Nonlinear analysis in synthesis of the resonant vibro-impact systems

Abstract

The paper considers differential equation of the vibro-impact resonance system with an asymmetric piecewise linear elastic characteristic. The time-instant of switching of elastic characteristic is determined on the basis of equality of oscillation period to average value of the corresponding eigenfrequencies. Then, expansion of the asymmetric piecewise linear elastic characteristic into Fourier series was made. The initial differential equation was reduced to a kind of parametric equations of Hill’s andMathieu’s type with taking into account the time of elastic characteristic change. Stability analysis of parametric Mathieu equation is shown for the analysis of natural oscillations. For stability analysis of the synthesized by various stiffness coefficients of vibro-impact system, dependencies of Mathieu equations coefficients on the parameter of synthesis are used. The solution of the initial equation with forced oscillations in the form of asymmetric two-frequency vibrations has been obtained by means of Bubnov-Galerkin and Levenberg-Marquardt methods for nonlinear algebraic systems of equations, also amplitude and phase frequency dependence was graphically drawn. The basic equation with an asymmetric elastic response characteristic feature is determined by the fixed natural frequency of oscillations independently of amplitude. Numerical solution of differential equations by means of Runge–Kutta method are presented for comparison. Comparison of the vibro-impact resonance system kinematics characteristics, synthesized by the elastic parameters and solved by the listed methods, is conducted. The feasibility of using nonlinear analysis presented in two harmonics in Fourier series asymmetric elastic characteristic is justified in the article. The suggested approach with Bubnov-Galerkin method for general Hill’s equation and correlation analysis of time kinematic characteristics was used. Acceleration frequency spectrum and harmonics are obtained on the basis of Runge–Kutta numerical method simulation of the initial nonlinear differential equation.