an updating method for damped structural dynamic models 1

AN UPDATING METHOD FOR DAMPED STRUCTURAL DYNAMIC MODELS A.Chouaki, P.Ladevkze, L.Proslier Labor&o& de M&anique et Technologie (E.N.S. de Cxhan / Universite’ Paris 6 / C.N.R.S.) 61 Avenue du Pr&ident Wilson 94235 CACHAN CEDEX

ABSTRACT : An important problem in mechanics is cited as examples. In this work, the mars and stiffness the control ofcomplexstructural models. Although there matrices are assumed to be updated in a preliminary are several methods to improve structural dynamic mod- stage. The error on the constitutive relation developed els, only a few of them deal with the damping improve- can deal with either the damping or the non-linearities ments. The problem studied herein concern8 the damping due to materials and contact. It can also utilize multiple updating using experimental Frequency Response Func- static loads and vibration tests. tions. The method introduced is based on an error mea- The major idea involved in the error on the constitutive 8ure on the constitutive relation. The tuning strategy is relation is to verify exactly the accurate equations and based on localisation-correction stages. The first one is experimental data assumed as being accurate. The less the localisation of the erroneous regions. The second step accurate equations and experimental results are weighted and are “fairly” verified. is the correction of the parameter8 belonging to these re gions. Since the updating problem is an ill-posed problem, the Several examplesillustrate the effectiveness of the method proposed approach is an iterative process. Each iteration to update the damping and to identify it in structural consists of two steps. joints. The first one is the localisation of the erroneous regions. These ones could be mis-mcdelled joints in structural assemblies. This step is performed using local indicator8 built on the error on the constitutive relation, very dif1 INTRODUCTION ferent from the classical sensitivity indicators of the op. The control of complex structural models is a growing timisation tools. The second step is the correction of the preoccupation especially in the spatial field. The prob- few parameter8 belonging to these regions. lem studied herein concerns the damping updating, using After recalling the notion of the error on the constitutive experimental Frequency Response Functions. relation, several examples illustrate the effectiveness of The major idea is to define a quality measure of a model the method to update the damping and to identify it in with rape& to experimental results. Many propositions the connection8 between structural components. The exhave been made. The most significant are referenced in amples dealt with are assumed linear, and the experimen111, 121 and 131. The proposed method ha.8 a clear me- tal data used are noisy incomplete Frequency Response cha&al foundation; it is-based on an error measure on Functions. the constitutive relation. The first investigations on the error on the constitutive relation have focused on the mass and stiffness updating using identified real modes [4]. The approach has been developed and applied in [5] and [6]. In a collaboration with the C.N.E.S [7], a software M.A.T has been 2 THE ERROR ON THE CONdeveloped and applied in relation with MSC/NASTRAN. More information about these initial investigation8 can be STITUTIVE RELATION found in [l] and [El. The study introduced herein concerns the damping up dating. Although there are several methods to improve Before introducing the error on the constitutive relation, structural dynamic models, only a few of them deal with we will describe the reference problem for small perturdamping improvements. [9] [lo, 111 and [12, 131 can be bations.

558

2.1 Reference problem Let’s consider a structure described by a domain a, during the time interval [0, T]. On the boundary an of the structure, the displacements (Id and the forces & are described on &R and &R respectively. Body forces L, are given in 0. Then, the reference problem during [0, T] can be describe as follows : find the displacement u(AJ,1), the stress a@, 1) and the density IJM, t), t E [0, T], M E fl such that they satisfy : . the boundary equations and the initial conditions

(14 (lb) UC) (14

The corresponding space is denoted $i*]. Hence, the reference problem can be written as : Find s E SriT1 satisfying the constitutive relations :

mlt= d(i(U) Ir; r 5 t)

(4)

a2u+_a(QI,;r_