A Singularity Analysis on Dynamics of Symmetric Cross-Ply Composite Sandwich Plates Under 1∶2 Resonance
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摘要: 内共振是一种典型的非线性动力学行为,点阵夹芯板在航空航天领域中有着广泛的应用背景.研究点阵夹芯板的内共振问题具有重要的理论及工程意义.在横向激励与面内激励联合作用下,基于四边简支点阵夹芯板的动力学方程,利用多尺度法得到极坐标形式的平均方程,进而化简成稳态形式的代数方程,研究其在1∶2内共振情况下的非线性动力学行为.该文利用推广的奇异性理论研究分叉问题,基于稳态形式的代数方程,计算出含有两个调谐参数、一个横向激励和一个面内激励这4个参数的限制切空间;在强等价的条件下,简化了稳态形式的代数方程;在非退化的情况下,计算出简化的代数方程的正规形;对于含有两个状态变量和4个分叉参数的一般非线性动力学方程的奇异性理论进行了推广;利用推广的奇异性理论得到正规形余维4的18个普适开折的表达式;计算出普适开折转迁集的表达式;这样清楚了点阵夹芯板受到小扰动,当分叉、滞后和双极限点产生时,调谐参数和激励参数之间的关系,数值仿真了转迁集和分叉图,结果表明在不同的分叉区域有不同的振动形式.Abstract: Inner resonance is a typical nonlinear dynamic behavior, and the symmetric crossply composite sandwich plates have been widely used in aerospace. The studies about inner resonance of such sandwich plates have both theoretical and engineering significances. Based on the dynamic equations for the sandwich-plates, of which the boundary conditions were simply supported on 4 sides, the transverse and inplane excitations were both considered. The average equations in the polar form were obtained with the multiscale method, and the algebraic equations in the steady state form were derived through the average equations. The singularity theory was utilized to investigate 1∶2 resonant bifurcations of the symmetric crossply sandwich plates. Based on the algebraic equations in the steady state form, the restricted tangent space was obtained for the bifurcation equations with 2 tuning parameters, an inplane excitation and a transverse excitation. Then the algebraic equations were simplified under strong equivalence, and the normal form of the algebraic equations were obtained in nondegenerate cases. The singularity theory were generalized for the general nonlinear dynamic equations with 2 state variables and 4 bifurcation parameters, and the 18 universal unfoldings of bifurcation equations with codimension 4 were obtained in the case of 1∶2 internal resonance. The transition sets in the parameter plane and the bifurcation diagrams were depicted. The relationships between the tuning parameters and the exciting parameters were determined when bifurcation, hysteresis, and double limit points happened. The numerical results indicate that the vibration modes in different bifurcation regions are different.
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Key words:
- bifurcation equation /
- singularity theory /
- universal unfolding /
- transition set
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[1] XU X F, QIAO P Z. Homogenized elastic properties of honeycomb sandwich with skin effect[J]. International Journal of Solids and Structures,2002,39(8): 2153-2188. [2] RUZZENE M. Vibration and sound radiation of sandwich beam with honeycomb trusscore[J]. Journal of Sound and Vibration,2004,277(4/5): 741-763. [3] CHEN A, DAVALOS J F. A solution including skin effect for stiffness and stress field of sandwich honeycomb core[J]. International Journal of Solids and Structures,2005,42(9/10): 2711-2739. [4] FROSTIG Y, THOMSEN O T, SHEINMAN I. On the non-linear high-order theory of unidirectional sandwich panels with a transversely flexible core[J]. International Journal of Solids and Structures,2005,42(5/6): 1443-1463. [5] YU S D, CLEGHORN W L. Free flexural vibration analysis of symmetric honeycomb panels[J]. Journal of Sound and Vibration,2005,284(1/2): 189-204. [6] CIELECKA I, JEDRYSIAK J. A non-asymptotic model of dynamics of honeycomb lattice-type plates[J]. Journal of Sound and Vibration,2006,296(1/2): 130-149. [7] LUO Y J, XIE S L, ZHANG X N. The actuated performance of multi-layer piezoelectric actuator in active vibration control of honeycomb sandwich panel[J]. Journal of Sound and Vibration,2008,317(3/5): 496-513. [8] LI Y Q, ZHU D W. Free flexural vibration analysis of symmetric rectangular honeycomb panels using the improved Reddy’s third plate theory[J]. Composite Structures,2009,88(1): 33-39. [9] LIU J, CHEN Y S, LI R F. A semi-analytical method for bending, buckling, and free vibration analyses of sandwich panels with square-honeycomb cores[J]. International Journal of Structural Stability and Dynamics,2010,10(1): 127-151. [10] LI Yongqiang, LI Feng, ZHU Dawei. Geometrically nonlinear free vibrations of the symmetric rectangular honeycomb sandwich panels with simply supported boundaries[J]. Composite Structures,2010,92(5): 1110-1119. [11] BURLAYENKO V N, SADOWSKI T. Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates[J]. International Journal of Non-Linear Mechanics,2010,45(10): 959-968. [12] SHARMA R S, RAGHUPATHY V P. Influence of core density, core thickness, and rigid inserts on dynamic characteristics of sandwich panels with polyurethane foam as core[J]. Journal of Reinforced Plastics and Composites,2010,29: 3226-3236. [13] BURLAYENKO V N, SADIWSKI T. Dynamic behaviour of sandwich plates containing single/multiple debonding[J]. Computational Materials Science,2011,50(4): 1263-1268. [14] ZHANG J H, ZHANG W. Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a honeycomb sandwich plate[J]. Acta Mechanica,2012,223(5): 1047-1066. [15] GOLUBITSKY M, LANGFORD W F. Classification and unfoldings of degenerate Hopf bifurcations[J]. Journal of Differential Equations,1981,41(3): 375-415. [16] GOLUBITSKY M, GUILLEMIN V. Stable Mapping and Their Singularities [M]. New York: Springer-Verlag, 1973. [17] MARTINET J. Singularities of Smooth Functions and Maps[M]. Landon: Cambridge University Press, 1982. [18] GOLUBISTKY M S, SCHAEFFER D G. Singularities and Groups in Bifurcation Theory [M]. New York: Springer-Verlag, 1988. [19] FUTER J E, SITTA A M, STEWART I. Singularity theory and equivariant bifurcation problems with parameter symmetry[J]. Mathematical Proceedings of the Cambridge Philosophical Society,1996,120(3): 547-578. [20] SITTA A M. Singularity theory and equivariant bifurcation problems with parameter symmetry[D]. PhD Thesis. University of Warwick and USP-sao Carlos, 1993. [21] 郭瑞芝. 等变分歧问题研究[D]. 博士学位论文. 长沙: 中南大学, 2006. (GUO Ruizhi. Study on equivariant bifurcation problems[D]. PhD Thesis. Changsha: Central South University, 2006.(in Chinese)) [22] 崔登兰, 李养成. 含两组状态变量且参数具有对称性的等变分歧问题及其开折的稳定性[J]. 应用数学和力学, 2007,28(2): 209-215.(CUI Denglan, LI Yangcheng. Equivariant bifurcation problems and the stability of open fold under contains two sets of state variables and parameters which have symmetry[J]. Applied Mathematics and Mechanics,2007,28(2): 209-215.(in Chinese)) [23] 胡凡努, 李养成. 关于两状态变量组的等变分歧问题的通用开折[J]. 数学理论与应用, 2000,20(3): 50-57.(HU Fannu, LI Yangcheng. Versal unfolding of equivariant bifurcation problems about two sets of state variables[J]. Mathematical Theory and Applications,2000,20(3): 50-57.(in Chinese)) [24] 高守平, 李养成. 多参数等变分歧问题关于左右等价的开折[J]. 数学年刊, 2003,24(3): 341-348.(GAO Shouping, LI Yangcheng. Open fold of equivariant bifurcation problems with multiparameter under the left and right equivalent group[J]. Annals of Mathematics,2003,24(3): 341-348.(in Chinese)) [25] 郭瑞芝, 李养成. 含两组状态变量的等变分歧问题在左右等价群下的开折[J]. 应用数学和力学, 2005,26(4): 489-496.(GUO Ruizhi, LI Yangcheng. Open fold of equivariant bifurcation problems with two sets of state variables under the left and right equivalent group[J]. Applied Mathematics and Mechanics,2005,26(4): 489-496.(in Chinese)) [26] CHEN F Q, LIANG J S, CHEN Y S, et al. Bifurcation analysis of an arch structure with parametric and forced excitation[J]. Mechanics Research Communication,2007,34: 213-221. [27] QIN Z H, CHEN Y S. Singular analysis of bifurcation systems with two parameters[J]. Acta Mechanica Sinica,2010,26(3): 501-507. [28] QIN Z H, CHEN Y S. Singular analysis of a two-dimensional bifurcation systems[J]. Science in China Series,2010,53(3): 608-611. [29] QIN Z H, CHEN Y S, LI J. Singular analysis of two-dimensional elastic cable with 1: 1 internal resonance[J]. Applied Mathematics and Mechanics,2010,31(2): 143-150. [30] 秦朝红. 两状态变量、两分叉参数系统的分叉分析及其工程应用[D]. 博士学位论文. 哈尔滨: 哈尔滨工业大学, 2010.(QIN Zhaohong. Singularity method for nonlinear dynamical analysis of systems with two parameters and its application in engineering[D]. PhD Thesis. Harbin: Harbin Institute of Technology, 2010.(in Chinese)) [31] 陈建恩. 轻质材料层合板的非线性动力学理论分析与实验研究[D]. 博士学位论文. 北京: 北京工业大学, 2013.(CHEN Jianen. Theoretical and experimental investigations on nonlinear dynamics of light-weight sandwich plate[D]. PhD Thesis. Beijing: Beijing University of Technology, 2013.(in Chinese))
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