Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems

A new technique which employs both Picard iteration and expansion in shifted Chebyshev polynomials is used to symbolically approximate the fundamental solution matrix for linear time-periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time. As in previous studies, the integration and product operational matrices associated with the Chebyshev polynomials are employed. However, the need to algebraically solve for the Chebyshev coefficients of the fundamental solution matrix is completely avoided as only matrix multiplications and additions are utilized. Since these coefficients are expressed as homogeneous polynomials of the system parameters, closed form approximations to the true solutions may be obtained. Also, because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. Two formulations are proposed. The first is applicable to general time periodic systems while the second approach is useful when the system equations contain a constant matrix. Three different example problems, including a double inverted pendulum subjected to a periodic follower force, are included and CPU times and convergence results are discussed.