自参数动力系统临界状态减振的非光滑方法

系统在达到稳态运动前通常要经历瞬态振动过程。瞬态振动对系统的影响是不可忽略的。在这个过程中作用于系统的激振力频率实际上是由零开始增加到某一工作频率。当激振力频率接近或超过系统的临界频率。系统发生共振，同时振幅达到峰值，并且此共振振幅通常要比系统的稳态振幅大的多。考虑到上述问题。文中分析了两种通过调制激振频率来实现降低临界振幅的方法，即让激振频率按单调线性增加或者分段线性增加到系统工作频率，通过系统相位的调制来实现减振。数值仿真结果表明上述方法能有效地降低临界振幅。同时上述方法工程实际应用中易于实现。     

一类滞回系统混沌振动的数值分析

在不同参数下,Brouc-Wen滞回模型使系统具有软或硬式响应特性,导致系统有非线性振动特性。利用数值方法,本文给出单自由度滞回系统 稳态振动最大振幅与频率之间的关系曲线。分析了滞回参数对硬式响应特性滞回振动系统的分叉与混沌的影响,发现一些新的现象。     

地基土—多自由度非线性结构相互作用体系的非平稳随机动力响应分析

本文应用一个改进的等效线性化方法探讨了地基土——多自由度非线性结构相互作用体系在随机地震荷载作用下的非平稳动力响应问题。对非线性恢复力模型采用一个具有三线性滞回特性的非线性系统。最后将在平稳Gauss过滤白噪声激励下的非平稳随机响应与Monte-carlo法的统计结果进行了比较,得出了较为满意的结果。     

光纤传感器和压电作动片对薄板振动控制的分析

利用双模模态域光纤传感器的模间干涉感知薄板结构中应变的变化、获得薄板结构振动变化状况，进一步利用压电体作为动元件，减小振幅，实现振动的主动控制，采用振型叠加法和直接积分相结合的方法，从理论上论证薄板分别受到正弦波激振力，脉冲激振力和方波激振力时的振动主动控制，并由脉冲激振下的响应得到控制前后的幅频特性曲线和相频特性曲线．。     

汽轮机转子在气流力和油膜力作用下的非线性动力学特性

为了分析转子在油膜力和气流激振力共同作用下的非线性振动特性，本文以短轴承支撑的不平衡刚性对称Jeffcott转子系统为研究对象，首先分析转子在非稳态油膜力作用下的振动特性，然后分析转子在油膜力和气流激振力共同作用下的非线性振动特性。采用数值模拟的方法研究了系统的分岔和混沌特性，计算结果表明，考虑气流激振力和油膜力共同作用下的转子系统与仅考虑油膜力的转子系统相比，在相对进气速度v=30m／s时，随着无量纲转速ω的增加。二者都出现了周期运动和混沌运动多次交替出现的复杂运动特性，但是前者首次出现倍周期分岔和混沌运动时的转运提前，在定转速情况下，随着v的增大，系统最终在经历周期运动之后进入混沌运动，而且圆盘中心的最大振幅随着v的增大而增大。     

变截面阻抗桩纵向振动问题积分变换解

利用拉普拉斯（Laplace)变换,将变截面阻抗桩振动的时域定解问题转换到频域的形式,求得了关于桩顶位移响应及速度响应的传递函数,然后利用求留数的方法求得了桩土系统的脉冲响应函数,利用脉冲响应函数求得了多种不同激振波形条件下（稳态正弦激励,瞬态半正弦激振及叠加有高频干扰波的瞬态半正弦激振等）的桩顶速度响应的表达式,并对桩土系统的频域特性作了深入的研究,得到了许多重要的结论。此外还将稳态正弦激振、态半正弦激振条件下的解与有关文献所采用分离变量法得到的解进行了对比,为两者的正确性提供了重要的证明。     

变化外形的水线对斜拉索风雨激振的激发作用

讨论变化外形的水线对斜拉桥拉索风雨激振的激发作用.首先,建立考虑水线外形变化的斜拉桥拉索风雨激振计算模型,并将由采用变化水线外形的模型得到的拉索最大振幅与固定水线外形模型的计算结果及试验实测结果对比,对模型进行验证;在模型得到验证的基础上,进一步讨论拉索和变化水线外形的水线振动稳定性及水线对拉索风雨激振贡献.计算结果表明,在四个典型工况下,考虑水线外形变化的模型得到的拉索计算振幅固定水线外形模型能更好地与试验实测振幅吻合;理论分析发现拉索速度和水线速度的耦合作用对水线运动的激发起到主导作用;水线的加速发散发生在拉索大幅激振之前.没有水线加速发散的时程中,拉索均不发生大幅激振.     

接地式三要素型动力吸振器性能分析

利用固定点理论优化接地类型的动力吸振器得到的结果可能不是全局最优参数,在选择其他参数时主系统可以获得更小的振幅, 接地类型动力吸振器的优化问题值得进一步研究. 因此,以一种接地式三要素型动力吸振器为对象,通过研究系统参数变化对固 定点位置与主系统最大振幅的影响,得到了此吸振器的局部最优参数并分析了它的性能. 首先建立了此系统模型的运动微分方程, 得到了主系统振幅放大因子,发现系统存在3个与阻尼无关的固定点. 固定点中幅值较大点随系统参数变化的趋势可以代表最大振 幅随系统参数变化的趋势,因此利用盛金公式得到了固定点幅值的表达式. 为了更加精确,进一步使用数值算法得到了最大振幅与 系统参数的关系图,发现系统中存在局部最优参数. 通过对比接地式吸振器与接地三要素型吸振器的最大振幅随系统参数变化的趋 势,得到了接地式三要素型吸振器的局部最优参数,并发现当固有频率比小于局部最优频率比时,接地式三要素型吸振器模型主系 统的最大振幅要远小于接地式动力吸振器模型.      

接地式三要素型动力吸振器性能分析

利用固定点理论优化接地类型的动力吸振器得到的结果可能不是全局最优参数,在选择其他参数时主系统可以获得更小的振幅,接地类型动力吸振器的优化问题值得进一步研究.因此,以一种接地式三要素型动力吸振器为对象,通过研究系统参数变化对固定点位置与主系统最大振幅的影响,得到了此吸振器的局部最优参数并分析了它的性能.首先建立了此系统模型的运动微分方程,得到了主系统振幅放大因子,发现系统存在3个与阻尼无关的固定点.固定点中幅值较大点随系统参数变化的趋势可以代表最大振幅随系统参数变化的趋势,因此利用盛金公式得到了固定点幅值的表达式.为了更加精确,进一步使用数值算法得到了最大振幅与系统参数的关系图,发现系统中存在局部最优参数.通过对比接地式吸振器与接地三要素型吸振器的最大振幅随系统参数变化的趋势,得到了接地式三要素型吸振器的局部最优参数,并发现当固有频率比小于局部最优频率比时,接地式三要素型吸振器模型主系统的最大振幅要远小于接地式动力吸振器模型.     

考虑非稳态风荷载斜拉索风雨激振响应分析

为研究自然风荷载对斜拉桥拉索风雨激振的影响,将数值模拟的非稳态风荷载作用到拉索振动微分方程中,对拉索振动响应进行了详细分析。首先,针对水线初始位置,使用最小二乘法拟合得到水线初始位置方程;接着,采用四阶Runge-Kutta法求解拉索振动响应。通过比较在非稳态风和稳态平均风作用下的拉索响应,发现在非稳态风荷载下拉索最大振幅的变化趋势并没有发生较大改变,皆是随着风速的增大先增大后减小;但拉索的整个振动过程发生了变化,伴随着节拍改变,其最大振幅也出现在不同振动周期内。此外,从风速-振幅曲线知,对频率为1Hz,2Hz和3Hz的拉索,在一定风速范围内,考虑非稳态风荷载的拉索振幅反而更大,而且此时的风速范围也更大。     

Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations

In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.     

Duffing 系统的主-亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.      

Steady-state responses of a belt-drive dynamical system under dual excitations

The stable steady-state periodic responses of a belt-drive system with a one-way clutch are studied. For the first time, the dynamical system is investigated under dual excitations. The system is simultaneously excited by the firing pulsations of the engine and the harmonic motion of the foundation. Nonlinear discrete–continuous equations are derived for coupling the transverse vibration of the belt spans and the rotations of the driving and driven pulleys and the accessory pulley. The nonlinear dynamics is studied under equal and multiple relations between the frequency of the firing pulsations and the frequency of the foundation motion.Furthermore, translating belt spans are modeled as axially moving strings. A set of nonlinear piecewise ordinary differential equations is achieved by using the Galerkin truncation.Under various relations between the excitation frequencies,the time histories of the dynamical system are numerically simulated based on the time discretization method. Furthermore, the stable steady-state periodic response curves are calculated based on the frequency sweep. Moreover, the convergence of the Galerkin truncation is examined. Numerical results demonstrate that the one-way clutch reduces the resonance amplitude of the rotations of the driven pulley and the accessory pulley. On the other hand, numerical examples prove that the resonance areas of the belt spans are decreased by eliminating the torque-transmitting in the opposite direction. With the increasing amplitude of the foundation excitation, the damping effect of the one-way clutch will be reduced. Furthermore, as the amplitude of the firingpulsations of the engine increases, the jumping phenomena in steady-state response curves of the belt-drive system with or without a one-way clutch both occur.     

内共振条件下轴向运动梁磁弹性超谐共振研究

以载流导线激发的磁场中轴向运动梁为研究对象,同时考虑外激励力作用,推导出梁的磁弹性非线性振动方程．通过位移函数的设定和伽辽金积分法,将非线性振动方程离散为常微分方程组．采用多尺度法得到系统的近似解析解．应用Matlab 和Mathematica 软件求解幅频响应方程,并对稳态解进行稳定性判定．通过具体算例得到前两阶假设模态的响应幅值随不同参数的变化规律．结果发现：系统在内共振条件下发生超谐波共振时,二阶假设模态幅值明显小于一阶;随着外激励的增大,多值解区域范围明显缩小;随着电流强度增加,振动幅值减小,表明载流导线能够起到控制共振的作用．     

Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations

The resonant resonance response of a single-degree-of-freedom non-linear vibro-impact oscillator, with cubic non-linearity items, to combined deterministic harmonic and random excitations is investigated. The method of multiple scales is used to derive the equations of modulation of amplitude and phase. The effects of damping, detuning, and intensity of random excitations are analyzed by means of perturbation and stochastic averaging method. The theoretical analyses verified by numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under certain conditions, impact system may have two steady-state responses. One is a non-impact response, and the other is either an impact one or a non-impact one.     

多尺度法的推广及在非线性黏弹性系统的应用

黏弹性材料在航空、机械、土木等领域具有广阔的应用前景,而具有1.5自由度的非线性Zener模型能更好地描述其特性.因此,研究多尺度法的推广和应用具有重要意义.在传统多尺度法的基础上,推广并利用多尺度法对非线性奇数阶微分方程进行研究,解决非线性奇数阶系统的动力学求解问题.以非线性Zener模型为例,首先通过推广的多尺度法...     

Dynamics of a Jeffcott rotor-magnetic bearing system with time delays

A Jeffcott rotor with an additional magnetic bearing locating at the disc is employed to investigate the effect of time delays on the non-linear dynamical behavior of the system. The time delays are presented in the proportional and derivative feedback, respectively. For the corresponding autonomous system, a linear stability analysis is performed for the system with two identical time delays in the control loop. The nature of a single Hopf bifurcation is determined by constructing a center manifold. For the non-autonomous system, the primary resonance response is studied for its small non-linear motions using the method of averaging. The effects of time delays and control gains, as well as excitation amplitude, on the amplitude of the steady-state response are investigated. Finally, experiments are carried out to validate the theoretical predictions.     

One-to-Two Internal Resonance in Two-Degrees-of-Freedom Nonlinear System with Narrow-Band Excitations

Response of two-degrees-of-freedom nonlinearsystem to narrow-band random parametric excitation isinvestigated. The method of multiple scales is used todetermine the equations of modulation of amplitude andphase. The effect of detunings and amplitude areanalyzed. Theoretical analyses and numerical simulationsshow that the nontrivial steady-state solution may changeform a limit cycle to a diffused limit cycle as theintensity of the random excitation increase. Under someconditions, the system may have two steady-statesolutions.     

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.     

Forced response of quadratic nonlinear oscillator: comparison of various approaches

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of different analytical approximate approaches. The forced vibration of snap-through mechanism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincaré method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are compared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method predicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the LindstedtPoincaré method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.