Application of Normal Mode Theory and Pseudoforce Methods to Solve Problems With Nonlinearities

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Abstract
In the design of structural systems such as nuclear reactor coolant loops consisting of piping, supports, bumpers, and tie rods, the basic structure is linear. For transient analysis of piping loops under conditions of earthquake and hypothetical accident of pipe rupture, the linear system becomes nonlinear because of forces due to bottoming in gaps, plastic action in the bumper stops or tie rods, etc. The dynamic analysis of such a structure normally employs the direct integration of the governing nonlinear equations of motion. A technique is presented in this paper where conventional normal mode theory is used even though there are nonlinearities. Nonlinearities such as bumper-gap elements, plasticity, etc., are defined as functions of motion and incorporated as generalized pseudoforces. This approach can, to a considerable degree, preserve the benefits of modal type analysis such as physical understanding in terms of frequencies and modes, and adequate and economical solutions using a reduced number of modes.

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