计入基础运动的环状周期结构振动特性分析
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摘要
研究了一类工程领域广泛应用的环状周期结构的弹性振动特性,重点分析了基础运动对弹性振动稳定性和固有频率分裂的影响.首先在随动坐标系下采用Hamilton原理建立了计入基础运动和面内切向及径向弹性振动的偏微分形式的动力学模型.然后,应用伽辽金方法将其离散得到一组常微分动力学方程.根据经典振动理论,得到了系统特征值的数学表达.最后采用数值方法计算了系统的特征值.根据特征值的实虚部取值预测了不稳定域和固有频率分裂规律,并用Runge-Kutta法给出稳定性的数值验证.该研究为陀螺仪等呈现平面或空间基础运动的环状周期结构的动态性能的改善提供了理论借鉴.
The elastic vibration characteristic of a ring-shaped periodic structure intensively used in engineering practice was examined,where the focus was on the effect of basic movement on elastic vibration stability and natural frequency splitting.Firstly,a partial differential dynamic model incorporating basic movement and in-plane tangential as well as radial elastic vibrations was established in moving frame using Hamilton's principle.Then,a set of ordinary differential dynamic equations were formulated using Galerkin method.The mathematical expressions of eigenvalues of the system were derived according to the classical vibration theory.Finally,the eigenvalues were calculated by use of numerical method.Unstable areas and the rules of natural frequency splitting were predicted by means of real and imaginary parts of eigenvalues.The stability was verified by numerical calculation with RungeKutta method.This research provides theoretical reference for the dynamic performance improvement of microgyroscope or other ring-shaped periodic structures undergoing two-or three-dimensional basic movement.
The elastic vibration characteristic of a ring-shaped periodic structure intensively used in engineering practice was examined,where the focus was on the effect of basic movement on elastic vibration stability and natural frequency splitting.Firstly,a partial differential dynamic model incorporating basic movement and in-plane tangential as well as radial elastic vibrations was established in moving frame using Hamilton's principle.Then,a set of ordinary differential dynamic equations were formulated using Galerkin method.The mathematical expressions of eigenvalues of the system were derived according to the classical vibration theory.Finally,the eigenvalues were calculated by use of numerical method.Unstable areas and the rules of natural frequency splitting were predicted by means of real and imaginary parts of eigenvalues.The stability was verified by numerical calculation with RungeKutta method.This research provides theoretical reference for the dynamic performance improvement of microgyroscope or other ring-shaped periodic structures undergoing two-or three-dimensional basic movement.
引文
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[10]Yoon S W,Lee S W,Najafi K.Vibration sensitivity analysis of MEMS vibratory ring gyroscopes[J].Sensors and Actuators:Physical,2011,171:163-177.
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[12]Xi X,Wu Y L,Wu X M,et al.Investigation on standing wave vibration of the imperfect resonant shell for cylindrical gyro[J].Sensors and Actuators A:Physical,2012,179:70-77.
[13]胡海岩.应用非线性动力学[M].北京:航空工业出版社,2000.Hu Haiyan.Applied Nonlinear Dynamics[M].Beijing:Aviation Industry Press,2000(in Chinese).
[14]Wu X H,Parker R G.Vibration of rings on a general elastic foundation[J].Journal of Sound and Vibration,2006,295(1/2):194-213.
[15]Wang S Y,Sun W J,Wang Y Y.Instantaneous mode contamination and parametric combination instability of spinning cyclic symmetric ring structures with expanding application to planetary gear ring[J].Journal of Sound and Vibration,2016,375:366-385.
[16]Xia Y,Wang S Y,Sun W J,et al.Analytical estimation on divergence and flutter vibrations of symmetrical three-phase induction stator via field-synchronous coordinates[J].Journal of Sound and Vibration,2016,386:407-420.
[17]Huang S C,Soedel W.Effects of Coriolis acceleration on the free and forced in-plane vibration of rotating rings on elastic foundation[J].Journal of Sound and Vibration,1987,115(2):253-274.
[2]Canchi S V,Parker R G.Effect of ring-planet mesh phasing and contact ratio on the parametric instabilities of a planetary gear ring[J].ASME Journal of Mechanical Design,2008,130:014501-1-014501-6.
[3]Canchi S V,Parker R G.Parametric instability of a circular ring subjected to moving springs[J].Journal of Sound and Vibration,2006,293(1/2):360-379.
[4]Canchi S V,Parker R G.Parametric instability of a rotating circular ring with moving,time-varying springs[J].ASME Journal of Vibration and Acoustics,2006,128(1/2):231-243.
[5]Zhang D S,Wang S Y,Liu J P.Analytical prediction for free response of rotationally ring-shaped periodic structures[J].ASME Journal of Vibration and Acoustics,2014,136:041016-1-041016-12.
[6]Wang Y Y,Wang S Y,Zhu D H.Dual-mode frequency splitting elimination of ring periodic structures via feature shifting[J].Journal of Mechanical Engineering Science,2016,230(18):3347-3357.
[7]徐进友,刘建平,王世宇,等.环状旋转周期结构模态摄动分析[J].天津大学学报,2010,43(11):1015-1019.Xu Jinyou,Liu Jianping,Wang Shiyu,et al.Modal perturbation analysis for annular rotationally periodic structures[J].Journal of Tianjin University,2010,43(11):1015-1019(in Chinese).
[8]Zhang D S,Wang S Y.Spatial-dependent resonance mode and frequency of rotationally periodic structures subjected to standing wave excitation[J].Mechanical Systems and Signal Processing,2017,94:482-498.
[9]Esmaeili M,Durali M,Jalili N.Ring microgyroscope modeling and performance evaluation[J].Journal of Vibration and Control,2006,12(5):537-553.
[10]Yoon S W,Lee S W,Najafi K.Vibration sensitivity analysis of MEMS vibratory ring gyroscopes[J].Sensors and Actuators:Physical,2011,171:163-177.
[11]张薇薇,胡业发.磁力轴承-转子-基础系统的耦合动力学分析[J].机械制造,2009,47(534):4-6.Zhang Weiwei,Hu Yefa.Coupling dynamics analysis of magnetic bearing-rotor-foundation system[J].Machine Manufacturing,2009,47(534):4-6(in Chinese).
[12]Xi X,Wu Y L,Wu X M,et al.Investigation on standing wave vibration of the imperfect resonant shell for cylindrical gyro[J].Sensors and Actuators A:Physical,2012,179:70-77.
[13]胡海岩.应用非线性动力学[M].北京:航空工业出版社,2000.Hu Haiyan.Applied Nonlinear Dynamics[M].Beijing:Aviation Industry Press,2000(in Chinese).
[14]Wu X H,Parker R G.Vibration of rings on a general elastic foundation[J].Journal of Sound and Vibration,2006,295(1/2):194-213.
[15]Wang S Y,Sun W J,Wang Y Y.Instantaneous mode contamination and parametric combination instability of spinning cyclic symmetric ring structures with expanding application to planetary gear ring[J].Journal of Sound and Vibration,2016,375:366-385.
[16]Xia Y,Wang S Y,Sun W J,et al.Analytical estimation on divergence and flutter vibrations of symmetrical three-phase induction stator via field-synchronous coordinates[J].Journal of Sound and Vibration,2016,386:407-420.
[17]Huang S C,Soedel W.Effects of Coriolis acceleration on the free and forced in-plane vibration of rotating rings on elastic foundation[J].Journal of Sound and Vibration,1987,115(2):253-274.