随机Bagley-Torvik方程的非平稳解析解

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关键词：随机振动；分数阶导数；Bagley-Torvik方程；完全非平稳；Mittag-Leffler函数 中图分类号：0324；0321 文献标志码：A DOI：10.16385/j.cnki.issn.1004-4523.202309028

Non-stationary analytic solution of the stochastic Bagley-Torvik equation

KONG Fan1，XU Yijian1，GUO Wenjie²，HONG Xu1，CAO Hongyou³ （1.College of Civil Engineering，Hefei University of Technology，Hefei 23ooo9，China; 2.School of Transportation Engineering，East China Jiaotong University，Nanchang 33Ool3，China; 3.School of Civil Engineering and Architecture，Wuhan Universityof Technology，Wuhan 43Oo7o，China）

Abstract：TheBagley-Torvik（B-T）equationisadiferentialequationofmotionwithfractional（3/2）orderderivativeermsthat is applied todescribethemotioofarigidplateinNewtonian，viscousfluid.Inthispaper，wedevelop nonstationaryanalytic solu tionsoftheB-Tequationwhoseinhomogeneous termisastochasticprocess.TheB-Tequationistransformedintoahalforder state-space equation inmatrix formandeigen-analysis is performedtoobtaincomplex eigenvaluesand eigenvectors.Subsequently， the generalized coordinate transformation is introduced to decouple the equation intoa system of independent 1/2 -orderdifferential equations whicharesolvedbyLaplace transformtoobtainthesolutioningeneralizedcoordinates；Thegeneralizedcoordinate solu tionisconvertedintoanaturalcoordinatesolutiontoobtaintheimpulseorstepresponsefunction.Whentheinhomogeneous term oftheequationisastochasticprocess，theLaplace transformcanbeused toderivethetime-varying frequencyresponsefunction fromwhichtheanalytical solutionofthe nonstationarystochasticresponsecanbeobtainedbyrelyingontherelationship between theexcitationandtheresponse powerspectraldensity.Thecorectnessofthemethodisverifiedbynumericalcasesusingthe Spanos-Solomos fully non-statoionary stochastic excitation as an example.

Keywords：randomvibration;fractionalderivative;Bagley-Torvik equation;fullynon-stationary;Mittag-Leflerfunction

Bagley-Torvik（以下简称为B-T）方程是一种带3/2分数阶导数项的微分方程，由BAGLEY和TORVIK提出[1]，用于描述刚性板在牛顿流体中的振动状态。（剩余13391字）