Sensitivity analysis for non-selfadjoint problems

Abstract

The problems of stability for non-conservative systems are connected with the questions concerning stability of vibrations. These problems are described by non-selfadjoint operators or, alternatively, by non-symmetric matrices. The condition of stability depends on the parameters of the problem, i.e. on the design parameters, the boundary conditions, the load distribution, etc. Therefore it is important to obtain quantitative information about this dependence.

For these non-conservative problems it is shown, in general, how the different sensitivity analyses can be performed without introducing any new concepts of eigenvalue analysis. In the primary analysis as well as in the sensitivity analysis, the integrated treatment of the adjoint problem is of major importance because it admits a stationarity principle for these nonselfadjoint problems. Stated in another way, it means that if the variations are related to the "mutual energies", then the variations of eigenvector and adjoint eigenvector are not implicitly involved. One of the main questions asked in this paper relates to the change in the flutter load as a function of the change in stiffness, mass, boundary conditions or in the load distribution.

The extended "Beck column" is treated in a non-discretized analysis. Then we concentrate on the viscoelastic vibrating columns, which have to be treated by a discretized model. The results of our analysis, which clearly show the mutual effects of external and internal damping, are presented. Finally the sensitivities are computed for these rather complicated structural models.