Method of Polynomial Vectors for Solving Nonlinear Vibrations with Multiple Time Scales

The perturbation method of solving nonlinear vibrations has already been presented in textbooks for different nonlinear systems. The perturbation methods, however, lead to solving a set of second-order ordinary differential equations (ODEs). One of the main drawbacks of the ODEs-based methods is of low efficiency for those with multiple degrees of freedom.

A new method which uses the polynomial vectors for solving nonlinear vibration systems of multiple degrees of freedom is proposed. The nonlinear equations of motion of ordinary differential form are put into the form of state equations first. Then, a set of non-homogeneous perturbation equations is obtained using multiple time scales.

The approximate solutions of the non-homogeneous equations are directly given by the linearized equations and their nonlinear parts using just one-step calculation, which provides all solutions of system. All the secular terms related to resonances are determined by the integral part of the approximate solutions. A two-dimension oscillator of cubic nonlinearity and a four-dimension spring-pendulum are taken as examples to demonstrate the solution process of the proposed method.

This paper proposes a complete method to tackle the calculation problem of approximate solutions of higher-dimension nonlinear systems. The proposed method is based on the first-order state equations. All the approximate solutions can be obtained directly from the non-homogeneous perturbation equations by just one-step calculation. The result shows that the secular terms with explicit time arise from the integral part. Compare with the ODEs-based method of multiple time scales, the proposed method has a concise solution structure and computational process.