功能梯度矩形板的非线性振动
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摘要
功能梯度材料是一种新型的复合材料,通过各组分材料的巧妙结合,可广泛应用于各种要求的材料领域。功能梯度板壳非线性振动问题在工程实际中往往常见,其非线性振动微分方程在数学上的求解具有很大难度,因此研究其板壳的非线性振动具有重要的理论和实际意义。
本文致力于研究功能梯度矩形板在外界激励作用下的非线性强迫振动问题。重点研究功能梯度板的主共振和参数振动等问题。首先,根据薄板理论和功能梯度材料的物性参数,应用伽辽金原理导出了功能梯度矩形板在横向简谐激励和面内参数激励分别作用下的非线性振动方程。
其次,针对四边固支功能梯度矩形板的主共振问题和四边简支功能梯度矩形板的主参数共振问题,应用多尺度法进行求解,分别导出了系统相应的幅频响应方程,并基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件;针对四边简支功能梯度矩形板的强非线性主共振问题,基于参数变换的思想,应用改进的多尺度法对系统的达芬方程进行了求解,得到了系统的幅频响应方程。
最后,通过编程计算,对几种典型的功能梯度材料板进行了分析。分别给出了在不同参数情况下,四边固支功能梯度矩形板主共振和四边简支功能梯度矩形板主参数共振下的幅频曲线及主共振下的动相平面相轨迹图和主参数共振下的解的存在区域图。对于四边简支功能梯度矩形板强非线性主共振的分析,给出其时间历程图、相图及幅频曲线。讨论了阻尼系数、材料组分指数以及板厚等参数对系统非线性振动特性的影响。
Functionally Graded Material is a new type of composite materials, which can be widely applied to material areas of various requirements through the ingenious combination of component materials. The nonlinear vibration problems of functionally graded plates and cylindrical shells is particularly common in engineering practice, the solution in mathematics of the nonlinear governing partial equation is very difficult, therefore, the study on the nonlinear vibration of functionally graded plates and cylindrical shells has important theoretical and practical significance.
In this paper, nonlinear forced vibration of rectangular functionally graded plates subjected to external excitation is studied. Principal resonance and parameter vibration of FGM plates and other issues are focused discussed. Firstly, according to the classical plate theory and the physical properties parameters of functionally graded materials, the nonlinear forced vibration equations of rectangular functionally graded plates respectively in the harmonic excitation and parametric excitation were deduced by using Galerkin method.
Secondly, for principal resonance of FGM rectangular plate with four clamped edges and principal parametric resonance of simply supported FGM rectangular plate, the amplitude-frequency response equations were respectively derived using multi-scale method. Based on the Lyapunov stable theory, the critical conditions of steady-state solutions’stability were gotten. For the strongly nonlinear principal resonance of simply supported FGM rectangular plate, based on parameter transformation, the Duffing equation of system was studied by using modified multi-scale method. The amplitude-frequency response equation was gotten.
Finally, several typical functionally graded materials were analyzed. In principal resonance, the amplitude-frequency curves, amplitude-excitation curves of FGM rectangular plate with four clamped edges, the amplitude- frequency curves and time history curves and phase trajectory of simply supported FGM rectangular plate were derived under different situations. In principal parametric resonance, the amplitude-frequency curves and amplitude- damping response curves and the solution’s existence regional diagram of the simply supported FGM rectangular plate were derived under different situations. The influence of different parameters on nonlinear resonance properties of system was discussed.
本文致力于研究功能梯度矩形板在外界激励作用下的非线性强迫振动问题。重点研究功能梯度板的主共振和参数振动等问题。首先,根据薄板理论和功能梯度材料的物性参数,应用伽辽金原理导出了功能梯度矩形板在横向简谐激励和面内参数激励分别作用下的非线性振动方程。
其次,针对四边固支功能梯度矩形板的主共振问题和四边简支功能梯度矩形板的主参数共振问题,应用多尺度法进行求解,分别导出了系统相应的幅频响应方程,并基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件;针对四边简支功能梯度矩形板的强非线性主共振问题,基于参数变换的思想,应用改进的多尺度法对系统的达芬方程进行了求解,得到了系统的幅频响应方程。
最后,通过编程计算,对几种典型的功能梯度材料板进行了分析。分别给出了在不同参数情况下,四边固支功能梯度矩形板主共振和四边简支功能梯度矩形板主参数共振下的幅频曲线及主共振下的动相平面相轨迹图和主参数共振下的解的存在区域图。对于四边简支功能梯度矩形板强非线性主共振的分析,给出其时间历程图、相图及幅频曲线。讨论了阻尼系数、材料组分指数以及板厚等参数对系统非线性振动特性的影响。
Functionally Graded Material is a new type of composite materials, which can be widely applied to material areas of various requirements through the ingenious combination of component materials. The nonlinear vibration problems of functionally graded plates and cylindrical shells is particularly common in engineering practice, the solution in mathematics of the nonlinear governing partial equation is very difficult, therefore, the study on the nonlinear vibration of functionally graded plates and cylindrical shells has important theoretical and practical significance.
In this paper, nonlinear forced vibration of rectangular functionally graded plates subjected to external excitation is studied. Principal resonance and parameter vibration of FGM plates and other issues are focused discussed. Firstly, according to the classical plate theory and the physical properties parameters of functionally graded materials, the nonlinear forced vibration equations of rectangular functionally graded plates respectively in the harmonic excitation and parametric excitation were deduced by using Galerkin method.
Secondly, for principal resonance of FGM rectangular plate with four clamped edges and principal parametric resonance of simply supported FGM rectangular plate, the amplitude-frequency response equations were respectively derived using multi-scale method. Based on the Lyapunov stable theory, the critical conditions of steady-state solutions’stability were gotten. For the strongly nonlinear principal resonance of simply supported FGM rectangular plate, based on parameter transformation, the Duffing equation of system was studied by using modified multi-scale method. The amplitude-frequency response equation was gotten.
Finally, several typical functionally graded materials were analyzed. In principal resonance, the amplitude-frequency curves, amplitude-excitation curves of FGM rectangular plate with four clamped edges, the amplitude- frequency curves and time history curves and phase trajectory of simply supported FGM rectangular plate were derived under different situations. In principal parametric resonance, the amplitude-frequency curves and amplitude- damping response curves and the solution’s existence regional diagram of the simply supported FGM rectangular plate were derived under different situations. The influence of different parameters on nonlinear resonance properties of system was discussed.
引文
1沈惠申.功能梯度复合材料板壳结构的弯曲、屈曲和振动.力学进展, 2004,34(1):53-60
2李进,田兴华.功能梯度材料的研究现状及应用.宁夏工程技术, 2007,6(1):80-83
3赵军,艾兴,张建华.功能梯度材料的发展及展望.材料导报, 1997,11(4):57-60
4田兴华.功能梯度材料及其强度分析. [浙江大学博士论文]. 2006:1-6
5黄旭涛,严密.功能梯度材料回顾与展望.材料科学与工程, 1997,15(4):35-38
6王引真,孙永兴,阎国超.功能梯度材料的研究动态.机械工程材料, 1997,21(4):35-37
7刘延柱,陈立群编著.非线性振动.北京:高等教育出版社, 2001
8陈树辉.强非线性振动的定量方法.广州:广东科技出版社, 2004
9陈予恕.非线性振动.北京:高等教育出版社, 1983
10 V. K. Melnikov. On the Stability of the Centre for Time-periodic Perturbations. Transactions of Moscow Mathematical Society, 1963,12(1):1-57
11 Ali Hasan Nayfeh, Dean T Mook. Nonlinear Oscillations. New York: John Wiley& Sons, 1979
12戴世强,庄峰青.一类非线性振动系统的渐进解.中国科学, 1986,16(1):34-40
13 J. P. Otty. A Perturbation Method for a Set of Purely Non-linear Differential Equations. International Journal of Control, 1979,30(4):587-595
14李骊.强非线性振动系统的定性理论与定量方法.北京:科学出版社, 1997
15边祖光.功能梯度材料板壳结构的耦合问题研究. [浙江大学博士学位论文]. 2005:2-10
16叶开沅.力学发展简史.北京:知识出版社, 1987
17 J. N. Reddy, C. M. Wang, S. Kitipornchai. Axisymmetric Bending of Functionally Grade Circular and Annular Plates. European Journal of Mechanics A/Solids, 1999,18(2):185-195
18 Z. Q. Cheng, S. Kitipornchai. Exact Bending Solution of Inhomogeneous Plates fromHomogeneous Thin-plate Deflection. AIAA Journal, 2000,38(7):1289-1291
19 H. S. Shen. Nonlinear Bending Response of Functionally Graded Plates Subjected to Transverse Loads and in Thermal Environments. International Journal of Mechanical Sciences, 2002,44(3):561-584
20刘进,武兰河,张晓炜.功能梯度材料板的弯曲问题.石家庄铁道学院学报, 2003,16(2):1-5
21 Z. Q. Cheng. Nonlinear Bending of Inhomogeneous Plates. Engineering Structures, 2001,23(10):1359-1363
22 N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material. Transactions of Japan Society of Mechanical Engineers, Series A, 1991,57(533):98-103
23 Y. Obata, N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material (Influence of Boundary Conditions). Transactions of Japan Society of Mechanical Engineers, Series A, 1992,58(553):1689-1695
24杨杰,沈惠申.热/机械荷载下功能梯度材料矩形厚板的弯曲行为.固体力学学报, 2003,24(1):119-124
25 G. N. Praveen, J. N. Reddy. Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates. International Journal of Solids and Structures, 1998,35(33):4457-4476
26 R. Javaheri, M. R. Eslami. Buckling of Functionally Fraded Plates under in-plane Compressive Loading. ZAMM, 2002,82(4):277-283
27 R. Javaheri, M. R. Eslami. Thermal Buckling of Functionally Graded Plates. AIAA Journal, 2002,40(1):162-169
28 R. Javaheri, M. R. Eslami. Thermal Buckling of Functionally Graded Plates Based on Higher Order Theory. Journal of Thermal Stresses, 2002,25(7):603-625
29孙建亮,周振功,王彪.功能梯度压电压磁材料中断裂问题的分析.力学学报, 2005,37(1):9-14
30李春雨,邹振祝,段祝平.正交各向异性功能梯度材料反平面裂纹尖端力场.固体力学学报, 2001,22(1):81-84
31李永东,贾斌,张男,戴耀等.功能梯度材料有限宽板的反平面断裂问题研究.应用数学与力学, 2006,27(6):683-689
32 Zhen-Gong Zhou, Lin-Zhi Wu. Non-local Theory Solution for the Anti-plane Shear of two Collinear Permeable Cracks in Functionally Graded Piezoelectric Materials. International Journal of Engineering Science, 2006,44(18-19):1366-1379
33曹志远.功能梯度板的非线性动力分析.固体力学学报, 2006,27(1):21-25
34曹志远.各种边界条件平行四边形功能梯度板的动力特性解析解.力学季刊, 2006,27(2):255-261
35夏贤坤,沈惠申.功能梯度材料剪切板屈曲后的自由振动.固体力学学报, 2008,29(2):129-133
36黄小林,沈惠申.热环境下功能梯度材料板的自由振动和动力响应.工程力学, 2005,22(3):224-228
37 Chun-Sheng Chen. Nonlinear Vibration of a Shear Deformable Functionally Graded Plate. Composite Structures, 2005,68(3):295-302
38 Chun-Sheng Chen, Tsyr-Jang Chen, Rean-Der Chien. Nonlinear Vibration of Initially Stressed Functionally Graded Plates. Thin-Walled Structures, 2006,44(8):844-851
39 Xiao-Lin Huang, Hui-Shen Shen. Nonlinear Vibration and Dynamic Response of Functionally Graded Plates in Thermal Environments. International Journal of Solids and Structures, 2004,41(9-10):2403-2427
40 Chin-Ping Fung, Chun-Sheng Chen. Imperfection Sensitivity in the Nonlinear Vibration of Functionally Graded Plates. European Journal of Mechanics. A, Solids, 2006,25(3):425-436
41 Y. X. Hao, L.H. Chen, W. Zhang et_al. Nonlinear Oscillations, Bifurcations and Chaos of Functionally Graded Materials Plate. Journal of Sound and Vibration, 2008,312(4-5):862-892
42 A. Allahverdizadeh, M. H. Naei, M. Nikkhah Bahrami. Nonlinear Free and Forced Vibration Analysis of Thin Circular Functionally Graded Plates. Journal of Sound and Vibration, 2008,310(4-5):966-984
43 Da-Guang Zhang, You-He Zhou. A Theoretical Analysis of FGM Thin Plates Based on Physical Neutral Surface. Computational Materials Science, 2008,44(2):716-720
44 Da-Guang Zhang, You-He Zhou. Nonlinear Free Vibration of Piezoelectric Laminated Composite Plate. Finite Elements in Analysis and Design, 2009,45(10):686-694
45 J. Yang, Hui-Shen Shen. Dynamic Response of Initially Stressed Functionally Graded Rectangular Thin Plates. Composite Structures, 2001,54(4):497-508
46 J. Yang, Hui-Shen Shen. Vibration Characteristics and Transient Response of Shear-deformable Functionally Graded Plates in Thermal Environments. Journal of Sound and Vibration, 2002,255(3):579-602
47陈伟球,叶贵如,蔡金标等.横观各向同性功能梯度材料矩形板的自由振动.振动工程学报, 2001,14(3):263-267
48陈伟球,叶贵如,蔡金标等.球面各向同性功能梯度球壳的自由振动.力学学报, 2001,33(4):768-775
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50赵凤群,王忠民,刘宏昭.非保守力作用下FGM矩形板的稳定性分析.应用力学学报, 2007,24(1):318-322
51 T. Y. Ng, K. Y. Lam, K. M. Liew. Effects of FGM Materials on the Parametric Resonance of Plate Structures. Computer Methods in Applied Mechanics and Engineering, 2000,190(8-10):953-962
52 T. Y. Ng, K. Y. Lam, K. M. Liew, et_al. Dynamic Stability Analysis of Functionally Graded Cylindrical Shells under periodic axial loading. International Journal of Solids and Structures, 2001,38(8):1295-1309
53 J. Yang, Hui-Shen Shen. Free Vibration and Parametric Resonance of Shear Deformable Functionally Graded Cylindrical Panels. Journal of Sound and Vibration, 2003,261(5):871-893
54伍晓红,沈亚鹏.压电功能梯度板自由振动的三维解.固体力学学报, 2003,24(1):75-82
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56李世荣,范亮亮.热环境中功能梯度材料圆板的自由振动.振动工程学报,2007,20(4):353-360
57郝育新,张伟.四边简支FGM矩形板非线性振动中的内共振.振动与冲击, 2009,28(6):153-155
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60 T. D. Burton, Z. Rahman. On the Multi-scale Analysis of Strongly Non-linear Forced Oscillators. International Journal of Non-linear Mechanics, 1986,21(2):135-146
61 S. H. Chen, Y. K. Cheung. A Modified Lindstedt-Poincare Method for a Strongly Nonlinear System with Quadratic and Cubic Nonlinearities. Shock and Vibration, 1996,3(4):279-285
62 Y. K. Cheung, S. H. Chen, S. L. Lau. A Modified Lindstedt-Poincare Method for Certain Strongly Nonlinear Oscillators. International Journal of Nonlinear Mechanics, 1991,26(3/4):367-378
63钱长照.求解一类含阻尼的强非线性系统的改进的多尺度法.振动与冲击, 2006,26(5):93-95
2李进,田兴华.功能梯度材料的研究现状及应用.宁夏工程技术, 2007,6(1):80-83
3赵军,艾兴,张建华.功能梯度材料的发展及展望.材料导报, 1997,11(4):57-60
4田兴华.功能梯度材料及其强度分析. [浙江大学博士论文]. 2006:1-6
5黄旭涛,严密.功能梯度材料回顾与展望.材料科学与工程, 1997,15(4):35-38
6王引真,孙永兴,阎国超.功能梯度材料的研究动态.机械工程材料, 1997,21(4):35-37
7刘延柱,陈立群编著.非线性振动.北京:高等教育出版社, 2001
8陈树辉.强非线性振动的定量方法.广州:广东科技出版社, 2004
9陈予恕.非线性振动.北京:高等教育出版社, 1983
10 V. K. Melnikov. On the Stability of the Centre for Time-periodic Perturbations. Transactions of Moscow Mathematical Society, 1963,12(1):1-57
11 Ali Hasan Nayfeh, Dean T Mook. Nonlinear Oscillations. New York: John Wiley& Sons, 1979
12戴世强,庄峰青.一类非线性振动系统的渐进解.中国科学, 1986,16(1):34-40
13 J. P. Otty. A Perturbation Method for a Set of Purely Non-linear Differential Equations. International Journal of Control, 1979,30(4):587-595
14李骊.强非线性振动系统的定性理论与定量方法.北京:科学出版社, 1997
15边祖光.功能梯度材料板壳结构的耦合问题研究. [浙江大学博士学位论文]. 2005:2-10
16叶开沅.力学发展简史.北京:知识出版社, 1987
17 J. N. Reddy, C. M. Wang, S. Kitipornchai. Axisymmetric Bending of Functionally Grade Circular and Annular Plates. European Journal of Mechanics A/Solids, 1999,18(2):185-195
18 Z. Q. Cheng, S. Kitipornchai. Exact Bending Solution of Inhomogeneous Plates fromHomogeneous Thin-plate Deflection. AIAA Journal, 2000,38(7):1289-1291
19 H. S. Shen. Nonlinear Bending Response of Functionally Graded Plates Subjected to Transverse Loads and in Thermal Environments. International Journal of Mechanical Sciences, 2002,44(3):561-584
20刘进,武兰河,张晓炜.功能梯度材料板的弯曲问题.石家庄铁道学院学报, 2003,16(2):1-5
21 Z. Q. Cheng. Nonlinear Bending of Inhomogeneous Plates. Engineering Structures, 2001,23(10):1359-1363
22 N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material. Transactions of Japan Society of Mechanical Engineers, Series A, 1991,57(533):98-103
23 Y. Obata, N. Noda, T. Tsuji. Steady Thermal Stresses in a Plate of Functionally Gradient Material (Influence of Boundary Conditions). Transactions of Japan Society of Mechanical Engineers, Series A, 1992,58(553):1689-1695
24杨杰,沈惠申.热/机械荷载下功能梯度材料矩形厚板的弯曲行为.固体力学学报, 2003,24(1):119-124
25 G. N. Praveen, J. N. Reddy. Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates. International Journal of Solids and Structures, 1998,35(33):4457-4476
26 R. Javaheri, M. R. Eslami. Buckling of Functionally Fraded Plates under in-plane Compressive Loading. ZAMM, 2002,82(4):277-283
27 R. Javaheri, M. R. Eslami. Thermal Buckling of Functionally Graded Plates. AIAA Journal, 2002,40(1):162-169
28 R. Javaheri, M. R. Eslami. Thermal Buckling of Functionally Graded Plates Based on Higher Order Theory. Journal of Thermal Stresses, 2002,25(7):603-625
29孙建亮,周振功,王彪.功能梯度压电压磁材料中断裂问题的分析.力学学报, 2005,37(1):9-14
30李春雨,邹振祝,段祝平.正交各向异性功能梯度材料反平面裂纹尖端力场.固体力学学报, 2001,22(1):81-84
31李永东,贾斌,张男,戴耀等.功能梯度材料有限宽板的反平面断裂问题研究.应用数学与力学, 2006,27(6):683-689
32 Zhen-Gong Zhou, Lin-Zhi Wu. Non-local Theory Solution for the Anti-plane Shear of two Collinear Permeable Cracks in Functionally Graded Piezoelectric Materials. International Journal of Engineering Science, 2006,44(18-19):1366-1379
33曹志远.功能梯度板的非线性动力分析.固体力学学报, 2006,27(1):21-25
34曹志远.各种边界条件平行四边形功能梯度板的动力特性解析解.力学季刊, 2006,27(2):255-261
35夏贤坤,沈惠申.功能梯度材料剪切板屈曲后的自由振动.固体力学学报, 2008,29(2):129-133
36黄小林,沈惠申.热环境下功能梯度材料板的自由振动和动力响应.工程力学, 2005,22(3):224-228
37 Chun-Sheng Chen. Nonlinear Vibration of a Shear Deformable Functionally Graded Plate. Composite Structures, 2005,68(3):295-302
38 Chun-Sheng Chen, Tsyr-Jang Chen, Rean-Der Chien. Nonlinear Vibration of Initially Stressed Functionally Graded Plates. Thin-Walled Structures, 2006,44(8):844-851
39 Xiao-Lin Huang, Hui-Shen Shen. Nonlinear Vibration and Dynamic Response of Functionally Graded Plates in Thermal Environments. International Journal of Solids and Structures, 2004,41(9-10):2403-2427
40 Chin-Ping Fung, Chun-Sheng Chen. Imperfection Sensitivity in the Nonlinear Vibration of Functionally Graded Plates. European Journal of Mechanics. A, Solids, 2006,25(3):425-436
41 Y. X. Hao, L.H. Chen, W. Zhang et_al. Nonlinear Oscillations, Bifurcations and Chaos of Functionally Graded Materials Plate. Journal of Sound and Vibration, 2008,312(4-5):862-892
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