Forced harmonic response of viscoelastic structures by an asymptotic numerical method

This work presents an asymptotic numerical method for forced harmonic vibration analyses of viscoelastic structures. A mathematical formulation that may account for various viscoelastic models is presented. Power series expansions and Padé approximants of the displacement and frequency are developed and the finite element method is used for numerical solution. Only some matrix inversions and a few iterations are needed for large frequency ranges. Iterations of the process lead to a powerful continuation method for harmonic responses of viscoelastic structures with constant and frequency dependent coefficients. For numerical tests, undamped, viscoelastic and sandwich viscoelastic beams and plates are considered. Passive control, response curves and equivalent damping characteristics are obtained for various frequency ranges, excitation amplitudes and viscoelastic models.     

几何非线性损伤粘弹性中厚板的动力学行为分析

根据Timoshenko几何变形假设和Boltzmann叠加原理,推导出控制损伤粘弹性Timoshenko中厚板的非线性动力方程以及简化的Galerkin截断方程组;然后利用非线性动力系统中的数值方法求解了简化方程组．通过分析可知,板在谐载荷的作用下,具有非常丰富的动力学特性．同时研究了板的几何参数、材料参数及载荷参数对损伤粘弹性中厚板动力学行为的影响．     

蜂窝夹层板的非线性动力学研究

研究了正六角形蜂窝夹层板的非线性动力学问题．考虑高阶横向剪切变形和横向阻尼的影响,建立了面内激励和横向外激励联合作用下的四边简支蜂窝夹层板的非线性偏微分运动控制方程．综合运用Galerkin方法和数值方法,模拟不同激励作用下的混沌运动,得到二维相图、二维波形图和频谱图．研究结果表明：随着激励的增加,系统会重复呈现周期运动、混沌运动、周期运动的变化规律．     

旋转粘弹性夹层梁非线性自由振动特性研究

对旋转粘弹性夹层梁的非线性自由振动特性进行了分析.基于Kelvin-Voigt粘弹性本构关系和大挠度理论,建立了旋转粘弹性夹层梁的非线性自由振动方程,并使用Galerkin法将偏微分形式振动方程化为常微分振动方程.采用多重尺度法对非线性常微分振动方程进行求解,通过小参数同次幂系数相等获得微分方程组,并通过求解方程组及消除久期项来获得旋转粘弹性夹层梁非线性自由振动的一次近似解.用数值方法讨论了粘弹性夹层厚度、转速和轮毂半径对梁固有频率的影响.结果表明:固有频率随转速增大而增大,随夹层厚度增大而减小,随轮毂半径的增大而增大.     

Nonlinear dynamics of nano-resonators: an analytical approach

Prior to the design and fabrication of MEMS/NEMS devices, analysis of static and dynamic behaviors of these systems is necessary. In the present study, the nonlinear dynamic behavior of micro- and nano-mechanical resonators is investigated and classified based on the resonator’s physical parameters for first time. The Galerkin method is used to convert the distributed-parameter model to a nonlinear ordinary differential equation where mid-plane stretching, axial stress, DC electrostatic and AC harmonic voltages are taken into account. To obtain the analytical frequency response of the micro resonator near its primary resonance, the second order multiple scales method is applied to the general equation of motion with cubic, quadratic and parametric nonlinearities. It is demonstrated that variation of the micro resonator’s physical parameters strongly affects its dynamic behavior by changing equilibrium points and their stability properties and complex behaviors appear in its frequency and phase responses. The global dynamics of the resonator is classified into four different categories in terms of the system parameters in this paper. The dynamic characteristics and frequency response of each class are analyzed numerically as well as analytically. Comparison of the obtained closed-form solution with the numerical simulation results confirms its validity. A striking point of the obtained closed-form solution is that it predicts some complex nonlinear behaviors of the resonator. This paper presents a quick and efficient method for determining the global dynamics of the micro-resonators and can be useful in design and analyses of these devices.     

Multi-objective optimal design of hybrid viscoelastic/composite sandwich beams by using the generalized differential quadrature method and the non-dominated sorting genetic algorithm II

In this study, the multi-objective optimal design of hybrid viscoelastic/composite sandwich beams for minimum weight and minimum vibration response is aimed. The equation of motion for linear vibrations of a multi-layer beam is derived by using the principle of virtual work in the most general form. These governing equations together with the boundary conditions are discretized by the generalized differential quadrature method (GDQM) in the frequency domain for the first time. Also, the time and temperature dependent properties of the viscoelastic materials are taken into consideration by a novel ten-parameter fractional derivative model that can realistically capture the response of these materials. The material variability is accounted for by letting an optimization algorithm choose a material freely out of four fiber-reinforced composite materials and five viscoelastic damping polymers for each layer. The design parameters, i.e., the orientation angles of the composites, layer thicknesses and the layer materials that give the set of optimal solutions, namely the Pareto frontier, is obtained for the three and nine-layered clamped-free sandwich beams by using a variant of the non-dominated sorting genetic algorithms (NSGA II).     

几何非线性的损伤粘弹性Timoshenko梁的动力学行为

从考虑损伤的粘弹性材料的一种卷积型本构关系出发,建立了在有限变形下损伤粘弹性Timoshenko梁的控制方程．利用Galerkin方法对该组方程进行简化。得到一组非线性积分一常微分方程．然后应用非线性动力学数值分析方法,如相平面图,Poincare截面分析了载荷参数对非线性损伤粘弹性Timoshenko梁动力学性能的影响．特别考察了损伤对粘弹性梁的动力学行为的影响．     

The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

粘弹性地基上损伤弹性Timoshenko梁动力学行为

建立了粘弹性地基上损伤弹性Timoshenko梁在有限变形情况下的运动微分方程,这是一组非线性偏微分方程.为了便于分析,首先利用Galerkin方法对该方程组进行简化,得到一组非线性常微分方程.然后利用Matlab软件进行数值模拟,考察了载荷参数、地基粘性参数和弹性参数、损伤对梁振动的影响.采用非线性动力学中的各种数值方法,如时程曲线、相平面图、Poincare截面和分叉图,发现增大地基的粘弹性参数,有利于增强结构运动的稳定性,而损伤会降低结构运动的稳定性.     

轴向运动粘弹性夹层梁的横向振动

研究了小扰度下轴向匀速运动粘弹性夹层梁的振动模态和固有频率.基于Kelvin粘弹性本构方程,建立了轴向运动粘弹性夹层梁横向振动控制方程.分别采用Galerkin截断和复模态分析方法,研究两端简支的粘弹性夹层梁的固有频率和模态函数,讨论了轴向运动速度、夹心层与约束层厚度比、初始轴力等参数对夹层梁固有频率、临界速度及稳定性的影响.     

Nonlinear forced vibration of damped plates by an asymptotic numerical method

This work deals with damped nonlinear forced vibrations of thin elastic rectangular plates subjected to harmonic excitation by an asymptotic numerical method. Using the harmonic balance method and Hamilton’s principle, the governing equation is converted into a static formulation. A mixed formulation is used to transform the problem from cubic nonlinearity to quadratic one sequence. Displacement, stress and frequency are represented by power series with respect to a path parameter. Equating the like powers of this parameter, the nonlinear governing equation is transformed into a sequence of linear problems with the same stiffness matrix. Through a single matrix inversion, a considerable number of terms of the perturbation series can easily be computed with a limited computation time. The starting point, corresponding to a regular solution, is obtained by the Newton–Raphson method. In order to increase the step length, Padé approximants are used. Numerical tests are presented and compared with numerical and analytical results in the literature, for different boundary conditions, excitations and damping coefficients.     

Viscoelastic-stiffness tensor of anisotropic media from oscillatory numerical experiments

A finely layered media behaves as an anisotropic medium when the dominat wavelengths are much larger than the layer thickness. If the constituent are anelastic, a generalization of Backus averaging predicts that the medium is effectively a transversely isotropic viscoelastic (TIV) medium. To test and validate the theory, we present a novel procedure to determine the complex and frequency-dependent stiffness components of a TIV medium. The methodology consists in performing numerical compressibility and shear harmonic tests on a representative sample of the material. These tests are described by a collection of non-coercive elliptic boundary-value problems formulated in the space-frequency domain, which are solved using a Galerkin finite-element procedure. Results on the existence and uniqueness of the continuous and discrete problems as well as optimal error estimates for the Galerkin finite-element method are derived. Numerical examples illustrates the implementation of the numerical oscillatory tests to determine the set of complex and frequency-dependent effective TIV coefficients and the associated phase velocities and quality factors for a periodic sequence of epoxy and glass thin layers. The results are compared to the analytical phase velocities and quality factors predicted by the Backus/Carcione theory.     

悬臂式蜂窝夹层板的非线性动力学建模及分析

蜂窝夹层结构因其良好的力学特性,在众多工程领域具有非常广泛的应用.本文建立了悬臂边界条件下,蜂窝夹层板的动力学模型并研究其非线性动力学行为.选取文献中更加接近实体有限元解的等效弹性参数公式对蜂窝芯层进行等效简化,得到六角形蜂窝芯的等效弹性参数.基于Reddy高阶剪切变形理论,应用Hamilton原理建立悬臂式蜂窝夹层板在受到面内激励和横向激励联合作用下的偏微分运动方程.然后利用Galerkin方法得到两自由度非自治常微分形式运动方程.在此基础上,通过对悬臂式蜂窝夹层板进行数值模拟分析系统的非线性动力学.结果表明面内激励和横向激励对系统的动力学特性有着重要影响,在不同激励作用下系统会出现周期运动、概周期运动以及混沌运动等复杂的非线性动力学响应.     

An integral-equation solution for the finite deflection of clamped skew sandwich plates

The large-deflection behaviour of skew sandwich plates is governed by a system of five coupled nonlinear partial differential equations which are highly complex in nature. In the reported study, this problem is analysed using an integral-equation approach. The integral equations of beams along the skew directions is used with appropriate boundary conditions to transform the governing nonlinear partial differential equations into a set of nonlinear algebraic equations. These equations are then solved using an iterative scheme suggested by Brown. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load-deflection characteristics have been presented for clamped skew sandwich plates.     

基于Kelvin模型的粘弹性浅拱的动力稳定性

研究了外激励作用下非线性粘弹性浅拱的动力行为．通过达朗贝尔原理和欧拉一贝努利假定建立了浅拱的动力学控制方程,其中采用Kelvin模型来表示非线性粘弹性材料的本构关系,并利用Galerkin法将方程简化用于数值分析．分析了粘弹性材料参数、浅拱矢高、外激励幅值和频率对系统分岔和混沌等非线性动力学行为的影响,结果表明各种参数条件下系统的非线性动力特性十分复杂,周期运动、准周期运动和混沌运动窗口在一定条件下交替出现．     

Component mode synthesis and polynomial chaos expansions for stochastic frequency functions of large linear FE models

This paper presents a methodological approach for the numerical investigation of frequency transfer functions for large FE systems with linear and nonlinear stochastic parameters. The component mode synthesis methods are used to reduce the size of the model and are extended to stochastic structural vibrations. The statistical first two moments of frequency transfer functions are obtained by an adaptive polynomial chaos expansion. Free and fixed interface methods with and without reduction of interface dof are used. The coupling with the first and second order polynomial chaos expansion is elaborated for beams and assembled plates with linear and nonlinear stochastic parameters.     

轴向运动薄板非线性振动及其稳定性研究

应用增量谐波平衡法(IHB法)研究轴向运动薄板横向非线性振动特性及其稳定性.通过Hamilton原理推导出了非惯性参考系下四边简支轴向运动薄板的横向振动微分方程,然后利用Galerkin方法离散运动方程.对离散后的非线性方程组应用IHB法进行非线性振动分析,研究了在固有频率之比ω20/ω10接近于3:1情况下,外激励频率ω在ω10附近的具有内部共振的基谐波响应.最后用多元Floquet理论分析了系统周期解的稳定性,其中采用Hsu方法来计算转移矩阵.通过对具体例子的数值计算,分别得到了自由振动和不同外激励下的频幅相应曲线,通过对比运动梁模型和运动薄板模型的计算结果,分析了各种模型的适用范围.     

The effect of nonlinear elasticity on the large amplitude free vibration behavior of elastic plates at small scale

The effect of nonlinear elasticity on the free vibration behavior of elastic plates has been evaluated by employing continuum mechanics model. The second-order non-linear stress–strain relationship has been considered and the Kirchhoff’s hypothesis has been applied on the elastic plate. The large deformation during vibration has also been considered. By using the Hamilton principle, the governing equations of the free vibration of the plate under different boundary condition have been obtained. In order to get the explicit solutions of the governing equations, the Galerkin’s method and the harmonic balance method have been utilized. The relationship between the vibration frequency and the vibration amplitude has been discussed and the vibration frequencies of different shaped plate have been compared. It is perceived that the nonlinear elasticity has a distinct effect on the free vibration of the plate.

     

Chaotic oscillations of viscoelastic microtubes conveying pulsatile fluid

As the first endeavour, the influence of a pulsatile flow on the large-amplitude bifurcation behaviour of viscoelastic microtubes subject to longitudinal pretention is studied with special consideration to chaos. The viscoelastic microtube is surrounded by a nonlinear spring bed. A modified size-dependent nonlinear tube model is developed based on a combination of the couple stress theory and the Euler–Bernoulli theory. Hamilton’s principle, as an equation derivation technique, and Galerkin’s procedure, as a discretisation technique, are used. Finally, the discretised differential equations of the pulsatile fluid-conveying viscoelastic microscale tube are solved using a time-integration approach. It is investigated that how the bifurcation response for both motions along the axial and transverse axes is highly dependent of the mean value and the amplitude of the speed of the pulsatile flow.