Analysis of Dynamic Systems With Periodically Varying Parameters Via Chebyshev Polynomials

In this paper a general method for the analysis of multidimensional second-order dynamic systems with periodically varying parameters is presented. The state vector and the periodic matrices appearing in the equations are expanded in Chebyshev polynomials over the principal period and the original differential problem is reduced to a set of linear algebraic equations. The technique is suitable for constructing either numerical or approximate analytical solutions. As an illustrative example, approximate analytical expressions for the Floquet characteristic exponents of Mathieu’s equation are obtained. Stability charts are drawn to compare the results of the proposed method with those obtained by Runge-Kutta and perturbation methods. Numerical solutions for the flap-lag motion of a three-bladed helicopter rotor are constructed in the next example. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton, and Gear algorithms. The results obtained in both the examples indicate that the suggested approach is extremely accurate and is by far the most efficient one.