旋转径向矩形截面悬臂梁的建模与耦合振动分析

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振动与冲击 ›› 2021, Vol. 40 ›› Issue (14) : 60-68.

论文

武祥林，焦映厚，陈照波

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Modeling and coupling vibration analysis of rotating radial rectangular section cantilever beams

WU Xianglin，JIAO Yinghou，CHEN Zhaobo

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摘要

该研究提出了一种建立旋转径向悬臂梁动力学模型的分析方法。基于这种方法，建立了考虑弯曲变形，轴向变形和扭转变形之间耦合效应的矩形截面旋转悬臂梁的动力学模型。该动力学模型可以考虑由于旋转所引起的科里奥利效应，旋转软化效应，应力刚化效应，同时可以考虑梁的剪切变形、转动惯量以及截面翘曲。采用瑞利-里兹法对动力学方程进行了求解，计算得到了不同参数下旋转径向悬臂梁的固有频率及模态振型，将计算得到的固有频率与三维有限元方法和文献中的方法所获得的固有频率进行了对比。结果表明，采用该分析模型计算所得到的结果与三维有限元方法所获得的结果吻合较好，并且比文献中方法所得到结果具有更高的准确性。此外，还详细研究了旋转径向悬臂梁不同模态振型中模态组分的耦合形式，提出了一种改进的表格型模态振型的表示方法，并且深入研究了转速以及安装角对于模态组分的影响规律。

Abstract

An analytical method was proposed for establishing the dynamic model of a rotating radial cantilever beam.Based on the method, the dynamic model of a rotating cantilever beam with rectangular cross section considering the coupling effect among bending, axial and torsional deformations was established.In the dynamic model the Coriolis effect, rotation softening effect, and stress stiffening effect were considered altogether.Moreover, the influences of the beam’s shear deformation, moment of inertia and cross-section warping were also included.The Rayleigh-Ritz method was used to solve the dynamic equations.The natural frequencies and modal shapes of the rotating radial cantilever beam with different parameters were calculated.The calculated natural frequencies were compared with those by the three-dimensional finite element method as well as by the methods in the literature.The results show that the results obtained by the model are in good agreement with the results obtained by the finite element method, and have higher accuracy than the results obtained by the methods in the literature.In addition, the coupling form of the modal components in different modal shapes of the rotating radial cantilever beam was also studied in detail.An improved form-mode modal shape representation method was proposed, and the effects of rotational speed and setting angle on modal components were deeply studied.

导出引用

武祥林，焦映厚，陈照波. 旋转径向矩形截面悬臂梁的建模与耦合振动分析[J]. 振动与冲击, 2021, 40(14): 60-68

WU Xianglin，JIAO Yinghou，CHEN Zhaobo. Modeling and coupling vibration analysis of rotating radial rectangular section cantilever beams[J]. Journal of Vibration and Shock, 2021, 40(14): 60-68

参考文献

［1］STAFFORD R O, GIURGIURTIU V.Semi-analytic methods for rotating Timoshenko beams［J］.International Journal of Mechanical Sciences, 1975,17: 719-727.

［2］YOKOYAMA T.Free vibration characteristics of rotating Timoshenko beams［J］.Journal of Mechanical Sciences, 1988,30(10): 743-755.

［3］YU S D, CLEGHORN W L.Free vibration of a spinning stepped Timoshenko beam［J］.Journal of Applied Mechanics, 2000,67(4): 839-841.

［4］BAZOUNE A, KHULIEF Y A, STEPHEN N G, et al.Dynamic response of spinning tapered Timoshenko beams using modal reduction［J］.Finite Element in Analysis and Design, 2001,37(3): 199-219.

［5］HASHEMI S M, RICHARD M J.Natural frequencies of rotating uniform beams with Coriolis effects［J］.Journal of Vibration and Acoustics, 2001,123(4): 444-455.

［6］OGUAMANAM D C D, HEPPLER G R.Centrifugal stiffening of Timoshenko beams［C］//Structures, Structural Dynamics, and Materials and Co-located Conferences.Reston: AIAA, 1996.

［7］FUNG E H K, YAU D T W.Effects of centrifugal stiffening on the vibration frequencies of a constrained flexible arm［J］.Journal of Sound and Vibration, 1999,224(5): 809-841.

［8］BERZERI M, SHABANA A A.Study of the centrifugal stiffening effect using the finite element absolute nodal coordinate formulation［J］.Multibody System Dynamics, 2002,7: 357-387.

［9］ADAIR D, JAEGER M.Simulation of tapered rotating beams with centrifugal stiffening using the adomian decomposition method［J］.Applied Mathematical Modelling, 2016,40(4): 3230-3241.

［10］ROY P A, MEGUID S A.Analytical modeling of the coupled nonlinear free vibration response of a rotating blade in a gas turbine engine［J］.Acta Mechanica, 2018,229: 3355-3373.

［11］TIAN J J, SU J P, ZHOU K, et al.A modified variational method for nonlinear vibration analysis of rotating beams including Coriolis effects［J］.Journal of Sound and Vibration, 2018,426(21): 258-277.

［12］HODGES D H, DOWELL E H.Nonlinear equations of motion for the elastic and torsion of twisted nonuniform rotor blades［R］.Washington, D.C.: NASA, 1974.

［13］SURACE G, ANGHEL V, MARES C.Coupled bending-bending-torsion vibration analysis of rotating pretwisted blades: an integral formulation and numerical examples［J］.Journal of Sound and Vibration, 1997,206(4): 473-486.

［14］AVRAMOV K V, PIERRE C, SHYRIAIEVA N.Flexural-flexural-torsional nonlinear vibrations of pre-twisted rotating beams with asymmetric cross-sections［J］.Journal of Vibration and Control, 2007,13(4): 329-364.

［15］AVRAMOV K V, PIERRE C, SHYRIAIEVA N.Nonlinear equations of flexural-flexural-torsional oscillations of rotating beams with arbitrary cross-section［J］.International Applied mechanics, 2008,44: 123-132.

［16］OZGUMUS O O, KAYA M O.Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending-torsion coupling［J］.International Journal of Engineering Science, 2007,45(2): 562-586.

［17］OZGUMUS O O, KAYA M O.Energy expressions and free vibration analysis of a rotating Timoshenko beam featuring bending-bending-torsion coupling［J］.Archive of Applied Mechanics, 2013,83: 97-108.

［18］YUTAEK O, HONG H Y.Vibration analysis of a rotating pre-twisted blade considering the coupling effects of stretching, bending, and torsion［J］.Journal of Sound and Vibration, 2018,431(29): 20-39.

［19］NOVOZHILOV V V.Foundations of the nonlinear theory of elasticity［M］.3rd ed.Rochester: Graylock Press, 1963.

［20］TIMOSHENKO S P, GOODIER J N.Theory of elasticity［M］.3rd ed.New York: McGraw-Hill Education, 1970.

［21］SINHA S K.Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end［J］.International Journal of Non-linear Mechanics, 2005,40(1): 113-149.

［22］RAO S S.Mechanical vibrations［M］.4th ed.New York: Pearson Education Inc., 2004.

［23］SINHA S K.Combined torsional-bending axial dynamics of a twisted rotating cantilever Timoshenko beam with contact-impact loads at the free end［J］.Journal of Applied Mechanics, 2007,74(3): 505-522.