Receptance Based Normalization of Operational Mode Shapes D. Bernal Northeastern University, Civil and Environmental Engineering Department, Center for Digital Signal Processing, Boston, MA 02192. ABSTRACT Normalization of modes obtained from operational modal analysis has received significant attention since the seminal paper on the use of mass perturbations by Parloo et.al., in 2002. The present paper presents a formulation where the square of the scaling constants are obtained from an over-determined linear system of equations. The formulation is exact for arbitrary changes in the modal model when the modal basis is complete, as well as for all conditions where existing uncoupled solutions are exact, independently of truncation. The approach is designated as the Receptance Based Normalization (RBN) scheme because the coefficients in the system of equations are obtained from evaluations of the modal series expression for the Receptance matrix. INTRODUCTION The fundamental contribution in resolving the scaling issue for operational modes is credited to Parloo et. al., [1], who showed that the information required is encoded in the derivatives of the eigenvalues with respect to known perturbations. An issue of some concern with the sensitivity-based normalization resides in the fact that the finite difference estimate of the eigenvalue derivatives, which have to be obtained from changes that are sufficiently large to offset variability, may be inaccurate due to nonlinearity in the eigenvalue vs mass perturbation relationships. The potential for significant error from this source has long been recognized and has prompted development of variants of the perturbation approach wherein the scaling constants depend on total eigenvalue changes, instead of the derivatives [2,3.4]. This paper presents a normalization approach that does not call for specific distributions of the perturbation to attain accuracy and applies in the case of complex eigenvectors. The formulation, designated as the Receptance Based Mode Normalization (RBN) method, computes the (square) of the scaling constants from a complex valued over-determined system of equations obtained by evaluating the pole residue form of the Receptance at the eigenvalues of the perturbed condition. A version of RBN restricted to the case of real modes appears in [5]. THE RECEPTANCE APPROACH TO EIGENVECTOR NORMALIZATION Let 'M be the mass perturbation andO I j j and be the pole and arbitrarily scaled (complex) eigenvector in the mass perturbed condition for the jth mode. In the derivation that follows it is convenient to treat the eigenvectors as if they are available at all the coordinates, although it will be apparent at the end that only measured coordinates are needed. From the polynomial eigenvalue problem, focusing on the perturbed condition, one has 2 ( ) 0 ª º ¬ ' O O ¼I j j j M M C K (1) from where it follows that 1 2 2 j j j j j M C K M ª º ¬O O 'OI I ¼ (2) T. Proulx (ed.), Linking Models and Experiments, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 5, 393 DOI 10.1007/978-1-4419-9305-2_29, © The Society for Experimental Mechanics, Inc. 2011
RkJQdWJsaXNoZXIy MTMzNzEzMQ==