Useful polynomials for the transient response of solids under mechanical and thermal excitations

Integral- and fractional-order polynomials capable of describing the transient response of mechanical and thermal systems to a generalized excitation of displacement or temperature were derived using convolution theory. Excellent agreement between the derived polynomials and an existing closed-form solution was then seen for the mechanical case where a single-degree of freedom system without damping was subjected to an asymptotic-exponential and half-sine displacement excitations. Excellent agreement was also seen with the thermal case of a slab subjected to an asymptotic-exponential temperature rise on the exposed surface. However, in order to avoid the use of functions involving complex arguments that arise during a portion of the integration, the Laplace Transform identity of the convolution integral and a ten-term Gaver-Stehfest inversion formula were used. The use of polynomials in the solution was intended to allow the incorporation of empirical date not easily represented by standard functions. The resulting relationships are easily programmed and can be used to solve a variety of problems in engineering, as well as verify numerical calculations involving more sophisticated methods such as finite-element analysis.