分数导数粘弹性模型的矩形板的振动分析

利用分数阶Kevin粘弹性模型,建立矩形薄板的动力学方程,并利用拉普拉斯变换及其逆变换给出四边简支粘弹性薄板的解析解,并着重分析在常值荷载作用下,分数阶Kevin粘弹性模型的分数阶参数、粘性参数和模量参数对挠度的影响。结果表明,随着粘性参数和分数阶参数的增大,粘弹性板的挠度变小;随着模量参数增大,粘弹性板的挠度变大。     

分数导数粘弹性土层模型中桩基竖向振动特性研究

粘弹土体的应力-应变关系利用分数导数粘弹性模型进行描述,建立了分数导数粘弹性土层的竖向振动控制方程。在考虑三维波动的条件下,利用势函数和分离变量的方法求解了分数导数粘弹性土层的竖向振动。考虑桩土边界条件和接触条件对分数导数粘弹性土中桩基的竖向振动进行了研究,分析了主要桩土力学参数对桩顶复刚度和导纳的影响规律。研究表明：...     

具有分数导数本构关系的粘弹性浅拱的非线性动力学行为

利用分数导数本构模型描述材料的粘弹性特性,建立了粘弹性浅拱在横向荷载作用下的动力学方程。利用Galerkin截断法并结合边界条件分别得到了一阶和二阶Galerkin系统的控制微分方程。通过数值计算,分析了简谐激励下一阶Galerkin系统的非线动力学行为。研究表明:随着外激励幅值的变化,粘弹性浅拱系统可以通过倍周期分岔或阵发性两条路径进入混沌;固定外激励幅值、频率以及阻尼系数等状态参数,不同初始条件下,系统可以出现多周期解共存、周期解与混沌解共存的现象。     

高速光驱粘弹性阻尼减振框架结构的动力学方程及有限元数值解

对安装有粘弹性阻尼器的高速光驱框架结构进行了理论建模，粘弹性阻尼器采用分数Kelvin固体模型，建立了高速光驱粘弹性阻尼减振框架结构的分数阶动力学有限元方程，并利用Newmark数值积分法得到了数值解     

分数导数微分算子描述的粘弹性土层中群桩的水平振动研究

将桩周土体视为粘弹性介质,利用分数导数粘弹性模型描述土体的力学特性,在Novak 平面应变假定的基础上,借助于势函数并考虑土体边界条件求得了分数导数微分算子描述的粘弹性土层水平位移的衰减函数以及分数导数粘弹性土层的刚度和阻尼系数。利用Winkler 动力弹簧-阻尼器模型模拟桩-土之间的动力相互作用,并在此基础上利用初始参数法求解了分数导数粘弹性土层中桩-桩水平动力相互作用和群桩的水平振动问题。以数值算例的形式讨论了分数导数微分算子的阶数和土体的模型参数对分数导数微分算子描述的粘弹性土层水平位移的衰减函数和群桩的水平动力阻抗的影响。研究表明:分数导数微分算子的阶数对土层水平位移的衰减函数的影响与桩间距和荷载方向角有关;分数导数粘弹性土中群桩的动力阻抗可以退化到经典粘弹性和弹性情况;分数导数微分算子的阶数和土体模型参数对群桩水平动力阻抗有较大影响。     

基于刚柔耦合的液压挖掘机机械臂非线性动力学研究

为准确描述液压挖掘机机械臂动动力学模型,根据柔性多体动力学理论,采用模态函数描述臂架的弹性变形,利用LAGRANGE定理和虚功原理建立挖掘机臂架系统刚柔耦合的非线性动力学方程。对已建立的动力学方程利用MATLAB进行数值求解, 运用仿真软件ADAMS及NASTRAN建立液压挖掘机机械臂刚柔耦合模型并进行仿真分析,通过对比二者结果表明动力学方程建模方法的正确性。运用数值求解的方法进行模态计算和动力学响应分析,求解相关几何参数的一阶固有频率灵敏度,分析了影响机械臂动力学特性的主要模态参数,为进一步研究其结构优化及运动精度控制提供依据。     

路面基础激励下分数导数型包装系统 核心部件振动特性研究

研究分数导数型包装系统关键部件在路面激励下的动力学响应,建立了二自由度分数导数型包装系统动力学方程并求解,得到了核心部件位移的解析解和振幅比的表达式。并讨论了质量比、频率比、阻尼比等结构参数对隔振系统隔振效果的影响。     

基础竖向多频参数激励下圆弧拱平面内动力失稳研究

地震波、冲击波、环境振动激励会通过地基基础传递到拱上,致使拱发生动力失稳失去承载能力。为深入研究拱在基础竖向激励下的动力稳定性,该文基于能量法,建立了基础竖向激励下圆弧拱平面内动力稳定能量方程,利用哈密顿原理得到了拱面内径向和切向振动的耦合控制方程,求解了圆弧拱平面内失稳前的动轴力与动弯矩解析解。引入拱轴线不可压缩假设,解决了圆弧拱平面内动力控制方程的解耦问题。利用伽辽金法建立了基础竖向多频激励下圆弧拱平面内二阶常微分动力稳定方程,运用多尺度法推导了基础竖向多频激励下圆弧拱平面内动力失稳的临界激励频率解析公式,得到了圆弧拱同时发生一阶反对称参数共振和二阶正对称共振失稳的动力不稳定域,并利用有限元数值分析验证了理论解析解的正确性。进一步分析了拱矢跨比、长细比和圆心角对动力不稳定域的影响。     

饱和分数导数型粘弹性土层竖向振动放大效应

将土骨架视为具有分数阶导数本构关系的粘弹性体,采用Biot动力固结方程,在频率域内研究了饱和分数导数粘弹性土层的土骨架粘性、土层厚度等对竖向振动放大系数的影响。通过动力控制方程解耦和边界条件束缚,给出了经典弹性饱和土、分数导数型粘弹性饱和土和经典粘弹性饱和土三种情况下饱和土层的位移、应力和孔压解析表达式。考察了饱和土各物理和几何参数对竖向振动放大系数的影响,结果表明:在不同土层厚度时,经典弹性饱和土、分数导数型粘弹性饱和土及经典粘弹性饱和土的竖向振动放大系数各不相同;分数导数模型的材料参数对振动放大系数有较大影响。     

脉动流作用下粘弹性直管动力学特性分析

在Hamilton体系中应用精细积分法分析脉动流作用下粘弹性管道的动力学响应特性。首先建立了Hamilton体系下输送振荡流粘弹性直管的辛对偶正则方程组,接着推导了求解非线性运动方程的线性插值精细积分法,最后分析了不同流速、不同激振频率下输流直管非线性动力学特性。数值计算结果表明该方法能快速高效求解输流管道运动方程,计算精度令人满意。     

Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam with Finite Deformation

The motion equations governing the dynamical behavior of a viscoelasticTimoshenko beam with finite deformation are derived and simplified byGalerkin method. The viscoelastic material is assumed to obey thethree-dimensional fractional derivative constitutive relation. Thedynamical behaviors of the simplified systems with order 1 and order 2are numerically computed and compared by using the computational methodpresented by the authors. The dynamical behaviors of the systems areuniform qualitatively, but there is a little deviation quantitatively.And the truncated system with order 1 is safer than the one of order 2.It is also shown that the lower order system is reasonable. Theinfluences of the load parameter and the fractional derivative parameter(material parameter) on the deflection of the beam are consideredrespectively. The numerical methods in nonlinear dynamics, such as phasediagram, and Poincaré section, are applied to reveal dynamical behaviorsof the nonlinear viscoelastic Timoshenko beam. There are plenty ofdynamical behaviors, such as periodicity, bifurcation, quasi-periodicityand chaos in the dynamical system.     

基于时滞惯性流形的浅拱动力屈曲研究

从动力学观点,浅拱受冲击是一种无穷维或者连续的动力系统,论文针对抛物线浅拱,应用有关薄壁结构的基本理论和非线性几何关系推导并建立其控制微分方程。然后,利用时滞惯性流形的思想,提出一种求解这类强非线性偏微分方程的新方法,即基于时滞惯性流形的非线性Galerkin方法。通过这种方法,把原始方程的解投影到由控制方程中线性算子的特征函数所张成的完备空间内,并构造出无限维子空间内的动力行为与有限维子空间内的动力行为之间的耦合作用,该耦合作用认为高低阶分量间的相互作用并不是一种简单的瞬时行为,而是与模态发展的历史有关。通过数值分析得到：系统存在两个稳定平衡位置,与传统的Galerkin方法相比,所提出的基于时滞惯性流形的非线性Galerkin方法可以大幅度地降低方程的维数,提高计算速度,有效地降低对计算机内存的需求和减少计算时间。某种程度上,时滞惯性流形为系统的非线性动力行为如屈曲、分岔、突跳等动态模拟和数值分析提供了一个新的更为合理的研究手段。     

具有几何和物理非线性粘弹性梁的混沌运动

建立了描述具有几何和物理非线性均匀梁动力学行为的偏微分－积分方程,梁的材料满足Leaderman非线性本构关系。对于两端简支的情形,采用Galerkin方法简化为常微分－积分方程;然后通过引进附加变量的方法进一步简化为常微分方程;最后利用相平面图、功率谱和Lyapunov指数等非线性动力学中的数值方法识别梁的动力学行为。结果表明梁的运动呈现混沌性态。     

非线性粘弹性桩耦合运动中的混沌分析

研究轴向周期载荷作用下非线性粘弹性桩纵横向耦合运动中的混沌运动。桩体材料满足Leaderman非线性粘弹性本构关系和近似的非线性几何关系，考虑桩体发生纵横向运动的耦合，得到的方程为耦合的非线性偏微分一积分方程；利用Galerkin方法将方程简化并进行数值计算，揭示非线性粘弹性桩的混沌运动和分岔等动力学行为。     

轴压粘弹性圆柱壳在横向扰动下的混沌行为

基于大挠度薄壳的Donnell-Kármán理论和Kelvin–Voigt粘弹性本构关系,对轴压粘弹性圆柱壳在横向扰动下的混沌行为进行了研究。导出了关于挠度和应力函数的控制方程,借助Galerkin原理将粘弹性圆柱壳的控制方程转化为二阶三次非线性微分动力系统,用Melnikov函数给出了系统发生Smale马蹄型混沌的临界条件。数值计算分析了轴压载荷和粘性阻尼系数对混沌运动的影响。通过分岔图、位移时程曲线、相平面图和Poincaré映射描述了系统的运动行为。研究表明：当轴压载荷与圆柱壳的材料参数满足一定关系时,系统才有可能发生Smale马蹄型混沌;随着轴压载荷的增大,混沌运动区域逐渐减小;随着粘性系数与外阻尼系数比值的增大,混沌运动区域逐渐减小;轴压粘弹性圆柱壳在横向扰动下既会发生定常运动也会发生混沌运动     

粘弹性阻尼隔振体的非线性振动分析

 研究了由基础振动激励、粘弹性材料隔离的被动隔振体的非线性动力响应.用变形的三次多项式函数表征隔振材料的非线性刚度,用分数阶算子表征阻尼,建立被动隔振体的分数阶非线性动力学方程.用谐波平衡法研究其非线性动力响应特性,得出频率响应方程和幅频曲线,分析了非线性因素对系统的影响.最后,用Floquet理论讨论了周期解的稳态性和稳定区间.分析结果表明,含分数阶算子的动力学模型能够准确地描述粘弹性材料隔振器的动态特性.忽略隔振材料的阻尼非线性、刚度非线性将导致隔振设计和隔振效果分析的明显误差.分析方法和结论为精确进行粘弹性材料的被动隔振设计和隔振效果评价提供理论参考.     

Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

黏弹性地基中基于虚土桩模型的桩顶纵向振动阻抗研究

基于虚土桩模型,对均质粘弹性地基中桩土纵向耦合振动问题进行了研究。首先,假定桩侧土为各向同性的线性粘弹性材料,并考虑土体的竖向波动效应,结合Euler-Bernoulli杆件理论,建立了桩土纵向耦合振动的定解问题;其次,采用分离变量法求解桩侧土纵向振动的控制方程,得到了桩侧土与桩身接触面上的剪切动刚度,将所得的剪切动刚度代入到桩身振动控制方程,采用Laplace变换技术,进一步求得了任意荷载作用下桩顶纵向振动阻抗的解析解。基于所得解,详细讨论了不同桩身设计参数时桩端土厚度对桩顶纵向振动阻抗的影响。最后,将虚土桩模型与其他桩端土支承模型进行了对比研究,结果表明,对虚土桩模型选用合适的材料参数和桩端土厚度,其得到的桩端支承复刚度值介于现有多种模型的计算值之间。     

Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models

The dynamic response of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. The mechanical behavior of the viscoelastic material is described by differential constitutive equations with fractional order derivatives. The governing equations, which are derived by considering the dynamic equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacements, whose order is in general greater than two with respect to time derivatives. A method is presented to establish the additional required initial conditions beside the described initial displacements and velocities. Using the Analog Equation Method (AEM) in conjunction with the Domain Boundary Element Method (D/BEM) the governing equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using the numerical method for multi-term FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the dynamic response of plane bodies described by different viscoelastic models.     

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.