The 1/3rd subharmonic and 3rd superharmonic resonance of a shape memory alloy (SMA) laminated beam
Xia Hui Zhang 1,   Ming Gao 2, 3  
,   Ying Hao 4
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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
Ming Gao   

College of Mechanical & Electronical Engineering, Shandong Agriculture University, 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
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.
Akhavan-Rad B., Kheirikhah M.M., 2019, Static analysis of sandwich plates embedded with shape memory alloy wires using active strain energy tuning method, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 41,160, 1-17.
Asadi H., Bodaghi M., Shakeri M., 2013, On the free vibration of thermally pre/post-buckled shear deformable SMA hybrid composite beams, Aerospace Science and Technology, 31, 73-86.
Collet M., Foltête E., Lexcellent C., 2001, Analysis of the behavior of a shape memory alloy beam under dynamical loading, European Journal of Mechanics – A/Solids, 20, 4, 615-630.
Fu S., Lu Q., 2012, Nonlinear dynamics and vibration reduction of a dry friction oscillator with SMA restraints, Nonlinear Dynamics, 69, 3, 1365-1381.
Ghaznavi A., Shariyat M., 2017, Non-linear layerwise dynamic response analysis of sandwich plates with soft auxetic cores and embedded SMA wires experiencing cyclic loadings, Composite Structures, 171, 185-197.
Lu P., Cui F.S., Tan M.J., 2009, A theoretical model for the bending of a laminated beam with SMA fiber embedded layer, Composite Structures, 90, 4, 458-464.
Machado L.G., Savi M.A., Pacheco P.M.C.L., 2003, Nonlinear dynamics and chaos in coupled shape memory oscillators, International Journal of Solids and Structures, 40, 19, 5139-5156.
Nassiri-Monfared A., Baghani M., Zakerzadeh M.R., Fahimi P., 2018, Developing a semianalytical model for thermomechanical response of SMA laminated beams, considering SMA asymmetric behavior, Meccanica, 53, 4-5, 957-971.
Odeny D.M. Jr, Donadon M.V., Castro S.G.P., 2017, Aeroelastic behavior of stiffened composite laminated panel with embedded SMA wire using the hierarchical Rayleigh-Ritz method, Composite Structures, 181, 26-45.
Paiva A., Savi M.A., 2006, An overview of constitutive models for shape memory alloys, Mathematical Problems in Engineering, 2006, 4, 1-30.
Razavilar R., Fathi A., Dardel M., Hadi J.A., 2018, Dynamic analysis of a shape memory alloy beam with pseudoelastic behavior, Journal of Intelligent Material Systems and Structures, 29, 9, 1835-1849.
Ren Y., Liu Y., Yang S., Wang X., 2010, Active deformation models of SMA fiber hybrid thin-walled laminated beams, Chinese Journal of Solid Mechanics, 31, 3, 228-236.
Ren Y., Shao B., 2001, Material damping of SMA fiber hybrid laminated beam, Mechanics in Engineering, 23, 16-20.
Samadpour M., Asadi H., Wang Q., 2016, Nonlinear aero-thermal flutter postponement of supersonic laminated composite beams with shape memory alloys, European Journal of Mechanics – A/Solids, 57, 18-28.
Savi M.A., Pacheco P.M.C.L., Braga A.M.B., 2002, Chaos in a shape memory two-bar truss, International Journal of Non-Linear Mechanics, 37, 8, 1387-1395.
Shao B., Ren Y., 2003, Analysis of free vibrations of shape memory alloy hybrid composite beams, Engineering Mechanics, 20, 4, 183-187.
Shao B., Ren Y., 2004, The semi-active control of shape memory alloy composite beam, Mechanics in Engineering, 26, 16-19.
Yu H., 2018, New application of shape memory alloy on variable thickness wing, Internal Combustion Engine and Parts, 17, 234-235.
Zeng S., Wan X., 2011, Shape memory alloys and their application in aviation industry (1), Heat Treatment Technology and Equipment, 32, 3, 1-5.
Zhang X., Wu Z., 2017, Bifurcation analysis of shape memory alloy laminated beam, Chinese Journal of Applied Mechanics, 34, 3, 397-403.
Zhang Z., 2012, Dynamic bifurcation and control of the structures with Shape Memory Alloy (SMA), Ph.D. Thesis, Tianjin University, Tianjin, China.