On exact and approximated formulations for scaling-mode shapes in operational modal analysis by mass and stiffness change

Abstract

When operational modal analysis (OMA) is used to estimate modal parameters, mode shapes cannot be mass normalized. In the past few years, some equations have been proposed to scale mode shapes using the mass-change method, which consists of repeating modal testing after changing the mass at different points of the structure where the mode shapes are known. In this paper, the structural-dynamic-modification theory is used to derive a set of equations, from which all the existing formulations can be derived. It is shown that the known equations can be divided into two types, the exact and the approximated equations, where the former type does in fact fulfill the equations derived from the theory of structural modification, whereas the remaining equations do not, mainly because the change of the mode shapes of the modified structure is not properly taken into account. By simulations, the paper illustrates the large difference in accuracy between the approximate and the exact formulations. The paper provides two new exact formulations for the scaling factors, one for the non-modified structure and – for the first time in the literature – one for the modified structure. The simulations indicate the influence of errors from the measured natural frequencies and mode shapes on the estimation of the scaling factors using the two exact formulations from the literature and the new exact formulation proposed in this paper. In addition, the paper illustrates statistics of the errors on mode-shape scaling. All simulations were carried out using a plate with closely spaced modes.