Isotropic layered soil–structure interaction caused by stationary random excitations

Soil–structure interaction emanating from seismic stationary random excitations is studied using the pseudo-excitation method in combination with the precise integration method. The soil considered is a viscoelastic, transversely isotropic and layered half space and the structure which it supports is modelled by the finite element method. The excitation sources are random field ones that are stationary in the time domain and are located in the soil. The pseudo-excitation method is used to transform this stationary random soil–structure interaction problem into a series of deterministic harmonic response analyses and the precise integration method is used to integrate the ordinary differential equations in the frequency–wavenumber domain. The power spectral densities of the soil–structure interaction responses caused by the stationary random excitations are investigated.     

多点非均匀调制演变随机激励下结构地震的响应

针对大跨度结构在非均匀调制演变随机激励作用下,考虑行波效应时的非平衡随机地震响应问题,应用虚拟激励法进行了分析,由于虚拟激励法自动计及了参振振型的互相关项以及激励之间的互相关项,理论上是精确解,时变功率谱的计算采用精细逐步积分格式,使计算效率进一步得到提高。     

含有界随机参数的双势井Duffing-Van der pol系统的对称破裂分岔

讨论谐和激励作用下含有界随机参数的双势井Duffing-Van der pol系统的对称破裂分岔现象。首先用Chebyshev多项式逼近法将随机系统化成与其等价的确定性系统,然后通过等价确定性系统来探索随机Duffing-Van der pol系统的对称破裂分岔现象。数值模拟显示随机Duffing-Van der pol系统与确定性均值参数系统有着类似的对称破裂分岔行为,文中的主要数值结果表明Chebyshev多项式逼近法是研究非线性随机参数系统动力学问题的一种有效方法。     

Random vibration of viscoelastic system under broad-band excitations

An analytical method is proposed to study the response of a viscoelastic system with strongly non-linear stiffness force and under broad-band random excitations. The random excitations can be additive, or multiplicative, or both, and they can be stationary or non-stationary with evolutionary spectra. With the proposed method, contributions of the viscoelastic force to both damping and stiffness are taken into account separately, and then the extended version of the stochastic averaging, called the quasi-conservative averaging, is applied to the system to derive the averaged equation of energy envelope. Probability density functions of system responses, such as the total energy, the amplitude, and the state variables, can then be obtained analytically. The accuracy of the method is substantiated by comparing the analytical results with those from Monte Carlo simulations. Effects of parameters in the viscoelastic force and in the non-linear stiffness force on the system responses are also investigated.     

Necessary and sufficient conditions for existence of stationary solutions of some nonlinear stochastic systems

At the state of statistical stationarity, the response of a nonlinear system under multiplicative random excitations can be either trivial or non-trivial, depending on the spectral levels of the excitations and the values of certain system parameters. Assuming that the random excitations are Gaussian white noises, the two types of response may be investigated by way of their stationary densities, which are obtainable for first order dynamical systems and for higher order dynamical systems belonging to the class of generalized stationary potential. Alternatively, the Lyapunov exponents can be computed for perturbation from either the trivial or non-trivial solution, since a negative sign for the greatest Lyapunov exponent provides both the necessary and sufficient conditions for the stability of sample functions with probability one. It is shown in two specific examples, that the boundary at which the greatest Lyapunov exponent changes its sign coincides with the boundary for regularity (or being normalizable) for the probability density in both the trivial and non-trivial solutions. Thus, the stability conditions in the strong sense of probability one and the weak sense in distribution are identical in these cases.     

Unified Analysis of Complex Nonlinear Motions via Densities

A unified approach of using densities to analyze bothdeterministic and stochastic complex responses including chaotic andrandom motions of nonlinear engineering systems is illustrated in thisstudy. Motivations to examine deterministic nonlinear dynamical systemsvia densities are first discussed. Essential mathematical background andtechniques pertinent to the analyses of both deterministic chaos andrandom chaotic processes are briefly summarized. Densities of nonlinearresponses are computed by numerically solving the Fokker–Planckequation to examine stochastic properties of random chaotic responses.It is demonstrated that, by introducing random perturbations in anotherwise deterministic excitation, the existence of attractors can beefficiently and clearly depicted by the evolution of a uniqueprobability density over the physical phase space. Two distinctasymptotic behaviors of densities: (i) invariance and (ii) sweeping, ofcomplex motions and their relationship to response stabilities predictedby the Foguel Alternative Theorem are numerically demonstrated.Applications using the probability densities to compute reliabilityindices of an engineering system are demonstrated.     

确定性谐和与随机噪声联合作用下二自由度系统的响应

研究了二自由度非线性系统在确定性谐和与随机噪声联合激励下的主共振响应。用多尺度法分离了系统的快变项 ,讨论了系统的阻尼项、随机项等对系统响应的影响。在一定条件下 ,系统具有两个均方响应值和跳跃现象 ,饱和现象也存在。数值模拟表明本文提出的方法是有效的     

Effects of excitation probability distribution on system responses

For a system subjected to a random excitation, the probability distribution of the excitation may affect behaviors of the system responses. Such effects are investigated for a variety of dynamical systems, including a linear oscillator, an oscillator of cubic non-linearity in both damping and stiffness, and a non-linear oscillator of the van der Pol type. The random excitations are assumed to be stationary stochastic processes, sharing the same spectral density, but with different probability distributions. Each excitation process is generated by passing a Brownian motion process through a non-linear filter, which is governed by an Ito stochastic differential equation. Monte Carlo simulations are carried out to obtain the transient and stationary properties of the system response in each case. It is shown that, under different excitations, the transient behaviors of the system response can be markedly different. The differences tend to reduce, however, as time of exposure to the excitations increases and the system reaches the stationary state.     

随机杆系结构几何非线性分析的递推求解方法

建立了随机静力作用下考虑几何非线性的随机杆系结构的随机非线性平衡方程. 将和 位移耦合的随机割线弹性模量以及随机响应量表示为非正交多项式展开式，运用传统的摄动方法获 得了关于非正交多项式展式的待定系数的确定性的递推方程. 在求解了待定系数后，利用非 正交多项式展开式和正交多项式展开式的关系矩阵，可以很方便地得到未知响应量的二阶统计矩. 两杆结构和平面桁架拱的算例结果表明，当随机量涨落较大时，递推随机有限元方法比基于 二阶泰勒展开的摄动随机有限元方法更逼近蒙特卡洛模拟结果，显示了该方法对几何非线性 随机问题求解的有效性.     

Response of flexure–torsion coupled composite thin-walled beams with closed cross-sections to random loads

A study of the flexure–torsion coupled random response of the composite beams with solid or thin-walled closed-sections subjected to various types of concentrated and distributed random excitations is dealt with in this paper. The effects of flexure–torsion coupling, shear deformation and rotary inertia are included in the present formulations. The random excitations are assumed to be stationary, ergodic and Gaussian. Analytical expressions for the displacement response of the composite beams are obtained by using normal mode superposition method combined with frequency response function method. The present method can produce the effective solutions for the composite Timoshenko beams with circumferentially antisymmetric (CAS) configuration and more general beam assemblages of connected beams. The influences of flexure–torsion coupling, shear deformation and rotary inertia on the random response of an appropriately chosen composite beam from the literature are demonstrated and discussed.