# Wave Propagation in Structural Elements - Direct and Inverse Problems in the Time Domain Doctoral thesis, 1998

The subject of this thesis is direct and inverse wave propagation problems in structural elements. The considered structures are beams, plates and rods. The models for the beams and the plates are based on linear theory, while the rods are assumed to be nonlinear. The greater part of the work concerns the Timoshenko beam. The analysis is performed in the time domain using wave splitting. Wave splitting in conjunction with scattering operator techniques is used for the linear problems. For the nonlinear problems, a finite difference scheme is adopted. The basis for the solution of the scattering problems considered in this thesis is the wave splitting concept. Wave splitting is a change of the dependent variables that diagonalizes the dynamic equations for a nonvarying medium. The diagonalization implies that the transformed variables propagate independently in definite directions. In a varying medium, these fields couple as the waves scatters. Information on the scattered fields is the main subject for both the direct and the inverse problems. The scattering operators that are used for the linear problems are the imbedding technique, the Green function approach and the propagator formalism. The two former are special cases of the latter. In general, the different operators map the incident field to the scattered fields in the varying region. The operators are represented by integral kernels that are convolved with the incident field. These kernels are independent of the wave fields, and depend only on the properties of the scatterer. The integral kernels satisfy integro-differential equations. The key to the direct and the inverse problems is to derive and solve these equations. This is done numerically, using the method of characteristics. The direct wave propagation problems on Timoshenko beams are studied using Green functions for a free, homogeneous beam, and imbedding techniques as well as propagator formalism for a viscoelastically restrained, inhomogeneous beam. The inverse problem is addressed for a homogeneous beam on a viscoelastic layer, using the imbedding technique. Both direct and inverse problems on a Mindlin plate with varying thickness in one direction are presented. Green functions are used for the direct problem, while the imbedding approach is used for the inverse problem. Concerning the nonlinear rod, both direct and inverse problems are studied for various types of viscoelastic, inhomogeneous rods. The inverse algorithm is based on a minimization of a functional, using iterative procedures.

integro-differential equations

non-uniform beams

wave splitting

wave propagator

finite difference

Timoshenko beam

imbedding

Mindlin plate

inverse problem

viscoelastic damping

nonlinear rod

transient wave propagation

Green function

## Author

#### Peter Folkow

Department of Theoretical and Applied Mechanics

#### Subject Categories

Mechanical Engineering

#### ISBN

91-7197-700-7

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1427