Differential and difference sensitivities of natural frequencies and mode shapes of mechanical structures

Abstract

A method, based upon the theory of the adjoint structures, is formulated for calculating the derivatives of natural frequencies and normalized mode shapes with respect to structural parameter changes in terms of local mass, stiffness, or damping, starting with data obtained by experimental processing techniques such as modal analysis. The method applies for statically or kinematically undeterminate structures, which is not the case for most classical methods of sensitivity analysis. The method is extended to obtain large-change sensitivities and frequency response sensitivities to structural nonparameter changes (e.g., the addition of a damped vibration absorber). Two examples demonstrate the procedure and the usefulness of the sensitivity analysis. HE dynamic of complex mechanical equipment and the prediction of the dynamic behavior of a modified mechanical structure has turned out to be a difficult way of reaching the objectives of such analysis. However, modal techniques and computer-interfaced testing equip- ment have contributed to a solution of those problems. Furthermore, modal results (complex modal displacements, natural frequencies, damping values) may be used in system synthesis methods1>2 to predict mathematically the effect of structural changes on the dynamic behavior. The objective of this paper is to present a * 'sensitivity analysis using experimental data, obtained by modal analysis, for computational assessment of the most effective parameter change in order to obtain a desired dynamic behavior. The sensitivities give the influence of the building element parameters on the natural frequencies and mode shapes of the mechanical structures. They provide us with an answer to the question of where to change, e.g., to obtain a maximum shift of a specific natural frequency or to reduce most effectively the modal displacements in certain points for a specific mode. The assumption of linear and statically or kinematically determinate structures simplifies the calculation of the sen- sitivities.35 Van Belle6'7 developed a more general method, called the theory of adjoint structures, for the sensitivity of mechanical structures, yielding equations still valuable for statically or kinematically undeterminate structures. In this paper, the method based upon the theory of adjoint structures is extended to obtain sensitivities in the case of viscously damped systems and expressions for the sensitivities are derived, using finite instead of infinitesimal changes.