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Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises
Chaos, Solitons and Fractals 87 (2016) 125–135
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises Guidong Yang a, Wei Xu a,∗, Xudong Gu b, Dongmei Huang a a b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072,China Department of Engineering Mechanics, Northwestern Polytechnical University, Xian, 710129, PR China
a r t i c l e
i n f o
Article history: Received 24 November 2015 Revised 26 February 2016 Accepted 10 March 2016 Available online 28 March 2016 Keywords: Vibroimpact Duffing system Bilateral barriers Energy loss Averaging method External and parametric excitations
a b s t r a c t In this paper, a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system is considered for the cases of external and parametric Gaussian white noise random excitations. According to the levels of the system energy, the motions of the unperturbed vibroimpact system are divided into two types: oscillations without impacts and oscillations with alternate impacts on both sides. Then, under the assumption that the vibroimpact Duffing system is quasiconservative, the stochastic averaging method for energy envelope is applied to obtain the averaged drift and diffusion coefficients for the two types of motions, respectively. The Probability Density Functions (PDFs) of stationary responses are derived by solving the corresponding Fokker-Plank-Kolmogorov (FPK) equation. Lastly, results obtained from the proposed procedure are validated by directly numerical simulation. Meanwhile, effects of the position of bilateral barriers and the random excitations on the PDFs of the stationary responses are also discussed. © 2016 Published by Elsevier Ltd.
1. Introduction Vibroimpact system, as a specific class of nonlinear systems, is a kind of mechanical system in which certain masses may collide with others or with rigid barriers during their oscillations [1]. The differential equation of its motion is supplemented by an impact condition. In the field of engineering and physics, behaviors of numerous devices and apparatuses involve impact. Some examples may include the interaction of gears in a transmission, inverted pendulum oscillating against two-sided barriers, ship roll motion against one-sided barrier, heat exchanger tube fretting due to adjacent tubes interaction, automotive braking systems and so on [28]. Therefore, it is of great importance to investigate the dynamic properties of this kind of systems. In fact, the vibroimpact system has received considerable attention for the past decades. Different types of modeling techniques such as the power-law phenomenological modeling, Zhuravlev and Ivanov non-smooth coordinate transformations, Hertzian contact force, point wise mapping, and saw-tooth-time-transformation, were developed to model the interacting forces associated with the vibroimpact process [9, 10]. Based on the above models, the dynamics of vibroimpact system were investigated numerically [11, 12] and some special phenomena, such as grazing bifurcations [13-15], chatter and sticking ∗
Corresponding author. Tel.: +862988492393. E-mail address: [email protected] (W. Xu).
http://dx.doi.org/10.1016/j.chaos.2016.03.017 0960-0779/© 2016 Published by Elsevier Ltd.
motions [16, 17] and border collision bifurcations [18], were examined extensively. Furthermore, the stability [19], the multi-valued response [20], the mean response and the energy loss due to impact [21, 22] were also studied. The response PDF is an important characteristic of vibroimpact system under random excitations. However, many methods which used in the smooth system may not be applicable due to the existence of the non-smooth factors and the random perturbations. Hence, this increases the difficulty to obtain the analytical results. By using the Hertzian contact models, Jing [23] obtained the closed form solutions for the stationary response of a single-degree-of-freedom vibroimpact system. Huang [24] developed the stochastic averaging method to study the stationary responses of a multi-degree-of-freedom vibroimpact system under white noise excitations. Xu [25] investigated the random vibration problems of vibroimpact systems with inelastic impact by a modified Hertzian contact model. The Zhuravlev non-smooth coordinate transformation [26] is a quite effective method to deal with the vibroimpact system in which the impact condition is represented as just finite relations between the impact and rebound velocities. Together with this method, considerable results are obtained by various methods. Dimentberg and Iourtchenko [27] presented a comprehensive review on random vibrations of vibroimpact systems. Feng and Li [28-30] studied the stationary responses of several strongly nonlinear vibroimpact systems with a barrier at the position of the static equilibrium under different random
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Fig. 1. Vibroimpact system with two symmetric distributed rigid barriers.
2. Vibroimpact Duffing system with bilateral barriers 2.1. Motion of the vibroimpact Duffing oscillator A single-degree-of-freedom vibroimpact Duffing system with bilateral barriers is shown in Fig. 1. Herein M represents the mass and can be simply taken as unity. The rigid barriers are located at both sides of the mass, respectively. denotes the distance of one barrier from the static equilibrium position of the system. The corresponding differential equation of motion of the vibroimpact Duffing system can be formulated by Fig. 2. The schematic drawing of the phase orbits.
excitations. Yang [31] employed a stochastic averaging method for energy envelope to explore the effects of the parametric excitations on the response PDFs of a Rayleigh -Van der Pol vibroimpact system. Zhu [32, 33] proposed a solution procedure for vibroimpact Duffing systems with a unilateral zero or non-zero offset barrier by utilizing the Exponential-Polynomial Closure (EPC) method. Subsequently, Zhu [34] extended the solution procedure to deal with the case of external Poisson impulses. Sri Namachchivaya and Park [35] used a stochastic averaging method to study the dynamics of a vibroimpact system excited by random perturbations. Besides, the path integration method [36] was also applied to obtain the response PDFs of a vibroimpact system with high energy loss. Recently, Gu [37] presented a stochastic averaging method for analyzing vibroimpact systems under Gaussian white noise excitations. The advantage is that this method did not need the assistance of the Zhuravlev coordinate transformation and proved to be effective when the position of the barrier was non-zero. As mentioned above, most of the work focuses on the stationary responses of vibroimpact system with a unilateral zero offset barrier under random excitations since the analysis is relatively easy at this time. The cases for a unilateral non-zero offset barrier are rarely reported, less than those for bilateral barriers. However, many engineering applications have to face with impact problems with bilateral barriers, for instance, tubes of a heat exchanger that usually have small gaps in their intermediate supports. In this paper, the stationary responses of a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system are considered for the cases of external and parametric Gaussian white noise random excitations. The layout of the paper is as follows. The vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises is presented and the unperturbed vibroimpact system is analyzed in Section 2. Under the assumption that the vibroimpact Duffing system is quasi-conservative, the stationary responses PDFs are derived by the stochastic averaging method for energy envelope in Section 3. Results obtained from the proposed procedure are verified by directly numerical simulation results in Section 4. Furthermore, the effects of the position of bilateral barriers and the random excitations on the stationary responses PDFs are also investigated. A conclusion is given in the last section.
x¨ + ε 2 cx˙ + ax + bx3 = ε ξ1 (t ) + ε xξ2 (t ), |x| < ,
(1a)
x˙ + = −r x˙ − , x = ±.
(1b)
Where ɛ is a small parameter; c denotes the damping coefficient; a and b represent linear and nonlinear stiffness coefficients, respectively. ξ 1 (t) and ξ 2 (t) are zero-mean independently Gaussian white noises, and their intensities are 2D11 and 2D22 , respectively. x˙ + and x˙ − are the velocities of system after and before the instant of impacts, respectively. r is the restitution coefficient which is used to characterize the energy losses due to impact determined by the condition x = ±. When r equals to one, the vibroimpact system reduces to an elastic one; when r equals to zero, the velocity of vibroimpact system after impact vanishes. Therefore, the value range of r is 0 < r ≤ 1. From Eq. (1a), the system will oscillate between x = − and x = . At this time, the motion of system is “freedom”. Then the impact condition in Eq. 1(b) is imposed to Eq. (1a) when the system oscillates at the barrier. It is obvious from Eq. (1b) that the velocities of the oscillator have a jump at the instant of impact. Thus, the system governed by Eq. (1a) and (1b) is a non-smooth system and the analytical study of random processes in this system becomes very difficult to deal with. 2.2. Unperturbed vibroimpact system In the absence of damping and random perturbation, the unperturbed Duffing vibroimpact system takes the form
x¨ + ax + bx3 = 0, |x| < ,
(2a)
x˙ + = −r x˙ − , x = ±.
(2b)
The total energy of the Duffing vibroimpact system is
H=
1 2 x˙ + G(x ), 2
(3)
where
G (x ) =
x au + bu3 du = 0
a 2 b 4 x + x . 2 4
(4)
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Fig. 3. Stationary PDFs of total energy obtained numerically and analytically for different values of restitution coefficient.
Note that the potential energy function G(x) is an even one, the potential energy at the position x = − and x = is the same. That is
G(−) = G() =
a 2 b 4 + . 2 4
(5)
The schematic drawing of the phase orbits for the unperturbed Duffing vibroimpact system is shown in Fig. 2. Let the system be an initial energy level, i.e., H. When H < G(), the oscillator cannot reach at the rigid barrier, so there is no impact occurrence. The Duffing system will oscillate between −A and A. The period of oscillation is
T (H ) = 2
A
−A
1 2H − 2G ( x )
A dx = 2 −A
1 2H − ax2 − bx4 /2
1 2 x˙ 2 1−
dx.
(6)
(7)
where H1− denotes the total energy of the unperturbed Duffing vibroimpact system before impacting with the barrier; x˙ 1− denotes the velocity of oscillator before the instants of impact. After impact, the velocity of oscillator becomes from x˙ 1− to x˙ 1+ = −r x˙ 1− by means of the impact condition Eq. (1b) . The total energy is
H1+ = G() +
1 2 x˙ 2 1+
δ1 (H ) = H1− − H1+ = 1 − r2 (H1− − G() )
(8)
(9)
After that, the system moves like a free oscillator until it reaches at the position x = −. Then the impact occurs. Before the instant of this impact, the total energy of the system is
H2− = H1− − δ1 (H ) = G(−) +
1 2 x˙ 2 2−
(10)
where x˙ 2− denotes the velocity of the oscillator at the position x = −before the impact. After the instant of this impact, the total energy becomes
H2+ = G(−) +
where A is the amplitude of the oscillator determined by H = G(A ). Since H = G(A ) is a monotone increasing function, it is obvious that A < at this time. When H > G(), the oscillator will impact with the rigid barriers. Without loss of generality, assume that the occurrence of the first impact is at the position . Then the energy before impact can be derived as
H1− = G() +
Therefore, the energy loss during this impact is calculated as
1 2 x˙ 2 2+
(11)
where x˙ 2+ = −r x˙ 2− . Then the energy loss at the position x = − is
δ2 (H ) = H2− − H2+ = 1 − r2 (H1− − δ1 (H ) − G(−) )
(12)
If we define the impact quasi-period as the time from the first impact at the position x = to the second impact at the same position, then the energy loss in one quasi-period can be obtained as
δ (H ) = δ1 (H ) + δ2 (H ) = (1 − r4 )(H1− − G() )
(13)
The quasi-period is
δ (t ) = t2 − t1 = 2
−
1 2H − 2G ( x )
dx
(14)
where t1 and t2 represents the time of the first impact at the position x = and the second one, respectively. From Eq. (13), it can be seen that the energy loss for the unperturbed Duffing vibroimpact system relies on the restitution coefficient, the energy before impact and the position of the rigid barriers. Besides, it is worth noting that if we choose the impact
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Fig. 4. Joint PDFs of the displacement x and velocity y for r = 0.98 (a) Monte-Carlo results for bilateral barriers; (b) analytical results for bilateral barriers.
quasi-period from the position x = −, the process of analysis is similar and the result is the same.
Letting x˙ = y, Eq. (1a) can be transformed as the following Itô stochastic differential equations:
dx = ydt, 3. Stochastic Response
dy = m(x, y )dt + σ (x, y )dW (t ), |x| <
(15)
where W(t) is a unit Weiner process and
3.1. Stochastic averaging of energy envelop
m(x, y ) = −ε 2 cy − ax − bx3 ,
(16)
Note that ɛ is a small parameter, the vibroimpact Duffing system formulated by Eq. (1a) and (1b) is a lightly damped system excited by weakly Gaussian white noises. Assume that 1 − r 4 1, the vibroimpact system is quasi-conservative.
σ 2 (x, y ) = 2ε 2 D11 + 2ε 2 x2 D22 .
(17)
Substituting Eq. (3) and Eq. (4) into Eq. (15) as well as using the Itô differential rule, the following equation about displacement
G. Yang et al. / Chaos, Solitons and Fractals 87 (2016) 125–135
129
Fig. 5. Joint PDFs of the displacement x and velocity y for r = 0.92 (a) Monte-Carlo results for bilateral barriers; (b) analytical results for bilateral barriers; (c) Monte-Carlo results for unilateral non-zero offset barrier.
x and energy H is obtained:
dx = ydt y=±√2H−ax−bx4 /2 dH = −ε 2 cy2 + ε 2 D11 + ε 2 x2 D22 dt +y 2ε 2 D11 + 2ε 2 x2 D22 dW (t )y=±√2H−ax−bx4 /2
According to the Khasminskii theorem [38], H(t)can be approximated by a Markov process satisfying the following mean Itô stochastic differential equation:
dH = m(H )dt + σ (H )dW (t ) (18)
Under the assumption that the damping coefficient, the noise intensities and 1 − r 4 are small, the energy H(t) is a slowly varying random process while the displacement x(t) is a fast one. The random vibration during each cycle would be close to the natural oscillations of the corresponding conservative system.
(19)
where m(H) and σ (H) are the mean drift coefficient and the mean diffusion coefficient, respectively. As there exist two types of possible motion: oscillations without impacts and oscillations with alternate impacts on both sides, the calculation of the mean drift coefficient and the mean diffusion coefficient should be worked out in two cases.
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Fig. 5. Continued
Fig. 6. Stationary PDFs of total energy obtained numerically and analytically for different values of the position of the rigid barriers.
When H < G(), there is no impact occurrence. The mean drift coefficient and the mean diffusion coefficient of the vibroimpact system can be obtained as those of systems without impact
2 m (H ) = T (H )
A
−ε c 2
−A
2H −
ax2
−
bx4 /2 +
ε 2 D + ε 2 D22 x2 dx 11 2H −ax2 −bx4 /2
σ
2
2 (H ) = T (H )
T (H ) = 2
A
−A
A
2H − ax2 − bx4 /2 2D11 ε 2 + 2D22 ε 2 x2 dx
−A
1 2H − ax2 − bx4 /2
dx
(20)
G. Yang et al. / Chaos, Solitons and Fractals 87 (2016) 125–135
Fig. 7. Joint PDFs of the displacement x and velocity y for = 1.0 (a) Monte-Carlo results; (b) analytical results.
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Fig. 8. Stationary PDFs of total energy H obtained numerically and analytically for different values of external and parametric Gaussian white noises intensities.
where A is the positive root of H = G(A ), and can be calculated as
the stationary solution of Eq. (22) can be obtained analytically as
1/2
A = [(a2 + 4bH ) − a]/b. When H > G(), the system will oscillate with alternate impacts on both sides. Considering the energy loss during one quasi-period, the mean drift coefficient and the mean diffusion coefficient of the vibroimpact system are obtained as
m (H ) =
2 T (H )
−ε 2 c
2H − ax2 − bx4 /2
2 T (H )
T (H ) = 2
−
2H − ax2 − bx4 /2 2D11 ε 2 + 2D22 ε 2 x2 dx 1
2H − ax2 − bx4 /2
dx.
(21)
3.2. PDFs of stationary response The corresponding mean FPK equation of Eq. (19) is (22)
The exact solution of Eq. (22) is very difficult to obtain due to the existence of nonlinear factors. We mainly seek the stationary response. When the boundary conditions are
0 ≤ p < ∞, H = 0, dp p → 0, dH → 0, H → ∞,
exp 0
2m ( u ) du . σ 2 (u )
(24)
where C is a normalization constant. By means of Eq. (3), the joint probability density of the displacement x and the velocity y is easy to derive as
p( x ) =
where −δ (H )/T (H ) represents the energy loss during one impact period.
∂p ∂ 1 ∂2 2 =− [m(H ) p] + σ (H ) p . ∂t ∂H 2 ∂ H2
σ 2 (H )
H
p( H ) y2 T (H ) H= 2 +G(x ),|x|≤
(25)
Further, the corresponding marginal probability density functions of the displacement x and the velocity y are obtained as follows, respectively,
−
p( H ) =
p(x, y ) =
−
ε 2 D11 + ε 2 D22 x2 δ (H ) + dx − T 2 4 (H ) 2H − ax − bx /2 σ 2 (H ) =
C
(23)
p( y ) =
∞ −∞
−
p(x, y )dy,
(26)
p(x, y )dx.
(27)
4. Analysis of response PDFs In this section, results obtained by the stochastic averaging method for energy envelope is shown and compared with those from directly numerical simulation to assess the effectiveness of the proposed procedure. Since the stochastic averaging method for energy envelope is only effective for weak damping, small perturbations and energy loss, we choose the corresponding parameters as order of ɛ2 . To investigate the effectiveness of the proposed procedure onto the stationary response of the vibroimpact Duffing system, numerical and analytical results of PDFs for the total energy H and those of the joint PDFs for the displacement x and velocity y are obtained and shown in Fig. 3-5, respectively, with the parameters a = 1.0, b = 0.2, ε 2 c = 0.025, D11 = 0.01, D22 = 0.01, = 0.6, and
G. Yang et al. / Chaos, Solitons and Fractals 87 (2016) 125–135
Fig. 9. Stationary PDFs of displacement x obtained numerically and analytically for different values of external and parametric Gaussian white noises intensities.
Fig. 10. Stationary PDFs of velocity y obtained numerically and analytically for different values of external and parametric Gaussian white noises intensities.
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two different restitution coefficients (r = 0.98, 0.92). In Fig. 3, the solid line represents the analytical results obtained by the stochastic averaging method while the • denotes the directly numerical results from Monte Carlo simulation. It can be seen that they are in good agreement. The response curves of PDFs of the total energy are not smooth which is different from systems without impact. The occurrence of the non-smooth position is at H = G(). In Fig. 4(a), the Monte Carlo simulation result of the joint PDFs for the displacement x and velocity y is shown to compare with the analytical result in Fig. 4(b), when r = 0.98. The similar results are presented in Fig. 5(a) and (b) while r = 0.92. Obviously, good agreement can be found for different restitution coefficients. Comparing Fig. 4 (b) with Fig. 5 (b), it can be seen that the decrease of the restitution coefficient leads to the increase of the peak value of the joint PDFs. That is, smaller restitution coefficient leads to larger probability that the system stay close to the equilibrium state. Furthermore, in order to analyze different effects on the joint PDFs for the displacement x and velocity y between unilateral non-zero offset barrier and bilateral barriers, Fig. 5 (c) shows the Monte Carlo results for unilateral non-zero offset barrier with the same parameters in Fig. 5 (a). It is clear that the joint PDFs is truncated in one side from Fig. 5(c), whereas the joint PDFs is truncated in both sides from Fig. 5(a). As discussed above in Section. 2, the position of the rigid barriers is an important factor involving energy loss. To investigate its effect on the stationary response of the vibroimpact Duffing system, Fig. 6 shows PDFs curves of total energy H for r = 0.98 and two different values of ( = 0.6, 1.0), other parameters are the same as those in Fig. 4. Herein, the solid line represents the analytical results and the • denotes the Monte Carlo simulation results. It can be seen that the increase of the parameter leads to lower value of peak and smaller rate of decay. It would be relatively easier to understand from Eqs. (5) and (13). The reason is that G() is a monotone increasing function, hence δ (H)would be a monotone decreasing function for given energy H and restitution coefficient r meaning that larger leads to smaller energy loss and then yields to smaller probability that system stay close to the equilibrium state. To more clearly understand the phenomena, Fig. 7 provides the joint PDFs for the displacement x and velocity y when = 1.0. Comparing with Fig. 4, it can be seen that the peak values become smaller with the increase of . To illustrate the effects of external and parametric Gaussian white noises, Figs. 8-10 demonstrate the stationary PDFs of the total energy H, displacement x and velocity y numerically and analytically for different excitation intensities, respectively. Similar to Fig. 4, the solid line is the results obtained from the proposed procedure and the • denotes those from Monte Carlo simulation. As can be seen from Fig. 8, the curves of the total energy H are not smooth at H = G(). When the parametric Gaussian white noise intensity is fixed as D22 = 0.01, the increase of external Gaussian white noise intensity D11 leads to lower peak values. Similarly, when the external Gaussian white noise intensity is fixed as D11 = 0.01, the peak values also become lower with the increase of D22 . It can be seen from Fig. 9 that the curves of the stationary PDFs of the displacement x are truncated due to the existence of the double side barriers. In Fig. 10, the stationary PDFs of the velocity y present lower peaks and become more fat when the excitation intensities of external and parametric Gaussian white noises increase. Furthermore, both the PDFs of the displacement x and velocity y are symmetry and smooth. 5. Conclusions In this paper, the stochastic response problems of a vibroimpact Duffing system with bilateral barriers are considered. The random excitations are external and parametric Gaussian white noises.
Firstly, the unperturbed vibroimpact system is analyzed according to the levels of the system energy. When the total energy of the system is less than the potential energy at the position , the system will moves like a system without impact; when total energy of the system is larger than the potential energy at the position , the system will oscillate with alternate impacts on both sides. The energy loss due to impact is discussed. Secondly, under the assumption that the vibroimpact Duffing system is quasiconservative, the stochastic averaging method for energy envelope is applied to derive the drift and diffusion coefficients of the averaged Itô stochastic differential equation for the two types of motions. Thirdly, the corresponding FPK equation is solved to obtain the PDFs of stationary responses. At last, results by directly numerical simulation are shown to compare with those obtained from the proposed procedure. The good agreement between the two results shows that the proposed procedure is quite effective. Besides, effects of the position of bilateral barriers and the intensities of the external and parametric Gaussian white noises random excitations on the PDFs of the stationary responses are discussed. It is worth noting that the proposed procedure may be further generalized to other vibroimpact systems with bilateral barriers. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011and 11502201) and and the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2014JQ1001). References [1] Song L. Nonlinear random vibration of vibroimpact systems. Ph.D. thesis. Florida: University of Miami; 2006. [2] Ibrahim RA. Recent advances in vibro-impact dynamics and collision of ocean vessels. J Sound Vib 2014;333:5900–16. [3] Babitsky VI. Theory of vibro-impact systems and applications. Berlin: Springer; 1998. [4] Brogliato B. Impacts in mechanical systems: analysis and modelling. Dordrecht: Springer Science & Business Media; 20 0 0. [5] Dimentberg MF. Statistical dynamics of nonlinear and time-varying systems. Research Studies Press, Wiley; 1988. [6] Ibrahim RA. Vibro-impact dynamics: modeling, mapping and applications. Dordrecht: Springer Science & Business Media; 2009. [7] Luo AC, Guo Y. Vibro-impact dynamics. New York: John Wiley & Sons; 2012. [8] Ibrahim RA, Grace IM. Modeling of ship roll dynamics and its coupling with heave and pitch. Math Probl Eng 2010;2010, Article ID 934714, 32 pages. [9] Goldsmith W, Frasier JT. Impact: the theory and physical behavior of colliding solids. J Appl Mech 1961;28(4):639. [10] Di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Tost GO, et al. Bifurcations in nonsmooth dynamical systems. SIAM Rev 2008:629–701. [11] Shaw S, Holmes P. A periodically forced impact oscillator with large dissipation. J Appl Mech 1983;50:849–57. [12] Luo G. Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops. Phys Lett A 2004;323:210–17. [13] de Weger J, Binks D, Molenaar J, van de Water W. Generic behavior of grazing impact oscillators. Phys rev lett 1996;76:3951. [14] Chin W, Ott E, Nusse HE, Grebogi C. Grazing bifurcations in impact oscillators. Phys Rev E 1994;50:4427. [15] Luo AC, Chen L. Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts. Chaos Soliton Fract 2005;24:567–78. [16] Wagg D. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator. Chaos, Soliton Frac 2004;22:541–8. [17] Luo AC, O’CONNOR D. Periodic motions and chaos with impacting chatter and stick in a gear transmission system. Int J Bifurcat Chaos 2009;19:1975–94. [18] Di Bernardo M, Budd C, Champneys A. Corner collision implies border-collision bifurcation. Physica D: Nonlinear Phenomena 2001;154:171–94. [19] Janin O, Lamarque C-H. Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dynam 2002;28:231–41. [20] Huang D, Xu W, Liu D, Han Q. Multi-valued responses of a nonlinear vibroimpact system excited by random narrow-band noise. JVib Control 2014. doi:10.1177/1077546314546512. [21] Dimentberg M, Iourtchenko D. Towards incorporating impact losses into random vibration analyses: a model problem. Probabilist eng mech 1999;14:323–8. [22] Iourtchenko D., Iwankiewicz R. Analysis of a vibroimpact system under a Poisson impulse process excitation’. Fifth European Conference on Structural Dynamics EURODYN 20 0220 02.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises Guidong Yang a, Wei Xu a,∗, Xudong Gu b, Dongmei Huang a a b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072,China Department of Engineering Mechanics, Northwestern Polytechnical University, Xian, 710129, PR China
a r t i c l e
i n f o
Article history: Received 24 November 2015 Revised 26 February 2016 Accepted 10 March 2016 Available online 28 March 2016 Keywords: Vibroimpact Duffing system Bilateral barriers Energy loss Averaging method External and parametric excitations
a b s t r a c t In this paper, a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system is considered for the cases of external and parametric Gaussian white noise random excitations. According to the levels of the system energy, the motions of the unperturbed vibroimpact system are divided into two types: oscillations without impacts and oscillations with alternate impacts on both sides. Then, under the assumption that the vibroimpact Duffing system is quasiconservative, the stochastic averaging method for energy envelope is applied to obtain the averaged drift and diffusion coefficients for the two types of motions, respectively. The Probability Density Functions (PDFs) of stationary responses are derived by solving the corresponding Fokker-Plank-Kolmogorov (FPK) equation. Lastly, results obtained from the proposed procedure are validated by directly numerical simulation. Meanwhile, effects of the position of bilateral barriers and the random excitations on the PDFs of the stationary responses are also discussed. © 2016 Published by Elsevier Ltd.
1. Introduction Vibroimpact system, as a specific class of nonlinear systems, is a kind of mechanical system in which certain masses may collide with others or with rigid barriers during their oscillations [1]. The differential equation of its motion is supplemented by an impact condition. In the field of engineering and physics, behaviors of numerous devices and apparatuses involve impact. Some examples may include the interaction of gears in a transmission, inverted pendulum oscillating against two-sided barriers, ship roll motion against one-sided barrier, heat exchanger tube fretting due to adjacent tubes interaction, automotive braking systems and so on [28]. Therefore, it is of great importance to investigate the dynamic properties of this kind of systems. In fact, the vibroimpact system has received considerable attention for the past decades. Different types of modeling techniques such as the power-law phenomenological modeling, Zhuravlev and Ivanov non-smooth coordinate transformations, Hertzian contact force, point wise mapping, and saw-tooth-time-transformation, were developed to model the interacting forces associated with the vibroimpact process [9, 10]. Based on the above models, the dynamics of vibroimpact system were investigated numerically [11, 12] and some special phenomena, such as grazing bifurcations [13-15], chatter and sticking ∗
Corresponding author. Tel.: +862988492393. E-mail address: [email protected] (W. Xu).
http://dx.doi.org/10.1016/j.chaos.2016.03.017 0960-0779/© 2016 Published by Elsevier Ltd.
motions [16, 17] and border collision bifurcations [18], were examined extensively. Furthermore, the stability [19], the multi-valued response [20], the mean response and the energy loss due to impact [21, 22] were also studied. The response PDF is an important characteristic of vibroimpact system under random excitations. However, many methods which used in the smooth system may not be applicable due to the existence of the non-smooth factors and the random perturbations. Hence, this increases the difficulty to obtain the analytical results. By using the Hertzian contact models, Jing [23] obtained the closed form solutions for the stationary response of a single-degree-of-freedom vibroimpact system. Huang [24] developed the stochastic averaging method to study the stationary responses of a multi-degree-of-freedom vibroimpact system under white noise excitations. Xu [25] investigated the random vibration problems of vibroimpact systems with inelastic impact by a modified Hertzian contact model. The Zhuravlev non-smooth coordinate transformation [26] is a quite effective method to deal with the vibroimpact system in which the impact condition is represented as just finite relations between the impact and rebound velocities. Together with this method, considerable results are obtained by various methods. Dimentberg and Iourtchenko [27] presented a comprehensive review on random vibrations of vibroimpact systems. Feng and Li [28-30] studied the stationary responses of several strongly nonlinear vibroimpact systems with a barrier at the position of the static equilibrium under different random
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Fig. 1. Vibroimpact system with two symmetric distributed rigid barriers.
2. Vibroimpact Duffing system with bilateral barriers 2.1. Motion of the vibroimpact Duffing oscillator A single-degree-of-freedom vibroimpact Duffing system with bilateral barriers is shown in Fig. 1. Herein M represents the mass and can be simply taken as unity. The rigid barriers are located at both sides of the mass, respectively. denotes the distance of one barrier from the static equilibrium position of the system. The corresponding differential equation of motion of the vibroimpact Duffing system can be formulated by Fig. 2. The schematic drawing of the phase orbits.
excitations. Yang [31] employed a stochastic averaging method for energy envelope to explore the effects of the parametric excitations on the response PDFs of a Rayleigh -Van der Pol vibroimpact system. Zhu [32, 33] proposed a solution procedure for vibroimpact Duffing systems with a unilateral zero or non-zero offset barrier by utilizing the Exponential-Polynomial Closure (EPC) method. Subsequently, Zhu [34] extended the solution procedure to deal with the case of external Poisson impulses. Sri Namachchivaya and Park [35] used a stochastic averaging method to study the dynamics of a vibroimpact system excited by random perturbations. Besides, the path integration method [36] was also applied to obtain the response PDFs of a vibroimpact system with high energy loss. Recently, Gu [37] presented a stochastic averaging method for analyzing vibroimpact systems under Gaussian white noise excitations. The advantage is that this method did not need the assistance of the Zhuravlev coordinate transformation and proved to be effective when the position of the barrier was non-zero. As mentioned above, most of the work focuses on the stationary responses of vibroimpact system with a unilateral zero offset barrier under random excitations since the analysis is relatively easy at this time. The cases for a unilateral non-zero offset barrier are rarely reported, less than those for bilateral barriers. However, many engineering applications have to face with impact problems with bilateral barriers, for instance, tubes of a heat exchanger that usually have small gaps in their intermediate supports. In this paper, the stationary responses of a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system are considered for the cases of external and parametric Gaussian white noise random excitations. The layout of the paper is as follows. The vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises is presented and the unperturbed vibroimpact system is analyzed in Section 2. Under the assumption that the vibroimpact Duffing system is quasi-conservative, the stationary responses PDFs are derived by the stochastic averaging method for energy envelope in Section 3. Results obtained from the proposed procedure are verified by directly numerical simulation results in Section 4. Furthermore, the effects of the position of bilateral barriers and the random excitations on the stationary responses PDFs are also investigated. A conclusion is given in the last section.
x¨ + ε 2 cx˙ + ax + bx3 = ε ξ1 (t ) + ε xξ2 (t ), |x| < ,
(1a)
x˙ + = −r x˙ − , x = ±.
(1b)
Where ɛ is a small parameter; c denotes the damping coefficient; a and b represent linear and nonlinear stiffness coefficients, respectively. ξ 1 (t) and ξ 2 (t) are zero-mean independently Gaussian white noises, and their intensities are 2D11 and 2D22 , respectively. x˙ + and x˙ − are the velocities of system after and before the instant of impacts, respectively. r is the restitution coefficient which is used to characterize the energy losses due to impact determined by the condition x = ±. When r equals to one, the vibroimpact system reduces to an elastic one; when r equals to zero, the velocity of vibroimpact system after impact vanishes. Therefore, the value range of r is 0 < r ≤ 1. From Eq. (1a), the system will oscillate between x = − and x = . At this time, the motion of system is “freedom”. Then the impact condition in Eq. 1(b) is imposed to Eq. (1a) when the system oscillates at the barrier. It is obvious from Eq. (1b) that the velocities of the oscillator have a jump at the instant of impact. Thus, the system governed by Eq. (1a) and (1b) is a non-smooth system and the analytical study of random processes in this system becomes very difficult to deal with. 2.2. Unperturbed vibroimpact system In the absence of damping and random perturbation, the unperturbed Duffing vibroimpact system takes the form
x¨ + ax + bx3 = 0, |x| < ,
(2a)
x˙ + = −r x˙ − , x = ±.
(2b)
The total energy of the Duffing vibroimpact system is
H=
1 2 x˙ + G(x ), 2
(3)
where
G (x ) =
x au + bu3 du = 0
a 2 b 4 x + x . 2 4
(4)
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Fig. 3. Stationary PDFs of total energy obtained numerically and analytically for different values of restitution coefficient.
Note that the potential energy function G(x) is an even one, the potential energy at the position x = − and x = is the same. That is
G(−) = G() =
a 2 b 4 + . 2 4
(5)
The schematic drawing of the phase orbits for the unperturbed Duffing vibroimpact system is shown in Fig. 2. Let the system be an initial energy level, i.e., H. When H < G(), the oscillator cannot reach at the rigid barrier, so there is no impact occurrence. The Duffing system will oscillate between −A and A. The period of oscillation is
T (H ) = 2
A
−A
1 2H − 2G ( x )
A dx = 2 −A
1 2H − ax2 − bx4 /2
1 2 x˙ 2 1−
dx.
(6)
(7)
where H1− denotes the total energy of the unperturbed Duffing vibroimpact system before impacting with the barrier; x˙ 1− denotes the velocity of oscillator before the instants of impact. After impact, the velocity of oscillator becomes from x˙ 1− to x˙ 1+ = −r x˙ 1− by means of the impact condition Eq. (1b) . The total energy is
H1+ = G() +
1 2 x˙ 2 1+
δ1 (H ) = H1− − H1+ = 1 − r2 (H1− − G() )
(8)
(9)
After that, the system moves like a free oscillator until it reaches at the position x = −. Then the impact occurs. Before the instant of this impact, the total energy of the system is
H2− = H1− − δ1 (H ) = G(−) +
1 2 x˙ 2 2−
(10)
where x˙ 2− denotes the velocity of the oscillator at the position x = −before the impact. After the instant of this impact, the total energy becomes
H2+ = G(−) +
where A is the amplitude of the oscillator determined by H = G(A ). Since H = G(A ) is a monotone increasing function, it is obvious that A < at this time. When H > G(), the oscillator will impact with the rigid barriers. Without loss of generality, assume that the occurrence of the first impact is at the position . Then the energy before impact can be derived as
H1− = G() +
Therefore, the energy loss during this impact is calculated as
1 2 x˙ 2 2+
(11)
where x˙ 2+ = −r x˙ 2− . Then the energy loss at the position x = − is
δ2 (H ) = H2− − H2+ = 1 − r2 (H1− − δ1 (H ) − G(−) )
(12)
If we define the impact quasi-period as the time from the first impact at the position x = to the second impact at the same position, then the energy loss in one quasi-period can be obtained as
δ (H ) = δ1 (H ) + δ2 (H ) = (1 − r4 )(H1− − G() )
(13)
The quasi-period is
δ (t ) = t2 − t1 = 2
−
1 2H − 2G ( x )
dx
(14)
where t1 and t2 represents the time of the first impact at the position x = and the second one, respectively. From Eq. (13), it can be seen that the energy loss for the unperturbed Duffing vibroimpact system relies on the restitution coefficient, the energy before impact and the position of the rigid barriers. Besides, it is worth noting that if we choose the impact
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Fig. 4. Joint PDFs of the displacement x and velocity y for r = 0.98 (a) Monte-Carlo results for bilateral barriers; (b) analytical results for bilateral barriers.
quasi-period from the position x = −, the process of analysis is similar and the result is the same.
Letting x˙ = y, Eq. (1a) can be transformed as the following Itô stochastic differential equations:
dx = ydt, 3. Stochastic Response
dy = m(x, y )dt + σ (x, y )dW (t ), |x| <
(15)
where W(t) is a unit Weiner process and
3.1. Stochastic averaging of energy envelop
m(x, y ) = −ε 2 cy − ax − bx3 ,
(16)
Note that ɛ is a small parameter, the vibroimpact Duffing system formulated by Eq. (1a) and (1b) is a lightly damped system excited by weakly Gaussian white noises. Assume that 1 − r 4 1, the vibroimpact system is quasi-conservative.
σ 2 (x, y ) = 2ε 2 D11 + 2ε 2 x2 D22 .
(17)
Substituting Eq. (3) and Eq. (4) into Eq. (15) as well as using the Itô differential rule, the following equation about displacement
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Fig. 5. Joint PDFs of the displacement x and velocity y for r = 0.92 (a) Monte-Carlo results for bilateral barriers; (b) analytical results for bilateral barriers; (c) Monte-Carlo results for unilateral non-zero offset barrier.
x and energy H is obtained:
dx = ydt y=±√2H−ax−bx4 /2 dH = −ε 2 cy2 + ε 2 D11 + ε 2 x2 D22 dt +y 2ε 2 D11 + 2ε 2 x2 D22 dW (t )y=±√2H−ax−bx4 /2
According to the Khasminskii theorem [38], H(t)can be approximated by a Markov process satisfying the following mean Itô stochastic differential equation:
dH = m(H )dt + σ (H )dW (t ) (18)
Under the assumption that the damping coefficient, the noise intensities and 1 − r 4 are small, the energy H(t) is a slowly varying random process while the displacement x(t) is a fast one. The random vibration during each cycle would be close to the natural oscillations of the corresponding conservative system.
(19)
where m(H) and σ (H) are the mean drift coefficient and the mean diffusion coefficient, respectively. As there exist two types of possible motion: oscillations without impacts and oscillations with alternate impacts on both sides, the calculation of the mean drift coefficient and the mean diffusion coefficient should be worked out in two cases.
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Fig. 5. Continued
Fig. 6. Stationary PDFs of total energy obtained numerically and analytically for different values of the position of the rigid barriers.
When H < G(), there is no impact occurrence. The mean drift coefficient and the mean diffusion coefficient of the vibroimpact system can be obtained as those of systems without impact
2 m (H ) = T (H )
A
−ε c 2
−A
2H −
ax2
−
bx4 /2 +
ε 2 D + ε 2 D22 x2 dx 11 2H −ax2 −bx4 /2
σ
2
2 (H ) = T (H )
T (H ) = 2
A
−A
A
2H − ax2 − bx4 /2 2D11 ε 2 + 2D22 ε 2 x2 dx
−A
1 2H − ax2 − bx4 /2
dx
(20)
G. Yang et al. / Chaos, Solitons and Fractals 87 (2016) 125–135
Fig. 7. Joint PDFs of the displacement x and velocity y for = 1.0 (a) Monte-Carlo results; (b) analytical results.
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Fig. 8. Stationary PDFs of total energy H obtained numerically and analytically for different values of external and parametric Gaussian white noises intensities.
where A is the positive root of H = G(A ), and can be calculated as
the stationary solution of Eq. (22) can be obtained analytically as
1/2
A = [(a2 + 4bH ) − a]/b. When H > G(), the system will oscillate with alternate impacts on both sides. Considering the energy loss during one quasi-period, the mean drift coefficient and the mean diffusion coefficient of the vibroimpact system are obtained as
m (H ) =
2 T (H )
−ε 2 c
2H − ax2 − bx4 /2
2 T (H )
T (H ) = 2
−
2H − ax2 − bx4 /2 2D11 ε 2 + 2D22 ε 2 x2 dx 1
2H − ax2 − bx4 /2
dx.
(21)
3.2. PDFs of stationary response The corresponding mean FPK equation of Eq. (19) is (22)
The exact solution of Eq. (22) is very difficult to obtain due to the existence of nonlinear factors. We mainly seek the stationary response. When the boundary conditions are
0 ≤ p < ∞, H = 0, dp p → 0, dH → 0, H → ∞,
exp 0
2m ( u ) du . σ 2 (u )
(24)
where C is a normalization constant. By means of Eq. (3), the joint probability density of the displacement x and the velocity y is easy to derive as
p( x ) =
where −δ (H )/T (H ) represents the energy loss during one impact period.
∂p ∂ 1 ∂2 2 =− [m(H ) p] + σ (H ) p . ∂t ∂H 2 ∂ H2
σ 2 (H )
H
p( H ) y2 T (H ) H= 2 +G(x ),|x|≤
(25)
Further, the corresponding marginal probability density functions of the displacement x and the velocity y are obtained as follows, respectively,
−
p( H ) =
p(x, y ) =
−
ε 2 D11 + ε 2 D22 x2 δ (H ) + dx − T 2 4 (H ) 2H − ax − bx /2 σ 2 (H ) =
C
(23)
p( y ) =
∞ −∞
−
p(x, y )dy,
(26)
p(x, y )dx.
(27)
4. Analysis of response PDFs In this section, results obtained by the stochastic averaging method for energy envelope is shown and compared with those from directly numerical simulation to assess the effectiveness of the proposed procedure. Since the stochastic averaging method for energy envelope is only effective for weak damping, small perturbations and energy loss, we choose the corresponding parameters as order of ɛ2 . To investigate the effectiveness of the proposed procedure onto the stationary response of the vibroimpact Duffing system, numerical and analytical results of PDFs for the total energy H and those of the joint PDFs for the displacement x and velocity y are obtained and shown in Fig. 3-5, respectively, with the parameters a = 1.0, b = 0.2, ε 2 c = 0.025, D11 = 0.01, D22 = 0.01, = 0.6, and
G. Yang et al. / Chaos, Solitons and Fractals 87 (2016) 125–135
Fig. 9. Stationary PDFs of displacement x obtained numerically and analytically for different values of external and parametric Gaussian white noises intensities.
Fig. 10. Stationary PDFs of velocity y obtained numerically and analytically for different values of external and parametric Gaussian white noises intensities.
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two different restitution coefficients (r = 0.98, 0.92). In Fig. 3, the solid line represents the analytical results obtained by the stochastic averaging method while the • denotes the directly numerical results from Monte Carlo simulation. It can be seen that they are in good agreement. The response curves of PDFs of the total energy are not smooth which is different from systems without impact. The occurrence of the non-smooth position is at H = G(). In Fig. 4(a), the Monte Carlo simulation result of the joint PDFs for the displacement x and velocity y is shown to compare with the analytical result in Fig. 4(b), when r = 0.98. The similar results are presented in Fig. 5(a) and (b) while r = 0.92. Obviously, good agreement can be found for different restitution coefficients. Comparing Fig. 4 (b) with Fig. 5 (b), it can be seen that the decrease of the restitution coefficient leads to the increase of the peak value of the joint PDFs. That is, smaller restitution coefficient leads to larger probability that the system stay close to the equilibrium state. Furthermore, in order to analyze different effects on the joint PDFs for the displacement x and velocity y between unilateral non-zero offset barrier and bilateral barriers, Fig. 5 (c) shows the Monte Carlo results for unilateral non-zero offset barrier with the same parameters in Fig. 5 (a). It is clear that the joint PDFs is truncated in one side from Fig. 5(c), whereas the joint PDFs is truncated in both sides from Fig. 5(a). As discussed above in Section. 2, the position of the rigid barriers is an important factor involving energy loss. To investigate its effect on the stationary response of the vibroimpact Duffing system, Fig. 6 shows PDFs curves of total energy H for r = 0.98 and two different values of ( = 0.6, 1.0), other parameters are the same as those in Fig. 4. Herein, the solid line represents the analytical results and the • denotes the Monte Carlo simulation results. It can be seen that the increase of the parameter leads to lower value of peak and smaller rate of decay. It would be relatively easier to understand from Eqs. (5) and (13). The reason is that G() is a monotone increasing function, hence δ (H)would be a monotone decreasing function for given energy H and restitution coefficient r meaning that larger leads to smaller energy loss and then yields to smaller probability that system stay close to the equilibrium state. To more clearly understand the phenomena, Fig. 7 provides the joint PDFs for the displacement x and velocity y when = 1.0. Comparing with Fig. 4, it can be seen that the peak values become smaller with the increase of . To illustrate the effects of external and parametric Gaussian white noises, Figs. 8-10 demonstrate the stationary PDFs of the total energy H, displacement x and velocity y numerically and analytically for different excitation intensities, respectively. Similar to Fig. 4, the solid line is the results obtained from the proposed procedure and the • denotes those from Monte Carlo simulation. As can be seen from Fig. 8, the curves of the total energy H are not smooth at H = G(). When the parametric Gaussian white noise intensity is fixed as D22 = 0.01, the increase of external Gaussian white noise intensity D11 leads to lower peak values. Similarly, when the external Gaussian white noise intensity is fixed as D11 = 0.01, the peak values also become lower with the increase of D22 . It can be seen from Fig. 9 that the curves of the stationary PDFs of the displacement x are truncated due to the existence of the double side barriers. In Fig. 10, the stationary PDFs of the velocity y present lower peaks and become more fat when the excitation intensities of external and parametric Gaussian white noises increase. Furthermore, both the PDFs of the displacement x and velocity y are symmetry and smooth. 5. Conclusions In this paper, the stochastic response problems of a vibroimpact Duffing system with bilateral barriers are considered. The random excitations are external and parametric Gaussian white noises.
Firstly, the unperturbed vibroimpact system is analyzed according to the levels of the system energy. When the total energy of the system is less than the potential energy at the position , the system will moves like a system without impact; when total energy of the system is larger than the potential energy at the position , the system will oscillate with alternate impacts on both sides. The energy loss due to impact is discussed. Secondly, under the assumption that the vibroimpact Duffing system is quasiconservative, the stochastic averaging method for energy envelope is applied to derive the drift and diffusion coefficients of the averaged Itô stochastic differential equation for the two types of motions. Thirdly, the corresponding FPK equation is solved to obtain the PDFs of stationary responses. At last, results by directly numerical simulation are shown to compare with those obtained from the proposed procedure. The good agreement between the two results shows that the proposed procedure is quite effective. Besides, effects of the position of bilateral barriers and the intensities of the external and parametric Gaussian white noises random excitations on the PDFs of the stationary responses are discussed. It is worth noting that the proposed procedure may be further generalized to other vibroimpact systems with bilateral barriers. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011and 11502201) and and the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2014JQ1001). References [1] Song L. Nonlinear random vibration of vibroimpact systems. Ph.D. thesis. Florida: University of Miami; 2006. [2] Ibrahim RA. Recent advances in vibro-impact dynamics and collision of ocean vessels. J Sound Vib 2014;333:5900–16. [3] Babitsky VI. Theory of vibro-impact systems and applications. Berlin: Springer; 1998. [4] Brogliato B. Impacts in mechanical systems: analysis and modelling. Dordrecht: Springer Science & Business Media; 20 0 0. [5] Dimentberg MF. Statistical dynamics of nonlinear and time-varying systems. Research Studies Press, Wiley; 1988. [6] Ibrahim RA. Vibro-impact dynamics: modeling, mapping and applications. Dordrecht: Springer Science & Business Media; 2009. [7] Luo AC, Guo Y. Vibro-impact dynamics. New York: John Wiley & Sons; 2012. [8] Ibrahim RA, Grace IM. Modeling of ship roll dynamics and its coupling with heave and pitch. Math Probl Eng 2010;2010, Article ID 934714, 32 pages. [9] Goldsmith W, Frasier JT. Impact: the theory and physical behavior of colliding solids. J Appl Mech 1961;28(4):639. [10] Di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Tost GO, et al. Bifurcations in nonsmooth dynamical systems. SIAM Rev 2008:629–701. [11] Shaw S, Holmes P. A periodically forced impact oscillator with large dissipation. J Appl Mech 1983;50:849–57. [12] Luo G. Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops. Phys Lett A 2004;323:210–17. [13] de Weger J, Binks D, Molenaar J, van de Water W. Generic behavior of grazing impact oscillators. Phys rev lett 1996;76:3951. [14] Chin W, Ott E, Nusse HE, Grebogi C. Grazing bifurcations in impact oscillators. Phys Rev E 1994;50:4427. [15] Luo AC, Chen L. Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts. Chaos Soliton Fract 2005;24:567–78. [16] Wagg D. Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator. Chaos, Soliton Frac 2004;22:541–8. [17] Luo AC, O’CONNOR D. Periodic motions and chaos with impacting chatter and stick in a gear transmission system. Int J Bifurcat Chaos 2009;19:1975–94. [18] Di Bernardo M, Budd C, Champneys A. Corner collision implies border-collision bifurcation. Physica D: Nonlinear Phenomena 2001;154:171–94. [19] Janin O, Lamarque C-H. Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dynam 2002;28:231–41. [20] Huang D, Xu W, Liu D, Han Q. Multi-valued responses of a nonlinear vibroimpact system excited by random narrow-band noise. JVib Control 2014. doi:10.1177/1077546314546512. [21] Dimentberg M, Iourtchenko D. Towards incorporating impact losses into random vibration analyses: a model problem. Probabilist eng mech 1999;14:323–8. [22] Iourtchenko D., Iwankiewicz R. Analysis of a vibroimpact system under a Poisson impulse process excitation’. Fifth European Conference on Structural Dynamics EURODYN 20 0220 02.
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