机械振动数值分析的重心插值配点法
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摘要
重心Lagrange插值具有数值稳定性好、计算精度高的优点。
本文首先分析了重心Lagrange插值的基本性质,给出了采用重心Lagrange插值逼近任意连续函数的数值算法。
采用重心Lagrange插值近似未知函数,推导了未知函数各阶导数微分矩阵的显式表达式,建立了求解二阶常微分方程初值问题的重心Lagrange插值配点法。提出了一种新的初值条件施加方法,将初始导数条件离散后的代数方程,附加到控制微分方程离散代数方程组,采用最小二乘法求解代数方程。数值算例表明,本文所提出的初值条件施加方法的计算精度高于传统的初值条件置换方法。
利用重心Lagrange插值配点法分析了周期激励和一般激励下线性振动问题。
数值计算得到位移值后,采用微分矩阵可以方便地得到速度和加速度值。重心Lagrange插值配点法不但对位移具有极高的计算精度,而且对于速度和加速度也就有极高的计算精度。
采用重心Lagrange插值配点法分析非线性振动问题。
采用重心Lagrange插值配点法离散非线性振动方程及其初始条件,得到一组非线性代数方程,利用Newton法求解非线性方程,得到非线性振动的位移。同时,利用微分矩阵和重心Lagrange插值可以方便地求出非线性振动的速度、加速度和振动周期。
线性和非线性振动问题的数值算例表明,重心Lagrange插值配点法具有公式简单、数值实施方便和计算精度高的优点。
Barycentric Lagrange interpolation has merits of good numerical stability and higher computational accuracy. In this thesis, the basic properties of barycentric Lagrange interpolation are analysis firstly. The numerical algorithm of approximating continuous functions by barycentric Lagrange interpolation is given.
Using barycentric Lagrange interpolation to approximate unknown function, the explicit formulations of element of differentiation matrix are constructed. The barycentric Lagrange interpolation collocation method (BLICM) for solving initial value problems of differential equation is introduced. Put the discrete algebraic equations of initial derivative conditions and algebraic equation system of governing differential equation together to form a new algebraic system. The least-square method is adopted to solve them. The numerical examples indicate that the computational accuracy of treatment of initial conditions proposed in this thesis is higher than traditional replace method.
The linear vibration problems under periodic exciting force and general exciting force are numerical analysis by barycentric Lagrange interpolation collocation method. After obtaining the displacement values of vibrating, the velocities and accelerations are computing directly by differentiation matrices. Not only the displacement but also velocity and acceleration have higher accurate in barycentric Lagrange interpolation collocation method.
In the procedure of barycentric Lagrange interpolation collocation method to analyze the nonlinear vibration problems, the nonlinear vibration governing equation and initial conditions are translated into a set of nonlinear algebraic equations. The nonlinear algebraic equations system is solved using Newton method to obtain the displacement of vibration. At the same time the velocity, acceleration and period of vibrating can be calculated by differentiation matrix and barycentric Lagrange interpolation, respectively.
The numerical examples of linear and nonlinear vibration demonstrate that barycentric Lagrange interpolation collocation method have advantages of simple formulations, easier to program and higher computational accuracy.
本文首先分析了重心Lagrange插值的基本性质,给出了采用重心Lagrange插值逼近任意连续函数的数值算法。
采用重心Lagrange插值近似未知函数,推导了未知函数各阶导数微分矩阵的显式表达式,建立了求解二阶常微分方程初值问题的重心Lagrange插值配点法。提出了一种新的初值条件施加方法,将初始导数条件离散后的代数方程,附加到控制微分方程离散代数方程组,采用最小二乘法求解代数方程。数值算例表明,本文所提出的初值条件施加方法的计算精度高于传统的初值条件置换方法。
利用重心Lagrange插值配点法分析了周期激励和一般激励下线性振动问题。
数值计算得到位移值后,采用微分矩阵可以方便地得到速度和加速度值。重心Lagrange插值配点法不但对位移具有极高的计算精度,而且对于速度和加速度也就有极高的计算精度。
采用重心Lagrange插值配点法分析非线性振动问题。
采用重心Lagrange插值配点法离散非线性振动方程及其初始条件,得到一组非线性代数方程,利用Newton法求解非线性方程,得到非线性振动的位移。同时,利用微分矩阵和重心Lagrange插值可以方便地求出非线性振动的速度、加速度和振动周期。
线性和非线性振动问题的数值算例表明,重心Lagrange插值配点法具有公式简单、数值实施方便和计算精度高的优点。
Barycentric Lagrange interpolation has merits of good numerical stability and higher computational accuracy. In this thesis, the basic properties of barycentric Lagrange interpolation are analysis firstly. The numerical algorithm of approximating continuous functions by barycentric Lagrange interpolation is given.
Using barycentric Lagrange interpolation to approximate unknown function, the explicit formulations of element of differentiation matrix are constructed. The barycentric Lagrange interpolation collocation method (BLICM) for solving initial value problems of differential equation is introduced. Put the discrete algebraic equations of initial derivative conditions and algebraic equation system of governing differential equation together to form a new algebraic system. The least-square method is adopted to solve them. The numerical examples indicate that the computational accuracy of treatment of initial conditions proposed in this thesis is higher than traditional replace method.
The linear vibration problems under periodic exciting force and general exciting force are numerical analysis by barycentric Lagrange interpolation collocation method. After obtaining the displacement values of vibrating, the velocities and accelerations are computing directly by differentiation matrices. Not only the displacement but also velocity and acceleration have higher accurate in barycentric Lagrange interpolation collocation method.
In the procedure of barycentric Lagrange interpolation collocation method to analyze the nonlinear vibration problems, the nonlinear vibration governing equation and initial conditions are translated into a set of nonlinear algebraic equations. The nonlinear algebraic equations system is solved using Newton method to obtain the displacement of vibration. At the same time the velocity, acceleration and period of vibrating can be calculated by differentiation matrix and barycentric Lagrange interpolation, respectively.
The numerical examples of linear and nonlinear vibration demonstrate that barycentric Lagrange interpolation collocation method have advantages of simple formulations, easier to program and higher computational accuracy.
引文
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[2]Hannah J,Stephens R C.Advanced Mechanics of Machines[M].London:Edward Arnold(Publishers)Ltd,1963.
[3]Weideman,J A C,Reddy,S C.A Matlab differentiation matrix suite[J].ACM Transactions on Mathematical Software,2000,26(4):465-519.
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[5]王鑫伟.微分求积法在结构力学中的应用[J].力学进展,1995,25(2):232-240.
[6]Bert C W,Malik M.Differential quadrature method in computational mechanics:a review[J].Applied Mechanics Reviews,1996,49:1-27.
[7]林成森.数值分析[M].北京:科学出版社,2006.1.
[8]John P.Boyd.Chebyshev and Fourier Spectral Methods[J].New York:Dover Publications,Inc.,2000.
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[10]Raymond A Adomaitis,Yi-hung Lin.A collocation/quadrature-based sturm-liouville problem solver[J].Applied Mathematics and Computation,2000,110(2-3):205-223.
[11]白凤兰,尹丽,邹永魁.近似求解Cahn-Hilliard方程的拟谱方法[J].吉林大学学报(理学版).2003,41(3):262-268.
[12]J-P Berrut and H D Mittelmann.The linear rational pseudospectral method with iteratively optimized poles for two-point boundary-value problems[J].SIAM Journal of Scientific Computation,2001,23:961-975.
[13]Daniele Funaro,Wilhelm Heinrichs.Some results about the pseudospectral approximation of one-dimensional fourth-order problems[J].Numer.Math.,1990,58:399-418.
[14]Tobin A.Driscoll,Bengt Fornberg.A block pseudospectral method for maxwell's equations Ⅰ.One-dimensional case[J].Journal of Computational Physics,1998,140:47-65.
[15]麻剑锋,沈新荣,章本照,陈华军.极坐标系下泊松方程的拟谱方法[J].空气动力学学报,2006,24(2):243-245.
[16]王健.梁振动问题的多辛Fourier拟谱方法[J].上海交通大学学报,2004,38(4):658-663.
[17]Baltensperger R,Berrut J-P.The errors in calculating the pseudospectral differentiation matrices for Chebyshev-Gauss_Lobatto points[J].Comp.Math.Applic.,1999,37:41-48.
[18]Bayliss A,Class A,Matkowsky B J.Roundoff error in computing derivatives using the Chebyshev differentiation matrix[J].J.Comp.Phys.,1994,116:380-383.
[19]Bert C W,Malik M.Differential quadrature method in computational mechanics:a review[J].Applied Mechanic Review,1996,49(1):1-28.
[20]李萍,陈殿云.微分求积法与有限元相结合分析带弹性安置质量圆板的横向振动[J].工程力学,2006,23(5):52-55.
[21]刘洋,杨永波,梁枢平.轴向均布载荷下压杆稳定问题的DQ解[J].力学与实践,2005,27:44-47.
[22]梁枢平,刘洋,杨永波,郑文衡.梁的轴力非线性静力问题的进一步研究[J].华中科技大学学报(自然科学版),2005,33(2):116-118.
[23]倪樵,黄玉盈,陈贻平.微分求积法分析具有弹性支承输液管的临界流速[J].计算力学学报,2001,18(2):146-149.
[24]陈贻平,兰箭,董湘怀,李志刚.正交异性变厚度圆薄板微分求积分析[J].华中科技大学学报,1999,27(8):52-53,62
[25]聂国隽,仲政.用微分求积法求解梁的弹塑性问题[J].工程力学,2005,22(1):59-62,27.
[26]聂国隽,仲政.变截面门式刚架结构分析的微分求积单元法[J].力学季刊,2005,26(2):198-203.
[27]聂国隽,仲政.框架结构P-Δ效应分析的微分求积单元法[J].力学季刊,2004,25(2):195-200.
[28]Shu C,Wang C M.Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates[J].Engineering Structures,1999,21:125-134.
[29]王鑫伟,何柏庆.调和微分求积法权系数矩阵的一种显式计算式[J].南京航空航天大学学报,1995,27(4):496-501.
[30]王琳,倪樵.用微分求积法分析输液管道的非线性的动力学行为[J].动力学与控制学报,2004,2(4):56-61.
[31]周凯红,王元勋,李春植.微分求积法在单摆非线性振动分析中的应用[J].力学与实践,2003,25:50-52.
[32]Faruk C,Sliepcevich C M.Application of Differential Quadrature to Solution of Pool Boiling in Cavities[J].Proceedings of the Oklahoma Academy of Science,1985,65:73-78.
[33]Moinuddin Malik,Faruk Civan.A comparative study of differential quadrature and cubature methods vis-a-vis some conventional techniques in context of convection-diffusion-reaction problems[J].Chemical Engineering Science,1995,50(3):531-547.
[34]Shu C.Application of differential quadrature method to simulate natural convection in a concentric annulus[J].Int.J.Numer.Meth.Fluids,1999,30:977-993.
[35]Tanaka M,Chen W.Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems[J].Applied Mathematical Modelling,2001,25:257-268.
[36]O'Mahoney D C.A differential quadrature solution of the two-dimensional inverse heat conduction problem[J].International Communications in Heat and Mass Transfer,2003,30(8):1061-1070.
[37]Jules Kouatchou.Comparison of time and spatial collocation methods for the heat equation[J].Journal of Computational and Applied Mathematics,2003,150:129-141.
[38]Berrut J-P,Baltensperger R,Mittelmann H D.Recent developments in barycentric rational interpolation,Trends and Applications in Constructive Approximation[A].In De Bruin M G,D Mache H,Szabados J,eds.,International Series of Numerical Mathematics[C],Birkhauser:Verlag Basel,2005,151,27-51.
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