13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- formula; the response of the structure was reconstructed by using the identified forces for comparison; and the objective function between the identified and measured values were minimized. The dynamic programming technique possesses inherent limitations that cannot be avoided, however, it still effectively solves these problems during the identification process, and greatly reduces the influence from insufficient known qualities and improper boundary conditions, and obtain decent results comparable with the exact forces. The mathematical model is then applied to estimate the wheel-rail vertical load of a high-speed train, and the inversion results are compared with the rolling and vibrating test-bed and the very detailed SIMPACK model simulation results. 2. Basis of load identification theory The finite element model of an n-DOFs linear elastic time-invariant structure, the dynamic governing equation is given by: ( ) ( ) ( ) ( ) 0 MXt CXt KXt Ft (1) where M,C, andKare the system mass, damping, and stiffness matrices, respectively; ( ) X t is the displacement vectors of the structure; and Fis the vector of the input excitation forces. Using the state space formulation, Eq. (1) is converted into a set of first order differential equations as follows: x Ax Bf (2) For the load identification problem, the known responses of the system M,C, andK are used to solve the unknown input vector ( ) f s which is in discrete form. In order to facilitate the computer solution, these differential equations are then rewritten as discrete equations using the standard exponential matrix representation. 1 i i i x Cx Df (3) i i y Qx (4) where, Ah C e is the exponential matrix, and together with matrix 1( ) D A C I is the input influence matrix which represents the dynamics of the system and associates with load. Qis a 2 m n selection matrix related the measurements to the state variables. 1 ix denotes the values at the ( 1) i th time step of the computations. The goal is to find the unknown forcing term f that will cause the system described in Eq.(3) to best match the measurements ˆiy . The mathematical representation of a best match is to minimize the least squares error between ˆiy and iy . This is expressed in matrix-vector notation with the inner product of two vectors (.,.). The least error squares are now expressed as: 1 2 1 ˆ ˆ ( ) ( ) ( ) ( ) N T T i i i i i i i E y y y y f f (5) whereT is the transpose of a matrix, iy and ˆiy are the output variables of the system for the identification formula and measurement, respectively. 1and 2are symmetric positive definite matrices that provide the flexibility of weighting the measurement and the forcing terms. The second term is known as the regularization parameter and the method is called the Tikhonov method. The value of 2is very important for the result, fortunately, there exists a method that can be used to estimate the optimum value of 2, see the reference [13].
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