The 1/3rd subharmonic and 3rd superharmonic resonance of a shape memory alloy (SMA) laminated beam
Ming Gao 2,3
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Department of Civil Engineering, Hebei Jiaotong Vocational and Technical College, Shijiazhuang, China
College of Mechanical and Electronical Engineering, Shandong Agriculture University, Taian, China
Tianjin Key Laboratory of Microgravity and Hypogravity Environment Simulation Technology, Tianjin, China
College of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao, China
Submission date: 2019-05-21
Final revision date: 2020-01-06
Acceptance date: 2020-08-20
Online publication date: 2020-11-06
Publication date: 2021-01-15
Corresponding author
Ming Gao   

College of Mechanical & Electronical Engineering, Shandong Agriculture University, China
Journal of Theoretical and Applied Mechanics 2021;59(1):27-41
This paper examines the 1/3rd subharmonic resonance and the 3rd superharmonic resonance of simply-supported shape memory alloy (SMA) laminated beams. First, the dynamic equation for SMA laminated beams under transverse load is established using physical equations, force equilibrium conditions, the compatibility equation of deformation, and a constitutive model of SMA polynomial functions. Then, a differential equation for transverse vibration of the SMA laminated beams is derived by the Galerkin process assuming the boundary conditions for simply-supported beams. Next, the amplitude-frequency response equations for the 1/3rd subharmonic resonance and the 3rd superharmonic resonance of these beams are derived by an averaging method before their respective transition sets are calculated, and their amplitude-frequency response diagrams were plotted using singularity theory. The results show two different types of amplitude-frequency responses to nonlinear vibration under the 1/3rd subharmonic resonance and the 3rd superharmonic resonance: quasi-linear and hard characteristic. In the quasi-linear area, SMA thickness A does not make much difference to the response of the system, whereas in the hard-characteristics area, under the same excitation amplitude f, the resonance frequency increases with A. In the nonlinear area, SMA can obviously reduce vibration amplitude, but it is not obvious for the 1/3rd subharmonic resonance. The nonlinear solution of both the 1/3rd subharmonic resonance and 3rd superharmonic resonance are stable.
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