Bifurcation Analysis Of Shape Memory Alloy Laminated Beam

SMA is widely applied in structural vibration control.Usually,SMA is added into structures in various forms.SMA makes the system's dynamic behavior more complex.In this article,bifurcation analysis of SMA laminated beam and influence of parameters on the system are analyzed,which can provide theoretical references for structural control and design.1)The vibration equation is established.SMA laminated beam is a beam with SMA thin layer on its upper and lower surfaces.With the continuous polynomial stress-strain model,the dimensionless vibration equation of the system is obtained by physical equations,force balance equations and displacement coordation conditions.With Galerkin method,the discrete beam's high-order modal vibration equation can be solved.By average method,the amplitude-frequency response equation of the beam's high-order modal vibration can be obtained.2)Singularity analysis of first-order modal vibration.To analyze the effects of temperature,excitation amplitude and the parameter connected with SMA layer thickness on the system's vibration.Two groups of parameters are chosen,with Maple,the transition sets and regional bifurcation diagrams can be obtained.Results show that the response is quasi-linear or the right side is hysteretic,the response amplitude is greater as excitation amplitude is larger,temperature only affects the resonant frequency but has no effect on the response amplitude,and when the response is quasi-linear,SMA nearly has no vibration attenuation effect,when the response is nonlinear,the vibration attenuation effect of SMA layer is more obvious as excitation amplitude is greater and SMA layer is thicker,and the system's resonance is more difficult to occur.The theoretical and numerical results are basically the same.3)Singularity analysis of higher-order modal vibration.When excitation frequency is higher,there will be higher-order modal vibration of the beam.The dynamic behavior of second-order and third-order modal vibration of SMA beam will be discussed.The influence of the structural parameter releated with SMA layer thickness and the excitation amplitude on the system is discussed.Respective transition sets are solved,and results show that the response is quasi-linear,the right side is hysteretic,the left side is hysteretic or the both two sides are hysteretic.The theoretical and numerical results are nearly the same.