SYMBOLIC COMPUTATION OF FUNDAMENTAL SOLUTION MATRICES FOR LINEAR TIME-PERIODIC DYNAMICAL SYSTEMS

• Published in: Journal of sound and vibration 1997-09, Vol.206 (1), p.61-85
• Main Authors: Sinha, S.C., Butcher, E.A.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid dynamics (ballistics, collision, multibody system, stabilization...); Solid mechanics
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1006/jsvi.1997.1079

A new technique which employs bothPicard iterationand expansion inshifted Chebyshev polynomialsis 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, theintegrationandproduct operational matricesassociated withe 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 isnotbased 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 time and convergence results are discussed.