Solution and stability of a set of PTH order linear differential equations with periodic coefficients via Chebyshev polynomials

Chebyshev polynomials are utilized to obtain solutions of a set of pth order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the ‘differential state space formulation’ and the ‘differential direct formulation’, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.

[1]  Application of Chebyshev polynomials to the optimal control of time-varying linear systems , 1985 .

[2]  S. C. Sinha,et al.  Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov—Floquet transformation , 1994 .

[3]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[4]  J. J. Stoker.  Nonlinear Vibrations in Mechanical and Electrical Systems , 1950 .

[5]  G. Hill.  On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon , 1886 .

[6]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[7]  G. Gaonkar,et al.  On computing Floquet transition matrices of rotorcraft , 1981 .

[8]  S. Barnett.  Multiplication of Generalized Polynomials, with Applications to Classical Orthogonal Polynomials , 1984 .

[9]  Designing Linear Optimal Regulators Via Chebyshev Polynomials , 1993, 1993 American Control Conference.

[10]  S. Sinha,et al.  A new approach in the analysis of linear systems with periodic coefficients for applications in rotorcraft dynamics , 1994, The Aeronautical Journal (1968).

[11]  S. Sinha,et al.  Optimal control of mechanical systems subjected to periodic loading via chebyshev polynomials , 1993 .

[12]  S. Sinha,et al.  A new numerical technique for the analysis of parametrically excited nonlinear systems , 1993 .

[13]  R. V. Dooren,et al.  A Chebyshev technique for solving nonlinear optimal control problems , 1988 .

[14]  C. Hsu,et al.  On approximating a general linear periodic system , 1974 .

[15]  Martin Avery Snyder.  Chebyshev methods in numerical approximation , 1968 .

[16]  S. Sinha,et al.  An efficient computational scheme for the analysis of periodic systems , 1991 .

[17]  Vimal Singh,et al.  Perturbation methods , 1991 .

[18]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[19]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[20]  S. Sinha,et al.  Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems , 1996 .

[21]  S. Sinha,et al.  Analysis of dynamic systems with periodically varying parameters via chebyshev polynomials , 1993 .

[22]  S. Sinha,et al.  An approximate analysis of transient response of time-dependent linear systems by use of orthogonal polynomials☆ , 1976 .

[23]  S. C. Sinha,et al.  Control of General Dynamic Systems With Periodically Varying Parameters Via Liapunov-Floquet Transformation , 1994 .

[24]  K. G. Lindh,et al.  Infinite determinant methods for stability analysis of periodic-coefficient differential equations , 1970 .

[25]  H. Denman,et al.  Stability analysis of systems with periodic coefficients: An approximate approach , 1979 .

[26]  S. C. Sinha,et al.  A new computational technique for the stability analysis of slender rods , 1992, Archive of Applied Mechanics.