Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

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.

PERIODIC VIBRO-IMPACTS AND THEIR STABILITY OF A DUAL COMPONENT SYSTEM

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.

Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems

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.

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

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.

Analysis of force and energy density transferred to barrier in a single degree of freedom vibro-impact system

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.

Multi-valued responses and dynamic stability of a nonlinear vibro-impact system with a unilateral non-zero offset barrier

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.

Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation

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.

Dynamical behaviour of a controlled vibro-impact system

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.

Optimization for vibro-impact nonlinear energy sink under randomexcitation

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.

COEXISTING PERIODIC ORBITS IN VIBRO-IMPACTING DYNAMICAL SYSTEMS

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.

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

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.

Stationary response of colored noise excited vibro-impact system

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.

Comparison of the performance and dynamics of the asymmetric single-sided and symmetric double-sided vibro-impact nonlinear energy sinks with optimized designs

The operation of symmetric double-sided and asymmetric single-sided vibro-impact nonlinear energy sinks(DSVI NES and SSVI NES)is considered in this study.The methodology of optimization procedures is described.It is emphasized that the execution of optimization procedures is ambiguous,allows for a great deal of arbitrariness,and requires experience and intuition on the part of the implementer.There are a lot of damper parameter sets providing similar attenuation of the primary structure(PS)vibrations.It is shown that the efficiency of such mitigation for both VI NES types with optimized parameters is similar.However,their dynamic behavior differs significantly.The system with the attached DSVI NES exhibits calm dynamics with periodic motion and symmetrical bilateral impacts on both obstacles.The system with attached SSVI NES exhibits rich complex dynamics when the exciting force frequency is varied.Periodic modes of different periodicity with different numbers of asymmetric impacts per cycle on the PS directly and on the obstacle alternate with various irregular regimes,namely,chaotic mode,intermittency,and crisis-induced intermittency.The regions of bilateral impacts are narrow and located near resonance;they are narrower for a system with an attached DSVI NES.In a system with an attached SSVI NES,there are wider areas of asymmetric unilateral impacts.

单自由度含干摩擦碰撞振动系统周期运动转迁及共存特征

在参数平面和状态平面内,研究了一类单自由度含干摩擦和弹性约束碰撞振动系统周期运动的分布、转迁及共存特征。首先,给出了系统各运动状态(滑动、黏着及碰撞)的衔接条件,并利用响应微扰计算方法得到系统的Floquet矩阵。其次,由相轨线与黏着面的关系将系统在参数平面内的运动划分为六种类型,根据Floquet乘子和滑移分岔发生的几何条件判定相邻周期运动在转迁时的稳定性和分岔类型,并基于胞映射和打靶法分析稳定及不稳定周期运动共存及转迁特征。研究发现,在相邻的多态共存区边界线交点处,系统在不同初始状态下存在分岔共存现象。由于间隙和干摩擦广泛存在于实际机械系统中,研究此类系统的动力学特性,对后续系统参数的设计及各种动力学行为的控制具有重要的意义。

一类双边碰撞振动系统的随机响应研究

碰撞过程凭借瞬时冲击模型得以简化,但采用经典的Newton恢复系数模型为计算带来了误差。本研究引入修正的恢复系数模型取代经典的Newton恢复系数模型,预测了一类宽带噪声激励下双边碰撞振动系统的随机动力学响应。位移的碰撞条件借助自由振动系统被转化为能量的碰撞条件。根据系统的能量水平,系统的运动可被分为双边碰撞和无碰撞振动两类。进一步地,两类运动的平均漂移系数和扩散系数可借助能量包线随机平均法求解获得。在此基础上,建立并求解对应的Fokker-Plank-Kolmogorov (FPK)方程,进而得到系统的稳态响应。最后,将所提方法应用于Duffing振子,讨论了屈服速度、挡板间距和噪声激励对响应的影响,并用蒙特卡罗模拟验证了所述方法的有效性。

宽带噪声激励下碰撞摩擦系统的随机响应和稳定性的研究

研究了宽带噪声激励下碰撞摩擦系统的随机响应和概率为1渐近稳定性.基于非光滑变换和随机平均法得到了碰撞摩擦系统响应的稳态概率密度,并通过与Monte Carlo数值模拟结果对比,验证了上述解析方法的有效性.讨论了摩擦力和碰撞恢复系数对系统稳态概率密度的影响.基于平均Itô微分方程,得到其线性化方程的最大Lyapunov指数的表达式,通过Lyapunov指数确定系统平凡解的稳定性.结果表明,改变碰撞恢复系数和摩擦系数能调整系统的随机稳定性.

THE CRITERION OF HIGHER PERIOD ON CONTROLLING CHAOS IN A VIBRO-IMPACT SYSTEM

针对工程实际中一类碰撞振动系统的混沌行为，设计了一种动态间歇反馈控制策略，仅对不碰撞时刻进行动态反馈，该方法易于操作，选择参数范围较广，能有效节约控制能量，理论分析表明混沌吸引子中的不稳定周期轨道可以稳定化，并以此控制方法为例，采用Poincare映射和功率谱相结合技术能有效判别倍化出的各种形式的时·稳定高周期轨道和nH概周期轨道。该方法能准确识别混沌控制后高倍周期数和概周期吸引不变圈数，它对于碰撞振动系统的优化设计、振动控制及安全运行提供了理论参考。

Reliability and control of strongly nonlinear vibro-impact system under external and parametric Gaussian noises

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.

冲击减振器等效刚度的一种新型设计方法

以某航空电源控制器机箱为研究对象,经分析由于加速度响应量值超过额定加速度限值导致器件出现故障。为了设计合适的减振器等效刚度,有效降低传递到该器件的冲击响应量值,提出一种新型的冲击减振器刚度设计方法。首先将安装减振器后的系统等效为单自由度的质量弹簧系统,计算冲击载荷下加速度响应的解析解,然后借助Matlab的Simulink平台进行联合仿真分析,绘制出冲击响应的传递率随频率比的变化曲线,根据该曲线确定出满足目标传递率的频率设计值;其次借助ANSYS的模态分析功能,研究减振器三向刚度耦合特性对主振动频率的影响,最后筛选出能兼顾目标传递率和系统稳定性的最优减振器刚度设计方案,显著提升了减振器刚度设计效率。获得的主要结论有:1)当目标传递率β_(max)设定为0.9时,根据β_(max)-R曲线确定主频率应设计在25.33 Hz或109.09 Hz;2)当减振器三向刚度设计为X=Z=50 N/mm,Y=663 N/mm时,系统能够满足目标传递率且整体刚性最优。

含间隙双频激励碰撞振动系统的动力学分析

文章研究了一类单自由度双频激励碰撞振动系统的动力学模型,建立了2个在不同频率正弦激励作用下的碰撞振动运动微分方程,给出了含间隙系统双频激励下Poincaré映射的建立方法,推导出该映射的Jacobian矩阵,从而得到相应的特征值。同时,研究了单个频率变化时对周期运动的影响,以及在一定的频率比下不同间隙和频率对周期分布的影响。进一步利用特征值研究了系统随频率变化出现的周期运动、分岔、擦边以及混沌现象,通过Poincaré映射图和相图对分岔点处的特性进行了分析。