变刚度Winkler地基上受压非均质矩形板的自由振动与屈曲特性
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摘要
基于经典薄板理论,利用Hamilton原理建立变刚度Winkler地基上受压非均质矩形板自由振动与屈曲问题的控制微分方程并进行无量纲化。通过一种半解析方法-微分变换法(DTM)研究其无量纲固有频率和屈曲临界载荷特性。采用DTM将其无量纲控制微分方程及边界条件变换为等价的代数方程,得到含有频率和屈曲载荷的特征方程。将该问题退化为面内变刚度矩形板情形,其DTM解与精确解进行对比,结果表明DTM具有非常高的精度和很强的适用性。计算出在不同边界条件下屈曲临界载荷并分析地基刚度变化参数、弹性模量变化参数、密度变化参数、面内载荷和长宽比对矩形板无量纲固有频率的影响,给出了不同边界条件下变刚度Winkler地基上受压非均质矩形板的前三阶振型。
Based on the classical thin plate theory, the governing differential equation of free vibration and buckling of a compressed non-homogeneous rectangular plate on Winkler foundation with variable stiffness was established by using Hamilton's principle, and then its dimensionless form was obtained. The characteristics of the plate's dimensionless natural frequencies and buckling critical loads were studied with a semi-analytical method called the differential transformation method(DTM). DTM was used to convert dimensionless governing differential equation and boundary conditions into equivalent algebraic equations, and derive characteristic equations of frequencies and buckling loads. Then, the problem was degenerated into the case of an in-plane variable stiffness rectangular plate, and its DTM solution was compared with the analytical solution. The results showed that DTM have very higher accuracy and stronger applicability. Finally, the buckling critical loads were calculated under different boundary conditions, and the effects of foundation stiffness parameters, elastic modulus parameters, density parameter, in-plane loads and length-width ratio on the plate's dimensionless natural frequencies were analyzed. The first three modal shapes of the compressed non-homogeneous rectangular plate on Winkler foundations with variable stiffness were deduced under different boundary conditions.
Based on the classical thin plate theory, the governing differential equation of free vibration and buckling of a compressed non-homogeneous rectangular plate on Winkler foundation with variable stiffness was established by using Hamilton's principle, and then its dimensionless form was obtained. The characteristics of the plate's dimensionless natural frequencies and buckling critical loads were studied with a semi-analytical method called the differential transformation method(DTM). DTM was used to convert dimensionless governing differential equation and boundary conditions into equivalent algebraic equations, and derive characteristic equations of frequencies and buckling loads. Then, the problem was degenerated into the case of an in-plane variable stiffness rectangular plate, and its DTM solution was compared with the analytical solution. The results showed that DTM have very higher accuracy and stronger applicability. Finally, the buckling critical loads were calculated under different boundary conditions, and the effects of foundation stiffness parameters, elastic modulus parameters, density parameter, in-plane loads and length-width ratio on the plate's dimensionless natural frequencies were analyzed. The first three modal shapes of the compressed non-homogeneous rectangular plate on Winkler foundations with variable stiffness were deduced under different boundary conditions.
引文
[1] ELISHAKOFF I,PENTARAS D,GENTILINI C.Mechanics of functionally graded material structures[M].Singapore:World Scientific,2016.
[2] 何建璋,褚福运,仲政,等.面内变刚度矩形薄板自由振动问题的辛弹性分析[J].同济大学学报(自然科学版),2013,41(9):1310-1317.HE Jianzhang,CHU Fuyun,ZHONG Zheng et al.Symplectic elasticity approach for free vibration of a rectangular plate with in-plane material inhomogeneity[J].Journal of Tongji University (Natural Science),2013,41(9):1310-1317.
[3] 李尧臣,聂国隽,杨昌锦.面内功能梯度矩形板的近似理论与解答[J].力学学报,2013,45(4):560-567.LI Yaochen,NIE Guojun,YANG Changjin.Approximate theory and analytical solution for rectangular plates with in-plane stiffness gradient[J].Chinese Journal of Theoretical and Applied Mechanics,2013,45(4):560-567.
[4] LAL R,SAINI R.Buckling and vibration analysis of non-homogeneous rectangular Kirchhoff plates resting on two-parameter foundation[J].Meccanica,2015,50(4):893-913.
[5] LONG N V,QUOC T H,TU T M.Bending and free vibration analysis of functionally graded plates using new eight-unknown shear deformation theory by finite-element method[J].International Journal of Advanced Structural Engineering,2016,8(4):391-399.
[6] 蒲育,赵海英,滕兆春.四边弹性约束FGM矩形板面内自由振动的DQM求解[J].振动与冲击,2016,35(17):58-65.PU Yu,ZHAO Haiying,TENG Zhaochun.In-plane free vibration of FGM rectangular plates with elastically restrained edges by differential quadrature method[J].Journal of Vibration and Shock,2016,35(17):58-65.
[7] LAL R,SAINI R.On the use of GDQ for vibration characteristic of non-homogeneous orthotropic rectangular plates of bilineary varying thickness[J].Acta Mechanica,2015,226(5):1605-1620.
[8] DHANPATI R L.Transverse vibrations of non-homogeneous orthotropic rectangular plates of variable thickness:A spline technique[J].Journal of Sound and Vibration,2007,306(1/2):203-214.
[9] UYMAZ B,AYDOGDU M,FILIZ S.Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method[J].Composite Structures,2012,94(4):1398-1405.
[10] HATAMI M,GANJI D D,SHEIKHOLESLAMI M.Differential transformation method for mechanical engineering problems[M].London:Elsevier,2017.
[11] 赵家奎.微分变换及其在电路中的应用[M].武汉:华中理工大学出版社,1988.
[12] 滕兆春,王晓婕,付小华.弹性地基上变截面梁自由振动的DTM分析[J].兰州理工大学学报,2016,42(1):166-169.TENG Zhaochun,WANG Xiaojie,FU Xiaohua.Free vibration analysis of variable cross-section beams on elastic foundation by using DTM[J].Journal of Lanzhou University of Technology,2016,42(1):166-169.
[13] ATTARNEJAD R,SHAHBA A.Application of differential transform method in free vibration analysis of rotating non-prismatic beams[J].World Applied Science Journal,2008,5(4):441-448.
[14] KAYA M O,OZGUMUS O O.Flexural-torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM[J].Journal of Sound and Vibration,2007,306(3/4/5):495-506.
[15] KAYA M O.Free vibration analysis of a rotating Timoshenko beam by differential transform method[J].Aircraft Engineering and Aerospace Technology,2006,78(3):194-203.
[16] ?ZDEMIR ?,KAYA M O.Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli-Euler beam by differential transform method[J].Journal of Sound and Vibration,2006,289(1/2):413-420.
[17] OZGUMUS O O,KAYA M O.Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method[J].Meccanica,2006,41(6):661-670.
[18] KACAR A,TAN H T,KAYA M O.Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method[J].Mathematical and Computational Applications,2011,16(3):773-783.
[19] ?ATAL S.Solution of free vibration equations of beam on elastic soil by using differential transform method[J].Applied Mathematical Modelling,2008,32(9):1744-1757.
[20] TALEBI S,UOSOFVAND H,Ariaei A.Vibration analysis of a rotating closed section composite Timoshenko beam by using differential transform method[J].Journal of Applied and Computational Mechanics,2015,1(4):181-186.
[21] YALCIN H S,ARIKOGLU A,OZKOL I.Free vibration analysis of circular plates by differential transformation method[J].Applied Mathematics and Computation,2009,212(2):377-386.
[22] LAL R,AHLAWAT N.Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method[J].European Journal of Mechanics-A/Solids,2015,52:85-94.
[23] SHARIYAT M,ALIPOUR M M.Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations[J].Archive of Applied Mechanics,2011,81(9):1289-1306.
[24] KUMAR Y.Free vibration analysis of isotropic rectangular plates on Winkler foundations using differential transform method[J].International Journal of Applied Mechanics and Engineering,2013,18(2):589-597.
[25] KUMAR Y.Differential transform method to study free transverse vibration of monoclinc rectangular plates on Winkler foundation[J].Applied and Computational Mechanics,2013,7(2):145-154.
[26] SEMNANI S J,ATTARNEJAD R,FIROUZJAEI R K.Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method[J].Acta Mechanica,2013,224(8):1643-1658.
[27] MUKHTAR F M.Free vibration analysis of orthotropic plates by differential transform and Taylor collocation methods based on a refined plate theory[J].Archive of Applied Mechanics,2017,87(1):15-40.
[2] 何建璋,褚福运,仲政,等.面内变刚度矩形薄板自由振动问题的辛弹性分析[J].同济大学学报(自然科学版),2013,41(9):1310-1317.HE Jianzhang,CHU Fuyun,ZHONG Zheng et al.Symplectic elasticity approach for free vibration of a rectangular plate with in-plane material inhomogeneity[J].Journal of Tongji University (Natural Science),2013,41(9):1310-1317.
[3] 李尧臣,聂国隽,杨昌锦.面内功能梯度矩形板的近似理论与解答[J].力学学报,2013,45(4):560-567.LI Yaochen,NIE Guojun,YANG Changjin.Approximate theory and analytical solution for rectangular plates with in-plane stiffness gradient[J].Chinese Journal of Theoretical and Applied Mechanics,2013,45(4):560-567.
[4] LAL R,SAINI R.Buckling and vibration analysis of non-homogeneous rectangular Kirchhoff plates resting on two-parameter foundation[J].Meccanica,2015,50(4):893-913.
[5] LONG N V,QUOC T H,TU T M.Bending and free vibration analysis of functionally graded plates using new eight-unknown shear deformation theory by finite-element method[J].International Journal of Advanced Structural Engineering,2016,8(4):391-399.
[6] 蒲育,赵海英,滕兆春.四边弹性约束FGM矩形板面内自由振动的DQM求解[J].振动与冲击,2016,35(17):58-65.PU Yu,ZHAO Haiying,TENG Zhaochun.In-plane free vibration of FGM rectangular plates with elastically restrained edges by differential quadrature method[J].Journal of Vibration and Shock,2016,35(17):58-65.
[7] LAL R,SAINI R.On the use of GDQ for vibration characteristic of non-homogeneous orthotropic rectangular plates of bilineary varying thickness[J].Acta Mechanica,2015,226(5):1605-1620.
[8] DHANPATI R L.Transverse vibrations of non-homogeneous orthotropic rectangular plates of variable thickness:A spline technique[J].Journal of Sound and Vibration,2007,306(1/2):203-214.
[9] UYMAZ B,AYDOGDU M,FILIZ S.Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method[J].Composite Structures,2012,94(4):1398-1405.
[10] HATAMI M,GANJI D D,SHEIKHOLESLAMI M.Differential transformation method for mechanical engineering problems[M].London:Elsevier,2017.
[11] 赵家奎.微分变换及其在电路中的应用[M].武汉:华中理工大学出版社,1988.
[12] 滕兆春,王晓婕,付小华.弹性地基上变截面梁自由振动的DTM分析[J].兰州理工大学学报,2016,42(1):166-169.TENG Zhaochun,WANG Xiaojie,FU Xiaohua.Free vibration analysis of variable cross-section beams on elastic foundation by using DTM[J].Journal of Lanzhou University of Technology,2016,42(1):166-169.
[13] ATTARNEJAD R,SHAHBA A.Application of differential transform method in free vibration analysis of rotating non-prismatic beams[J].World Applied Science Journal,2008,5(4):441-448.
[14] KAYA M O,OZGUMUS O O.Flexural-torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM[J].Journal of Sound and Vibration,2007,306(3/4/5):495-506.
[15] KAYA M O.Free vibration analysis of a rotating Timoshenko beam by differential transform method[J].Aircraft Engineering and Aerospace Technology,2006,78(3):194-203.
[16] ?ZDEMIR ?,KAYA M O.Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli-Euler beam by differential transform method[J].Journal of Sound and Vibration,2006,289(1/2):413-420.
[17] OZGUMUS O O,KAYA M O.Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method[J].Meccanica,2006,41(6):661-670.
[18] KACAR A,TAN H T,KAYA M O.Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method[J].Mathematical and Computational Applications,2011,16(3):773-783.
[19] ?ATAL S.Solution of free vibration equations of beam on elastic soil by using differential transform method[J].Applied Mathematical Modelling,2008,32(9):1744-1757.
[20] TALEBI S,UOSOFVAND H,Ariaei A.Vibration analysis of a rotating closed section composite Timoshenko beam by using differential transform method[J].Journal of Applied and Computational Mechanics,2015,1(4):181-186.
[21] YALCIN H S,ARIKOGLU A,OZKOL I.Free vibration analysis of circular plates by differential transformation method[J].Applied Mathematics and Computation,2009,212(2):377-386.
[22] LAL R,AHLAWAT N.Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method[J].European Journal of Mechanics-A/Solids,2015,52:85-94.
[23] SHARIYAT M,ALIPOUR M M.Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations[J].Archive of Applied Mechanics,2011,81(9):1289-1306.
[24] KUMAR Y.Free vibration analysis of isotropic rectangular plates on Winkler foundations using differential transform method[J].International Journal of Applied Mechanics and Engineering,2013,18(2):589-597.
[25] KUMAR Y.Differential transform method to study free transverse vibration of monoclinc rectangular plates on Winkler foundation[J].Applied and Computational Mechanics,2013,7(2):145-154.
[26] SEMNANI S J,ATTARNEJAD R,FIROUZJAEI R K.Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method[J].Acta Mechanica,2013,224(8):1643-1658.
[27] MUKHTAR F M.Free vibration analysis of orthotropic plates by differential transform and Taylor collocation methods based on a refined plate theory[J].Archive of Applied Mechanics,2017,87(1):15-40.