Approximation methods for inverse problems involving the vibration of beams with tip bodies

In this short paper we briefly outline two cubic spline based approximation schemes for the solution of inverse problems involving the vibration of flexible beams with attached tip bodies. The identification problem is formulated as the least squares fit to data of a hybrid system of coupled partial and ordinary differential equations describing the dynamics of the beam and tip bodies. The resulting optimization problem is infinite dimensional and as such, necessitates the use of some form of approximation. The schemes we have developed are based upon the construction of a sequence of approximating identification problems in which the underlying constraining state equations are semi-discrete finite dimensional approximations to the infinite dimensional distributed system which governs the original identification problem. Our study includes both theoretical convergence results and numerical testing.