Nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to mono-frequency excitation

In this paper, the nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to a mono-frequency excitation is considered. The model of the viscoelastic guide is a parallel combination of springs and viscous dampers. The governing equation of motion is developed using Hamilton’s principle. Applying the method of multiple scales to the governing partial differential equation, the solvability condition and approximate solutions are derived. Three cases, namely primary, subharmonic and superharmonic resonances are studied and appropriate analytical solutions are obtained. The effect of mean value velocity, force amplitude, guide stiffness and viscosity coefficient of the string on the frequency-response and bifurcation points is investigated. Findings are in good agreement with results extracted from numerical modeling.     

轴向变速黏弹性Rayleigh梁非线性参数振动稳态响应

研究非线性轴向运动黏弹性Rayleigh梁因速度周期变化产生的亚谐波共振。轴向运动速度在平均速度附近做简谐周期性脉动。通过取物质导数的Kelvin本构关系描述Rayleigh梁的黏弹性。运用多尺度近似解析方法,构建轴向运动Rayleigh梁的非线性偏微分方程的可解性条件,分析参数振动稳态响应的振幅与扰动速度频率关系。并运用微分求积方法直接离散非线性Rayleigh梁的控制方程,以验证近似解析方法分析。通过数值算例,分析了系统参数对稳态响应曲线的影响。     

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton’s principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli–Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin’s method and the fourth-order Runge–Kutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.     

Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity

Two-to-one parametric resonance in transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity is investigated. The transport speed is assumed to be a constant mean speed with small harmonic variations. The nonlinear partial differential equation that governs transverse vibration of the string is derived from Newton's second law. The method of multiple scales is applied directly to the equation, and the solvability condition of eliminating secular terms is established. Closed-form solutions for the amplitude of the vibration and the existence conditions of nontrivial steady-state response in two-to-one parametric resonance are obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation are presented. Lyapunov's linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions for two-to-one parametric resonance. Some numerical examples highlighting the effects of the related parameters on the stability conditions are presented.     

温度场中简谐激励斜梁的1/3次亚谐共振分析

以温度场中简谐激励斜梁的非线性振动方程为研究对象,应用多尺度法,求得非线性振动系统1/3次亚谐共振的一次近似解。对该解进行数值计算,分析温度、激励、几何尺寸等参数对1/3次亚谐共振幅频响应曲线的影响。随着初始温度和激励幅值的增加,1/3次亚谐共振的振幅和共振区增大。随着温度影响系数和长高比的增加,1/3 亚谐共振的振幅和共振区减小。     

基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析

基于欧拉-伯努利梁理论,利用Lagrange法建立了楔形和锥形截面梁在外激作用下的非线性微分方程.提出了一种基于Bessel函数和Meijer-G函数线性组合的无需迭代及近似截断的振型函数,且该振型函数不依赖于楔形和锥形变截面梁的弯曲振动的运动方程是否为标准的Bessel形式,该方法能快速求解线性基频和模态函数.随后将...     

三缸发动机式增程器扭转振动模糊PID主动抑制

针对增程器系统在发动机转矩波动激励下的多谐波扭转振动问题,提出一种模糊PID控制算法,利用发电机输出扭矩主动抑制扭转振动。建立8自由度轴系强迫振动模型,然后以角加速度以及转速为控制目标,采用模糊PID控制器主动抑制扭转振动。以均平方角加速度为扭转振动的评价指标。分析稳态和瞬态工况下扭转振动主动抑制效果。仿真结果表明,稳态工况下相比只以转速为控制目标的PID控制方式,均平方角加速度降低68.3%;相比传统的PID控制方法,均平方角加速度降低29.0%。所提出的方法可以有效抑制增程器轴系的扭转振动。     

非线性弹性地基上矩形薄板的非线性振动与奇异性分析

研究非线性弹性地基上小挠度矩形薄板的非线性振动，应用弹性力学理论建立非线性弹性地基上小挠度矩形薄板受简谐激励作用的动力学方程，利用Galerkin方法将其转化为非线性振动方程。根据非线性振动的多尺度法求得系统主参数共振-主共振情况的一次近似解，并进行数值计算。分析了阻尼系数、地基系数、激励参数等对系统主参数共振-主共振的影响。系统主参数共振-主共振曲线均具有跳跃现象。随着阻尼、地基系数的改变，系统响应曲线具有“类软刚度特征”。随着参数激励幅值的改变，系统响应曲线具有“类硬刚度特征”。应用奇异性理论得到系统主参数共振-主共振稳态响应的转迁集和分岔图。     

受机械力作用金属／陶瓷功能梯度板主共振奇异性

研究机械力作用下金属，陶瓷功能梯度薄板主共振奇异性问题。按照功能梯度薄板的非线性动力学方程，得到金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统主共振幅频响应分岔方程并进行奇异性分析，求得幅频响应分岔方程在开折参数平面的转迁集和分岔图。     

基础简谐激励下匀速旋转运动电流变夹层梁的振动稳定性

摘 要：建立了基础激励和定轴转动联合作用时电流变夹层梁的运动微分方程,着重研究了基础简谐激励和匀速旋转运动作用时电流变夹层梁的振动稳定性。采用多尺度法获得了梁的一次近似解析解和参激振动失稳的条件。通过对电流变夹层梁在不同激励参数、控制电场和旋转角速度时的振动响应时间历程曲线和对应相图的数值分析,探讨了电场作用下电流变夹层梁的参激振动稳定性。仿真结果表明,在一定的条件下,可以通过控制作用于电流变夹层梁的电场强度来改变系统出现运动不稳定的临界激励幅值,提高结构的动力稳定性。     

粘弹性变速运动梁稳定性的直接多尺度分析

研究粘弹性轴向加速运动梁横向振动的稳定性。将多尺度法直接应用于系统控制方程,导出了消去长期项的可解性条件。利用该条件得到了次谐共振和组合共振的稳定边界。给出的数值算例说明了粘弹性系数对稳定性边界的影响。     

纵向振动粘弹性桩的分叉和混沌运动

研究了轴向周期载荷作用下非线性粘弹性嵌岩桩纵向振动的混沌运动。假定桩和土体分别满足Leaderman非线性粘弹性本构关系和线性粘弹性本构关系,得到的运动方程为非线性积分-偏微分方程;利用Galerkin方法将方程简化,并进行了数值计算。数值结果表明纵向振动的非线性粘弹性桩可以通过准周期分叉的方式进入混沌运动。     

非线性弹性地基上圆形薄板主参数共振-主共振研究

研究非线性地基上圆形薄板受简谐激励的非线性振动问题。按照弹性力学理论建立非线性地基上圆形薄板受简谐激励的动力学方程。利用Galerkin方法将其转化为非线性振动方程,它是达芬-马休型方程。应用非线性振动的多尺度法求得系统主参数共振-主共振条件的一次近似解,并进行数值计算。分析阻尼、地基系数、几何参数等对共振响应曲线的影响。比较了两种地基的计算结果。     

窄带随机激励带集中质量梁的主参激共振

以受窄带随机激励带集中质量柔性梁为研究对象,采用多尺度法导出系统单模态主参激共振调谐方程组,结合FPK方程分析了力-幅特性下联合概率密度随激励幅值所出现的随机跳跃现象,结果显示:在三值响应范围内存在一外扇形峰及一内火山口峰,随着激励幅值减小,外扇形峰减弱而内火山口峰增强,表明响应逐渐从非平凡稳态响应向平凡稳态响应跳跃。     

Nonlinear dynamics of composite laminated cantilever rectangular plate subject to third-order piston aerodynamics

This paper presents the analysis of the nonlinear dynamics for a composite laminated cantilever rectangular plate subjected to the supersonic gas flows and the in-plane excitations. The aerodynamic pressure is modeled by using the third-order piston theory. Based on Reddy’s third-order plate theory and the von Kármán-type equation for the geometric nonlinearity, the nonlinear partial differential equations of motion for the composite laminated cantilever rectangular plate under combined aerodynamic pressure and in-plane excitation are derived by using Hamilton’s principle. The Galerkin’s approach is used to transform the nonlinear partial differential equations of motion for the composite laminated cantilever rectangular plate to a two-degree-of-freedom nonlinear system under combined external and parametric excitations. The method of multiple scales is employed to obtain the four-dimensional averaged equation of the non-automatic nonlinear system. The case of 1:2 internal resonance and primary parametric resonance is taken into account. A numerical method is utilized to study the bifurcations and chaotic dynamics of the composite laminated cantilever rectangular plate. The frequency–response curves, bifurcation diagram, phase portrait and frequency spectra are obtained to analyze the nonlinear dynamic behavior of the composite laminated cantilever rectangular plate, which includes the periodic and chaotic motions.     

Steady-state response of the parametrically excited axially moving string constituted by the Boltzmann superposition principle

Summary.  The steady-state transverse vibration of a parametrically excited axially moving string with geometric nonlinearity is investigated in this paper. The Boltzmann superposition principle is employed to characterize the material property of the string. The method of multiple scales is applied directly to the governing equation, which is a nonlinear partial-differential-integral equation. The solvability condition of eliminating the secular terms is established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the summation resonance are obtained. Some numerical examples showing effects of the viscoelastic parameter, the amplitude of excitation, the frequency of excitation, and the transport speed are presented. Received February 12, 2002; revised October 25, 2002 Published online: May 8, 2003 The research was supported by the National Natural Science Foundation of China (Project No. 10172056).     

受机械力作用金属／陶瓷功能梯度板的超谐共振研究

研究机械力作用下金属／陶瓷功能梯度薄板3次超谐共振问题．按照功能梯度薄板的非线性动力学方程，得到金属／陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统3次超谐共振近似解并进行数值计算。分析阻尼、激励、几何尺寸等参数对系统3次超谐共振幅频响应曲线的影响．     

定轴转动与基础激励下梁的非线性动力学

采用Kane方程，建立了含耦合的几何及惯性非线性项的定轴转动与轴向基础激励联合作用下柔性梁的非线性动力学控制方程组，该方程组不仅包含二次及三次非线性项，而且体现了参数激励与外激励的联合作用。运用多尺度法，研究了匀转速，顺臂安装下悬臂梁的一阶模态主参激共振与外激励1/2次亚谐共振同时作用时梁的一阶近似稳态响应。结果表明，梁的一阶模态幅频特性将受到转速，旋转半径和激励幅值等参数变化的显著影响。     

索-梁组合结构中拉索的非线性响应

研究由桥面振动引起的斜拉索参数共振和亚谐波共振问题。首先,建立索-梁组合结构力学模型,推导了考虑拉索初始垂度的索-梁组合结构非线性动力学方程。然后利用多尺度方法研究斜拉索的参数共振和亚谐波共振,并对稳态解的稳定性进行了分析。最后对斜拉索参数共振和亚谐波共振进行数值模拟,得到不同阻尼及不同初始条件下的拉索时间历程曲线。数值模拟结果表明斜拉索振幅与阻尼有关,但不受拉索初始条件影响。     

受机械力作用金属／陶瓷功能梯度板的亚谐共振研究

研究机械力作用下金属，陶瓷功能梯度薄板1／3次亚谐共振问题。按照功能梯度薄板的非线性动力学方程，得到金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统1／3次亚谐共振近似解并进行数值计算。分析阻尼、激励、几何尺寸等参数对系统1／3次亚谐共振幅频响应曲线的影响。