微分本构粘弹性轴向运动弦线横向振动分析的差分法

给出了微分本构粘弹性轴向运动弦线横向振动数值仿真的一种差分法．文中建立了具有微分本构的粘弹性运动弦线的横向振动模型;通过对系统的控制方程和本构方程在不同的分数节点离散,得到一种新的差分方法．利用这一方法,弦线振动方程的数值计算过程可以交替地显式进行,且有较小的截断误差和好的数值稳定性．与通用的方法比较,新的方法计算简单、方便．文中利用方程的不变量检验了数值结果的可靠性,并利用这一方法给出了一类弦线模型的参数振动分析．     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

末端质量作用下变长度轴向运动梁的振动特性

对带集中质量,变长度(或速度)轴向运动梁的振动特性采用两种精确方法求解.首先,对变长度轴向运动Euler(欧拉)梁横向自由振动方程进行化简,通过复模态分析得到本征方程,并在有集中质量的边界条件下得到频率方程,用数值方法求解固有频率和模态函数.然后,采用有限元方法建立运动梁自由振动的方程,求解矩阵方程得到复特征值和复特征向量,结合形函数得到复模态位移.最后,将两种方法的计算结果进行了分析和对比.数值算例的结果表明:不同的轴向运动速度和集中质量对变长度轴向运动梁的振动特性有显著影响,两种计算方法的结果接近且均有效.     

一类具积分项非线性梁方程整体解的存在性

讨论了一类在轴向载荷作用下,有限长粘弹性梁的非线性振动问题,用Galerkin方法证明了问题的整体强解和整体经典解的存在性.     

利用模糊系统求解非线性Fredholm-Ⅱ积分方程

提出了一种利用模糊系统求解非线性Fredholm-Ⅱ积分方程解析解的方法:首先将积分方程转化为模糊系统,然后利用具有紧支集和正规性的尺度函数构造模糊基函数和积分方程的模糊解,最后构造能量误差函数,通过最小化误差函数学习模糊系统的参数.数值实验表明:用模糊系统求得的便于运算的解析解比用Galerkin方法求得的数值解精度高.     

非线性粘弹性柱的稳定性和混沌运动

研究了受轴向周期力作用的各向同性简支柱的动力学稳定性。假定粘弹性材料满足Lea-derman非线性本构关系。导出运动方程为非线性偏微分-积分方程,并利用Galerkin方法简化为非线性微分-积分方程。应用平均法进行了稳定性分析,并用数值结果进行验证。数值结果还表明系统可能存在混沌运动。     

大挠度圆柱壳在温度场中的热弹耦合振动分析

对温度场与与应力场耦合时的圆柱壳的非线性热弹耦合的振动问题,推导得到了基本的振动方程,热传导方程和协调方程,对短圆柱壳运用伽辽金(Galerkin)法求解,得出振幅随时间变化的数值解,得到一些有价值的结论．即随着温度幅值和耦合系数的增大,振动衰减的速度变缓,热弹耦合效应减弱．随着长径比、长厚比的增大,振幅衰减的速度变快,同时热振动频率也随之增大,即热弹耦合效应增强．耦合系数越大,轴向应力、轴向力以及轴向弯矩越小．     

平带驱动系统耦合振动的模态分析

使用模态分析的方法研究平带驱动系统的耦合振动．平带驱动系统是由连续的可模型化为弦线的平带,离散的滑轮和一个张紧臂组成的混合系统．从控制方程推导得到了系统的特征方程．通过数值计算,研究了轴向运动速度和初始张力对系统频率的影响．     

隔水套管波流联合作用下非线性动力响应

考虑流及波流联合作用,研究了深水套管的涡激非线性振动．将套管简化为梁模型,计及Morison非线性流体动力和涡激荷载,建立套管的涡激振动方程．采用Korolov函数求解套管的固有频率和模态,提出了计算涡激非线性动力响应的Galerkin方法,计算了160 m水深中170 m长套管的固有频率和模态,研究了流引起的主共振和波流联合引起的组合共振．计算结果表明波流联合作用下套管的动力响应明显增大,结果也揭示了波流联合激励下套管复杂的动力响应特性．     

3∶1内共振下超临界输液管受迫振动响应

首次研究了超临界流速输液管在3∶1内共振条件下的稳态幅频响应.考虑超临界速度引起的管道屈曲位形,建立描述连续体非线性振动的偏微分-积分方程.通过Galerkin截断方法,将连续体方程离散化.对于同时含有平方与立方非线性的多自由度系统,发展高阶多尺度法建立可解性条件.稳态幅频响应曲线揭示了内共振条件下,不同模态间能量的转移.最后,数值仿真结果验证了近似解析分析的有效性.     

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity

Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.     

Bifurcation and chaos of an axially accelerating viscoelastic beam

This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin–Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial–differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincaré map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam.     

Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation

This paper investigates bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams. The Kelvin model is used to describe the viscoelastic property of the beam material, and the Lagrangian strain is used to account for geometric nonlinearity due to small but finite stretching of the beam. The transverse motion is governed by a nonlinear partial-differential equation. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. When the Galerkin truncation is based on the eigenfunctions of a linear non-translating beam subjected to the same boundary constraints, a computation technique is proposed by regrouping nonlinear terms. The scheme can be easily implemented in practical computations. When the transport speed is assumed to be a constant mean speed with small harmonic variations, the Poincaré map is numerically calculated based on 4-term Galerkin truncation to identify dynamical behaviors. The bifurcation diagrams are present for varying one of the following parameter: the axial speed fluctuation amplitude, the mean axial speed and the beam viscosity coefficient, while other parameters are unchanged.     

Stabilization for the Kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control

The first objective of this paper is to make the mathematical model for vibration suppression of an axially moving heterogeneous string. In order to describe the geometrical nonlinearity due to finite transverse deformation, the exact expression of the strain is used. The mathematical modeling is derived first by using Hamilton’s principle and variational lemma and the derived nonlinear PDE system is the Kirchhoff type equation with boundary feedback control. Next, we show the existence and uniqueness of strong solutions of the PDE system via techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of the Faedo–Galerkin method and estimate a decay rate for the energy. The theoretical results are assured by numerical results of the solution’s shape and asymptotic behavior for the system.     

粘弹性运动带动力响应分析

基于Kelvin粘弹性材料本构模型及带运动方程,建立了运动带非线性动力学分析模型.基于该模型和Lie群分析方法推导了匀速运动及简谐运动带线性问题的解析解;基于该非线性模型的数值仿真讨论了运动带材料参数、带稳态运动速度、扰动速度对系统动态响应的影响.结果表明:1)当带匀速运动时,无论系统是线性还是非线性,运动带横向振动"频率"都随着带运动稳态速度增加而减小.2)随着材料粘性增加,系统耗散能力逐渐增强,动态响应逐渐减小.3)当带运动速度简谐波动时,系统动态响应随扰动速度增大而增大.扰动频率对带横向振动影响较大.     

轴向运动三参数黏弹性梁弱受迫振动的渐近分析

研究了轴向运动三参数黏弹性梁的弱受迫振动．建立了轴向运动三参数黏弹性梁受迫振动的控制方程．使用多尺度法渐近分析了运动梁的稳态响应,导出了解稳定性边界方程、稳态振幅的表达式以及稳态响应非零解的存在条件．依据Routh-Hurwitz定律决定了非线性稳态响应非零解的稳定性．     

车路耦合非线性振动高阶Galerkin截断研究

将移动车辆模型化为运动的两自由度质量-弹簧-阻尼系统,道路模型化为立方非线性黏弹性地基上的弹性梁,并将路面不平度设定为简谐函数．通过受力分析,建立车路非线性耦合振动高阶偏微分方程．采用高阶Galerkin截断结合数值方法求解耦合系统的动态响应．首次研究不同截断阶数对车路耦合非线性振动动态响应的影响,确定Galerkin截断研究车路耦合振动的收敛性．研究结果表明,对于软土地基的沥青路面,耦合振动的动态响应,需要150阶以上的截断才能达到收敛效果．并通过高阶收敛的Galerkin截断研究了系统参数对车路耦合非线性振动动态响应的影响.     

Calculation of nonlinear vibrations of a viscoelastic bar

The problem of forced nonlinear vibrations of a viscoelastic bar is reduced by the Bubnov—Galerkin method to a nonlinear integro-differential equation. Transverse vibrations with small amplitude are considered, and therefore the approximate solution is constructed by the small-parameter method.