Journal of Vibration Testing and System Dynamics

Volume 1 Issue 1 March 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 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, Heilongjiang, P. R. 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 Physics Universität Bielefeld Bielefeld Nordrhein-Westalen, Germany 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 1, March 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(1) (2017) 1-14

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

The Use of the Fitting Time Histories Method to Detect the Nonlinear Behaviour of Laminated Glass S. Lenci†, L. Consolini, F. Clementi Department of Civil and Buildings Engineering, and Architecture, Polytechnic University of Marche, via Brecce Bianche, I-60131 Ancona, Italy Submission Info Communicated by C.S. Suh Received 20 December 2016 Accepted 20 January 2017 Available online 1 April 2017 Keywords Experimental vibrations Laminated glass Natural frequencies Damping factors Nonlinear behaviour Fitting Time History method

Abstract The experimental free vibrations of a laminated glass beam are investigated with the aim of extracting the nonlinear characteristics of the dynamical behaviour, by an appropriate post-processing of data ensuing from the tests. An updated version of the Fitting Time History (FTH) technique is used. It is based on the least square approximation of the measured damped free vibrations of the laminated glass, and provides the optimal values of the natural frequencies and damping coefficients. While in a previous work attention was mainly devoted to the determination of the linear dynamical properties, here the focus is on the nonlinear behaviour, in particular on the nonlinear relationship between the excitation amplitude and: (i) the natural frequencies, a fact that is commonly encountered in nonlinear dynamics and known as ‘backbone curve’; (ii) the damping coefficient, a fact that is somehow unexpected and commonly not reported in the literature. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In previous papers [1, 2] the authors addressed the problem of the experimental determination of the dynamical properties of laminated glass beam, in particular natural frequencies and modal damping coefficients. Those works were motivated (i) by the growing interest of laminated glass in structural engineering [3, 4], with a lot of new applications ranging in many field of engineering up to architecture, and (ii) by the fact that, while in the static case the literature is relatively abundant [5–8], the experimental determination of dynamical properties of laminated glass was less investigated [9,10], and mainly in conjunction with impact [11, 12] and acoustic [13, 14] problems. An experimental campaign was performed on different specimens [1, 2], and the results were postprocessed by the Fitting Time History (FTH) technique, which is based on the least square approximation of the measured free damped vibrations of the laminated glass (but the technique can be used for the vibration analysis of any kind of structure and material). The principal natural frequencies and the related modal damping coefficients have been determined. † 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.2017.03.001

2

S. Lenci, L. Consolini, F. Clementi / Journal of Vibration Testing and System Dynamics 1(1) (2017) 1–14

In this work the problem of determining the nonlinear features of the dynamical parameters is addressed, extending to the dynamical case the work [15] where the nonlinear behaviour of laminated glass plates in the static case is investigated. An improved use of the FTH technique permits to reveal how the natural frequencies as well as the modal damping coefficients depend nonlinearly on the excitation amplitude. The former behaviour is common and somehow expected, being a consequence of the nonlinear behaviour of the interlayer (observed experimentally for example in [6] and investigated analytically in [16]) and possibly - to a minor extent - of the glass layers, although it was not previously highlighted in the literature dealing with the dynamical behaviour of laminated glass. The latter behaviour, on the other hand, was unexpected since often the damping coefficient is assumed to be constant (while other types of nonlinearity are introduced to model the nonlinear damping, such as the dependence on the quadratic and cubic powers of the velocity [17], etc.). Only recently, in fact, it has been realized that the damping can depend on the excitation amplitude [18], and this work provides a further experimental observation of this behaviour. The nonlinearities are shown to be not predominant, but they can be clearly seen in the dynamics of the investigated system. This is a consequence of the fact that experimentally only small amplitudes have been considered, where of course the behaviour is mainly linear. As a matter of fact, the capability of detecting the nonlinear behaviour even for small amplitudes is an appealing property of the FTH method. 2 Experiments In [1, 2] various two-layer laminated glass elements have been tested experimentally. The length of the layers was 1000 mm, the thickness of each layer 10 mm, and the thickness of the PVB interlayer 1.52 mm. Two different widths are considered, and it is shown, as expected, that the width has no influence on the investigated transversal vibrations. The specimens have free-free boundary conditions, rest on a sponge substrate, and have been hit by an impact hammer. During the free vibrations following the hitting (used for identification purposes), the accelerations at the corners are measured by 4 different accelerometers. The sampling frequency was fs = 16384 Hz (ωs = 102943.708), i.e. one measure each 0.000061035 s is taken, and 8192 measures were collected, so that each registration lasts 0.5 s. This frequency is much larger that the identified natural frequencies, so that we have not aliasing errors. An extended identification analysis is performed addressing the effect of ageing, imperfections, etc. More details can be found in [1, 2]. Before proceeding with the identification, the recorded signal must be cut at the beginning, to eliminate the transient, and at the end, to eliminate the low amplitude oscillations that are strongly subjected to noise. In the initial part, we individuate the first point where the signal is zero after the first half oscillation, and we eliminate the registrations before this point. Then, we consider 4096 points (out of the 8192 measured), so that only half of the total length of the signal in considered [1,2]. We have found that this is a good balance between accuracy and computational efforts in the next identification phase. 3 Identification In order to extract the dynamical characteristics (natural frequency and damping), an improvement of the Fitting Time History (FTH) technique is used. It starts from the observation that, in free vibrations, the single i-th mode system response (here acceleration, but the same holds for velocity and


3

displacement) satisfies a(t) ¨ + 2ξi ωi a(t) ˙ + ωi2 a(t) = 0,

(1)

and thus it is given by a(t) = Ai (t) sin(ωiD t + φi ),

ωiD = ωi

q

1 − ξi2 ,

Ai (t) = Bi e−ξi ωit ,

(2)

where Ai (t) is the time varying amplitude of the oscillation, ωi the i-th (circular) natural frequency (the period is Ti = 2π /ωi and the natural frequency fi = ωi /2π ), ξi the damping coefficient, Bi the initial amplitude and φi the phase. Bi and φi depend on the initial conditions, and are not properties of the system; thus, they are not quantities of mechanical interest, although they appear in the identification process and must be considered. To identify ωi and ξi , a least square fitting between the measured signal and the approximation (2) is performed. At each recorded time instant t j , let aˆ j be the recorded acceleration and a j = a(t j ) its analytical approximation given by (2). Then, the normalized L2 -norm of the difference q 4096 I(ωi , ξi , Bi , φi ) =

∑ j=1 (a j −aˆ j )2 4096

(3)

max j {aˆ j }

is minimized by varying ωi , ξi , Bi and φi . I is named the quality index. The first improvement to the previous identification consists of considering the possible nonlinear behaviour of the system (which is actually the goal of this paper). Since it is known that in the nonlinear regime the frequency depends on the square of the amplitude [19], we assume that

ωi (t) = ωi0 + ωi2 A2i (t),

(4)

where ωi0 is the linear (circular) natural frequency and ωi2 the nonlinear correction term; both are quantities of mechanical interest, to be identified. Practically, the expression (4) is implemented at each recorded time instant t j as

ωi, j = ωi (t j ) = ωi0 + ωi2 A2i (t j ) = ωi0 + ωi2 A2i, j ,

Ai, j = Bi e−ξi ωi, j t j .

(5)

At each t j , and for given values of ωi0 , ωi2 , Bi and ξi , the previous are two transcendental equations in the two unknowns ωi, j and Ai, j . Solving these equations, while possible, is time consuming, and thus, to speed up the process, we use the frequency computed at the previous time instant to compute Ai, j ; with this Ai, j we then compute ωi, j : Ai, j = Bi e−ξi ωi, j−1t j

ωi, j = ωi0 + ωi2 A2i, j .

(6)

We have checked that the approximations introduced by this assumption are negligible since the time step is sufficiently small. Note that Ai,0 = Bi . The second improvement, which is an element of novelty not well investigated in the literature, consists of assuming that also the damping is a nonlinear function of the amplitude:

ξi (t) = ξi0 + ξi1 Ai (t) + ξi2 A2i (t),

(7)

where ξi0 is the linear damping coefficient and ξi1 and ξi2 are the linear and nonlinear correction terms, to be identified. Note that a generic polynomial dependence is assumed, since there are no theoretical reasons suggesting that there are no even powers of the amplitude, as instead occurs for the frequency. Eliminating the amplitude Ai between (4) and (7) we obtain that the damping can be seen as a function of the frequency. This agree with the findings of [1,2] in which it is shown that this dependence is approximately linear (see Fig. 14 and eq. (19) of [1, 2]).

4


Again, to eliminate solving online transcendental equations, (4) and (7) are implemented as Ai, j = Bi e−ξi, j−1 ωi, j−1t j

ωi, j = ωi0 + ωi2 A2i, j ,

ξi, j = ξi0 + ξi1 Ai, j + ξi2 A2i, j .

(8)

With the above improvements the quality index I becomes a function of the 7 parameters ωi0 , ωi2 , ξi0 , ξi1 , ξi2 , Bi and φi . ωi0 and ξi0 describe the linear behaviour, ωi2 , ξi1 , ξi2 the nonlinear behaviour, while Bi and φi are technical parameters, as said before. In the case of multi-modal response, the approximate solution (2) is sought after in the form q N (9) Ai (t) = Bi e−ξi ωit , a(t) = ∑ Ai (t) sin(ωiD t + φi ), ωiD = ωi 1 − ξi2 , i=1

where the sum is extended to all the N involved modes, not necessarily an ordered sequence, and where nonlinear corrections (4) and (7) are considered for each modal frequency and damping coefficient. This entails assuming that there is no nonlinear coupling between different modes (each frequency depends - quadratically - on the amplitude of its mode only, and not on the amplitudes of the other modes; the same for the damping coefficients), which is reasonable if we are far enough from internal resonances. The quality index becomes a function of 7 × N parameters, 5 × N of mechanical interest. Its minimization becomes more complicated from a mathematical point of view, but allows simultaneous identification of various linear and nonlinear modal properties. In this work the minimization is done by the built-in minimization tool of Excel©, possibly combining different algorithms and with a sequence of operator driven minimization steps. 3.1

Excitation in the middle point

We start to consider the case in which the hammer hits the specimen in the middle point, so that only symmetric modes (first, third, etc.) are theoretically excited. We analyze the acceleration time history of Accelerometer 1 of the test T1 (see Tab. 1 and Figs. 3, 4 and 5 of [2]). The time history, after cutting the initial and the final parts, is reported in Fig. 1a. Figure 1a shows an amplitude modulation of the main oscillation amplitude, suggesting the presence of a low frequency component in the signal. This is clearly visible - at ω ∼ = 125.66 - in the Fast Fourier Transform (FFT) of Fig. 1c, and is basically due to the rigid body oscillation above the sponge substrate. It is not of interest from a mechanical point of view, but it has to be taken into account in the identification process since it affects the recorded signal. The enlargement of Fig. 1b, on the other hand, shows that there is an higher frequency oscillation in the initial part of the time history. The peak corresponding to this frequency is barely visible at ω ∼ = 3568.85 - in the FFT of Fig. 1c. It is the natural frequency of the third mode (the second symmetric). Before starting the identification by means of the FTH method, the FFT of the signal is analyzed (Fig. 1c). In addition to the secondary frequencies yet discussed, we have that the main frequency, corresponding to the first mode, is ω ∼ = 753.982. The associated damping, computed by the half-power bandwidth method [20], is given by ξ ∼ = 730.386 and ω2 ∼ = 768.642). These values are = 2.537% (ω1 ∼ important because they are used as a starting point for the minimization of I. It must be remarked that they come from the FFT, which is a linear operator, so that they poorly describe the nonlinear behaviour, if any. In order to have a term of comparison for the improvement obtained by the subsequent identification, we first compute the normalized L2 -norm of the given signal, which is equivalent to I for Bi = 0. It is the Case 1 in Tab. 1. 3.1.1

One-mode identifications

According to the fact that the FFT clearly highlights a dominant frequency (the first mode), we initially consider a one mode approximation of the given signal. We start using ω = 753.982 and ξ = 2.537%


a)

(a)

b)

5

(b)

c) (c)

Fig. 1 Hammer hitting in the middle point. (a) Acceleration time history, (b) an enlargement in the initial part, and (c) the FFT.

obtained by the FFT, and minimize I by varying only B1 and φ1 . The results are reported in the second column (Case 2) of Tab. 1, and shows that the quality index I is yet high. This is a proof that the frequencies and damping computed by the FFT are not accurate. The next step consists of identifing the linear modal parameters ω10 and ξ10 . This is obtained by minimizing I(ω10 , ξ10 , B1 , φ1 ), and it is the Case 3 of Tab. 1. The results show the major improvement obtained by the FTH method: the quality index strongly decrease (from 17.323 % to 3.193 %), and we have a much more accurate estimation. The differences (i.e. the improvements) with respect to the FFT identification are 4.21 % for the natural frequency (from 753.982 to 723.548) and 15.63 % for the damping coefficient (from 2.537 % to 2.194 %). We then consider the nonlinear correction of the frequency only. The results are reported as Case 4 in Tab. 1. There is only a minor improvement of I, according to the fact that the nonlinear correction ω12 is small. Finally, we also allow for amplitude dependent modifications of the damping, and identify all the 7 parameters (Case 5 of Tab. 1). Again, I decreases only slightly. Summing up Cases 4 and 5 we can conclude that nonlinear effects are clearly present in the backbone curve (Fig. 2a), that show an hardening behaviour of the laminated glass. They are also present in the damping (Fig. 2b), that decreases for increasing amplitudes. Since |ξ11 | is negligible with respect to |ξ12 |, we have found that also damping, as frequency, practically has a quadratic dependence of the vibration amplitude. These nonlinear effects are however quantitatively modest, as shown by the fact the nonlinear

6


a)

b)

(a)

(b)

c)

(c)

Fig. 2 (a) The backbone curve A(ω ) and (b) the amplitude dependent damping ξ (A) for the first natural frequency; (c) the backbone curves for the first and the third frequencies.

corrections do not provide major reduction to I. This is confirmed by the fact that for A = 120 m/s2 the (nonlinear) frequency is only 0.32 % larger than the linear one, and the damping decreases from 2.200 % to 1.715 %. The modest bending of the backbone curve for the first frequency can also be appreciated in Fig. 2c where the same A(ω ) of Fig. 2a is shown in a different scale. As said in the Introduction, the low influence of the nonlinear terms is due to the fact that only low intensity hammer hitting have been used (to not brake the glass), so that small amplitudes are obtained and the large amplitudes nonlinear regime is not fully experienced by the specimen. It is worth to note that passing from Case 3 to Case 5 both ω10 and ξ10 remain practically unchanged, meaning that the nonlinear terms do not affect the linear estimation or, in other words, that the simpler linear estimation (Case 3) is adequate to have a reliable identification of the linear frequency and linear damping coefficient. The comparison between the experimental time history and that obtained in the Case 5 is reported in Fig. 3. Finally, we have also tried an amplitude dependent phase, φi (t) = φi0 + φi1 Ai (t) + φi2 A2i (t). This however leads to a negligible improvements (the quality factor decreases only to the value I = 3.117%), and thus this possible improvement will be no longer considered.


7

Table 1 Results of more and more refined identifications. Hammer hitting in the middle point. Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

I

26.464 %

17.323 %

3.193 %

3.168 %

3.118 %

2.027 %

0.880 %

ω00

0

0

0

0

0

88.687

88.882

ω02

0

0

0

0

0

-0.094

-0.036

ξ00

0

0

0

0

0

7.441 %

7.533 %

ξ01

0

0

0

0

0

-0.007 ×10−8

-0.007 ×10−8

ξ02

0

0

0

0

0

-0.001

×10−8

-0.001 ×10−8

B0

0

0

0

0

0

8.938

φ0

0

0

0

0

0

0.147

0.022

9.125

ω10

0

(753.982)

723.548

723.568

723.538

723.571

723.494

ω12

0

0

0

0.000207

0.000159

0.000186

0.000123

ξ10

0

(2.537 %)

2.194 %

2.197 %

2.200 %

2.202 %

2.200 %

ξ11

0

0

0

0

ξ12

0

0

0

0

0.862 ×10−8

0.763 ×10−8

0.763 ×10−8

B1

0

129.153

152.267

152.623

φ1

0

2.942

3.655

ω30

0

0

0

-33.690 ×10−8

-31.378 ×10−8

-33.252 ×10−8

3.615

3.625

3.618

3.634

0

0

0

3594.025

145.935

146.557

146.001

ω32

0

0

0

0

0

0

0.555

ξ30

0

0

0

0

0

0

5.228 %

ξ31

0

0

0

0

0

0

ξ32

0

0

0

0

0

0

0.00003 ×10−8

B3

0

0

0

0

0

0

φ3

0

0

0

0

0

0

3.1.2

-0.00032 ×10−8 36.909

1.212

Multi-mode identifications

The next improvement involves adding more harmonics, those suggested by the FFT of Fig. 1c, in the analytical approximation of the free vibration. We start by adding to the first (yet identified) frequency that of the rigid body motion. The results of the minimization of I are the Case 6 of Tab. 1. While the mechanical quantities of the first mode remain practically unchanged, thus confirming the robustness of the previous one mode identification, the simple addition of the rigid body motion permits an important decrement of I. However, we note that the identified rigid motion frequency ω00 = 88.687 is quite different from the estimation obtained from the FFT (ω ∼ = 125.66, see above), showing the low quality of this latter. We then add the frequency of the third mode, and report the results in Tab. 1, Case 7. The identified frequency ω30 = 3594.025 is close to the FFT identified ω ∼ = 3568.85. The quality index further decreases to the value I = 0.880% that is excellent from a practical point of view. The addition of two modes practically does not change the values ω10 , ω12 , ξ10 , ξ11 and ξ12 identified with the single mode approximation. This shows that this latter is robust, even if it is not able to detect higher order natural frequencies. The nonlinearity of the rigid body motion is negligible in all multi-mode approximations (and in any case out of mechanical interest); the nonlinearity of the first frequency is practically unchanged with respect to that identified by the single mode approximation, as perceived by comparing the two curves in Fig. 2a; the nonlinearity of the backbone curve of the third frequency is yet of the hardening type and, contrary to previous cases, seems to be important, as graphically illustrated in Fig. 2d and

8


Fig. 3 Comparison between the experimental and the identified time histories.

as confirmed by the fact that ω32 is much larger that ω12 . A word of caution must however be used in this latter case, since the overall systems experience small displacements, and within this framework the third mode has a small amplitude (in fact, B3 is about 4 times smaller than B1 ), so that it is unlike that the third mode really experience fully nonlinear regime. Thus, in the present case it is not unlike that ω32 is strongly affected by noise. In any case, it is an initial, possibly rough, identification of the nonlinearity of the third mode. 3.2

Excitation at about one-third of the beam length

In the second case herein investigated the hammer hits the specimen at about one-third of its lenght, so that now all modes, symmetric and anti-symmetric, are excited. We analyze the acceleration time history of Accelerometer 1 of the test T8 (see Tab. 7 of [2]). The experimental time history, after cutting the initial and the final parts, is reported in Fig. 4a and enlarged in Fig. 4b. Again, the pre-requisite for the FTH analysis is constituted by the FFT of the recorded signal, which is reported in Fig. 4c. In addition to the first (at ω ∼ = 100.531 - rigid body), second (at ω ∼ = 728.849 ∼ first mode) and forth (at ω = 3493.451 - third mode), now there is clearly a third peak corresponding to the second, anti-symmetric, mode (at ω ∼ = 1859.823) that was not excited in case considered in Sect. 3.1. The special choice of the hitting point has as consequence that the first and the second modes have about the same magnitude, the former reducing from about 36.161 m/s2 (Sect. 3.1) to about 5.254 m/s2 (present case), the latter increasing from about 0 m/s2 (Sect. 3.1) to about 4.193 m/s2 (present case). The rigid body oscillation and the third mode, on the other hand, keep similar peak amplitudes (4.342 m/s2 vs 4.183 m/s2 the former, 0.907 m/s2 vs 0.3648 m/s2 the latter). The damping coefficients, computed by the half-power bandwidth method [20], are given by ξ ∼ = ∼ ∼ ∼ ∼ ∼ 6.647% (ω1 = 673.302 and ω2 = 770.205) and ξ = 6.458% (ω1 = 1726.617 and ω2 = 1966.831) for the first and second mode, respectively. As in Sect. 3.1, the damping of the third mode is not computed √ on the FFT since the peak amplitude is too small. The amplitude levels corresponding to a peak / 2 are highlighted in Fig. 4c for the first and second natural frequencies (second and third peaks of the FFT, respectively). The Case 8 in Tab. 2 corresponds to the normalized L2 -norm of the given signal (equivalent to I for Bi = 0) and it is used as a reference to measure the improvements of I. Having yet investigated in Sect. 3.1 the effects of various parameters in the one-mode identification (Cases 1 to 5 in Tab. 1), in the following we consider directly all the 7 parameters for each single mode.


(a)

9

(b)

c) (c)

Fig. 4 Hammer hitting at one third. (a) Acceleration time history, (b) an enlargement in the initial part, and (c) the FFT.

3.2.1

One-mode identifications

We starting using the one-mode approximation (2) to identify the first natural frequency. We thus minimize I(ω10 , ω12 , ξ10 , ξ11 , ξ12 , B1 , φ1 ) using as a starting point the parameters obtained from the FFT: ω10 = 728.849, ω12 = 0, ξ10 = 6.647%, ξ11 = 0, ξ12 = 0, B1 = 5.254 m/s2 and φ1 = 0. The results are the Case 9 in Tab. 2. We repeat the same procedure for the second and the third mode, and the single-mode identified values are reported as Cases 10 and 11 in Tab. 2, respectively. In all three cases the quality index is still high, so that now the single mode approximation cannot be expected to be accurate, especially for the identified nonlinear parameters. This is basically a consequence of the fact that the signal is intrinsically multi-modal, and has not a predominant frequency, as shown by the FFT of Fig. 4c, contrarily to what happens for the case of Sect. 3.1. In this case a multi-modal approximation is certainly necessary. 3.2.2

Multi-mode identifications

We start by considering the first mode and the rigid body oscillations. The results of the minimization of I are the Case 12 of Tab. 2. We see that I is still large, and accordingly the identified values of the first mode do not change that much from Case 9 to Case 12. We conclude that the identification is not yet reliable. The three-mode (rigid body, first and second) identification is the Case 13 of Tab. 2. Now the

10


Table 2 Results of more and more refined identifications. Hammer hitting at one third of the length. Case 8 I

Case 9

14.499 % 11.330 %

ω00 0

0

Case 10

Case 11

Case 12

Case 13

Case 14

8.804 %

11.468 %

10.632 %

1.201 %

0.763 %

0

0

83.220

81.810

81.815

ω02 0

0

0

0

0.265

-0.022

-0.017

ξ00 0

0

0

0

8.097 %

6.317 %

6.318 %

ξ01 0

0

0

0

ξ02 0

0

0

0

-0.000066 ×10−8 -0.000066 ×10−8 -0.000066 ×10−8 -0.000040 ×10−8 -0.000040 ×10−8 -0.000040 ×10−8

B0 0

0

0

0

11.231

9.101

9.113

φ0 0

0

0

0

-0.722

0.116

0.102

ω10 0

687.387

0

0

685.776

710.716

710.541

ω12 0

-0.121

0

0

-0.128

0.016

0.015

ξ10 0

6.102 %

0

0

6.049 %

5.123 %

5.126 %

ξ11 0

0.001361 ×10−8 0

0

0.001359 ×10−8 0.001359 ×10−8 0.001359 ×10−8

63.942

0

ξ12 0 B1 0

4954.630 ×10−8 0

0

0

4168.742 ×10−8 -1013.286 ×10−8 -1013.361 ×10−8 60.275

44.812

44.832

φ1 0

-1.554

0

0

-1.458

-3.274

-3.260

ω20 0

0

1835.446

0

0

1826.206

1824.847 -0.00452

ω22 0

0

0.00169

0

0

-0.00300

ξ20 0

0

6.465 %

0

0

6.308 %

×10−8

6.273 % ×10−8

0.000059 ×10−8

ξ21 0

0

0.000059

0

0

0.000059

ξ22 0

0

0

B2 0

0

-0.003242 ×10−8 0

-0.003189 ×10−8 -0.003192 ×10−8

φ2 0

0

-3.240

0

0

-3.083

-3.046

ω30 0

0

0

3476.665

0

0

3585.038

ω32 0

0

0

-0.802

0

0

21.227

ξ30 0

0

0

3.097 %

0

0

6.387 %

ξ31 0

0

0

0

0

ξ32 0

0

0

-35.716 ×10−8

0

B3 0

0

0

-1645.130 ×10−8 0

0.000100 ×10−8

43.377

0

0

φ3 0

0

0

2.472

0

0

125.116

0

0

122.355

121.305

-0.000735 ×10−8 16.384

-0.0718

quality index strongly reduces, down to a value (1.201 %) that is small enough and guarantees that the identified parameters are reliable. This agrees with the fact that the first three peaks in the FFT of Fig. 4c are dominant. There is an important quantitative improvement (with respect to Cases 9 and 12) in the identified ω10 , while the improvement in ω20 has a minor extent. The major qualitative improvement consists of changing of the sign of both ω12 and ω22 , meaning that in the present case one-mode identifications are not able to correctly capture the hardening/softening behaviour of the nonlinear oscillations. As a matter of fact, the first mode is shown to be hardening (ω12 > 0), confirming the results of Sect. 3.1, while the second mode softening (ω22 < 0). We have that |ξ11 | 0 and Eq.(5) are feasible. Notice that the modes ordered by Eq.(5) are different from the one ordered in the usual way in which the modes are ordered by the magnitudes of natural frequencies. Let m be fixed and denote by Pm the orthogonal projection in H onto the space spanned by the first L m eigenvectors of A in the sequence as listed in Eq.(5), and Qm = I − Pm , namely, H = Pm H Qm H. For the sake of simplicity, hereafter Pm H and Qm H are termed as lower and higher subspaces, respectively. From Eq.(5), it is clear that the modes in lower subspace are slightly damped and can be considered as ‘active’ modes, and the modes in higher subspace are strongly damped and ‘inactive’.

56

Yan Liu, et al. / Journal of Vibration Testing and System Dynamics 1(1) (2017) 53–63

Following Mode Analysis Technique in linear structural dynamics, let x = Φn a, here vector a is the modal co-ordinates rather than specified physical co-ordinates, and substitutes vector x into Eq.(1), then pre-multiplying by the matrix ΦTn derived from the eigenvectors of operator A, yields a set of n equations in terms of modal co-ordinates, a¨ +Ca˙ + Ka = ΦTn f .

(6)

Because the system studied is dissipative, that means there exists an absorbing set B = {Φn a ∈ D(A) |, |a| ≤ ρ }. As t ≥ T , a(t) ∈ B, ∃T . In other words, the modal coordinates remain bounded. Applying Pm and Qm to Eq.(1), produces a set of equations in the following form, p¨ +C1 p˙ + K1 p = h(p + q, p˙ + q), ˙

(7)

˙ q¨ +C2 q˙ + K2 q = g(p + q, p˙ + q),

(8)

where h = ΦTm f , g = ΦTn−m f . Indeed, the solution to Eq.(6) can be written as a(t) = p(t) + q(t). One can easily prove that h(p + q, p˙ + q) ˙ and g(p + q, p˙ + q) ˙ also satisfy the Lipschitz condition listed above. For measuring the distance between the original and reduced systems in phase space spanned by a(t) and a(t), ˙ a distance will be given. ˙ be a point in the phase space defined in modal co-ordinates, the Definition 1. Let y(t) = {a(t), a(t)} distance between two points can be then defined as 1

dist|·| (y2 (t), y1 (t)) = [|a2 (t) − a1 (t)|2 + |a˙2 (t) − a˙1 (t)|2 ] 2 .

(9)

Lemma 1 (Gronwall’s Lemma). Let y be a positive locally integrable function on (0, +∞) such that dy dt is locally integrable on (0, +∞), and which satisfies dy (τ ) ≤ −ay(τ ) + b, dt

∀τ > 0;

then y(τ ) ≤ y(0)exp(−aτ ) +

a, b = const > 0, b a

∀τ ≥ 0.

(10)

(11)

3 Interaction between Lower and Higher Modes based on Approximate Inertial Manifolds In this section, the interaction between lower and higher modes will be derived based on Approximate Inertial Manifolds, in order to improve the distance between the original and reduced systems as the traditional Galerkin method is applied to the system. For the model reduction based on the traditional Galerkin method by which the influence of higher modes on the solution is ignored, the influence of model reduction on the long-term behaviors of the system can be obtained by estimating the distance between a(t) and Pm H in phase space as described by Theorem 2 [2]. Theorem 2. [2] Assuming |a| ≤ ρ , |g| ≤ M0 , then as t → +∞, dist|·| (a(t), Pm H) ≤ ( where M12 =

M0 2 2ωm+1 ξm+1

1 M1 2 )2 , 2ωm+1 ξm+1

+ 2ωm+1 ξm+1 ωl2 ρ 2 , and ωl = max{ωm+1 , ωm+2 , . . . , ωn } ≥ 1.

(12)


57

It is clear that as m is increasing, the distance between the original and reduced systems will become more ‘shorter’, as expected in sense of dynamics. It is well known that the evolution of the modes in higher subspace ordered by Eq.(5) are generally quasi-static with respect to that of the modes in lower subspace in some dissipative dynamic systems with high damping, after a certain temporal interval. Indeed, the asymptotic behavior of high dimensional dissipative dynamical systems can be described by the deterministic flow on a low dimensional attractor. Such systems and the properties have been proved theoretically and experimentally [1, 30]. On the other hand, for some dynamic systems with strongly nonlinearities, if the modes in higher subspace are completely ignored, it will lead to a mistake for studying the long-term behaviors due to the nonlinear coupling between ‘active’ and ‘inactive’ modes. As stated above, the perturbation from the truncation of modes in the higher subspace has a nonnegligible influence on the long-term behaviors of some original dynamic systems and can be considered as a correction to the approximate solution obtained by the traditional Galerkin method [2]. The nonlinear Galerkin methods, which stems from the theory of dynamical systems and Inertial Manifolds, have attracted some attention, and one of their main advantages is that they show remarkable accurate results with higher convergence rates than traditional Galerkin method in the numerical analysis of the long-term behaviors. There are several nonlinear Galerkin methods, and the main differences among them are the constructions of the interactions between ‘active’ and ‘inactive’ modes [12]. In this study, the following type of AIMs, which is used frequently, is constructed. For the systems with high damping governed by Eqs.(7) and (8), the following properties can then be assumed for the modes in higher subspace defined by Eq.(5) in this study, q(t) ¨ → 0, and q(t) ˙ → 0. (13) Eq.(8) can then be reduced to the following form, K2 q = g(p + q, p). ˙

(14)

Eq.(14) can be further written in the viewpoint of mapping as q = Φ(p, p) ˙ = K2

−1

g(p + Φ(p, p), ˙ p), ˙

∀p ∈ Pm H

\

B.

(15)

The graph G = GraphΦ can be referred to as the Approximate Inertial Manifolds for such second order in time nonlinear dissipative autonomous dynamic systems. The method presented above is frequently used to construct AIMs with rapid convergence rates in the analysis of the long-term behaviors. Generally, the following form is named for Nonlinear Galerkin Approach for nonlinear dissipative evolution equation, p¨ +C1 p˙ + K1 p = h(p + Φ(p, p), ˙ p). ˙ (16) ˙ Notice that Φ(p, p) ˙ is not considered in this kind of AIMs. As mentioned above, this kind of AIMs can present remarkable accurate results with higher convergence rates than traditional Galerkin method in the numerical analysis of the long-term behaviors, though the initial behaviors may be somewhat different from the original systems. Furthermore, the relationship between lower and higher modes, namely the fixed point in mapping governed by Eq.(15), can be obtained under the same assumptions as stated in Theorem 3. Theorem 3. Assuming

k1 ωk 2

< 1, then there exists a Lipschitzian mapping Φ: Pm H ∩ B −→ Qm H ∩ B, K2 Φ(p, p) ˙ = g(p + Φ(p. p), ˙ p), ˙

where ωk = min{ωm+1 , ωm+2 , . . . , ωn }.

(17)

58


Proof. A mapping is constructed as the following q 7−→ K2

−1

g(p + q, p). ˙

(18)

Let q1 , q2 ∈ Qm H ∩ B, then |K2

−1

g(p + q1 , p) ˙ − K2

−1

−1

g(p + q2 , p)| ˙ ≤ |K2 ||g(p + q1 , p) ˙ − g(p + q2 , p)| ˙ k1 ≤ 2 |q1 − q2 |. ωk

(19)

Following the Banach contraction mapping theorem, a mapping in inner product space H defined in Section 2, can be obtained as Φ(p, p) ˙ = K2

−1

(20)

g(p + Φ(p, p), ˙ p). ˙

The mapping Φ(p, p) ˙ is a Lipschtzian mapping, since −1

−1

|Φ(p1 , p˙1 ) − Φ(p2 , p˙2 )| = |K2 g(p1 + Φ(p1 , p˙1 ), p˙1 ) − K2 g(p2 + Φ(p2 , p˙2 ), p˙2 )| 1 ≤ 2 |g(p1 + Φ(p1 , p˙1 ), p˙1 ) − g(p2 + Φ(p2 , p˙2 ), p˙2 )| ωk k1 ≤ 2 (|p1 + Φ(p1 , p˙1 ) − p2 − Φ(p2 , p˙2 )| + | p˙1 − p˙2 |) ωk k1 ≤ 2 (|p1 − p2 | + |Φ(p1 , p˙1 ) − Φ(p2 , p˙2 )| + | p˙1 − p˙2 |). ωk

(21)

Further, |Φ(p1 , p˙1 ) − Φ(p2 , p˙2 )| ≤ It is easy to verify that Theorem 3 is proved.

k1 ωk 2 −k1

> 0 if

k1 ωk2

k1 (|p1 − p2 | + | p˙1 − p˙2 |). 2 ωk − k1

(22)

< 1.

4 Error Estimate for the Reduction based on AIMs in Long-term Behaviors As for the model reduction based on Approximate Inertial Manifolds defined above, which the influence of higher modes on the solution is considered by, and the influence of model reduction on the long-term behaviors of the system can be estimated as Theorem 4. In phase space, estimating the distance between a(t) and G can be obtained by estimating the distance between q(t) and Φ. Further, a comparison will be made between the model reductions based on traditional Galerkin method and Approximate Inertial Manifolds, respectively. Theorem 4. Assuming ωk12 < 1, then there exists a Lipschitzian mapping Φ: Pm H ∩ B −→ Qm H ∩ B, and k the distance between the original and the reduced systems based on the Approximate Inertial Manifolds presented is 1 M2 2 )2 , (23) dist|·| (a(t), G) ≤ ( 2ωm+1 ξm+1 where M22 =

k12 M1 2 2(ωm+1 ξm+1 )2

k2

+ (2ωm+1 ξm+1 ωl2 + ωm+11ξm+1 )ρ 2 , and ωl = max{ωm+1 , ωm+2 , . . . , ωn } ≥ 1.


59

Proof. Assuming a(t) ∈ B as t ≥ T . Subtracting Eq.(14) from Eq.(8) as t ≥ T , yields, ˙ = g(p + q, p˙ + q) ˙ − g(p + Φ(p. p), ˙ p). ˙ q¨ +C2 q˙ + K2 (q − Φ(p, p))

(24)

˙ Pre-multiplying Eq.(24) by q˙ − Φ(p, p), ˙ then d ˙ ˙ ˙ p)) ˙ T C2 q˙ + (q˙ − Φ(p, (q˙ − Φ(p, p)) ˙ T q˙ + (q˙ − Φ(p, p)) ˙ T K2 (q − Φ(p, p)) ˙ dt ˙ =(q˙ − Φ(p, p)) ˙ T (g(p + q, p˙ + q) ˙ − g(p + Φ(p. p), ˙ p)). ˙

(25)

˙ Due to the fact that Φ(p, p) ˙ is not considered in the AIMs constructed in this paper, as described by Eq.(16), then d ˙ ˙ 2 ] + 2|q|2 =2(q˙ − Φ(p, p)) ˙ T (g(p + q, p˙ + q) ˙ − g(p + Φ(p. p), ˙ p)) ˙ [|q| ˙ 2 + |q − Φ(p, p)| dt ¯ + q, p˙ + q) ˙ − g(p ¯ + Φ(p. p), ˙ p)| ˙ 2, ≤|q|2 + |g(p

(26)

where q − Φ(p, p) ˙ ωm+2 (q2 − Φ2 (p, p)), ˙ . . . , ωn (qn−m − Φn−m (p, p))} ˙ T, ˙ = {ωm+1 (q1 − Φ1 (p, p)), p p p T q = { 2ωm+1 ξm+1 q˙1 , 2ωm+2 ξm+2 q˙2 , . . . , 2ωn ξn q˙n−m } , g1 , √ 1 g2 , . . . , √ 1 gn−m }T . g¯ = { √ 1 2ωm+1 ξm+1

2ωm+2 ξm+2

2ωn ξn

Thanks to Eq.(5), Eq.(26) can be written further as

d [|q| ˙ 2 + |q − Φ(p, p)| ˙ 2 ] + |q|2 ≤ |g(p ¯ + q, p˙ + q) ˙ − g(p ¯ + Φ(p. p), ˙ p)| ˙ 2 dt (27) k1 2 2 ≤ (|q − Φ(p, p)| ˙ + |q˙ − 0|) . 2ωm+1 ξm+1 p p p T With Eq.(5) and q = { 2ωm+1 ξm+1 q˙1 , 2ωm+2 ξm+2 q˙2 , . . . , 2ωn ξn q˙n−m } , Eq.(27) can be rewritten as, d [|q| ˙ 2 + |q − Φ(p, p)| ˙ 2 + 2ωm+1 ξm+1 |q − Φ(p, p)| ˙ 2 ] + 2ωm+1 ξm+1 |q| ˙ 2 dt k1 2 ≤2ωm+1 ξm+1 |q − Φ(p, p)| ˙ 2+ (|q − Φ(p, p)| ˙ + |q|) ˙ 2 2ωm+1 ξm+1 k1 2 ˙ 2+ (2|q − Φ(p, p)| ˙ 2 + 2|q| ˙ 2) ≤2ωm+1 ξm+1 |q − Φ(p, p)| 2ωm+1 ξm+1 ≤(with ωl = max{ωm+1 , ωm+2 , . . . , ωn } ≥ 1)

(28)

k1 2 (2|q − Φ(p, p)| ˙ 2 + 2|q| ˙ 2) 2ωm+1 ξm+1 k1 2 k1 2 )|q − Φ(p, p)| ˙ 2+ |q| ˙ 2. =(2ωm+1 ξm+1 ωl 2 + ωm+1 ξm+1 ωm+1 ξm+1

≤2ωm+1 ξm+1 ωl 2 |q − Φ(p, p)| ˙ 2+

Since, |q| ˙ 2 ≤ |q(t)| ˙ 2 + |q(t)|2 ≤ (due to Eq.(12)) M1 2 , 2ωm+1 ξm+1

(29)

|q − Φ(p, p)| ˙ ≤ ρ , as t → +∞.

(30)

≤ and

60


Therefore, following Eq.(28), yields, d ˙ 2 + 2ωm+1 ξm+1 |q − Φ(p, p)| ˙ 2 ] + 2ωm+1 ξm+1 |q| ˙ 2 [|q| ˙ 2 + |q − Φ(p, p)| dt k1 2 k12 M1 2 ≤(2ωm+1 ξm+1 ωl 2 + )ρ 2 + ωm+1 ξm+1 2(ωm+1 ξm+1 )2 . =M2 2 .

(31)

Following Gronwall Lemma, ˙ 2 |q| ˙ 2 + |q − Φ(p, p)| 2 2 ˙ + |q(0) − Φ(p(0), p(0))| ≤e−2ωm+1 ξm+1t (|q(0)| ˙ )+

ˆ

t

M2 2 e−2ωm+1 ξm+1 (τ −t) d τ

0

(32)

2

2 2 ˙ + |q(0) − Φ(p(0), p(0))| =e−2ωm+1 ξm+1t (|q(0)| ˙ )+

M2 (1 − e−2ωm+1 ξm+1t ) 2ωm+1 ξm+1

By virtue of Eq.(5), as t → +∞, then 2 2 ˙ |q(t)| ˙ 2 + |q(t) − Φ(p(t), p(t))| ˙ =|q(t) ˙ − Φ(p, p)| ˙ 2 + |q(t) − Φ(p(t), p(t))| ˙

≤|q| ˙ 2 + |q − Φ(p, p)| ˙ 2

(33)

M2 2 ≤ . 2ωm+1 ξm+1

Further, dist|·| (a(t), G) ≤ (

1 M2 2 )2 . 2ωm+1 ξm+1

(34)

Theorem 4 is proved. That means the distance between G and original system in phase space remains bounded in an absorbing set, as t → +∞. It is clear that the distance between the original system and the AIMs will become ‘short’, as m is increasing. Following Theorem 1 and Theorem 2, the comparison between dist|·| (a(t), Pm H) and dist|·| (a(t), G) can be made. For convenience, dist 2|·| (a(t), Pm H) and dist 2|·| (a(t), G) will be used as the following, instead of dist|·| (a(t), Pm H) and dist|·| (a(t), G). dist 2|·| (a(t), Pm H) ≤ dist 2|·| (a(t), G) ≤

M0 2 4ωm+1 2 ξm+1 2

+ 2ωl 2 ρ 2 ,

k1 2 [M0 2 + 4ωm+1 2 ξm+1 2 ρ 2 (1 + ωl 2 )] 8ωm+1 4 ξm+1 4

+ 2ωl 2 ρ 2 .

(35)

(36)

It can be seen that dist|·| (a(t), G) is much more closer to the solution of the original systems compared 2

M0 is satisfied. For some dynamic to dist|·| (a(t), Pm H) as t → +∞, if condition k1 2 ≤ 2 M0 +4ωm+1 2 ξm+1 2 ρ 2 (1+ωl 2 ) systems in practice, such a condition can be satisfied.

5 Explicit Iterative Scheme to Approach AIMs An explicit iterative scheme will be proposed for obtaining the fixed point of mapping governed by Eq.(18), due to the implicit.


Theorem 5. Assuming Denote by

k1 ωk2

< 1, and let Tp, p˙ (q) = K2

−1

61

g(p + q, p), ˙ ∀q ∈ Qm H ∩ B for every p ∈ Pm H ∩ B,.

Φ0 (p(t), p(t)) ˙ = 0, Φ1 (p(t), p(t)) ˙ = Tp, p˙ (Φ0 (p(t), p(t))), ˙ ··· Φn+1 (p(t), p(t)) ˙ = Tp, p˙ (Φn (p(t), p(t))), ˙ ··· .

(37)

Then there exists a unique Lipschitzian mapping Φ: Pm H ∩ B −→ Qm H ∩ B, and the distance between Φn (p(t), p(t)) ˙ and Φ(p(t), p(t)) ˙ is |Φn (p(t), p(t)) ˙ − Φ(p(t), p(t))| ˙ ≤ [(

k1 )n /(1 − 2 ωk − k1

k1 M0 )] 2 . 2 ωk − k1 ωk

Proof. Similar as the Proof in Theorem 2, the mapping Tp (q) = K2 contraction mapping, and |Φn (p(t), p(t)) ˙ − Φ(p(t), p(t))| ˙ ≤ [( ≤ [( ≤ [(

k1 )n /(1 − 2 ωk − k1 k1 )n /(1 − 2 ωk − k1 k1 )n /(1 − 2 ωk − k1

−1

(38)

g(p + q, p) ˙ is a Lipschitzian

k1 )]|Φ1 (p(t), p(t)) ˙ − Φ0 (p(t), p(t))| ˙ 2 ωk − k1 k1 )]|K2 −1 g(p + Φ(p, p))| ˙ 2 ωk − k1

(39)

k1 M0 )] 2 . 2 ωk − k1 ωk

Theorem 5 is proved. 6 Concluding Remarks In combination of mode analysis technique in structure dynamics and Approximate Inertial Manifolds stemmed from nonlinear dynamics, a method is presented to reduce the second order in time nonlinear dissipative autonomous dynamic systems with higher dimension, which are encountered frequently in engineering. In comparison to the traditional Galerkin method, the presented method can improve the distance between the original and reduced systems on the long-term behaviors, since it takes into account the interaction between the lower and higher modes. Indeed, such interaction is common and plays an important role in the complicated nonlinear phenomena of nonlinear dissipative mechanical dynamic systems. The results show that the numerical method presented can provide a better and acceptable approximation to the long-term behaviors of the second order in time nonlinear dissipative autonomous dynamic systems with many degrees-of-freedom. As a further study, some numerical analysis will be given in order to testify that this method can present a remarkable accurate result with higher convergence rates than traditional Galerkin method in the numerical study of the long-term behaviors. 7 Acknowledgments The research is supported by the National Natural Science Foundation of China (Grant No. 51305355), the National Fundamental Research Program of China (973 Program, Grant No. 2012CB026002) and the National Key Technology R&D Program of China (Grant No. 2013BAF01B02).

62


References [1] Steindl, A. and Troger, H. (2001), Methods for dimension reduction and their application in nonlinear dynamics, International Journal of Solids and Structures, 38, 2131-2147. [2] Zhang, J. Z., Liu, Y., and Chen, D. M. (2005), Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours, Applied Mathematics and Mechanics, 26, 938-943. [3] Guckenheimer, J. and Holmes, P. (1983), Nonlinear oscillations, dynamical system, and bifurcations of vector fields, Springer-Verlag, New York. [4] Wiggins, S. (1990), Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag, New York. [5] Seydel, R. (1994), Practical bifurcation and stability analysis: from equilibrium to chaos, Springer-Verlag, New York. [6] Friswell, M. I., Penny, J. E. T., and Garvey, S. D. (1996), The application of the IRS and balanced realization methods to obtain reduced models of structures with local nonlinearities, Journal of Sound and Vibration, 196, 453-468. [7] Fey, R. H. B., van Campen, D. H., and de Kraker, A. (1996), Long term structural dynamics of mechanical system with local nonlinearities, ASME Journal of Vibration and Acoustics, 118, 147-163. [8] Kordt, M. and Lusebrink, H. (2001), Nonlinear order reduction of structural dynamic aircraft models, Aerospace Science and Technology, 5, 55-68. [9] Slaats, P. M. A., de Jongh, J., and Sauren, A. A. H. J. (1995), Model reduction tools for nonlinear structural dynamics, Computer and Structures, 54,1155-1171. [10] Zhang, J. Z. (2001), Calculation and bifurcation of fluid film with cavitation based on variational inequality, International Journal of Bifurcation and Chaos, 11, 43-55. [11] Murota, K. and Ikeda, K. (2002), Imperfect Bifurcation in Structures and Materials, Springer-Verlag, New York. [12] Marion, M. and Temam, R. (1989), Nonlinear Galerkin methods, Siam Journal on Numerical Analysis, 26, 1139-1157. [13] Temam, R. (1997), Infinite-dimentional dynamical system in mechanics and physics, Springer-Verlag, New York. [14] Titi, E. S. (1989), On approximate inertial manifolds to the Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 149, 540-557. [15] Jauberteau, F., Rosier, C., and Temam, R. (1990), A nonlinear Galerkin method for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 80, 245-260. [16] Olaf Schmidtmann (1996), Modelling of the interaction of lower and higher modes in two-dimensional MHDequations, Nonlinear Analysis, Theory, Methods and Applications, 26, 41-54. [17] Chueshov, I. D. (1996), On a construction of approximate inertial manifolds for second order in time evolution equations, Nonlinear Analysis, Theory, Methods and Applications, 26, 1007-1021. [18] Rezounenko, A. V. (2002), Inertial manifolds for retarded second order in time evolution equations, Nonlinear Analysis, 51, 1045-1054. [19] Carlo R. Laing, Allan McRobie, and Thompson, J. M. T. (1999), The Post-processed Galerkin method applied to non-linear shell vibrations, Dynamics and Stability of Systems, 14, 163-181. [20] Zhang, J. Z., van Campen, D. H., Zhang, G. Q., Bouwman, V., and ter Weeme, J. W. (2001), Dynamic stability of doubly curved orthotropic shallow shells under impact, AIAA Journal, 39, 956-961. [21] Foias, C., Sell, G. R., and Temam, R. (1985), Varietes Inertielles des Equations Differentielles Dissipatives, C. R. Acad. Sci. Paris Ser. I Math., 301, 139-141. [22] Chow, S. N. and Lu, K. (2001), Invariant Manifolds for Flows in Banach Space. Journal of Differential Equations, 74, 285-317. [23] Foias, C., Manley, O., and Temam, R. (1987), On the Interaction of Small and Large Eddies in Turbulent Flows, C. R. Acad. Sci. Paris Ser. I Math., 305, 497-500. [24] Foias, C., Sell, G. R., and Titi, E. S. (1989), Exponential Tracking and Approximation of Inertial Manifolds for Dissipative Nonlinear Equations, Journal of Dynamics and Differential Equations, 1, 199-244. [25] Zhang, J. Z., Ren, S., and Mei, G. H. (2011), Model reduction on inertial manifolds for N-S equations approached by multilevel finite element method, Communications in Nonlinear Science and Numerical Simulation, 16, 195-205. [26] Zhang, J. Z., Liu, Y., and Feng, P. H. (2011), Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application. Communications in Nonlinear Science and Numerical Simulation, 16, 4666-4670. [27] Lei, P. F., Zhang, J. Z., Li, K. L., and Wei, D. (2015), Study on the transports in transient flow over


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impulsively started circular cylinder using Lagrangian coherent structures. Communications in Nonlinear Science and Numerical Simulation, 22, 953-963. [28] Rezounenko, A. V. (2004), Investigations of Retarded PDEs of Second Order in Time using the Method of Inertial Manifolds with Delay, Ann. Inst. Fourier (Grenoble), 54, 1547-1564. [29] Zhang, J. Z., Chen, L. Y., and Ren, S. (2013), Model reduction of nonlinear continuous shallow arch and dynamic buckling simulations on approximate inertial manifolds with time delay, Journal of Applied Nonlinear Dynamics, 2, 343-354. [30] Holmes, P., Lumley, J. L., and Berkooz, G. (1996), Turbulence, coherent structures, dynamical systems and symmetry, Cambridge University Press.



Redundant Control of A Bipedal Robot Moving from Sitting to Standing Jing Lu2 , Xian-Guo Tuo1,2†, Yong Liu2 , Tong Shen2 1 School

of Automation and Information and Engineering, Sichuan University of Science and Engineering, Zigong 643000, China 2 Fundamental Science on Nuclear Wastes and Environmental Safety Laboratory, Southwest University of Science and Technology, Mianyang 621010, China Submission Info Communicated by C. S. Suh Received 29 January 2017 Accepted 27 February 2017 Available online 1 April 2017 Keywords Bipedal robot Sitting to standing planning Hierarchical redundant control Constrained control Quadratic least square algorithm

Abstract To control a multi-joints bipedal robot, it is necessary to choose a proper law of control that ensures the safe and stable motions of the robot. Even for the simplest postures, such as sitting and standing, there are constraints to respect so that the inverse geometric model (MGI) would not have infinitive solutions. In this paper, a particular task of sitting-to-standing is simulated for a bipedal robot using MATLAB. The articulation evolution and the variation of the shoulder position are evaluated using typical control and hierarchical redundant control in which the cases of with and without constraints are both considered. The simulation results show that the hierarchical redundant control has a better effect than the typical one, and when considering the constraints, the articulations’ angular velocities vary more smoothly so that the motor is better protected. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Bipedal robot has a typical multi-articulation structure that consists of several joints. It is a fact that the control of a bipedal robot is to control the movement of its articulation [1]. Actually, the bipedal robot’s movement in practice usually includes some simple motions such as walking, standing, and picking up an object [2]. However, because of the absence of the conditions for the angles calculation, the inverse-kinematics model may have infinitive solutions [3]. It may cause some problems like the intern interruption in robots, passing singular points, losing the center of gravity or even breaking the robot. It is necessary to impart specific conditions of constraint or an effective method before the robot launches the motion. To control a bipedal robot with stability, several biped gait models have been developed taking inspiration in, for example, inverted pendulum, passive gait model, and mass-spring model. Imposing control constraints including stability criterion and energy constraints has been shown to have helped improve the control effect [4]. In this paper, a bipedal robot is presented as a simplified 4 degrees-of-freedom configuration for the investigation of sitting-to-standing motion. In the first † 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.2017.03.005

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Jing Lu, el al. / Journal of Vibration Testing and System Dynamics 1(1) (2017) 65–71

section, all the associated parameters of the simplified bipedal robot are defined. The theoretical bases of the following 4 different cases; namely, typical control without constraint, hierarchical redundant control without constraint, typical control with constraint, and hierarchical redundant control with constraint are reviewed in the section that follows where quadratic least square algorithm is used to solve these problems with constraint. Lastly, the bipedal robot model is simulated in MATLAB and the performances of the typical control and redundant control are compared. Furthermore, the comparisons of the control with and without constraints are discussed.

2 Simplified Bipedal Robot Model The configuration shown in Fig. 1 is a 4 degrees of freedom model of linkages, with variables [θ1 , θ2 , θ3 , gθ4 ] corresponding to the articulations of the feet, the knee, the hip, and the shoulder, respectively. The point CoM is the current center of gravity and the variable Shoulder y is the height of the shoulder. Variables li , mi , and ci correspond to the length, the mass and the position of the center of gravity of the different branches i. The angles use absolute values instead of relative values to simplify T T the calculation. TInT this paper, the objective is to control the height of the shoulder, Shoulder y, on the axis-Y with a satisfied position of the robot’s center of gravity CoM on the axis-X. Because the 4th angle θ4 does not affect Shoulder y by changing it, it is ignored for simplification. Thus, there are 2 T objectives and 3 variables for the control which constitute a sub-determine system whose solution sets are therefore infinitive.

Fig. 1 A bipedal robot model

Fig. 1 A bipedal robot model

3 Control Schemes 3.1 3.1.1

Control without constraints Typical control

T

T T T T T

Only the degree angles are considered in this control. With the direct geometric model, the relation between the height of the shoulder and the values of the angle is yshoulder = f (θ )

(1)

where yshoulder is Shoulder y and θ is the vector of degree angles [θ1 , θ2 , θ3 , gθ4 ]. With the robot model in Fig. 1, yshoulder can be written as yshoulder = l1 sin(θ1 ) + l2 sin(θ2 ) + l3 sin(θ3 )

(2)


67

Therefore the motion depending on the angular velocity of the articulation motion can be determined by considering the deviation of Eq. (1) in the followings y˙shoulder = Jshoulder θ˙

(3)

where J is a Jacobian matrix. With Eq. (2), J can be written as [ ] Jshoulder = l1 cos(θ1 ), l2 cos(θ2 ), l3 cos(θ3 ), 0

(4)

Transform Eq. (3) and the control of each angular velocity can be obtained as + θ˙ = Jshoulder y˙shoulder

(5)

+ in which Jshoulder is the pseudo-inverse matrix of Jshoulder .

3.1.2

Hierarchical redundant control

To ensure that the robot does not fall down and collapse, the influence of the center of gravity cannot be ignored. As defined previously, this robotic system has 2 constraints and 3 variables for the control which constitute a sub-determine system whose general solutions are therefore

θ˙ = J + y˙ + (I − J + J)Z

(6)

with Z being any vector. The position of the center of gravity of the robot on the X-axis can be written as CoMx = r1 cos(θ1 ) + r2 cos(θ2 ) + r3 cos(θ3 ) + r4 cos(θ4 ) (7) where

 r1 = (m1 c1 +m2 l1 +m3 l1 +m4 l1 )/M        r2 = (m2 c2 +m3 l2 +m4 l2 )/M r3 = (m3 c3 +m4 l3 )/M    r4 = (m4 c4 )/M     M = m1 + m2 + m3 + m4

Moreover, the following two equations can be obtained { + ˙ + (I − J + JCoMx )Z θ˙ = JCoMx CoMx CoMx y˙ = Jshoulder θ˙

(8)

(9)

˙ is the deviation of the position of the center of gravity on axis X (CoMx ), Jshoulder is the where CoMx Jacobian matrix of the shoulder position yshoulder , JCoMx is the Jacobian matrix of CoMx as follows [ ] JCoMx = −r1 sin(θ1 ), −r2 sin(θ2 ), −r3 sin(θ3 ), −r4 sin(θ4 )

(10)

For a safe motion, the control of the center of gravity must have a prior level in the hierarchical structure. Thus, integrating Eq. (7) and Eq. (8) one can calculate the angular velocity as + + ˙ + (I − J + JCoMx )J˜+ ˙ θ˙ = JCoMx CoMx CoMx shoulder (y˙shoulder − Jshoulder JCoMxCoMx)

(11)

+ + J˜shoulder = Jshoulder (I − JCoMx JCoMx )

(12)

where

68


3.2

Control with constraints

In practice, the motions of the articulations are normally limited by some physical constraints such as the max acceleration of the motors, the maximum angular velocity of the motors, among others. Thus, a proper control scheme must respect these constraints in order to protect the robot. In the present section the quadratics least square algorithm (noted QP) is employed to calculate the optimized solution for the control that respects the physical constraints. The common form of the QP algorithm is written as [5]:

{ Find θ˙ ∗ ∈ Arg min J · θ˙ − y˙ (13) s.t in which θ˙ ∗ is the solution to be determined and s.t are the constraints. To find the solution is equivalent to minimizing the objective function as follows. 3.2.1

Typical control

When the height of the shoulder is commanded, the objective function is

min Jshoulder θ˙ − y˙ 2 The corresponding QP expression is constructed as the set below

{

˙ min Jshoulder θ − y˙ 2 Sub ject to θ˙ ≤ θ˙max 3.2.2

(14)

(15)

Hierarchical redundant control

Even though the QP algorithm can calculate the solution for a sub-determine system, it can only satisfy one of these objectives each time. If a hierarchical control is demanded, the QP calculates twice for the two objectives as the result includes two independent solutions [6]. Thus, it is necessary to determine the relation between these two objectives. As shown in Eq. (6), the vector Z is uncertain and used as the objective function. Refer to Eq. (10) and Eq. (11), the vector Z can be written as + ˙ Z = J˜shoulder (y˙shoulder − Jshoulder · JCoMx · CoMx)

(16)

The expression of Z can be simplified to render the following QP expression of the hierarchical control,  ˙ − JCoMx θ˙shoulder )∥2  min ∥JCoMx θ˙CoMx − (CoMx min θshoulder < θshoulder < max θshoulder (17)  min θCoMx < θCoMx < max θCoMx where θshoulder and θCoMx are the solutions that abide to the constraints of the height of shoulder and the center of gravity.

4 Simulations and Experimental Results In this section, the experimental results for the four different control cases are presented. MATLAB is used as the platform for building the bipedal robot model and running the simulations. Moreover, the convex optimization tool CVX is used to calculate the resolution of the QP expressions [7]. The model is initially set up using the following conditions: • The initial posture of the robot is sitting and it takes its arms and lines them up. In this case, the θInitial are [π /2, 0, π /2, π ] (rad). • The length of each branch [l1 , l2 , l3 , l4 ] is given, respectively, as [0.45, 0.65, 0.7, 0.7] (m).

T

T

T

T

69


• The distribution of the center of gravity [c1 , c2 , c3 , c4 ] is defined at [0.175, 0.35, 0.35, 0.35] (m). • The mass of each branch [m1 , m2 , m3 , m4 ] is [15, 25, 35, 15] (kg). • Under CoMx 4.1

ʌ

ʌ ʌ

ʌʌ

T

ʌ

T

these conditions, the initial height of shoulder is calculated to be 1.15m and the initial of is 0.4m.

Control without constraints

1.5m is chosen as the desired height of the shoulder. For the case of typical control, the evolution of the shoulder position on y-axis is given in Fig. 2. The figure shows that the shoulder successfully reaches the desired height after 1s and the surpassing height reaches to 1.55m. Furthermore, with the evolution Tthat only the θ2 works and the of the articular commands seen in Fig. 2(b), it is a simple movement T angular velocity is 0.5rad/s.

(a) (a)

(b) (b)

Fig. 2 Evolution of the shoulder position and angular velocity of the 4 articulations that correspond to typical control without constraints

For the case of the hierarchical redundant control, the results of the evolution of the center of gravity and the position of the shoulder are given in Fig. 3.

(a) (a)

(b) (b)

Fig. 3 Evolution of the center position and shoulder position that correspond to hierarchical redundant control without constraints

Fig. 3(a) shows that at first the position of CoMx moves toward the negative orientation of the Xaxis and then returns to position 0 after 1s. The largest surpassing value of CoMx is -0.1m. Comparing Fig. 3(b) to Fig. 2 one sees that the times needed to reach the desired locations are nearly the same.

T

T T T

TT T

T

T

T

70


(b) (b)

(a) (a)

Fig. 4 Evolution of angular velocity of the 4 articulations and shoulder position that correspond to typical control with constraints

The surpassing height in Fig. 3(b) is 1.525m which is less than its counterpart in Fig 2. The time to reach the highest point one the other hand is longer in Fig. 3(b). The evolution of the shoulder position is more sliding in the hierarchical control. 4.2

Control with constraints

For the typical control, the desired height is always set to be 1.5m and the limitations of the angular velocity θ˙i is specified to be [-0.5rad/s, 0.5rad/s]. The result in Fig. 4 shows the evolution of the angular velocity and the height variation of the shoulder. Fig. 4(a) shows that the shoulder reaches the desired location in 1s while the evolution in Fig. 4(b) indicates that θ1 , θ2 and θ3 vary while θ4 stays standstill. The angular velocity respects the constraints and has a saturation between -0.5rad/s and 0.5rad/s. The objective function evidently works. Nevertheless, θ˙1 and θ˙2 vary rather abruptly between a maximum and a minimum values - a scenario that is physically harmful to the proper operation of the motors.

T

(a) (a)

T

(b) (b)

Fig. 5 Comparison of the center position and the shoulder height between the QP hierarchical redundant control and typical hierarchical redundant control

An interval of [-0.2rad/s, 0.2rad/s] is then chosen for the hierarchical redundant control case. The T T comparisons of the center of gravity and the height, with and without considering constraints are shown in Fig. 5. It is evident that the maximum surpassing point of the center position for the QP method is less than that of the typical method, while the maximum surpassing points of the height are opposite.


71

Fig. 6 Comparison of the angular velocities between the QP hierarchical redundant

Fig. 6 Comparison of the angular velocities between the QP hierarchical redundant control and typical hierarchical redundant control

This indicates that the robot motion under the QP control is safer. However, more time is needed to reach the desired height of the shoulder. In Fig.6, the variation of the angular velocity under the QP method is less smooth than the case without constraints. The angular velocity θ4 as seen in Fig. 6 no longer stays at 0 but is saturated at between -0.2rad/s and 0.2rad/s. This is a rather interesting observation.

5 Concluding Remarks A simple viable control law was considered for the control of a bipedal robot. It was shown to be effective in optimizing the control of the system when it is redundant. Simple transition of posture from being sitting to being standing subject to particular physical constraints were considered numerically using several different experiments. The simulation result of the case of the typical control without constraint showed that the robot utilized the least amount of joints to meet the objective. This was a case of the simplest movement. In the case of hierarchical redundant control with constraints, the robot was seen to move more smoothly. This was particularly so with the variation of the center position, thus promising the stable motion and safety operation of the robot. The logical next step would be to validate the control law design on a physical bipedal robot of the identical configuration by evaluating against the virtual model. Once the control design demonstrates satisfactory performance, more complex posture plannings will be researched and simulated.

References [1] Bao, Z.J., Ma, P.S., Jiang S., et al. (1999). The problem and the research history from biped robot to humanoid robot, ROBOT, 21(4), 312-320. [2] Pu C.J., Wang, Y.J., Li, Z.Z., et al. (2008). A method for trajectory planning in the stability of getting up for biped walking robots, Journal of Southwest University (Natural Science Edition), 30(11), 125-130. [3] Sok, K. W., Kim, M., and Lee, J. (2007). Simulating biped behaviors from human motion data, ACM Transactions on Graphics, 26(3), 107. [4] Hu, L.Y. and Sun, Z.Q. (2005). Survey on gait control strategies for biped robot, Journal of Computer Research and Development, 42(5), 728-733. [5] Ben-Tal, A. and Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(23), 769-805. [6] Megretski, A. and Rantzer, A. (1997). System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, 42(6), 819-830. [7] Grant, M.C and Boyd S.P (2007). CVX: A system for disciplined convex programming, Global Optimization, 2007, 155-210.



Time-delay effects on periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator Albert C.J. Luo †, Siyuan Xing Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805, USA Submission Info Communicated by S. Lenci Received 5 February 2017 Accepted 2 March 2017 Available online 1 April 2017 Keywords Time-delayed Duffing oscillator nonlinear frequency-amplitudes Bifurcation tree Implicit mapping Mapping structures Period-1 motions to chaos

Abstract In this paper, time-delay effects on periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are discussed. One often considers the time-delay interval is very small compared to the oscillation period. In engineering application, the time-delay interval is often very large. Bifurcation trees of periodic motions to chaos varying with time-delay are presented for such a time-delayed, Duffing oscillator. Using the finite discrete Fourier series, harmonic amplitude varying with time-delay for stable and unstable solutions of period-1 to period-4 motions are developed. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, time-delay effects on period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In Luo and Xing [1,2], consider a time-delayed Duffing oscillator as x¨ + δ x˙ + α1 x − α2 xτ + β x3 = Q0 cos Ωt

(1)

where x = x(t) and xτ = x(t − τ ). The constants (δ , α1 , α2 and β ) are the damping coefficient, linear stiffness, linear displacement time-delay term coefficient, and nonlinear term coefficient, respectively. Ω and Q0 are excitation frequency and amplitude, respectively. In Luo and Xing [1], period-1 motions experiencing the complicated trajectories were presented through the semi-analytical method, such period-1 motions cannot be obtained from the traditional perturbation analysis. Further, Luo and Xing [2] analytically predicted the bifurcation trees of period-1 motions to chaos. In the previous papers, the time-delay is specifically chosen. When one investigated the time-delay system, the timedelay term is approximated as xτ = x(t − τ ) ≈ x(t) − x(t) ˙ τ (2) † Corresponding

author. Email address: [email protected], [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2017.03.006

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A.C.J. Luo, S. Xing/Journal of Vibration Testing and System Dynamics 1(1) (2017) 73–91

If the time-delay is very small, such an approximation in the dime-delay systems can be used as an approximate study of time-delay systems. If the time-delay becomes large, such an approximation cannot be used to investigate time-delay systems. Therefore, in this paper, without such an approximation of time-delay terms, periodic motions in the time-delayed Duffing oscillator will be investigated. In recent years, periodic motions in time-delayed dynamical systems were investigated. One employed the perturbation method to investigate the approximate solutions of periodic motion in the time-delayed nonlinear oscillators (e.g., Hu et al [3], Wang and Hu [4]). In addition, the traditional harmonic balance method was also used for approximate solutions of periodic motions in delayed nonlinear oscillators (e.g., MacDonald [5]; Liu and Kalmar-Nagy [6]; Lueng and Guo [7]). However, such approximate solutions of periodic motions in the time-delayed oscillators are based on one or two harmonic terms without enough accuracy. In 2013, Luo [8] presented the generalized harmonic balance method for periodic motions in time-delayed, nonlinear dynamical systems. Luo and Jin [9] applied such a generalized harmonic balance method to investigate the time-delayed, quadratic nonlinear oscillator, and the analytical bifurcation trees of period-1 motions to chaos were obtained. Luo and Jin [10] used such a generalized harmonic balance method to find complex period-1 motions of the periodically forced Duffing oscillator with a time-delayed displacement, which cannot be obtained from the traditional harmonic balance and perturbation methods. Furthermore, Luo and Jin [11] analytically determined the bifurcation trees of the period-1 motions to chaos in the time-delayed Duffing oscillator, and complex period-m motions were observed in such a time-delayed Duffing oscillator. Since the generalized harmonic method cannot be used the nonlinear dynamical systems with non-polynomial nonlinearity. In 2015, Luo [12] developed a semi-analytical method to determine periodic motions in nonlinear dynamical systems through discrete implicit maps. The discrete maps were obtained by the discretization of differential equations. From a specific mapping structure, periodic motions in the nonlinear dynamical systems can be determined, and from such solutions of discrete nodes, the frequency-amplitudes can be determined. Luo and Guo [13] applied such a semi-analytical method to investigate bifurcation trees of periodic motions to chaos in the Duffing oscillator. In addition, the semi-analytical method in Luo [12] was also developed for the time-delay systems. Luo and Xing [1,2] used such a method for complicated period-1 motions and the corresponding bifurcation trees to chaos. Herein, the time-delay effects on periodic motions in the time-delayed Duffing oscillator will be discussed. In this paper, time-delay effects on bifurcation trees of period-1 motions to chaos in the timedelayed, hardening Duffing oscillator will be investigated. The semi-analytical method will be employed to determine periodic motions. The discrete implicit maps will be obtained by discretization of the corresponding differential equations of the time-delayed dynamical systems. Periodic motions varying with time-delay in the time-delayed Duffing oscillator will be presented, and the corresponding stability and bifurcation analysis of periodic motions will be completed. From the finite Fourier series, nonlinear harmonic amplitude varying with time-delay will be discussed. Numerical illustrations of periodic motions will be performed to demonstrate time-delay effects on periodic motions.

2 Method From Luo [12], a period-m flow in a time-delayed, nonlinear dynamical system can be described through mN discrete nodes for period-mT . To determine a period-m motion in the time-delay dynamical systems, the following theorem is presented herein for a better reference. Theorem 1. Consider a time-delay nonlinear dynamical system x˙ = f(x, xτ ,t, p) ∈ R n , with x(t0 ) = x0 , x(t) = Φ (x0 ,t − t0 , p) for t ∈ [t0 − τ , ∞).

(3)

75


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 fields f(x(m) (tk ), xτ (m) (tk ),tk , p) are exact. Suppose discrete (m) τ (m) (m) nodes xk and xk are on the approximate solution of the periodic flow under ||x(m) (tk ) − xk || ≤ εk τ (m) 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

(4) (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)

(xk , xk τ (m) xj

=

τ (m)

(m)

(m) (m) h j (xr j −1 , xr j , θr j ), j

τ (m)

(e.g., xr

(m)

τ (m)

(m)

τ (m)

)=Pk (xk−1 , xk−1 ) with gk (xk−1 , xk ; xk−1 , xk

(m)

, p) = 0,

= k, k − 1; r j = j − l j , k = 1, 2, · · · , mN;

(m)

)

(5)

lr j

1 [τ − ∑ hr j +i ]). hr j i=1

(m)

= xsr + θr (xrr −1 − xrr ), θr =

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 ; τ (m)

(m)

τ (m)

(m)

with Pk : (xk−1 , xk−1 ) → (xk , xk (m)

(m)

τ (m)

For xmN = P(x0 , x0

(m)∗

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

(m)∗

gk (xk−1 , xk τ (m)∗

xj

(m)∗

τ (m)∗

τ (m)∗

; xk−1 , xk

(m)∗

τ (m)∗

, xk

, p) = 0,

(m)∗

 

(m)∗

(m)∗

(m)∗

(m)∗

τ (m)∗

= xmN and x0

(6)

) (k = 0, 1, · · · , mN) computed by

= h j (xr j −1 , xr j , θr j ), j = k, k − 1

xr j −1 = x mod (r j −1+mN,mN) , xr j x0

)(k = 1, 2, · · · , mN).

(k = 1, 2, · · · , mN) (7)

(m)∗

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

τ (m)∗

= xmN .

τ (m)∗

Then the points xk and xk (k = 0, 1, · · · , mN) are the approximation of points x(m) (tk ) and xτ (m) (tk ) (m)∗ τ (m)∗ (m) (m)∗ (m) τ (m) of periodic solutions. In the neighborhoods of xk and 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)

(

τ (m)

∂xj

(m) ∆xr j + (m) ∂ xr j

∂xj

(m) ∂ xr j −1

(m)

∆xr j −1 ) = 0

(8)

with r j = j − l j , j = k − 1, k; (k = 1, 2, · · · , mN). The resultant Jacobian matrices of the periodic flow are [ (m) ] ∂ yk = Ak Ak−1 · · · A1 (k = 1, 2, · · · , mN), DPk(k−1)...1 = (m) ∂ y0 (y(m)∗ ,··· ,y(m)∗ ) 0 [ k (m) ] ∂ ymN and DP = DPmN(mN−1)...1 = = AmN AmN−1 · · · A1 . (m) ∂ y0 (y(m)∗ ,··· ,y(m)∗ ) 0

mN

(9)

76


[

where (m) ∆yk

[

and (m) Ak (m) Bk

=

=

(m) (m) (m) Ak ∆yk−1 , Ak

(m)

Bk

(m)

=

] (10)

(m)

∂ yk−1

(m)∗ (m)∗ (yk−1 ,yk )

]

(m)

(ak(rk−1 −1) )

, s = 1 + lk−1

n×n

(m)

Ik

≡

(m)

∂ yk

0k

n(s+1)×n(s+1)

(m) [(ak(k−1) )n×n , 0n×n , · · ·

(m)

, (ak(rk −1) )n×n ],

(11)

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

= (∆xk , ∆xk−1 , · · · , ∆xrk−1 )T ,

(m)

(m)

(m)

(12)

(m)

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

ak j = [ (m)

akr j = [ (m)

∂ gk (m) ∂ xk

]−1

∂ gk (m)

∂xj

, τ (m)

∂ gk

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

ak(r j −1) = [

∂ gk (m) ∂ xk

]−1

j

∑

∂ gk

τ (m) α = j−1 ∂ xα

(13)

τ (m) ∂ xα

∂ xr j −1

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). (14) The eigenvalues of DP for such a periodic flow are determined by |DP − λ In(s+1)×n(s+1) | = 0,

(15)

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 ]).

(16)

(i) If the magnitudes of all eigenvalues of PD are less than one (i.e., |λi | < 1, i = 1, 2, · · · , n(s + 1)), the approximate periodic solution is stable. (ii) If at least the magnitude of one eigenvalue of DP is greater than one (i.e., |λi | > 1, i ∈ {1, 2, · · · , 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 [12].


77

3 Discretization of dynamical systems Let x = (x, y)T and xτ = (xτ , yτ )T . For discrete time tk = kh (k = 0, 1, 2, ...), xk = (xk , yk )T and xτk = (xkτ , yτk )T . Using a midpoint scheme for the time interval t ∈ [tk−1 ,tk ] (k = 1, 2, ...), the foregoing differential equation in Eq.(1) is discretized to form an implicit map Pk : Pk : (xk−1 , xτk−1 ) → (xk , xτk )

(17)

⇒ (xk , xτk ) = Pk (xk−1 , xτk−1 ).

In other words, the implicit map can be expressed as 1 xk =xk−1 + h(yk + yk−1 ) 2 h 1 (18) yk =yk−1 + h[Q0 cos Ω(tk−1 + ) − δ (yk + yk−1 ) 2 2 1 1 1 τ − α1 (xk + xk−1 ) + α2 (xkτ + xk−1 ) − β (xk + xk−1 )3 ]. 2 2 8 τ The time-delay node xk ≈ x(tk−τ ) of xk ≈ x(tk ) lies between xk−lk and (xk−lk −1 ). The time-delay node is determined by an interpolation function of two points xk−lk and xk−lk −1 . For a time-delay node xτj ( j = k − 1, k), the following relation can be hold. xτj = h j (xr j −1 , xr j , θr j )forr j = j − l j .

(19)

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 (20) τ yτj = y j−l j −1 + (1 − + l j )(y j−l j − y j−l j −1 ). h Thus, the time-delay nodes are expressed by non-time-delay nodes. The discretization of differential equation for the time-delayed, hardening Duffing oscillator is completed. xτj = x j−l j −1 + (1 −

4 Period-m motions A period-m motion in the time-delayed, hardening Duffing oscillator is described by the following discrete mapping structure: (m)

τ (m)

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

(m)

τ (m)

) → (xmN , xmN ) (21)

mN−actions (m) τ (m) = P(x0 , x0 )

with

τ (m)

(m)

(m)

τ (m)

Pk : (xk−1 , xk−1 ) → (xk , xk

)

(k = 1, 2, · · · , mN). (m)∗

The points xk determined by

(22)

(k = 1, 2, · · · , mN) on the period-m motion for the time-delayed Duffing oscillator are (m)∗

(m)∗

τ (m)∗

τ (m)∗

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

x0

(m)∗

τ (m)∗

= xmN and x0

τ (m)∗

= xmN .

} (k = 1, 2, · · · , mN)

(23)

78


With gk = (gk1 , gk2 )T , the algebraic equations for period-m motion are (m)

(m)

(m)

(m)

(m)

(m)

gk1 = xk − [xk−1 + 21 h(yk + yk−1 )] = 0, (m)

(m)

gk2 = yk − {yk−1 + h[Q0 cos Ω(tk−1 + 12 h) − 12 δ (yk + yk−1 ) τ (m)

(m)

(m)

− 21 α1 (xk + xk−1 ) + 12 α2 (xk

τ (m)

(m)

(m)

+ xk−1 ) − 81 β (xk + xk−1 )3 ]}

(24)

=0 (k = 1, 2, · · · , mN) τ (m)

Time-delay node x j

( j = k, k − 1) are from Eq.(20)

τ (m) (m) + l j )(xk−l j − xk−l j −1 ), h τ τ (m) (m) (m) (m) y j = yk−l j −1 + (1 − + l j )(yk−l j − yk−l j −1 ). h τ (m)

(m)

= xk−l j −1 + (1 −

xj

(25)

From Eqs.(23) – (25), discrete nodes of periodic motions in the time-delayed Duffing oscillator are (m)∗ obtained by the 2(mN + 1) equations. If the discrete nodes xk (k = 1, 2, . . . , mN) of the period-m motion is achieved, the corresponding stability of the period-m motion is discussed by the eigenvalue analysis of the Jacobian matrix of the mapping structure based on the corresponding discrete nodes. (m)∗ τ (m)∗ (m) (m)∗ (m) τ (m) τ (m)∗ τ (m) and xk For xk , xk = xk + ∆xk and xk = xk + ∆xk . The linearized equation of implicit mapping is k

∑

∂ gk

(m) ∆x j + (m) j=k−1 ∂ x j

τ (m)

∂ gk τ (m) ∂xj

(

τ (m)

∂xj

∂xj

τ (m) ∆xr j + (m) ∂ xr j

(m)

(m) ∂ xr j −1

∆xr j −1 ) = 0

(26)

withr j = j − l j , j = k − 1, k; (k = 1, 2, . . . , mN). Define (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)

(27)

(m)

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

(m)

(m)

∆yk−1 = (∆xk−1 , ∆xk−2 , . . . , ∆xrk−1 −1 )T . The resultant Jacobian matrix of the period-m motion is [ DP = DPmN(mN−1)···1 = =

(m) (m) (m) AmN AmN−1 · · · A1

(m)

∂ ymN

(m)

∂ y0

[ =

(m)∗

(y0

(m)∗

,y1

(m)∗

, ··· ,yN

)

(28)

= A(m)

where (m) ∆yk

]

(m) (m) (m) Ak ∆yk−1 , Ak

≡

(m)

∂ yk

] (29)

(m)

∂ yk−1

(m)∗

(m)∗

(yk−1 ,yk

)


and (m)

ak j = [ (m)

akr j = [

∂ gk

−1 ∂ gk ] , (m) (m) ∂ xk ∂xj

τ (m)

∂ gk

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

(m)

ak(r j −1) = [

τ (m)

∂ gk

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

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

Bk

(m)

(m)

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

(m)

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

0k

(30)

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

(m)

Ik

79

s

where

[ ] ] ∂ gk −1 − 21 h 1 − 12 h = = , , (m) ∆ 12 hδ − 1 ∂ x(m) ∆ 12 hδ + 1 ∂ xk−1 k [ ] [ ] τ (m) τ (m) ∂xj ∂xj 0 0 0 0 = , = , (m) ( τh − l j ) 0 ∂ x(m) (1 − τh + l j ) 0 ∂ xr j −1 rj [ ] ∂ gk 00 , = (m) 0 − 21 hα2 ∂xj

∂ gk

[

(31)

∆ = 18 h[4α1 + 3β (xk + xk−1 )2 ]. The eigenvalues of DP for the period-m motion in the time-delayed Duffing oscillator are computed by |DP − λ I2(s+1)×2(s+1) | = 0.

(32)

(i) If the magnitudes of all eigenvalues of PD are less than one (i.e., |λi | < 1, i = 1, 2, . . . , n(s + 1)), the approximate periodic solution is stable. (ii) If at least the magnitude of one eigenvalue of DP is greater than one (i.e., |λi | > 1, i ∈ {1, 2, . . . , 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. The bifurcation conditions are given as follows. (i) If λi = 1 with |λ j | < 1 ( i, j ∈ {1, 2, . . . , 2(s + 1)} and i ̸= j), the saddle-node bifurcation (SN) occurs. (ii) If λi = −1 with |λ j | < 1 ( i, j ∈ {1, 2, . . . , 2(s + 1)} and i ̸= j), the period-doubling bifurcation (PD) occurs. (iii) If |λi, j | = 1 with |λl | < 1 (i, j, l ∈ {1, 2, . . . , 2(s + 1)} and λi = λ¯ j , l ̸= i, j), Neimark bifurcation (NB) occurs.

80


5 Bifurcation trees varying time-delay In this section, the complete bifurcation tree of period-1 motion to chaos for the periodically forced, time-delayed, damped, Duffing oscillator will be presented through the analytical predictions of period1 to period-4 motions. Periodic motions for such time-delayed system will be illustrated. Consider a set of system parameters as

δ = 0.5, α1 = 10.0, α2 = 5.0, β = 10, Ω = 1.8, Q0 = 200.0 (33) and where T = 2π /Ω. The bifurcation trees of period-1 to period-4 motions varying with time-delay in the time-delayed Duffing oscillator are predicted analytically through the semi-analytical method. The discrete nodes are analytically determined by the implicit mapping. The bifurcation trees varying with time-delay are illustrated by displacement and velocity of the periodic nodes with mod (k, N) = 0, as shown in Fig.1. 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 period2 motions appear from the period-doubling bifurcations of the period-1 motions, and period-4 motion appear from the period-doubling bifurcation of the period-2 motion. The global view of the bifurcation trees is presented in Fig.1(a) and (b) for τ /T ∈ [0, 1] and the zoomed views of the bifurcation trees are presented for a specific frequency ranges in Figs.1(c)-(f). The ranges of time-delay for the stable and unstable periodic motions are listed in Table 1, and the bifurcation points are tabulated in Table 2. In Fig.1(a)-(b), the global view of the bifurcation trees of period-1 to period-4 motion are presented for τ /T ∈ [0, 1]. For τ = 0, the time-delay Duffing oscillator become the non-time-delayed Duffing oscillator. For τ /T = 1, the time-delay is the same as the excitation period. There is an unstable period1 motion for the entire range of τ /T ∈ [0, 1]. For τ /T ∈ [0, 0.0074), there is a period-2 motion in the first bifurcation tree. At τ /T ≈ 0.0074, the saddle-node bifurcation of the period-2 motion exists, and the period-2 motion vanishes, and the period-1 motion from unstable to stable. In other words, the perioddoubling bifurcation occurs at τ /T ≈ 0.0074. The period-1 motion is stable for τ /T ∈ (0.0074, 0.0645). For τ /T ∈ (0.0645, 0.1418), there is the second bifurcation tree from period-1 to period-4 motions are presented as shown in Fig.1(c) and(d). The period-doubling bifurcation of period-1 motions are at τ /T ≈ 0.0645, 0.1418. The period-2 motion exists in τ /T ∈ (0.0645, 0.1418). At τ /T ≈ 0.0762, 0.1298, the period-doublings of period-2 motions occur, and the onset of period-4 motions takes place. In the range of τ /T ∈ (0.0762, 0.1298), the period-4 motion exists and the corresponding period-doubling bifurcations occur at τ /T ≈ 0.0794, 0.1266. The unstable period-4 motion lies in τ /T ∈ (0.0794, 0.1266) where the period-8 motion exists. For the third branch, the period-doubling bifurcations of the period-1 motion are atτ /T ≈ 0.2278, 0.2765. Only the stable period-2 motion exists at τ /T ∈ (0.2278, 0.2765). Similarly, for the fourth branch, the period-doubling bifurcations of the period-1 motion are at τ /T ≈ 0.6824, 0.7334. Only the stable period-2 motion exists at τ /T ∈ (0.6824, 0.7334). For τ /T ∈ (0.8213, 0.8853), there is the fifth bifurcation tree from period-1 to period-4 motions are presented, as shown in Fig.1(e) and(f). The period-doubling bifurcation of period-1 motions are at τ /T ≈ 0.8213, 0.8853. The period2 motion exists in τ /T ∈ (0.8213, 0.8853). At τ /T ≈ 0.8331, 0.8738, the period-doublings of period-2 motions occur for onsets of period-4 motion. For τ /T ∈ (0.8331, 0.8738), the period-4 motion exists and the corresponding period-doubling bifurcation occurs at τ /T ≈ 0.8368, 0.8706. The unstable period-4 motion lies in τ /T ∈ (0.8368, 0.8706) where the period-8 motion also exists.


(i)

(ii)

(iii)

(iv)

(v)

81

(vi)

Fig. 1: Bifurcation trees of period-1 motions to chaos varying with time-delay: (i) periodic node displacement, (ii) periodic node velocity. The first zoomed view: (iii) periodic node displacement, (iv) periodic node velocity, the second zoomed view: (v) periodic node displacement, (vi) periodic node velocity.(α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8)

6 Harmonic amplitudes and phases (m)

(m)

(m)

Consider the predicted nodes of period-m motions as xk = (xk , yk )T for k = 0, 1, 2, . . . , mN in the timedelayed, hardening Duffing oscillator. The approximate expression of a period-m motion is determined by the finite Fourier series as

82


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

(34)

Table 1: Bifurcations for periodic motions.(α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8)

Over all range

τ /T (stable)

τ /T (unstable)

Motion type

(0.0074,0.0645)

(0.0,0.0074)

P-1

(0.1418,0.2278)

(0.0645,0.1418)

(0.2765,0.6284)

(0.2278,0.2765)

(0.7334,0.8213)

(0.6824,0.7334)

(0.8853,0.9869)

(0.8213,0.8853) (0.9869,1.0)

1st branch

(0.0,0.0074)

—

P-2

2nd branch

(0.0645,0.0762)

(0.0762,0.1298)

P-2

(0.0794,0.1266)

P-4

(0.1298,0.1418) (0.0762,0.0794) (0.1266,0.1298) 3rd branch

(0.2278,0.2765)

—

P-2

4th branch

(0.6824,0.7334)

—

P-2

5th branch

(0.8213,0.8331)

(0.8331,0.8738)

P-2

(0.8368,0.8706)

P-4

—

Quasi-periodic

(0.8738,0.8853) (0.8331,0.8368) (0.8706,0.8738) 6th branch

(0.9869,1.0)

Table 2: Bifurcations for symmetric period-1 motions. (α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8) τ /T

Bifurcations

Motion type

0.0074

PD

For P-2

2nd branch

0.0645, 0.1418

PD

For P-2

(0.0645,0.1418)

0.0762, 0.1298

PD

For P-4

0.0794, 0.1266

PD

For P-8

0.2278, 0.2765

PD

For P-2

0.6824, 0.7334

PD

For P-2

5th branch

0.8213, 0.8853

PD

For P-2

(0.8213,0.8853)

0.8331, 0.8738

PD

For P-4

0.8368, 0.8706

PD

For P-8

0.9869

NB

Quasi-periodic

1st branch (0.0,0.0074)

3rd branch (0.2278,0.2765) 4th branch (0.6824,0.7334)

6th branch (0.9869,1.0)


83

(m)

There are (2M + 1) unknown vector coefficients of a0 , b j/m , c j/m ( j = 1, 2, . . . , M). From the given (m)

nodes xk (k = 0, 1, 2, . . . , mN), such unknowns (2M + 1 ≤ mN + 1) can be determined. The predicted (m) nodes xk on the period-m motion is expressed by the finite Fourier series as for tk ∈ [0, mT ] (m)

x(m) (tk ) ≡ xk

mN/2

(m)

= a0 +

∑

j=1 (m)

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

mN/2

= a0 +

∑

b j/m cos(

j=1

j 2kπ j 2kπ ) + c j/m sin( ) m N m N

(35)

(k = 0, 1, . . . , mN − 1) where

2kπ 2π = 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=

(36)

From discrete nodes on the period-m motion, equation (35) gives (m)

a0 =

1 mN−1 (m) ∑ xk , N k=0

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

(37)

Thus the approximate expression for period-m motion in Eq.(1) is determined by (m)

x

(t) ≈

(m) a0 +

mN/2

∑

j=1

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

(38)

The foregoing equation can be rewritten as {

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

{

}

(m)

}

x1 (t) (m) x2 (t)

≡

{ ≈

(m)

a01 (m) a02

}

    j   mN/2  A  j/m1 cos( Ωt − φ j/m1 ) m + ∑ j   A  j=1  j/m1 cos( Ωt − φ j/m2 )  m

(39)

where the harmonic amplitudes and harmonic phases for period-m motion are √

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

b2j/m1 + c2j/m1 ,φ j/m1 = arctan

(40)

For simplicity, harmonic amplitudes of displacement x(m) (t) for period-m motions will be presented only. Thus the displacement can be expressed as x

(m)

(t) ≈

(m) a0 +

mN/2

∑

j=1

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

(41)

84


and (m)

x(m) (t) ≈ a0 +

mN/2

∑

j=1

where

√ A j/m =

j A j/m cos( Ωt − φ j/m ) m

b2j/m + c2j/m ,φ j/m = arctan

c j/m . b j/m

(42)

(43)

To determine time-delay effects on periodic motion in the time-delayed Duffing oscillator, harmonic amplitudes varying with time-delay are presented, as shown in Fig.2. The selected harmonic amplitudes (m) are constant term a0 (m = 1, 2, 3, 4) and harmonic amplitudes Ak/m (m = 4, k = 1, 2, 3, 4, 8, 12, 280, 281, · · · , 284). The saddle-node, period-doubling and Neimark bifurcations of period-m motions (m = 1, 2, 3, 4) are listed in Table 2. The constant terms versus excitation frequency is presented in Fig.2(i). For unstable symmetric (m) (1) period-1 motion, a0 = a0 = 0. For a pair of asymmetric period-m motions, a0 ̸= 0. The bifurcation (m) tree is clearly observed and a0 ̸= 0 is off the origin points of displacements. The maximum values (m) of centers are around a0 ≈ ±0.2. The asymmetric period-1 to period-2 motion, and the period-2 to period-4 motion are observed. In the six bifurcation trees, two of them possess more complete bifurcation trees but the three of them possesses from asymmetric period-1 to period-2 motions. One of the asymmetric period-1 motion goes to the quasi-periodic motion. To observe the multiple bifurcation trees in τ /T ∈ [0, 1], harmonic amplitude A1/4 varying with time-delay is presented in Fig.2(ii) for period-4 motions. For period-1 and period-2 motions, A1/4 = 0. For the two branches, both stable and unstable period-4 motions exist. Because of period-doubling bifurcations, the period-8 motion will be developed on such two branches. The maximum quantity level of A1/4 is about A1/4 ∼ 0.06. Harmonic amplitude A1/2 varying with time-delay is presented in Fig.2(iii) for period-2 and period-4 motions. For period-1 motions, A1/2 = 0. The five branches of bifurcation trees have period-2 motions. The maximum quantity level of harmonic amplitude A1/2 is about A1/2 ∼ 0.075. Similarly, harmonic amplitude A3/4 versus time-delay is presented in Fig.2(iv) 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 ∼ 4 × 10−3 . Harmonic amplitude A1 varying with time-delay is presented in Fig. 2(v) for period-1 to period-4 motions plus unstable symmetric period-1 motion. The quantity of the primary harmonic amplitude is about A1 ∈ (2.7, 2.9). To avoid abundant illustrations, the main primary harmonic amplitudes are presented herein. Thus, harmonic amplitude A2 varying with time-delay is presented in Fig.2(vi) for period-1 to period-4 motions. For symmetric period-1 motions, A2 = 0. Asymmetric period-1 motions and the corresponding period-2 and period-4 motions on the bifurcation trees are presented. The quantity of harmonic amplitude A2 . is about A2 ∈ (0.15, 0.35). The six bifurcation branches are observed, and the corresponding bifurcations are labeled. Compared to the primary harmonic amplitude A1 , the quantity level of the second primary harmonic amplitude drops dramatically. Harmonic amplitude A3 varying with timedelay is presented in Fig. 2(vii) for period-1 to period-4 motions, which are similar to the primary harmonic amplitude A3 . The quantity of the primary harmonic amplitude is about A3 ∈ (0.35, 0.55). To avoid abundant illustrations, one more set of harmonic amplitudes varying with time-delay are presented. Harmonic amplitude A70 varying with time-delay is presented in Fig.2(viii). The quantity level of harmonic amplitude A70 is in A70 ∈ (10−11 , 10−9 ). Harmonic amplitude A281/4 varying with time-delay is presented in Fig.2(ix) for period-4 motion, which is similar to the harmonic amplitude A1/4 . However, the quantity levels for the four bifurcation trees are different. The quantity levels of harmonic amplitudes are A281/4 ∈ (0, 10−10 ). Harmonic amplitude A141/2 varying with time-delay is presented in Fig.2(x) for period-2 and period-4 motions, which is similar to the harmonic amplitude A1/2 . The quantity levels for the two bifurcation trees are also different. The quantity levels of harmonic


85

amplitudes are A141/2 ∈ (0, 10−9 ). Harmonic amplitude A119/4 varying with time-delay is presented in Fig.2(xi) for period-4 motions. The quantity levels of harmonic amplitudes are in A283/4 ∈ (0, 10−10 ), respectively. Harmonic amplitude A71 varying with time-delay is presented in Fig.2(xii), similar to the harmonic amplitude A1 . The quantity levels of harmonic amplitudes are A71 ∈ (10−12 , 2 × 10−10 ). (m)L (m)R L However, the harmonic phases are different. a0 = −a0 (m = 2l , l = 0, 1, 2, · · · ), and φk/2 l = mod R + ((k + 2r)/2l + 1)π , 2π ) (r = 0, 1, . . . , 2l − 1; r = 0, 1, . . . , 2l − 1) for t = rT . (φk/2 0 l

(i)

(ii)

(iii)

(iv)

(v)

(vi) (m)

Fig. 2: Harmonic amplitudes varying with time-delay: (i) a0 (m = 1, 2, 3, 4), (iii)-(xii) A j/m ( j = 1, 2, 3, 4, 8, 12, 280, 281, 282, 283, 284). (α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8)).

86


(vii)

(viii)

(ix)

(x)

(xi)

(xii)

Fig. 2: Continued.

7 Numerical simulations In this section, time-delay effects on periodic motions will be further discussed through time displacement responses, trajectory, harmonic amplitude, and phase distributions. To illustrate complexity of periodic motions in the time-delayed Duffing oscillator, 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


(i)

(ii)

(iii)

(iv)

(v)

(vi)

87

Fig. 3: Period-1 motion with large time-delay (τ = 3.2): (i) displacement (left), (ii) displacement (right); (iii) trajectory (left), (iv) trajectory (right), (v) harmonic amplitudes, (vi) harmonic phases. (I.C.: x0 = 3.617569, y0 = −1.414469(left); x0 = 3.050657, y0 = −5.592911(right)). (α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8).

effects on periodic motions. The system parameters in Eq.(33) are used. Numerical and analytical results are presented by solid curves and circular symbols, respectively. The initial time-delay is presented through blue circular symbols. The delay-initial-starting and delay-initial finishing points are ’D.I.S.’ and ’D.I.F.’, respectively. Consider asymmetric period-1, period-2 and period-4 motions on the same bifurcation tree. From the fifth branch of bifurcation tree from period-1 motion to chaos, consider an asymmetric period-

88


(i)

(ii)

(iii)

(iv)

(v)

(vi)

Fig. 4: Period-2 motion with large time-delay (τ = 3.07): (i) displacement (left), (ii) displacement (right); (iii) trajectory (left), (iv) trajectory (right), (v) harmonic amplitudes, (vi) harmonic phases. (I.C.: x0 = 3.534165, y0 = 0.147268 (left); x0 = 2.911555, y0 = −5.503532(right)). (α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8).

1 motion with τ = 3.2, and the initial conditions are computed for a pair of the two asymmetric period-1 motions from the analytical prediction (i.e., x0 = 1.772412, y0 = 9.408279(left); x0 = 2.545040, y0 = 6.247176(right)). τ /T ≈ 0.916732 implies that the time-delay takes about 92% excitation period. The time-delay is very large, which cannot be treated in Eq. (2). The initial time-delay is presented by green symbols. The numerical solution of the period-1 motion is presented by solid curves and the analytical prediction is depicted by red and black symbols. Displacements for the two asymmetric


(i)

(ii)

(iii)

(iv)

(v)

(vi)

89

Fig. 5: Period-4 motion with large time-delay (τ = 3.043): (i) displacement (left), (ii) displacement (right); (iii) trajectory (left), (iv) trajectory (right), (v) harmonic amplitudes, (vi) harmonic phases. (I.C.: x0 = 3.556379, y0 = 0.844449(left); x0 = 2.811085, y0 = −4.859946(right)). (α1 = 10.0, α2 = 5.0, β = 10.0, δ = 0.5, Q0 = 200, Ω = 1.8).

period-1 motions are presented in Fig.3(i) and (ii). The two trajectories with time-delay for the two period-1 motions are shown in Fig.3 (iii) and (iv). The trajectories have two small cycles on both sides plus a big circle to connect small cycles on the both sides. The corresponding harmonic amplitudes and phases are presented in Fig.3(v) and (vi), respectively. A0 = aR0 = −aL0 ≈ 0.2027, A1 ≈ 2.8545, A2 ≈ 0.2294, A3 ≈ 0.4234, A4 ≈ 0.2378, A5 ≈ 0.2067, A6 ≈ 0.2500, A7 ≈ 0.7041, A8 ≈ 0.2140, A9 ≈ 0.2715, A10 ≈ 0.0228, A11 ≈ 0.0205, A12 ≈ 0.0156, A13 ≈ 0.0168, A14 ≈ 0.0254, A15 ≈ 0.0357, A16 ≈ 0.0196 and A17 ≈ 0.0120.

90


Other harmonic amplitudes lie in Ak ∈ (10−15 , 10−3 )(k = 18, 19, . . . , 100 ) and A100 ≈ 2.7643e-14 With increasing harmonic orders, the harmonic amplitudes decrease. Thus, one can use 100 harmonic terms to approximate the two asymmetric period-1 motions. The harmonic phases changes with harmonic orders from 0 to 2π with φkL = mod (φkR + π , 2π ). For an asymmetric period-2 motion with τ = 3.07, the initial conditions for a pair of two asymmetric period-2 motions are x0 = 3.534165, y0 = 0.147268 (left); and x0 = 2.911555, y0 = −5.503532(right). τ /T ≈ 0.87949 and the time-delay is also very large. The initial time-delay is still presented by green symbols. Displacements for the two asymmetric period-2 motion are presented in Fig.4(i) and (ii). The time for period-2 motion is doubled from the period-1 motion, and the period-2 motion is almost repeated the period-1 motion responses two times. The two trajectories with time-delay for the two period-2 motions are shown in Fig.4 (iii) and (iv). The trajectories have four small cycles on both sides plus two big circles to connect small cycles on the both sides. The harmonic amplitudes and phases for (2)R (2)L period-2 motions are presented in Fig.4(v) and (vi), respectively. A0 = a0 = −a0 ≈ 0.1599, A1/2 ≈ 0.0278, A1 ≈ 2.8876, A3/2 ≈ 4.9731e-3, A2 ≈ 0.1865, A5/2 ≈ 0.0278, A3 ≈ 0.4509, A7/2 ≈ 1.3442e-3, A4 ≈ 0.1992, A9/2 ≈ 0.0508, A5 ≈ 0.1549, A11/2 ≈ 5.5952e-3, A6 ≈ 0.2210, A13/2 ≈ 0.0742, A7 ≈ 0.6233, A15/2 ≈ 0.0110, A8 ≈ 0.1935, A17/2 ≈ 0.0538, A9 ≈ 0.2275, A19/2 ≈ 3.5800e-3, A10 ≈ 0.0167, A21/2 ≈ 1.1879e-3, A11 ≈ 0.0174, A23/2 ≈ 1.8651e-3, A12 ≈ 0.0113, A25/2 ≈ 2.8108e-3, A13 ≈ 0.0153, A27/2 ≈ 1.8776e-3, A14 ≈ 0.0222, A29/2 ≈ 7.2705e-3, A15 ≈ 0.0280, A31/2 ≈ 2.1347e-3, A16 ≈ 0.0148, A33/2 ≈ 4.0593e-3 and A17 ≈ 8.3640e-3. Other harmonic amplitudes lie in Ak/2 ∈ (10−15 , 10−3 ) (k = 35, 36, . . . , 184) and A92 ≈ 3.6909e-14. With increasing harmonic orders, the harmonic amplitudes also decrease. Harmonic terms Ak/2 ( mod (k, 2) ̸= 0) for period-2 motion only is much smaller than Ak/2 ( mod (k, 2) = 0). Thus, one can use 184 harmonic terms to approximate the two asymmetric period-2 motions. The harmonic phases from 0 to 2π are L = mod (φ R + ((k + 2r)/2 + 1)π , 2π ) (k = 1, 2, · · · ; r = 0) for t = rT . with φk/2 0 k/2 For asymmetric period-4 motion with τ = 3.043, the initial conditions for a pair of two asymmetric period-4 motions are x0 = 3.556379, y0 = 0.844449 (left); x0 = 2.811085, y0 = −4.859946(right). τ /T ≈ 0.871756. Displacements for the two asymmetric period-4 motion are presented in Fig.5(i),(ii). The time for period-4 motion is doubled from the period-2 motion, and the period-4 motion almost repeats the period-1 motion responses four times. The two trajectories with time-delay for the two period-4 motions are shown in Fig.5 (iii),(iv). The trajectories have eight small cycles on both sides plus four big circles to connect small cycles on the both sides. The harmonic amplitudes and phases are determined the (4)R (4)L configurations of period-4 motions, as shown in Fig.5(v),(vi), respectively. A0 = a0 = −a0 ≈ 0.1602, A1/4 ≈ 0.0165, A1/2 ≈ 0.0470, A3/4 ≈ 1.1720e-3, A1 ≈ 2.8869, A5/4 ≈ 1.0541e-3, A3/2 ≈ 9.2195e-3, A7/4 ≈ 6.9794e-3, A2 ≈ 0.1832, A9/4 ≈ 0.0120, A5/2 ≈ 0.0465, A11/4 ≈ 2.0944e-3, A3 ≈ 0.4532, A13/4 ≈ 2.1644e-3, A7/2 ≈ 1.7849e-3, A15/4 ≈ 6.6001e-3, A4 ≈ 0.1996, A17/4 ≈ 0.1996, A9/2 ≈ 0.0825, A19/4 ≈ 4.8229e-3, A5 ≈ 0.1339, A21/4 ≈ 3.7822e-3, A11/2 ≈ 7.4525e-3, A23/4 ≈ 6.6448e-3, A6 ≈ 0.2276, A25/4 ≈ 0.0191, A13/2 ≈ 0.1180, A27/4 ≈ 8.1634e-3, A7 ≈ 0.6048, A29/4 ≈ 4.4439e-3, A15/2 ≈ 0.0158, A31/4 ≈ 6.8376e-3, A8 ≈ 0.1970, A33/4 ≈ 0.0147, A17/2 ≈ 0.0847, A35/4 ≈ 5.2900e-3, A9 ≈ 0.2192, A37/4 ≈ 2.1003e-3, A19/2 ≈ 5.2590e-3, A39/4 ≈ 9.6538e-4, A10 ≈ 0.0155, A41/4 ≈ 7.6987e-4, A21/2 ≈ 2.0035e-3, A43/4 ≈ 1.9215e-4, A11 ≈ 0.0165, A45/4 ≈ 3.2691e-4, A23/2 ≈ 3.0490e-3, A47/4 ≈ 3.7773e-4, A12 ≈ 9.7068e-3, A49/4 ≈ 1.0006e-3, A25/2 ≈ 4.0860e-3, A51/4 ≈ 2.1320e-4, A13 ≈ 0.0148, A53/4 ≈ 4.5249e-4, A27/2 ≈ 2.9097e-3, A55/4 ≈ 6.5572e-4, A14 ≈ 0.0219, A57/4 ≈ 1.8797e-3, A29/2 ≈ 0.0112, A59/4 ≈ 7.1085e-4, A15 ≈ 0.0269, A61/4 ≈ 3.9270e-4, A31/2 ≈ 3.4989e-3, A63/4 ≈ 5.7719e-4, A16 ≈ 0.0138, A65/4 ≈ 1.1417e-3, A32/2 ≈ 6.0483e-3, A67/4 ≈ 3.7905e-4 and A17 ≈ 8.3640e-3. Other harmonic amplitudes lie in Ak/2 ∈ (10−15 , 10−3 ) (k = 69, 70, . . . , 368) and A92 ≈ 1.8864e-14. With increasing harmonic orders, the harmonic amplitudes also decrease. Harmonic terms Ak/4 ( mod(k, 4) ̸= 0 and mod(k, 2) ̸= 0) for period-4 motion only is much smaller than Ak/4 ( mod (k, 2) = 0 or mod (k, 4) = 0 ). Thus, one can use 368 harmonic terms to approximate the two asymmetric period-4 motions. The relations of harmonic phases between the two asymmetric period-4 L = mod (φ R + ((k + 4r)/4 + 1)π , 2π )(k = 1, 2, · · · ; r = 0) for t = rT . motions from 0 to 2π are with φk/4 0 k/4


91

8 Conclusions In this paper, the time-delay effects of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator were investigated. The time-delay nodes of periodic motions were interpolated by the non-time-delayed nodes to obtain the implicit mappings. From mapping structures of periodic motion, bifurcation trees of period-1 motions to chaos, varying with time-delay, were predicted analytically, and the corresponding stability and bifurcation were determined through eigenvalue analysis. From the discrete finite Fourier series, nonlinear harmonic amplitudes varying with time-delay for 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. Period-1 motions of the timedelayed Duffing oscillator did not change too much for τ /T ∈ [0, 1] with Ω = 1.8. With period-doubling bifurcation, period-2 and period-4 motions in the time-delayed Duffing oscillator were not away too much from the period-1 motions in τ /T ∈ [0, 1] with Ω = 1.8, which can be observed through the quantity levels of harmonic amplitudes of periodic motions.

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, 1141-1186. [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, Chaos, Solitons & Fractals, 89, 405-434. [3] Hu, H.Y., Dowell, E.H., and Virgin, L.N. (1998), Resonance of harmonically forced Duffing oscillator with time-delay state feedback, Nonlinear Dynamics, 15(4), 311-327. [4] Hu, H.Y. and Wang, Z.H. (2002), Dynamics of Controlled Mechanical Systems with Delayed Feedback (Springer, Berlin). [5] MacDonald, N., 1995, Harmonic balance in delay-differential equations, Journal of Sounds and Vibration, 186 (4), 649-656. [6] Leung, A.Y.T. and Guo, Z. (2014), Bifurcation of the periodic motions in nonlinear delayed oscillators, Journal of Vibration and Control, 20, 501-517. [7] Liu, L. and Kalmar-Nagy, T. (2010), High-dimensional harmonic balance analysis for second-order delaydifferential equations, Journal of Vibration and Control, 16(7-8), 1189-1208. [8] 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. [9] 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, Non-linearity, and Complexity, 3, 87-107. [10] 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(4), 325-340. [11] 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) [12] Luo, A.C.J. (2015), Periodic flows in nonlinear dynamical systems based on discrete implicit maps, International Journal of Bifurcation and Chaos, 25(3), Article No: 1550044 (62 pages). [13] 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.

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Mohamed Belhaq Laboratory of Mechanics University Hassan II-Casablanca Casablanca, Morocco Email: [email protected]

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Frank Z. Feng Department of Mechanical and Aerospace Engineering University of Missouri Columbia, MO 65211, USA Email: [email protected]

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Jihong Wen Institute of Mechatronical Engineering National University of Defense Technology Changsha, Hunan, P.R. China Email: [email protected]

Lianhua Wang College of Civil Engineering Hunan University Changsha, Hunan, P.R. China Email: [email protected]

Hiroaki Watanabe Department of Mechanical Engineering Kyushu University Nishi-ku, Fukuoka 819-0395, Japan Email: [email protected]

Xingzhong Xiong School of Automation & Information Engineering Sichuan University of Science and Engineering Zigong, Sichuan, P. R. China Email: [email protected]

Guozhi Yao Modine Manufacturing Company Racine, WI 53403, USA Email: [email protected]

Weinian Zhang School of Mathematics, Sichuan University Chengdu, Sichuan, P.R. China Email: [email protected]

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

March 2017

Contents The Use of the Fitting Time Histories method to detect the nonlinear behavior of laminated glass S. Lenci, L. Consolini, F. Clementi………………..…………..…..……..……......

1－14

Dynamics of Turning Operation Part I:Experimental Analysis Using Instantaneous Frequency E.B. Halfmann, C.S. Suh, W.N.P. Hung….….…………………………….………

15－33

Dynamics of Turning Operation Part II:Model Validation and Stability at High Speed E.B. Halfmann, C.S. Suh, W.N.P. Hung...................................................................

35－52

Model Reduction for Second Order in Time Nonlinear Dissipative Autonomous Dynamic Systems Yan Liu, Jiazhong Zhang, Jiahui Chen, Yamiao Zhang…………………..….……

53－63

Redundant Control of A Bipedal Robot Moving from Sitting to Standing Jing Lu, Xian-Guo Tuo, Yong Liu, Tong Shen………...………....………........…

65－71

Time-delay Effects on Periodic Motions in a Periodically Forced, Time-delayed, Hardening Duffing Oscillator Albert C.J. Luo, Siyuan Xing………...…………..…...…………....…………...…

73－91

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

Printed in USA

Journal of Vibration Testing and System Dynamics

Journal of Vibration Testing and System Dynamics

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