Chebyshev Expansion of Linear and Piecewise Linear Dynamic Systems With Time Delay and Periodic Coefficients Under Control Excitations

Abstract

In this paper, a new efficient method is proposed to obtain the transient response of linear or piecewise linear dynamic systems with time delay and periodic coefficients under arbitrary control excitations via Chebyshev polynomial expansion. Since the time domain can be divided into intervals with length equal to the delay period, at each such interval the fundamental solution matrix for the corresponding periodic ordinary differential equation (without delay) is constructed in terms of shifted Chebyshev polynomials by using a previous technique that reduces the problem to a set of linear algebraic equations. By employing a convolution integral formula, the solution for each interval can be directly obtained in terms of the fundamental solution matrix. In addition, by combining the properties of the periodic system and Floquet theory, the computational processes are simplified and become very efficient. An alternate version, which does not employ Floquet theory, is also presented. Several examples of time-periodic delay systems, when the excitation period is equal to or larger than the delay period and for linear and piecewise linear systems, are studied. The numerical results obtained via this method are compared with those obtained from Matlab DDE23 software (Shampine, L. F., and Thompson, S., 2001, “Solving DDEs in MATLAB,” Appl. Numer. Math., 37(4), pp. 441–458.) An error bound analysis is also included. It is found that this method efficiently provides accurate results that find general application in areas such as machine tool vibrations and parametric control of robotic systems.