Superharmonic and Subharmonic Resonance Analysis of A Rectangular Hyperelastic Membrane Resting on Nonlinear Elastic Foundation Using The Method of Multiple Scales

Document Type : Research Article

Authors

Shahrood University of Technology

Abstract

In this paper, the nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic Winkler-Pasternak foundation subjected to uniformly distributed hydrostatic pressure are investigated. The membrane is composed of an incompressible, homogeneous, and isotropic material. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Also, the kinetic energy, the work of uniform distributed force and pressure, and the effects of damping are determined, according to the strain energy function for neo-Hookean hyperelastic constitutive law. By applying Galerkin’s method, the nonlinear partial differential equation of motion in the transversal direction is transformed to the ordinary differential equations. Then, utilizing the method of multiple scales, the superharmonic and subharmonic resonances including the 1:3 superharmonic and 3:1 subharmonic, 1:5 superharmonic, and 5:1 subharmonic, 1:7 superharmonic, and 7:1 subharmonic are analyzed. Also, the analytical results are compared with those presented by other researchers. Finally, the effect of the Winkler and Pasternak stiffness, the material properties, and various geometrical characteristics on the superharmonic and subharmonic resonances of the vibration behavior of a rectangular hyperelastic membrane is investigated.

Keywords

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References

[1] J.J. Thomsen, Vibrations and stability: advanced theory, analysis, and tools, Springer, Berlin, 2003.
[2] Y. Ishida, T. Yamamoto, T. Ikeda, Nonlinear forced oscillations caused by quartic nonlinearity in a rotating shaft system, Journal of Vibration and Acoustics, 112(3) (1990) 288-297.
[3] F.M. Li, G. Yao, 1/3 Subharmonic resonance of a nonlinear composite laminated cylindrical shell in subsonic air flow, Composite Structures, 100 (2013) 249-256.
[4] D. Zou, Z. Rao, N. Ta, Coupled longitudinal-transverse dynamics of a marine propulsion shafting under superharmonic resonances, Journal of Sound and Vibration, 346 (2015) 248-264.
[5] J.C. Ji, A.Y.T. Leung, Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions, International Journal of Non-Linear Mechanics, 38(6) (2003) 829-835.
[6] S.L. Tsyfansky, V.I. Beresnevich, Detection of fatigue cracks in flexible geometrically non-linear bars by vibration monitoring, Journal of Sound and Vibration, 213(1) (1998). 159-168.
[7] A.P. Bovsunovskii, Vibrations of a nonlinear mechanical system simulating a cracked body, Strength of materials, 33(4) (2001) 370-379.
[8] A.P. Bovsunovsky, C. Surace, R. Ruotolo, The effect of damping on the non-linear dynamic behaviour of a cracked beam at resonance and super-resonance vibrations, In Key Engineering Materials, 245 (2003) 97-106.
[9] A. Selvadurai, Deflections of a rubber membrane, Journal of the Mechanics and Physics of Solids, 54(6) (2006) 1093-1119.
[10] G. Saccomandi, R.W. Ogden, Mechanics and thermomechanics of rubberlike solids, Springer, New York, 2004.
[11] I.D. Breslavsky, M. Amabili, M. Legrand, Nonlinear vibrations of thin hyperelastic plates, Journal of Sound and Vibration, 333(19) (2014) 4668-4681.
[12] S. Razavi, A. Shooshtari, Analytical investigation of nonlinear free vibration of Magneto-electro-elastic rectangular thin plate resting on a nonlinear elastic foundation, Amirkabir Journal of Mechanical Engineering, 49(2) (2017) 317-324. (In Persian)
[13] M. Forsat, Investigating nonlinear vibrations of higher-order hyper-elastic beams using the Hamiltonian method, Acta Mechanica, 231(1) (2020) 125-138.‏
[14] W. Chen, L. Wang, H. Dai, Nonlinear free vibration of hyperelastic beams based on neo-Hookean model, International Journal of Structural Stability and Dynamics, 20(1) (2020) 2050015.‏
[15] G. Varzandian, S. Ziaei, Analytical solution of non-Linear free vibration of thin rectangular plates with various boundary conditions based on non-Local theory, Amirkabir Journal of Mechanical Engineering, 48(4) (2017). (In Persian)
[16] J. Zhang, J. Xu, X. Yuan, H. Ding, D. Niu, W. Zhang, Nonlinear vibration analyses of cylindrical shells composed of hyperelastic materials, Acta Mechanica Solida Sinica, 32(4) (2019) 463-482.
[17] Z. Zhao, X. Yuan, W. Zhang, D. Niu, H. Zhang, Dynamical modeling and analysis of hyperelastic spherical shells under dynamic loads and structural damping, Applied Mathematical Modelling,‏ 95 (2021) 468-483.
[18] H. Koivurova, A. Pramila, Nonlinear vibration of axially moving membrane by finite element method, Computational Mechanics, 20(6) (1997) 573-581.
[19] J.M. Wu, Z. Tian, Y. Wang, X.X. Guo, Nonlinear vibration characteristics analysis of variable density printing moving membrane, in: 2016 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications (SPAWDA), IEEE, 2016, pp. 407-410.
[20]      J. Wu, M. Shao, Y. Wang, Q. Wu, Z. Nie, Nonlinear vibration characteristics and stability of the printing moving membrane, Journal of Low Frequency Noise, Vibration and Active Control, 36(3) (2017) 306-316.
[21] F. Khan, F. Sassani, B. Stoeber, Nonlinear behaviour of membrane type electromagnetic energy harvester under harmonic and random vibrations, Microsystem Technologies, 20(7) (2014) 1323-1335.
[22] X. Sun, J.Z. Zhang, Nonlinear vibrations of a flexible membrane under periodic load, Nonlinear Dynamics, 85(4) (2016) 2467-2486.
[23] C.J. Liu, Z.L. Zheng, X.Y. Yang, H. Zhao, Nonlinear damped vibration of pre-stressed orthotropic membrane structure under impact loading, International Journal of Structural Stability and Dynamics, 14(1) (2014) 1350055.
[24] P.B. Gonçalves, R.M. Soares, D. Pamplona, Nonlinear vibrations of a radially stretched circular hyperelastic membrane, Journal of Sound and Vibration, 327(1-2) (2009) 231-248.
[25] R.M. Soares, P.B. Gonçalves, Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane, International Journal of Solids and Structures, 49(3-4) (2012) 514-526.
[26] R.M. Soares, P.F. Amaral, F.M. Silva, P.B. Gonçalves, Nonlinear breathing motions and instabilities of a pressure-loaded spherical hyperelastic membrane, Nonlinear Dynamics, 99(1) (2020) 351-372.‏
[27] R.M. Soares, P.B. Gonçalves, Nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic foundation, Meccanica, 53(4-5) (2018) 937-955.
[28] D. Pamplona, D. Mota, Numerical and experimental analysis of inflating a circular hyperelastic membrane over a rigid and elastic foundation, International Journal of Mechanical Sciences, 65(1) (2012) 18-23.
[29] A. Kerr, On the formal development of elastic foundation models, Ingenieur-Archiv, 54(6) (1984) 455-464.
[30] R.D. Chien, C.S. Chen, Nonlinear vibration of laminated plates on a nonli ,near elastic foundation, Composite structures, 70(1) (2005) 90-99.
[31] S. Esfahani, Y. Kiani, M. Eslami, Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations, International Journal of Mechanical Sciences, 69 (2013) 10-20.
[32] P. Malekzadeh, A. Setoodeh, Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM, Composite Structures, 80(4) (2007) 569-579.
[33] D.M. Santee, P.B. Gonçalves, Oscillations of a beam on a non-linear elastic foundation under periodic loads, Shock and Vibration, 13(4-5) (2006) 273-284.
[34] S. Karimi, H. Ahmadi, K. Foroutan, Nonlinear vibration analysis of rectangular hyperelastic membrane resting on nonlinear elastic foundation using the method of multiple scales, Iranian Society of Acoustics and Vibration, December (2019). (In Persian)
[35] A.E. Green, J.E. Adkins, Large elastic deformations and non-linear continuum mechanics, Clarendon Press, Oxford, 1960.
[36] A.H. Nayfeh, D.T. Mook, Nonlinear Oscilations, John Wiley and Sons, New York, 1995.
Amirkabir Journal of Mechanical Engineering
Volume 53, Issue 8
November 2021
Pages 4535-4564

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