Analytical solution and modulation of time periodic parameter for dynamic systems

Pseudo closed form complete solution for nonlinear dynamical systems subjected to inhomogeneous generalized forcing using state transition matrices was developed, based on the Fourier expansion of periodic matrices and substitution of matrix exponential terms via Lagrange-Sylvester theorem. With a prescribed frequency spectrum, a jump modulation procedure was employed to avoid the aliasing effect when constructing approximation for equivalent stairstep stiffness function, yielding the appropriate constant value over each subinterval. A Hill-Messiner type of two degree-of-freedom system was presented afterwards to demonstrate methods for the forced response and the procedure for the modulation of time periodic parameter. The obtained results are presented and discussed.