Primary resonance of traveling viscoelastic beam under internal resonance

Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The unde- termined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.

Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

A homotopy analysis method（HAM）is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces.The validity of the HAM is independent of the existence of small parameters in the considered equation.The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter.Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar＇e method and the incremental harmonic balance method.

Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation

The chaotic dynamic snap-through and complex nonlinear vibrations are investigated in a rectangular asymmetric cross-ply bistable composite laminated cantilever shell,in cases of 1:2 inter-well internal resonance and primary resonance.The transverse foundation excitation is applied to the fixed end of the structure,and the other end is in a free state.The first-order approximate multiple scales method is employed to perform the perturbation analysis on the dimensionless two-degree-of-freedom ordinary differential motion control equation.The four-dimensional averaged equations are derived in both polar and rectangular coordinate forms.Deriving from the obtained frequency-amplitude and force-amplitude response curves,a detailed analysis is conducted to examine the impacts of excitation amplitude,damping coefficient,and tuning parameter on the nonlinear internal resonance characteristics of the system.The nonlinear softening characteristic is exhibited in the upper stable-state,while the lower stable-state demonstrates the softening and linearity characteristics.Numerical simulation is carried out using the fourth-order Runge-Kutta method,and a series of nonlinear response curves are plotted.Increasing the excitation amplitude further elucidates the global bifurcation and chaotic dynamic snap-through characteristics of the bistable cantilever shell.

Bifurcation control of nonlinear oscillator in primary and secondary resonance

A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.

Optimal Delayed Control of Nonlinear Vibration Resonances of Single Degree of Freedom System

The primary resonance of a single-degree-of-freedom(SDOF)system subjected to a harmonic excitation is mitigated by the method of optimal time-delay feedback control.The stable regions of the time delays and feedback gains are obtained from the stable conditions of eigenvalue equation.Attenuation ratio is applied for evaluating the performance of the vibration control by taking aproportion of peak amplitude of primary resonance for the suspension system with or without controllers.Taking the attenuation ratio as the objective function and the stable regions of the time delays and feedback gains as constrains,the optimal feedback gains are determined by using minimum optimal method.Finally,simulation examples are also presented.

气动力及外部荷载下覆冰输电线共振特性分析

为进一步研究覆冰输电线的非线性振动问题,在考虑气动荷载和外部激励的前提下分析输电线的舞动特征,建立覆冰输电线在气动荷载及外部激励作用下的力学模型,推导模型的非线性舞动方程。基于Galerkin方法将推导的非线性舞动方程转换成常微分方程,利用多尺度法得到模型的主共振及谐波共振的幅频响应函数。数值分析结果表明:输电线产生主共振时,响应幅值随着调谐参数的变化会出现同步与失步现象,外部激励荷载或风速的增大均会使系统响应幅值增大并呈多值性及不稳定性;张力的增大使系统响应幅值减小,并减弱其非线性特性;谐波共振的响应幅值随外部激励荷载或风速的增大而增大,呈现多值性和跳跃性;共振峰向调谐参数右方向偏移,系统呈硬弹簧特性。

有界窄带激励下手臂系统主共振分析

应用多尺度法,对随机窄带激励下人体手臂与振动机械耦合系统的第1阶主共振进行分析,建立手臂系统的二自由度非线性动力学方程。依次求解系统第1阶主共振的幅频响应方程和Ito随机微分方程。为研究随机振动下变量的分布规律,使用矩估计法计算第1阶主共振随机均方响应期望和均方值的近似表达式。通过数值分析得到人体手臂系统内参数对主共振的影响。结果表明,第1阶主共振的均方响应曲线与它们的幅频响应曲线结构相近具有拓扑关系,增大外激励的幅值主共振振幅的均方值和共振区域均增大,而系统阻尼的增大则会使均方值和共振区域减小。非线性刚度系数的改变对系统影响很小。

空间索面悬索桥的非线性主共振参数分析

针对空间索面悬索桥的非线性振动问题,采用一种基于连续介质理论的广义模型开展了非线性主共振参数分析.该模型同时考虑了悬索桥的桥面竖向运动和缆索竖向-水平运动的几何非线性特点.基于该模型和时间多尺度法,推导了求解空间索面悬索桥的竖弯模态主共振响应的调制方程.研究表明,调制方程中的非线性系数会显著影响系统主共振的最大响应幅值,并影响竖弯振动模态的硬化或软化特性.此外,主缆与吊杆的倾角、主缆垂跨比、主梁与主缆的抗拉刚度比等系统参数对非线性系数同样存在显著影响,特别是当系统参数位于奇异点附近时,系统的动力特性会随梁体和主缆的对称振动模态对应的非线性系数突变而剧烈变化,在系统设计中应避免这种情况.该研究能够为该类桥梁的结构设计提供参考.

Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations

The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin’s method and the multi-scale method are used to obtain ordinary differential equations (ODEs) of the system and their modulation equations, respectively. Frequency- and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin’s integration are conducted. The analysis shows that the two results have a slight difference;however, they both have sufficient accuracy to solve the proposed dynamic system.

Nonlinear resonances phenomena in a modified Josephson junction model

In this paper,the equivalent circuit of the non-autonomous Josephson junction(JJ)is presented and the effect of the proper frequency on the phaseφis studied.We also study nonlinear resonance phenomena in the oscillations of a modified Josephson junction(MJJ).These oscillations are probed through a system of nonlinear differential equations and the multiple time scale method is employed to investigate all different types of resonance that occur.The results of primary,superharmonic and subharmonic resonances are obtained analytically.We show that the system exhibits hardening and softening behaviors,as well as hysteresis and amplitude hopping phenomena in primary and superharmonic resonances,and only the hysteresis phenomenon in subharmonic resonance.In addition,the stabilities and the steady state solutions in each type of resonances are kindly evaluated.The number of equilibrium points that evolve with time and their stabilities are also studied.Finally,the equations of motion are numerically integrated to check the correctness of analytical calculations.We further show that the dynamics of the MJJ is strongly influenced by its parameters.

时滞反馈下非正交面齿轮主共振特性的多尺度法研究

非正交面齿轮传动可以满足轴交角在0°到180°之间任意角的非正交传动形式,建立了含时滞反馈的非正交面齿轮传动系统的非线性动力学模型,考虑了时变啮合刚度、传动误差、齿侧间隙和输入扭矩波动等因素。此外,采用多尺度法对系统的主共振特性进行分析,判定了系统的主共振稳定性条件。用数值方法分析了时滞控制参数、啮合阻尼、时变啮合刚度波动幅值和载荷波动对系统幅频特性的影响。结果表明:在控制过程中应合理选择控制参数以避免主共振振幅过大和产生不稳定分支;适当的啮合阻尼有利于抑制系统主共振的振幅和缩减不稳定分支;过高的激励频率易产生主共振的不稳定分支;主共振的不稳定分支随着啮合刚度的波动的增加逐渐缩减,但是在激振频率接近主共振频率时,较小的啮合刚度波动也会导致系统失稳;载荷波动的增加会导致系统主共振幅值增加,对系统的稳定性造成损害。

矩形薄板热电磁弹耦合作用下的主共振

本文研究了考虑热、电、磁、力多物理场作用下的矩形薄板主共振。根据冯卡门大挠度理论和哈密顿原理得到模型的非线性控制方程,使用谐波平衡法分析得到了系统的一阶近似响应及幅频函数,分析表明模型表现出硬非线性、跳跃等现象。对不同温差、阻尼、板厚、磁场、模态、长宽比对主共振的影响进行了分析,得到了幅频响应曲线,并使用MATLAB的龙格库塔方法对其进行了数值验证。结果表明,温差是影响矩形薄板主共振的最大因素,磁场是影响矩形薄板主共振最小因素,这是由于材料的压磁系数和热弹耦合系数的差异所致。

立方速度反馈控制的准零刚度隔振器动力学特性分析

针对传统线性被动隔振系统中存在降低固有频率及提高承载力、抑制共振峰值及改善高频传递特性这两对矛盾,本文采用斜置弹簧作为负刚度调节机构与正刚度弹性元件并联组成准零刚度隔振器,并将立方速度反馈控制策略引入到准零刚度隔振器的主动控制中。采用多尺度法求得受控系统的主共振、亚谐共振响应,利用Sylvester结式法获得系统不发生跳跃现象的临界激励幅值和跳跃频率并研究系统的传递特性。与线性隔振系统相比,准零刚度隔振器具有高静低动刚度特性,能提高系统的承载力并降低固有频率。与被动隔振系统相比,采用立方速度反馈控制的准零刚度隔振器主共振峰处的幅值及亚谐共振存在区域逐渐减小,力传递率共振峰处的幅值降低且高频传递特性得到有效改善,对提高准零刚度隔振器的隔振性能具有理论指导意义。

机电非线性振动系统参、强联合激励的分岔

建立机电耦联非线性电机模型的数学方程。利用非线性振动理论中的非固定参数的渐近法 ,研究由电磁力激发的参数激励、强迫激励联合作用下非线性振动系统的动力学特性 ,由理论分析、数值计算得到系统的分岔转迁集和四种分岔响应曲线。在实验的基础上揭示各种电磁参数及机械参数对主参数共振的振幅、运动稳定性及共振曲线的拓扑结构的影响 ,同时显示系统的运动状态及其稳定域 ,分析结果为有效地控制电机的稳定运行提供依据。

基于三次迟滞模型的超声电机圆环定子主共振响应

将改进的三次多项式迟滞模型用于描述压电材料的弹性迟滞非线性特性,建立了压电材料二维非线性本构关系。根据Hamilton原理和Rayleigh-Ritz假设模态方法,建立了不考虑界面力时旋转行波超声电机圆环定子驻波振动的非线性动力学模型。用多尺度法求解定子的一次近似主共振响应,通过定常解分析,发现定子主共振响应中存在振幅跳跃和多解现象,着重分析了压电材料弹性迟滞非线性参数对主共振响应的影响。结果表明,迟滞参数α使幅频响应曲线左偏并出现多解现象,迟滞参数α和β同时影响系统响应振幅的大小。数值计算验证了解析解的正确性。从理论上揭示了压电材料弹性迟滞非线性对圆环定子主共振响应的影响,为超声电机的优化设计和控制提供了理论依据。

悬臂梁智能结构主共振响应的最优化控制

应用最优化控制方法研究悬臂梁主共振响应的减振控制。由主共振振动定常解的稳定条件得到反馈控制增益的参数范围。以主共振有控制振动峰值与无控制振动峰值的比值为减振的衰减率,以该衰减率为目标函数,反馈控制增益参数范围为约束条件,利用最优化方法计算得到衰减率最小的速度反馈控制参数。计算发现速度反馈控制参数数值越大,衰减率越小,控制效果越好。以非线性振动系统的能量函数为目标函数,计算得到能量函数值最小的位移反馈控制参数,实现智能结构体主共振响应的控制器设计。由悬臂梁主共振振动实验实测数据计算出最优化控制参数,设计非线性主共振振动实验控制系统,进行减振控制实验研究。

两端水平激励斜拉索面内非线性振动响应

为研究两端水平激励作用下斜拉索面内非线性振动,利用Galerkin方法得到斜拉索的离散运动方程。利用多尺度法得到亚谐波共振和主共振时面内一阶振动幅值的表达式。由方程的推导过程可知:两端水平激励斜拉索在亚谐波共振时可等效为一端水平激励斜拉索,将上、下端激励幅值叠加后按一端激励计算;在主共振时则与一端水平激励有本质区别,不能按上述方式等效。算例分析结果较好地验证了该结论。

主共振与内共振下纵横耦合轴系动力学分析

考虑Von Karman非线性位移-应变关系,利用Hamilton原理建立了轴系纵横耦合下的动力学模型。利用Galerkin法对偏微分方程进行离散,采用多尺度法求解了离散方程。研究了纵向主共振并伴随内共振(由纵向第一阶固有频率近似等于横向第一阶正进动与反进动频率之和而产生)联合激励时轴系的动力学响应。研究表明随着系统参数以及激励载荷的不同,轴系出现不同的动力学特性。当激励载荷小于一临界值时,纵向激励力只能激起纵向振动,系统响应与线性系统一样;当载荷超过临界值时,纵向激励力同时激起了轴系的横向正进动与反进动频率,此时纵向振动出现能量饱和现象,能量从纵向渗透到横向。能量在正反进动模态间的分配与其正反进动频率成反比,从而使反进动幅值大于正进动幅值。同时响应中也出现跳跃现象。数值分析结果与摄动分析结果一致。

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。

非线性弹性地基上矩形薄板的主参数共振

分析非线性弹性地基上受参数激励小挠度矩形薄板的主参数共振问题,由冯卡门方程和伽辽金方法得到系统的非线性振动方程,它是杜分-马休型方程,应用非线性振动的多尺度法得到平均方程。数值计算结果表明:阻尼系数、地基系数、几何参数对主参数共振曲线影响明显。