基于改进Chebyshev级数的层合结构振动分析新理论
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摘要
提出了一种基于改进Chebyshev级数的层合结构高阶分层建模理论.该理论位移场由线性位移场和高阶位移场组成,线性位移场控制位移场的总体分布趋势,高阶位移场进行局部修正.高阶位移场由具有统一表达式的改进Chebyshev级数表示,通过改变高阶截断阶数可实现高阶位移场快速配置,能够满足不同建模精度需求.采用该高阶分层理论和广义谱方法推导了层合结构的自由振动特征方程,研究了一般边界条件下层合梁、板、壳的自由振动特性,并将计算结果与其他文献数据对比.结果表明:基于改进Chebyshev级数的层合结构高阶分层理论具有较高的建模精度和计算效率.
A new layerwise theory for vibration analysis of laminated structures based on modified Chebyshev polynomials was proposed. The displacement field in each discrete layer was composed of a global linear component introduced under the layerwise strategy and a local high-order counterpart considered to improve the accuracy of the theory. In each discrete layer,the high-order displacement field distribution through the laminate thickness was determined with the modified Chebyshev polynomials. Therefore,the proposed theory offers an easy analysis operation to realize different modeling precision requirements only by changing the truncation order without the need for reprogramming from case to case. The theory also has the ability of achieving arbitrary modeling precision according to practical requirements. Based on the proposed theory,the general spectral method was combined to formulate the vibration equations of laminated beams,plates and shells. To test the efficiency and accuracy of the present theory,dynamic properties of laminated beams,plates and shells with different dimensions,boundary conditions and lamination schemes were studied. The numerical results obtained from the present theory are in good agreement with exact elasticity solutions published previously.
A new layerwise theory for vibration analysis of laminated structures based on modified Chebyshev polynomials was proposed. The displacement field in each discrete layer was composed of a global linear component introduced under the layerwise strategy and a local high-order counterpart considered to improve the accuracy of the theory. In each discrete layer,the high-order displacement field distribution through the laminate thickness was determined with the modified Chebyshev polynomials. Therefore,the proposed theory offers an easy analysis operation to realize different modeling precision requirements only by changing the truncation order without the need for reprogramming from case to case. The theory also has the ability of achieving arbitrary modeling precision according to practical requirements. Based on the proposed theory,the general spectral method was combined to formulate the vibration equations of laminated beams,plates and shells. To test the efficiency and accuracy of the present theory,dynamic properties of laminated beams,plates and shells with different dimensions,boundary conditions and lamination schemes were studied. The numerical results obtained from the present theory are in good agreement with exact elasticity solutions published previously.
引文
[1]刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展[J].力学学报,2017,49(3):487-506.(LIU R enhuai,XU E Jianghong.Development of nonlinear mechanics for laminated composite plates and shells[J].Chinese Journal of Theoretical and A pplied Mechanics,2017,49(3):487-506.(in Chinese))
[2]缪旭弘,郭凤水,贾地.国外潜艇吸隔声材料研究现状及关键技术分析[C]//船舶水下噪声学术讨论会.西安,2007.(MIU Xuhong,GUO Fengshui,JIA Di.Research on development and key technical of submarine sound insulation materials in foreign countries[C]//A cademic Seminar on Underwater Noise of Ships.Xi’an,2017.(in Chinese))
[3]QATU MS.Vibration of Laminated Shells and Plates[M].San Diego:Elsevier,2004.
[4]REDDY J N.Mechanics of Laminated Composite Plates and Shells:Theory and Analysis[M].Florida:CR CPress,2003.
[5]WU Z,MA R,CHENW.A refined three-node triangular element based on the HWvariational theorem for multilayered composite plates[J].Composite Structures,2017,161:132-144.
[6]WU Z,CHENW.A global higher-orderzig-zag model in terms of the HWvariational theorem for multilayered composite beams[J].Composite Structures,2016,158:128-136.
[7]LIU B,XING Y F,QATU MS,et al.Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells[J].Composite Structures,2012,94(2):484-493.
[8]XING Y,LIU B,XU T.Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions[J].International Journal of Mechanical Sciences,2013,75(4):178-188.
[9]JING,YE T,CHENY,et al.An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions[J].Composite Structures,2013,106:114-127.
[10]JING,YE T,MA X,et al.A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions[J].International Journal of Mechanical Sciences,2013,75(10):357-376.
[11]YE T,JING,SHI S,et al.Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations[J].International Journal of Mechanical Sciences,2014,84:120-137.
[12]徐坤,陈美霞,谢坤.正交各向异性功能梯度材料平板振动分析[J].噪声与振动控制,2016,36(4):14-20.(XU Kun,CHENMeixia,XIE Kun.Vibration analysis of orthotropic functionally graded plates[J].Noise and V ibration Control,2016,36(4):14-20.(in Chinese))
[13]丁皓江,陈伟球,徐荣桥.横观各向同性层合矩形板弯曲、振动和稳定的三维精确分析[J].应用数学和力学,2001,22(1):16-22.(DING Haojiang,CHENWeiqiu,XU Rongqiao.On the bending,vibration and stability of laminated rectangular plates with transversely isotropic layers[J].A pplied Mathematics and Mechanics,2001,22(1):16-22.(in Chinese))
[14]吕朝锋.基于状态空间架构的微分求积法及其应用[D].博士学位论文.杭州:浙江大学,2006.(LChaofeng.State-space-based differential quadrature method and its applications[D].PhDT hesis.Hangzhou:Z hejiang U niversity,2006.(in Chinese))
[15]SAADA A S.Elasticity:Theory and Applications[M].2nd ed.Florida:Ross Publishing Inc,2009.
[16]YE T,JING,SU Z.Three-dimensional vibration analysis of sandwich and multilayered plates with general ply stacking sequences by a spectral-sampling surface method[J].Composite Structures,2017,176:1124-1142.
[17]REDDY J N.Energy Principles and Variational Methods in Applied Mechanic[M].New York:John Wiley&Sons,1984.
[18]YE T,JING,SU Z.A spectral-sampling surface method for the vibration of 2-D laminated curved beams with variable curvatures and general restraints[J].International Journal of Mechanical Sciences,2016,110:170-189.
[19]JING,YE T,SU Z.Elasticity solution for vibration of 2-D curved beams with variable curvatures using a spectral-sampling surface method[J].International Journal for Numerical Methods in Engineering,2017,111(11):1075-1100.
[20]YE T,JING,ZHANG Y.Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation,rotary inertia and initial curvature[J].Composite Structures,2015,133:202-225.
[21]瞿叶高,华宏星,谌勇,等.复合材料旋转壳自由振动分析的新方法[J].力学学报,2013,45(1):139-143.(QU Yegao,HUA Hongxing,CHENYong,et al.A new method for free vibration analysis of composite laminated shelles of revolution[J].Chinese Journal of Theoretical and A pplied Mechanics,2013,45(1):139-143.(in Chinese))
[22]瞿叶高,孟光,华宏星,等.基于区域分解的薄壁回转壳自由振动分析[J].应用力学学报,2013,30(1):1-6.(Q U Yegao,MENG Guang,HUA Hongxing,et al.Free vibration analysis of thin shells of revolution based on domain decomposition method[J].Chinese Journal of A pplied Mechanics,2013,30(1):1-6.(in Chinese))
[23]YE T,JING.Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method[J].Computers&Structures,2016,167:115-130.
[24]YE T,JING,YE X,et al.A series solution for the vibrations of composite laminated deep curved beams with general boundaries[J].Composite Structures,2015,127:450-465.
[25]PAGANO NJ.Exact solutions for rectangular bidirectional composites and sandwich plates[J].Journal of Composite Materials,1970,4(1):20-34.
[26]QU Y,WU S,LI H,et al.Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds:beams,plates and solids[J].Composites Part B:Engineering,2015,73:96-110.
[27]CHENWQ,LCF.3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported[J].Composite Structures,2005,69(1):77-87.
[28]YE T,JING,SU Z,et al.A modified Fourier solution for vibration analysis of moderately thick laminated plates with general boundary restraints and internal line supports[J].International Journal of Mechanical Sciences,2014,80:29-46.
[29]NOSIER A,KAPANIA R,REDDY J.Free vibration analysis of laminated plates using a layer-wise theory[J].A IA A Journal,2013,31(12):2335-2346.
[30]QU Y,MENG G.Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory[J].Composite Structures,2014,112:378-396.
[2]缪旭弘,郭凤水,贾地.国外潜艇吸隔声材料研究现状及关键技术分析[C]//船舶水下噪声学术讨论会.西安,2007.(MIU Xuhong,GUO Fengshui,JIA Di.Research on development and key technical of submarine sound insulation materials in foreign countries[C]//A cademic Seminar on Underwater Noise of Ships.Xi’an,2017.(in Chinese))
[3]QATU MS.Vibration of Laminated Shells and Plates[M].San Diego:Elsevier,2004.
[4]REDDY J N.Mechanics of Laminated Composite Plates and Shells:Theory and Analysis[M].Florida:CR CPress,2003.
[5]WU Z,MA R,CHENW.A refined three-node triangular element based on the HWvariational theorem for multilayered composite plates[J].Composite Structures,2017,161:132-144.
[6]WU Z,CHENW.A global higher-orderzig-zag model in terms of the HWvariational theorem for multilayered composite beams[J].Composite Structures,2016,158:128-136.
[7]LIU B,XING Y F,QATU MS,et al.Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells[J].Composite Structures,2012,94(2):484-493.
[8]XING Y,LIU B,XU T.Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions[J].International Journal of Mechanical Sciences,2013,75(4):178-188.
[9]JING,YE T,CHENY,et al.An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions[J].Composite Structures,2013,106:114-127.
[10]JING,YE T,MA X,et al.A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions[J].International Journal of Mechanical Sciences,2013,75(10):357-376.
[11]YE T,JING,SHI S,et al.Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations[J].International Journal of Mechanical Sciences,2014,84:120-137.
[12]徐坤,陈美霞,谢坤.正交各向异性功能梯度材料平板振动分析[J].噪声与振动控制,2016,36(4):14-20.(XU Kun,CHENMeixia,XIE Kun.Vibration analysis of orthotropic functionally graded plates[J].Noise and V ibration Control,2016,36(4):14-20.(in Chinese))
[13]丁皓江,陈伟球,徐荣桥.横观各向同性层合矩形板弯曲、振动和稳定的三维精确分析[J].应用数学和力学,2001,22(1):16-22.(DING Haojiang,CHENWeiqiu,XU Rongqiao.On the bending,vibration and stability of laminated rectangular plates with transversely isotropic layers[J].A pplied Mathematics and Mechanics,2001,22(1):16-22.(in Chinese))
[14]吕朝锋.基于状态空间架构的微分求积法及其应用[D].博士学位论文.杭州:浙江大学,2006.(LChaofeng.State-space-based differential quadrature method and its applications[D].PhDT hesis.Hangzhou:Z hejiang U niversity,2006.(in Chinese))
[15]SAADA A S.Elasticity:Theory and Applications[M].2nd ed.Florida:Ross Publishing Inc,2009.
[16]YE T,JING,SU Z.Three-dimensional vibration analysis of sandwich and multilayered plates with general ply stacking sequences by a spectral-sampling surface method[J].Composite Structures,2017,176:1124-1142.
[17]REDDY J N.Energy Principles and Variational Methods in Applied Mechanic[M].New York:John Wiley&Sons,1984.
[18]YE T,JING,SU Z.A spectral-sampling surface method for the vibration of 2-D laminated curved beams with variable curvatures and general restraints[J].International Journal of Mechanical Sciences,2016,110:170-189.
[19]JING,YE T,SU Z.Elasticity solution for vibration of 2-D curved beams with variable curvatures using a spectral-sampling surface method[J].International Journal for Numerical Methods in Engineering,2017,111(11):1075-1100.
[20]YE T,JING,ZHANG Y.Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation,rotary inertia and initial curvature[J].Composite Structures,2015,133:202-225.
[21]瞿叶高,华宏星,谌勇,等.复合材料旋转壳自由振动分析的新方法[J].力学学报,2013,45(1):139-143.(QU Yegao,HUA Hongxing,CHENYong,et al.A new method for free vibration analysis of composite laminated shelles of revolution[J].Chinese Journal of Theoretical and A pplied Mechanics,2013,45(1):139-143.(in Chinese))
[22]瞿叶高,孟光,华宏星,等.基于区域分解的薄壁回转壳自由振动分析[J].应用力学学报,2013,30(1):1-6.(Q U Yegao,MENG Guang,HUA Hongxing,et al.Free vibration analysis of thin shells of revolution based on domain decomposition method[J].Chinese Journal of A pplied Mechanics,2013,30(1):1-6.(in Chinese))
[23]YE T,JING.Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method[J].Computers&Structures,2016,167:115-130.
[24]YE T,JING,YE X,et al.A series solution for the vibrations of composite laminated deep curved beams with general boundaries[J].Composite Structures,2015,127:450-465.
[25]PAGANO NJ.Exact solutions for rectangular bidirectional composites and sandwich plates[J].Journal of Composite Materials,1970,4(1):20-34.
[26]QU Y,WU S,LI H,et al.Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds:beams,plates and solids[J].Composites Part B:Engineering,2015,73:96-110.
[27]CHENWQ,LCF.3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported[J].Composite Structures,2005,69(1):77-87.
[28]YE T,JING,SU Z,et al.A modified Fourier solution for vibration analysis of moderately thick laminated plates with general boundary restraints and internal line supports[J].International Journal of Mechanical Sciences,2014,80:29-46.
[29]NOSIER A,KAPANIA R,REDDY J.Free vibration analysis of laminated plates using a layer-wise theory[J].A IA A Journal,2013,31(12):2335-2346.
[30]QU Y,MENG G.Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory[J].Composite Structures,2014,112:378-396.