复合柔性结构全局模态函数提取与状态空间模型构建
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摘要
随着航空航天等领域中实际工程结构的大型化和柔性化,结构的非线性振动和主动振动控制问题越来越凸显.分析和处理此类结构出现的复杂振动问题的关键在于建立系统的非线性动力学模型与状态空间模型.对于由柔性部件、刚体、连接部件构成的复合柔性结构,由于各部件之间的振动耦合效应,单个柔性部件在悬臂、简支和自由等静定边界下的模态与结构的真实模态有较大差异.为此,本文提出复合柔性结构全局模态的解析提取方法,通过全局模态离散得到系统非线性动力学模型,从而构建状态空间模型.该方法采用笛卡尔坐标描述系统的运动,建立系统的运动方程;结合描述柔性部件的偏微分方程、刚体的常微分运动方程、连接界面处力、力矩、位移和转角的匹配条件以及系统的边界条件,利用分离变量法给出统一形式的频率方程,获取系统的固有频率和解析函数表征的全局模态.这里提出的全局模态提取方法不仅便于复合柔性结构固有频率和全局模态的参数化分析,而且为建立复合柔性结构低维非线性动力学模型和状态空间模型提供了有效的途径,对于推进这类结构的非线性动力学分析与主动振动控制研究具有重要意义.
With the scale enlarging and flexibility of the actual engineering structures utilized in aerospace and other fields, the issues on the study of nonlinear vibration and active vibration control of the structure become more and more important. The key process of dealing with the vibration and control for such a kind of structure is to establish the nonlinear dynamic model and formulate the state space model of the system. For composite flexible structures composed of flexible components, rigid bodies and flexible joints, because of the vibration coupling between each part of the structure, the modes of an individual flexible component with the cantilever, simply supported and free stationary boundary are different from the real mode of the structure. In this paper, an analytic extraction method of global modes of composite flexible structures is presented, and the nonlinear dynamic model and the state-space model of the system can be obtained by the global mode discretization. Adopting the Cartesian coordinates to describe the motion of the system, establishing the motion equations of the system, and combining with the partial differential equation of the flexible part, the ordinary differential motion equation of the rigid body, the matching condition of force, moment, slope of the deflection curve and displacement at the interface, and the boundary condition of the system, the frequency equation of uniform form is given by using the separating variable method. Consequently, the natural frequencies and the global mode representation of the analytic function of the system are obtained. The global mode extraction method presented here not only facilitates the parametric analysis of the natural frequencies and global modes of composite flexible structures, but also provides an effective way to establish the low dimensional nonlinear dynamic model and the state space model of the composite flexible structure, which is of great significance for the study of nonlinear dynamic responses and the design of active vibration control of this kind of structures.
With the scale enlarging and flexibility of the actual engineering structures utilized in aerospace and other fields, the issues on the study of nonlinear vibration and active vibration control of the structure become more and more important. The key process of dealing with the vibration and control for such a kind of structure is to establish the nonlinear dynamic model and formulate the state space model of the system. For composite flexible structures composed of flexible components, rigid bodies and flexible joints, because of the vibration coupling between each part of the structure, the modes of an individual flexible component with the cantilever, simply supported and free stationary boundary are different from the real mode of the structure. In this paper, an analytic extraction method of global modes of composite flexible structures is presented, and the nonlinear dynamic model and the state-space model of the system can be obtained by the global mode discretization. Adopting the Cartesian coordinates to describe the motion of the system, establishing the motion equations of the system, and combining with the partial differential equation of the flexible part, the ordinary differential motion equation of the rigid body, the matching condition of force, moment, slope of the deflection curve and displacement at the interface, and the boundary condition of the system, the frequency equation of uniform form is given by using the separating variable method. Consequently, the natural frequencies and the global mode representation of the analytic function of the system are obtained. The global mode extraction method presented here not only facilitates the parametric analysis of the natural frequencies and global modes of composite flexible structures, but also provides an effective way to establish the low dimensional nonlinear dynamic model and the state space model of the composite flexible structure, which is of great significance for the study of nonlinear dynamic responses and the design of active vibration control of this kind of structures.
引文
1张军徽,崔洋洋,佟安.条带式太阳帆的结构动力学分析.力学学报,2019,51(1):237-244(Zhang Junhui,Cui Yangyang,Tong An.Structural dynamic and stability analysis of a stripped solar sail.Chinese Journal of Theoretical and Applied Mechanics,2019,51(1):237-244(in Chinese))
2胡璐,闫寒,张文明等.黏性流体环境下V型悬臂梁结构流固耦合振动特性研究.力学学报,2018,50(3):643-653(Hu Lu,Yan Han,Zhang Wenming et al.Analysis of flexural vibration of V-shaped beams immersed in viscous fluids.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):643-653(in Chinese))
3陶连金,刘春晓,边金等.大跨度Y形柱地铁车站结构地震反应研究.力学学报,2017,49(1):55-64(Tao Lianjin,Liu Chunxiao,Bian Jin et al.Seismic response of subway station with large span and Y-shaped column.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):55-64(in Chinese))
4曹登庆,白坤朝,丁虎等.大型柔性航天器动力学与振动控制研究进展.力学学报,2019,51(1):1-13(Cao Dengqing,Bai Kuncha,Ding Hu et al.Advances in dynamics and vibration control of large-scale flexible spacecraft.Chinese Journal of Theoretical and Applied Mechanics,2019,51(1):1-13(in Chinese))
5 Ibrahimbegovi′c A,Mamouri S,Taylor RL,et al.Finite element method in dynamics of flexible multibody systems:Modeling of holonomic constraints and energy conserving integration schemes.Multibody System Dynamics,2000,4(2-3):195-223
6和兴锁,顾致平,邓峰岩.大型空间柔性梁的有限元动力学建模方法研究.机械科学与技术,2007,26(9):1194-1197(He Xingsuo,Gu Zhiping,Deng Fengyan.Dynamics modeling of large spatial flexible beam with finite element method.Mechanical Science and Technology for Aerospace Engineering,2007,26(9):1194-1197(in Chinese))
7 Al-Bedoor BO,Almusallam AA.Dynamics of flexible-link and flexible-joint manipulator carrying a payload with rotary inertia.Mechanism&Machine Theory,2000,35(6):785-820
8刘锦阳,李彬,洪嘉振.作大范围空间运动柔性梁的刚-柔耦合动力学.力学学报,2006,38(2):276-282(Liu Jingyang,Li Bin,Hong Jiazhen.Rigid-flexible coupling dynamics of a flexible beam with three-dimensional large overall motion.Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):276-282(in Chinese))
9 Cai GP,Hong JZ,Yang SX.Dynamic analysis of a flexible hubbeam system with tip mass.Mechanics Research Communications,2005,32(2):173-190
10陈萌,马艳红,刘书国等.航空发动机整机有限元模型转子动力学分析.北京航空航天大学学报,2007,33(9):1013-1016(Chen Meng,Ma Yanhong,Liu Shuguo et al.Rotor dynamic analysis of whole aero-engine models based on finite element method.Journal of Beijing University of Aeronautics and Astronautics,2007,33(9):1013-1016(in Chinese))
11邱志成,姜衡.一种柔性铰链连接的柔性板结构振动模态分析.空间控制技术与应用,2011,37(5):1-7(Qiu Zhicheng,Jiang Heng.Vibration modal analysis of flexible plate structure connected by a flexible hinge.Aerospace Control and Application,2011,37(5):1-7(in Chinese))
12代颖,崔淑梅,宋立伟.车用电机的有限元模态分析.中国电机工程学报,2011,31(9):100-104(Dai Ying,Cui Shumei,Song Liwei.Finite element method modal analysis of driving motor for electric vehicle.Proceedings of the CSEE,2011,31(9):100-104(in Chinese))
13宾光富,曾求洪,王钢等.基于有限元模型的电机零部件与整机结构振动关联特性.电机与控制学报,2016,20(11):101-106(Bin Guangfu,Zeng Qiuhong,Wang Gang,et al.Structural vibration correlation feature analysis for parts and whole-machine of motor based on finite element model.Electric Machines and Control,2016,20(11):101-106(in Chinese))
14 Silva C.A reduced-order analytical model for the nonlinear dynamics of a class of flexible multi-beam structures.International Journal of Solids&Structures,1998,35(25):3299-3315
15 Hu Q,Ma G.Adaptive variable structure maneuvering control and vibration reduction of three-axis stabilized flexible spacecraft.European Journal of Control,2006,12(6):654-668
16 Hu Q.Input shaping and variable structure control for simultaneous precision positioning and vibration reduction of flexible spacecraft with saturation compensation.Journal of Sound Vibration,2008,318(1):18-35
17 Gennaro SD.Active vibration suppression in flexible spacecraft attitude tracking.Journal of Guidance Control&Dynamics,1998,21(3):400-408
18程磊,王天舒,李俊峰.挠性多体卫星姿态动力学与控制.清华大学学报(自然科学版),2005,45(11):1506-1509(Cheng Lei,Wang Tianshu,Li Junfeng.Attitude dynamics and control of a flexible multi-body satellite.Journal of Tsinghua University(Science and Technology),2005,45(11):1506-1509(in Chinese))
19 Pan KQ,Liu JY.Investigation on the choice of boundary conditions and shape functions for flexible multi-body system.Acta Mechanica Sinica,2012,28(1):180-189
20 Schwertassek R,Wallrapp O,Shabana AA.Flexible multibody simulation and choice of shape functions.Nonlinear Dynamics,1999,20(4):361-380
21 Hablani HB.Constrained and unconstrained modes-some modeling aspects of flexible spacecraft.Journal of Guidance Control Dynamics,1982,5(2):164-173
22 Wei J,Cao D,Liu L,et al.Global mode method for dynamic modeling of a flexible-link flexible-joint manipulator with tip mass.Applied Mathematical Modelling,2017,48:787-805
23 Gattulli V,Morandini M,Paolone A.A parametric analytical model for non-linear dynamics in cable-stayed beam.Earthquake Engineering&Structural Dynamics,2002,31(6):1281-1300
24 Gattulli V,Lepidi M.Nonlinear interactions in the planar dynamics of cable-stayed beam.International Journal of Solids&Structures,2003,40(18):4729-4748
25 Zhao Y,Kang H.In-plane free vibration analysis of cable-arch structure.Journal of Sound&Vibration,2008,312(3):363-379
26赵跃宇,杨相展,刘伟长等.索-梁组合结构中拉索的非线性响应.工程力学,2006,23(11):153-158(Zhao Yueyu,Yang Xiangzhan,Liu Weichang,et al.Nonlinear response of cables in cablestayed bean structure.Engineering Mechanics,2006,23(11):153-158(in Chinese))
27 Wang Z,Sun C,Zhao Y,et al.Modeling and nonlinear modal characteristics of the cable-stayed beam.European Journal of Mechanics-A/Solids,2014,47:58-69
28王志搴,赵珧冰,孙测世等.索-梁组合结构的建模及面内动力特性分析.固体力学学报,2014,35(3):308-312(Wang Zhiqian,Zhao Yaobing,Sun Ceshi et al.Modeling and in-plane dynamics analysis of the cable-stayed beam.Chinese Journal of Solid Mechanics,2014,35(3):308-312(in Chinese))
29 Hablani HB.Modal analysis of gyroscopic flexible spacecraft:a continuum approach.Journal of Guidance Control&Dynamics,1982,5(5):448-457
30 Hablani HB.Hinges-free and hinges-locked modes of a deformable multibody space station-a continuum analysis.Journal of Guidance Control Dynamics,1990,13(2):286-296
31 Hablani HB.Modal identities for multibody elastic spacecraft.Journal of Guidance Control&Dynamics,1991,14(14):294-303
32曲广吉.复合柔性结构航天器动力学建模研究.中国工程科学,1999,1(2):52-56(Qu Guangji.Dynamics modeling of spacecraft with composite flexible structures.Engineering Science,1999,1(2):52-56(in Chinese))
33缪炳祺,曲广吉,夏邃勤等.关于柔性航天器动力学模型降阶问题.中国工程科学,2001,3(11):60-64(Liao Bingqi,Qu Guangji,Xia Suiqin,et al.On the order reduction of dynamics models of flexible spacecraft.Engineering Science,2001,3(11):60-64(in Chinese))
34缪炳祺,曲广吉,夏邃勤.柔性航天器的模态综合-混合坐标动力学建模.浙江工业大学学报,2002,30(2):139-142(Miao Bingqi,Qu Guangji,Xia Suiqin.Modal synthesis-hybrid coordinate dynamics modeling of flexible spacecrafts.Journal of Zhejiang University of Technology,2002,30(2):139-142(in Chinese))
35 Song MT,Cao DQ,Zhu WD.Dynamic analysis of a micro-resonator driven by electrostatic combs.Communications in Nonlinear Science&Numerical Simulation,2011,16(8):3425-3442
36 Cao DQ,Song MT,Zhu WD,et al.Modeling and analysis of the inplane vibration of a complex cable-stayed bridge.Journal of Sound&Vibration,2012,331(26):5685-5714
37 Wei J,Cao D,Huang H,et al.Dynamics of a multi-beam structure connected with nonlinear joints:modelling and simulation.Archive of Applied Mechanics,2018(1):1-16
38 Wei J,Cao D,Wang L,et al.Dynamic modeling and simulation for flexible spacecraft with flexible jointed solar panels.International Journal of Mechanical Sciences,2017,130:558-570
39杨海峰,王晓军,邱志平.压电柔性结构振动的鲁棒控制.北京航空航天大学学报,2009,35(8):957-961(Yang Haifeng,Wang Xiaojun,Qiu Zhiping.Robust vibration control of flexible piezoelectric structure.Journal of Beijing University of Aeronautics and Astronautics,2009,35(8):957-961(in Chinese))
40吴爽,赵寿根,吴大方等.太阳翼铰链结构的动力学实验与非线性动力学建模.宇航学报,2013,34(12):1550-1556(Wu Shuang,Zhao Shougen,Wu Dafang,et al.Dynamic experiments and nonlinear dynamics modeling of joints in solar array.Journal of Astronautics,2013,34(12):1550-1556(in Chinese))
2胡璐,闫寒,张文明等.黏性流体环境下V型悬臂梁结构流固耦合振动特性研究.力学学报,2018,50(3):643-653(Hu Lu,Yan Han,Zhang Wenming et al.Analysis of flexural vibration of V-shaped beams immersed in viscous fluids.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):643-653(in Chinese))
3陶连金,刘春晓,边金等.大跨度Y形柱地铁车站结构地震反应研究.力学学报,2017,49(1):55-64(Tao Lianjin,Liu Chunxiao,Bian Jin et al.Seismic response of subway station with large span and Y-shaped column.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):55-64(in Chinese))
4曹登庆,白坤朝,丁虎等.大型柔性航天器动力学与振动控制研究进展.力学学报,2019,51(1):1-13(Cao Dengqing,Bai Kuncha,Ding Hu et al.Advances in dynamics and vibration control of large-scale flexible spacecraft.Chinese Journal of Theoretical and Applied Mechanics,2019,51(1):1-13(in Chinese))
5 Ibrahimbegovi′c A,Mamouri S,Taylor RL,et al.Finite element method in dynamics of flexible multibody systems:Modeling of holonomic constraints and energy conserving integration schemes.Multibody System Dynamics,2000,4(2-3):195-223
6和兴锁,顾致平,邓峰岩.大型空间柔性梁的有限元动力学建模方法研究.机械科学与技术,2007,26(9):1194-1197(He Xingsuo,Gu Zhiping,Deng Fengyan.Dynamics modeling of large spatial flexible beam with finite element method.Mechanical Science and Technology for Aerospace Engineering,2007,26(9):1194-1197(in Chinese))
7 Al-Bedoor BO,Almusallam AA.Dynamics of flexible-link and flexible-joint manipulator carrying a payload with rotary inertia.Mechanism&Machine Theory,2000,35(6):785-820
8刘锦阳,李彬,洪嘉振.作大范围空间运动柔性梁的刚-柔耦合动力学.力学学报,2006,38(2):276-282(Liu Jingyang,Li Bin,Hong Jiazhen.Rigid-flexible coupling dynamics of a flexible beam with three-dimensional large overall motion.Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):276-282(in Chinese))
9 Cai GP,Hong JZ,Yang SX.Dynamic analysis of a flexible hubbeam system with tip mass.Mechanics Research Communications,2005,32(2):173-190
10陈萌,马艳红,刘书国等.航空发动机整机有限元模型转子动力学分析.北京航空航天大学学报,2007,33(9):1013-1016(Chen Meng,Ma Yanhong,Liu Shuguo et al.Rotor dynamic analysis of whole aero-engine models based on finite element method.Journal of Beijing University of Aeronautics and Astronautics,2007,33(9):1013-1016(in Chinese))
11邱志成,姜衡.一种柔性铰链连接的柔性板结构振动模态分析.空间控制技术与应用,2011,37(5):1-7(Qiu Zhicheng,Jiang Heng.Vibration modal analysis of flexible plate structure connected by a flexible hinge.Aerospace Control and Application,2011,37(5):1-7(in Chinese))
12代颖,崔淑梅,宋立伟.车用电机的有限元模态分析.中国电机工程学报,2011,31(9):100-104(Dai Ying,Cui Shumei,Song Liwei.Finite element method modal analysis of driving motor for electric vehicle.Proceedings of the CSEE,2011,31(9):100-104(in Chinese))
13宾光富,曾求洪,王钢等.基于有限元模型的电机零部件与整机结构振动关联特性.电机与控制学报,2016,20(11):101-106(Bin Guangfu,Zeng Qiuhong,Wang Gang,et al.Structural vibration correlation feature analysis for parts and whole-machine of motor based on finite element model.Electric Machines and Control,2016,20(11):101-106(in Chinese))
14 Silva C.A reduced-order analytical model for the nonlinear dynamics of a class of flexible multi-beam structures.International Journal of Solids&Structures,1998,35(25):3299-3315
15 Hu Q,Ma G.Adaptive variable structure maneuvering control and vibration reduction of three-axis stabilized flexible spacecraft.European Journal of Control,2006,12(6):654-668
16 Hu Q.Input shaping and variable structure control for simultaneous precision positioning and vibration reduction of flexible spacecraft with saturation compensation.Journal of Sound Vibration,2008,318(1):18-35
17 Gennaro SD.Active vibration suppression in flexible spacecraft attitude tracking.Journal of Guidance Control&Dynamics,1998,21(3):400-408
18程磊,王天舒,李俊峰.挠性多体卫星姿态动力学与控制.清华大学学报(自然科学版),2005,45(11):1506-1509(Cheng Lei,Wang Tianshu,Li Junfeng.Attitude dynamics and control of a flexible multi-body satellite.Journal of Tsinghua University(Science and Technology),2005,45(11):1506-1509(in Chinese))
19 Pan KQ,Liu JY.Investigation on the choice of boundary conditions and shape functions for flexible multi-body system.Acta Mechanica Sinica,2012,28(1):180-189
20 Schwertassek R,Wallrapp O,Shabana AA.Flexible multibody simulation and choice of shape functions.Nonlinear Dynamics,1999,20(4):361-380
21 Hablani HB.Constrained and unconstrained modes-some modeling aspects of flexible spacecraft.Journal of Guidance Control Dynamics,1982,5(2):164-173
22 Wei J,Cao D,Liu L,et al.Global mode method for dynamic modeling of a flexible-link flexible-joint manipulator with tip mass.Applied Mathematical Modelling,2017,48:787-805
23 Gattulli V,Morandini M,Paolone A.A parametric analytical model for non-linear dynamics in cable-stayed beam.Earthquake Engineering&Structural Dynamics,2002,31(6):1281-1300
24 Gattulli V,Lepidi M.Nonlinear interactions in the planar dynamics of cable-stayed beam.International Journal of Solids&Structures,2003,40(18):4729-4748
25 Zhao Y,Kang H.In-plane free vibration analysis of cable-arch structure.Journal of Sound&Vibration,2008,312(3):363-379
26赵跃宇,杨相展,刘伟长等.索-梁组合结构中拉索的非线性响应.工程力学,2006,23(11):153-158(Zhao Yueyu,Yang Xiangzhan,Liu Weichang,et al.Nonlinear response of cables in cablestayed bean structure.Engineering Mechanics,2006,23(11):153-158(in Chinese))
27 Wang Z,Sun C,Zhao Y,et al.Modeling and nonlinear modal characteristics of the cable-stayed beam.European Journal of Mechanics-A/Solids,2014,47:58-69
28王志搴,赵珧冰,孙测世等.索-梁组合结构的建模及面内动力特性分析.固体力学学报,2014,35(3):308-312(Wang Zhiqian,Zhao Yaobing,Sun Ceshi et al.Modeling and in-plane dynamics analysis of the cable-stayed beam.Chinese Journal of Solid Mechanics,2014,35(3):308-312(in Chinese))
29 Hablani HB.Modal analysis of gyroscopic flexible spacecraft:a continuum approach.Journal of Guidance Control&Dynamics,1982,5(5):448-457
30 Hablani HB.Hinges-free and hinges-locked modes of a deformable multibody space station-a continuum analysis.Journal of Guidance Control Dynamics,1990,13(2):286-296
31 Hablani HB.Modal identities for multibody elastic spacecraft.Journal of Guidance Control&Dynamics,1991,14(14):294-303
32曲广吉.复合柔性结构航天器动力学建模研究.中国工程科学,1999,1(2):52-56(Qu Guangji.Dynamics modeling of spacecraft with composite flexible structures.Engineering Science,1999,1(2):52-56(in Chinese))
33缪炳祺,曲广吉,夏邃勤等.关于柔性航天器动力学模型降阶问题.中国工程科学,2001,3(11):60-64(Liao Bingqi,Qu Guangji,Xia Suiqin,et al.On the order reduction of dynamics models of flexible spacecraft.Engineering Science,2001,3(11):60-64(in Chinese))
34缪炳祺,曲广吉,夏邃勤.柔性航天器的模态综合-混合坐标动力学建模.浙江工业大学学报,2002,30(2):139-142(Miao Bingqi,Qu Guangji,Xia Suiqin.Modal synthesis-hybrid coordinate dynamics modeling of flexible spacecrafts.Journal of Zhejiang University of Technology,2002,30(2):139-142(in Chinese))
35 Song MT,Cao DQ,Zhu WD.Dynamic analysis of a micro-resonator driven by electrostatic combs.Communications in Nonlinear Science&Numerical Simulation,2011,16(8):3425-3442
36 Cao DQ,Song MT,Zhu WD,et al.Modeling and analysis of the inplane vibration of a complex cable-stayed bridge.Journal of Sound&Vibration,2012,331(26):5685-5714
37 Wei J,Cao D,Huang H,et al.Dynamics of a multi-beam structure connected with nonlinear joints:modelling and simulation.Archive of Applied Mechanics,2018(1):1-16
38 Wei J,Cao D,Wang L,et al.Dynamic modeling and simulation for flexible spacecraft with flexible jointed solar panels.International Journal of Mechanical Sciences,2017,130:558-570
39杨海峰,王晓军,邱志平.压电柔性结构振动的鲁棒控制.北京航空航天大学学报,2009,35(8):957-961(Yang Haifeng,Wang Xiaojun,Qiu Zhiping.Robust vibration control of flexible piezoelectric structure.Journal of Beijing University of Aeronautics and Astronautics,2009,35(8):957-961(in Chinese))
40吴爽,赵寿根,吴大方等.太阳翼铰链结构的动力学实验与非线性动力学建模.宇航学报,2013,34(12):1550-1556(Wu Shuang,Zhao Shougen,Wu Dafang,et al.Dynamic experiments and nonlinear dynamics modeling of joints in solar array.Journal of Astronautics,2013,34(12):1550-1556(in Chinese))