The Earthquake Engineering Online Archive

Solution Methods for Damped Linear Dynamic Systems

Chen, Harn-Ching

UCB/SEMM-1987/08, Dept. of Civil Engineering, University of California, Berkeley, 1987-11, 127 pages

This dissertation considers solution methods for damped linear structural systems which are subjected to transient loading. To accommodate a range of possible applications, the damping is assumed to be non-proportional to mass and stiffness and may also be large and/or lumped. Some important characteristic properties of the damped system are presented. The second-order equations of motion are reduced to a first-order set by doubling the size of the problem to facilitate the subsequent analysis and computation. The Rayleigh-Ritz method is generalized for the matrix pencil associated with a damped system. A projection method is discussed as an alternative. Both provide a basis for computing partial eigensolutions of a large damped dynamic system. The subspace interation method is modified to extract eigensolutions of a damped dynamic system. The iteration vectors are arranged in such a way that only real arithmetic is required to describe the complex solution vectors. The algorithmn is implemented to solve some typical numerical examples. The Lanczos method also is extended to find eigensolutions of a damped dynamic system. The loss of orthogonality between Lanczos vectors is investigated and two schemes are presented to restore the required orthogonality. The algorithm presented can take full advantage of the symmetry and sparsity of the associated matrices and also involves only real arithmetic during the solution process. The algorithm is implemented and tested with the numerical examples introduced in the subspace solution. Within the framework of mode superposition, a number of choices are discussed for finding the response of a dynamic system subjected to initial conditions and external loadings. In particular, the eigenvectors are used to transform the equations of motion into a set of uncoupled equations. A closed form solution and a numerical solution to the uncoupled equations are presented. An example is shown to illustrate the mode displacement method for the analysis of transient dynamic problem.

Available online: http://nisee.berkeley.edu/documents/SEMM/SEMM-87-08.pdf (4 MB)