Full-Text Search：
《Acta Mechanica Sinica》 2002-03

## DYNAMIC STABILITY OF A SLENDER BEAM WITH INTERNALRESONANCE UNDER A LARGE LINEAR MOTION

Feng Zhihua Hu HaiyanInstitute of Vibration Engineering Researeh, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China
Dynamic modeling of a flexible beam undergoing a large linear motion is presentedin this paper at first. The equations of motion for the beam are derived by using Kane's equationand then simplified through the Rayleigh-Ritz method. Different from the linear modeling methodwhere the generalized inertia forces and the generalized active forces are linearized in the model-ing process, the present model takes the coupled cubic non-linearities of geometrical and inertialtypes into consideration. In the case of a simply supported slender beam under certain averageacceleration of base, the second natural frequency of the beam may approximate to the tripled firstone so that the condition of 3:1 internal resonance of the beam holds true. The method of multi-ple scales is used to solve directly the nonlinear differential equations and to derive the nonlinearmodulation equation for either the principal parametric resonance or the combination parametricresonance with 3:1 internal resonance between the first two modes of the beam. The dynamicstability of the trivial state of the system is investigated by using Cartesian transformation indetail. The equations of approximate transition curves in the plane of dimensionless frequency andexcitation parameter that separate stable from unstable solution are derived. For the case of prin-cipal parametric resonance of the first mode, in addition to the principal instability region, thereexist several new narrow instability regions because of the presence of internal resonance. Thesenarrow regions move from the left of the principal instability region to the right of it when thefrequency detuning parameter of internal resonance increases from the negative to the positive. Incontrast with the case of principal parametric resonance of the first mode, single mode equilibriumsolution is possible for the principal parametric resonance of the second mode. Furthermore, thestability of the trivial solution for the third mode or the higher ones has not been affected by theinternal resonance between the first two modes. Finally,the modulation equations are reduced toa two-dimensional system and the type of a Hopf bifurcation is determined in the vicinity of thebifurcation via the center manifold theorem and a limit cvcle is found.
【Fund】： 国防基础科研计划项目(10172005)资助．
【CateGory Index】： O326
CAJViewer7.0 supports all the CNKI file formats; AdobeReader only supports the PDF format.
 【References】
 Chinese Journal Full-text Database 10 Hits
 1 Feng Zhihua Hu Haiyan (Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016);NONLINEAR DYNAMIC MODELING AND PERIODIC VIBRATION OF A CANTILEVER BEAM SUBJECTED TO AXIAL MOVEMENT OF BASEMENT[J];Acta Mechanica Solida Sinica;2002-04 2 Bo Wang1 Liqun Chen1,2(1Shanghai institute of Applied Mathematics and Mechanics,Shanghai University,Shanghai,200072)(2Department of Mechanics,Shanghai University,Shanghai,200072);ASYMPTOTIC ANALYSIS ON STABILITY OF AXIALLY ACCELERATING VISCOELASTIC BEAMS WITH NUMERICAL VERIFICATION[J];Chinese Journal of Solid Mechanics;2009-02 3 DING Hu CHEN LiQun(Shanghai Institute of Applied Mathematics and Mechanics,Department of Mechanics at ShanghaiUniversity,Shanghai 200072,China);ANALYSIS OF STABILITY IN PARAMETRIC RESONANCE OF AN AXIALLY ACCELERATING VISCOELASTIC BEAM[J];Journal of Mechanical Strength;2008-06 4 YANG XiaoDong TANG YouQi(Department of Engineering Mechanics,Shenyang Institute of Aeronautical Engineering,Shenyang 110136,China);DYNAMICS OF TRANSVERSE VIBRATIONS OF AXIALLY MOVING BEAM FOR TIMOSHENKO MODEL[J];Journal of Mechanical Strength;2008-06 5 YANG XiaoDong1 TANG YouQi1 GE XinSheng2(1.Department of Engineering Mechanics,Shenyang Institute of Aeronautical Engineerin,Shenyang 110136,China)(2.Department of Mechanical Engineering,Beijing Institute of Mechinary,Beijing 100085,China);NATURAL FREQUENCY ANALYSIS OF AXIALLY MOVING,SIMPLY SUPPORTED TIMOSHENKO BEAM[J];Journal of Mechanical Strength;2009-02 6 FENG Zhi-hua (School of Mechatronic Eng.,Suzhou University, Suzhou 215021,China);Dynamic stability of rigid rapiers I:nonlinear dynamic modeling[J];Machine Design;2002-12 7 FENG Zhi hua,QIAN Zhi liang,JI Ying dong,ZHU Xiao dong (School of Mechatronic Eng., Suzhou University,Suzhou 215021,China);Dynamic stability of rigid rapiers II: stability analysis[J];Machine Design;2003-05 8 Han Wei Jin Dongping Hu Haiyan (Institute of Vibration Engineering Research, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China);DYNAMICS AND STABILITY OF A NON-LINEAR CONTROLLED SYSTEM SUBJECT TO PARAMETRIC EXCITATION[J];Acta Mechanica Sinica;2003-03 9 Chen Shuhui~2) Huang Jianliang (Department of Applied Mechanics and Engineering,Zhongshan University,Guangzhou 510275,China);ON INTERNAL RESONANCE OF NONLINEAR VIBRATION OF AXIALLY MOVING BEAMS[J];Acta Mechanica Sinica;2005-01 10 Yao Zhigang Zhang Wei Chen Lihua (College of Mechanical Engineering,Beijing University of Technology,Beijing 100124,China);BIFURCATION,CHAOTIC DYNAMICS AND CONTROL OF PIEZOELECTRIC LAMINATED COMPOSITE BEAM[J];Chinese Journal of Theoretical and Applied Mechanics;2009-01
 China Proceedings of conference Full-text Database 4 Hits
 1 Chen Shuhui,Huang Jianliang (Department of Applied Mechanics and Engineering, Sun Yat-sen University, 510275 Guangzhou);NONLINEAR VIBRATION OF AXIALLY MOVING BEAMS[A];[C];2004 2 Ji Ying1,Zou Yong1, Pan Jianqiang2, Bi Qinsheng1 (1 Department of Mechanics, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013) (2 Central Institute of Shipping Science );COMPLEXITY ANALYSIS IN A TWO DEGREE OF FREEDOM SYSTEM WITH 1:1 INTERNAL RESONANCE[A];[C];2004 3 Chen Shuhui1, Huang Jianliang1 Huang Lingling2 1. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275 2. Experiment Center of Rudimentary Education, Sun Yat-sen University, Guangzhou 510000;STABILITY ANALYSIS OF NONLINEAR VIBRATION OF AXIALLY MOVING BEAMS[A];[C];2007 4 Yang Xiaodong Department of Engineering Mechanics,Shenyang Institute of Aeronautical Engineering;Free Vibrations of Moving Angle-Ply Composite Plates[A];[C];2007
 【Co-citations】
 Chinese Journal Full-text Database 10 Hits
 1 Peng Zhen~(1),Yang Zhian~(1,2)(1.Key Lab of Structure and Vibration Engineering of Tangshan and Tangshan College,Tangshan 063000,China;2.School of Mechanical Engineering,Tianjin University 300072,China);Analysis of primary resonance of a beam externally excited on the Winkler foundation in temperature field[J];Earthquake Engineering and Engineering Vibration;2006-03 2 Peng Zhen~(1),Yang Zhian~(1,2)(1.Key Laboratory of Structure and Vibration Engineering of Tangshan and Tangshan College,Tangshan 063000,China;2.School of Mechanical Engineering,Tianjin University,Tianjin 300072,China);Analysis of 1/3 subharmonic resonance of a beam externally excited on Winkler foundation in temperature field[J];Earthquake Engineering and Engineering Vibration;2006-04 3 Peng Zhen,Yang Zhian (Key Lab of Structural and Vibration Engineering of Tangshan and Tangshan College,Tangshan 063000,China);Analysis of 3 superhamonic resonance of a beam externally excited on the Winkler foundation in temperature field[J];Earthquake Engineering and Engineering Vibration;2006-05 4 Mao Lijun~1 Li Aiqun~2(~1Department of Civil Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)(~2College of Civil Engineering, Southeast University, Nanjing 210096, China);Decomposition method of building system of sliding base isolation based on the elementary sliding base-isolation system[J];Journal of Southeast University (Natural Science Edition);2005-02 5 Xu Zhaotang Zhu Rupeng(College of Mechanical and Electrical Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China);Analysis of main resonance of drive shaft with rigid multi-supports[J];Journal of Southeast University(Natural Science Edition);2006-01 6 Nie Chunyan 1,2 Shi Yaowu1 Yi Wensuo1 (1Comunication Institute, Jilin Univ.Chang chun,China,130025; 2Electronic Information Institute,Changchun Univ.Chang chun,China,130022);Determining Chaotic Solution to Measure Arbitrary Periodic Signals[J];Journal of Electronic Measurement and Instrument;2005-04 7 SHEN Hong-liang1,2,CHEN Yong-liang1,LIU Chang1(1.Department of Aerodynamics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;2.Workstation of Postdoctor,Shenyang Aircraft Design & Research Institute,Shenyang 110035,China);Application of the Extended Bifurcation Analysis Methods in Flight Dynamics[J];Flight Dynamics;2008-02 8 YANG Zhi-an1 , XI Xiao-yan2 , LI Wen-lan3 (1. Key Lab of Structure and Vibration Engineering of Tangshan, Tangshan College, Tangshan 063000, China; 2. School of Mechanical Engineering of Hebei Polytechnic University, Tangshan 063000, China; 3. Library of Tianjin University, Tianjin 30072, China);SINGULARITIES AND NONLINEAR VIBRATION OF ELASTIC STRAIGHT BARS IN THERMAL FIELD UNDER HARMONIC EXCITATION[J];Engineering Mechanics;2006-06 9 Chen Liqun Wu Jun (Department of Mechanics, Shanghai University, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, 200436);BIFURCATION IN TRANSVERSE VIBRATION OF AXIALLY ACCELERATING VISCOELASTIC STRINGS[J];Acta Mechanica Solida Sinica;2005-01 10 Li Jian Zhang Sijin(College of Mechanics and Aerospace,Hunan University,Changsha,410082);CELL-MAPPING COMPUTATION METHOD FOR NON-SMOOTH DYNAMICAL SYSTEMS[J];Acta Mechanica Solida Sinica;2007-01
 China Proceedings of conference Full-text Database 4 Hits
 1 CHEN Yu-shu (School of Astrnautics, Harbin Institute of Technology, Harbin, 150001) (School of Mechnical Engineering, Tianjin University, Tianjin, 300072);Structures of Dynamics and Non-linear Dynamics in Engineering[A];[C];2007 2 LI Xin-ye, ZHANG Hua-biao, YANG Yan-long ( Hebei University of Technology, Tianjin 300130, China);The Principal Parametric Resonance of Coupled van der Pol Oscillators with 1:1 Internal Resonance and Its Feedback Control[A];[C];2007 3 Li Xinye Zhang Huabiao Hebei University of Technology;The Principal Resonance of Coupled van der Pol Oscillators with 1:1 internal resonance and Its Feedback Control[A];[C];2007 4 Liu Lili Jin Dongping Hu Haiyan Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,210016 Nanjing,P.R.China;Chaotic oscillations of a tethered satellite in the station keeping phase[A];[C];2007
 【Co-references】
 Chinese Journal Full-text Database 10 Hits
 1 Chen Lichun(Dept. of Machanical Engineering);Traveling and Stationary Waves in a Vibrating Uniform String[J];JOURNAL OF ANSHAN INSTITUTE OF IRON AND STEEL TECHNOLOGY;1996-01 2 XIAO Shifu 1),2) CHEN Bin 2) LIU Caishan 2) ( 1) Southwest Institute of Structural Mechanics, P.O.Box 919-401, Sichuan, Mianyang,621900 2) Department of Mechanics and Engineering Science, Peking University, Beijing, 100871);Dynamic Behavior and Buckling Analysis of Thin Rectangular Plate Rotating around Its Symmetrical Axis[J];Acta Scicentiarum Naturalum Universitis Pekinesis;2005-03 3 ZHANG Wei ( College of Mechanical Engineering and Applied Electronics Technology, Beijing Polytechnic University, Beijing 100022, China );Nonlinear Dynamics and Local Bifurcations in the Flexible Beam[J];Journal of Beijing Polytechnic University;2001-04 4 Bao Yingbin;Li Runpei;Gu yongning;Preliminary Optimum Design of TLP[J];Ship Engineering;1999-01 5 ZENG Xiao-hui1, SHEN Xiao-peng1, WU Ying-xiang1, LIU Jie-ming2 (1.Division of Engineering Sciences, Institute of Mechanics, CAS, Beijing 100080, China; 2.China Offshore Oil Research Center, CNOOC, Beijing 100027, China);Key Mechanical Problems in Analysis and Design of Deep-sea Platform[J];Ship Engineering;2005-05 6 MA Chi;DONG Yan-qiu; YANC Li-ting (Tianjin University,Tianjin 300072);Comparative Study of Vortex-induced Nonlinear Oscillation of TLP Tethers under Two Different Boundary Conditions[J];JOURNAL OF SHIP MECHANICS;2000-01 7 Shen Xudong (Dongfang Electric Machinery Works);Analysis on Vibration of Stator Core Caused by Subharmonics[J];Large Electric Machine and Hydraulic Turbine;1988-01 8 Hu Minqiang Chen Xianzhen Zhou keding (Huazhong University of Science and Technology);Analytical Solution of the Three Dimensional Magnetic Field in the End Region of the Turbogenerator[J];Large Electric Machine and Hydraulic Turbine;1988-03 9 Qu Dazhuang (Harbin Research Institute of Large Electrical Machinery);The Finite Element Method for the Calculating Natural Frequency of Turbogenerator Stator End Windings[J];Large Electric Machine and Hydraulic Turbine;1988-05 10 Luo Rongjie(Zhejiang University);Three Dimensional Magnetic Field Analysis in the End Region of Hydrogenerator by the Separation of Variables[J];Large Electric Machine and Hydraulic Turbine;1989-03
 【Secondary References】
 Chinese Journal Full-text Database 10 Hits
 1 WEI Hai-jun(Marine Engineering College,Dalian Maritime University,Dalian 116026,China);Study on some problems of calculating ship shafting torsional vibration[J];Journal of Ship Mechanics;2006-05 2 WEI Hai-jun (Marine Engineering College,Dalian Maritime University,Dalian 116026,China);Effects of marine fire angle and connecting rod mass on diesel engine shaft torsional vibration[J];Journal of Dalian Maritime University;2007-03 3 Yin Zhenkun Chen Shuhui(Department of Applied Mechanics and Engineering,Sun Yat-sen University,Guangzhou 510275,China);STUDY ON NONLINEAR VIBRATION AND STABILITY OF AXIALLY MOVING PLATES[J];Journal of Dynamics and Control;2007-04 4 HU Yu-da1 , QIU Jia-jun2(1. School of Constructional Engineering and Mechanics ,Yanshan University, Qinhuangdao 066004, China; 2. Department of Mechanics, Tianjin University, Tianjin 300072, China);A STIFFENED LAMINATED COMPOSITE CONICAL SHELL MODEL FOR VIBRATION OF LARGE GENERATOR END WINDING[J];Engineering Mechanics;2005-02 5 Hu Yuda 1 Qiu Jiajun 2 ( 1 Department of Engineering Mechanics, Yanshan University, Qinhuangdao, 066004) ( 2Department of Mechanics, Tianjin University, Tianjin, 300072);TWO-DEGREE-OF-FREEDOM PRINCIPAL RESONANCE AND BIFURCATION OF GENERATOR END WINDING UNDER ELECTROMAGNETIC FORCE AND FRICTION FORCE[J];Acta Mechanica Solida Sinica;2004-04 6 Bo Wang1 Liqun Chen1,2(1Shanghai institute of Applied Mathematics and Mechanics,Shanghai University,Shanghai,200072)(2Department of Mechanics,Shanghai University,Shanghai,200072);ASYMPTOTIC ANALYSIS ON STABILITY OF AXIALLY ACCELERATING VISCOELASTIC BEAMS WITH NUMERICAL VERIFICATION[J];Chinese Journal of Solid Mechanics;2009-02 7 WANG Chun-Ni,LI Shi-Rong(School of Science,Lanzhou Univ.of Tech.,Lanzhou 730050,China);Primary parametric resonance and control of beams subjected to thermal shock load[J];Journal of Lanzhou University of Technology;2009-04 8 FAN Guo-min,ZHENG Xiao-hong(North China Institate of Science and Technology,Yanjiao Beijing-East 101601);Dimensional Analysis of axially moving string[J];Journal of North China Institute of Science and Technology;2006-02 9 FAN Guo-min(North China Institution of Science and Technology,Yanjiao Beijing-East 101601);Chaotic Motions of an Axially Moving Viscoelastic Beam[J];Journal of North China Institute of Science and Technology;2007-04 10 LI Gao-feng~1 YANG Zhi-an~2 (1.College of Mechanical Engineering,Hebei Polytechnic University,Tangshan Hebei 063009,China 2.Key Lab of Structure and Vibration of Tangshan College and Tangshan College,Tangshan 063009,China);Research Development on Nolinear Belt Driver Mechanism and the Coupled Mechanical and Electrical Dynamics Model[J];Journal of Hebei Institute of Technology;2006-04
 China Proceedings of conference Full-text Database 5 Hits
 1 Ji Ying1,Zou Yong1, Pan Jianqiang2, Bi Qinsheng1 (1 Department of Mechanics, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013) (2 Central Institute of Shipping Science );COMPLEXITY ANALYSIS IN A TWO DEGREE OF FREEDOM SYSTEM WITH 1:1 INTERNAL RESONANCE[A];[C];2004 2 LI Xin-ye, ZHANG Hua-biao, YANG Yan-long ( Hebei University of Technology, Tianjin 300130, China);The Principal Parametric Resonance of Coupled van der Pol Oscillators with 1:1 Internal Resonance and Its Feedback Control[A];[C];2007 3 Li Xinye Zhang Huabiao Hebei University of Technology;The Principal Resonance of Coupled van der Pol Oscillators with 1:1 internal resonance and Its Feedback Control[A];[C];2007 4 Yang Xiaodong Department of Engineering Mechanics,Shenyang Institute of Aeronautical Engineering;Free Vibrations of Moving Angle-Ply Composite Plates[A];[C];2007 5 Fan Guo-min Zhang Wei College of Mechanical Engineering,Beijing University of Technology,Beijing 100022,China North china university of science and technology,Yanjiao Beijing 101601,China;PERIODIC AND CHAOTIC OSCILLATION OF A VISCOELASTIC AXIALLY MOVING BEAM WITH 1:2 INTERNAL RESONANCE[A];[C];2007
Similar Journals
 > Chinese Journal of Applied Mechanics > Journal of Agricultural Mechanization Research > Journal of Huaqiao University(Natural Science) > Mechanics in Engineering > Journal of Vibration Engineering > Chinese Journal of Theoretical and Applied Mechanics > Journal of Yancheng Institute of Technology(Social Science Edition) > Journal of Fuzhou University(Natural Science Edition) > Journal of Liaoning Institute of Science and Technology > Journal of Dynamics and Control