses are stable, their instantaneous frequency plots show bifurcations. Tool vibration amplitude corresponding to the higher feed rate is of a few nanometers. The lower feed rate case has its vibration amplitudes in Fig. 14 in [1] that is 1,000 times higher. The tool instantaneous frequency response of the lower feed rate case is well behaved, while the higher feed rate case is evidently a highly bifurcated state. Cutting force oscillations for both cases in Fig. 21 are in the same range and of the similar characteristics as in Figs. 17 and 19. Responses associated with DOC = 2.00mm and t0 = 0.125mm in Fig. 22 show stability in both the time and frequency domains. Their corresponding Lyapunov spectra also confirm the same. Moreover, the corresponding cutting forces in Fig. 9, 10 show very large oscillations of hundreds of Newtons


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in magnitude. Forces of the higher feed rate have lower oscillation amplitudes. Extreme machining instability is seen in Fig. 23 where higher feed rate and DOC = 2.49mm are considered. This implies that the stability limit for t0 = 0.125mm is in between DOC = 2.00mm and 2.49mm. Fig. 25 shows the corresponding X- and Z-direction cutting force components. It can be seen that the cutting forces oscillate randomly with very high amplitudes, thus signifying a state of instability. 5 Concluding Remarks Effects of tool geometry and too feed rate on cutting stability were investigated using a comprehensive model whose feasibility was physically demonstrated in [34] and [35]. Manufacturing industry has long learned to employ tool inserts with complex geometry to achieve better product surface condition. However, most models developed for understanding machining dynamics and cutting stability ignore the various effects attributable to tool geometry. One of the reasons for this is the fact that 1D machining modeling is inherently infeasible for incorporating various tool angles that are inherently 3-dimensional. Numerical investigations presented in the paper deem neglecting tool geometry improbable. It was observed that variations in tool geometry can significantly impact cutting stability. A machining process can be unstable for a particular DOC using one set of tool geometry and, become stable through careful selection of proper tool inserts with different set of tool geometry, at the same DOC. This raises the question over if true dynamical stability could be identified without considering tool geometry. It is essential that tool geometry is also considered in modeling 3D turning operation to

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ensure proper interpretation of cutting dynamics. The effects of two feed rates (or undeformed chip thicknesses) on cutting dynamics involving whirling were studied. It was shown that tool feed impacted cutting force and workpiece and tool vibrations. X/Z-direction cutting force magnitude was 20%/15% higher for higher feed rate. Numerical experiment performed in the section supported that increasing chip thickness induces higher chip load, and higher chip load in turn results in larger cutting forces. These higher forces induced larger workpiece deflection, thus higher vibration magnitudes. Nonlinearity of the tool structure had a dominant effect on the vibration amplitude of the tool. At high DOCs, it was seen that high feed rate contributed to stability, thus indicating that feed rate was among the parameters that impact cutting stability. It should be cautioned that these results, which were obtained by considering only two feed rates, are insufficient to conclude that higher feeds would bring about higher stability limit. Further study on investigating manipulating feed rate to achieve higher stability limit is therefore needed. References [1] Dassanayake, A.V. and Suh, C.S. (2008), On Nonlinear Cutting Response and Tool Chatter in Turning Operation, Communications in Nonlinear Science and Numerical Simulations, 13, 979-1001. [2] Dassanayake, A.V. and Suh, C.S. (2007), Machining Dynamics Involving Whirling. Part I: Model Development and Validation, Journal of Vibration and Control, 13(5), 475-506. [3] Dassanayake, A.V. and Suh, C.S. (2007), Machining Dynamics Involving Whirling. Part II: Machining States Described by Nonlinear and Linearized Models, Journal of Vibration and Control, 13(5), 507-526. [4] Arnold, R.N. (1946), The Mechanism of Tool Vibration in the Cutting of Steel, Proceedings of the Institution of Mechanical Engineering, 154, 267-284. [5] Statan, T.E. and Hyde, J.H. (1925), An Experimental Study of the Forces Exerted on the Surface of the Cutting Tool, Proceedings of The Institution of Mechanical Engineering, 1(2), 175-195. [6] Merit, H.E. (1965), Theory of Self Excited Machine Tool Chatter: Contribution to Machine Tool Chatter,


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ASME Journals of Engineers for Industry, 84, 447-454. [7] Suh, C.S., Khurjekar, P.P., and Yang, B. (2002), Characterization and Identification of Dynamic Instability in Milling Operation, Mechanical Systems and Signal Processing, 16(5), 829-848. [8] Clancy, B.E. and Shin, Y.C. (2002), A Comprehensive Chatter Prediction Model for Face Turning Operation Including Tool Wear Effect, International Journal of Machine Tools and Manufacture, 42, 1035-1044. [9] Volger, M.P., DeVor R.E., and Kapoor, S.G. (2002), Nonlinear Influence of Effective Lead Angle in Turning Process Stability, Journal of Manufacturing Science and Engineering, 124, 473-475. [10] Rao, B.C. and Shin, Y.C. (1999), A Comprehensive Dynamic Cutting Force model for Chatter Prediction in Turning, International Journal of Machine Tools and Manufacture, 9(10), 1631-1654. [11] Kim, J.S. and Lee, B.H. (1990), An Analytical Model of Dynamic Cutting Forces in Chatter Vibration, International Journal of Machine Tools and Manufacture, 31(3), 371-381. [12] Grzesik, W. (1996), A Revised Model for Predicting Surface Roughness in Turning, Wear, 194, 143-148. [13] Sahin, Y. and Motorcu, A.R. (2005), Surface Roughness Model for Machining Mild Steel with Coated Carbide Tool, Materials and Design, 26(4), 321-326. [14] Thomas, M., Beauchamp, Y., Youssef, A.Y., and Masounave, J. (1996), Effect of Tool Vibration on Surface Roughness during Lathe Dry Turning Process, Computers and Industrial Engineering, 31(3-4), 637-644. [15] Hanna, N.H. and Tobias, S.A. (1974), Theory of Non-linear Regenerative Chatter, ASME Journal of Engineering for Industry, 96, 247-255. [16] Litak, G. (2002), Chaotic Vibrations in a Regenerative Cutting Process, Chaos, Solutions, and Fractals, 13(7), 1531-1535. [17] Gouskov, A.M., Voronov, S.A., Paris, H., and Batzer, S.A. (2002), Nonlinear Dynamics of a Machining System with Two Independent Delays, Communications in Nonlinear Science and Numerical Simulations, 7, 207-221. [18] Wiercigroch, M. and Cheng, A.H.D. (1997), Chaotic and Stochastic Dynamics of Orthogonal Metal Cutting, Chaos, Solutions, and Fractals, 8(4), 715-726. [19] Wang, X.S., Hu, J., and Gao, J.B. (2006), Nonlinear Dynamics of regenerative Cutting Process – Comparison of Two Models, Chaos, Solitons, and Fractals, 29(5), 1219-1228. [20] Chandiramani, N.K. and Pothala, T. (2006), Dynamics of 2DOF Regenerative Chatter during Turning, Journal of Sound and Vibration, 290, 448-464. [21] Deshpande, N. and Fofana, M.S. (2001), Non-linear Regenerative Chatter in Turning, Robotics and Computer Integrated Manufacturing, 17(1-2), 107-112. [22] Nayfeh, A.H., Chin, C., and Pratt, J. (1997), Perturbation Methods in Non-linear Dynamics- Applications to Machining Dynamics, Journal of Manufacturing Science and Engineering, 119, 485-492. [23] Grabec, I. (1988), Chaotic Dynamics of the Cutting Process, International Journal of Machine Tools and Manufacture, 28(1), 19-32. [24] Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., et al. (1998), The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non-stationary Time Series Analysis, Proceedings of the Royal Society of London Series A, 454, 903-995. [25] Yang, B. and Suh, C.S. (2003), Interpretation of Crack-Induced Rotor Nonlinear Response Using Instantaneous Frequency, Mechanical Systems and Signal Processing, 18, 491-513. [26] Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A. (1985), Determining Lyapunov Exponents from a Time Series Analysis, Physica, 16D, 285-317. [27] Hopkins, R.B. (1970), Design Analysis of Shafts and Beam, Robert E. Krieger Publishing Company, Inc., Malabar, FL. [28] Mabie, H.H. and Ocvirk, F.W. (1975), Mechanisms and Dynamics of Machinery, John Wiley and Sons Inc., New York, NY. [29] Moon, F.C. (editor), (1998), Dynamics and Chaos in Manufacturing Processes, John Wiley and Sons Inc., New York, NY. [30] Olgac, N. and Hosek, M. (1998), A New Perspective and Analysis of Machine Tool Chatter , International Journal of Machine Tools and Manufacture, 38, 783-798. [31] Beachley, N.H. and Harrison, H.L. (1978), Introduction to Dynamic System Analysis, Harper and Row Publishers, New York, NY. [32] Childs, T.H.C., Maekawa, K., Obikawa, T., and Yamane, Y. (2000), Metal Machining: Theory and Applications, Arnold Publishers, London, Great Britain. [33] Trent, E.M. and Wright, P.K. (2000), Metal Cutting: Fourth Edition Butterworth Heinemann, Boston, MA. [34] Halfmann, E.B., Suh, C.S., and Hung, N.P. (2017), Dynamics of Turning Operation Part I: Experimental Analysis Using Instantaneous Frequency, Vibration Testing and System Dynamics, 1(1), 15-33. [35] Halfmann, E.B., Suh, C.S., and Hung, N.P. (2017), Dynamics of Turning Operation Part II: Model Validation and Stability at High Speed, Vibration Testing and System Dynamics, 1(1), 35-52.



The Ackerman Steered Car Non-Holonomic Lagrangian Mechanics System: Mathematics Problem Treatment of the Geometrical Theory Soufiane Haddout1†, Zhiyi Chen2 , Mohamed Ait Guennoun1 1 2

Department of Physics, Faculty of Science, Ibn Tofail University, B.P 242, 14000 Kenitra, Morocco University of British Columbia, 1935 Lower Mall, Vancouver, BC V6T 1X1 Canada Submission Info Communicated by J. Zhang Received 18 June 2017 Accepted 3 September 2017 Available online 1 January 2018 Keywords Ackerman Steered Car Lagrangian systems Nonholonomic constraints KReduced equations of motion Numerical solution

Abstract Mechanical systems have traditionally provided a fertile area of study for researchers interested in nonlinear control, due to the inherent nonlinearities and the Lagrangian structure of these systems. Recently, a great deal of emphasis has been placed on studying systems with nonholonomic constraints, including mobile wheeled robots and multiple-trailer vehicles, where the wheels provide a no-slip velocity constraint. In this paper, a new methods in non-holonomic mechanics are applied to a problem of an ackerman steered car motion for the first time. This method of the geometrical theory of general nonholonomic constrained systems on fibered manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by Krupkov´ a in 1990’s. The relevance of this theory for general types of nonholonomic constraints, not only linear or affine ones, was then verified on appropriate models. Frequently considered constraints on real physical systems are based on rolling without sliding, i.e. they are holonomic, or semi-holonomic, i.e. integrable. Moreover, there exist some practical examples of systems subjected to true (non-integrable) nonholonomic constraint conditions. On the other hand, the equations of motion of an ackerman steered car are highly nonlinear and rolling without slipping condition can only be expressed by nonholonomic constraint equations. In this paper, the geometrical theory is applied to the above mentioned mechanical problem using the above mentioned Krupková approach. The results of numerical solutions of constrained equations of motion, derived within the theory, are presented and thus they open the possibility of direct application of the theory to practical situations in engineers. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

† Corresponding


ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2017.12.003

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Soufiane Haddout, et al. / Journal of Vibration Testing and System Dynamics 1(4) (2017) 319–331

1 Introduction In some mechanical and engineering problems one encounters different kinds of additional conditions, constraining and restricting motions of mechanical systems. Such conditions are called constraints [1]. Constraints may be given by algebraic equations connecting coordinates (holonomic or geometric constraints), or by differential equations, which restrict coordinates and components of velocities. Non integrable kinematic constraints, which cannot be reduced to holonomic ones, are called non-holonomic constraints [1]. In addition, the motion of mechanical systems is frequently subjected to various constraint conditions, holonomic or nonholonomic. Nonholonomic constraints lead typically to nonlinear equations of motion of the constrained system. While theories of holonomic or some special types of linear non-holonomic constraints are already well elaborated for quite general situations, various theoretical approaches to general non-holonomic mechanics occur up to now, from the physical point of view on the one hand, and from the geometrical point of view on the other. On the other hand, in last decades numerous physical and engineerings applications make necessary to profound research and complete the theory of the nonholonomic systems and numerical aspects of solutions are presented. Therefore problems of nonholonomics mechanics are intensively studied in many recent papers, e.g., [2–10], in which are used modern methods and concepts of differential geometry and global analysis and which contribute to the essential advance in both from the theoretical and application aspects. The geometrical theory used in the presented paper was presented for first order mechanical problems in [11] and then generalized for higher order case in [12] brings an appropriate tool for constructing certain type of equations of motion of nonholonomic mechanical systems subjected to quite general constraints (an application to typically non-linear constraint see in [13]). The theory is developed on fibered manifolds and their jet prolongations as underlying geometrical structures, naturally related to the character of physical problems. The main physical idea of the theory is based on the concept of Chetaev-type constraint forces introduced in analogy to “classical” Chetaev forces (see [14]). Equations of corresponding unconstrained motion are related to the so-called dynamical form and they define the components of this form. Using equations of constraints a special canonical distribution on the first jet prolongation of the underlying manifold (corresponds to the phase space) can be constructed. Then first prolongations of admissible trajectories of the constrained motion are just integral sections of this distribution. By adding Chetaev-type forces (with Lagrange multipliers) to equations of motion, a dynamical form of the constrained problem is obtained and deformed equations of motion are constructed. These equations together with constraint conditions give the system of differential equations for unknown constrained trajectories and Lagrange multipliers. Another possible approach to the problem within the same theory starts from its description by the so-called Lepage class of forms instead the dynamical form itself. The Lepage class is, of course, closely related to the dynamical form, and it is obtained by the factorization of modules of forms by special submodules irrelevant from the point of view of the problem. This procedure leads to the so-called reduced equations of motion containing no Lagrange multipliers and giving the system of differential equations for constrained trajectories only. Nevertheless, constraint forces can be then obtained from deformed equations. In addition, on the base of the geometrical theory with Chetaev-type constraint forces, one can formulate a constraint variational principle and solve the corresponding constraint inverse variational problem (see e.g., [15]), as well as study symmetries of constrained systems. Symmetries and arising first integrals may then essentially simplify integration of the resulting constrained equations of motion, see e.g., [16]. (Nevertheless, in the present paper no attention is paid to higher order theories, field theories and the constraint variational problem.) Of course, the calculation procedure itself is made in coordinates. Its practical advantage lies in the possibility to choose appropriate coordinates, and also in two equivalent alternatives of solving the problem. The first of them is based on the solution of reduced equations of motion free of Lagrange multipliers and additional computation of these multipliers and corresponding constraint forces from


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dynamical equations, or alternatively, the direct solution of dynamical equations containing Lagrange multipliers. The decision between these two procedures is influenced by the concrete physical problem. Even though the corresponding constraint is semi-holonomic and thus it could be in principle treated by classical methods of Lagrange multipliers (for details concerning the method in general see e.g. the classical textbook of analytical mechanics [17]), the direct application of Krupkov´ a’s geometrical theory is very effective in this situation. On the other hand, a great interest has been devoted towards ackerman steered car modeling as it is a mechanical system characterized by nonholonomic constraints. On the other hand, many researchers have tried to find proper equations to describe the dynamic of this system. Mainly, it is possible to distinguish between two different approaches: the first obtains the motion equations using the Newton’s laws, while the second studies the system from a Lagrangian or Hamiltonian point of view ([19, 20]). So far, the greatest part of the existing literature has been dedicated to models with lots of simplifications, even if these have been capable to explain the dynamical characteristics of the ackerman steered car. For example, linearized equations of motion are commonly introduced in order to cope more easily with the problem. The aim of this paper is to use the geometrical theory for obtaining non-linear equations of motion of the above exposed mechanical problem, using the above mentioned Krupková approach for a practical mechanical system and find their solution in some particular cases, any simplification are not used. This is made in the last section, where the sets of equations of motion i.e., reduced, are derived. The numerical solution of reduced equation is presented. We arrange the remaining parts as follows: In Section II, we introduce the geometrical theory of nonholonomic mechanical systems. Further, Section III presents the nonlinear dynamics of ackerman steered car. After that, the proposed the reduced equations for an Ackerman steered car and numerical solution in Section VI. Finally, the paper is wrapped up with conclusions in Section V.

2 Geometrical concept of nonholonomic mechanical systems In this section, we recall basic geometrical concepts of the theory we will use. For more details and proofs see [11]. As underlying geometrical structures of the theory fibred manifolds and their jet prolongations are considered. Key geometrical objects adapted to the fibred structure are sections and their jet prolongations, projectile and vertical vector fields, as well as horizontal and contact differential forms. The detailed theoretical background can be noted in [20]. The geometrical theory of non-holonomic mechanical systems is developed on an (m+1)-dimensional underlying fibred manifold (Y, π , X ) with the one-dimensional base X is considered, ((t ∈ X ) being time in non-relativistic mechanics), m-dimensional fibers (configuration space), and its jet prolongations (J sY , πs , X ) with s = 1, 2 for typical physical cases (a fiber of J 1Y over (t ∈ X ) represents the phase space). We denote (V, ξ ), ξ = (t, qσ ), 1 ≤ σ ≤ m, a fibred chart on Y , (U, ζ ), U = π (V ), ζ = (t), the associated chart on X and (Vs , ξs ), Vs = πs−1 (U ), ξs = (t, qσ , qσs ) the associated fibred chart on J sY , where qσ1 = q˙σ and qσ2 = q¨σ . Moreover, denote by πr,s : J rY → J sY , 0 ≤ s ≺ r ≤ 2, J 0Y = Y , canonical projections. A section of fibered manifold (Y, π , X ) is a smooth mapping γ : I → Y , such that γ ◦ π = idI , I ⊂ X being an open set. Analogously sections of (J rY, πr , X ) are defined. A section δ of (J rY, πr , X ) is called holonomic if it is of the form δ = J rY , where γ is a section of (Y, π , X ). Recall, that a vector field η on J rY is called πr -projectile if there exists a vector field η0 on X such that T πr η = η0 ◦ πr . A vector field η is called πr -vertical if T πr η = 0. A form ρ on J rY is called πr -horizontal if its contraction by an arbitrary chosen πr -vertical vector field η vanishes, i.e. it holds iη ρ = 0. A form ρ is called contact if J r γ ∗ ρ = 0 for all sections γ of (Y, π , X ). Concepts of πr,s -projectile vector field, πr,s -vertical vector field, and πr,s -horizontal form are defined by the quite analogous way. Moreover, for every k-form ρ on J rY there exists the unique decomposition into its q-contact components, 0 ≤ q ≤ k, ∗ πr+1,r ρ = ∑kq=0 pq ρ , the 0-contact components pq ρ = hρ are called also the horizontal one.

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Fig. 1 Schematic general of geometry of the ackerman steered Car structure [Available at http://www.technicopedia.com/8880.html].

From the point of view of physics, all possible trajectories of so-called first order unconstrained mechanical system on a fibred manifold are given just by section γ of (Y, π , X ) such that they are solution of the system of m second order ordinary differential equations of motion: Eσ ◦ J 2 γ = 0,

Eσ = Aσ (t, qλ , q˙λ ) + Bσ ν (t, qλ , q˙λ ) q¨ν

(1)

where 1 ≤ λ ≤ m and Einstein summation are used. Consider the 1-contact π2,0 -horizontal 2-form on J 2 γ = 0, E = Eσ ω σ ∧ dt = 0, called dynamical form. A solution γ of Eqs (1) is called a path of E. We define the Lapage class [α ] of E by the requirement p1 α = E (see [11, 21]). The class [α ] is named also the mechanical system. Every representative of this class is of the form:

α = Aσ ω σ ∧ dt + Bσ ν ω σ ∧ dq˙σ + Fσ ν ω σ ∧ ω ν

(2)

where ω σ = dqσ − q˙σ dt are contact 1-forms forming the basis of 1-forms (dt, ω σ , q˙σ ) on J 1Y adapted to the contact structure. So, [α ] = α mod 2-contact forms. The following proposition was proved (see [11, 21]): Proposition 1. A section γ of (Y, π , X ) is a path of the dynamical from E if and only if J 1 γ ∗ iη α = 0 For every π1 -vertical vector field η on J 1Y .

(3)


323

3 Non-holonomic dynamics A non-holonomic constrained mechanical system is defined on the (2m + 1 − k)-dimensional constrained sub-manifold ℘ ⊂ J 1Y fibered over Y and given by k equations (1 ≤ k ≤ m − 1) f i (t, qσ , q˙σ ) = 0 such that rank(

∂ fi )=k ∂ q˙σ

1≤i≤k

Or in the explicit normal form q˙m−k+i − gi (t, qσ , q˙l ),

1 ≤ l ≤ m−k

(4)

It is evident that only admissible trajectories for a non-holonomic mechanical system are such sections γ : I t → Y for which J 1 γ (t) ∈ ℘ for all t ∈ I, i.e, f i ◦ J 1 γ = 0 for 1 ≤ i ≤ k (the so-called ℘-admissible sections). The constraint (4) leads to the canonical distribution ℑ of codimension k on ℘. Its annihilator is of the form ∂ gi (5) ℑ0 = span ϕ i , ϕ i = − l ω l + ι ∗ ω m−k+i ∂ q˙ where ι : ℘ → J 1Y is the canonical embedding. The canonical distribution is closely related to the constraint ideal Θ(ℑ0 ) (6) Θ(ℑ0 ) = ϕ i ∧ χi |χi is a form on ℘ where ϕ i are 1-forms on ℘ called canonical constraint 1-forms. The importance of the canonical distribution is evident from its following property (see [16, 21]): A section γ of Y is ℘–admissible if and only if J 1 γ is an integral section of the canonical distribution. We have already mentioned in the first part that there are two possible equivalent approaches to the description of non-holonomic mechanical system-one of them, called physical, is based on deformed equations with constraint forces and Lagrange multipliers and the other, geometrical one, uses reduced equations. Geometrical approach introduces the constrained mechanical system related to the mechanical system [α ] by the equivalence relation: [α℘] = [ι ∗ α ] mod Θ(ℑ0 )

(7)

A ℘ -admissible section γ of (Y, π , X ) is called a path constrained system [α℘] if for every π1 -vertical vector field η belonging to the canonical distribution it holds J 1 γ ∗ iη α = 0

(8)

The following proposition can be formulated (see again [22]): Proposition 2. A section γ of (Y, π , X ) is a path of the deformed system [αΦ ] if and only if for every π1 -vertical vector field η belonging to ℑ holds f i ◦ J 1 γ = 0, (A l + B ls q¨s ) ◦ J 2 γ = 0 k

A l = (Al + ∑ Am−k+ j j=1

k

k k ∂gj ∂gj ∂ gi ∂ gi σ + σ q˙ )) ◦ ι + (B + B )×( l,m−k+i m−k+ j,m−k+i ∑ ∑ ∂ q˙l i=1 ∂ q˙l ∂t ∂q j=1

B ls = (Bls + ∑ [Bl,m−k+i i=1

∂ gi ∂ gi ∂ g j ∂ gi + B ] + (B )) ◦ ι m−k+i,s m−k+ j,m−k+i ∂ q˙s ∂ q˙l ∂ q˙l ∂ q˙s

(9) (10)

(11)

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Relations (8) represent the system of reduced equations for m unknown functions qσ γ (k of them are first order and (m − k) second order ordinary differential equations). Physical approach is based on Chetaev-type constraint forces. Such a force is given by the constraint itself, in analogy with holonomic situations. It is expressed by the dynamical form [9]: Φ = Φσ ω σ ∧ dt = μi

∂ fi ∧ dt 1 ≤ i ≤ k ∂ q˙σ

(12)

where functions μi (t, qλ , q˙λ ) are Lagrange multipliers. Note that such dynamical form satisfies the generalized principle of virtual workiη Φ|U = 0 for every π1 -vectical vector field η belonging to the constraint distribution ℑU , ℑU0 = span ϕ i , d f i , 1 ≤ i ≤ k , U , U ∩ Q = 0/ being an open set of a chart on J 1Y , [9]. Denote [αΦ ] = [α − Φ],

αΦ = [Aσ − μi

∂ fi σ ]ω ∧ dt + Bσ ν ω σ ∧ dq˙σ + Fσ ν ω σ ∧ ω ν ∂ q˙σ

(13)

The equivalence class [αΦ ] is called the deformed mechanical system. A ℘-admissible section γ of (Y, π , X ) is called a path of [αΦ ] if (Eσ − Φσ ) ◦ J 2 γ . The following proposition holds (see [21, 22]): Proposition 3. A section γ of Y is a path of the deformed system [αΦ if and only if for every π1 -vectical vector field η on J 1Y it holds J 1 γ ∗ iη αΦ = 0 Or equivalently Aσ + Bσ ν q¨v = μi

∂ fi and f i ◦ J 1Y = 0 ∂ q˙σ

(14)

System (13) is given by k first order and m second order ordinary differential equations for unknown functions μi and qσ γ and it represents the deformed equation. Semiholonomic constraints: Let us now describe a special type of non-holonomic constraints, called semi-holonomic. Such conditions usually take place for rolling of rigid bodies without slipping. A system of constraints (3) is called semi-holonomic if the constraint ideal (5) is differential, i.e the canonical distribution (4) is completely integrable (see. e.g [11]). This means that d ϕ ∈ Θ(ℑ0 ) and thus following conditions hold:

∂c gi dc ∂ gi ∂ 2 gi ( − ) = 0, = 0, 1 ≤ l, s ≤ m − k, 1 ≤ i ≤ k ∂ ql dt ∂ q˙l ∂ q˙s ∂ q˙l where

(15)

dc

∂c ∂c ∂gj ∂ ∂c ∂ ∂ = + q˙l l + g j m−k+ j = + ( ) = 0, l l l m−k+ j ∂q ∂q ∂ q˙ ∂ q dt ∂t ∂q ∂q

In the following section we apply the obtained equations (8 and 13) obtained for general non-holonomic mechanical system to the example of ackerman steered car system.

4 Lagrange’s equation of Ackerman Steered Car We will consider the ackerman steered car shown in Figure 2. The ackerman steered car enables five degrees of freedom. The Lagrange equation determination can be determined by defining total potential and kinetic energy of the system as a function of generalized coordinates: x and y are the locations of the midpoint of the rear axle, θ is the orientation of the body with respect to the x-axis. To simplify the


325

Fig. 2 Coordinate systems of the Ackerman steered car [23].

equations system is modeled as a single drive wheel located along the medial axis of the vehicle. This imaginary wheel rotates about its axle by an angle ψ . The steering angle of each wheel is controlled by a single steering wheel, with a mechanical linkage coupling each front wheel. In the simple model, a single front wheel, with a steering angle, φ is used to represent the steering input. Because the distance l between contact points has not changed and instantaneous center of rotation has not changed, the motion of the two-wheeled vehicle is kinematically equivalent to the motion of the fourwheeled car vehicle ([23, 24]). The Lagrange function of unconstrained mechanical system is given by relation: L = T −V 1 1 1 (16) L = m(x˙2 + y˙2 ) + J θ˙ 2 + J θ˙ φ˙ + J φ˙ 2 + JK ψ˙ 2 2 2 2 where using m we refer total weight of Ackerman steered car including the wheels, J will be moment of inertia of Ackerman steered car around the center of mass and Jk will be moment of inertia of wheel.


326

5 Ackerman Steered Car dynamic motion Using the geometrical theory of nonholonomic system mentioned in the second section we define the structure of the mechanical system as follows. In cases where the number of degrees of freedom is greater than the number of generalized coordinates, additionally defined coordinates are not independent of the present generalized coordinates. Equations with terms in the time derivatives of the generalized coordinates and which cannot be integrated are called nonholonomic constraint equations. There are five degrees of freedom of the unconstrained mechanical system. Thus, the fibered corresponding manifold of the problem is R × R5 , pr1 , R where pr1 is the cartesian projection on the first factor. We choose the fibered chart on Y as (V, ξ ) where V is an open set V ⊂ Y and ξ1 = t, q1 , q2 , q3 , q4 , q5 = (t, x, y, θ , ψ , φ ). The associated chart on the base is (pr1 , Φ), Φ = (t) where t is the time coordinate, and associated fibered chart on J 1Y = R × R5 × R5 is (V1 , ξ1 ), V1 = pr−1 (V, ξ ), ξ1 = (t, qσ , q˙σ ), 1 ≤ σ ≤ 5, i.e. ξ1 = t, x, y, θ , ψ , φ , x, ˙ y, ˙ θ˙ , ψ˙ , φ˙ . The basic parameters used to specify the ackerman steered car geometry are illustrated in Figure 2. On the other hand, the Euler-Lagrange equations of system motion are: E1 ≡ −mx¨ = 0 E2 ≡ −my¨ = 0 E3 ≡ −J θ¨ − J φ¨ = 0

(17)

E4 ≡ −2JK ψ¨ = 0 E5 ≡ −J θ¨ − J φ¨ = 0 The Lepage class of the unconstrained mechanical system is thus given by the representative: ¨ 5 ∧ dt + .. ¨ 3 ∧ dt − my¨ω 4 ∧ dt − J φω α = − mx¨ω 1 ∧ dt − J θ¨ ω 2 ∧ dt − 2JK ψω .. [−m] ω 1 ∧ d x˙ + [−J] ω 2 ∧ d θ˙ + [−4JK ] ω 3 ∧ d ψ˙ + [−m] ω 4 ∧ d y˙ + [−J]ω 5 ∧ d φ˙

(18)

Where:

ω 1 = dx − xdt, ˙ ω 2 = d θ − θ˙ dt, ω 3 = d ψ − ψ˙ dt, ω 4 = dy − ydt, ˙ ω 5 = d φ − φ˙ dt 5.1

The constraint

The condition that the Ackerman steered car rolls without sliding on the plane means that the instantaneous velocity of the point of contact of the Ackerman steered car-wheel is equal to zero at all times. This gives rise to the following nonholonomic constraints: ˙ θ) = 0 f 1 ≡ (x˙ sin θ − ycos 2 ˙ ˙ θ + φ )] = 0 f ≡ θ l cos(φ )[x˙ sin(θ + φ ) − ycos(

(19)

3

f ≡ Rψ˙ [x˙ cos θ − y˙ sin θ ] = 0 Or in normal form cos θ y˙ sin θ sin(θ + φ ) cos(θ + φ ) θ˙ ≡ g2 = x˙ + y˙ l cos φ l cos φ sin θ cos θ + y˙ ψ˙ ≡ g3 = x˙ R R x˙ ≡ g1 =

(20)

These three nonholonomic conditions define the constraint submanifold ℘ of dimension dimJ 1Y −2 = 8. Constraint (20) obeys condition (15), i.e. it is semiholonomic. The geometric theory allows us to


solve such a problem immediately, without integrating the constraint. ⎤ ⎡ cos θ 1 0 0 0 sin θ ∂ fi ⎥ ⎢ θ +φ ) cos(θ +φ ) rank[ σ ] = rank ⎣ sin( l cos φ l cos φ 1 0 0 ⎦ = 3 ∂ q˙ sin θ cos θ 010 R R 5.2

327

(21)

Constrained mechanical system-reduced equations

The geometrical approach described in the first section applied to our problem leads to the constrained mechanical system [α℘] related to the unconstrained mechanical system [α ]. The class [α℘] is generated e.g. by the following representative: 2

α℘ = A 1 ω 1 ∧ dt + A 2 ω 2 ∧ dt + ∑ B l1 ω 1 ∧ d y+B ˙ l2 ω 2 ∧ d φ˙

(22)

l=1

Where:

ω 1 = dy − ydt, ˙ ω 2 = d φ − φ˙ dt

Computing the coefficients A l according to equation (9) we obtain following expressions: A 1 = A 2 = 0 B 11 = −m − 4Jk B 22 B 12

cos2 θ sin2 (θ + φ ) − m l 2 cos2 φ R2

(23)

= −m = B 21 = 0

The reduced equations are of the form: A 1 + B 11 y¨ + B 12 φ¨ = 0 A 2 + B 22 φ¨ + B 21 y¨ = 0 cos θ y˙ sin θ sin(θ + φ ) cos(θ + φ ) θ˙ ≡ g2 = x˙ + y˙ l cos φ l cos φ sin θ cos θ + y˙ ψ˙ ≡ g3 = x˙ R R

x˙ ≡ g1 =

(24)

There is no analytical solution of reduced equations of motion, in general situation. 5.3

Numerical solution of reduced equations of motion

The reduced equations of the Ackerman steered car motion derived in Section above (Eq.24) were numerically solved for following example values of parameters characterizing the Ackerman steered car: m = 3.4kg; R = 0.04m; J = 0.327kg.m2 , Jk =0.1kg.m2 , and, l = 0.12m. A numerical solution was made with the help of the program Maple.13. In Figs. 3-7, the graphical outputs x(t), y(t), θ (t), ψ (t) and φ (t) of calculations are presented. A real Ackerman steered car is a complex space structure. The Governing equations for the dynamic response of an Ackerman steered car (i.e., reduced equations) are derived based on Krupková approach. These equations are essentially representing the coupled engineering problem of structural dynamics and multi-body dynamics are difficult to solve analytically. The numerical studies of reduced equations of motion are presented and we find it as effective and applicable for problems in physics and engineering for preliminary visualization.

Orientation of the body [rad]


1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3

4

Time [s] Fig. 3 Solution of Ackerman steered car orientation θ (t).

Angle of imaginary wheel rotation [rad]

328

2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3

Time [s] Fig. 4 Solution of Ackerman steered car of imaginary wheel rotation ψ (t).

4


1.5

Steering angle[rad]

1.0

0.5

0.0

-0.5

-1.0

-1.5

0

1

2

3

4

Time [s] Fig. 5 Solution of steering angle rotation φ (t).

4.0 3.5 3.0

x[m]

2.5 2.0 1.5 1.0 0.5 0.0

0

1

2

3

4

Time [s] Fig. 6 Solution of the location of the midpoint of the rear axle x(t).

5

329

330


2.0 1.5 1.0 0.5

y[m]

0.0 -0.5 -1.0 -1.5 -2.0

0

1

2

3

4

5

Time [s] Fig. 7 Solution of the location of the midpoint of the rear axle y(t).

6 Conclusions In this paper we studied a real practical system subjected to a true nonholonomic constraint conditionan Ackerman steered car model for first time. Moreover, the presented results formulation indicate the effectiveness of the geometrical theory (Krupková approach) of nonholonomic constraints for formulating of motion of concrete nonholonmoic constraints systems with constraints based on the assumption of rolling without slipping. For semiholoonomic constraints an advantage of the theory lies also in the possibility to include the constraint directly into equations of motion. Then it appears that the model of Chetaev-type forces is appropriate for describing the studied type of the constraint from the point of view of physics.

References [1] Swaczyna, M. (2011), Several examples of nonholonomic mechanical systems, Communications in Mathematics, 19(1), 27-56. [2] Bullo, F. and Lewis, A.D. (2004), Geometric Control of Mechanical Systems, Springer Verlag, New York, Heidelberg, Berlin. [3] Cardin, F. and Favreti, M. (1996), On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18, 295-325. [4] Carinena, J.F. and Raada, M.F. (1993), Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers, J. Phys. A: Math. Gen., 26, 1335-1351. [5] Cortes, J., de Leon, M., Marrero, J.C., and Mart’ınez, E. (2009), Nonholonomic Lagrangian systems on Lie algebroids, Discrete Contin. Dyn. Syst. A, 24, 213-271. [6] de Leon, M., Marrero, J.C., and de Diego, D.M. (1997), Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A: Math. Gen., 30, 1167-1190. [7] de Leon, M., Marrero, J.C., and de Diego, D.M. (1997), Mechanical systems with nonlinear constraints, Int.


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Journ. Theor. Phys., 36(4), 979-995. [8] Giachetta, G. (1992), Jet methods in nonholonomic mechanics, J. Math. Phys., 33, 1652-1655. [9] Janov´ a, J. and Musilová, J. (2009), Non-holonomic mechanics mechanics: A geometrical treatment of general coupled rolling motion, Int. J. Non-Linear Mechanics, 44, 98-105. [10] Czudkov´ a, L. and Musilová, J. (2013), Nonholonomic mechanics. A practical application of the geometrical theory on fibred manifolds to a planimeter motion, International Journal of Non-Linear Mechanics, 50, 19-24. [11] Krupková, O. (1997a), Mechanical systems with nonholonomic constraints, J. Math. Phys., 38, 5098. [12] Krupková, O. (2000), Higher order mechanical systems with constraints, J. Math. Phys., 41, 5304. [13] Krupková, O. and Musilov´ a, J. (2001), The relativistic particle as a mechanical system with nonlinear constraints, J. Phys. A: Math. Gen., 34, 3859. [14] Chetaev, N.G. (1932-1933), On the Gauss principle, Izv. Kazan. Fiz.-Mat. Obsc., 6, 323 (in Russian). [15] Krupková, O. and Musilov´ a, J. (2005), Non-holonomic variational systems, Rep. Math. Phys., 55, 211. [16] Swaczyna, M. (2005), Variational aspects of non-holonomic mechanical systems. Ph.D. Thesis, Palack´ y University, Olomouc. [17] Brdicka, M. and Hlad´ık, J. (1987), Theoretical Mechanics (Teoretická mechanika, in Czech), Academia, Praha. [18] Neimark, J.I. and Fufaev, N.A. (1972), Dynamics of nonholonomic systems, 33, of Translations of Mathematical Monographs. Providence, Rhode Island: AMS. [19] Koon, W.S. and Marsden, J.E. (1997), The Hamiltonian and Lagrangian approach to the dynamics of nonholonomic systems, Reports in Mathematical Physics, 40, 21-62. [20] Krupková, O. (1998), On the geometry of non-holonomic mechanical systems, in: Proceedings of the Conference on Differential Geometry and its Applications, 533. [21] Krupková, O. (1997b), The Geometry of Ordinary Variational Equations, Lecture Notes in Mathematics, Springer, Berlin. [22] Conner, D. (2007), Integrating Planning and Control for Constrained Dynamical Systems [thesis], Robotic Institute, Carnegie Mellon University, Pittsburgh, Pennsylvania, December. [23] Lipták, T., Kelemen, M., Gmiterko, A., Virgala, I., Mikov´ a, L’, and Hroncov´ a, D. (2016), Comparison of Different Approaches of Mathematical Modelling of Ackerman Steered Car-like System, Journal of Automation and Control, 4(2), 15-21.



Flow-induced Vibration of Flexible Bottom Wall in a Lid-driven Cavity Xu Sun†, Wenxin Li, Zehua Ye National Engineering Laboratory for Pipeline Safety/MOE Key Laboratory of Petroleum Engineering, China University of Petroleum-Beijing, 102249, China Submission Info Communicated by S. Lenci Received 2 July 2017 Accepted 21 September 2017 Available online 1 January 2018 Keywords Flow-induced vibration Fluid-structure interaction Membrane Lid-driven Cavity Finite element method

Abstract Flow-induced vibration of the flexible bottom wall in a lid-driven cavity is investigated numerically using a well-validated finite element scheme for fluid-structure interaction (FSI). First, the mechanical and mathematical models of a lid-driven cavity with flexible bottom are presented, and the corresponding FSI solution procedure is introduced briefly. Then, the accuracy and stability of the developed FSI scheme and code are examined and a grid independence test is carried out. Finally, using very fine increment, bifurcations of the flow-induced vibration (FIV) of the flexible bottom with respect to the structure rigidity and Reynolds number are studied in detail. The results could reveal more details of the benchmark FSI model involving a lid-driven cavity with flexible bottom, gaining a better understanding on other FIVs caused by the internal unsteady flows. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Flow and heat transfer characteristics in an enclosure have received a considerable attention in the past decades, due to their importance in widely engineering applications, such as heat dissipation of electronic devices, heat transfer in solar ponds, cooling of lubrication systems and thermal-hydraulics of nuclear reactors [1–4]. To improve forced and/or natural convection as well as enhance heat transfer in an enclosure, various methods such as using oscillating, wavy, or flexible wall have been studied [5–11]. Khanafer et al. [5] investigated the effect of a sliding lid in a sinusoidal fashion on the mixed convection in a square cavity. They found that the horizontally oscillating speed of the lid have a profound effect on the structure of fluid flow and heat transfer fields. Sowayan [6] studied the effect of the bottom wall vibration on the heat transfer enhancement of a square cavity. He reported that the heat transfer rate of the cavity is increased with the oscillating frequency of the bottom wall. Al-Amiri et al. [7] investigated the effect of sinusoidal wavy bottom wall on the mixed convection heat transfer in a lid driven cavity. Their results show that the heat transfer in a square lid driven cavity can be enhanced using sinusoidal wavy bottom surface, and the enhancement is affected significantly by the amplitude of the wavy surface. Cho et al. [8] carried out a study on the mixed convection heat transfer † Corresponding



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Xu Sun, Wenxin Li, Zehua Ye / Journal of Vibration Testing and System Dynamics 1(4) (2017) 333–341

of water-based nanofluids in a lid-driven cavity with right and left wavy surfaces. Their results show that, for a given nanofluid, the mean Nusselt number can be optimized if the wavy surface geometry parameters are tuned appropriately. Al-Amiri and Khanafer [9] proposed a novel method for the heat transfer enhancement in a lid-driven cavity using the flexible bottom wall. In their investigation, the bottom wall of the lid-driven cavity is replaced by a flexible plate, which is allowed to deform under the loads imposed from the fluid in the cavity. Using a fully coupled fluid-structure interaction (FSI) simulation, they found that flow-induced deformation of the flexible bottom wall could improve the lid-driven cavity to obtain higher rate of mixed convection heat transfer. Khanafer [10] studied the effect of a flexible wall on the natural convection flow and heat transfer in a cavity filled with porous medium. It was also found that the flexible wall could improve the heat transfer rate in a cavity. Subsequently, Khanafer [11] compared the mixed convection flow and heat transfer characteristics in a lid-driven cavity with flexible and wavy bottom walls. They found that the flexible bottom wall exhibits more significant heat transfer enhancement than the wavy wall at high Grashof numbers. In the existing works of Al-Amiri and Khanafer [9], Khanafer [10] and Khanafer [11], the flow in the cavity is always steady, and the flexible wall will eventually stay at an equilibrium state, and the dynamic behavior of the flexible wall as well as its effect on the flow and heat transfer in the cavity has not been investigated yet. Besides the applications mentioned above, the lid-driven cavity with a flexible bottom has also been used widely as a benchmark for validating the FSI numerical methods and codes [12–16]. Different from those in [9–11], in the literatures of [12–16] the process of energy transportation is not considered, and the velocity of the lid is supposed to be sinusoidal to induce the unsteady flow in the cavity as well as the flow-induced vibration (FIV) of the flexible bottom. Since the primary motivation in [12–16] is code validation, few attention has been paid to the dynamic response of the flexible bottom wall. In this paper, FIV of the flexible bottom in a lid-driven square cavity is studied using a wellvalidated FSI code based on the characteristic-based split (CBS) finite element method (FEM). Using very fine increment for each parameter, bifurcation characteristics of the FIV of the flexible bottom with respect to the structure rigidity and Reynolds number are analyzed in detail.

2 Mechanical model and numerical methods The model of a lid-driven cavity with a flexible bottom reported by Bathe and Zhang [12] is used. As shown in Fig.1, different from the classical model of lid-driven cavity [17], the rigid bottom wall is replaced by a flexible membrane and an inlet and an outlet are added near the two top corners to allow the fluid to flow in or out of the square cavity. An oscillating motion in the horizontal direction is imposed on the lid to induce unsteady flow in the cavity, which further results in the vibration of the flexible bottom. The flow in the cavity is supposed to be two-dimensional, incompressible and laminar, and the effect of body force such as gravity is not considered. Based on these assumptions, the governing equations of the flow domain can be expressed as

∂ ui =0 ∂ xi

(1)

1 ∂ 2 ui ∂ ui ∂ ui ∂p + uj =− + ∂t ∂xj ∂ xi Re ∂ x j ∂ x j

(2)

where ui denotes the velocity components normalized by velocity amplitude of the lid U , xi the coordinate components normalized by the cavity width L, t the time normalized by L/U , p the pressure normalized by ρF U 2 and Re = U L/υ , where ρF and υ is the density andkinetic viscosity of the fluid, respectively. The motion of the lid is governed by u1 = U [1 − cos(2π t 5)]. The no-slip boundary


335

Fig. 1 A lid-driven cavity with a flexible bottom wall [12].

condition is imposed onall boundaries except the inlet and outlet, where the boundary conditions are taken as p = 0 and ∂ u1 ∂ n = ∂ u2 ∂ n = 0. On the fluid-membrane interface, the fluid is moved with the flexible membrane. The mathematical model proposed by Smith and Shyy [18,19] and Gordnier [20] is used to describe the FIV of the flexible membrane. In their model, the membrane is supposed to be driven by the pressure difference between the lower and upper surfaces, namely Δp = plower − pupper , and the vibration is restricted in z direction, as seen in Fig.1. Moreover, because the thickness is much smaller compared with the initial length of the membrane, the shear forces and bending moments at each membrane section as well as the effect of gravity are ignored. Based on these assumptions, the dynamic response of the membrane bottom can be described by

ρS h

3 ∂ 2z ∂z ∂ 2z ∂z − T + ρ C [1 + ( )2 ]− 2 = Δp S d 2 2 ∂t ∂t ∂x ∂x

(3)

where ρS is the membrane density normalized by ρF , h the thickness of the membrane normalized by L, z the displacement normalized by L and Cd the damping normalized by U . In Eq.(3), the tension T can be computed by (4) T = Eh δ0 + δ¯ where E is the elastic modulus normalized by ρF U 2 , δ0 the pre-strain and δ¯ the strain, which can be obtained by ˆ L L − L ∂z S , LS = δ¯ = 1 + ( )2 dx (5) L ∂x 0 A well-validated FSI solution procedure proposed in our previous studies [21–23] is used to simulate the FIV of the flexible bottom wall in the lid-driven cavity. In this method, a modified CBS FEM for moving mesh [24,25] is combined with the segment spring analogy method [26] and dual-time stepping (DTS) method [27] to form an implicit flow solver, the membrane vibration is solved implicitly by the Galerkin FEM and generalized-α method [28, 29], and the flow and structure solvers are coupled by the loosely-coupled partitioned approach. The flexible membrane is divided equally by Hermite elements, while the flow domain is divided by unstructured triangular grids with linear shape functions for both velocity and pressure. Moreover, the grid nodes of the structure and fluid are overlapping each other on the fluid-membrane interface, and the same computational time step is used for the flow and structure solvers. For the fluid-membrane interaction problems in laminar flow regime, this FSI solution procedure has shown good accuracy and stability. More details about the numerical methods can be found in Zienkiewicz et al. [30], Sunet al. [23–25], and Sun and Zhang [21, 22, 26] and will not be presented here.

336


(a) Mesh-1

(b) Mesh-2

Fig. 2 Computational meshes.

Fig. 3 Transient displacements at the membrane centre (t = 0 ∼ 100).

3 Results and discussions 3.1

Grid independence test

To examine the accuracy and stability of the proposed FSI solution procedure, flow-induced vibration of the flexible bottom wall in Fig.1 at U = 1, Re = 100, ρS = 500, Cd = 0, E = 25000, δ0 = 0 and h = 0.002, which are the same with those reported in Bathe and Zhang [12], is computed first. Two types of unstructured meshes with different grid densities are applied for computation. Mesh-1 has 4560 grid nodes and 8829 triangular elements in the flow domain and 50 equally divided elements on the membrane, while Mesh-2 has 9247 grid nodes and 18100 triangular elements in the flow domain and 101 equally divided elements on the membrane, as shown in Fig.2. The time steps for Mesh-1 and Mesh-2 are 0.01 and 0.005, respectively. In Fig.3, the computed displacements at the centre point of the flexible bottom in t = 0 ∼ 100 are listed. As seen in the figure, the flexible bottom exhibits a periodic response after a transient process at t = 0 ∼ 20. Moreover, the numerical results computed from Mesh-1 and Mesh-2 are very close and agree very well with those provided by Bathe and Zhang [12]. Therefore, Mesh-1 with dt = 0.01 is taken as the computed mesh in the rest of this paper.


Fig. 4 Bifurcation diagram of the flow-induced vibration at membrane centre with respect to Eh.

(a) Eh=5

(b) Eh=15

(e) Eh=2600

(f) Eh=2700

(g) Eh=3100

(h) Eh=3200

(i) Eh=3700

(j) Eh=4400

(k) Eh=5000

(l) Eh=5020

(m) Eh=5080

(n) Eh=5260

(c) Eh=600

(o) Eh=5280

(d) Eh=2200

(p) Eh=10000

Fig. 5 Phase portraits of the membrane centre at different structure rigidities.

337

338


3.2

Effect of membrane rigidity

Fixing Re at 100, ρS at 500, Cd at 0 and δ0 at 0 while changing Eh from 5 to 10000, effect of the membrane rigidity is investigated firstly. Fig.4 shows the bifurcation diagram of FIV at the membrane centre with respect to Eh. As shown in Fig.4, the vibrating amplitude at the membrane centre is increased first in Eh ∈ [5, 15] while then decreased in Eh ∈ [15, 10000], and the averaged deformation of the flexible bottom is always downward and approaches to zero with the increase of Eh. It seems that the resonance happens near Eh=15, resulting in the peak of the vibrating amplitude. Besides the vibrating amplitude, the vibrating state of the flexible membrane is also changed significantly with the rigidity. As shown in Figs.4 and 5(a), when the rigidity is very small, namely Eh ∈ [5, 10], the membrane exhibits a period-1 response. Near Eh = 15, however, the vibrating state at the membrane centre is changed from period 1 to period 2, as shown in Figs.4 and 5(b). Then, the period-2 response is kept with further increase of Eh in [15, 500]. Near Eh = 600, the vibrating state bifurcates from period 2 to period 3, which becomes more apparent when Eh is further increased from 600 to 2100, as shown in Figs.4 and 5(c). Then, the structure response becomes more complicated when the rigidity is increased from 2100 into Eh ∈ [2200, 5260]. As seen in Figs.4 and 5(d) to 5(n), in Eh ∈ [2100, 5260] the vibration is changed among several types of multiple-period and quasi-period states. Finally, when Eh is increased slightly from 5260 to 5280, the membrane response turns back to period 2, which is no longer changed when Eh is further increased up to 10000. 3.3

Effect of Reynolds number

Fixing ρS at 500, Cd at 0, Eh at 50 and δ0 at 0 while changing Re from 50 to 300, effects of the Reynolds number are studied. Fig.6 is the bifurcation diagram of the flow-induced vibration at the membrane centre, and Fig.7 shows the phase portraits at several representative points. As seen in Fig.6, the vibrating amplitude at the membrane centre is increased first in Re ∈ [50, 175] while then not changed much in Re ∈ [175, 300], and the time-averaged deformation is downward and decreased slightly with Re. Similar with the membrane rigidity, the Reynolds number also has significant effect on the vibration state of the flexible bottom of the cavity. When Re is less than 70, the flexible bottom exhibits a period-1 response, as shown in Figs.6 and 7(a). Near Re=70, the upper branch of the bifurcation diagram is changed from one to three, which results in a period-2 response, as shown in Figs.6 and 7(b). Near Re=100, however, the period-2 response turns back to period 1. Comparing Fig.7(c) with Fig.7(a), it can be found that the period-1 response at Re=100 is very different from that at Re < 70, which indicates that there are two stable limit cycle oscillation (LCO) states on the phase plane. Then, this period-1 vibration state also becomes unstable and turns to quasi-period via perioddoubling bifurcations near Re = 136, as seen in Figs.6 and 7(d). After that, the quasi-period response is maintained in Re ∈ [136, 150]. When Re is increased slightly from 150 to 151, however, the dynamic response of the membrane is changed greatly. Not only the vibrating amplitude is enlarged a lot, but also the vibrating state jumps suddenly from quasi-period to multiple-period, as seen in Fig.7(e) and 7(f). At Re=151, the orbits is tangled not only with the LCO having lager amplitude (see Fig.7(c)) but also with the LCO with smaller amplitude (see Fig.7(a)). Then, the stability of multiple-period response of the membrane is decreased gradually when Re is further increased in [151, 169], as seen in Fig.7(g). Near Re=170, the orbit is no longer affected by the LCO with smaller amplitude (see Fig.7(a)) and the vibrating state is changed from multiple-period to period-3. In Re ∈ [174, 190], the period-3 state becomes unstable, and the membrane exhibits quasi-period or chaotic response, depending on whether the orbit is tangled with the LCO with smaller amplitude (see Fig.7(a)), as seen in Figs.6 and 7(i) to 7(m). Finally, the quasi-period response is not changed any more in Re ∈ [190, 300], as seen in Figs.6 and 7(n).


Fig. 6 Bifurcation diagram of the flow-induced vibration at membrane centre with respect to Re.

(a) Re=50

(e) Re=150

(i) Re=173

(b) Re=70

(f) Re=151

(j) Re=174

(m) Re=190

(c) Re=100

(d) Re=136

(g) Re=169

(h) Re=170

(k) Re=176

(l) Re=178

(n) Re=300

Fig. 7 Phase portraits of the membrane centre at different Reynolds numbers.

339

340


4 Conclusions As a benchmark FSI problem, the lid-driven cavity with a flexible bottom is usually used for code validation. As a result, few attention has been paid to the dynamic behavior of the flexible bottom in previous studies. In this paper, effects of the structure rigidity and the Reynolds number on the FIV of the flexible bottom in a lid-driven cavity are investigated using a well-validated FSI scheme based on the CBS FEM. From the computed results, several conclusions could be drawn. First, for the model of a lid-driven cavity with a flexible bottom employed widely by many literatures [12–16], the mean deformation of the flexible bottom is downward in most cases, and the mean deformation is decreased with the increase of the structure rigidity or Reynolds number. Second, the vibrating amplitude of the flexible bottom is increased first and then decreased due to the appearance of resonance with the increase of the structure rigidity, but it will be decreased monotonously with the Reynolds number. Finally, both of the structure rigidity and Reynolds number have significant effects on the state of FIV of the flexible bottom. With change of the structure rigidity or Reynolds number, the flexible bottom of the lid-driven cavity could exhibit period-1, period-2, period-3, multiple-period, quasi-period or chaotic response under the excitation of the unsteady flow in the cavity. If the thermal field in the cavity is also taken into account, these vibration states must result in different heat transfer features, which should be further investigated in the future research.

Acknowledgments This work is supported by the National Natural Science Foundation of China (No.51506224), Opening fund of State Key Laboratory of Nonlinear Mechanics and Science Foundation of China University of Petroleum-Beijing (No.C201602). The authors would like to thank for the kindly support of these foundations.

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Fast Unbalancing of Rotating Machines by Combination of Computer Vision and Vibration Data Analysis A. Najedpak, C. Yang† Department of Mechanical Engineering, University of North Dakota, Grand Forks, ND 58202-8359, USA Submission Info Communicated by A.C.J. Luo Received 2 June 2017 Accepted 28 September 2017 Available online 1 January 2018 Keywords Unbalance correction Rotating machines Computer vision Vibration data analysis

Abstract Unbalance is one of the most common mechanical faults in rotating machines. Although different balancing methods have been developed, most of them require balancing machine to perform unbalance correction. A method using accelerometers data and intricate vibration theories can eliminate the need of balancing machine, and the amplitude and phase of the machine’s vibrations can be identified. However it needs numerous measurements, and in some cases it is even impossible to be implemented. To overcome this problem, a novel approach with reduced number of measurements is presented in this paper. The proposed method requires only two measurements: one from original unbalanced condition, and the other from modified situation after adding an arbitrary trial mass to a marked location. The rotating rotor is being video recorded under original unbalanced and modified situations. The position of the marked area is identified when the amplitude of the sinusoidal vibration response reaches the maximum. The correction mass and its adding location are calculated using proposed method. To demonstrate the effectiveness of our method, an experiment is setup. Vibrations under healthy, unbalanced and balanced conditions are analyzed. The results demonstrated that the developed method is more cost effective with the same accuracy as the other contested balancing techniques. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Rotating machines are wildly used in industry. To avoid any unexpected failure and costly downtime due to emergency shutdowns and to provide a safe working environment, the condition monitoring methods are important to keep track of machines health at all times. The most common condition monitoring methods include vibration analysis, infrared thermography, electric current and voltage analysis, acoustic and ultrasound testing, and lubricant analysis. Vibration signals have provided the most valuable information on the condition of the machine and is the most cost effective monitoring procedure available, while other methods suffer from high condition monitoring costs. Nejadpak and † Corresponding



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Yang has investigated the effectiveness of current and voltage analysis to find the healthy operating condition of the electric motor [1]. It was demonstrated that it is possible to find mechanical and electrical failures in a rotating machine. However, their method is focused on the electrical faults of the motor and fails to detect the particular mechanical faults of the system. Researchers in [2–7] have applied vibration analysis to detect fault in the rotating machines. They have demonstrated that with accelerometer data, it is possible to identify the exact type of faults at earlier stages of fault development in the machines during operation. Copping [4] has defined four stages of bearing faults: Stage one mainly includes complications with lack of lubrication and microscopic failures in the bearings. The signature fault frequencies would appear at stages two and three. Sage four is the risk zone when the bearing has to be immediately replaced. As mentioned, the current sensors would have failed to identify the bearing fault at early stages, while the accelerometers have detected the smallest vibrations caused by the faulty condition. Bearing faults, misaligned shaft, cracked shaft, and unbalanced rotor are common mechanical faults of the rotating machines. Each fault has its own specific vibration response. For example, the system is under unbalanced rotor condition if the frequency response of the system shows high vibrations at 1X of the operating speed, when bearing is at stage 4 of fault, the system experiences noticeable random vibrations at higher frequencies between 2-5 KHz [1], and the vibrations would increase at multiples of the operating speed in loose foundation. There are new methods implemented for fault detection based on machine learning algorithms and data analysis to categorize different faults. The authors in [8] applied principal components analysis to categorize different faults based on their specific vibration response in frequency domain, and the severity of the faults can also be classified Unbalance is one of the most common mechanical faults in rotating machines. When machine is operated at high speed, small unbalance mass would generate large unbalanced force, then result in inaccuracies in manufacturing process, thermal distortions and deformations. Static unbalance and dynamic unbalance are different situation. Static unbalance occurs when the principal axis is parallel to the axis of rotation, while dynamic unbalance concurs when the principal axis does not intersect with the axis of rotation. Dynamic unbalance is the most common type of unbalanced condition. Once the faults are detected and identified, correction action should be taken. There are several methods which are mostly based on adding at least two trial massed to the system and reading the vibration changes at bearings. The required balancing mass and location are computed through mathematical equations. Despite being accurate, the conventional methods often require more measurements and accessibility to the rotor that is bothersome in some cases. In this paper, the commonly used correction methods for unbalanced condition are investigated and a novel approach based on computer vision methods is presented. The remainder of the paper is organized as follows: First, the available balancing methods and their accuracy and suitability are discussed, followed by the experiment setup and the required image processing techniques for our balancing technique. Then, different edge detection algorithms are applied and the best detector is implemented for line detection in the video frames captured by the phone’s camera. Finally, the accuracy of the proposed method and further details are presented

2 Balancing method Theoretically, given information on magnitudes and locations of all uneven distributed masses, the correction mass needed to add to or remove from the unbalanced rotor can be precisely calculated for both magnitude and location. However how to find magnitude and location of unbalanced mass remains challenge. In practice, balancing machine is required. To balance the unbalanced rotating equipment, the magnitude and phase of vibration caused by original unbalanced force are measured first, then after a trial mass is added on the reference plane, the magnitude and phase of vibration resulted from both original unbalanced and trial mass are measured again. This general used balancing method requires


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complex instruments. Nisbett proposed a new technique for measuring phases of vibration in [9]. Their method does not require the complex instruments. The phase of vibration is calculated mathematically. The governing equations to find the trial mass are [9]:

ϕA = tan−1 [ ΔA1 = ΔA2 =

ΔB1 = ΔB2 =

(1)

2 2 (A01 )2 + (A180 1 ) − 2A 2

(2)

2 2 (A02 )2 + (A180 2 ) − 2A 2

(3)

ϕB = tan−1 [

2 2 2 (A90 1 ) − (ΔA1 ) − A ] (A01 )2 − (ΔA1 )2 − A2

2 2 2 (B90 1 ) − (ΔB1 ) − B ] (B01 )2 − (ΔB1 )2 − B2

(4)

2 2 (B01 )2 + (B180 1 ) − 2B 2

(5)

2 2 (B02 )2 + (B180 2 ) − 2B 2

(6)

where A represents the vibration magnitude on bearing A for the unbalanced rotor, ϕA is the phase angle on bearing A for the unbalanced rotor, ΔA1 is the additional vibration magnitude when the trial mass is added to plane A, ΔA2 is the additional vibration magnitude when the trial mass is added to plane B. Similarly, B, ϕB , ΔB1 , ΔB2 are defined. The correction factors are defined by the following equations: (7) RAX ΔA1 + RBX ΔA2 = −A cos ϕA RAX ΔB1 + RBX ΔB2 = −B cos ϕB

(8)

RAY ΔA1 + RBY ΔA2 = −A sin ϕA

(9)

RAY ΔB1 + RBY ΔB2 = −B sin ϕB

(10)

The required balancing mass is the product of correction factors and the trial mass. The method proposed in [9] can provide accurate results, however, it requires eight measurements because the trial mass must be added to four different angles on each of the two planes of the rotor. Therefore, the method presented in [9] requires taking more intricate measurements. If there is a lack of accessibility to the rotor, that method cannot be implemented. To overcome above mentioned problems, we propose a new novel methodology in this paper, the phase angle is measured based on combination of the accelerometer data along with a video recording of the rotating flywheel. Gao et al. proposed a method to edge detection on images through combination of Sobel edge detection operator and soft-threshold wavelet de-noising. Sobel edge detectors use the following 3-by-3 kernels to convolve the image and calculate the horizontal and vertical changes [11]. ⎡ ⎤ ⎡ ⎤ −1 0 1 −1 −2 −1 Gy = ⎣ 0 0 0 ⎦ (11) Gx = ⎣−2 0 2⎦ , −1 0 1 1 2 1 G=

G2x + G2y

(12)

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Fig. 1 Original image of unbalanced flywheel.

(a)

(b)

(c)

Fig. 2 Detected edges (a) Sobel edges, (b) Roberts edges, (c) Canny edges.

θ = atan(

Gy ) Gx

(13)

where Gx is the horizontal kernel, Gy is the vertical kernel, G is the gradient magnitude, θ is the gradient’s direction. Roberts edge detectors use the following two kernels [12]: 1 0 0 1 , Gy = (14) Gx = 0 −1 −1 0 The Canny edge detector [10] use the following procedures: 1) implement a Gaussian filter to smooth the image, 2) find the intensity gradients of the image, 3) apply a non-maximal suppression, 4) define specific thresholds to determine the edges, and 5) track edges using hysteresis. The edge detection procedures are accomplished via MATLAB. One frame of the original video is shown in Fig. 1. The detected edges using these three edge detectors are illustrated in Fig. 2. (a) Sobel edge detectors, (b) Roberts edge detectors, and (c) Canny edge detectors, respectively. From Fig. 2, it’s clear that Canny edge detectors on (c) have detected more edges compared to Sobel (a) or Roberts (b) edge detectors. This is verified visually and numerically as canny edge detectors have identified more white pixels. Therefore, after compared with the original video of the rotating flywheel, Canny edge detectors is the best edge detection method. To detect the marker position located on the flywheel, a line detection algorithm is implemented. RANSAC and Hough transform are the most common line detection algorithms available. Here, Hough transform described in [13] is implemented to detect the marker line and subsequently the slope of this line. Note that to achieve the perfect line detection algorithm we can change and optimize


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Fig. 3 The Hough lines detected on the marker.

Fig. 4 The experiment setup.

the thresholds for Hough peaks. The threshold is modified to show one single line on the marker and avoid the unnecessary noises. The marker lines detected on the image were taken from the flywheel. Later on, the lines will be detected on the frames of the captured video of rotating flywheel. Fig. 3 displays the Hough line detected on the marker. The corresponding phase angle here is 0 degrees, which is considered as a reference point (It will be called 12 o’clock as a reference position).

3 Experiment setup The experiment setup includes one three-phase induction motor with an AC drive for the motor, three accelerometers mounted along vertical, horizontal and axis directions, an adapter, and a data acquisition device containing a NI-PXI 4498 module with the sampling rate at 10 KHz. The picture of experiment setup is shown in Fig. 4. The measured accelerometer data is analyzed using NI Sound and Vibration Assistant software. The motor is set to run at 20 Hz (1200 RPM). The operating speed is selected rather low to make sure the features detected on the marker have strong intensity values and can be used for a proper edge detection scheme. In this experiment for capturing the video of the rotating shaft the IPhone 6 camera is used in slow motion mode which results in capturing 240 frames per second. The motor is operating at 20 Hz (1200 RPM) which means we will see 20 peaks in the vibration spectrum per

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Fig. 5 Vibration response of the healthy condition.

Fig. 6 Vibration response of unbalanced condition.

Fig. 7 Frequency response of the healthy condition.

Fig. 8 Frequency response of unbalanced condition.

second. Moreover, it means at each round (one full rotating of the flywheel) there will be 12 frames recorded which will be referred to as the 12 hours of the clock. If more accuracy is desired, the operating speed of the motor could be reduced. In next section the video frames are analyzed and the results are combined with the accelerometer data. Based on these results, the location and amount of trial mass is originated.

4 Balancing results and discussions To demonstrate the effective and efficiency of the proposed method, the vibration response of wellbalanced flywheel attached to a motor in vertical axis and of unbalanced rotor with an external object attached to the flywheel in time domain are measured and shown in Fig. 5 and Fig. 6, respectively. Comparing Figs. 5 and 6, it is clear that by adding the external mass, the overall vibrations of the system has increased significantly (it’s almost tripled) as the machine is affected by an increased force of: (15) F = mω 2 e where F represents the unbalanced force, m is the unbalanced mass, e represents the distance from the unbalanced mass to the center of rotation, ω is the operating speed of the machine. Using signal processing method of Fast Fourier transform (FFT) the vibration response of both healthy and unbalanced system are converted into frequency domain. Initially the Discrete Fourier Transform (DFT) was applied, however, as the number of samples increases, FFT is more computationally intensive. In this study the corresponding vibration signal due to the unbalanced condition is


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Fig. 9 Frequency response of unbalanced condition in Decibel scale.

Fig. 10 Phase angle of the unbalanced flywheel.

Fig. 11 The trial mass added on the flywheel.

Fig. 12 Vibration response of the system when a trial mass is added.

more noticeable, therefore, the vibration response is analyzed in linear format. Figs. 7, and 8 display the frequency response of the healthy and unbalanced conditions in linear format. Comparing Figs.7 and 8, we notice that the unbalanced load has affected the frequency response at 1X operating speed (20 Hz) significantly. The peaks observed at 40, 80, 120 Hz are related to other machine characteristics and are not considerably affected by the unbalanced condition. Analyzing the data in linear form usually completely neglects the small vibrations in frequency domain. Since logarithmic scale may capture small changes, the vibration response of the unbalanced condition in frequency domain in logarithmic scale (Decibel scale) is shown in Fig. 9. Comparing Figs. 8 and 9, it is observed that even the smallest vibrations are revealed in logarithmic scale. In this paper only the large vibrations due to the unbalanced condition are investigated. To demonstrate the proposed new balancing method of rotating machines by processing video images of a flywheel, the recorded video and the vibration response in time domain are investigated together to

350


Fig. 13 The orientation vectors on polar coordinates.

Fig. 14 The balanced flywheel vibration response.

determine the position of the marker when the sinusoidal vibration response reaches maximum. The marker location is shown in Fig.10. From Fig.10, we observed that the Hough line is pointing toward 11 o’clock meaning the phase angle is about 30 degrees. In this situation, the maximum amplitude of the vibrations is 0.133 g derived from Fig. 6. Next, a trial mass weighing 63.6 gram is added to a random location on the flywheel as seen of Fig. 11, then the vibration of the system is video recorded for the second time. The vibration data is shown in Fig. 12. Fig.12 shows that the maximum vibration when a trial mass is added to the unbalanced rotor is 0.213 g. The phase angle is measured 120 degrees (line pointing toward 8 o’clock). Based on the values − derived from the videos and accelerometer data, two vectors are defined. → a is defined by displacement → − value and phase angle of the original unbalanced flywheel. b is defined with the same variables after the trial mass is added to the flywheel. Fig. 13 demonstrates the vectors on the polar coordinates. → − → → In Fig. 13, − c is defined as subtraction of − a from b here the objective is to find the amount of → → a and − c and it denotes how much the trial mass has to be angle ϕ which is the angle shown between − rotated with respect to the center. Based on the values measured adjustment angle of ϕ is calculated 60 degrees. Therefore, the trail mass is moved 60 degrees and the following vibration response shown in Fig. 14 is achieved. Fig.14 demonstrates the vibration response of the system when the trail mass is added to the unbalanced flywheel. Notice that the vibrations amplitude has decreased compared to both previous unbalanced conditions. Fig. 15 shows the frequency response of all investigated conditions. Fig. 15 shows how the vibrations of the system have reduced significantly when it’s stabilized by adding the trial mass to the correct location on the flywheel. This system can become completely balanced when


351

Fig. 15 The frequency response of the investigated conditions.

the correct mass of M derived from (16) is added to the determined location. a (16) M = × mtrial c 0.133 × 63.6 = 33.68(gram) (17) M= 0.2511 Therefore, the adjusted trial mass of 33.68 gram would have resulted in making the rotor completely balanced. It is important to note that the original unbalanced mass was 30.77 gram. This means the method provided here has achieved a great level of accuracy.

5 Conclusions and future work In this paper, a novel method for balancing and reducing the vibrations of a rotor affected by an unbalanced load torque is presented. The conventional balancing techniques were investigated and their drawbacks were discussed. The developed method has used a combination of the accelerometer data with simple video recordings of the rotating flywheel on an induction motor. Implementing functions from image processing toolbox of MATLAB, an automatic balancing methodology is developed. The results from the vibration spectrum demonstrated that this method has a high level of accuracy while it has decreased the amount of required experiments and is more computationally intensive. This method will help specify the amount and location of balancing mass with a good precision and few number of measurements compared to the conventional method. This method is developed to create a smartphone application. The reliability of the application is depending on the quality of the camera and sensitivity of mobile phone’s accelerometers. More case studies will be conducted to further demonstrate the effectiveness of the proposed balancing method.

Acknowledgement The work reported in this paper was funded by UND ME department and UND VPAA New Faculty Start-up Award 43700-2725-UND0031020.

References [1] Nejadpak, A. and Yang, C. (2016), A vibration-based diagnostic tool for analysis of superimposed failures in electric machines, IEEE International Conference on Electro Information Technology (EIT). [2] Renwick, J.T. and Babson, P.E. (1985), Vibration Analysis—A Proven Technique as a Predictive Maintenance Tool, IEEE Transactions on Industry Applications, 2(1985), 324-332.

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[3] Tsypkin, M. (2011), Induction motor condition monitoring: Vibration analysis technique-A practical implementation, IEEE International Electric Machines & Drives Conference. [4] Copping, M. (2015), Vibration analysis reporting–bearing failure stages and responses, Reliabilityweb.com. [5] Plante, T., Nejadpak, A., and Yang, C. (2015), Faults detection and failures prediction using vibration analysis, IEEE AUTOTESTCON. [6] Mehala, N. (2010), Condition monitoring and fault diagnosis of induction motor using motor current signature analysis, Diss. National Institute of Technology Kurukshetra, India. [7] Ebersbach, S. and Peng, Z. (2018), Expert system development for vibration analysis in machine condition monitoring, Expert Systems with Applications, 34(1), 291-299. [8] Plante, T., Stanley, L, Nejadpak, A., and and Yang, C. (2016), Rotating machine fault detection using principal component analysis of vibration signal, IEEE AUTOTESTCON. [9] Nisbett, K. (1996), Dynamic balancing of Rotating Machinery Experiment, Technical Manual. http://web.mst.edu/∼stutts/ME242/LABMANUAL/DynamicBalancingExp.pdf. Accessed on April 5, 2017. [10] Canny, J. (1986), A computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 679-698. [11] Gao, W., Zhang, X., Zhang, L., and Liu, H. (2010), An improved Sobel edge detection, 2010 3rd IEEE International Conference on Computer Science and Information Technology, 5, IEEE, 2010. [12] Shrivakshan, G.T. and Chandrasekar, C. (2012), A comparison of various edge detection techniques used in image processing, International Journal of Computer Science Issues, 9(5), 272-276. [13] Duda, R.O. and Hart, P.E. (1972), Use of the Hough transformation to detect lines and curves in pictures, ommunications of the ACM, 15(1), 11-15.



Towards Infinite Bifurcation Trees of Period-1 Motions to Chaos in a Time-delayed, Twin-well Duffing Oscillator Siyuan Xing, Albert C.J. Luo† Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA Submission Info Communicated by C.S. Suh Received 1 September 2017 Accepted 3 October 2017 Available online 1 January 2018 Keywords Time-delayed Duffing oscillator Twin-well Duffing oscillator Period-1 motions to chaos Bifurcation tree Implicit mapping Mapping structures Nonlinear frequency-amplitudes

Abstract In this paper, bifurcation trees of periodic motions to chaos in a periodically forced, time-delayed, twin-well Duffing oscillator are predicted by a semi-analytical method. The twin-well Duffing oscillator is extensively used in physics and engineering. The bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is very significant for determine motion complexity. Thus, the bifurcation trees for periodic motions to chaos in such a time-delayed, twin-well Duffing oscillator are obtained analytically. From the finite discrete Fourier series, harmonic frequency-amplitude characteristics for period-1 to period-4 motions are analyzed. The stability and bifurcation behaviors of the time-delayed Duffing oscillator are different from the non-time-delayed Duffing oscillator. From the analytical prediction, numerical illustrations of periodic motions in the time-delayed, twin-well Duffing oscillator are completed. The complexity of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. As a slowly varying excitation becomes very slow, the excitation amplitude will approach infinity for the infinite bifurcation trees of period-1 motion to chaos. Thus infinite bifurcation trees of period-1 motion to chaos can be obtained. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In 2016, Luo and Xing [1, 2] studied in the stability and bifurcation of periodic motion. From such studies, the motion complexity caused by time-delay was observed, which was different from the nontime-delayed hardening Duffing oscillator in Luo and Guo [3]. Because the twin-well Duffing oscillator has been extensively adopted in physics and engineering, the control of such an oscillator becomes very important in application. Thus, the time-delay term has been used to control the Duffing oscillator. However, the time-delay term introduced to the Duffing oscillator changes the stability and bifurcation of motions in such an oscillator. In 2016, Guo and Luo [4] investigated the dynamics of the twin-well Duffing oscillator. From such an investigation, the complex periodic motions and bifurcation trees to † Corresponding

author. Email address: [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2017.12.006

354

Siyuan Xing, Albert C.J. Luo / Journal of Vibration Testing and System Dynamics 1(4) (2017) 353–392

chaos were very complicated, which were much beyond the traditional analysis of periodic motions in the twin-well Duffing oscillator. To investigate the time-delay effects on stability and bifurcation of periodic motions in the time-delayed Duffing oscillator, the semi-analytical method will be used to investigate the bifurcation trees of period-1 motions to chaos herein. In the traditional analysis of periodic motions in nonlinear systems used were the perturbation methods, and harmonic balance method. In 1788, Lagrange [5] used the method of averaging to investigate periodic motions of three-body problem through a perturbation of the two-body problem. The method of averaging is based on the periodic motions of linear dynamical systems with slowly time-varying coefficients. In the end of the 19th century, Poincare [6] developed the perturbation theory for periodic motions in nonlinear dynamical systems, and the nonlinear systems were expressed with linear solutions parts plus the perturbed terms. Using the order of perturbation parameters, the perturbed term solutions were determined. In 1920, van der Pol [7] studied the periodic solutions of oscillation systems in circuits by the method of averaging, which were experimentally observed. In 1928, Fatou [8] gave the proof of the asymptotic validity through the solution existence theorems of differential equations, which provided the mathematical foundation for such a method. In 1935, Krylov and Bogoliubov [9] further developed the method of averaging for nonlinear oscillations in nonlinear vibration systems. Since then, one thought the perturbation method as a unique way for periodic solutions in nonlinear dynamical systems. In 1964, Hayashi [10] used the perturbation and simple harmonic balance method for periodic motions in nonlinear systems. To obtain the better approximate solutions of periodic motion in nonlinear dynamical systems, the multiscale perturbation method was adopted extensively (e.g., Nayfeh [11]; Nayfeh and Mook [12]), and the perturbation method has been employed for periodic motions in the Duffing oscillator. In addition, one used the perturbation method for the time-delayed Duffing oscillators (e.g., Hu and Wang [13]). In recent years, chaotic motions in nonlinear systems were investigated through the perturbation analysis. In fact, such a way for chaotic motions in nonlinear dynamical systems was inadequate. In 2012, Luo [14] developed the generalized harmonic balance method for periodic motions in nonlinear dynamical systems. Luo and Huang [15] applied such a method for approximate analytical solutions of periodic motions in the Duffing oscillator, and Luo and Huang [16] gave the analytical bifurcation trees of period-m motions to chaos in the Duffing oscillator (also see, Luo and Huang [17, 18]). From the generalized harmonic balance method, Wang and Liu [19] developed a numerical scheme for coefficients in the finite Fourier series expression of period motions in nonlinear dynamical systems. In 2013, Luo [20] proposed an analytical method for analytical solutions of periodic motions in time-delayed, non-linear dynamical systems. Luo and Jin [21–23] studied bifurcation trees of periodic motions to chaos in the time-delayed, periodically forced Duffing oscillator. From the analytical method, the time-delay causes the difficulty to determine the stability. Herein, the semi-analytical method will be employed for the time-delayed twin-well Duffing oscillator. In this paper, bifurcation trees of period-1 motions to chaos in the time-delayed, twin-well Duffing oscillator will be studied through the semi-analytical method. Periodic motions in the time-delayed twin-well Duffing oscillator will be described by specific mapping structures of implicit maps, and solving algebraic equations in mapping structures will produce periodic motions in the time-delayed Duffing oscillator, and the corresponding stability and bifurcation of periodic motions will be analyzed by the eigenvalue analysis. The bifurcation trees of periodic motion to chaos will be predicted analytically. From the finite Fourier series, nonlinear harmonic frequency-amplitude characteristics of the bifurcation trees will be discussed. Numerical illustrations of periodic motions will be carried out.


355

2 Semi-analytical method To determine period-m motion in the time-delay dynamical systems, the following theorem presented herein is from Luo [24]. Theorem 1. Consider a time-delay nonlinear dynamical system x˙ = f(x, xτ ,t, p) ∈ R n ,

(1)

with x(t0 ) = x0 , x(t) = Φ (x0 ,t − t0 , p) for t ∈ [t0 − τ , ∞).

If such a time-delay dynamical system has a period-m flow x(m) (t) with finite norm ||x(m) || and period mT (T = 2π Ω), there is a set of discrete time tk (k = 0, 1, · · · , mN) with (N → ∞) during m-periods (mT ), and the corresponding solution x(m) (tk ) and vector field f(x(m) (tk ), xτ (m) (tk ),tk , p) are exact. Suppose τ (m) (m) are on the approximate solution of the periodic flow under ||x(m) (tk ) − discrete nodes xk and xk (m) τ (m) xk || ≤ εk and ||xτ (m) (tk ) − xk || ≤ εkτ with small εk , εkτ ≥ 0 and τ (m)

||f(x(m) (tk ), xτ (m) (tk ),tk , p) − f(xk , xk (m)

tk , p)|| ≤ δk

(2) (m)

τ (m)

(m)

τ (m)

with a small δk ≥ 0. During a time interval t ∈ [tk−1 ,tk ], there is a mapping Pk : (xk−1 , xk−1 ) → (xk , xk (k = 1, 2, · · · , mN) as τ (m)

(m)

τ (m)

(m)

(m)

(m)

τ (m)

τ (m)

(xk , xk ) =Pk (xk−1 , xk−1 ) with gk (xk−1 , xk ; xk−1 , xk , p) = 0, τ (m) (m) (m) x j = h j (xr j −1 , xr j , θr j ), j = k, k − 1; r j = j − l j , k = 1, 2, · · · , mN; τ (m)

(e.g., xr

(m)

(m)

(m)

= xsr + θr (xrr −1 − xrr ), θr =

1 hr j

)

lr j

(3)

[τ − ∑ hr j +i ]). i=1

where gk is an implicit vector function and h j is an interpolation vector function. Consider a mapping structure as (m) (m) P = PmN ◦ PmN−1 ◦ · · · ◦ P2 ◦ P1 : x0 → xmN ; (4) τ (m) (m) τ (m) (m) with Pk : (xk−1 , xk−1 ) → (xk , xk ) (k = 1, 2, · · · , mN). (m)

(m)

τ (m)

For xmN = P(x0 , x0

(m)∗

), if there is a set of points (xk (m)∗

(m)∗

τ (m)∗

τ (m)∗

, xk

τ (m)∗

gk (xk−1 , xk ; xk−1 , xk , p) = 0, τ (m)∗ (m)∗ (m)∗ = h j (xr j −1 , xr j , θr j ), j = k, k − 1 xj (m)∗

(m)∗

(m)∗

xr j −1 = x mod (r j −1+mN,mN) , xr j (m)∗

x0

(m)∗

τ (m)∗

= xmN and x0

) (k = 0, 1, · · · , mN) computed by

) (k = 1, 2, · · · , mN) (5)

(m)∗

= x mod (r j +mN,mN) ;

τ (m)∗

= xmN .

τ (m)∗

(k = 0, 1, · · · , mN) are the approximation of points x(m) (tk ) and xτ (m) (tk ) and xk Then the points xk (m)∗ (m) (m)∗ (m) τ (m)∗ τ (m) and xk of periodic solutions. In the neighborhoods of xk , with xk = xk + ∆xk and xk = τ (m)∗ τ (m) xk + ∆xk , the linearized equation is given by (m)∗

k

∑

∂ gk

(m) ∆x j + (m) j=k−1 ∂ x j

∂ gk τ (m) ∂xj

τ (m)

(

∂xj

(m) ∆xr j + (m) ∂ xr j

τ (m)

∂xj

(m) ∂ xr j −1

with r j = j − l j , j = k − 1, k; (k = 1, 2, · · · , mN).

(m)

∆xr j −1 ) = 0

(6)

356


The resultant Jacobian matrices of the periodic flow are (m)

DPk(k−1)...1 = [

∂ yk

]

(m)∗

(m) (y0

∂ y0

(m)∗

,··· ,yk

)

= Ak Ak−1 · · · A1 (k = 1, 2, · · · , mN), (7)

(m)

and DP = DPmN(mN−1)...1 = [

∂ ymN

]

where (m) ∆yk

and

" (m) Ak (m)

Bk

=

=

(m) (m) Ak ∆yk−1 ,

(m)

Bk

(m)

= AmN AmN−1 · · · A1 .

(m)∗ (m) (y(m)∗ ,··· ,ymN ) 0

∂ y0

(m) Ak

(m)

(ak(rk−1 −1) )n×n

≡

(m)

∂ yk

(m)

∂ yk−1

(m)∗

(8) )

# , s = 1 + lk−1

(m)

0k

Ik

(m)∗

(yk−1 ,yk

(m)

= [(ak(k−1) )n×n , 0n×n , · · ·

(m)

n(s+1)×n(s+1) (m) , (ak(rk −1) )n×n ],

(9)

Ik = diag(In×n , In×n , · · · , In×n )ns×ns , (m) 0k = (0n×n , 0n×n · · · , 0n×n )T ; {z } | s (m)

(m)

(m)

(m)

yk = (xk , xk−1 , · · · , xrk−1 )T , (m) (m) (m) (m) yk−1 = (xk−1 , xk−2 , · · · , xrk−1 −1 )T , (m)

(m)

(m)

(m)

∆yk = (∆xk , ∆xk−1 , · · · , ∆xrk−1 )T , (m) (m) (m) (m) ∆yk−1 = (∆xk−1 , ∆xk−2 , · · · , ∆xrk−1 −1 )T ; (m)

ak j = [ (m) akr j (m)

=[

∂ gk (m) ∂ xk

∂ gk (m)

∂ xk

ak(r j −1) = [

]−1

∂ gk (m)

∂xj

,

]

τ (m)

∂ gk ∂ x ∑ τ (m) ∂ xαr , j α = j ∂ xα j+1

−1

(10)

∂ gk

(11) τ (m)

j ∂ gk ∂ x −1 ] ∑ τ (m) ∂ xrα −1 (m) j ∂ xk α = j−1 ∂ xα

with r j = j − l j , j = k − 1, k. (m)

The properties of discrete points xk (k = 1, 2, · · · , mN) can be estimated by the eigenvalues of DPk(k−1)···1 as |DPk(k−1)···1 − λ¯ In(s+1)×n(s+1) | = 0 (k = 1, 2, · · · , mN). (12) The eigenvalues of DP for such periodic motion are determined by |DP − λ In(s+1)×n(s+1)| = 0,

(13)

and the stability and bifurcation of the periodic flow can be classified by the eigenvalues of DP(y∗0 ) with o m o ([nm 1 , n1 ] : [n2 , n2 ] : [n3 , κ3 ] : [n4 , κ4 ]|n5 : n6 : [n7 , l, κ7 ]).

(14)

(i) If the magnitudes of all eigenvalues of DP are less than one (i.e.,|λi | < 1, i = 1, 2, · · · , n(s + 1)), the approximate periodic solution is stable.


357

(ii) If at least the magnitude of one eigenvalue of DP is greater than one (i.e., |λi | > 1, i ∈ {1, 2, · · · , n(s+ 1)}), the approximate periodic solution is unstable. (iii) The boundaries between stable and unstable periodic flow with higher order singularity give bifurcation and stability conditions with higher order singularity. Proof. See Luo [24]. 3 Discretization of dynamical systems Consider a time-delayed, twin-well Duffing oscillator as x¨ + δ x˙ − α1 x − α2 xτ + β x3 = Q0 cos Ωt

(15)

where x = x(t) and xτ = x(t − τ ). The first order form of the above equation is x˙ = y, y˙ = Q0 cos Ωt − δ y + α1 x + α2 xτ − β x3 .

(16)

Set x = (x, y)T and xτ = (xτ , yτ )T . For tk = kh (k = 0, 1, 2, ...), discrete points and time-delay points are xk = (xk , yk )T and xτk = (xτk , yτk )T , respectively. The midpoint scheme is used for discretization of Eq.(16) for t ∈ [tk−1 ,tk ] (k = 1, 2, ...), and an implicit map Pk is given by Pk : (xk−1 , xτk−1 ) → (xk , xτk ) ⇒ (xk , xτk ) = Pk (xk−1 , xτk−1 ).

(17)

The two implicit equations of the implicit map are 1 xk =xk−1 + h(yk + yk−1 ) 2 h 1 yk =yk−1 + h[Q0 cos Ω(tk−1 + ) − δ (yk + yk−1 ) 2 2 1 1 1 + α1 (xk + xk−1 ) + α2 (xτk + xτk−1 ) − β (xk + xk−1 )3 ]. 2 2 8

(18)

If the time-delay node xτk ≈ x(tk−τ ) of xk ≈ x(tk ) lies between xk−lk and xk−lk −1 (lk = int(τ /h)), the time-delay node can be approximated by an interpolation function of two points xk−lk and xk−lk −1 . For a time-delay node xτj ( j = k − 1, k), we have xτj = h j (xr j −1 , xr j , θr j ) for r j = j − l j .

(19)

For instance, using the simple Lagrange interpolation, the time-delay discrete node xτj = h j (xr j −1 , xr j , θr j ) ( j = k, k − 1) becomes

τ + l j )(x j−l j − x j−l j −1 ), h τ yτj = y j−l j −1 + (1 − + l j )(y j−l j − y j−l j −1 ). h xτj = x j−l j −1 + (1 −

Thus, the time-delay nodes are approximated by non-time-delay nodes.

(20)

358


4 Period-m motions and stability For a period-m motions in time-delay dynamical systems, a discrete mapping structure is defined as τ (m)

(m)

P = PmN ◦ PmN−1 ◦ · · · ◦ P2 ◦ P1 : (x0 , x0 {z } | mN−actions (m)

τ (m)

τ (m)

(m)

(xmN , xmN ) = P(x0 , x0 with

τ (m)

(m)

(m)∗

(21)

) τ (m)

(m)

Pk : (xk−1 , xk−1 ) → (xk , xk Points xk

τ (m)

(m)

) → (xmN , xmN )

) (k = 1, 2, · · · , mN).

(22)

on the periodic motion of the time-delayed, twin-well, Duffing oscillator are determined by ) (m)∗ (m)∗ τ (m)∗ τ (m)∗ gk (xk−1 , xk ; xk−1 , xk , p) = 0 (k = 1, 2, · · · , mN) τ (m)∗ (m)∗ (m)∗ = h j (xr j −1 , xr j , θr j ), j = k, k − 1 xj (23) (m)∗

x0

τ (m)∗

(m)∗

τ (m)∗

= xmN and x0

= xmN .

With gk = (gk1 , gk2 )T , from Eq.(18), algebraic equations for the period-m motion for Pk are (m)

(m)

(m)

(m)

gk1 =xk − [xk−1 + 12 h(yk + yk−1 )] = 0, (m)

(m)

(m)

(m)

gk2 =yk − {yk−1 + h[Q0 cos Ω(tk−1 + 21 h) − 21 δ (yk + yk−1 ) τ (m)

(m)

(m)

+ 21 α1 (xk + xk−1 ) + 21 α2 (xk

τ (m)

(m)

(m)

+ xk−1 ) − 18 β (xk + xk−1 )3 ]}

(24)

=0 (k = 1, 2, · · · , mN) τ (m)

From Eq.(20), the corresponding algebraic equations for time-delay node x j ( j = k, k − 1) in Eq.(23) are τ τ (m) (m) (m) (m) x j = xk−l j −1 + (1 − + l j )(xk−l j − xk−l j −1 ), h (25) τ τ (m) (m) (m) (m) y j = yk−l j −1 + (1 − + l j )(yk−l j − yk−l j −1 ). h From Eqs.(23)-(25), discrete nodes of periodic motions in the time-delayed, twin-well, Duffing nonlinear (m)∗ oscillator are determined by 2(mN + 1) equations. If discrete nodes xk (k = 1, 2, · · · , mN) of the periodm motion are achieved, the stability of the period-m motion can be determined through the Jacobian (m)∗ and matrix of the mapping structure with the corresponding discrete nodes. In neighborhood of xk τ (m)∗ (m) (m)∗ (m) τ (m) τ (m)∗ τ (m) xk , with xk = xk + ∆xk and xk = xk + ∆xk , the linearized equation is k

∑

∂ gk

(m) ∆x j + (m) j=k−1 ∂ x j

τ (m)

∂ gk τ (m) ∂xj

(

∂xj

τ (m) ∆xr j + (m) ∂ xr j

τ (m)

∂xj

(m) ∂ xr j −1

(m)

∆xr j −1 ) = 0

(26)

with r j = j − l j , j = k − 1, k; (k = 1, 2, · · · , mN). Setting (m)

yk

(m)

(m)

(m)

= (xk , xk−1 , · · · , xrk−1 )T ,

(m)

(m)

(m)

(m)

yk−1 = (xk−1 , xk−2 , · · · , xrk−1 −1 )T , (m)

∆yk

(m)

(m)

(m)

(m)

= (∆xk , ∆xk−1 , · · · , ∆xrk−1 )T , (m)

(m)

(m)

∆yk−1 = (∆xk−1 , ∆xk−2 , · · · , ∆xrk−1 −1 )T ,

(27)


359

the resultant Jacobian matrix of the period-m motion is (m)

DP = DPmN(mN−1)···1 = [ =

∂ ymN

(m) (m) (m) AmN AmN−1 · · · A1

=A

where (m) ∆yk

(m) (m) = Ak ∆yk−1 ,

]

(m)∗ (m)∗ (m) (y(m)∗ ,y1 , ··· ,yN ) 0

∂ y0

(m) Ak

≡

(28)

(m)

(m)

∂ yk

(m)

∂ yk−1

(m)∗

(m)∗

(yk−1 ,yk

(29) )

and

(m) ak j

=

∂ gk

−1

∂ gk

(m) ,a (m) kr j ∂xj

∂ gk

−1

= (m) (m) ∂ xk ∂ xk τ (m) ∂ gk −1 j ∂ gk ∂ xα (m) ak(r j −1) = ∑ τ (m) (m) (m) ∂ xk ∂ xr j −1 α = j−1 ∂ xα

j+1

∑

α= j

τ (m)

∂ gk ∂ xα τ (m)

∂ xα

(m)

∂ xr j

with r j = j − l j , j = k − 1, k; # " (m) (m) Bk (ak(rk−1 −1) )2×2 (m) , s = 1 + lk−1 Ak = (m) (m) 0k Ik 2(s+1)×2(s+1) (m)

Bk

(m)

(m)

,

(30)

(m)

= [(ak(k−1) )2×2 , 02×2 , · · · , (ak(rk −1) )2×2 ],

Ik

= diag(I2×2 , I2×2 , · · · , I2×2 )2s×2s ,

(m) 0k

= (02×2 , 02×2 , · · · , 02×2 )T . {z } | s

where

∂ gk −1 − 12 h 1 − 21 h = = , , (m) (m) ∆ 12 hδ − 1 ∆ 21 hδ + 1 ∂ xk−1 ∂ xk τ (m) τ (m) ∂xj ∂xj 0 0 0 0 = τ = , , (m) (m) ( h − l j) 0 (1 − τh + l j ) 0 ∂ xr j −1 ∂ xr j ∂ gk 0 0 = , (m) 0 − 21 hα2 ∂xj 1 ∆ = h[−4α1 + 3β (xk + xk−1 )2 ]. 8

∂ gk

(31)

The eigenvalues of DP for the period-m motion in the time-delayed, twin-well, Duffing oscillator are computed by |DP − λ I2(s+1)×2(s+1)| = 0, (32) (i) For |λi | < 1 (i = 1, 2, · · · , 2(s + 1)), the approximate solution of the period-m motion is stable. (ii) For |λi | > 1 (i ∈ {1, 2, · · · , 2(s + 1)}), the approximate solution the period-m motion is unstable. (iii) The boundaries between the stable and unstable solutions with higher order singularity give bifurcation and stability conditions. (iv) For λi = 1 with |λ j | < 1(i, j ∈ {1, 2, · · · , 2(s + 1)} and i 6= j), the saddle-node bifurcation (SN) of the period-m motion occurs.

360


(v) If λi = −1 with |λ j | < 1(i, j ∈ {1, 2, · · · , 2(s + 1)} and i 6= j), the period-doubling bifurcation (PD) of the period-m motion occurs. (vi) If |λi, j | = 1 with |λl | < 1(i, j, l ∈ {1, 2, · · · , 2(s + 1)} and λi = λ¯ j , l 6= i, j), the Neimark bifurcation (NB) of the period-m motion occurs. 5 Finite Fourier series analysis (m)

(m)

(m)

For approximate node points xk = (xk , yk )T (k = 0, 1, 2, · · · , mN) of a period-m motion, the approximate solution of the period-m motion is expressed by the finite Fourier series as (m)

x(m) (tk ) ≈xk

mN/2

(m)

= a0 +

∑

j=1

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

mN/2

(m)

=a0 +

∑

b j/m cos(

j=1

j 2kπ j 2kπ ) + c j/m sin( ) m N m N

(33)

(k = 0, 1, · · · , mN − 1) where 2π 2kπ = N∆t; Ωtk = Ωk∆t = , Ω N (m) (m) (m) a0 = (a01 , a02 )T , b j/m = (b j/m1 , b j/m2 )T , c j/m = (c j/m1 , c j/m2 )T .

T=

(34)

From Eq.(33), the coefficients of the finite Fourier series are (m)

a0 =

1 mN−1 (m) ∑ xk , mN k=0

  2 mN−1 (m) 2 jπ  b j/m = xk cos(k ),  ∑ mN k=1 mN  2 mN−1 (m) 2 jπ c j/m = xk sin(k ) ∑ mN k=1 mN

    

(35) ( j = 1, 2, · · · , mN/2).

Thus the approximate expression of the period-m motion in Eq.(16) is determined by (m)

x(m) (t) ≈ a0 +

mN/2

∑

j=1

j j b j/m cos( Ωt) + c j/m sin( Ωt). m m

(36)

The foregoing equation can be rewritten as

where

x(m) (t) y(m) (t)

(

≡

(m) x1 (t) (m) x2 (t)

)

( ≈

(m) a01 (m) a02

 j  Ωt − φ j/m1 ) m j    A j/m1 cos( Ωt − φ j/m2 ) m 

)

mN/2   A j/m1 cos(

+

∑

j=1

q c j/m1 b2j/m1 + c2j/m1 , φ j/m1 = arctan , b j/m1 q c j/m2 A j/m2 = b2j/m2 + c2j/m2 , φ j/m2 = arctan . b j/m2

(37)

A j/m1 =

(38)


361

For simplicity, harmonic amplitudes of displacement x(m) (t) for period-m motions will be presented. Similarly, the harmonic amplitudes of velocity y(m) (t) for periodic motions can also be determined. Thus the displacement can be expressed as mN/2

(m)

x(m) (t) ≈ a0 +

∑

j=1

=

(m) a0 +

mN/2

∑

j=1

where A j/m =

q

j j b j/m cos( Ωt) + c j/m sin( Ωt) m m (39) j A j/m cos( Ωt − φ j/m ) m

b2j/m + c2j/m , φ j/m = arctan

c j/m . b j/m

(40)

6 Bifurcation trees of periodic motions In this section, the complete bifurcation tree of period-1 motion to chaos for the periodically forced, time-delayed, twin-well Duffing oscillator will be presented through the analytical predictions of period1 to period-4 motions. Illustration of periodic motions for such time-delayed system will be given. Consider a set of system parameters as

δ = 0.5, α1 = 10.0, α2 = 5.0, β = 10, Q0 = 100

(41)

and the time-delay is τ = T /4 with the excitation period T = 2π /Ω. To ensure computational accuracy of ε = 10−9 , due to the discretization with ε ∼ O(h3 ), h ∼ 10−3 is selected. Thus N = T /h = T /∆t. For instance, we have N = 1024 for Ω ≥ 7. N = 2048 for 3 ≤ Ω < 7. N = 3072 for 2 ≤ Ω < 3. N = 6144 for 1 < Ω < 2. N = 12288 for 0 < Ω < 1. However, for Ω < 0.5, the computational accuracy is not higher. For the stability analysis of periodic motions, the eigenvalues of the resultant Jacobian matrix are computed through the QR method in the following numerical illustrations. The parameters of Eq.(41) are chosen arbitrarily. 6.1

Bifurcation trees of periodic motions

The bifurcation trees of period-1 to period-4 motions in the time-delayed twin-well Duffing oscillator are predicted analytically through the implicit mapping. The bifurcation trees are illustrated by displacement and velocity of the periodic nodes x mod (k,N) with mod (k, N) = 0, as shown in Figs.1-15. The solid and dashed curves represent the stable and unstable motions, respectively. The acronyms ‘SN’, ‘PD’ and ‘NB’ represent the saddle node, period doubling, and Neimark bifurcations, respectively. The period-1, period-2, and period-4 motions are labeled by P-1, P-2, and P-4, respectively. The period-2 motions appear from the PD bifurcations of the period-1 motions, and period-4 motion appear from the PD bifurcation of the period-2 motion. The global view of the bifurcation trees is presented in Fig.1 for Ω ∈ (0, 20] and the zoomed views of the bifurcation trees are presented for a specific frequency ranges in Figs.2-15. The bifurcation points are tabulated in Table 1. In Fig.1, the global view of the bifurcation trees of period-1 to period-4 motions are presented for Ω ∈ (0, 20]. The stable and unstable symmetric period-1 motion exist for Ω ∈ (0, ∞). From Fig.1(b), for Ω ∈ (5.431, 18.94), on the upper branch, the symmetric period-1 motion is stable. For Ω ∈ (8.261, 18.94), on the middle branch, the symmetric period-1 motion is unstable. For Ω ∈ (8.261, 9.90), on the lower branch, the symmetric period-1 motion is stable. However, for Ω ∈ (9.90, ∞), the lower branch of the symmetric period-1 motion is unstable. The saddle-node bifurcations of the symmetric period-1 motion occur at Ω ≈ 18.94, 8.264 with jumping phenomena (catastrophe). The saddle-node bifurcation of the


Periodic Node Displacement, xmod(k,N)

7.0

SN

SN SN

SNSN P-1

3.5

S P-1

0.0 A -3.5

-7.0 0.0

5.0

10.0

15.0

SN

140.0

Periodic Node Velocity, ymod(k,N)

362

20.0

SN SN

SN

1.5

95.0 P-1

50.0

0.3 9.0

15.0

S

P-1

5.0

A -40.0 0.0

Excitation Frequency,

5.0

10.0

15.0

20.0


(a)

(a)

(b)

(0,20) y

0.5,

= 10, =T /4 = 10, 5.0, with 100, frequency Fig. 1 The global view of bifurcation trees of period-1 to period-4 motions varying excitation

mod( mod( , ,20)). ) =) = 0.0.(a) node displacement x mod (k,N) , (b) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5.0, β = 10, (Ω ∈ (0, SNSN Q0 = 100, τ = T /4). mod (k, N) = 0. SN

symmetric period-1 motion occurs at Ω ≈ 9.9 with the onset of an asymmetric period-1 motion relative to the twin-potential wells, and such an asymmetric period-1 motion is stable for Ω ∈ (9.90, ∞) and Q0 = 100. For Ω ∈ (0, 5.431), the stable and unstable symmetric period-1 motions vary with excitation frequency trees, and in such a frequency range, there are 18 bifurcation trees of period-1 to period-4 motion. In Fig.2 (a)-(d), the first bifurcation trees of period-1 to period-4 motion is presented for Ω ∈ (4.30, 5.54). For such a bifurcation tree, there are seven (7) saddle-node bifurcations and nine (9) period-doubling bifurcations. One (1) saddle-node bifurcation is for symmetric period-1 motions to the asymmetric period-1 bifurcation. Two (2) saddle-node bifurcations are for the jumping phenomena of the asymmetric period-1 motion. Four (4) saddle-node bifurcations are for the jumping phenomena of period-4 motions. Two (2) period-doubling bifurcations are for asymmetric period-1 to period2 motion. Four (4) period-doubling bifurcations are for period-2 to period-4 motion, and three (3) period-doubling bifurcations are for period-4 to period-8 motion. The zoomed view of bifurcation tree for Ω ∈ (4.70, 4.79) is presented in Fig.2(c) and (d) to show the details of period-2 to period-4 motions and the corresponding catastrophes of period-4 motions are observed clearly. In Fig.3 (a)-(d), the second bifurcation trees of period-1 to period-4 motion is presented for Ω ∈ (3.10, 4.34), which continues the asymmetric period-1 motion from the first bifurcation tree. For the second bifurcation tree, there are four (4) saddle-node bifurcations and six (6) period-doubling bifurcations. Two (2) saddle-node bifurcations are for the jumping phenomena of period-2 motion, and two (2) saddle-node bifurcations are for the jumping phenomena of period-4 motion. Two period-doubling bifurcations are for asymmetric period-1 to period-2 motions. Two (2) period-doubling bifurcations are for period-2 to period-4 motion, and two (2) period-doubling bifurcations are for period-4 to period-8 motion. The asymmetric period-1 motion continues to the third bifurcation tree of period-1 to period-4 motion for Ω ∈ (2.35, 3.35), as shown in Fig.4(a)-(f). Such a bifurcation tree has six (6) saddle-node bifurcations, six (4.3,4.54) (6) period-doubling bifurcations and three (3) Neimark bifurcations. Three (3) saddle-node bifurcations y 0.5, = 10, T /the 4 = 10, 5.0, 100, are for the jumping of the symmetric period-1 motion. Two (2) saddle-node bifurcations are =for mod( , ) = 0. mod( , ) =of0. period-2 motion. One (1) saddle-node bifurcation at Ω ≈ 2.4104 is for the symmetric to jumping asymmetric period-1 motion. Between such a saddle-node bifurcation and the saddle-node bifurcation of Ω ≈ 5.431, a large bifurcation tree of the asymmetric period-1 to period-4 motion exists, and the large bifurcation tree includes three bifurcation trees of asymmetric period-1 motion to chaos, which


363

mod( )= mod(,k , N ) =0.0. Periodic Node Displacement, xmod(k,N)

PD PD SN

SN SN

PD PD PD

P-2 P-2

P-4

P-1

3.2 P-1

P-1 S

P-4 A

1.8

A

0.4

-1.0 4.30

4.61

4.92

5.23

SN PDPD

32.0


SN PDPD

4.6

A P-1 S

4.0

-10.0

P-2

P-1

P-4

4.61

4.92

(b)

0.7 P-2

4.74

SN PD

18.0



PD SN

1.6

4.72

5.54


2.5

-0.2 4.70

5.23

(b)

PD

P-4

P-2

P-4

P-1

(a) SN

SN

18.0


3.4

PDPDPD

A

-24.0 4.30

5.54

PDPDSN

4.76

PD SN

16.0

14.0 -10.0 P-4 -24.0 4.70

4.72

P-2

4.74

4.76



(c)

y y

0.5, 0.5,

= 10, = 10,

(d) 5.0, 5.0,

= 10, = 10,

(4.3,4.54) (4.3,4.54) 100, = T / 4 100, = T / 4

mod( = Fig. mod(2, , ) The ) =0.0. first bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ mod( ,, ))==0. mod( 0. (4.3, 4.54)). (a,c) node displacement x mod (k,N) , (b,d) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5.0, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.

are named by the first, second and third bifurcation trees. In the third bifurcation trees, three pairs of period-doubling bifurcations are for period-1 to period-2, period-2 to period-4, and period-4 to period-8 motions. From Ω ≈ 2.4104 to 2.403 via Ω ≈ 3.1, there is a stable and unstable symmetric period-1 motion. In addition, three (3) Neimark bifurcations of the symmetric period-1 motion occur for quasi-periodic motions. From Ω ≈ 2.403 to 1.6948, there is a big bifurcation tree including two bifurcation trees (i.e., the fourth and fifth bifurcation trees). The fourth bifurcation tree of period-1 to period-4 motion for Ω ∈ (2.082, 2.422) is presented in Fig.5(a)-(f). For such a bifurcation tree, there are five (5) saddle-node bifurcations and ten (10) period-doubling bifurcations. One (1) saddle-node bifurcation at Ω ≈ 2.403 is for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the asymmetric period-1 motion, and two (2) saddle-node bifurcations are for the jumping phenomena of the period-2 motion. Two (2) period-doubling bifurcations are for the period-1 to period-2 motion. Four (4) period-doubling bifurcations are for the period-2 to period-4 motions, and also four (4) period-doubling bifurcations are for the period-4 to period-8 motion. In Fig.5(c)-(f), the zoomed views of the small local bifurcation trees are presented. With the connections of two asymmetric period-1 motions, the fifth bifurcation tree of the period-1 to period-4 motion for Ω ∈ (1.66, 2.14) is presented in Fig.6(a)-(f). This bifurcation tree has five (5) saddle-node bifurcations


SN PD


4.00

PD PD

PD

P-1 P-2

P-2 3.30

P-1

2.60

A

1.90

1.20 3.10

3.41

3.72

4.03

SNPD

26.0

P-4


364

P-1

8.0

-1.0

A

3.41

PD PD

23.4



4.34

(b)

P-4

3.30

3.24

2.40

2.10 3.15510

4.03

(b)

(a)

P-2

3.72


(a) SN

PD P-2 P-1


3.36

P-4

17.0

-10.0 3.10

4.34

PD P-2

3.15515

3.15520


23.1

SN

PD PD

P-2

P-4

12.0

8.0

4.0 3.15510

3.15515

3.15520


(c) (d) (3.10,3.43) 0.5, = 10, (3.10,3.43) = 10, = 5, = 100, (3.10,3.43) (3.10,3.43) 0.5, = 10, = 10, = 5, = 100, mod( , , ))== 0.0. mod( Fig.TT3//44The second zoomed view of bifurcation tree of period-1 to period-4 motions varying with excitation mod( , , )= TT/ 4/ 4 (Ω mod( ) 3.43)). =0.0. frequency ∈ (3.10, (a, c) node displacement x mod (k,N) , (b, d) node velocity y mod (k,N) . (δ = 0.5, α1 = 10,

α2 = 5, β = 10, 1.687 Q T /4). mod (k, N) = 0. 1.687 1.2338 0 = 100, τ =1.2338

1.687 1.2338 1.687 1.2338 (1.514,1.698) (1.514,1.698) and ten (10) period-doubling bifurcations. One (1) saddle-node bifurcation at Ω ≈ 1.6934 is for the (1.514,1.698) (1.514,1.698) symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the period-4 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the symmetric period-1 motion. Two (2) period-doubling bifurcations are for the asymmetric period1 to period-2 motion. Four (4) period-doubling bifurcations are for the period-2 to period-4 motion, and four (4) period-doubling bifurcation are for the period-4 to period-8 motion. From(1.22,1.54) Ω ≈ 1.6934 (1.22,1.54) to Ω ≈ 1.687, the symmetric period-1 motion has two (2) saddle-node bifurcations for the jumping phenomena of the symmetric period-1 motion. 1.2338 1.2338 From Ω ≈ 1.687 to Ω ≈ 1.2338, there is 1.2338 a1.2338 big bifurcation tree including two small bifurcation trees. 1.2338 1.22102, 1.22102, The two bifurcation trees of the period-1 to period-4 motion are named the sixth and1.2338 seventh bifurcation trees. In the sixth bifurcation tree for Ω ∈ (1.514, 1.698), there are five (5) saddle-node bifurcations and 10 period-doubling bifurcations, as shown in Fig. 7(a)-(f). One (1) saddle-node bifurcation is for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the asymmetric period-1 motions, and two (2) saddle-node bifurcations for the jumping phenomena of the period-2 motion. Two (2) period-doubling bifurcations are for the asymmetric period1 to period-2 motion. Four (4) period-doubling bifurcations are for the period-2 to period-4 motion, and four (4) period-doubling bifurcations are for the period-4 to period-8 motions. Continuing with


NB NB SN PD NB

SN A

1.2

3.9

P-2

P-1

S

SN

SN PD PD

P-1 P-2 P-4

-0.4

3.6 2.378

2.430

-2.0 2.35

2.60

2.85

3.10

NB NB SN PD NB

P-2

11.0 A

S

-4.0 P-1 -19.0

A P-1

-34.0 2.35

3.35

2.60

(a)



10.0

P-4 P-2

3.8 P-1 P-1

3.7 A

3.6 2.378

2.398

2.418

2.438

SN

P-4 P-2 8.0

A 6.0

P-1 S

4.0

2.0 2.378

2.458

P-1

2.398

2.418

(c) 17.6



SN

3.2

2.7

2.4 3.3130

3.3131

2.458

(d)

PD

P-4

2.438


(c) PD

3.35

SN PD PD


3.6

3.10

(b)

SN PD PD

S

SN

(b)

(a) SN

2.85



3.9

SN P-2

2.8 A

26.0SNPDPD

SNPD SNPD P-1



4.4 SNPDPD

365

A

3.3132

P-2

3.3133


3.3134

PD

SN

PD

17.4

6.0 P-4

P-2

1.0

-4.0 3.3130

3.3131

3.3132

3.3133

3.3134


(f) (e) (e) (f) (2.35,3.35) (2.35,3.35) (2.35,3.35) 0.5, (2.35,3.35) 0.5, 10, ==10, ==5,5, ==10, ==200, 10, 200, (2.35,3.35) 0.5, = 10, (2.35,3.35) = 10, = 5, = 200, mod( , ) = 0. T / 4 mod( = mod( ,, , ))=)=0. mod( 0.0. TTT//4 Fig. 44/ 4 The third zoomed view of bifurcation tree of period-1 to period-4 motions varying with excitation mod(, ,) =) 0. = 0. T /T4/ 4 mod(

frequency (Ω ∈ (2.35, 3.35)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.



4.0

SNPD PD

SN

P-4

2.8

P-2 P-1

P-1

S 1.6 A A 0.4 2.082

2.166

2.250

SNPDPD 14.0

SN SN

PDPD

2.334


366

SN

P-4

P-1

S

A -10.0 A

2.166

2.250

PD PD PD

12.0



(b)

A P-2

P-4

2.4 P-1 1.4 A

0.4 2.088

2.093

2.098

2.103

SN

PD PD PD

P-1

(c) (c)

-4.0

A -12.0

2.093

7.0

P-4

P-4

P-2

1.2

0.6 2.185

2.187

2.189

2.191



P-2

2.103

2.108

SN

(d)

2.4

1.8

2.098

Excitation Frequency, SN SN SN SN PDPDSN SN

SN PDPD

P-1 P-4

-20.0 2.088

2.108

P-2

4.0


3.0

2.418

(b)

(a)

P-1

2.334


(a)

SN

P-2

2.0


3.4

SN

P-1

-22.0 2.082

2.418

PDPDSN

SN PDPDSN

SNPDPD

3.5 P-4

P-2

0.0 P-2

P-4

-11.0

-19.0 2.185


2.187

2.189

2.191


(2.082,2.418) 0.5, (f) = 10, = 10, (2.082,2.418) = 5, = 100, (e) (2.082,2.418) (2.082,2.418) 0.5, = 10, = 10, = 5, = 100, mod( ,, ))==0. 0. TT(2.082,2.418) //44 mod( 0.5, = 10, = 10, = 5, = 100, (2.082,2.418) mod( mod( ,, ))==zoomed 0.0. Fig.TT5//44 The fourth view of bifurcation tree of period-1 to period-4 motions varying with excitation mod( , , ) )==0.0. TT/ /44 (Ω mod( frequency ∈ (2.082, 2.418)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ =

0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.


PD PD PDPD PD

SN P-1

1.95

A

S P-1

1.50

P-2

1.05 A 0.60 1.66

1.78

1.90

2.02

SN

14.0 SNSNPD

P-4



2.40 SNSNPD

S 6.0

P-1 P-2 -2.0

-10.0

A

1.78


1.90

(b) (b)

SN

-9.50

P-1 1.12 P-1 S A 1.00

0.94 1.692

1.697

1.702

PD PDPD P-2

P-4



PD PDPD P-2

1.06

-10.00

P-1

P-1

A

S -10.25

-10.50 1.692

1.707

1.697

1.707

(d) PD SN

14.0

SN PD

PD SN P-2



1.702


(c) (c) SN PD

P-4

-9.75


2.40

1.95

1.50 P-2

P-4

1.05

0.60 2.036

2.040

2.044

2.048

-2.0

-10.0

-18.0 2.040

2.041

mod( mod( ,, )) == 0. 0.

2.042


y y

P-4

6.0


(e)

2.14

(b)

(a)

SN

2.02


(a) 1.18

PDPDPD P-4

P-1

-18.0 1.66

2.14

367

0.5, 0.5, 0.5,

= 10, = 10, = 10,

= (f) 5, = 5, = 5,

= 10, = 10, = 10,

2.043

2.044

(1.66,2.14) (1.66,2.14) T /4 200, (1.66,2.14) 200, = T / 4 200, = T / 4

mod( mod(6,, )The )==0. 0. fifth bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ Fig. mod( , ) = 0.(a, c, e) node displacement x (1.66, 2.14)). mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.


SN

SN

PDPD PD P-2

P-4

P-1 P-1 2.4

P-1 S A

A

1.4

0.4 1.514

1.560

1.606

1.652

(a) Periodic Node Displacement, xmod(k,N)

0.8 A

1.524

(c)


SN PDPD

3.10

P-4

1.568

Excitation Frequency, :

(e)

1.652

1.698

PD

PDPD

P-2

-1.8

A

-2.3

1.521

1.524

1.527


(d)

SNPDPD

P-2

P-4

PD PDSN

P-4

6.0

P-2

2.0

-2.0 1.560

1.570

P-2

-1.3

10.0

1.15

1.565

1.606

P-1

PD PD SN

P-4

1.563

SN

1.80

0.50 1.560

1.560

-2.8 1.518

2.45 P-2

0.0

(b)

1.527


4.0

P-4

P-4

2.9

1.521

S

A

-0.8

3.0

0.6 1.518

8.0

PDPD P-2

P-1

P-1

A

PD

SN

3.1

P-2

P-4



SN

PDPDPD

SN P-1

-4.0 1.514

1.698


12.0 SN PD



3.4 SN PD


368

1.563

1.565

1.568


1.570

(f)

Fig. 7 The sixth bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (1.514, 1.698)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.


SN

P-4 P-1

2.4

A 1.4

P-2 0.4 1.22

1.30

1.38

(a)

SN

P-4 P-1

A

1.234

1.238

1.242


3.0

SNPDPD

P-2

P-4

1.476

1.478

1.480


P-2

(e)

1.540

SN

PDPDPD

S P-1

14.0 P-1 13.5

P-2 P-4

1.234

1.238

1.242

1.246


(d)

SN PDPD

SNPDPD

6.5

1.0

P-2

P-4

P-4

P-2

-4.5

-10.0 1.474

1.482

1.460

A 14.5

12.0

1.2

0.6 1.474

1.380

PDPDSN

P-4

SN

2.4

1.8

1.300

13.0 1.230


(c)

S

(b)

1.246


-5.0

15.0

S

2.00 1.230

P-1

2.42

2.14

2.0

P-2

2.28

A


PD PDPD

P-1

PDPDPD P-2 P-4

9.0

-12.0 1.220

1.54


2.56 SN

SN

P-1

1.46



SNSN PD 16.0

PDPDPD P-1

S



3.4SNSN PD

369

1.476

1.478

1.480


1.482

(f)

Fig. 8 The seventh bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (1.22, 1.54)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.



3.5

SN PDPDSN

PD PDPD

SN

P-4 P-1 2.7 A S P-2

A

P-4 P-1

1.9

1.1 1.131

1.155

1.179

1.203

(a) SN


3.5

PD

P-1

P-2 P-4

A

1.130

1.135

1.140

1.145


3.5

SNPD

PD PD

P-4

P-2

1.162

1.164


(e)

SN

PD P-1

6.0

A

0.0

P-1

1.227

PD PD

SN

P-2 P-4

A 1.135

1.140

(d)

1.145

1.150

SN SN

PD PD

PD SN

6.0

P-4

P-4

P-4

0.0

P-2

-6.0

-12.0 1.160

1.166

1.203

-6.0

12.0

P-4

1.179

(b)

1.9

1.1 1.160

1.155

S


PD SN

2.7 P-4

A

P-1


P-1

-6.0

-12.0 1.130

1.150


(c)

A

12.0

2.7

1.9

P-4

0.0

SN

P-2

P-4

PD PD

SN



6.0

-12.0 1.131

1.227


PD PD PD

12.0 SN PD PD SN P-2

P-2


370

1.162

1.164


1.166

(f)

Fig. 9 The eighth bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (1.131, 1.227)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.


PD PDPD

SN

3.4 SNPD SN



3.4 SNPD SN

2.8

2.2 A P-1 1.6 P-2 P-4

1.0 0.924

0.978

1.032

1.086


PD PD PD

SN

2.06

S 1.98 P-1 A 1.94

0.9350

0.9355

3.4

SN PDPD

2.8

P-2

2.2

1.6

1.0 1.110

1.112

1.114

1.116


(e)

PD PD PD

-14.2 P-1 P-1 -14.3

S A

-14.4

0.9345

0.9350

0.9355

0.9360


(d) SNPD PD

PDPD SN

-2.0

-8.0 P-2

-14.0 1.110

1.118

P-4

4.0

P-4 P-2

P-4

1.140

P-2

PDPDSN

1.086

SN



1.032

(b)

-14.5 0.9340

0.9360


(c)

0.978

-14.1

P-2

0.9345

P-2 P-4

P-1

1.90 0.9340

A P-1 1.6

P-4 2.02

2.2



(a)

PD PDPD

2.8

1.0 0.924

1.140


SN

371

P-2

P-4

P-4

1.112

1.114

1.116


1.118

(f)

Fig. 10 The ninth bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (0.924, 1.140)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.


2.8 SN SN PD A

PD PD


P-4 P-4 P-2

P-1

P-1

2.0

1.6

1.2 0.933

0.940

0.947

(a)

SN PDPD PD

SN

PD PD PD

PD PD

4.1

-0.4

P-4

P-1

P-2 P-1

P-2 -4.9

0.940

PD

P-4

0.947

0.954

(b) 12.0 SN SN

SN PDPD PD

PD PD

PD

P-1

P-4

P-2

P-4

2.3 P-2 P-1 1.7 S

A

P-1 A 1.1 0.848

0.866

0.884

0.902

(c)

PD

2.9

S

2.7

A

2.4

2.2

0.805

0.813

0.821


(e)

P-1

4.0

A

A

P-1 0.0

0.866

0.884

(d) PD

12.0

P-1

1.9 0.797

S

0.902

0.920


SN

P-1

P-1

P-2 P-1

PD

P-4

P-4

8.0

P-2

P-2

-4.0 0.848

0.920



PD PD




2.9 SN SN

SN PD

-9.4 0.933

0.954


8.6 SN A

P-2 2.4

SN

PD PD PD



372

PD

SN P-1

P-1 P-2 5.0

P-1 S -2.0 A

-9.0 0.797

0.829

0.805

0.813

0.821


(f)

0.829

Fig. 11 The tenth, eleventh and twelfth bifurcation trees of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (0.933, 0.954), (0.848, 0.920), (0.797, 0.829)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.


PDPDPD

14.0

P-1 2.8 A 2.4

2.0

P-1 S P-2

1.6

P-1

1.2 0.68

P-4

0.71

0.74

0.77

(a)

SN SN

S 0.0

P-1 P-1

-7.0

P-4 P-1

0.71

0.74

P-2

0.77

0.80


(b) SN SN

-7.0

PDPDPD

P-4

2.95

P-1 A P-1

2.85

S P-2

2.75

2.65 0.688

0.690

0.692

(c)

3.05

P-4

P-4 P-2

2.00

1.65 0.7760

0.7772

0.7784

0.7796


(e)

P-1

-10.0 P-4 0.690

0.692

0.694


(d) 11.0

P-2

P-4

S -9.0

PDSNPDSN

2.70

2.35

P-2

P-1

SN PD PD

A -8.0

-11.0 0.688

0.694



A 7.0

PD PD PD

PD PDPD

SN



3.05

SNSNPD

-14.0 0.68

0.80



SN



3.2 SNSN PD

373

PDSNPD SN

6.5 P-4 2.0

P-2

0.7772

0.7784

0.7796


P-2 P-4

P-4

-2.5

-7.0 0.7760

0.7808

SNPD PD

(f)

0.7808

Fig. 12 The thirteenth bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (0.68, 0.80)). (a, c, e) node displacement x mod (k,N) , (b, d, f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.

374


A

PD PD

SN

9.0



3.2 SN SN PD P-1

P-4 2.8 P-1 P-1 2.4 S S 2.0

A P-2

1.6 0.55

0.57

0.59

0.61

A -3.0

S

P-1

P-4 P-2

0.57

0.59

SN

0.61

0.63

SN PD PD PD

8.0

P-4


P-1

2.9

P-1 2.8 S A 2.7 P-2

0.558

0.562

(c)

2.90

SN

2.40

2.15

0.5932


(e)

3.0 P-1 P-2 0.5

0.558

0.562

0.566


(d) 9.0

P-2

0.5930

P-1

P-4

1.90 0.5928

S

5.5

PD SN PD

2.65

A

-2.0 0.554

0.566



P-1

3.0

(b)

PDPDPD P-4

2.6 0.554

P-1

S

SN




3.0

SN

SN

PD PD

P-1

-9.0 0.55

0.63


(a)

SN SN PD

5.5

PD SN PD

P-4 P-2

2.0

-1.5

-5.0 0.5928

0.5934

SN

0.5930

0.5932


(f)

0.5934

Fig. 13 The fifteenth bifurcation tree of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (0.55, 0.63)). (a,c,e) node displacement x mod (k,N) , (b,d,f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.



PD PD P-2

P-2 P-4

SN P-1

P-4

P-1 S

S A P-1 A 2.3

2.1 0.41

0.42

0.43

0.44

SN SNPDPDPD


2.80

P-1

2.40

P-1

A

0.376


(c)

-6.0 S P-4

P-1

0.42

0.43

SN SN PDPDPD

6.0

P-4

0.366

A

P-1

2.20 0.356

-4.0

(b)

SN

P-2 2.60

P-1 S

PD PD PD

P-1 P-2

P-2 P-4

0.45

PD PDPD

SN P-1

P-4 P-2

3.0 P-4

P-1

P-2 0.0 P-1

-3.0

P-1 A

-6.0 0.356

0.386

0.44



SN

P-2

-2.0

-8.0 0.41

0.45


(a)

PD PD P-1

2.7

2.5

SN SN PD PD

0.0


SN SN PD PD

2.9

375

0.366

0.376


0.386

(d)

Fig. 14 The seventeenth and eighteenth bifurcation trees of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (0.41, 0.45), (0.356, 0.385)) without small bifurcation trees. (a,c) node displacement x mod (k,N) , (b,d) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.

the two asymmetric period-1 motion, the seventh bifurcation tree of period-1 to period-4 motion are presented for Ω ∈ (1.22, 1.54) in Fig.8(a)-(f). For such a bifurcation tree, there are five (5) saddle-node bifurcations and ten (10) period-doubling bifurcations. One saddle-node bifurcation at Ω ≈ 1.2338 is for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the period-2 motion. From Ω ≈ 1.2338 to 1.22102, the symmetric period-1 motion has two (2) saddle-node bifurcations for the jumping. Similarly, two (2) period-doubling bifurcations are for the asymmetric period-1 to period-2 motion. Four (4) period-doubling bifurcations are for the period-2 to period-4 motion, and four (4) period-doubling bifurcations are for the period-4 to period-8 motions. From Ω ≈ 1.22102 to Ω ≈ 0.93427, there is a large bifurcation tree including three small bifurcation trees (e.g., eighth, ninth and ten bifurcation trees). The eighth bifurcation tree for Ω ∈ (1.131, 1.227) has five (5) saddle-node bifurcations and ten (10) period-doubling bifurcations, as shown in Fig.9(a)(f). One (1) saddle-node bifurcation at Ω ≈ 1.22102 is for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the asymmetric period-1 motion, and two (2) saddle-node bifurcations are for the jumping phenomena of the period-4 motion. Two (2) period-doubling bifurcations are for the asymmetric period-1 to period-2 motion. Four (4) period-doubling bifurcations are for the period-2 to period-4 motion, and four (4) period-doubling bifurcations are for the period-4 to period-8 motions. Continuing with the two period-1 motions, the

376


ninth bifurcation tree is presented for Ω ∈ (0.924, 1.140) in Fig.10(a)-(f). In such a bifurcation tree, there are seven (7) saddle-node bifurcations and fifteen (15) period-doubling bifurcations. Two (2) saddle-node bifurcations are for symmetric to asymmetric period-1 motion. At Ω ≈ 0.936938, the saddle-node bifurcation is from the symmetric period-1 to unstable asymmetric period-1 motion. Two (2) saddle-node bifurcation are for the jumping phenomena of the asymmetric period-1 motion, and two (2) saddle-node bifurcation are for the jumping phenomena of the asymmetric period-2 motion. From Ω ≈ 0.93327 to Ω ≈ 0.936938, the symmetric period-1 motion has two (2) saddle-node bifurcations of the jumping phenomena of symmetric period-1 motion. From Ω ≈ 0.936938 to Ω ≈ 0.8694, the tenth and eleventh bifurcation trees exist, as shown in Figs.11 (a,b) and (c,d), respectively. The tenth bifurcation tree lies in Ω ∈ (0.9333, 0.9540) with four (4) saddle-node bifurcation and six (6) period-doubling bifurcations, which can be observed in Fig.11(a) and (b). One saddle-node bifurcation is for the symmetric to asymmetric period-1 motion, and three saddle-node bifurcations are for the catastrophes of the asymmetric period-1 motion. Two (2) perioddoubling bifurcations are for period-1 to period-2 motion, and two (2) period-doubling bifurcations are for period-2 to period-4 motion, and two (2) period-doubling bifurcations are for period-4 to period-8 motion. With continuation of the two asymmetric period-1 motions, the eleventh bifurcation tree of the period-1 to period-4 motions is presented for Ω ∈ (0.848, 0.920) in Fig.11(c)-(d). Such a bifurcation tree has three (3) saddle-node bifurcations and six (6) period-doubling bifurcations. One (1) saddlenode bifurcation is for the symmetric to asymmetric period-1 motion. From Ω ≈ 0.8694 to Ω ≈ 0.8265, there are two (2) saddle-node bifurcations for the catastrophes of the symmetric perios-1 motions. The three pairs of six (6) period-doubling bifurcations are for asymmetric period-1 to period-2, period-2 to period-4, and period-4 to period-8 motion. At Ω ≈ 0.8265, the symmetric period-1 motion switches to the asymmetric period-1 motion to form another big bifurcation tree for Ω ∈ (0.68876, 0.82650), which includes the twelfth and thirteenth bifurcation trees. In the twelfth bifurcation tree, only one (1) saddlenode bifurcation for symmetric to asymmetric period-1 motion and two period-doubling bifurcations for period-1 to period motion are observed for Ω ∈ (0.797, 0.829) in Fig.11 (e) and (f). In such a bifurcation, the period-2 motion is stable, which will not be further developed for period-4 motion. With continuation of the asymmetric period-1 motion, the thirteenth bifurcation tree for Ω ∈ (0.68, 0.80) is presented in Fig.12(a)-(f). In such a range, there are seven (7) saddle-node bifurcations and twelve (12) period-doubling bifurcations. One saddle-node bifurcation is for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the symmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the period-2 motion, and two (2) saddle-node bifurcations are for the jumping phenomena of the period-4 motion. Two (2) period-doubling bifurcations are for asymmetric period-1 to period-2 motion, and four (4) perioddoubling bifurcations are for period-2 to period-4 motion, and four (4) period-doubling bifurcations are for period-4 to period-8 motion. With the continued extension of the symmetric period-1 motion, there is an undeveloped bifurcation tree for Ω ∈ (0.65, 0.68) with the two (2) saddle-node bifurcations at Ω ≈ 0.6752, 0.6611 for the symmetric to asymmetric period-1 motion. The asymmetric period-1 motion for this frequency range is stable. Such an undeveloped bifurcation tree is named the fourteenth bifurcation tree, as presented in Figs.15(a) and (b). From Ω ≈ 0.6611 to Ω ≈ 0.6247, only the symmetric period-1 motion exists. For Ω ∈ (0.55, 0.63), the fifteenth bifurcation tree of period-1 to period-4 motion is presented in Fig.13(a)-(f). There are six (6) saddle-node bifurcations and six (6) period-doubling bifurcations. Two (2) saddle-node bifurcations are for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping of period-4 motions. Two (2) saddle-node bifurcations are for the jumping phenomena of the symmetric period-1 motion. Three pairs of two (2) period-doubling bifurcations are for period-1 to period-2, period-2 to period-4, and period-4 to period-8 bifurcations. From Ω ≈ 0.5598 to Ω ≈ 0.491748, the symmetric period-1 motion exists. At Ω ≈ 0.5598 and Ω ≈ 0.561, the two jumping



A

1.7

P-1 S

P-1 S

1.6

1.5 0.65

0.66

0.67


SN


2.9

SN

2.1

0.49


2.60

S 2.50 A

2.40 0.3200

0.3225

0.3250

0.3275


(e)

A

2.0

0.47

0.49

(d)

0.51

SN

SN

3.0 P-1 P-1

S

1.0

A -1.0

-3.0 0.3200

0.3300

SN

S

5.0

P-1

SN


SN

P-1

5.0

SN

0.68

P-1

0.67

SN

-1.0 0.45

0.51


2.70

0.66

8.0

S

(c)

A

(b)

SN

P-1

S

S -4.0

2.5

0.47

P-1

A

1.7 0.45

P-1

0.0



(a)

P-1

4.0

-8.0 0.65

0.68



8.0

P-1

1.8

SN

SN

SN

SN

1.9

377

0.3225

0.3250

0.3275


0.3300

(f)

Fig. 15 The undeveloped fourteenth, sixteenth and nineteenth bifurcation trees of period-1 to period-4 motions varying with excitation frequency (Ω ∈ (0.65, 0.68), (0.45, 0.51), (0.32, 0.33)). (a,c,e) node displacement x mod (k,N) , (b,d,f) node velocity y mod (k,N) . (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). mod (k, N) = 0.

378


points of the symmetric period-1 motion are observed. For Ω ∈ (0.45, 0.51), the sixteenth undeveloped bifurcation tree exists, as presented in Fig.15(c) and (d). There are three (3) saddle-node bifurcations for the symmetric to asymmetric period-1 motion and catastrophe of the asymmetric period-1 motion. From Ω ≈ 0.481044 to Ω ≈ 0.4443, only a symmetric period-1 motion exists. For Ω ∈ (0.41, 0.45), the seventeenth bifurcation tree is presented in Figs.14(a) and (b), and four (4) saddle-node bifurcations and six (6) period-doubling bifurcations are observed. Two (2) saddle-node bifurcations are for the symmetric to asymmetric period-1 motion. Two (2) saddle-node bifurcations are for the jumping phenomena of the symmetric period-1 motion. Three pairs of two (2) period-doubling bifurcations are for period-1 to period-2, period-2 to period-4, and period-4 to period-8 bifurcations. From Ω ≈ 0.4174 to Ω ≈ 0.3808, only a symmetric period-1 motion exists. For Ω ∈ (0.356, 0.398), the eighteenth bifurcation tree is presented in Figs.14(c) and (d), and also three (3) saddle-node bifurcations and six (6) period-doubling bifurcations are obtained. Two (2) saddle-node bifurcations are for the symmetric to asymmetric period-1 motion, and two (2) saddle-node bifurcations are for the jumping phenomenon of the symmetric period-1 motion. Three pairs of two (2) period-doubling bifurcations are for period-1 to period-2, period-2 to period-4, and period-4 to period-8 bifurcations. With the continuation extension of the symmetric period-1 motion, the nineteenth undeveloped bifurcation tree for Ω ∈ (0.32, 0.33) is presented in Figs.15 (e) and (f). The two (2) saddle-node bifurcations are for the symmetric to asymmetric period-1 motions. The asymmetric period-1 motion are stable. Table 1: Bifurcations for symmetric period-1 to period-4 motions, (α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 100, τ = T /4) Frequency range

Ω

Bifurcations

Motion switching

Global view (0-20)

18.94

SN

P-1(S) Jumping

1st

2nd

branch (4.30-5.50)

branch (3.10-4.30)

9.90

SN

P-1 from S to A

8.264

SN

P-1(S) Jumping

5.431

SN

P-1 from S to A

5.431

SN

P-1 from S to A

5.109

PD

P-1 to P-2

5.065

PD

P-2 to P-4

5.048

PD

P-4 to P-8

4.81

SN

P-1 (A) Jumping

4.7508

PD

P-2 to P-4

4.7506

SN

P-4 Jumping

4.728

PD

P-2 to P-4

4.7225

SN

P-4 Jumping

4.7006

SN

P-4 Jumping

4.7005

PD

P-4 to P-8

4.69316

PD

P-4 to P-8

4.69315

SN

P-4 Jumping

4.472

PD

P-2 to P-4

4.453

PD

P-1 to P-2

4.413

SN

P-1 (A) Jumping

4.424

PD

P-1 to P-2

4.079

PD

P-2 to P-4

4.0726

SN

P-4 Jumping

4.0737

SN

P-4 Jumping


3rd

branch (2.35-3.35)

4th branch (2.082-2.422)

5th

branch (1.66-2.14)

4.07

PD

P-4 to P-8

3.641

PD

P-1 to P-2

3.629

SN

P-2 Jumping

3.155173

PD

P-4 to P-8

3.155165

PD

P-2 to P-4

3.15513

SN

P-2 Jumping

3.3135

SN

P-2 Jumping

3.31318

PD

P-2 to P-4

3.31309

PD

P-4 to P-8

3.1

SN

P-1 (S) Jumping

2.9612

NB

P-1 (S) Quasi-periodic

2.896

PD

P-1 to P-2

2.869

SN

P-2 Jumping

2.86

SN

P-1 (S) Jumping

2.778

NB


2.71

NB


2.4207

PD

P-4 to P-8

2.4203

PD

P-2 to P-4

2.418

PD

P-1 to P-2

2.4104

SN

P-1 S to A

2.3837

SN

P-1 (S) Jumping

2.403

SN

P-1 S to A

2.348

PD

P-1 to P-2

2.341

PD

P-2 to P-4

2.339

PD

P-4 to P-8

2.1863

PD

P-4 to P-8

2.1861

PD

P-2 to P-4

2.186

SN

P-2 Jumping

2.1899

PD

P-4 to P-8

2.19

PD

P-2 to P-4

2.1903

SN

P-2 Jumping

2.098

PD

P-4 to P-8

2.0976

PD

P-2 to P-4

2.096

PD

P-1 to P-2

2.089

SN

P-1(A) Jumping

2.178

SN

P-1(A) Jumping

2.102

PD

P-1 to P-2

2.082

PD

P-2 to P-4

2.078

PD

P-4 to P-8

2.0577

PD

P-4 to P-8

2.0573

PD

P-4 to P-8

2.04316

SN

P-4 Jumping

2.04307

PD

P-2 to P-4

2.04101

PD

P-2 to P-4

2.041

SN

P-4 Jumping

1.7011

PD

P-4 to P-8

379

380


6th branch (1.514-1.698)

7th

8th

branch (1.22-1.54)

branch (1.131-1.227)

1.7007

PD

P-2 to P-4

1.6995

PD

P-1 to P-2

1.6948

SN

P-1 S to A

1.6834

SN

P-1 (S) Jumping

1.819

SN

P-1(S) Jumping

1.687

SN

P-1 S to A

1.649

PD

P-1 to P-2

1.642

PD

P-2 to P-4

1.64

PD

P-4 to P-8

1.5615

PD

P-4 to P-8

1.5614

PD

P-2 to P-4

1.5612

SN

P-2 Jumping

1.5687

SN

P-2 Jumping

1.5683

PD

P-2 to P-4

1.5681

PD

P-4 to P-8

1.52352

PD

P-4 to P-8

1.52317

PD

P-2 to P-4

1.52194

PD

P-1 to P-2

1.5188

P1

P-1 (A) Jumping

1.5567

SN

P-1 (A) Jumping

1.5223

PD

P-1 to P-2

1.5097

PD

P-2 to P-4

1.5072

PD

P-4 to P-8

1.48078

SN

P-2 Jumping

1.48066

PD

P-2 to P-4

1.48054

PD

P-4 to P-8

1.47581

PD

P-4 to P-8

1.47568

PD

P-2 to P-4

1.47544

SN

P-2 Jumping

1.3146

SN

P-1 (S) Jumping

1.2386

PD

P-4 to P-8

1.2384

PD

P-2 to P-4

1.2375

PD

P-1 to P-2

1.2338

SN

P-1 S to A

1.2303

SN

P-1(S) Jumping

1.22102

SN

P-1 S to A

1.20599

PD

P1-P2

1.20215

PD

P2-P4

1.2009

PD

P4-P8

1.16389

SN

P-4 Jumping

1.16377

PD

P-2 to P-4

1.16238

PD

P-2 to P-4

1.16212

PD

P-4 to P-8

1.16096

PD

P-4 to P-8

1.16090

SN

P-4 Jumping

1.14453

SN

P-1 (A) Jumping


9th

branch (0.924-1.140)

10th

11th

12th

branch (0.9333-0.954)

branch (0.848-0.920)

branch (0.797-0.829)

1.14219

PD

P4-P8

1.14025

PD

P2-P4

1.13807

PD

P1-P2

1.13362

SN

P-1 (A) Jumping

1.1314

PD

P-1 to P-2

1.1274

PD

P-2 to P-4

1.1270

PD

P-4 to P-9

1.11595

SN

P-2 Jumping

1.11589

PD

P2-P4

1.11576

PD

P4-P8

1.11168

PD

P4-P8

1.11153

PD

P2-P4

1.11153

SN

P-2 Jumping

1.042

SN

P-1 Jumping

0.953638

SN

P-1 S to A (unstable)

0.9504

PD

P-1 to P-2

0.94875

PD

P-2 to P-4

0.94838

PD

P-4 to P-8

0.94081

PD

P-4 to P-8

0.94057

PD

P-2 to P-4

0.9368

SN

P-1 (A) Jumping

0.9352

PD

P-4 to P-8

0.93518

PD

P-2 to P-4

0.93497

PD

P-1 to P-2

0.93427

SN

P-1 S to A

0.93139

SN

P-1 (S) Jumping

0.9536

SN

P-1(A) Jumping

0.9504

PD

P-1 to P-2

0.94875

PD

P-2 to P-4

0.94838

PD

P-4 to P-8

0.94081

PD

P-4 to P-8

0.94057

PD

P-2 to P-4

0.936938

SN

P-1 S To A (unstable)

0.93658

PD

P-1 to P-2

0.935376

SN

P-1 (A) Jumping

0.9336

SN

P-1 (A) Jumping

0.915

PD

P-1 to P-2

0.903

PD

P-2 to P-4

0.9013

PD

P-4 to P-8

0.875

PD

P-4 to P-8

0.8741

PD

P-2 to P-4

0.8718

PD

P-1 to P-2

0.8649

SN

P-1 S to A

0.85372

SN

P-1 (S) Jumping

0.8525

SN

P-1 (S) Jumping

0.8265

SN

P-1 S to A

381

382


13th branch (0.68-0.80)

14th branch (0.65-0.68) 15th branch (0.55-0.63)

16th branch (0.45-0.51))

17th

branch (0.41-0.45)

0.8147

PD

P-1 to P-2

0.8009

PD

P-1 to P-2

0.7859

PD

P-1 to P-2

0.7832

PD

P-2 to P-4

0.7823

PD

P-4 to P-8

0.78025

SN

P-4 Jumping

0.78022

PD

P-4 to P-8

0.7798

PD

P-4 to P-8

0.779754

SN

P-4 Jumping

0.77991

SN

P-2 Jumping

0.77987

PD

P-2 to P-4

0.779827

PD

P-4 to P-8

0.7772

PD

P-4 to P-8

0.77703

PD

P-2 to P-4

0.77686

SN

P-2 Jumping

0.7299

SN

P-1 (S) Jumping

0.69073

PD

P-4 to P-8

0.6906

PD

P-2 to P-4

0.6903

PD

P-1 to P-2

0.68876

SN

P-1 S to A

0.68857

SN

P-1 (S) Jumping

0.6752

SN

P-1 S to A

0.6611

SN

P-1 S to A

0.6247

SN

P-1 S to A

0.5979

PD

P-1 to P-2

0.5932

PD

P-2 to P-4

0.59296

SN

P-4 Jumping

0.59316

SN

P-4 Jumping

0.59315

PD

P-4 to P-8

0.562

PD

P-4 to P-8

0.5618

PD

P-2 to P-4

0.5614

PD

P-1 to P-2

0.561

SN

P-1 (S) Jumping

0.5598

SN

P-1 S to A

0.5546

SN

P-1 (S) Jumping

0.491748

SN

P-1 S to A

0.481044

SN

P-1 S to A

0.470351

SN

P-1 (A) Jumping

0.4443

SN

P-1 S to A

0.4343

PD

P-1 to P-2

0.4325

PD

P-2 to P-4

0.4323

PD

P-4 to P-8

0.4222

SN

P-1 (S) Jumping

0.4198

PD

P-4 to P-8

0.4195

PD

P-2 to P-4

0.4188

PD

P-1 to P-2


18th branch (0.356-0.398)

19th

branch (0.32-0.33)

0.4174

SN

P-1 S to A

0.4148

SN

P-1 (S) Jumping

0.3808

SN

P-1 S to A

0.3722

PD

P-1 to P-2

0.3705

PD

P-2 to P-4

0.3702

PD

P-4 to P-8

0.3643

PD

P-4 to P-8

0.3641

PD

P-2 to P-4

0.3634

PD

P-1 to P-2

0.3616

SN

P-1 S to A

0.3596

SN

P-1 (s) Jumping

0.3588

SN

P-1 (S) Jumping

0.3275

SN

P-1 S to A

0.3220

SN

P-1 S to A

383

Note: S-Symmetric periodic motion. A-Asymmetric period-1 motions. PD-period-doubling bifurcation. SN-saddle-node bifurcation.

6.2

Frequency-amplitude characteristics

The discrete nodes of symmetric and asymmetric period-m motions in the time-delayed, twin-well Duffing oscillator were computed by the corresponding mapping structures. To understand the nonlinear dynamics of periodic motions in such a time-delayed, twin-well, Duffing oscillator, the nonlinear frequency-amplitude characteristics of period-m motions should be studied. The discrete Fourier series will be employed to obtain the harmonic amplitudes and phase angles of period-m motions. To avoid the abundant illustrations, the frequency-harmonic amplitude curves for selected order harmon(m) ics are presented in Fig.16. The selected harmonic amplitudes are constant term a0 (m = 1, 2, 3, 4) and harmonic amplitudes Ak/m (m = 4, k = 1, 2, 3, 4, 8, 12, 80, 81, · · · , 84, 120, 124) are presented, and the saddle-node and period-doubling bifurcation points of period-m motions (m = 1, 2, 3, 4) are listed in Table 1. A global view of constant terms versus excitation frequency in the finite Fourier series is presented (1) in Fig.16(i). For the symmetric period-1 motion, a0 = a0 = 0. For the asymmetric period-m motions, (m) (m) a0 6= 0. Thus, a0 6= 0 for a pair of asymmetric period-m motions are presented in such a range. (m) a0 6= 0 represents that the centers of the asymmetric period-m motions are off the origin points of (m) displacements. To clearly illustrate constant distributions for the bifurcation trees, constants a0 varying with frequencies of Ω ∈ (1.2, 6.0), (0.535, 0.120), (0.310, 0.51) are presented in Fig.16(ii)-(iv), (m) respectively. The magnitudes for nineteen bifurcation trees dramatically drop from |a0 | = 0.7 to (m) |a0 | = 0.02. For Ω ∈ (9.90, 20.0), there is a bifurcation tree relative to the twin-wells of the Duffing √ (m) oscillator. a0 → ± 1.5 as Ω → ∞. From the symmetric to asymmetric period-1motion is observed, and the asymmetric period-1 to period-2 motion, and the period-2 to period-4 motion are observed. The fifteen (15) complete developed bifurcation trees of symmetric period-1 to period-4 motions are observed. The three (3) undeveloped bifurcation trees only possess the asymmetric period-1 motion on the bifurcation trees. One (1) undeveloped bifurcation tree is from the symmetric period-1 to period-2 motion. Harmonic amplitude A1/4 versus excitation frequency is presented in Fig.16(v) for period-4 motions. The period-1 and period-2 motions possesses A1/4 = 0. Both stable and unstable period-4 motions


SN

SN

Harmonic Amplitude, a0

P-1 0.7

0.0 S

-0.7 A

-1.4 0.0

10.0

15.0

(i)

P-4

P-2

P-2

S

0.0 P-1

A -0.6 P-1

P-4 P-2

P-2 P-4

PD SN PDSN

P-1 P-4 P-2

0.10

-0.15 A

1.005

(iii)

6.0

0.05

P-4

P-1

SN PD SN

SN SN SN

P-2 P-4

P-1

P-1

S

P-1

-0.05

A

0.360

0.410

0.460

0.510


(iv)

0.40

SN PD SN

0.00

-0.10 0.310

P-2

1.240


SN SN

P-1

S

0.770

4.5

(ii)

0.00

-0.30 0.535

3.0


PDSNSN PD SN

0.15

1.5

(m)

SN SNSN

(m)

0.30

P-1 P-4

0.6

20.0


SN

SNSNPDSN PD SN PD PD SNPD SN PD SN

-1.2

5.0



1.2 SN

(m)

SN SN

(m)

1.4


384

0.8

P-4 PD

0.30

0.20

PD PD SN

P-4

0.10 SN

0.00 0.20

1.45

PD

(v)

PD 0.6 P-2 PD 0.4 P-4

PD

0.2

PD PD

PD PD PD PD SN PD SN 2.70 SN 3.95 SN SN 5.20


Harmonic Amplitude, A1/2


P-4

PD 0.0 0.20

1.45

2.70 SN

SN 3.95 SN SN


(vi)

SN 5.20

Fig. 16 The frequency-amplitude characteristics of bifurcation tree of period-1 to period-4 motions varying (m)

with excitation frequency (Ω ∈ (0, 30]) for the time-delayed Duffing oscillator. (i)-(iv) a0 (m = 4), (v)-(xvii) Ak/m (k = 1, 2, 3, 4; 4, 8, 12, 80, 81, · · · , 84, 120, 124)(δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4).


385

8.0

0.30


SN

0.20

Harmonic Amplitude, A1

PD PD P-4

0.10

PD

0.00 0.20

6.0

S 4.0 SN


(vii)

PD SN

2.0 A

SN PD PD PD PD SN PD SN 2.70 SN 3.95 SN SN 5.20

1.45

P-1

A

SN 0.0 0.0

10.0

15.0

20.0


5.0

(viii)

3.6

2.4

SN SN

SN SN

SN

2.4

PD PD

SN

A

1.2 SN

PD SN

SN

PD

SN P-1



SN

S

1.6

P-2 PD P-4 P-1

PD

0.8

A S

NB 1.5

3.0

4.5

(ix)

0.0 0.0

6.0




1.0e-1

SN

S

PD SN A

P-1 SN

1.0e-3

2.0

SN

4.0

6.0


1.0e+1

SN

P-1

A

SN 0.0 0.0

PD

PD

A

(x)

1.0e+1

1.0e-2 PD 1.0e-5

PD SN P-4 P-2 PD P-1

1.0e-5

1.0e-9 0.0

5.0

10.0

15.0


Fig.16 Continued.

(xi)

20.0

0.0 SN

SN SN 2.0 SN

4.0


(xii)

SN

6.0

386


1.0e+0



1.0e+0

1.0e-3

P-4

PD 1.0e-6

PD PD

PD PD

1.0e-3 PD P-2 PD P-4

1.0e-6

PD PD PD

PD 1.0e-9 0.20

1.45

SN 2.70


(xiii)

1.0e-9 0.20

3.95 SN SN 5.20

SN

(xiv)

5.20

1.0e+1


1.0e-3

P-4

PD 1.0e-6

PD PD

PD PD

1.0e-4

SN

1.0e-9 A

S P-1

PD 1.0e-9 0.20

1.45

SN 2.70

(xv)

1.0e-14 0.0

3.95 SN SN 5.20

SN


15.0

20.0


1.0e+1

1.0e-4 PD

1.0e-8

PD SN P-4

P-2

PD

P-1

1.0e-4

SN

1.0e-9 A

P-1

1.0e-12 0.0 SN

10.0

(xvi)

5.0

SN


1.0e+0


SN 3.95 SN SN


1.0e+0


SN 2.70 SN

1.45

SN SN 2.0 SN

4.0


(xvii)

SN

S

1.0e-14 0.0

6.0

2.0

4.0


SN 6.0

(xviii)

Fig.16 Continued. exist. Because of period-doubling bifurcations, the period-8 motion will be developed. The maximum quantity level of A1/4 is about A1/4 ∼ 0.3. In Fig.16(vi), presented is the harmonic amplitude A1/2 varying with excitation frequency for period-2 and period-4 motions. The symmetric and asymmetric period-1 motions possess A1/2 = 0. The maximum quantity level of harmonic amplitude A1/2 is about A1/2 ∼ 0.8. Harmonic amplitude A3/4 versus excitation frequency is presented in Fig.16(vi) for period-4 motions, which is similar to the harmonic amplitude A1/4 . The maximum quantity level of harmonic amplitude A3/4 is about A3/4 ∼ 0.25. Harmonic amplitude A1 varying with excitation frequency is


387

presented in Fig. 16(viii) for period-1 to period-4 motions. The main skeleton of frequency-amplitude curve is similar to the symmetric period-1 motion because period-2 and period-4 motions are close to the asymmetric period-1 motions after period-doubling bifurcations. The maximum quantity level of the primary harmonic amplitude is about A1 ∼ 7.0. For Ω > 6, the harmonic frequency-amplitude curves of symmetric period-1 motion is very simple. The asymmetric period-1 motion relative to the potential well are close to the symmetric period-1 motion. For Ω < 6, the frequency-amplitude curves are very crowed. For clear illustration, in such a frequency range, a zoomed view is given in Fig.16(ix), and the proper labels are placed for bifurcations. To avoid abundant illustrations, a few main primary harmonic amplitudes are presented herein. Thus, harmonic amplitude A2 varying with excitation frequency is presented in Fig.16(x) for period1 to period-4 motions. The symmetric period-1 motions experience A2 = 0. Asymmetric period-1 motions and the corresponding period-2 and period-4 motions on the bifurcation trees are presented in Fig.16(x). The maximum quantity level of harmonic amplitude A2 is about A2 ∼ 2.4. Harmonic amplitude A3 varying with excitation frequency is presented in Fig. 16(xi) for period-1 to period-4 motions. The maximum quantity level of the third harmonic amplitude is about A3 ∼ 4. For Ω > 8, the harmonic frequency-amplitude curves of symmetric period-1 motion is very simple and quantity level of harmonic amplitudes is about A3 ∼ 0.3. For Ω < 8, the frequency-amplitude curves are very crowed and the corresponding quantity level is about A3 ∼ 4. To avoid abundant illustrations, the last set of harmonic frequency-amplitudes is presented. Harmonic amplitude A20 varying with excitation frequency is presented in Fig.16(xii) for Ω < 6.0. The quantity level is from A20 ∼ 100 to A20 ∼ 10−8 . Harmonic amplitude A81/4 varying with excitation frequency is presented in Fig.16(xiii) for period-4 motions, which is similar to the harmonic amplitude A1/4 . The quantity level is from A81/4 ∼ 10−2 to A81/4 ∼ 10−8 . However, the quantity levels for the four bifurcation trees are different. Harmonic amplitude A41/2 varying with excitation frequency is presented in Fig.16(xiv) for period-2 and period-4 motions, which is similar to the harmonic amplitude A1/2 . The quantity level is from A41/2 ∼ 10−1 to A41/2 ∼ 10−7 . Harmonic amplitude A83/4 varying with excitation frequency is presented in Fig.16(xv) for period-4 motions, which is similar to the harmonic amplitude A81/4 . The quantity level is from A83/4 ∼ 10−2 to A83/4 ∼ 10−8 . Harmonic amplitude A21 versus excitation frequency is presented in Fig.16(xvi) for period-1 to period-4 motions, similar to the harmonic amplitude A1 . The quantity level is from A21 ∼ 100 to A21 ∼ 10−12 . For Ω being close to zero, the quantity levels of harmonic amplitudes for A20 to A21 are close to 10−1 . Thus, harmonic amplitude A30 varying with excitation frequency is presented in Fig.16(xvii). The quantity level is from A30 ∼ 10−10 to A30 ∼ 10−13 . Harmonic amplitude A31 versus excitation frequency is presented in Fig.16(xviii). The quantity level is from A31 ∼ 100 to A21 ∼ 10−14 . For low frequency, the more accurate numerical computation is needed to get periodic motions. However, the harmonic phases are different. (m)L (m)R L = mod (φ R + ((k + 2r)/2l + 1)π , 2π ) (k = 1, 2, · · · ; Thus, a0 = −a0 (m = 2l , l = 0, 1, 2, · · · ), and φk/2 l k/2l r = 0, 1, · · · , 2l − 1) for t0 = rT . 7 Numerical simulations The analytical prediction of period-1 to period-4 motions in the time-delayed, twin-well, Duffing oscillator was predicted analytically, and the corresponding nonlinear frequency-amplitude characteristics were discussed. Initial conditions from the analytical prediction will be used for numerical simulations of period-1 to period-4 motions in the bifurcation trees, and the corresponding harmonic amplitudes of periodic motions will be presented to show harmonic terms effects on periodic motions. The system parameters in Eq.(41) are used. Numerical and analytical results are presented by solid curves and symbols, respectively. The initial time-delay are presented through blue circular symbols. The

388


18.0

5.0 T

9.0

Velocity, y

Displacement, x

2.5 D.I.F 0.0 D.I.S

-18.0 -4.0

-5.0 0.0

1.6

3.2

4.8

-2.0

Time, t

(a)

1e+2

Harmonic Phase, Mk

1e-3

1e-7

1e-11

0.0

2.0

A63

(b)

S

S

M M M M

S

M S

M 1e-15 0.0

16.0

32.0

48.0

Harmonic Order, k

(c)

4.0

Displacement, x

A1

Harmonic Amplitude, Ak

D.I.F -9.0

-2.5

D.I.S 0.0

0.0

64.0

M 16.0

32.0

48.0

Harmonic Order, k

(d)

64.0

Fig. 17 Stable symmetric period-1 motion (Ω = 2.41). (a) displacement, (b) trajectory (c) harmonic amplitude, (d) harmonic phase with initial condition x0 ≈ 1.616225, x˙0 ≈ −6.142844 (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4).

delay-initial-starting and delay-initial finishing points are “D.I.S.” and “D.I.F.”, respectively. Consider a symmetric period-1 motion of Ω = 2.41, and the initial condition is computed from the analytical prediction (i.e., x0 ≈ 1.616225 and x˙0 ≈ 14.818644). Displacement and trajectory for such a symmetric period-1 motion are presented in Fig.17(a) and (b), respectively. The initial time-delay is depicted by green symbols. The numerical and analytical solutions of the stable period-1 motion are presented by solid curves and circular symbols, respectively. The corresponding harmonic amplitudes and phases are presented in Fig.17(c) and (d), respectively. The main harmonic amplitudes are A1 ≈ 2.5753, A3 ≈ 0.6456, A5 ≈ 0.5763, A7 ≈ 0.0444, A9 ≈ 0.0525, and A11 ≈ 0.0138. Other harmonic amplitudes lie in A2l−1 ∈ (10−15 , 10−3 ) (l = 7, 8, · · · , 32) and A63 ≈ 8.9e-15. Thus, one can use six harmonic terms to approximate such a simple period-1 motion on the upper branch. The harmonic phase distribution is shown in Fig.17(d). Consider a pair of asymmetric period-1 motions at Ω = 2.39, as shown in Fig.18. The initial conditions are obtained from the analytical prediction. (x0 ≈ 1.879788 and x˙0 ≈ −9.020353) are for the left asymmetric period-1 motion, and (x0 ≈ 1.38369 and x˙0 ≈ −3.194548) are for the right asymmetric period-1 motion. In Fig.18(a) and (b), two trajectories for the left and right asymmetric period-1 motions are presented. The trajectories are very asymmetric. The harmonic amplitudes and phases are presented in Fig.18(c) and (d), respectively. The center of the trajectory is far away from the origin compared to the previous asymmetric period-1 motion. That is, aR0 = −aL0 = A0 ≈ 0.01843. The

18.0

18.0

9.0

9.0

Velocity, y

Velocity, y


D.I.S 0.0

389

D.I.S

0.0 D.I.F -9.0

-9.0 D.I.F -18.0 -4.0

-2.0

0.0

2.0

-18.0 -4.0

4.0

Harmonic Amplitude, Ak

A1 A0

1e-9

2.0

(b) S

A2

1e-4

4.0

S

M M M M M

S

S

A60 1e-14 0.0

15.0

30.0

45.0

Harmonic Order, k

1e+1

0.0

Displacement, x

Harmonic Phase, Mk

(a)

-2.0

Displacement, x

(c)

0.0

60.0

15.0

30.0

45.0

Harmonic Order, k

(d)

60.0

Fig. 18 The paired two stable asymmetric period-1 motions (Ω = 2.39). (a) Trajectory (left) (x0 ≈ 1.879788,˙x0 ≈ −9.020353), (b) trajectory (right) (x0 ≈ 1.38369, x˙0 ≈ −3.194548), (c) harmonic amplitudes, (d) harmonic phases. (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4).

main harmonic amplitudes for the two asymmetric period-1 motions are A1 ≈ 2.5473, A2 ≈ 0.0936, A3 ≈ 0.62241, A4 ≈ 0.3324, A5 ≈ 0.5439, A6 ≈ 0.1304, A7 ≈ 0.0441, A8 ≈ 0.0360, A9 ≈ 0.0375, A10 ≈ 0.0298. Other harmonic amplitudes lie in Ak ∈ (10−15 , 10−2 ) (k = 11, 12, · · · , 63) and A63 ≈ 1.66e-14. The two asymmetric period-1 motions need about 63 harmonic terms in the finite Fourier series for an approximate analytical expression. In addition to A1 ≈ 2.5473, the harmonic terms of A3 , A4 , A5 play important roles on such asymmetric period-1 motions. Because of A2l 6= 0 (l = 1, 2, · · · ), such harmonic terms make the two period-1 motions be asymmetric. In addition, harmonic phase distribution varying with harmonic orders is clearly presented. The gray and blue circular symbols are for the harmonic phases of the left and right asymmetric period-1 motions, respectively. The harmonic phase relations (L) (R) between two asymmetric period-1 motions are φk = mod (φk + (k + 1)π , 2π ) for k = 0, 1, 2, · · · . From the fourth branch of bifurcation tree from period-1 motion to chaos, the paired period-2 motions are presented in Fig.19 for Ω = 2.347. The initial conditions for the left and right asymmetric period-2 motions are (x0 ≈ 2.193401, x˙0 ≈ −11.703740) and (x0 ≈ 3.227069, x˙0 ≈ 1.473406), respectively. Two trajectories of the right and left period-2 motions are presented in Fig.19 (a) and (b), respectively. The asymmetry of the two period-2 motions with the initial time-delay is clearly observed. For a better understanding of complexity of period-2 motions, the harmonic amplitude spectrum is also presented for the paired period-2 motions. Due to the asymmetry of the two period-2 motions, the harmonic amplitudes for both of period-2 motions are same, as shown in Fig.19(c), and the corresponding har-

390


18.0

18.0

9.0

9.0

0.0

Velocity, y

Velocity, y

D.I.S

D.I.S

-9.0

0.0 D.I.F -9.0

D.I.F -18.0 -4.0

-2.0

0.0

2.0

1e-4

1e-9 A64 1e-14 0.0

16.0

32.0

48.0

Harmonic Order, k/2

2.0

4.0

(b)

S

A1

Harmonic Phase, Mk/2

Harmonic Amplitude, Ak/2

A1/2 A0

0.0

Displacement, x

1e+1

-2.0

Displacement, x

(a)

-18.0 -4.0

4.0

(c)

S

S

S

0.0

64.0

16.0

32.0

48.0

Harmonic Order, k/2

(d)

64.0

Fig. 19 The paired two stable period-2 motions (Ω = 2.347). (a) Trajectory (left) (x0 ≈ 2.193401, x˙0 ≈ −11.703740), (b) trajectory (right) (x0 ≈ 3.227069, x˙0 ≈ 1.473406), (c) harmonic amplitude, and (d) harmonic phase, (left) (δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4).

monic phases are presented in Fig.19(d). Compared to the paired asymmetric period-1 motions, the (2)L (2)R harmonic phase distributions become complicated. a0 = −a0 = A0 ≈ 0.0438. The main harmonic amplitudes for the two asymmetric period-2 motions are A1/2 ≈ 0.0132, A1 ≈ 2.4572, A3/2 ≈ 5.0610e-3, A2 ≈ 0.1956, A5/2 ≈ 3.3205e-3, A3 ≈ 0.5689, A7/2 ≈ 0.0144, A4 ≈ 0.6935, A9/2 ≈ 0.0195, A5 ≈ 0.4155, A11/2 ≈ 2.3323e-3, A6 ≈ 0.2598, A13/2 ≈ 3.3454e-3, A7 ≈ 0.0356, A15/2 ≈ 4.8124e-4, A8 ≈ 0.0644, A17/2 ≈ 3.7180e-3, A9 ≈ 0.0123, A19/2 ≈ 1.8588e-3 and A10 ≈ 0.0471. Other harmonic amplitudes lie in Ak/2 ∈ (10−15 , 10−2 ) (k = 21, 22, · · · , 128) and A64 ≈ 6.47e-14. The two asymmetric period-2 motions need about 128 harmonic terms in the finite Fourier series for an approximate analytical expression. The harmonic phase relations L = mod (φ R + (k/2 + 1)π , 2π ) for k = 0, 1, 2, · · · . between the two asymmetric period-2 motions are φk/2 k/2 Because the contributions of the harmonic terms A(2l−1)/2 (l = 1, 2, · · · ) are very small, the trajectories of the period-2 motions will be close to the corresponding unstable asymmetric period-1 motions. From the fourth branch of bifurcation tree from period-1 motion to chaos, period-4 motions are presented in Fig.20 for Ω = 2.34. The initial conditions for the left and right asymmetric period4 motions are (x0 ≈ 2.120181, x˙0 ≈ −11.052207) and (x0 ≈ 1.246387, x˙0 ≈ −0.677714), respectively. Two trajectories with the initial time-delay for the left and right paired of period-4 motions are presented in Fig.20(a) and (b), respectively. The harmonic amplitude spectrum for both of period-2 motions is presented in Fig.20(c). The asymmetry of the two paired period-4 motions causes the same harmonic amplitudes. However, the harmonic phases for the paired asymmetric period-4 mo-


18.0

18.0

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Fig. 20 The two paired stable period-4 motion (Ω = 2.34). (a) Trajectory (left) (x0 ≈ 2.120181, x˙0 ≈ −11.052207), (b) trajectory (right) (x0 ≈ 1.246387, x˙0 ≈ −0.677714) (c) harmonic amplitude, (d) harmonic phase δ = 0.5, α1 = 10, α2 = 5, β = 10, Q0 = 100, τ = T /4). (4)R

(4)L

tions are different, as presented in Fig.20(d). That is, a0 = −a0 = A0 ≈ 0.0455. The main harmonic amplitudes for the two asymmetric period-4 motions are A1/4 ≈ 2.1852e-3, A1/2 ≈ 0.0664, A3/4 ≈ 2.3274e-3, A1 ≈ 2.4494, A5/4 ≈ 3.1616e-4, A3/2 ≈ 0.0252, A7/4 ≈ 1.9902e-3, A2 ≈ 0.1981, A9/4 ≈ 6.1647e-4, A5/2 ≈ 0.0166, A11/4 ≈ 6.7831e-4, A3 ≈ 0.5612, A13/4 ≈ 8.9516e-4, A7/2 ≈ 0.0715, A15/4 ≈ 5.6228e-3, A4 ≈ 0.7016, A17/4 ≈ 4.7200e-3, A9/2 ≈ 0.0978, A19/4 ≈ 3.8642e-3, A5 ≈ 0.4093, A21/4 ≈ 1.4205e-3, A11/2 ≈ 0.0119, A23/4 ≈ 1.6677e-3, A6 ≈ 0.2621, A25/4 ≈ 1.2049e-3, A13/2 ≈ 0.0168, A27/4 ≈ 6.5633e-4, A7 ≈ 0.0372, A29/4 ≈ 2.4495e-4, A15/2 ≈ 2.2386e-3, A31/4 ≈ 3.3577e-4, A8 ≈ 0.0636, A33/4 ≈ 5.1202e-4, A17/2 ≈ 0.0185, A35/4 ≈ 1.0822e-3, A9 ≈ 0.0149, A37/4 ≈ 5.6691e-4, A19/2 ≈ 9.5896e-3, A39/4 ≈ 1.0627e-4 and A10 ≈ 0.0463. Other harmonic amplitudes lie in Ak/4 ∈ (10−12 , 10−2 ) (k = 41, 42, · · · , 240) and A60 ≈ 1.72e-14. The two asymmetric period-4 motions need about 240 harmonic terms in the finite Fourier series for an approximate analytical expression. The harmonic phase relations between the two asymmetric period-4 L = mod (φ R + (k/4 + 1)π , 2π ) for k = 0, 1, 2, · · · . motions are φk/4 k/4 8 Conclusions In this paper, the analytical prediction of periodic motions in a periodically forced, time-delayed, twinwell Duffing oscillator were presented. From mapping structures of periodic motion, bifurcation trees of

392


period-1 motions to chaos were predicted analytically, and the corresponding stability and bifurcation were studied through eigenvalue analysis. The nonlinear harmonic frequency-amplitude characteristics of bifurcation trees of period-1 motion to chaos were discussed. From the analytical predictions of periodic motions, numerical simulations of period-1 to period-4 motions were illustrated. As a slowly varying excitation becomes very slow (i.e., Ω → 0), the excitation amplitude will approach infinity (i.e., Q0 → ∞) for the infinite bifurcation trees of period-1 motion to chaos. Thus infinite bifurcation trees of period-1 motion to chaos can be obtained. The possible bifurcation trees of asymmetric period-1 motion to chaos in the low branch can be observed, which is relative to the twin potential wells. References [1] 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. [2] 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, Chao, Solitons and Fractals, 89, 405-434. [3] Luo, A.C.J. and Guo, Y. (2015), A semi-analytical prediction of periodic motions in Duffing oscillator through mapping structures, Discontinuity, Nonlinearity, and Complexity, 4(2), 121-150. [4] Guo, Y. and Luo, A.C.J. (2016), Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete mappings, International Journal of Dynamics and Control, 5, 223-238. [5] Lagrange, J.L. (1788), Mecanique Analytique (2 vol.) (edition Paris: Albert Balnchard;1965). [6] Poincare, H. (1899), Methodes Nouvelles de la Mecanique Celeste Vol.3. Paris: Gauthier-Villars. [7] van der Pol, B. (1920), A theory of the amplitude of free and forced triode vibrations, Radio Review 1, 1, 701-710, 754-762. [8] Fatou, P. (1928), Sur le mouvement d’un systeme soumis ‘a des forces a courte periode. Bull. Soc. Math., 56, 98-139. [9] 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 (in French). [10] Hayashi, C. (1964), Nonlinear oscillations in Physical Systems, New York: McGraw-Hill Book Company. [11] Nayfeh, A.H. (1973), Perturbation Methods, New York: John Wiley. [12] Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillation, New York: John Wiley. [13] Hu, H.Y. and Wang, Z.H. (2002), Dynamics of Controlled Mechanical Systems with Delayed Feedback, Berlin: Springer. [14] Luo, A.C.J. (2012), Continuous Dynamical Systems. Beijing/Glen Carbon: HEP/L&H Scientific. [15] 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. [16] 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). [17] 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. [18] 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. [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, A.C.J. (2013), Analytical solutions of periodic motions in dynamical systems with/without time-delay, International Journal of Dynamics and Control, 1, 330-359. [21] Luo, A.C.J. and Jin, H.X. (2014), Bifurcation trees of period-m motion to chaos in a time-delayed, quadratic nonlinear oscillator under a periodic excitation, Discontinuity, Nonlinearity, and Complexity, 3, 87-107. [22] Luo, A.C.J. and Jin, H.X. (2015), Complex period-1 motions of a periodically forced Duffing oscillator with a time-delay feedback. International Journal of Dynamics and Control, 3, 325-340. [23] Luo, A.C.J. and Jin, H.X. (2014), Period-m motions to chaos in a periodically forced Duffing oscillator with a time-delay feedback, International Journal of Bifurcation and Chaos, 24(10), Article no: 1450126 (20 pages). [24] Luo, A.C.J. (2015), Periodic flows in nonlinear dynamical systems based on discrete implicit maps, International Journal of Bifurcation and Chaos, 25, Article No:1550044 (62 pages).

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Journal of Vibration Testing and System Dynamics Volume 1, Issue 4

December 2017

Contents A Time-Frequency PID Controller Design for Improved Anti-Interference Performance of a Solenoid Valve Applicable to Hydraulic Cylinder Actuation Xiu-Heng Wu, Zheng-He Song, Yue-Feng Du, En-Rong Mao, C. Steve Suh…......

281－294

Impact of Tool Geometry and Tool Feed on Machining Stability Achala V. Dassanayake, C. Steve Suh……..………………………..…….………

295－317

The Ackerman Steered Car Non-Holonomic Lagrangian Mechanics System: Mathematics Problem Treatment of the Geometrical Theory Soufiane Haddout, Zhiyi Chen, Mohamed Ait Guennoun…………........................

319－331

Flow-induced Vibration of Flexible Bottom Wall in a Lid-driven Cavity Xu Sun, Wenxin Li, Zehua Ye…………………..…………….……….…..….……

333－341

Fast Unbalancing of Rotating Machines by Combination of Computer Vision and Vibration Data Analysis A. Najedpak, C. Yang…………………….....………...………….....……........…

343－352

Towards Infinite Bifurcation Trees of Period-1 Motions to Chaos in a Time-delayed, Twin-well Duffing Oscillator Siyuan Xing, Albert C.J. Luo………...…………...………………………........…

353－392

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Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics

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