In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(...In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of themean value.Afterwards,the stochastic vibro-impact systemcan be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincarésections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the randomintensitymay result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.展开更多
A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole ...A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincare map of the system is constructed. Using the Poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.展开更多
A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the h...A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function （PDF） of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.展开更多
In this paper, multi-valued responses and dynamic properties of a nonlinear vibro-impact system with a unilateral nonzero offset barrier are studied. Based on the Krylov-Bogoliubov averaging method and Zhuravlev non-s...In this paper, multi-valued responses and dynamic properties of a nonlinear vibro-impact system with a unilateral nonzero offset barrier are studied. Based on the Krylov-Bogoliubov averaging method and Zhuravlev non-smooth trans- formation, the frequency response, stability conditions, and the equation of backbone curve are derived. Results show that in some conditions impact system may have two or four steady-state solutions, which are interesting and not mentioned for a vibro-impact system with the existence of frequency island phenomena. Then, the classification of the steady-state solutions is discussed, and it is shown that the nontrivial steady-state solutions may lose stability by saddle node bifurcation and Hopf bifurcation. Furthermore, a criterion for avoiding the jump phenomenon is derived and verified. Lastly, it is found that the distance between the system＇s static equilibrium position and the barrier can lead to jump phenomenon under hardening type of nonlinearity stiffness.展开更多
In this paper, we give a controlled two-degree-of-freedom （TDOF） vibro-impact system based on the damping control law, and then investigate the dynamical behaviour of this system. According to numerical simulation, ...In this paper, we give a controlled two-degree-of-freedom （TDOF） vibro-impact system based on the damping control law, and then investigate the dynamical behaviour of this system. According to numerical simulation, we find that this control scheme can suppress chaos to periodic orbit successfully. Furthermore, the feasibility and the robustness of the controller are confirmed, separately. We also find that this scheme cannot only suppress chaos, but also generate chaos in this system.展开更多
The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here...The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over ＂fast＂ variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the ease without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.展开更多
The generalized cell mapping(GCM) method is used to obtain the stationary response of a single-degree-of-freedom.Vibro-impact system under a colored noise excitation. In order to show the advantage of the GCM method, ...The generalized cell mapping(GCM) method is used to obtain the stationary response of a single-degree-of-freedom.Vibro-impact system under a colored noise excitation. In order to show the advantage of the GCM method, the stochastic averaging method is also presented. Both of the two methods are tested through concrete examples and verified by the direct numerical simulation. It is shown that the GCM method can well predict the stationary response of this noise-perturbed system no matter whether the noise is wide-band or narrow-band, while the stochastic averaging method is valid only for the wide-band noise.展开更多
This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is fur...This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called “time-2τ^2 map” of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T^1 （Hopf invariant circles）, tori 2T^1 and tori 2T^2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms＇ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.展开更多
A mass-spring-damper linear oscillator with a limiting stop barrier is presented. Modeling non-smooth processes in mechanical engineering is a complex problem. It is especially for the systems with more than a single ...A mass-spring-damper linear oscillator with a limiting stop barrier is presented. Modeling non-smooth processes in mechanical engineering is a complex problem. It is especially for the systems with more than a single degree of freedom. But recent studies in dynamical systems have been applied to single degree of freedom systems. The vibrating system, consisting of an oscillator with amplitude of motion limited by a barrier, is known as a vibro-impact system. The amount of force and kinetic energy transferred to a barrier has an important application in designing of engineering systems that undergo the vibro-impact phenomenon. The results show the effect of changing restitution coefficient of a barrier on the amount of force and energy absorbed.展开更多
This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact conditi...This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact condition is converted to the system energy is applied to obtain the averaged It? stochastic differential equation, by which the associated Backward Kolmogorov(BK)equation and Generalized Pontryagin(GP) equation are derived. Then, the dynamical programming equations are obtained based on the dynamical programming principle, by which the optimal bounded control for maximizing system reliability is devised.Finally, the effects of the bounded control and noise intensity on the reliability of the vibro-impact system are discussed in detail;meanwhile, the influence of impact conditions on the system's reliability is also studied. The feasibility and effectiveness of the proposed analytical method are substantiated by numerical results obtained from Monte-Carlo simulation.展开更多
A method is presented to seek for coexisting periodic orbits which may be stable or unstable in piecewise-linear vibro-impacting systems. The conditions for coexistence of single impact periodic orbits are derived, an...A method is presented to seek for coexisting periodic orbits which may be stable or unstable in piecewise-linear vibro-impacting systems. The conditions for coexistence of single impact periodic orbits are derived, and in particular, it is investigated in details how to assure that no other impacts will happen in an evolution period of a single impact periodic motion. Furthermore, some criteria for nonexistence of single impact periodic orbits with specific periods are also established. Finally, the stability of coexisting periodic orbits is discussed, and the corresponding computation formula is given. Examples of numerical simulation are in good agreement with the theoretic analysis.展开更多
The coexisting periodic impacting motions and their multiplicity of a kind of dual component systems under harmonic excitation are analytically derived. The stability condition of a periodic impacting motion is given ...The coexisting periodic impacting motions and their multiplicity of a kind of dual component systems under harmonic excitation are analytically derived. The stability condition of a periodic impacting motion is given by analyzing the propagation of small, arbitrary perturbation from that motion. In numerical simulations, the periodic impacting motions are classified according to the system states before and after an impact. The numerical results show that there exist many types of vibro-impacts and the bifurcation of periodic vibro-impacts is not smooth.展开更多
As a promising vibration control device, the vibro-impact nonlinear energy sink (VI-NES) gathered extensively attention in recent years. However, general optimization procedures have not been available forthe design o...As a promising vibration control device, the vibro-impact nonlinear energy sink (VI-NES) gathered extensively attention in recent years. However, general optimization procedures have not been available forthe design of VI-NES subjected to random excitations. To this end, this paper constitutes a research effortto address this gap. Specifically, the approximate analytical solution of the system stochastic responseis obtained in conjunction with non-smooth conversion and stochastic averaging methodology. Takingadvantages of this approximate solution, the variance of the system is defined and easily minimized tocalculate the optimal parameters for VI-NES. In addition, the results computed by this way fairly correlatewith direct numeric simulations.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.12172266,12272283)the Bilateral Governmental Personnel Exchange Project between China and Slovenia for the Years 2021-2023(Grant No.12)+2 种基金Slovenian Research Agency ARRS in Frame of Bilateral Project(Grant No.P2-0137)the Fundamental Research Funds for the Central Universities(Grant No.QTZX23004)Joint University Education Project between China and East European(Grant No.2021122).
文摘In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of themean value.Afterwards,the stochastic vibro-impact systemcan be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincarésections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the randomintensitymay result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.
基金supported by the National Natural Science Foundation of China (Grant No. 10972059)the Natural Science Foundation of the Guangxi Zhuang Autonmous Region of China (Grant Nos. 0640002 and 2010GXNSFA013110)+1 种基金the Guangxi Youth Science Foundation of China (Grant No. 0832014)the Project of Excellent Innovating Team of Guangxi University
文摘A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincare map of the system is constructed. Using the Poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11172233,10932009,and 11202160)the Natural Science Foundation of Shaanxi Province,China(Grant No.2012JQ1004)
文摘A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function （PDF） of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11472212,11532011,11302171,and 11302172)
文摘In this paper, multi-valued responses and dynamic properties of a nonlinear vibro-impact system with a unilateral nonzero offset barrier are studied. Based on the Krylov-Bogoliubov averaging method and Zhuravlev non-smooth trans- formation, the frequency response, stability conditions, and the equation of backbone curve are derived. Results show that in some conditions impact system may have two or four steady-state solutions, which are interesting and not mentioned for a vibro-impact system with the existence of frequency island phenomena. Then, the classification of the steady-state solutions is discussed, and it is shown that the nontrivial steady-state solutions may lose stability by saddle node bifurcation and Hopf bifurcation. Furthermore, a criterion for avoiding the jump phenomenon is derived and verified. Lastly, it is found that the distance between the system＇s static equilibrium position and the barrier can lead to jump phenomenon under hardening type of nonlinearity stiffness.
基金supported by the National Natural Science Foundation of China (Grant No 10472091)
文摘In this paper, we give a controlled two-degree-of-freedom （TDOF） vibro-impact system based on the damping control law, and then investigate the dynamical behaviour of this system. According to numerical simulation, we find that this control scheme can suppress chaos to periodic orbit successfully. Furthermore, the feasibility and the robustness of the controller are confirmed, separately. We also find that this scheme cannot only suppress chaos, but also generate chaos in this system.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10772046 and 50978058)Natural Science Foundation of Guangdong Province of China (Grant No. 102528000010000)
文摘The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over ＂fast＂ variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the ease without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.
基金supported by the National Natural Science Foundation of China (Grant No. 11772149)the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,China (Grant No. MCMS-I-19G01)the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD),China。
文摘The generalized cell mapping(GCM) method is used to obtain the stationary response of a single-degree-of-freedom.Vibro-impact system under a colored noise excitation. In order to show the advantage of the GCM method, the stochastic averaging method is also presented. Both of the two methods are tested through concrete examples and verified by the direct numerical simulation. It is shown that the GCM method can well predict the stationary response of this noise-perturbed system no matter whether the noise is wide-band or narrow-band, while the stochastic averaging method is valid only for the wide-band noise.
基金The project supported by the Nutional Natural Science Foundation of China(10472096)
文摘This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called “time-2τ^2 map” of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T^1 （Hopf invariant circles）, tori 2T^1 and tori 2T^2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms＇ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.
文摘A mass-spring-damper linear oscillator with a limiting stop barrier is presented. Modeling non-smooth processes in mechanical engineering is a complex problem. It is especially for the systems with more than a single degree of freedom. But recent studies in dynamical systems have been applied to single degree of freedom systems. The vibrating system, consisting of an oscillator with amplitude of motion limited by a barrier, is known as a vibro-impact system. The amount of force and kinetic energy transferred to a barrier has an important application in designing of engineering systems that undergo the vibro-impact phenomenon. The results show the effect of changing restitution coefficient of a barrier on the amount of force and energy absorbed.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11872305&11532011)Natural Science Basic Research Plan in Shanxi Province of China (Grant No. 2018JQ1088)。
文摘This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact condition is converted to the system energy is applied to obtain the averaged It? stochastic differential equation, by which the associated Backward Kolmogorov(BK)equation and Generalized Pontryagin(GP) equation are derived. Then, the dynamical programming equations are obtained based on the dynamical programming principle, by which the optimal bounded control for maximizing system reliability is devised.Finally, the effects of the bounded control and noise intensity on the reliability of the vibro-impact system are discussed in detail;meanwhile, the influence of impact conditions on the system's reliability is also studied. The feasibility and effectiveness of the proposed analytical method are substantiated by numerical results obtained from Monte-Carlo simulation.
文摘A method is presented to seek for coexisting periodic orbits which may be stable or unstable in piecewise-linear vibro-impacting systems. The conditions for coexistence of single impact periodic orbits are derived, and in particular, it is investigated in details how to assure that no other impacts will happen in an evolution period of a single impact periodic motion. Furthermore, some criteria for nonexistence of single impact periodic orbits with specific periods are also established. Finally, the stability of coexisting periodic orbits is discussed, and the corresponding computation formula is given. Examples of numerical simulation are in good agreement with the theoretic analysis.
基金Project supported in part by National Natural Science Foundation of China under the grant 59572024in part by Trans-century Training Program Foundation for the Talents by the State Education Commission of China
文摘The coexisting periodic impacting motions and their multiplicity of a kind of dual component systems under harmonic excitation are analytically derived. The stability condition of a periodic impacting motion is given by analyzing the propagation of small, arbitrary perturbation from that motion. In numerical simulations, the periodic impacting motions are classified according to the system states before and after an impact. The numerical results show that there exist many types of vibro-impacts and the bifurcation of periodic vibro-impacts is not smooth.
基金This work is supported by the National Natural Science Foundation of China(No.12072118)the National Natural Science Funds for Distinguished Young Scholar of the Fujian Province of China(No.2021J06024)the Project for Youth Innovation Fund of Xiamen(No.3502Z20206005).
文摘As a promising vibration control device, the vibro-impact nonlinear energy sink (VI-NES) gathered extensively attention in recent years. However, general optimization procedures have not been available forthe design of VI-NES subjected to random excitations. To this end, this paper constitutes a research effortto address this gap. Specifically, the approximate analytical solution of the system stochastic responseis obtained in conjunction with non-smooth conversion and stochastic averaging methodology. Takingadvantages of this approximate solution, the variance of the system is defined and easily minimized tocalculate the optimal parameters for VI-NES. In addition, the results computed by this way fairly correlatewith direct numeric simulations.
