EXTENSION OF MULTI-SCALE METHOD AND ITS APPLICATION TO NONLINEAR VISCOELASTIC SYSTEM

Viscoelastic materials have broad application prospects in aviation, machinery, civil engineering and other fields, and the nonlinear Zener model with 1.5 degrees of freedom can better describe their characteristics. Therefore, it is of great significance to study the extension and application of multi-scale method. Based on the traditional multi-scale method, the multi-scale method is extended to approximate the analytical solution of the nonlinear odd-order differential equation, and the dynamic problems of the nonlinear odd-order system are solved. Taking the nonlinear Zener model as an example, its dynamic behavior and stability condition under harmonic excitation are analyzed. Firstly, the approximate analytical solution of the nonlinear Zener model is obtained through the extended multi-scale method, and the analytical solution is verified by the numerical method. The results are in good agreement, which proves the correctness of the extended method. Then, the amplitude-frequency equation and phase-frequency equation of steady-state response are derived from the analytical solution, and it is found that there are multi-valued characteristics in a certain frequency range for weakly damped systems. Moreover, the stability condition of steady-state periodic solutions is obtained based on Lyapunov first method, and the system stability is analyzed by using this condition. Finally, the effects of nonlinear term, external excitation and the stiffness and damping coefficients of Maxwell elements on the dynamic behavior and system stability are analyzed by simulation. It is found that whether the stiffness is hardened or softened, the resonance amplitude can be gradually reduced and the multi-solution region can be expanded. The amplitude of external excitation has little influence on the backbone curve of amplitude frequency characteristics, but has a great influence on the shape of amplitude frequency curve. With the increase of the stiffness coefficient of Maxwell element, the resonance amplitude increases slightly. The increase of the damping of Maxwell element can reduce the resonance amplitude and the multi-solution region, and finally the multi-solution phenomenon can disappear. These results are of great significance to the study on dynamic characteristics of nonlinear viscoelastic systems in the future.