Construction,Solution And Dynamics Study Of Nonlinear Wave Motion Equations In Mechanical Components Bar And Beam

When parts(rods)in mechanical systems are impacted by dynamic loads,the dynamic study of impact mechanical systems based on the linear wave equation of classical elastic rods has been widely used in engineering,but it is only suitable for low-speed impact and the Poisson effect can be ignored.At present,finite deformation is used to describe and establish a mathematical model to obtain more accurate results for the beam and bar of high speed impact.In addition,compared with the traditional calculus model,it is found that the phenomenon in layered and porous media is explained by the physical and mechanical model based on the fractal derivative theory framework,which will be more comprehensive and profound.Therefore,in this paper,rods and functionally gradient beams subjected to impact loads in mechanical systems are taken as objects,and a more accurate nonlinear wave control model is established and its dynamics is studied.The main contents and results are as follows:(1)Considering the finite deformation and Poisson effect,the nonlinear wave governing equations of impact mechanical rod components are derived by using generalized Hamiltonian principle.On the basis of not reducing the form of the basic solution,the auxiliary equation is simplified and combined with the B?cklund transformation,the extended minimalist(G’/G)function expansion method of extension is proposed and analyzed theoretically.The precise solution of displacement gradient traveling wave of nonlinear wave control equation is obtained by using the extended minimalist(G’/G)function expansion method,and the results obtained by this example are in agreement with the theoretical analysis of the extended minimalist(G’/G)function expansion method.Combined with the axial strain formula expressed by displacement gradient,the corresponding nonlinear strain wave function is obtained,and the mutation mechanism of strain(stress)at the bar defect is revealed by numerical simulation method.(2)By means of the precise traveling wave solution of the nonlinear wave governing equation with poisson effect and finite deformation taken into account,the propagation characteristics of the nonlinear wave are studied,and then the dynamics equation of the unitary impact mechanical system based on the nonlinear wave is established.Based on this dynamic equation,the analytical formulas of the working efficiency,the maximum penetration and the maximum stress of the impact mechanical system in plastic and elastic working media with respect to the radius of the rod,the impact velocity,the mass ratio of the hammer rod to the hammer head,the medium loading stiffness or the plastic limit resistance and so on were obtained.It is found by numerical simulation that the increase of the impact velocity increases the maximum impact quantity,but the increase of the maximum stress caused by the increase of impact velocity cannot be reduced simply by increasing the rod radius.(3)The functional gradient beam with layered structure and porous characteristics is regarded as a fractal beam.Based on Euler-Lagrange formula and chain rule in the framework of fractal derivative theory,the propagation governing equation of fractal nonlinear bending wave in beam of mechanical components is derived,and the exact traveling wave solution of the equation is obtained by using the extended minimalist(G’/G)function expansion method.By means of numerical simulation,the variation of the waveform of the kink isolated wave solution with fractal dimension is analyzed,and the analysis results are compared with the results of modeling and analysis by different methods in other literatures.It is found that the fractal dimension and the parameters of functionally graded materials have the same effect on waveform change,which provides a new idea for the analysis of its dynamic properties.In this paper,the motion behavior of the loaded bar and beam members in the mechanical system is analyzed from the point of view of mathematics,and the deformation and motion models of the loaded bar and beam members are explored,which provides useful help for the establishment of the digital twin model of mechanical system.