A finite element method for nonlinear dynamic analysis using a modal transformation matrix

Abstract

A solution procedure is developed and evaluated for the solution of nonlinear dynamic structural problems. The nonlinear equations of motion are written using the finite element method with geometric nonlinearities and moderate rotations. The natural frequencies and mode shapes of the free vibration form of the equations are calculated and the mode shapes used to transform the equations of motion to another orthogonal coordinate system. In the transformed form the equations are uncoupled and may be integrated directly. This solution yields the displacements in the transformed coordinate system; these displacements are returned to the physical basis and used to evaluate the nonlinear terms for the next time. The process is repeated over the desired time range. The solution method is applied first to a simple geometrically nonlinear beam problem and then to a shell of revolution that has highly nonlinear response to the point load at its apex. The numerical results are evaluated for cases utilizing several numbers of mode shapes and are compared to a known convergent solution method. The conclusion reached is that economies in solution time and storage requirements can be effected by using a fraction of the full set of mode shapes and a larger time step than was hitherto permissible. The use of the solution method with existing computer codes and the convergence aspects of the method are discussed.