# Scattering on the Timoshenko Beam. Direct and Inverse Problems in the Time Domain

[Doktorsavhandling]

This thesis concerns the use of time domain methods for treating direct and inverse scattering problems for flexural waves on beams. The Timoshenko equation is used throughout to describe the behaviour of such waves. In the theoretical part of the work, the beam is considered as inhomogeneous as allowed by elementary beam theory, excluding side bending. Furthermore, the possibility of the beam to be resting on a viscoelastic foundation is allowed for. The damping influence of the foundation is characterised by constitutive relations involving the past history of beam rotation and deflection. Together, the suspension and the inhomogeneity of the beam constitute the region of scattering. The key to the direct and inverse problems lies in the scattering operators of this region. However, the concept of wave splitting is equally important. Basically this is a transformation of the equations governing the unrestrained and homogeneous beam, into equations that describe split wave fields propagating independently in definite directions. The wave splitting is necessary in order to decompose the flexural wave field at the boundary of incidence into its incoming and reflected parts.

Two different approaches to obtaining the scattering operators for the Timoshenko beam are used. First, there are the imbedding reflection and transmission operators that map an incident field, impinging on the boundary of the scattering region, to the field reflected at the same boundary, and the transmitted field at the back end of the region. This method essentially is a homotopy approach with the idea of imbedding the original problem in a family of related problems. The homotopy approach is used to imbed the scattering problem of the full region in a one-parameter family of sub-region scattering problems. Second, there is the wave propagator formalism which is more general in that the corresponding scattering operators, the wave propagators, map the incident field onto the split fields at any internal position of the scattering region. This formalism contains the imbedding method as a special case.

The scattering operators mentioned above have explicit representations in terms of integral kernels that satisfy matrix-valued integro-differential equations. These equations are derived and numerical methods for solving these equations are presented. Numerical solutions are obtained for two types of scatterers: a homogeneous beam on a viscoelastic foundation, modelled by exponential memory functions, and a non-uniform unrestrained beam. Moreover, the imbedding reflection equation is used to set up an explicit inverse algorithm in order to determine the variation of a non-uniform cross section from knowledge of reflection data. Examples of simulated noisy reconstructions are given for both circular and rectangular cross-sections.

**Nyckelord: **Timoshenko beam, viscoelastic damping, non-uniform beams, transient wave propagation, wave splitting, scattering operators, wave propagators, imbedding, inverse problems, integro-differential equations

Denna post skapades 2006-09-20. Senast ändrad 2013-09-25.

CPL Pubid: 934