Free response approximation of a 2-DOF parametric vibration system

For a 2-DOF parametric vibration system, using the concept of feedback modulation, its free response was expressed as a linear combination of vibrations with its natural frequencies and parametric excitation frequencies, and approximated with matrix trigonometric series. Adopting the harmonic balance method, the 2-DOF parametric vibration equation was converted into an infinite set of linear algebraic equations, the characteristic equation was obtained from the nonzero solution to homogeneous linear algebraic equations, and the system's main natural frequencies were achieved from numerical solutions to the characteristic equation. Introducing normalized modes, the system's modal coefficient matrix and the general solution of its free response were solved. With initial conditions, arbitrary constants of its free response's general solution were determined. A computation error function was defined and used to compare the proposed approach and Runge-Kutta algorithm. The computation results showed that when terms of the approximation series are larger than a certain number, the former's computational error is much smaller than the latter's one; the proposed method provides an effective analytical tool for free response approximation of a 2-DOF parametric system, and is valuable for theoretical study and engineering application. © 2019, Editorial Office of Journal of Vibration and Shock. All right reserved.