精细时程积分法及其数值衍生格式应用评估
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摘要
旋翼气动弹性耦合动力学方程本质上是一组刚性比较大的非线性偏微分方程。在有限元结构离散后,可改写为非齐次微分方程组,其中非齐次项是桨叶运动量(位移与速度)和气动载荷的函数。针对这类方程,本文尝试引入精细积分法及其衍生格式,借助数值方法计算Duhamel积分项。从积分精度与数值稳定性方面比较研究具有代表性的精细库塔法和高精度直接积分法。结合隐式积分算法,评估精细积分法应用于旋翼动力学方程的可行性。算例表明,精细积分法对矩形直桨叶动力学方程具有足够的求解精度。
Helicopter rotor aeroelasticity is essentially described by a set of stiff and nonlinear partial differential equations.They can be rewritten as non-homogeneous ordinary differential equations after discretion by finite element method.The non-homogeneous terms depend on time response and aerodynamic loads of the blades.This paper introduces a precise time integration method(PTI) and its derived formats to solve this kind of equations.The Duhamel integration term in the derived formats can be calculated using this numerical method.It also selects and compares the precise-Kutta method and high precision direct scheme(HPD) on integration precision and numerical stability.At last,an implicit integration method is used to comprehensively evaluate PTI on rotor dynamics.Numerical examples indicate that HPD scheme is precise evough to be used for rectangular blades.
Helicopter rotor aeroelasticity is essentially described by a set of stiff and nonlinear partial differential equations.They can be rewritten as non-homogeneous ordinary differential equations after discretion by finite element method.The non-homogeneous terms depend on time response and aerodynamic loads of the blades.This paper introduces a precise time integration method(PTI) and its derived formats to solve this kind of equations.The Duhamel integration term in the derived formats can be calculated using this numerical method.It also selects and compares the precise-Kutta method and high precision direct scheme(HPD) on integration precision and numerical stability.At last,an implicit integration method is used to comprehensively evaluate PTI on rotor dynamics.Numerical examples indicate that HPD scheme is precise evough to be used for rectangular blades.
引文
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[9] 富明慧,刘祚秋,林敬华.一种广义精细积分法[J].力学学报,2007,39(5):672-677.(FU Ming-hui,LIU Zuo -qiu,LIN Jing-hua.A generialized precise time step integration method[J].Chinese Journal of Theoretical and Applied Mechanics,2007,39(5):672-677.(in Chinese))
[10] 汪梦甫,周锡元.结构动力方程的高斯精细时程积分法[J].工程力学,2004,21(4):13-15.(WANG Meng-fu,ZHOU Xi-yuan.Gauss precise time -integration of structural dynamic analysis[J].Engineering Mecha-nics,2004,21(4):13-15.(in Chinese))
[11] 富明慧,梁华力.一种改进的精细-龙格库塔法[J].中山大学学报(自然科学版),2009,48(5):1-5.(FU Ming-hui,LIANG Hua-li.An improved precise Runge-Kutta integration[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2009,48(5):1-5.(in Chinese))
[12] Li K,Darby A P.A high precision direct integration scheme for nonlinear dynamic systems[J].Journal of Computational and Nonlinear Dynamics,2009,4(4):041008.
[13] Liu X M,Zhou G,Zhu S,et al.A modified highly precise direct integration method for a class of linear time -varying systems[J].Science China:Physics,Mechanics & Astronomy,2014,57(7):1382-1389.
[14] 吴杰.直升机旋翼气动弹性振动载荷研究[D].南京航空航天大学,2013.(WU Jie.Research on Helicopter Rotor Aeroelastic Vibratory Loads[D].Nanjing University of Aeronautics and Astronautics,2013.(in Chinese))
[15] Janin O,Lamarque C H.Comparison of several numerical methods for mechanical systems with impacts[J].International Journal for Numerical Methods in Engineering,2001,51(9):1101-1132.
[16] 钟万勰,杨再石.连续时间LQ控制主要本征对的算法[J].应用数学与力学,1991,12(1):45-50.(ZHONG Wan-xie,YANG Zai-shi.On the computation of the main eigen-pairs of the continuous-time linear quadratic control problem[J].Applied Mathematics and Mechanics,1991,12(1):45-50.(in Chinese))
[17] Leishman J G.Principles of Helicopter Aerodyna-mics (Second Edition)[M].Cambridge:Cambridge University Press,2007.
[18] Bathe K J,Wilson E L.Numerical Methods in Finite Element Analysis[M].Prentice-Hall,Englewood Cliffs,New Jersey,1976.
[19] Celaya E A ,Anza J J.BDF-α:A multistep method with numerical damping control[J].Univeral Journal of Computational Mathema-tics,2013,1(3):96-108.
[20] Ascher U M,Petzold L R.Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations[M].SIAM,Philadelphia,Pennsylvania,1998.
[21] Xie Y M.An assessment of time integration schemes for non-linear dynamic equations[J].Journal of Sound and Vibration,1996,192(1):321-331.
[22] Bathe K J,Noh G.Insight into an implicit time inte -gration scheme for structural dynamics[J].Computers & Structures,2012,98:1-6.
[2] Wang H W,Gao Z.Rotor vibratory load prediction based on generalized forces[J].Chinese Journal of Aeronautics,2004,17(1):28-33.
[3] 韩景龙,陈全龙,员海玮.直升机的气动弹性问题[J].航空学报,2015,36(4):1034-1055 (HAN Jing-long,CHEN Quan-long,YUAN Hai-wei.Aeroelasticity of helicopters[J].Acta Aeronautica et Astronautica Sinica,2015,36(4):1034-1055.(in Chinese))
[4] 钟万勰.结构动力方程的精细时程积分法[J].大连理工大学学报,1994,34(2):131-136.(ZHONG Wan-xie.On precise time-integration method for structural dyna-mics[J].Journal of Dalian University of Technology,1994,34(2):131-136.(in Chinese))
[5] 岳聪,任兴民,杨永锋,等.变速转子瞬时不平衡响应的精细算法[J].航空学报,2014,35(11):3046-3053.(YUE Cong,REN Xing-min,YANG Yong-feng,et al.A precise integration method on transient unbalance response of varying velocity rotor[J].Acta Aeronautica et Astronautica Sinica,2014,35(11):3046-3053.(in Chinese))
[6] 张瑞杰,李青宁,尹俊红.精细积分法应用于地震碰撞力反应谱计算研究[J].计算力学学报,2015,32(6):733-738.(ZAHNG Rui-jie,LI Qing-ning,YIN Jun-hong.Study of earthquake pounding force response spectrum computing using precise integration method[J].Chinese Journal of Computational Mechanics,2015,32(6):733-738.(in Chinese))
[7] 谭述君,吴志刚,钟万勰.矩阵指数精细积分方法中参数的自适应选择[J].力学学报,2009,41(6):961-966.(TAN Shu-jun,WU Zhi-gang,ZHONG Wan-xie.Adaptive selection of parameters for precise computation of matrix exponential based on Padé approximation[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(6):961-966.(in Chinese))
[8] 谭述君,钟万勰.非齐次动力方程Duhamel项的精细积分[J].力学学报,2007,39(3):374-381.(TAN Shu-jun,ZHONG Wan-xie.Precise integration me -thod for Duhamel terms arising from non-homogenous dynamic systems[J].Chinese Journal of Theoretical and Applied Mechanics,2007,39(3):374-381.(in Chinese))
[9] 富明慧,刘祚秋,林敬华.一种广义精细积分法[J].力学学报,2007,39(5):672-677.(FU Ming-hui,LIU Zuo -qiu,LIN Jing-hua.A generialized precise time step integration method[J].Chinese Journal of Theoretical and Applied Mechanics,2007,39(5):672-677.(in Chinese))
[10] 汪梦甫,周锡元.结构动力方程的高斯精细时程积分法[J].工程力学,2004,21(4):13-15.(WANG Meng-fu,ZHOU Xi-yuan.Gauss precise time -integration of structural dynamic analysis[J].Engineering Mecha-nics,2004,21(4):13-15.(in Chinese))
[11] 富明慧,梁华力.一种改进的精细-龙格库塔法[J].中山大学学报(自然科学版),2009,48(5):1-5.(FU Ming-hui,LIANG Hua-li.An improved precise Runge-Kutta integration[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2009,48(5):1-5.(in Chinese))
[12] Li K,Darby A P.A high precision direct integration scheme for nonlinear dynamic systems[J].Journal of Computational and Nonlinear Dynamics,2009,4(4):041008.
[13] Liu X M,Zhou G,Zhu S,et al.A modified highly precise direct integration method for a class of linear time -varying systems[J].Science China:Physics,Mechanics & Astronomy,2014,57(7):1382-1389.
[14] 吴杰.直升机旋翼气动弹性振动载荷研究[D].南京航空航天大学,2013.(WU Jie.Research on Helicopter Rotor Aeroelastic Vibratory Loads[D].Nanjing University of Aeronautics and Astronautics,2013.(in Chinese))
[15] Janin O,Lamarque C H.Comparison of several numerical methods for mechanical systems with impacts[J].International Journal for Numerical Methods in Engineering,2001,51(9):1101-1132.
[16] 钟万勰,杨再石.连续时间LQ控制主要本征对的算法[J].应用数学与力学,1991,12(1):45-50.(ZHONG Wan-xie,YANG Zai-shi.On the computation of the main eigen-pairs of the continuous-time linear quadratic control problem[J].Applied Mathematics and Mechanics,1991,12(1):45-50.(in Chinese))
[17] Leishman J G.Principles of Helicopter Aerodyna-mics (Second Edition)[M].Cambridge:Cambridge University Press,2007.
[18] Bathe K J,Wilson E L.Numerical Methods in Finite Element Analysis[M].Prentice-Hall,Englewood Cliffs,New Jersey,1976.
[19] Celaya E A ,Anza J J.BDF-α:A multistep method with numerical damping control[J].Univeral Journal of Computational Mathema-tics,2013,1(3):96-108.
[20] Ascher U M,Petzold L R.Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations[M].SIAM,Philadelphia,Pennsylvania,1998.
[21] Xie Y M.An assessment of time integration schemes for non-linear dynamic equations[J].Journal of Sound and Vibration,1996,192(1):321-331.
[22] Bathe K J,Noh G.Insight into an implicit time inte -gration scheme for structural dynamics[J].Computers & Structures,2012,98:1-6.
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