Solution of viscously damped linear systems using a set of load-dependent vectors

This paper considers a solution method for viscously damped linear structural systems which are subjected to transient loading. The equations of motion of such systems are written in a first-order form. A solution subspace is generated using the damped dynamic matrix and the static deflection from the first-order form of the equations of motion. Two convenient bases, Lanczos vectors and Ritz vectors, are constructed from this subspace. An approximate solution is then obtained by superposition of the Lanczos vectors or the Ritz vectors. In contrast to the traditional mode superposition method using complex eigenvectors, the Lanczos vectors or the Ritz vectors are less expensive to generate than the complex eigenvectors, yet yield comparable accuracy. In addition, there is no need for a static correction since the static deflection is already contained in our solution subspace. Numerical examples are presented to show the potential of using the Ritz vectors to compute responses of damped dynamic systems.     

Ritz vectors and generation criteria for mode superposition analysis

A generation procedure of Ritz vectors to control the inclusion of static effect and the number of vectors in mode superposition dynamic analysis is presented. The original algorithm of the Ritz vectors¹⁵ is modified to improve stability in the generation procedure and to include the use of static residual. To reject unimportant Ritz vectors, cut-off criteria, which are based on the participation of mass distribution and spatial load distribution, are proposed. Numerical examples are presented to illustrate the effectiveness of the derived Ritz vectors over the eigenvectors and the performance of the cutoff criteria in the mode superposition dynamic analysis.     

Method for solving dynamic equilibrium equations based on displacement excitation

Mode superposition is a widely used method for solving the dynamic equilibrium equation in structural dynamic analysis. However, the accuracy of this method may be reduced when the dynamic equilibrium equations are set up using displacement excitation. A new method for developing solutions for dynamic equilibrium equations based on displacement excitation is introduced. The dynamic equilibrium equation is decomposed into two parts, namely displacement excitation and velocity excitation, and precise integration and mode superposition methods are combined to solve the equation. Ritz vectors are then used to calculate the static response of the truncated modes of the structure, and a method for determining the number of participating modes is obtained. Using multi-degree-of-freedom systems as two computational examples, the differences in the structural responses obtained from the displacement excitation and acceleration excitation are compared and analyzed. It is shown that the new solution method generates consistent accuracy between the displacement excitation and acceleration excitation.     

A subspace modal superposition method for non-classically damped systems

For structures with non-proportional damping, complex eigenvectors or mode shapes must be used in order to decoe the equations of motion. The resulting equations can then be solved in a systematic way. The necessity of solvie complex eigenvalue problem of a large system remains an obstacle for the practical application of the method. This stres utilizes the fact that in practice only a small number of the complex modes are needed. Therefore, these complex modes be approximated by a linear combination of a small number of the undamped modes, which can be obtained by established methods with less cost. An additional eigenvalue problem is then solved in a subspace with a much sm dimension to provide the best combination coefficient for each complex mode. The method of solution for the decoue equations is then carried over, using the approximate complex modes expressed in undamped mode shapes, to resue simple formulas for the time- and frequency-domain solution. Thus, an efficient modal superposition method is develoe for non-proportionally damped systems. The accuracy of this approximate method is studied through an example. Comparing the frequency response result using the approximate method with that using the exact complex modes, found that the error is negligible.     

Frequency dependent ritz vectors

In order to reduce the size of problems involving analysis of the dynamic response of structural systems, a transformatio based on appropriately selected Ritz shapes is commonly employed. The lower mode shapes may at times serve a effective Ritz shapes. However, the computation of mode shapes is a time consuming task; in addition, the mode shapes may not form the best basis for representing the spatial distribution of loads. The recently developed load dependent vectors, which are derived from a static solution for the applied loads, address some of the problems inherent in the use of mode shapes. However, both the natural mode shapes and the load dependent vectors fail to account for the frequency content of the loading, a parameter that may influence strongly the response, particularly for loading with a high frequency content. A procedure is presented here for the generation of frequency dependent vectors. A combination of load dependent and frequency dependent vectors will often form a very efficient basis for the representation of the response, as illustrated by several examples presented here.     

Multiple‐support response spectrum analysis using load‐dependent Ritz vectors

Load‐dependent Ritz (LDR) vectors are used in conjunction with the multiple‐support response spectrum (MSRS) combination rule for analysis of structures subjected to spatially varying earthquake ground motions. The LDR vector approach for MSRS analysis is motivated by the fact that LDR vectors in general are more accurate and computationally simpler than eigenvectors in mode superposition analysis, and because many researchers and engineers are using LDR vectors in linear structural dynamic analysis. Mode truncation rules for the original MSRS method are modified to apply to LDR vectors. Two methods for selecting LDR vectors for multicomponent MSRS analysis are introduced. Idealized models of two real bridges with differing structural characteristics are used to investigate the accuracy and efficiency of the two LDR‐MSRS methods in comparison with results obtained by the original MSRS method as well as an extended version that accounts for the static contribution of truncated modes. The results show that the LDR‐MSRS method is generally more accurate than the original MSRS method and at least as accurate as the extended MSRS method. Copyright © 2014 John Wiley & Sons, Ltd.     

Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping

Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.     

Partial eigensolution of damped structural systems by Arnoldi's method

An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems. The algorithm can even be applied to structural systems with non-symmetric matrices. The algorithm is based on the use of Arnoldi's method to generate a Krylov subspace of trial vectors, which is then used to reduce a large eigenvalue problem to a much smaller one. The reduced eigenvalue problem is solved and the solutions are used to construct approximate solutions to the original large system. In the process, the algorithm takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, therefore, making it very economical for use in solving large problems. The numerical results from test examples are presented to demonstrate that a large fraction of the approximate solutions calculated are very accurate, indicating that the algorithm is highly effective for extracting a number of vibration modes for a large dynamic system, whether it is lightly or heavily damped.     

Use of ritz vectors in wave propagation and foundation response

The accuracy of a numerical method is demonstrated for the dynamic analysis of large complex finite element systems in which the spatial distribution of the loading is constant. The method is based on the use of a special class of Ritz vectors which were previously proposed and can be generated with minimum numerical effort. The purpose of this paper is to extend the use of these vectors to the solution of wave propagation and foundation response problems. The method is applied to one-, twoand three-dimensional problems in order to illustrate the efficiency and accuracy of the technique. Unless it is necessary to evaluate the very high-frequency behaviour of a structural system, it is shown that a small number of Ritz vectors will produce excellent results. Therefore, they can be very effective in the solution of three-dimensional soil-structure systems subjected to earthquake loading.     

A computationally efficient algorithm for the solution of eigenproblems for large structures with non-proportional damping using Lanczos method

In this paper, a solution method is presented to solve the eigenproblem arising in the dynamic analysis of non-proportional damping systems with symmetric matrices. The method is based on the Lanczos method to generate one pair of Krylov subspaces consisting of trial vectors, which is then used to reduce a large eigenvalue problem into a much smaller one. The method retains the n order quadratic eigenproblem, without employing the method of matrix augmentation traditionally used to cast the problem as a linear eigenproblem of order 2n. In this process, the method preserves the sparseness and symmetry of the system matrices and does not invoke complex arithmetic; thus making it very economical for use in solving large problems. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Approximate soil-structure interaction analysis by a perturbation approach: The case of stiff soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a structure on an elastic soil is derived through the application of a perturbation analysis. For stiff soils, the full solution is obtained as the sum of the solution for a rigid-soil and small perturbing terms related to the inverse of the soil shear modulus. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented by other authors for the fundamental mode. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

Response of non-classically damped structures in the modal subspace

The evaluation of the dynamic response of non-classically damped linear structures requires the solution of an eigenproblem with complex eigenvalues and modal shapes. Since in practice only a small number of complex modes are needed, the complex eigenvalue problem is solved in the modal subspace in which the generalized damping matrix is not uncoupled by classical real modes. It follows that the evaluation of the structural response requires in both cases the determination of complex modes by numerical techniques, which are not as robust as techniques currently used for the solution of the real eigenvalue problem, and the use of complex algebra. In the present paper an unconditionally stable step-by-step procedure is presented for the response of non-classically damped structures in the modal subspace without using complex quantities. The method is based on the evaluation of the fundamental operator in approximated form of the numerical procedure. In addition, the method can be easily modified to incorporate the modal superposition pseudo-static correction terms.     

Free and steady state vibration of non-linear structures using a finite element–non-linear eigenvalue technique

The problem of free vibration of non-linear structures is considered initially. It is shown that this problem can be represented as a non-linear eigenvalue problem. Variational principles for non-linear eigenvalue problems are defined. These variational principles are implemented with finite element models to define numerical approximations for the free vibration problem. The solution of these approximate equations provides a set of non-linear modal vectors and natural frequencies which vary with the amplitude of the solution. The non-linear eigenvalue parameters can be used in modal expansion approximations for the non-linear transient or steady state response of structural systems. To demonstrate the proposed techniques the free vibration and steady state vibration characteristics of a geometrically non-linear circular plate are determined.     

On the accuracy of mode superposition analysis in structural dynamics

It is pointed out that the number of modes which should be included in a mode superposition dynamic response analysis depends on both the frequency content and the distribution of the loading. If the loading frequency is low the effect of the higher modes can be approximated by a static analysis. A technique is described for calculating this static contribution from the higher modes; the total response is then represented by the sum of the lower mode dynamic response and the higher mode static effects. The effectiveness of the procedure is demonstrated by a numerical example.     

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

结构随机地震响应分析的复模态法

本文对多自由度基础平动结构随机地震响应问题进行了系统研究。针对用第1振型近似代表上部结构所得方程为非经典阻尼和非对称结构情况。用复模态法解耦。获得了以第1振型表示的结构地震响应的解析解。对单自由度体系。此解即为结构响应的精确解。本文方法也可用于带TMD减震结构等的随机地震响应分析与优化设计。     

Dynamic interaction of bridge–train system under non‐uniform seismic ground motion

The probability that an earthquake occurs when a train is running over a bridge in earthquake‐prone regions is much higher than before, for high‐speed railway lines are rapidly developed to connect major cities worldwide. This paper presents a finite element method‐based framework for dynamic analysis of coupled bridge–train systems under non‐uniform seismic ground motion, in which rail–wheel interactions and possible separations between wheels and rails are taken into consideration. The governing equations of motion of the coupled bridge–train system are established in an absolute coordinate system. Without considering the decomposition of seismic responses into pseudo‐static and inertia‐dynamic components, the equations of motion of the coupled system are formed in terms of displacement seismic ground motions. The mode superposition method is applied to the bridge structure to make the problem manageable while the Newmark‐β method with an iterative computation scheme is used to find the best solution for the problem concerned. Eight high‐speed trains running over a multi‐span steel truss‐arch bridge subject to earthquakes are taken as a case study. The results from the case study demonstrate that the spatial variation of seismic ground motion affects dynamic responses of the bridge–train system. The ignorance of pseudo‐static component when using acceleration seismic ground motions as input may underestimate seismic responses of the bridge–train system. The probability of separation between wheels and rails becomes higher with increasing train speed. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximate analysis methods for asymmetric plan base‐isolated buildings

An approximate method for linear analysis of asymmetric‐plan, multistorey buildings is specialized for a single‐storey, base‐isolated structure. To find the mode shapes of the torsionally coupled system, the Rayleigh–Ritz procedure is applied using the torsionally uncoupled modes as Ritz vectors. This approach reduces to analysis of two single‐storey systems, each with vibration properties and eccentricities (labelled ‘effective eccentricities’) similar to corresponding properties of the isolation system or the fixed‐base structure. With certain assumptions, the vibration properties of the coupled system can be expressed explicitly in terms of these single‐storey system properties. Three different methods are developed: the first is a direct application of the Rayleigh–Ritz procedure; the second and third use simplifications for the effective eccentricities, assuming a relatively stiff superstructure. The accuracy of these proposed methods and the rigid structure method in determining responses are assessed for a range of system parameters including eccentricity and structure flexibility. For a subset of systems with equal isolation and structural eccentricities, two of the methods are exact and the third is sufficiently accurate; all three are preferred to the rigid structure method. For systems with zero isolation eccentricity, however, all approximate methods considered are inconsistent and should be applied with caution, only to systems with small structural eccentricities or stiff structures. Copyright © 2001 John Wiley & Sons, Ltd.     

复合夹层结构频率及损耗因子的计算

频率和损耗因子是粘弹性材料复合夹层结构的两个重要的动力特性指标。本文采用复模量模型模拟夹层粘弹性材料特性的频率相关性,并基于大型通用有限元程序NASTRAN,提出采用模态应变能迭代法及复特征值迭代法求解复合夹层结构的各阶频率及损耗因子。该方法可以应用到大型复杂复合夹层结构中去,具有很好的实用性和较好的准确性。以复合夹层梁为例,进行了理论解析解和数值解的对比研究和优化设计分析。     

微分求积法在结构动力分析中的应用

微分求积法（DQM）是1种求解微分方程初（边）值问题的数值方法,通常以较小的计算工作量即可获得较高的数值精度。这种方法应用于工程领域时多用来解决梁、板等结构的静力分析或结构特征值分析等问题,即对边值问题的微分方程的求解。结构动力分析属于初值问题,荷载和结构反应都具有特殊性,直接套用DQM求解边值问题并不能获得问题的解。本文尝试利用微分求积原理建立求解结构动力反应的具体方法。借鉴单元法的思想,将荷载持时划分为若干个时步,在每个时步内对动态荷载和结构反应进行离散,然后用DQM对时步逐个进行求解,得到体系在整个时域内的反应过程。通过对3种不同自振周期的线弹性单自由度体系在不同频率简谐激励下反应的计算,阐释了本文方法的可行性以及高精度、高效率的特点,通过数值试验确定了时步内相对较优的节点数,并为时步长度的选取提供了建议。