Time Domain Methods for Load Identification of Linear and Nonlinear Systems
Doktorsavhandling, 2004

The automotive industry faces an increasing demand for virtual prototyping to simulate the operation conditions of vehicles and subsystems. Each component, as well as the complete vehicle, must meet the desired design requirements in a time- and cost-efficient development process. It is essential to obtain representative dynamic loads acting on these systems, since they will affect the quality of subsequent analysis. However, loads acting at interfaces between structural parts and their surroundings are frequently difficult to measure directly. This thesis investigates numerical methods of identifying loads acting at locations inaccessible for direct measurement. Time domain methods for load identification of linear and nonlinear mechanical systems are presented. The work focuses on situations in which the load locations are known a priori and the corresponding magnitudes are unknown. This particular class of load identification problem is solved using least squares formulations, since measurements of a finite number of local operational response quantities are considered to be at hand. For nonlinear systems, an iterative simulation error approach is proposed to solve the nonlinear least squares problem. The method is evaluated numerically by identifying different road profiles upon which a full-scale nonlinear automotive vehicle is travelling. For linear systems, several methods based on linear least squares formulations are surveyed. The emphasis is on formulations which take the form of structured block matrix problems. It is shown how an originally ill-posed problem can be reformulated to a well-posed problem using a sensor-specific time delay approach. But, even if the ill-posedness of the associated upper block triangular Toeplitz matrix is removed, the matrix may still be ill-conditioned. These difficulties are overcome by introducing numerical regularization strategies. A criterion for choosing the level of regularization is proposed. Moreover, two specific block algorithms utilizing the structure of the Toeplitz matrix are also derived.

inverse problem


upper block triangular Toeplitz matrix


wavelet parameterization

least squares problem

input estimation

time domain

linear and nonlinear systems

load identification


Patrik Nordberg

Chalmers, Tillämpad mekanik





Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 2170