The Earthquake Engineering Online Archive

Alternatives to standard mode superposition for analysis of non-classically damped systems

Kusainov, Amirlan A.; Clough, Ray W.

UCB/EERC-88/09, Earthquake Engineering Research Center, University of California, Berkeley, 1988-06, 64 pages (555.2/K87/1988)

Two alternative approximate procedures that parallel standard mode superposition are described. In the first, the mass matrix is transformed using eigenvectors based on the stiffness and damping matrices. In the second, the stiffness matrix is transformed using eigenvectors based on the damping and mass matrices; the transformed equation sets are then uncoupled by ignoring the coupling coefficients of the generalized mass and stiffness matrices, respectively. A third procedure is presented that involves the use of corrected diagonal terms in the transformed property matrices, taking into account contributions based on the off-diagonal terms. To demonstrate the accuracy of these approximations, a two degree-of-freedom system is solved in closed form using the standard method and each of the three alternatives; results are compared with the exact solution obtained using the complex eigenvectors. The analysis is then extended to multidegree-of-freedom systems; the appropriate conditions for the approaches to be valid are obtained. The response of a nine-story building undergoing a single sine wave impulse in the basement is calculated using the different approximate procedures.

Available online: http://nisee.berkeley.edu/documents/EERC/EERC-88-09.pdf (5 MB)