Application of the variable projection method for updating models of mechanical systems

To update the parameters of a mathematical model of a mechanical system usu ally a cost function is minimised which consists of the difference between calculated and measured quantities. This paper deals with the special case where the forces are unknown. Instead of following the usual way of handlin g this type of updating problem by assuming a model for the forces, in this paper the variable projection method is applied to estimate the unknown fo rces in addition to the model parameters. Under certain conditions this two -fold inverse problem can be solved by eliminating the force from the param eter estimation process. The remaining equation to estimate the model param eters consists of the projection of the response data, where the associated projector depends on the model parameters. This application of the variabl e projection method is essentially an extension of the output residual meth od and leads to an estimation equation which is non-linear with respect to the model parameters. The variable projection method is introduced and inve stigated for two general types of unknown forces. Two theoretical examples, wind excitation of a tower and a rotary machine under unknown unbalance co nfiguration, and the experimental case of a free-free steel beam tested by hammer excitation, are presented and discussed. (C) 1999 Academic Press.