Study On Multiple Steady States Of Piecewise Smooth Vibration System

Piecewise smoothing is a form of nonlinear mechanical systems,The multiplicity of solutions is an important characteristic of nonlinear systems,In many fields,such as mechanical system parameter optimization design,service performance and fault detection,Study on multiple solutions of piecewise smooth vibration system has practical engineering application value.The study on dynamic characteristics of piecewise smooth vibration system only focused on partial solutions in recent years.A large number of studies only focused on one or two steady-state solutions closest to one initial condition.One reason is that piecewise smooth system is difficult to be analyzed accurately.It is also an important reason to rely too much on analytic solution to establish Poincaré mapping in qualitative analysis.In this paper,the multiple steady states of piecewise smooth vibration system would been studied.Firstly,the mechanical model with single degree of freedom nonlinear vibration system has been established.the stiffness model of airbag which is difficult to analyze had been transformed into a polynomial stiffness model which can be analyzed approximately.On this basis,taking the numerical solution as the standard,the error of approximate analytical method is analyzed by statistical analysis.The results show that the accuracy of the approximate analytical method is unstable.It shouldn’t be used for qualitative analysis accurately.Therefore,it is not feasible to construct Poincare map with the idea of replacing analytical solution with approximate analytical solution.It is necessary to extend the existing qualitative analysis methods by using numerical solutions.In the theoretical work,based on the idea of numerical calculation instead of analytical solution.The dependence on analytical solution in the process of qualitative analysis is cast off.The systematic method for multiple steady states is determined for analysis of piecewise smooth vibration system which is difficult to analyze.The establishment of Poincaré mapping based on numerical calculation method,referred to as Poincaré section method,is the existing basis of theoretical research.Combined with the iterative principle and geometric principle of discrete system,a method to solve the approximate equilibrium point of the system near the bifurcation parameters is determined.Based on the similarity of difference principle and differential principle,an iterative method for solving Jacobian matrix and eigenvalue of Poincare map at approximate equilibrium point is determined.Based on the fact that the steady-state solution has the domain of attraction,a method of searching multiple solutions along different parameter paths is established.Through the simulation of the bifurcation behavior of nonlinear system with single degree of freedom,it is found that the main bifurcation processes are doubling bifurcation and hysteresis,and hysteresis is an accompanying characteristic of the existence of multiple steady states.At the same time,the coexistence forms of multiple stable states are explored,including the coexistence of different periodic solutions,the coexistence of periodic solutions and chaotic solutions,and the coexistence of symmetric phase trajectories.Then,a piecewise linear vibration model with two degrees of freedom is established,and the semi analytical method is optimized to study the bifurcation behavior of the piecewise linear system.Hopf bifurcation and doubling bifurcation found in the system are analyzed by using the numerical method.The approximate eigenvalues of the approximate Jacobian matrix of the Poincaré mapping at the approximate equilibrium point conform to the bifurcation theory,which proved the validity of the method.By analyzing nonclassical toroidal bifurcation,it is found that there must be an unstable attractor between two stable coexisting attractors.By analyzing the unstable limit cycle,the reason why the state transition can reflect multiple solutions is analyzed.Finally,a piecewise smooth vibration system model with two degrees of freedom is established.The influence of the second harmonic and the third harmonic on the dynamic behavior of the system is studied through comparative analysis.By studying the bifurcation behavior of the equilibrium point,another reason for the multiple steady state of the system is obtained.It is found that the coexistence of multiple solutions of periodic attractor,quasi periodic attractor and chaotic attractor could exist in the system.One of the bifurcations similar to Hopf flip bifurcation is studied.Finally,it is defined as the phase-2 phase-locked state of Hopf bifurcation.In this paper,the numerical solution is used to replace the analytical solution in the process of qualitative analysis,and the qualitative analysis method is expanded to some extent,so that it can be applied to the qualitative analysis of systems that are difficult to analyze or cannot be analyzed.On this basis,the multiple steady state problems of piecewise linear and piecewise smooth vibration systems are analyzed.At the same time,the causes of multiple steady states are analyzed.