Applying Homotopy Analysis To Solve Analytical Approximate Solutions Of Strong Nonlinear Vibration Systems

As nonlinear problems are widely present in the fields of mathematics and engineering,and have an important impact on both,due to the complexity of the nonlinear equations(groups),it is difficult to obtain accurate values.Classical perturbations such as the KBM method,the multi-scale method,and the averaging method are effective for studying several problems of weakly nonlinear systems,but are difficult to apply to strong nonlin ear systems.Therefore,quantitative analysis of strong nonlinear systems is a very difficult research topic.For the quantitative analysis method of strong nonlinear systems,Professor Liao Shijun of Shanghai Jiaotong University proposed a new calculation method---the homotopy analysis method in his doctoral thesis.This method is based on the continuously changing topological theory.The problem studied is independent of whether it contains small parameters,and the appropriate basis function can be selec ted according to the problem to better approximate the real solution.This method has been applied to solve strong nonlinear problems.In view of this,this paper takes the conductive cylindrical shell and the vehicle suspension system in the transverse magnetic field as the research object,and uses the homotopy analysis method to study the nonlinear dynamic behavior of the single-degree-of-freedom and two-degree-of-freedom systems respectively.The content and academic contributions are as follows:(1)Apply the homotopy analysis method to solve the forced vibration of the conductive cylindrical shell in the transverse magnetic field.Due to the influence of electromagnetic field,the electromagnetic elastic structure will exhibit complex mechanical behavior and directly affect the safety and reliability of the system operation.Therefore,studying the nonlinear characteristics of electromagnetic elastic structures is of great significance to ensure the stable operation of many high-tech devices.In this paper,the effects of thickness,excitation force and geometric nonlinear term on the amplitude-frequency curve of simply supported cylindrical shells are investigated.The results show that as the thickness of the shell increases,the resonance region becomes narrower and the hardening behavior of the system Significantly increase;as the amplitude of the excitation increases,the resonance region gradually becomes wider,and the range of the common amplitude value gradually becomes larger.The homotopy an alysis method has good precision and stability,and is especially suitable for strong nonlinear problems.It opens up a new way to solve nonlinear problems.(2)The forced vibration of the suspension system of the car was studied.The suspension system has a great influence on the stability of the car,the comfort of the ride and the dynamic performance of the car itself.Therefore,it is of great practical value to study the nonlinear vibration problem of automotive suspension systems.In this paper,the similar analytical solution of the system is obtained by homotopy analysis method,and the effects of road excitation,nonlinear factors and control parameters on the nonlinear vibration of the suspension system are analyzed.The results show that in the resonance region,the damping has great influence on the vehicle body.The larger damping can effectively reduce the relative displacement of the vehicle body.Outside the resonance region,the damping has little effect on the relative displacement of the vehicle body and the tire.In order to meet the ride comfort of the car,the choice of spring stiffness should not be too large,and the appropriate stiffness can be selected within a certain range of parameters.