On the properness condition for modal analysis of non-symmetric second-order systems

Abstract

To cite this version:Morvan Ouisse, Emmanuel Foltete. On the properness condition for modal analysis of non symmetric second order systems. Mechanical Systems and Signal Processing, Elsevier, 2011, 25 (2) Emmanuel FOLTÊTE AbstractNon symmetric second order systems can be found in several engineering contexts, including vibroacoustics, rotordynamics, or active control. In this paper, the notion of properness for complex modes is extended to the case of non self-adjoint problems. The properness condition is related to the ability of a set of complex modes to represent in an exact way the behavior of a physical second order system, meaning that the modes are the solutions of a quadratic eigenvalue problem whose matrices are those of a physical system. This property can be used to identify the damping matrices which may be difficult to obtain with mathematical modeling techniques. The first part of the paper demonstrates the properness condition for non symmetric systems in general. In the second part, the authors propose a methodology to enforce that condition in order to perform an optimal reconstruction of the "closest" physical system starting from a given basis complex modes. The last part is dedicated to numerical and experimental illustrations of the proposed methodology. A simulated academic test case is first used to investigate the numerical aspects of the method. A physical application is then considered in the context of rotordynamics. Finally, an experimental test case is presented using a structure with an active control feedback. An extension of the LSCF identification technique is also introduced to identify both left and right complex mode shapes from measured frequency response functions.

Model correlation and identification of experimental reduced models in vibroacoustical modal analysis

“…The approach consists in correcting the identified complex eigenvectors in order that they verify the required constraints, before applying the inverse procedure. In practice, this generally leads to small phase correction on the vectors and almost no change in amplitudes, which is coherent with experimental considerations [31].…”

“…In the field of experimental modal analysis, an experimental reduced model is built from complex modes, which are identified using FRFs (frequency response functions) with techniques adapted from those classically used for symmetric problems [31]. Here, for practical reasons and without loss of generality, one will assume that the eigenshapes are normalized such as ξ j ¼ 1; 8 j ¼ 1…2n.…”

“…Following the approach described in [31], it is possible to determine an experimental reduced model from the knowledge of N modes measured on N degrees of freedom. The general procedure is briefly recalled here.…”

“…The fact is that in general, the modes do not verify the constraints equations (30), which results in a bad conditioning of the inverse procedure. It has been shown [30,31] that imposing the properness condition on complex modes performs an indirect regularization of the problem. The approach consists in correcting the identified complex eigenvectors in order that they verify the required constraints, before applying the inverse procedure.…”

“…Balmès [30] has proposed a methodology to find optimal complex vectors which are as close as possible to the initial identified vectors, while verifying the so-called properness condition. This technique has been extended recently to non-self-adjoint problems [31]. However it is not applicable for vibroacoustics because of the relationship between cross terms as it will be detailed in this paper.…”