Keywords: moving force, dynamic programming, Tikhonov regularisation, L-curve, eigenvalue.

This paper describes a method to identify moving forces as they traverse a two-span continuous bridge. Applications include axle forces from traffic loading where the forces are varying due to the dynamic motion of the vehicle. The method is considerably more accurate at finding the mass corresponding to the oscillating force than existing Bridge Weigh-in-Motion algorithms and provides additional information about the time-varying nature of the force.

Moving forces are simulated using a finite element model of the bridge; the strains at four discrete points are used as measurements for the inverse problem. The inverse problem itself consists of three main parts. First, the equilibrium equation of motion is converted into a discrete time integration using the exponential matrix. Next, the inverse problem itself is formulated as a least squares minimisation problem with Tikhonov regularisation (smoothing) and then solved with dynamic programming. The final part of the inverse analysis is the choice of regularisation parameter - the optimal regularisation parameter is calculated using Hanson's L-curve [1].

A discrete time integration scheme is used to simulate moving forces on a bridge, the bridge being discretized into 160 Hermite beam elements. The equilibrium equation of motion is first converted into a vector matrix differential equation. Then, using the exponential matrix and a Pade approximation to the forcing function, the vector matrix differential equation is converted into a discrete time integration scheme. A formulation of this type provides a highly accurate solution to the equation of motion [2]. Simulated strains, contaminated with 3% Gaussian noise, are used as the measured data for the inverse problem.

For inverse problems in structural dynamics, the system matrices used for the dynamic simulation are too large for the inverse problem itself. In order to overcome this problem an eigenvalue reduction is employed to reduce the dimension of the system matrices. Furthermore, the boundary condition at the centre of the bridge results in a discontinuity in the inverse analysis. To counter this problem, the inner support is replaced with a very stiff spring. Once the original finite element model is modified, a modal analysis is preformed to calculate the eigenvalues and the eigenvectors of the system. The equilibrium equation of motion is then converted into a discrete time integration scheme using a finite number of modal co-ordinates.

The inverse problem is formulated as a least squares minimisation with Tikhonov regularisation, the regularisation term being added to provide bounds to the least squares formulation which is normally ill-conditioned. The recursive least squares problems are solved using dynamic programming and Bellman's principle of optimality [3]. The final part of the problem is the calculation of the regularisation parameter - an excessively small value leads to ill-conditioning while an excessively large value leads to "over-smoothing" and loss of accuracy. The optimal regularisation parameter is calculated using Hansen's L-curve method. The L-curve is a plot of the norm of the least squares error versus the norm of the solution. For every regularisation parameter these two norms are plotted and the corner of the L-curve is the optimal solution.

Numerical experiments were carried out for two forces moving at constant velocity with an axle spacing of 5m. Three different combinations of forcing functions were used to represent vehicle dynamics. It is shown that this approach and provides a good estimate of the variation of the forces with time.

1

Hansen, P.C., "Analysis of discrete ill-posed problems by means of the L-curve", SIAM Review., 34(4), 561-580, 1992. doi:10.1137/1034115

2

Trujillo D.M., "The direct numerical integration for linear matrix differential equations using Pade approximations", International Journal for Numerical Methods in Engineering., 9, 259-270, 1975. doi:10.1002/nme.1620090202

3

Bellman, R., Kalaba, R., "Introduction to mathematical theory of control process Volume II", New York: Academic Press, 1965.

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