Steady state response of undamped systems to excitations expressed as polynomials in time

Abstract

It is pointed out that an excitation, which expressed as a polynomial in time, a particular integral of the governing partial differential equation of an undamped continuous system may be assumed to be polynomial as well; this provides an alternative to the traditional use of the Duhamel integral. The particular integral in polynomial form is the steady state response corresponding to the polynomial excitation acting on the system since the infinite past. Frequency independent stiffness and mass matrices only are required to solve for the steady state exactly. Transients can be included by modal analysis. Recurrent formulae for piecewise linear and cubic forcing functions are given explicitly.