Research Of Strongly Nonlinear Vibration On Rod Structure,Arch Structure And Board Structure

In this paper,the theoretical research and numerical calculation for the "strongly nonlinear vibration problem of pole structure,arch structure and board structure",and the specific content is as follows:(1)Considering nonlinear elastic effect,the periodic action of concentrated load,mathematical model of strongly nonlinear partial differential equation of vibration of compressive pole structure is established.Strongly nonlinear partial differential equation simplified as Duffing-Mathieu equation by galerkin method.Duffing-Mathieu equation was solved principal oscillation analytic solution,the subharmonic solution and 3 times 21 superharmonic analytical solution by energy method.Numerical analysis of Dynamics to Duffing-Mathieu equations,finding out the influence of natural frequency of Duffing-Mathieu equation by the given numerical cases,and mapped the bifurcation diagram,time history response curve and phase diagram and poincare map;(2)Considering nonlinear damping effect,the periodic action of concentrated load,mathematical model of strongly nonlinear partial differential equation of vibration of semicircle arch structure is established.Strongly nonlinear partial differential equation simplified as Van der Pol-Mathieu equation by galerkin method.Van der Pol-Mathieu equation was solved principal oscillation analytic solution,the subharmonic solution 31 and 2 times superharmonic analytical solution by energy method;(3)Considering mutual influence of nonlinear elastic effect and nonlinear damping effect,the periodic action of distribution load,mathematical model of strongly nonlinear partial differential equation of vibration of laminated board structure is established.Strongly nonlinear partial differential equation simplified as Van der Pol-Duffing-Mathieu equation by galerkin method.Van der Pol-Duffing-Mathieu equation was solved principal oscillation analytic solution,the subharmonic solution and 3 times superharmonic 21 analytical solution by energy method.(4)Analog computation of Duffing-Mathieu equation,the Van der Pol-Mathieu equation and Van der Pol-Duffing-Mathieu equation,mapped the time history curve and phase diagram to compare analytical solution and numerical solution.Results show that calculation results to analytical solution match numerical solution are in good.