Approximate Analytical Solution Of Multi-frequency Dynamical Systems With Periodic Excitation

Multi-frequency coupled dynamic system with periodic excitation is a common mathematical model in various engineering fields.Due to the different frequencies,this kind of system will present a variety of complex dynamic behaviors,which has attracted the attention of many scholars.Since the analytical solution is one of the key problems in the study of system dynamics and vibration control,the solution of differential equations has been a hot topic in mathematics and other disciplines for many years.Because it is difficult to obtain the exact solution for complex system,approximately analytical solution is often used to the theoretical research in different fields.In this paper,based on the method of variation of constant and variable coefficient method,a new method named as Constant Substitution Method(CSM)is proposed to solve the approximate analytical solutions of multi-frequency dynamic systems with periodic excitations.The main research contents are as follows:Firstly,for single degree-of-freedom(DOF)system with constant coefficient,the relationship between analytical solution and initial value condition is studied for both homogeneous and non-homogeneous equation,and the conclusion that the value range of initial condition determines the value range of particular solution is given.Secondly,based on the single DOF system with two frequencies coupled,the Constant Substitution Method(CSM)is proposed.By numerical simulation,the approximately analytical solution based on CSM is in well agreement with the exact solution not only in steady-state response,but also in transient response.Thirdly,the transient process of approximate solution of Mathieu equation with forced periodic excitation is studied.The equation contains three kinds of frequencies:natural frequency,forced excitation frequency and parametric excitation frequency.The CSM is applied to the system,and the transient approximate analytical solution is obtained.By comparing the numerical solution with the analytical solution,it is found that the approximate solution obtained by CSM is more accurate.In addition,through parameters and variance analysis,it is found that the approximate analytical solution is suitable for a wide range of parameters.Finally,the steady-state process of the approximate analytical solution of Mathieu equation under forced periodic excitation is discussed.The steady-state solutions are obtained by CSM and classical first-order harmonic balance method respectively.The comparison between the two approximate analytical solutions and numerical solutions shows that the approximate analytical solution by CSM is better in agreement with the numerical solution than classical first-order harmonic balance method.At the same time,the parameter application range of constant substitution method with high precision is given for different parameters.