M.Sc Thesis
M.Sc Student Bogomolny Michael Approximate Vibration Reanalysis of Structures Department of Civil and Environmental Engineering Professor Emeritus Uri Kirsch

Abstract

The solution methods for eigenproblem analysis can be subdivided into several groups. Most iterative procedures require the solution of a set of linear equations in each iteration. The effectiveness of the methods depends largely on two factors: the possibility of a reliable use of the procedure, and the cost of the solution. These methods are not suitable for reanalysis, since the stiffness matrix must be decomposed for each iteration cycle or a change in the design.

The present research deals with the development, investigation and application of the Combined Approximation (CA) method for vibration reanalysis of structures. Initially, the CA method was developed for static reanalysis. It was later applied to non-linear, sensitivity and vibration problems. The method uses the combination of local (series) approximations and a reduced basis expression. In the solution process, the terms of the binomial series are used as basis vectors in a reduced basis expression. The basic idea of the reduced basis approach is that of transforming a problem with a large number of degrees of freedom into one with a much smaller number of degrees of freedom. The basis vectors are used as the columns of a transformation matrix for the reduced basis.
In the present research, the CA method is developed for two types of problems:

a)      Reanalysis of the vibration eigenproblem.

b)      Eigenproblem reanalysis by various iterative procedures.
In vibration reanalysis using the CA method we calculate each eigenpair (eigenvalue and eigenvector) separately. To improve the accuracy of the results in the calculation of higher modes, Gram-Schmidt orthogonalization has been used for all basis vectors.

The CA method for static reanalysis was used to improve the efficiency of some common iterative procedures for eigenproblem analysis. These procedures can then be used for efficient reanalysis of large scale structures. Effective solution with the CA method has been demonstrated for the following iterative procedures:

·   Inverse iteration

·   Inverse iteration with shifts

·   Lanczos iteration

·   Subspace iteration

Some numerical examples show that high accuracy of the results can be achieved efficiently by using the CA method.