Symplectic Analysis for Wave Propagation in One-Dimensional Nonlinear Periodic Structures
-
摘要: 利用辛数学方法分析了质量-弹簧非线性周期结构链中弹性波的传播问题.首先利用能量方法得到频域动力方程,随后通过小量变换将非线性动力方程线性化,得到辛矩阵,进而通过求解辛矩阵的本征值问题来研究波的传播性能.质量-弹簧模型中的弹簧刚度非线性对结构链的传播特性影响很大,研究发现非线性明显改变了周期结构的传播性能,而且不同于线性结构,非线性结构的传播特性与入射波强度有关.数值算例表明随着非线性强度及入射波强度的增大,传播通带宽度逐渐减小,禁带宽度逐渐增大.当入射波强度增大到一定值时,弹性波无法在结构中进行传播.与一般递归方法的比较分析,验证了辛数学方法在非线性周期结构波传播问题中的有效性与优越性.Abstract: The wave propagation problem in nonlinear periodic mass-spring structure chain was analyzed using the symplectic mathematical method. Firstly the energy method was applied to construct the dynamical equation and then the nonlinear dynamical equation was linearized using the small parameter perturbation method. The eigen-solutions of the symplectic matrix were applied to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain,arising from the nonlinear spring stiffness effect,has profound effects on the overall transmission of the chain. The wave propagation characteristics are not only altered due to the nonlinearity but also related with the incident wave intensity, which is a genuine nonlinear effect that is not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to a closing of the transmitting gaps. Comparison with the normal recursive approach demonstrates the effectiveness and superiority of the symplectic method in wave propagation problem for nonlinear periodic structures.
-
[1] Mead D M. Wave propagation in continuous periodic structures: research contributions from southampton, 1964-1995[J]. Journal of Sound and Vibration, 1996, 190(3): 495-524. doi: 10.1006/jsvi.1996.0076 [2] Yan Z Z,Wang Y S. Calculation of band structures for surface waves in two-dimensional phononic crystals with a wavelet-based method[J]. Physical Review B(Condensed Matter and Materials Physics), 2008, 78(9): 4306-4316. [3] Jensen J S. Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures[J]. Journal of Sound and Vibration, 2003, 266(5): 1053-1078. doi: 10.1016/S0022-460X(02)01629-2 [4] Zhang Y P, Wu B. Composition relation between gap solitons and Bloch waves in nonlinear periodic systems[J]. Physical Review Letters, 2009, 102(9): 3905-3908. [5] 刘志芳, 王铁锋, 张善元. 梁中非线性弯曲波传播特性的研究[J]. 力学学报, 2007, 39(2): 238-244. [6] Yagi D, Kawahara T. Strongly nonlinear envelope soliton in a lattice model for periodic structure[J]. Wave Motion, 2001, 34(1): 97-107. doi: 10.1016/S0165-2125(01)00062-2 [7] Richoux O, Depollier C, Hardy J. Propagation of mechanical waves in a one-dimensional nonlinear disordered lattice[J]. Physical Review E, 2006, 73(2): 6611-6621. [8] Marathe A, Chatterjee A. Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales[J]. Journal of Sound and Vibration, 2006, 289(4/5): 871-888. doi: 10.1016/j.jsv.2005.02.047 [9] Georgiou I T, Vakakis A F. An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators[J]. International Journal of Non-Linear Mechanics, 1996, 31(6): 871-886. doi: 10.1016/S0020-7462(96)00104-7 [10] Romeo F, Rega G. Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach[J]. Chaos, Solitons & Fractals, 2006, 29(3): 606-617. [11] 钟万勰. 应用力学的辛数学方法[M].北京:高等教育出版社, 2006. [12] Zhong W X, Williams F W, Leung A Y T. Symplectic analysis for periodical electro-magnetic waveguides[J]. Journal of Sound and Vibration, 2003, 267(2): 227-244. doi: 10.1016/S0022-460X(02)01451-7 [13] 冯康, 秦孟兆. 哈密尔顿系统的辛几何算法[M].杭州:浙江科学技术出版社, 2002. [14] Feng K. On difference schemes and symplectic geometry[C]Proceeding of the 1984 Beijing symposium on D D. Beijing: Science Press, 1984. [15] 张素英, 邓子辰. 非线性动力学系统的几何积分理论及应用[M].西安:西北工业大学出版社, 2005. [16] Elmaimouni L, Lefebvre J E, Zhang V, Gryba T. A polynomial approach to the analysis of guided waves in anisotropic cylinders of infinite length[J]. Wave Motion, 2005, 42(2): 177-189. doi: 10.1016/j.wavemoti.2005.01.005 [17] Wu C J, Chen H L, Huang X Q. Sound radiation from a finite fluid-filled/submerged cylindrical shell with porous material sandwich[J]. Journal of Sound and Vibration, 2000, 238(3): 425-441. doi: 10.1006/jsvi.2000.3086 [18] Bridges T J. Multi-symplectic structures and wave propagation[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1997, 121(1): 147-190. doi: 10.1017/S0305004196001429 [19] Reich S. Multi-symplectic Runge-Kutta collocation method for Hamiltonian wave equations[J]. Journal of Computational Physics, 2000, 157(2): 473-499. doi: 10.1006/jcph.1999.6372 [20] Marsden J E, Pekarsky S, Shkoller S, West M. Variational methods, multisymplectic geometry and continuum mechanics[J]. Journal of Geometry and Physics, 2001, 38(3/4): 253-284. doi: 10.1016/S0393-0440(00)00066-8 [21] Williams F W, Zhong W X, Bennett P N. Computation of the eigenvalues of wave propagation in periodic substructural systems[J]. Journal of Vibration and Acoustics, 1993, 115(4): 422-426. doi: 10.1115/1.2930367 [22] Zhou M, Zhong W X, Williams F W. Wave propagation in substructural chain-type structures excited by harmonic forces[J]. International Journal of Mechanical Sciences, 1993, 35(11): 953-964. doi: 10.1016/0020-7403(93)90032-P [23] 张洪武, 姚征, 钟万勰. 界带分析的基本理论和计算方法[J]. 计算力学学报, 2006, 23(3): 257-263. [24] Zhang H W, Yao Z, Wang J B, Zhong W X. Phonon dispersion analysis of carbon nanotubes based on inter-belt model and symplectic solution method[J]. International Journal of Solids and Structures, 2007, 44(20): 6428-6449. doi: 10.1016/j.ijsolstr.2007.02.033 [25] 姚征, 张洪武, 王晋宝. 基于界带模型的碳纳米管声子谱的辛分析[J]. 固体力学学报, 2008, 29(1): 13-22. [26] Hennig D,Tsironis G P. Wave transmission in nonlinear lattices[J]. Physics Reports, 1999, 307(5/6): 333-432. doi: 10.1016/S0370-1573(98)00025-8
点击查看大图
计量
- 文章访问数: 1870
- HTML全文浏览量: 130
- PDF下载量: 829
- 被引次数: 0