华侨大学陈振川全球求助翻译英语论文_(8).pdf

57 The Fifth International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY Huazhong University of Science and Technology Wuhan P R CHINA 27 28 August 2009 2009 Huazhong Universiti of Science and Technology Press A STOCHASTIC MULTI STEP TRANSVERSAL LINEARIZATION METHOD MTL IN ENGINEERING DYNAMICS M K Dasha D Royb M Moharanaa a Department of Civil Engineering Indira Gandhi Institute of Technology Sarang Dhenkanal Orissa 759146 India b Structural Engineering Division Indian Institute of Science Bangalore India E mail manoj orin yahoo co uk I INTRODUCTION An implicit linearization procedure referred to as a multi step transversal linearization MTL is proposed for strong numerical solutions of non linear SDE s acted upon by white noise excitations In this method the process of linearization of the stochastic vector field is done in such a way that the linearized solution manifold may have repeated transversal intersections with the targeted solution manifold of the non linear oscillator The linearization of the non linear part of the vector field is performed conditionally via a multi step Taylor like interpolation so as to be consistent with the typical characteristics of the white noise excitation for instance Oh Wiener increments over a time step of h etc Such an interpolating expansion of the non linear part of the operator over a set of discretization points results in a conditionally linearized and integrable set of SDE s whose exact solution may be explicitly constructed in terms of the discretized unknown state variables It may be mentioned here that the suggested functional expansion converts non linear part of the vector field into a linearized one with an explicit dependence on time and so the linearized vector field may be considered as a modified conditional forcing function However an expansion of the non linear terms needs the discretized values of the state variables at the grid interpolation discretization points Since these discretized values are not known a priori the expansion of the non linear vector field is only conditional i e it is conditioned on the anticipated possible knowledge of the discretized state variables at the grid points Such an anticipatory expansion of the non linear part of the vector field may include as many grid points forward along the time axis as desired and is so performed that the linearized and non linear vector fields remains instantaneously identical in form at these points of discretization Finally based on the condition of transversal intersections of the linearized and non linear solution manifolds at the points of discretization a set of coupled non linear algebraic equations in terms of the discretized state variables is established A limited numerical verification of the MTL procedure is provided for a few stochastically excited low dimensional non linear oscillators II THE METHODOLOGY Consider an n DOF stochastically driven oscillator of the following general form Communicating author 58 11 qq lenrrrr rr XA X Xf tA X X tt W tB X X t W t 1a or in an incremental form the above SDE s are written for its pth component as 12 21212 12 1 dd d d d pp pppp lne q pp rrr r XXt XAXXAXXtftt tBXXtW t 1b l A X X is the linear time invariant part of the drift vector field which admits a representation of the form l A X XKXDX where K and D are respectively the constant stiffness and damping matrices The deterministic external forces possibly time dependent are contained in the vector e f t The vector n A X X t represents non linear as well as parametrically excited parts of the drift vector field The field of diffusion vectors comprises of r t and 1 r B X X trq representing the additive and multiplicative parts of the field respectively The integer q denotes the number of independently evolving Wiener processes 1 r W t rq For convenience of discussion the 2n dimensional vector function X t would be used to denote 12 1 1 pp X tXpnXpn As with any other numerical technique the part of the time axis over which the SDE s are to be integrated needs to be discretized and ordered as 12011 NP II IIt tt 121 1 1 11 PPPNPNPNP tttttt where the integer N denotes the number of times the linearization procedure has to be applied so that an approximate solution over the entire interval I may be constructed and P denotes the number of grid discretization points including the point at which the initial conditions are defined covered by each application of the multi step linearization procedure To make the presentation of the method simpler without losing its generality 3P is chosen for discussion to follow In order to further motivate the development of the stochastic MTL method the next step is a functional representation of the response X tbased at 00 X ttt through an Ito Taylor expansion Towards this the same Lipschitz boundedness and differentiability criteria on the drift b velocity histories 123 0 25 1 0 0 2 45 1 0 0 0 0246810 6 4 2 0 2 4 6 MTL SHS Velocity Time sec 0246810 0 8 0 6 0 4 0 2 0 0 0 2 0 4 0 6 0 8 MTL SHS Displacement Time sec 62 a displacement histories b velocity histories c c Phase plot via MTL Fig 4 A Duffing oscillator under strong additive noise and a weak deterministic excitation 123 0 25 1 0 0 2 45 15 0 0 0 References 1 Dash M K and Roy D 2004 A Multiply Trasversal Linearization MTL Method and Application to Nonlinear and Stochastic Structural Dynamics Advances in Vibration Engineering 3 2 123 131 2 Gard T C 1988 Introduction to Stochastic Differential Equations Marcel Dekker Inc 3 Gikhman I I and Skorokhod A V 1972 Stochastic differential equations Springer Berlin 4 Kloeden P E and Platen E 1992 Higher order implicit strong numerical schemes for stochastic differential equations Jl Statistical Phy Vol 66 pp 283 314 5 Kloeden P E and Platen E 1999 Numerical solution of stochastic differential equations Springer Berlin 6 Milstein G N 1974 Approximate integrations of stochastic differential equations Theory Probab Appl Vol 19 pp 557 562 7 Milstein G