亚音速气流中复合材料悬臂板的非线性振动响应研究
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摘要
随着材料科学的发展,越来越多的新型材料应用到了工程实践中.在气流激励的作用下,对于以航空航天工程为背景、采用复合材料的板壳结构的非线性动力学问题仍是动力学领域的研究热点.本文研究了复合材料悬臂板在亚音速气流条件下的非线性振动和响应.根据理想不可压缩流体的流动条件和Kutta–Joukowski升力定理,基于升力面理论,利用涡格法计算了三维有限长平板机翼上的亚音速气动升力.将亚音速气动力施加到复合材料悬臂板上,利用Hamilton原理,考虑Reddy三阶剪切变形理论并引入冯·卡门非线性应变位移关系,建立了有限长平板的非线性动力学微分方程.利用有限元方法考察了不同几何参数下层合板悬臂板的固有特性,通过比较不同材料和几何参数的线性系统的固有频率,得到不同比例的内共振关系.利用Galerkin方法将偏微分方程截断为两自由度非线性常微分方程,在这里考虑了1:2的内部共振关系并利用多尺度法进行了摄动分析.对应多个选取参数,得到了频率响应曲线.结果展示了硬化弹簧型行为和跳跃现象.
With the development of materials science, more and more new materials have been applied to engineering practice. Under the action of airflow excitation, the nonlinear dynamics of plate and shell structures with composite materials based on aerospace engineering is still a hot research topic. In this work, the nonlinear vibrations and responses of a laminated composite cantilever plate under subsonic air flow are investigated. According to the flow condition of ideal incompressible fluid and the Kutta–Joukowski lift theorem, the subsonic aerodynamic lift on the three-dimensional finite length flat wing is calculated by using the Vertex Lattice(VL) method, which is applied to the cantilever plate. The finite length flat wing is modeled as a laminated composite cantilever plate based on the Reddy's third-order shear deformation plate theory, moreover the von Karman geometry nonlinearity is introduced. The nonlinear partial differential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamic force are established via Hamilton's principle. The partial differential equations are separated into two nonlinear ordinary differential equations via Galerkin method. Through comparing the natural frequencies of the system with different material and geometry parameters, the 1:2 internal resonance is considered here using multiple scales method. Corresponding to several selected parameters, the frequency-response characteristics are obtained. The hardening-spring-type behaviors and jump phenomena are exhibited.
With the development of materials science, more and more new materials have been applied to engineering practice. Under the action of airflow excitation, the nonlinear dynamics of plate and shell structures with composite materials based on aerospace engineering is still a hot research topic. In this work, the nonlinear vibrations and responses of a laminated composite cantilever plate under subsonic air flow are investigated. According to the flow condition of ideal incompressible fluid and the Kutta–Joukowski lift theorem, the subsonic aerodynamic lift on the three-dimensional finite length flat wing is calculated by using the Vertex Lattice(VL) method, which is applied to the cantilever plate. The finite length flat wing is modeled as a laminated composite cantilever plate based on the Reddy's third-order shear deformation plate theory, moreover the von Karman geometry nonlinearity is introduced. The nonlinear partial differential governing equations of motion for the laminated composite cantilever plate subjected to the subsonic aerodynamic force are established via Hamilton's principle. The partial differential equations are separated into two nonlinear ordinary differential equations via Galerkin method. Through comparing the natural frequencies of the system with different material and geometry parameters, the 1:2 internal resonance is considered here using multiple scales method. Corresponding to several selected parameters, the frequency-response characteristics are obtained. The hardening-spring-type behaviors and jump phenomena are exhibited.
引文
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4刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展.力学学报,2017,49(3):487-506(Liu Renhuai,Xue Jiang,Development of nonlinear mechanics for laminated composite plates and shells.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):487-506(in Chinese))
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7 Abe A,Kobayashi Y,Yamada G.Two-mode response of simply supported,rectangular laminated plates.International Journal of NonLinear Mechanics,1998,33(4):675-690
8 Abe A,Kobayashi Y,Yamada G.Three-mode response of simply supported,rectangular laminated plates.JSME International Journal Series C Mechanical Systems,Machine Elements and Manufacturing,1998,41(1):51-59
9 Abe A,Kobayashi Y,Yamada G.Analysis of subharmonic resonance of moderately thick antisymmetric angle-ply laminated plates by using method of multiple scales.Journal of Sound and Vibration,1998,217(3):467-484
10 Oh K,Nayfeh AH.Nonlinear combination resonances in cantilever composite plates.Nonlinear Dynamics,1996,11(2):143-169
11 Pai PF,Nayfeh AH.A nonlinear composite plate theory.Nonlinear Dynamics,1991,2(6):445-477
12 Pai PF,Nayfeh AH.A unified nonlinear formulation for plate and shell theories.Nonlinear Dynamics,1994,6(4):459-500
13 Librescu L,Reddy JN.A few remarks concerning several refined theories of anisotropic composite laminated plates.International Journal of Engineering Science,1989,27(5):515-527
14 Zhang W.Global and chaotic dynamics for a parametrically excited thin plate.Journal of Sound and Vibration,2001,239(5):1013-1036
15 Kitipornchai S,Yang J,Liew KM.Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections.International Journal of Solids and Structures,2004,41(9-10):2235-2257
16 Park T,Lee SY,Seo JW,et al.Structural dynamic behavior of skew sandwich plates with laminated composite faces.Composites Part B:Engineering,2008,39(2):316-326
17 Chien RD,Chen CS.Nonlinear vibration of laminated plates on an elastic foundation.Thin-Walled Structures,2006,44(8):852-860
18 Singh BN,Lal A,Kumar R.Nonlinear bending response of laminated composite plates on nonlinear elastic foundation with uncertain system properties.Engineering Structures,2008,30(4):1101-1112
19 Alijani F,Amabili M,Karagiozis K,et al.Nonlinear vibrations of functionally graded doubly curved shallow shells.Journal of Sound and Vibration,2011,330(7):1432-1454
20 Alijani F,Bakhtiari-Nejad F,Amabili M.Nonlinear vibrations of FGM rectangular plates in thermal environments.Nonlinear Dynamics,2011,66(3):251
21 Zhang W,Chen J,Zhang YF,et al.Continuous model and nonlinear dynamic responses of circular mesh antenna clamped at one side.Engineering Structures,2017,151:115-135
22 Liu T,Zhang W,Wang JF.Nonlinear dynamics of composite laminated circular cylindrical shell clamped along a generatrix and with membranes at both ends.Nonlinear Dynamics,2017,90(2):1393-1417
23 Sun Y,Zhang W,Yao MH.Multi-pulse chaotic dynamics of circular mesh antenna with 1:2 internal resonance.International Journal of Applied Mechanics,2017,9(4):1750060
24 Zhang W,Liu T,Xi A,et al.Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes.Journal of Sound and Vibration,2018,423:65-99
25 Dowell EH.Nonlinear oscillations of a fluttering plate.AIAA Journal,1966,4(7):1267-1275
26 Dowell EH.Nonlinear oscillations of a fluttering plate.II.AIAAJournal,1967,5(10):1856-1862
27 Amabili M,Pa?doussis MP.Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels,with and without fluid-structure interaction.Applied Mechanics Reviews,2003,56(4):349-381
28 Singha MK,Ganapathi M.A parametric study on supersonic flutter behavior of laminated composite skew flat panels.Composite Structures,2005,69(1):55-63
29 Haddadpour H,Navazi HM,Shadmehri F.Nonlinear oscillations of a fluttering functionally graded plate.Composite Structures,2007,79(2):242-250
30 Kuo SY.Flutter of rectangular composite plates with variable fiber pacing.Composite Structures,2011,93(10):2533-2540
31 Zhao MH,Zhang W.Nonlinear dynamics of composite laminated cantilever rectangular plate subject to third-order piston aerodynamics.Acta Mechanica,2014,225(7):1985-2004
32 Hao YX,Zhang W,Yang J,et al.Nonlinear dynamics of a functionally graded thin simply-supported plate under a hypersonic flow.Mechanics of Advanced Materials and Structures,2015,22(8):619-632
33 Drela M.Flight Vehicle Aerodynamics.MIT Press,2014
34 Nayfeh AH,Mook DT.Nonlinear Oscillations.John Wiley&Sons,2008
2夏巍,冯浩成.热过屈曲功能梯度壁板的气动弹性颤振.力学学报,2016,48(3):609-614(Xia Wei,Feng Haocheng.Aeroelastic flutter of post-buckled functionally graded panels.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):609-614(in Chinese))
3叶柳青,叶正寅.激波主导流动下壁板的热气动弹性稳定性理论分析.力学学报,2018,50(2):221-220(Ye Liuqing,Ye Zhengyin.Aeroelastic stability analysis of heated flexible panel in shockdominated flows.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):221-232(in Chinese))
4刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展.力学学报,2017,49(3):487-506(Liu Renhuai,Xue Jiang,Development of nonlinear mechanics for laminated composite plates and shells.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):487-506(in Chinese))
5 Noor AK,Burton WS.Assessment of shear deformation theories for multilayered composite plates.Applied Mechanics Reviews,1989,42(1):1-13
6 Noor AK,Burton WS.Stress and free vibration analyses of multilayered composite plates.Composite Structures,1989,11(3):183-204
7 Abe A,Kobayashi Y,Yamada G.Two-mode response of simply supported,rectangular laminated plates.International Journal of NonLinear Mechanics,1998,33(4):675-690
8 Abe A,Kobayashi Y,Yamada G.Three-mode response of simply supported,rectangular laminated plates.JSME International Journal Series C Mechanical Systems,Machine Elements and Manufacturing,1998,41(1):51-59
9 Abe A,Kobayashi Y,Yamada G.Analysis of subharmonic resonance of moderately thick antisymmetric angle-ply laminated plates by using method of multiple scales.Journal of Sound and Vibration,1998,217(3):467-484
10 Oh K,Nayfeh AH.Nonlinear combination resonances in cantilever composite plates.Nonlinear Dynamics,1996,11(2):143-169
11 Pai PF,Nayfeh AH.A nonlinear composite plate theory.Nonlinear Dynamics,1991,2(6):445-477
12 Pai PF,Nayfeh AH.A unified nonlinear formulation for plate and shell theories.Nonlinear Dynamics,1994,6(4):459-500
13 Librescu L,Reddy JN.A few remarks concerning several refined theories of anisotropic composite laminated plates.International Journal of Engineering Science,1989,27(5):515-527
14 Zhang W.Global and chaotic dynamics for a parametrically excited thin plate.Journal of Sound and Vibration,2001,239(5):1013-1036
15 Kitipornchai S,Yang J,Liew KM.Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections.International Journal of Solids and Structures,2004,41(9-10):2235-2257
16 Park T,Lee SY,Seo JW,et al.Structural dynamic behavior of skew sandwich plates with laminated composite faces.Composites Part B:Engineering,2008,39(2):316-326
17 Chien RD,Chen CS.Nonlinear vibration of laminated plates on an elastic foundation.Thin-Walled Structures,2006,44(8):852-860
18 Singh BN,Lal A,Kumar R.Nonlinear bending response of laminated composite plates on nonlinear elastic foundation with uncertain system properties.Engineering Structures,2008,30(4):1101-1112
19 Alijani F,Amabili M,Karagiozis K,et al.Nonlinear vibrations of functionally graded doubly curved shallow shells.Journal of Sound and Vibration,2011,330(7):1432-1454
20 Alijani F,Bakhtiari-Nejad F,Amabili M.Nonlinear vibrations of FGM rectangular plates in thermal environments.Nonlinear Dynamics,2011,66(3):251
21 Zhang W,Chen J,Zhang YF,et al.Continuous model and nonlinear dynamic responses of circular mesh antenna clamped at one side.Engineering Structures,2017,151:115-135
22 Liu T,Zhang W,Wang JF.Nonlinear dynamics of composite laminated circular cylindrical shell clamped along a generatrix and with membranes at both ends.Nonlinear Dynamics,2017,90(2):1393-1417
23 Sun Y,Zhang W,Yao MH.Multi-pulse chaotic dynamics of circular mesh antenna with 1:2 internal resonance.International Journal of Applied Mechanics,2017,9(4):1750060
24 Zhang W,Liu T,Xi A,et al.Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes.Journal of Sound and Vibration,2018,423:65-99
25 Dowell EH.Nonlinear oscillations of a fluttering plate.AIAA Journal,1966,4(7):1267-1275
26 Dowell EH.Nonlinear oscillations of a fluttering plate.II.AIAAJournal,1967,5(10):1856-1862
27 Amabili M,Pa?doussis MP.Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels,with and without fluid-structure interaction.Applied Mechanics Reviews,2003,56(4):349-381
28 Singha MK,Ganapathi M.A parametric study on supersonic flutter behavior of laminated composite skew flat panels.Composite Structures,2005,69(1):55-63
29 Haddadpour H,Navazi HM,Shadmehri F.Nonlinear oscillations of a fluttering functionally graded plate.Composite Structures,2007,79(2):242-250
30 Kuo SY.Flutter of rectangular composite plates with variable fiber pacing.Composite Structures,2011,93(10):2533-2540
31 Zhao MH,Zhang W.Nonlinear dynamics of composite laminated cantilever rectangular plate subject to third-order piston aerodynamics.Acta Mechanica,2014,225(7):1985-2004
32 Hao YX,Zhang W,Yang J,et al.Nonlinear dynamics of a functionally graded thin simply-supported plate under a hypersonic flow.Mechanics of Advanced Materials and Structures,2015,22(8):619-632
33 Drela M.Flight Vehicle Aerodynamics.MIT Press,2014
34 Nayfeh AH,Mook DT.Nonlinear Oscillations.John Wiley&Sons,2008