多自由度线性阻尼系统的频域及时域响应分析

本文从分析频率方程解的形式出发，通过设定响应的频域形式，得出自由振动时域响应的显式解，并导出任意强迫振动的Ｄｕｈａｍｅｌ积分．文中的分析方法适用于系统质量、阻尼、刚度阵均为非对称的情形，且属于精确求解．     

含立方刚度和滞回的隔振系统动力学特性分析

包含立方刚度和Bouc–Wen型滞回的隔振系统具有复杂的非线性动力学特性。系统无阻尼响应模型可基于无滞回恢复力建立,利用谐波平衡法和泰勒展开求得近似解析解。系统有阻尼响应模型可利用解析/数值联合方法求解,该方法基于谐波平衡法和Levenberg–Marquardt迭代算法,对于滞回产生的多值非光滑函数项,先计算时域响应再通过快速傅里叶变换求解谐波项系数。上述方法在含水平绞制梁的非线性隔振系统分析中得到有效应用。分析表明,在Bouc–Wen型滞回和立方刚度的综合影响下,隔振系统呈现渐软-渐硬特性,滞回阻尼和线性阻尼都可以有效抑制共振,但前者高频隔振效果优于后者。     

DYNAMIC CHARACTERISTICS OF VIBRATION ISOLATION SYSTEM WITH CUBIC STIFFNESS AND HYSTERESIS 1)

包含立方刚度和Bouc-Wen 型滞回的隔振系统具有复杂的非线性动力学特性。系统无阻尼响应模型可基于无滞回恢复力建立,利用谐波平衡法和泰勒展开求得近似解析解。系统有阻尼响应模型可利用解析/数值联合方法求解,该方法基于谐波平衡法和Levenberg-Marquardt 迭代算法,对于滞回产生的多值非光滑函数项,先计算时域响应再通过快速傅里叶变换求解谐波项系数。上述方法在含水平绞制梁的非线性隔振系统分析中得到有效应用。分析表明,在Bouc-Wen 型滞回和立方刚度的综合影响下,隔振系统呈现渐软–渐硬特性,滞回阻尼和线性阻尼都可以有效抑制共振,但前者高频隔振效果优于后者。     

基于频率相关黏性阻尼模型的复模态叠加法

黏性阻尼模型存在每周期耗散能量与外激励频率相关的缺陷, 复阻尼模型时域计算结果存在发散现象. 为克服上述两种阻尼模型的不足, 在复阻尼模型基础上, 依据时频域转化原则推导了频率相关黏性阻尼模型. 频率相关黏性阻尼模型不仅具有每周期耗散能量与外激励频率无关的优点, 还保证了结构位移时程的稳定收敛. 混合结构由具有不同阻尼特性的材料组成, 其阻尼矩阵为非比例矩阵, 无法直接采用实模态叠加法. 根据频率相关黏性阻尼模型与复阻尼模型的转换关系, 提出了适用于混合结构的基于频率相关黏性阻尼模型的复模态叠加法. 算例分析结果表明, 与基于黏性阻尼模型的复模态叠加法相比, 基于频率相关黏性阻尼模型的复模态叠加法不仅计算结果唯一, 且不增加矩阵维度, 具有较高的计算效率. 小阻尼情况下, 两种方法的计算结果近似相等, 且与复阻尼模型的频域计算结果一致. 当阻尼比较大时, 两种方法的计算结果差异增大, 但频率相关黏性阻尼模型的复模态叠加法与复阻尼模型的频域计算结果仍保持一致.      

基于频率相关黏性阻尼模型的复模态叠加法

黏性阻尼模型存在每周期耗散能量与外激励频率相关的缺陷,复阻尼模型时域计算结果存在发散现象.为克服上述两种阻尼模型的不足,在复阻尼模型基础上,依据时频域转化原则推导了频率相关黏性阻尼模型.频率相关黏性阻尼模型不仅具有每周期耗散能量与外激励频率无关的优点,还保证了结构位移时程的稳定收敛.混合结构由具有不同阻尼特性的材料组成,其阻尼矩阵为非比例矩阵,无法直接采用实模态叠加法.根据频率相关黏性阻尼模型与复阻尼模型的转换关系,提出了适用于混合结构的基于频率相关黏性阻尼模型的复模态叠加法.算例分析结果表明,与基于黏性阻尼模型的复模态叠加法相比,基于频率相关黏性阻尼模型的复模态叠加法不仅计算结果唯一,且不增加矩阵维度,具有较高的计算效率.小阻尼情况下,两种方法的计算结果近似相等,且与复阻尼模型的频域计算结果一致.当阻尼比较大时,两种方法的计算结果差异增大,但频率相关黏性阻尼模型的复模态叠加法与复阻尼模型的频域计算结果仍保持一致.     

变截面梁动力响应的频域传递矩阵计算方法

为提高变截面梁振动分析的计算效率,提出了基于频域传递矩阵法的动力计算算法．首先,选择线速度、角速度、弯矩和剪力作为求解变量,通过Laplace变换将变截面梁的动力响应时域偏微分方程转换为频域常微分方程;然后,通过求解频域方程并结合协调和边界条件建立变截面梁的频域传递矩阵;通过构造傅里叶级数展开形式的时域响应函数,对变截面梁传递矩阵方法求解的频响函数进行Laplace逆变换,建立了变截面梁的固有特性计算和时域瞬态响应计算方法,最后,借助数值仿真软件,开发了变截面梁动力响应分析的计算程序．完成对算例的仿真计算和分析,并与有限元计算结果进行对比,数值结果验证了该方法的正确性和有效性．     

一种中高频激励下动力系统响应的级数展开改进算法

针对频率响应函数的级数展开法在中高频激励时计算发散的问题,提出一种新的级数展开改进算法.将系统的结构模态划分为低阶和截断的高阶模态,在模态叠加分析的基础上,将频率响应函数进行泰勒级数展开.根据高低阶模态对质量矩阵和刚度矩阵的耦合特性,用低阶模态及系统矩阵表达高阶模态对响应的影响.研究结果表明,该算法将频率响应函数的级数展开法扩展到高频激励和中频激励范围阶段,在非完备模态条件下提高了频率响应函数的计算精度,数值计算检验了该方法准确可靠并有很好的收敛性.     

考虑弹簧阻尼作动器解析雅可比矩阵的多刚体动力学分析

弹簧-阻尼-作动器(spring-damper-actuator,SDA)是多体系统中常见的力元,在工程领域中有着广泛的应用.采用绝对坐标方法建立的多体系统动力学控制方程通常是复杂的非线性微分-代数方程组.为了保证数值解的精度和稳定性,通常需要采用隐式算法求解动力学方程,而雅可比矩阵的计算在隐式数值求解过程中至关重要.对于含有SDA的多体系统,SDA造成的附加雅可比矩阵是与广义坐标和广义速度相关的高度非线性函数.目前的很多研究工作专注于广义力向量的计算,然而对附加雅克比矩阵的计算则少有关注.针对含SDA的多刚体系统进行动力学分析,首先基于Newmark算法研究其在动力学方程求解中的雅可比矩阵的构成形式;然后推导SDA的广义力向量对应的附加雅可比矩阵,其中包括广义力向量对广义坐标和对广义速度的偏导数矩阵.最后通过两个数值算例研究附加雅可比矩阵对动力学分析收敛性的影响;数值分析表明:当SDA的刚度、阻尼和作动力数值较大时,SDA导致的附加雅可比矩阵对数值解的收敛性有重要影响;当考虑SDA对应的附加雅可比矩阵时,动力学分析可以以较少的迭代步实现收敛,从而减少分析时间.     

含分数阻尼特性元件的多体系统动力学研究

在绝对节点坐标体系下研究了具有分数导数阻尼特性元件的多体系统动力学建模、求解问题. 采用基于绝对节点坐标的无闭锁效应剪变梁单元离散柔性构件,建立了含常数质量矩阵的系统动力学方程, 并采用数值耗散可控的广义a方法求解. 通过数值算例计算,对比研究了算法参数与阻尼项的分数指数对系统动力学响应的影响规律.该方法可以进一步扩展到众多工程实际问题研究中.      

非参数概率系统非平稳随机响应研究

对复杂系统的测量和知识的有限性造成的不确定很难完全由随机参数模型进行描述,采用随机矩阵模型更具有一般性和合理性.本文应用随机矩阵模拟不确定线性动力系统有限元模型中质量阵、阻尼阵和刚度阵的概率不确定性.综合运用虚拟激励法和精细时程积分法建立了非参数概率系统非平稳随机响应的有效计算方法.数值模拟结果表明,对于高精度制造,模型的不确定性是不能忽略的.本文提出的算法为此类问题求解提供了一条有效途径.     

带非线性弹簧的两自由度准零刚度隔振系统力传递率特性

基于多自由度系统中的反共振特性,分别在传统线性隔振系统的上、下两层引入非线性倾斜弹簧负刚度机构,构成两自由度准零刚度隔振器。通过静态特性分析,推导出系统满足零刚度条件时,各参数之间的关系,分析了力学参数及结构参数对系统刚度特性的影响。建立两自由度准零刚度隔振系统的非线性动力学方程,利用平均法求解,推导出力传递率表达式,结合数值分析方法,探讨系统在不同的上、下层隔振器阻尼比、竖直刚度比及质量比情况下的力传递率特性,并与单自由度准零刚度隔振系统及线性斜弹簧两自由度准零刚度隔振系统进行对比研究。结果表明：当结构参数 （即：倾斜弹簧处于静平衡位置的长度与倾斜弹簧原长的比值）较小且倾斜弹簧为软化弹簧时,可在平衡位置附近获得较小的系统刚度及较大的低刚度区间;通过选择适当的上、下层隔振器阻尼比、竖直刚度比与质量比,可减小系统的起始隔振频率,增宽隔振频带,加快系统力传递率在特定频段内的衰减速率,改善系统的低频隔振性能。     

Flow reactor models for gas-liquid reactions based on the penetration theory

Flow reactor models for gas-liquid reaction systems are proposed in this paper based on the penetration theory in the isothermal case. The mass transfer mechanism accompanied by a chemical irreversible first-order reaction is mathematically treated in a new way in order to use its results to develop reactor design models conveniently. Analytical solutions can be obtained for the desgin equation system involving linear differential equations by using of either the eigenvalues or the Laplace-transformation and the superposition of the system. In addition, an iteration procedure is given to solve the nonlinear differential equation system numerically. Comparisons of the results from the analytical and numerical solutions are also made graphically.     

Relaxation time spectra from short frequency range small-angle dynamic rheometry

In this work, the influence of the experimental frequency range over the relaxation time spectrum is studied. The relaxation time spectra were calculated from dynamic moduli, a well-known ill-posed problem, using a regularization method. The method solves the ill-posed problem by simultaneous minimization of the regularized standard deviation and a restriction function. The solution was validated using a simulated spectrum. Truncated moduli data generated from simulated spectra were used to evaluate the method for smaller frequency range data. Finally, experimental data of a wormlike micellar system mixed in aqueous solution with a zwitterionic copolymer were used to validate the method. It was possible to obtain relaxation time spectra from short frequency range data if the relaxation time range is allowed to be higher than the inverse of the highest and lowest experimental frequencies. These spectra can be used qualitatively to describe complex systems when no time-temperature superposition experiments are feasible.     

Multi-scaled analysis of the damped dynamics of an elastic rod with an essentially nonlinear end attachment

We study multi-frequency transitions in the transient dynamics of a viscously damped dispersive finite rod with an essentially nonlinear end attachment. The attachment consists of a small mass connected to the rod by means of an essentially nonlinear stiffness in parallel to a viscous damper. First, the periodic orbits of the underlying hamiltonian system with no damping are computed, and depicted in a frequency–energy plot (FEP). This representation enables one to clearly distinguish between the different types of periodic motions, forming back bone curves and subharmonic tongues. Then the damped dynamics of the system is computed; the rod and attachment responses are initially analyzed by the numerical Morlet wavelet transform (WT), and then by the empirical mode decomposition (EMD) or Hilbert–Huang transform (HTT), whereby, the time series are decomposed in terms of intrinsic mode functions (IMFs) at different characteristic time scales (or, equivalently, frequency scales). Comparisons of the evolutions of the instantaneous frequencies of the IMFs to the WT spectra of the time series enables one to identify the dominant IMFs of the signals, as well as, the time scales at which the dominant dynamics evolve at different time windows of the responses; hence, it is possible to reconstruct complex transient responses as superposition of the dominant IMFs involving different time scales of the dynamical response. Moreover, by superimposing the WT spectra and the instantaneous frequencies of the IMFs to the FEPs of the underlying hamiltonian system, one is able to clearly identify the multi-scaled transitions that occur in the transient damped dynamics, and to interpret them as ‘jumps’ between different branches of periodic orbits of the underlying hamiltonian system. As a result, this work develops a physics-based, multi-scaled framework and provides the necessary computational tools for multi-scaled analysis of complex multi-frequency transitions of essentially nonlinear dynamical systems.     

Direct Stiffness Analysis of Lateral Buckling

Abstract

A direct stiffness method of analyzing the elastic ftexural-torsional buckling of rigid-jointed plane frames composed of l-section members and subjected to in-plane loads is presented. The in-plane stiffness matrix and the fixed-end resultants are obtained from the member stiffness matrices derived from the in-plane differential equations. These member stiffness matrices are assembled and solved, and their solutions are used to linearize the flexural-torsional buckling equations. The out-of-plane member stiffness matrices are then obtained numerically from the buckling equations by the method of finite integrals. The out-of-plane frame -stiffness matrix is assembled, and the critical loads are obtained when its determinant is zero. A computer program is developed which carries out either a first- or second-order in-plane analysis, and then determines the flexural-torsional buckling loads. The effects of in-plane deformations prior to buckling can be included. Very good agreement is obtained between the results computed by this program and known solutions, and its ability to analyze large complex frames is demonstrated.     

ANALYTICAL SOLUTIONS FOR RESPONSE OF COLLISION OF PARTICLE WITH CONICAL ROD CAUSED BY LONGITUDINAL VIBRATION

The objective is to present exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle and a new method is proposed for conical rod-particle impact analysis, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical example shows that one of the advantages of the present method is that the analytical form is very simple. The result is that mass ratio and some variables describing the geometrical shape of rods such as taper, length and radius play an important role in impact dynamic system.     

Partial pole assignment using rank-one control and receptance in second-order systems with time delay

In this note, a partial pole assignment approach is presented for second-order systems with time delay. The method uses the versatile system receptance for designing state-feedback, rank-one controllers for second-order systems with time delay in the measurements or actuation. The stability of the closed-loop system is pursued throughout an optimization problem formulated with basis on the classical frequency domain technique known as the Nyquist stability criterion. Besides the partial pole assignment, robustness measured in terms of phase and gain margins can be achieved using a genetic algorithm to solve the optimization problem. The proposed approach is shown to provide effective solutions for systems with different time delays in the measurements of displacements and velocities, and with singular mass matrix. Numerical examples are presented to illustrate the benefits of the approach.

     

多柔体系统数值分析的模型降噪方法

多柔体系统的动力学方程通常是一组刚性微分方程, 目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量, 其求解效率难以令人满意. 为了降低多柔体系统动力学方程的刚性, 从而可采用ODE45等常规微分方程求解器进行求解, 研究了在建模过程中滤除高频振荡分量的方法. 在以当前时刻为起点的短时间内对柔性体的应力进行均匀化, 用均匀化后的应力计算柔性体的变形虚功率, 由此得到的系统动力学方程的解中不含过高频率的弹性振动, 并且可以通过调节均匀化时间区间的长度参数控制滤波的范围. 数值算例表明: 这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器, 并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率. 本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.