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Mathematical Problems in Engineering Volume 2019 ,2019-07-25
Analysis of Nonlinear Vibrations and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh-Ritz Approach Combined with the Affine Transformation
Research Article
Wei Tian 1 , 2 Zhichun Yang 3 Tian Zhao 2
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DOI:10.1155/2019/9278069
Received 2019-04-28, accepted for publication 2019-07-04, Published 2019-07-04
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摘要

Nonlinear vibrations of a trapezoidal cantilever plate subjected to transverse external excitation are investigated. Based on von Karman large deformation theory, the Rayleigh-Ritz approach combined with the affine transformation is developed to obtain the nonlinear ordinary differential equation of a trapezoidal plate with irregular geometries. With the variation of geometrical parameters, there exists the 1:3 internal resonance for the trapezoidal plate. The amplitude-frequency formulations of the system in three different coupled conditions are derived by using multiple scales method for 1:3 internal resonance analysis. It is found that the strong coupling of two modes can change nonlinear stiffness behaviors of modes from hardening-spring to soft-spring characteristics. The detuning parameter and excitation amplitude have significant influence on nonlinear dynamic responses of the system. The bifurcation diagrams show that there exist the periodic, quasi-periodic, and chaotic motions for the trapezoidal cantilever plate in the 1:3 internal resonance cases and the nonlinear dynamic responses are dependent on the amplitude of excitation. The possible adverse dynamic behaviors and undesired resonance can be avoided by designing appropriate excitation and system parameters.

授权许可

Copyright © 2019 Wei Tian et al. 2019
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

通讯作者

Wei Tian.School of Aerospace, Xi’an Jiaotong University, Xi’an, China, xjtu.edu.cn;National Key Laboratory of Science and Technology on Liquid Rocket Engine, Xi’an Aerospace Propulsion Institute, Xi’an, China.twtp100@163.com

推荐引用方式

Wei Tian,Zhichun Yang,Tian Zhao. Analysis of Nonlinear Vibrations and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh-Ritz Approach Combined with the Affine Transformation. Mathematical Problems in Engineering ,Vol.2019(2019)

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参考文献
[1] H. Dai, X. Yue, S. Atluri. (2014). Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix. Journal of Mechanics of Materials and Structures.9(2):195-226. DOI: 10.1016/j.jsv.2014.10.038.
[2] S. Shokrollahi, F. Bakhtiari-Nejad. (2004). Limit cycle oscillations of swept-back trapezoidal wings at low subsonic flow. Journal of Aircraft.41(4):948-953. DOI: 10.1016/j.jsv.2014.10.038.
[3] W. Zhang, M. Zhao, X. Guo. (2013). Nonlinear responses of a symmetric cross-ply composite laminated cantilever rectangular plate under in-plane and moment excitations. Composite Structures.100:554-565. DOI: 10.1016/j.jsv.2014.10.038.
[4] E. H. Dowell. (1970). Panel flutter-a review of the aeroelastic stability of plates and shells. AIAA Journal.8(3):385-399. DOI: 10.1016/j.jsv.2014.10.038.
[5] W. Tian, Z. Yang, Y. Gu, X. Wang. et al.(2017). Analysis of nonlinear aeroelastic characteristics of a trapezoidal wing in hypersonic flow. Nonlinear Dynamics.89(2):1205-1232. DOI: 10.1016/j.jsv.2014.10.038.
[6] Y. X. Hao, L. H. Chen, W. Zhang, J. G. Lei. et al.(2008). Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. Journal of Sound and Vibration.312(4-5):862-892. DOI: 10.1016/j.jsv.2014.10.038.
[7] M. Sayed, A. Mousa. (2012). Second-order approximation of angle-ply composite laminated thin plate under combined excitations. Communications in Nonlinear Science and Numerical Simulation.17(12):5201-5216. DOI: 10.1016/j.jsv.2014.10.038.
[8] H. Dai, X. Yue, J. Yuan, S. N. Atluri. et al.(2014). A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity. Journal of Computational Physics.270:214-237. DOI: 10.1016/j.jsv.2014.10.038.
[9] W. Tian, Z. Yang, Y. Gu. (2017). Dynamic analysis of an aeroelastic airfoil with freeplay nonlinearity by precise integration method based on Padé approximation. Nonlinear Dynamics.89(3):2173-2194. DOI: 10.1016/j.jsv.2014.10.038.
[10] M. Ye, Y. Sun, W. Zhang, X. Zhan. et al.(2005). Nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation. Journal of Sound and Vibration.287(4-5):723-758. DOI: 10.1016/j.jsv.2014.10.038.
[11] K. M. Liew, K. C. Hung, M. K. Lim. (1995). Vibration of mindlin plates using boundary characteristic orthogonal polynomials. Journal of Sound and Vibration.182(1):77-90. DOI: 10.1016/j.jsv.2014.10.038.
[12] W. Tian, Z. Yang, T. Zhao. (2019). Nonlinear aeroelastic characteristics of an all-movable fin with freeplay and aerodynamic nonlinearities in hypersonic flow. International Journal of Non-Linear Mechanics.116:123-139. DOI: 10.1016/j.jsv.2014.10.038.
[13] H.-H. Dai, J. K. Paik, S. N. Atluri. (2011). The global nonlinear galerkin method for the solution of von karman nonlinear plate equations: an optimal and faster iterative method for the direct solution of nonlinear algebraic equations, using. Computers, Materials and Continua.23(2):155-185. DOI: 10.1016/j.jsv.2014.10.038.
[14] X. Wang, Z. Yang, J. Zhou, W. Hu. et al.(2016). Aeroelastic effect on aerothermoacoustic response of metallic panels in supersonic flow. Chinese Journal of Aeronautics.29(6):1635-1648. DOI: 10.1016/j.jsv.2014.10.038.
[15] A. Shooshtari, S. Razavi. (2015). Linear and nonlinear free vibration of a multilayered magneto-electro-elastic doubly-curved shell on elastic foundation. Composites Part B: Engineering.78:95-108. DOI: 10.1016/j.jsv.2014.10.038.
[16] F. X. An, F. Q. Chen. (2017). Multipulse orbits and chaotic dynamics of an aero-elastic fgp plate under parametric and primary excitations. International Journal of Bifurcation and Chaos.27(4, article 1750050). DOI: 10.1016/j.jsv.2014.10.038.
[17] H.-H. Dai, J. K. Paik, S. N. Atluri. (2011). The global nonlinear galerkin method for the analysis of elastic large deflections of plates under combined loads: a scalar homotopy method for the direct solution of nonlinear algebraic equations. Computers, Materials and Continua.23(1):69-99. DOI: 10.1016/j.jsv.2014.10.038.
[18] G. Anlas, O. Elbeyli. (2002). Nonlinear vibrations of a simply supported rectangular metallic plate subjected to transverse harmonic excitation in the presence of a one-to-one internal resonance. Nonlinear Dynamics.30(1):1-28. DOI: 10.1016/j.jsv.2014.10.038.
[19] M. Sayed, Y. S. Hamed, Y. A. Amer. (2011). Vibration reduction and stability of non-linear system subjected to external and parametric excitation forces under a non-linear absorber. International Journal of Contemporary Mathematical Sciences.6(21-24):1051-1070. DOI: 10.1016/j.jsv.2014.10.038.
[20] Z. X. Yang, Q. K. Han, Y. G. Chen. (2017). Nonlinear harmonic response characteristics and experimental investigation of cantilever hard-coating plate. Nonlinear Dynamics:1-12. DOI: 10.1016/j.jsv.2014.10.038.
[21] D. Younesian, H. Askari, Z. Saadatnia, M. KalamiYazdi. et al.(2010). Frequency analysis of strongly nonlinear generalized Duffing oscillators using HE's frequency-amplitude formulation and HE's energy balance method. Computers & Mathematics with Applications.59(9):3222-3228. DOI: 10.1016/j.jsv.2014.10.038.
[22] D. Younesian, H. Askari, Z. Saadatnia. (2011). Free vibration analysis of strongly nonlinear generalized Duffing oscillators using He's variational approach and homotopy perturbation method. Nonlinear Science Letters A.2(1):11-16. DOI: 10.1016/j.jsv.2014.10.038.
[23] C. Xue, E. Pan, Q. Han, S. Zhang. et al.(2011). Non-linear principal resonance of an orthotropic and magnetoelastic rectangular plate. International Journal of Non-Linear Mechanics.46(5):703-710. DOI: 10.1016/j.jsv.2014.10.038.
[24] H. Askari, Z. Saadatnia, E. Esmailzadeh, D. Younesian. et al.(2014). Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations. Journal of Sound and Vibration.333(22):5817-5835. DOI: 10.1016/j.jsv.2014.10.038.
[25] O. Doaré, S. Michelin. (2011). Piezoelectric coupling in energy-harvesting fluttering flexible plates: Linear stability analysis and conversion efficiency. Journal of Fluids and Structures.27(8):1357-1375. DOI: 10.1016/j.jsv.2014.10.038.
[26] F. Alijani, F. Bakhtiari-Nejad, M. Amabili. (2011). Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dynamics.66(3):251-270. DOI: 10.1016/j.jsv.2014.10.038.
[27] F. Alijani, M. Amabili. (2013). Non-linear dynamic instability of functionally graded plates in thermal environments. International Journal of Non-Linear Mechanics.50:109-126. DOI: 10.1016/j.jsv.2014.10.038.
[28] X. Wang, Z. Yang, W. Wang, W. Tian. et al.(2017). Nonlinear viscoelastic heated panel flutter with aerodynamic loading exerted on both surfaces. Journal of Sound and Vibration.409:306-317. DOI: 10.1016/j.jsv.2014.10.038.
[29] W. L. Ye, E. H. Dowell. (2011). Limit cycle oscillation of a fluttering cantilever plate. AIAA Journal.29(11):1929-1936. DOI: 10.1016/j.jsv.2014.10.038.
[30] H. Dai, X. Wang, M. Schnoor, S. N. Atluri. et al.(2017). Analysis of internal resonance in a two-degree-of-freedom nonlinear dynamical system. Communications in Nonlinear Science and Numerical Simulation.49:176-191. DOI: 10.1016/j.jsv.2014.10.038.
[31] D. Xie, M. Xu, H. Dai, E. H. Dowell. et al.(2015). Proper orthogonal decomposition method for analysis of nonlinear panel flutter with thermal effects in supersonic flow. Journal of Sound and Vibration.337:263-283. DOI: 10.1016/j.jsv.2014.10.038.
[32] F. Bakhtiari-Nejad, M. Nazari. (2009). Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate. Nonlinear Dynamics.56(4):325-356. DOI: 10.1016/j.jsv.2014.10.038.
[33] Y. Hao, W. Zhang, J. Yang. (2011). Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method. Composites Part B: Engineering.42(3):402-413. DOI: 10.1016/j.jsv.2014.10.038.
[34] W. Zhang, Z. Liu, P. Yu. (2001). Global dynamics of a parametrically and externally excited thin plate. Nonlinear Dynamics.24(3):245-268. DOI: 10.1016/j.jsv.2014.10.038.
[35] D. Xie, M. Xu, H. Dai, T. Chen. et al.(2017). New look at nonlinear aerodynamics in analysis of hypersonic panel flutter. Mathematical Problems in Engineering.2017-13. DOI: 10.1016/j.jsv.2014.10.038.
[36] W. Zhang. (2001). Global and chaotic dynamics for a parametrically excited thin plate. Journal of Sound and Vibration.239(5):1013-1036. DOI: 10.1016/j.jsv.2014.10.038.
[37] D. Xie, M. Xu, H. H. Dai, E. H. Dowell. et al.(2014). Observation and evolution of chaos for a cantilever plate in supersonic flow. Journal of Fluids and Structures.50:271-291. DOI: 10.1016/j.jsv.2014.10.038.
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