Stability analysis of parametrically excited systems with time-delay

This paper presents a new technique for studying the stability properties of parametrically excited dynamic systems with time delay modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the "infinite-dimensional Floquet transition matrix U". Two different formulas for the computation of the approximate U, whose size is determined by the number of polynomials employed, are given. The first one, which results in a numerical stability matrix, uses the direct integral form of the original system in state space form while the second, which can give a symbolic stability matrix in terms of parameters, uses a convolution integral (variation of parameters) formulation. An extension of the method to the case where the delay and parametric periods are commensurate is also available. Numerical and symbolic stability charts are produced for several examples of time-periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance-modulated turning. The results indicate that this method is a effective way to study the stability of periodic DDEs.