Stability Analysis Of Axially Excited Beam With Elastic Boundary

Beams are one of the most common structures in engineering.They are widely used in aerospace,machinery,building and other fields,such as railway tracks,bridges,pipes conveying fluid,drive shafts,frame beams.In mechanical,they can be simplified as a beam for analysis.When the structure works,the end connections of these engineering structures may periodically move and excite the structure.Because of the external axial excitation,the dynamic instability may occur and the beam may have a parametric resonance,which can result in a large structural vibration.It will not only affect the service life of the structure,but also cause fatal accidents and injuries.Therefore,it is of great practical significance to study the parametric vibration characteristics of the beam with elastic support.Parametric vibration is indirectly caused by the periodically changing of system parameters due to the external excitation.As the system parameters are time-varying,the parametric vibration system is a non-autonomous system.When the system parameters have a great variation.The linear system may be unstable and the nonlinear system may have complex nonlinear dynamic responses such as periodic vibration,period doubling bifurcation and chaotic motion.Therefore,parametric vibration has always been a hot topic in the field of vibration mechanics and nonlinear dynamics,which has highly theoretical and practical value.In this dissertation,the dynamic model of anxial excited beam with elastic supports is established.By using the multi-scale method,the stability boundary of parametric vibration of the linear beam system and the steadystate response of nonlinear beam system are obtained.The influence of the elastic support on the parametric vibration characteristics of the beam is also investigated.In the past,the research on the parametric vibration of beam structure mostly focused on the fixed boundary and the simply supported boundary.However,the elastic support is a more general support condition,which is closer to the practical project.Therefore,the parametric vibration of the elastically supported beams is a scientific problem worthy of studying.The detailed research contents are as follows:Firstly,the nonlinear dynamic model of viscoelastic beam with elastic supports is established and the characteristics of its free vibration are analyzed.Based on the Euler-Bernoulli beam theory and Hamilton's principle,the nonlinear dynamic governing equation of an axially excited beam supported by springs on both sides is established.Considering the stress caused by the small and finite bending deformation of the beam,the cubic nonlinear term is introduced.Moreover,the governing equation is simplified by using the quasi-static assumption so the equation has an integral term.In order to describe the viscoelasticity of the material,the Kelvin Voigt model is used.Based on the linear derived system,the natural frequencies and modal functions of the system with different support stiffnesses are obtained.At the same time,the modal orthogonality of the beam with elastic support boundary is proved.By analyzing the static equilibrium equation of the beam,the critical axial forces of the beams with different supporting stiffness are derived.It is found that the symmetrical spring stiffness does not affect the critical axial force of the system.Secondly,the stability of parametric vibration of the linear system is studied.Based on the linear control equation,the partial differential equation is discretized into ordinary differential equations by the Galerkin truncation.The semi-analytical solutions of the steady-state response are obtained by using the multi-scale method.According to the Routh-Hurwitz stability criterion,the expression of the parametric stability boundary of the system is obtained.The influences of the stiffness of the supporting spring on the stability of the parametric resonance are emphatically discussed.It is found that the stability boundary of the axially excited beam can be effectively increased by decreasing the support stiffness.Finally,the nonlinear response of the anxial excited beam with elastic support is studied.The Galerkin method is used to discretize the continuum structure and the multi-scale method is used for the semi-analytical solution.The convergence of truncation is analyzed by numerical method.It is found that the fourth-order truncation meets the accuracy requirement so that the semi-analytical solution can be verified by the Runge-Kutta method.Then,the effects of the stiffness of the supporting spring and the average axial force on the nonlinear response of the system are discussed.The results show that: Although reducing the support stiffness can reduce the range of resonance frequency,it will also increase the resonance amplitude of the system to a certain extent.In brief,the elastic support boundary studied in this dissertation is a more general situation between the simply supported boundary and the free boundary.The research shows that the stiffness of the support spring can significantly change the parametric stability boundary and nonlinear response of the axially excited beam.Therefore,the research results will provide the guidance for the design of structures subjected to axial excitation.