梁类结构的非线性动力学分析
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摘要
梁是工程中最基本的构件或部件,其非线性振动的研究具有理论和工程意义。梁的研究对板类、复杂结构的研究具有借鉴意义,另外梁还可以近似认为是管道的固化模型,对管道问题的研究也有一定的借鉴意义。近年来,除各向同性材质外,工程应用的梁材质性能多样。目前常见梁问题的研究大多集中在匀质梁的非线性动力学特性,复合材料层合结构梁的非线性动力学特性的研究则相对较少,本文首先对受横向简谐激励的匀质两端铰支梁的动力学特性进行分析,在此基础上研究两端简支复合材料层合梁的非线性动力学特性。
本文首先以基础受简谐激励的两端铰支匀质梁为研究对象,在多尺度法理论基础上,分别采用模态叠加法和渐近展开法分析了系统的主共振现象,并获得非线性方程的一次和二次近似解,同时比较了基于模态叠加和渐近展开两种方法在求解非线性方程的近似解时的不同,得出渐近展开法比模态叠加法表述更严谨,且计算过程更清晰的结论。
其次,采用渐近展开法对上述系统的超谐共振和次谐共振现象进行分析,结果表明系统在产生第一阶模态次谐共振的同时,产生第三阶模态的超谐共振现象。本文通过计算得出稳态响应的一次近似解,并得到系统相应的幅频响应曲线。
最后,基于Von-Karman应力应变关系和Reddy高阶剪切变形理论,利用Hamilton原理推导了受横向简谐激励的两端简支复合材料层合梁的非线性动力学方程。研究了两端简支复合材料层合梁,在横向载荷作用下的非线性动力学特性。
As one of the most basic components in engineer, the study of beam's nonlinear vibration has an important theoretical significance and engineering practical value. Researches on beams dynamics have provided a reference for the research on structures such as panel and other complex structures. Meanwhile, as considered similar to the curing model of pipeline, the results of beams'research have certain significance for plumbing problems. In recent years, in addition to isotropic materials, the properties of a beam material in engineering are diversiform. Currently, most common studies are focused on the nonlinear dynamic behaviors of homogeneous beam, while the nonlinear dynamics of composite laminated structures are merely studied. In this dissertation, the nonlinear dynamics of a homogeneous simply supported slender beam is taken into consideration, which, is subjected to harmonic base excitation. Based on the homogeneous beams, the nonlinear dynamics of composite laminated beam is also studied. The main contents of this dissertation are given as below:
Firstly, based on the multiple scales method, the primary resonance of the beam is analyzed, using modal superposition method and asymptotic expansion method, with the homogeneous simply supported slender beam as research object, which is subjected to transverse harmonic base excitation. The first-order and second-order approximation solutions of the nonlinear dynamic equation are obtained, and the differences between these two methods are compared. Based on the comparison, conclusion has been made that the asymptotic expansion method has significant advantages of reasonable expression and clearer calculation process.
Secondly, the super-harmonic resonance and sub-harmonic resonance of the homogeneous simply supported slender beams are analyzed by using asymptotic expansion method. The result shows that the system will produce super-harmonic resonance on the third mode when it produces sub-harmonic resonance on the first mode. The first-order approximate solution of the steady-state responses is calculated, and the amplitude-frequency response curves of the system are obtained.
Finally, according to Von-Karman type equations for geometric nonlinearity and Reddy's high-order shear deformation theory, the nonlinear governing partial differential equations of the motion are derived by using the Hamilton's principle, which are suitable for composite laminated beam subjected to the transverse excitations. The nonlinear dynamics behavior of the composite laminated beam subjected to transverse harmonic excitations is also analyzed in this dissertation.
本文首先以基础受简谐激励的两端铰支匀质梁为研究对象,在多尺度法理论基础上,分别采用模态叠加法和渐近展开法分析了系统的主共振现象,并获得非线性方程的一次和二次近似解,同时比较了基于模态叠加和渐近展开两种方法在求解非线性方程的近似解时的不同,得出渐近展开法比模态叠加法表述更严谨,且计算过程更清晰的结论。
其次,采用渐近展开法对上述系统的超谐共振和次谐共振现象进行分析,结果表明系统在产生第一阶模态次谐共振的同时,产生第三阶模态的超谐共振现象。本文通过计算得出稳态响应的一次近似解,并得到系统相应的幅频响应曲线。
最后,基于Von-Karman应力应变关系和Reddy高阶剪切变形理论,利用Hamilton原理推导了受横向简谐激励的两端简支复合材料层合梁的非线性动力学方程。研究了两端简支复合材料层合梁,在横向载荷作用下的非线性动力学特性。
As one of the most basic components in engineer, the study of beam's nonlinear vibration has an important theoretical significance and engineering practical value. Researches on beams dynamics have provided a reference for the research on structures such as panel and other complex structures. Meanwhile, as considered similar to the curing model of pipeline, the results of beams'research have certain significance for plumbing problems. In recent years, in addition to isotropic materials, the properties of a beam material in engineering are diversiform. Currently, most common studies are focused on the nonlinear dynamic behaviors of homogeneous beam, while the nonlinear dynamics of composite laminated structures are merely studied. In this dissertation, the nonlinear dynamics of a homogeneous simply supported slender beam is taken into consideration, which, is subjected to harmonic base excitation. Based on the homogeneous beams, the nonlinear dynamics of composite laminated beam is also studied. The main contents of this dissertation are given as below:
Firstly, based on the multiple scales method, the primary resonance of the beam is analyzed, using modal superposition method and asymptotic expansion method, with the homogeneous simply supported slender beam as research object, which is subjected to transverse harmonic base excitation. The first-order and second-order approximation solutions of the nonlinear dynamic equation are obtained, and the differences between these two methods are compared. Based on the comparison, conclusion has been made that the asymptotic expansion method has significant advantages of reasonable expression and clearer calculation process.
Secondly, the super-harmonic resonance and sub-harmonic resonance of the homogeneous simply supported slender beams are analyzed by using asymptotic expansion method. The result shows that the system will produce super-harmonic resonance on the third mode when it produces sub-harmonic resonance on the first mode. The first-order approximate solution of the steady-state responses is calculated, and the amplitude-frequency response curves of the system are obtained.
Finally, according to Von-Karman type equations for geometric nonlinearity and Reddy's high-order shear deformation theory, the nonlinear governing partial differential equations of the motion are derived by using the Hamilton's principle, which are suitable for composite laminated beam subjected to the transverse excitations. The nonlinear dynamics behavior of the composite laminated beam subjected to transverse harmonic excitations is also analyzed in this dissertation.
引文
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[3]胡海岩,孙久厚,陈怀海.机械振动与冲击[M].南京：航空工业出版社,2002
[4]刘燕.梁的非线性振动的进展[J].大众科技,2009,(10)：57-58
[5]刘善亮.轴向力和支承运动作用下梁的参数共振和内共振分析[D].沈阳：沈阳航空工业学院,2009
[6]鞠学滨.梁的非线性振动内共振模态研究[D].长沙：国防科技大学,2005
[7]Kane T R, Ryan R R, Banerjee A K. Dynamics of a cantilever beam attached to a moving base [J]. Journal of Guidance, Control and Dynamics,1987,10(2):139-151
[8]Yamaki N, Mori A. Non-linear vibrations of a clamped beam with initial defection and initial axial displacement, Part Ⅰ Theory [J]. Journal of Sound and Vibration, 1980,71(3),333-346
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[10]Nayfeh A H, Mook D T. Nonlinear oscillations [J]. Wiley Interscience, New York, 1979,304-321
[11]Chin C M, Nayfeh A H. Three-to-one internal resonances in hinged-clamped beams [J]. Nonlinear Dynamics,1997,17(2):129-154
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[13]Kar R C, Dwivedy S K. Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances [J]. International Journal of Non-linear Mechanics,1999,34(3):515-529
[14]Dwivedy S K, Kar R C. Dynamics of a slender beam with an attached mass under combination parametric and internal resonances. Part Ⅰ. Steady state response [J]. Journal of Sound and Vibration,1999,221 (5):823-848
[15]Dwivedy S K, Kar R C. Dynamics of a slender beam with an attached mass under combination parametric and internal resonances. Part II. Periodic and Chaotic Response [J]. Journal of Sound and Vibration,1999,222(2):283-305
[16]Dwivedy S K, Kar R C. Simultaneous combination, principal parametric and internal resonances in a slender beam with a lumped mass:Three-mode interactions [J]. Journal of Sound and Vibration,2001,242 (1):27-46
[17]Abou-Rayan A M, Nayfeh A H, Mook D T. Nonlinear response of a parametrically excited buckled beam[J]. Nonlinear Dynamics,1993(4):499-525
[18]Lee Y Y. R K L Su. Hui C K. The effect of the modal energy transfer on the sound radiation and vibration of a curved panel:Theory and Experiment [J]. Journal of Sound and Vibration,2009,324(02):1003-1015
[19]冯志华,胡海岩.受轴向基础激励悬臂梁非线性动力学建模及周期振动[J].固体力学学报,2002,23(4)：373-379
[20]张伟.简支柔性梁的非线性动力学和分叉[J].北京工业大学学报,2001,27(4)：400-405
[21]姚明辉,张伟.柔性悬臂梁非线性非平面运动的多脉冲轨道分析[J].动力学与控制学报,2004,2(2)：11-14
[22]易壮鹏,赵跃宇,朱克兆.梁的纵向与横向耦合振动的动力学分析[J].应用力学学报,2008,25(4)：687-692
[23]蒋丽忠,洪嘉振.作大范围平动弹性梁的动力学性质[J].振动与冲击,2002,21(2)：11-15
[24]刘桂斋.轴向初压力和横向初偏移对梁非线性振动的影响[J].振动工程学报,1989,2(3)：91-94
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