ABSTRACT: In this study, an updating method based on the concept of the error on the constitutive relation is presented. This concept has already been ...
A POSTERIORI CONSTITUTIVE RELATION ERROR FOR DYNAMIC MODEL UPDATING A.Chouaki, P.Ladeveze Laboratoire de l'r!ecanique et Technologie (E.N.S. de Cachan I Universite Paris 6 I C.N.R.S.) 61 A venue du President Wilson 9423.5 CACHAN CEDEX
ABSTRACT: In this study, an updating method based on the concept of the error on the constitutive relation is presented. This concept has already been exploited in the field of a posterion error estimation to qualify the quality of a Finite Element computation. This paper illustrates how the a posteriori error aspect of the method is used to update dynamic models. It enhances the capabilities of detecting the mis-modelled regions or the erroneous substructures. Examples dealing with lightly damped structures and structural joints, using noisy frequency response functions are presented.
1
INTRODUCTION
Nowadays, in order to predict the dynamic behaviour of structures, models are becoming more accurate and more sophisticated. Consequently, the validation of these models using experimental information remains, a crucial stage. Therefore, the methods for updating structural dynamic models are undergoing increased attention, and intensive development. Two main categories of methods can be distinguished. The first one consists of the "direct" methods. In this category, we can find methods that minimise a correcting matrix norm ( "Frobenius" norm) such as [1]. Also found are methods very close in nature to the Control Theory, such as the "Eigenstructure Assignment Techniques" ([2]) and the "Minimum Rank Perturbation Methods" ([3]). The second category includes the parametric methods (or "indirect" methods), in which the models are corrected by acting on the structural parameters. Typically, these methods are based either on the input equation such as in [4] or on the output equation such as in [5]. These methods are commonly called "Sensitivity Methods" because they make use of the sensitivity terms of the eigenfrequencies, of the eigenmodes or of cost functions, with respect to the design parameters. Generally, the selection of the parameters to update is based on the dynamic residuals of the equilibrium equation.
Lastly, another family of methods exists wherein the approach we are developing can be positioned. It is an extension to the work conducted in the field of "a posteriori" error estimators, in an effort to quantify the quality of a finite element computation (see [6]). The method we are developing herein is based on the concept of the error on the constitutive relation. After initial investigations [7], this method has been theoretically extended to a wide range of problems in [8]. It can incorporate either the damping or the non-linearities due to both materials and contact. Moreover, it is able to assimilate different types of experimental information (responses to static loads, or modal and forced vibrations). The major characteristic of our method, with respect to those cited previously, is that the experimental measurements are not the only reference in quantifying the dynamic model's quality. First, the "reliable" equations (such as the equilibrium equation) are distinguished from the "less reliable" ones (such as the constitutive relations). Therefore, the equilibrium equation enters into what is called the reference. Concerning the experimental data, the more accurate information is also to be distinguished from the less accurate information; for example, the frequencies and the sensor's locations are assumed to be more accurate data. A global error is defined, taking into account both the modelling knowledge and all the experimental information. Then, the regions that contributes significantly to the error are the most erroneous ones. The updating process is iterative. Each iteration consists of two steps: the first one is the localization of the most erroneous regions. This step is performed using local indicators built on the error on the constitutive relation, which are very different from the classical sensitivity indicators of optimization tools. The second step involves the. correction of the few parameters belonging to these regwns. In this paper, the concept of the error on the constitutive relation is recalled succinctly. More attention is given to define an a posteriori error which allows us to control
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the quality of the updated model with respect to a set of measurements. It is achieved by introducing a scaling stage and a weighting stage. Examples concentrating on lightly-damped structures are presented. These examples display damping, in both a diffuse form and a localized form (such as in the joints between substructures). The models dealt with are linear; the experimental data are frequency response functions, contaminated with noise and collected at a few locations. Finally, some hints are made for future applications of the concept of the a posteriori constitutive relation error estimator.
2
THE ERROR ON THE CONSTITUTIVE RELATION
In this paper, only the principles of the method are recalled. For more details, see [9). Let us consider a structure given by a domain fl during the time interval [0 T]. On the boundary of the structure, the displacements Ud and the forces F d are described on 8 1 n and 8 2 0, respectively. Body forces [_d are given in fl. The reference problem during [0, T) can then be written as follows: find the displacement U(M, t), the stress ar(M, t) and the density [(M, t), t E [0, T), ME fl such that they satisfy:
an
I
• the boundary equations and the initial conditions • the equilibrium equation • the constitutive relations \:It E [0, T], \:1M E fl,
arit= A(i(U) \,.; T s; t) ()2[!
1 is a parameter belonging to [0, 1] that depends on the reliability of the relations ( 1a) and ( 1b) of the analytical model. Its current value is 0.5. In [11], it is shown that the triplet s(U,ar,I.), solution of the reference problem, is also given by the problem below:
Find s E
3
ar = 1\it + Bd:
1
1'" )nr
.
{(1- 1HL- L) o(~- u.) +1 tr[(arc- ars)(it(U c)- d:.)]}dtdfl
(2)
()2U
.
= p--= ot 2 +aU-
(4)
~I
lw +~ l 2
17~ (s)
=
(p + Ta)(Uc- rL,)(*)
o
(Uc- U.) dfl
tr[(I\ + Tw 2 B)(~t( U c) -Its)(*) (IE(U c)- Its)] dfl (5)
All the quantities involved in (5) are complex amplitudes. For the sake of simplicity, they are denoted as the associated quantities. (*) designates the conjugate quantities. T is a time that depends on the structure being dealt with. Then, the reference problem can be rewritten as follows:
I 3.1
,.~t o
-r
and
These models allow treating the cases of proportional, non-proportional, viscous and hysteretic damping. In the relation (4), K is the Hooke operator, a and B are real operators, linear, symmetric and positive definite to satisfy the Drucker stability conditions. Then, we assume that the data of the problem are harmonic [9]. Therefore, the error on the constitutive relation becomes:
~t({L)
1J;(s) =sup
3
( )
Let's now write the error on the constitutive relation in the frequency domain. First, the constitutive relations of the problem are rewritten as follows:
(lb)
Let's now define the quantity:
S~0/l
MODEL IMPROVEMENTS
(la)
= [grad!lJsym, pis the density, assumed here to be constant with respect to time. The constitutive relations are assumed to satisfy the Drucker stability conditions [10]. These conditions ensure the uniqueness aspect of the reference problem, and they are satisfied by a large class of materials. To define the error on the constitutive relation, we introduce the triplets, displacement-stress-force (U, ar, I), inn X [0, T]. Such a triplet involves two kinds of quantities: "statical" quantities (I, ar), denoted (r 5 , ar 8 ), and a "kinematical" quantity U, denoted~· Using the constitutive relations and the initial conditions, the triplets can be related to (arc, fc) and (~t.,rL,) such that:
with s' E
s admissible (s E S~~T]) means that the triplet satisfies exactly the reliable equations of the problem: the boundary equations and the initial conditions and the equilibrium equation. Then, we seek to satisfy as much as possible the less reliable equations: the constitutive relations.
.
I It= P at 2 + !:!(u \,.; T s; t)
s[O,T]
I minimizinga~;(s')
Find Sw E Sad w minimising 1J~(s') where s' E
( )
Sad,w
6
Comparing the model with experimental data
Until now, we have known that the solution to the reference problem (related to the assumed model) is also given by the minimization of the error on the constitutive relation in (6). Here, the problem is to define a quality measure for comparing the model with the experimental information. Hence, experimental results must be judiciously incorporated, into the error on the constitutive relation, in order to overcome two limitations.
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The first one is the incompleteness of the experimental measurements. To overcome this limitation, several updating methods rely on regularization techniques (see: [12]). Others authors recommended two ways to proceed. The first one is to enlarge the experimental knowledge by introducing intentional modifications to the structure being tested ([13]). It is also important to be able to exploit data generated from either static loads or vibration tests, under different boundary conditions. The second one concerns the M.A.M. (Multiple Analytical Model) concept introduced by Ibrahim in [14]. Given that conducting multiple tests can be very time consuming and expensive, it is recommended to make use of multiple analytical models. All of the propositions cited above can be applied to the error on the constitutive relation. Another limitation is the unavoidable additive measurement noise. Thus, it becomes necessary to distinguish the more accurate information from less accurate information. The experimental results consist of: the angular frequency w, the force};, dw and the measured displacements ~ denoted: II~. II is a projection operator that details the measured part of the displacements. C) characterises the measured quantities. Thus, a global modified error on the constitutive relation can now be defined which enables us to compare the analytical model with the experimental results; for a given frequency w, this error can be written, in the case of one excitation, as follows: ')
r
~,
e: (s)
= TJ:(s) + 1 _
r
II TilL,-
-
II~
'>
11-
3.2.1
The aim of this stage is to scale the error on the constitutive relation (7) in such a way as to maintain the same levels of error throughout the frequency range [wmin, Wmax]. First, let us recall that the error on the constitutive relation is a function which has an energetic interpretation. Given that the shapes and levels of the peak resonances are mainly affected by the damping properties of the structure, the error on the constitutive relation can be thought of as a dissipation error in the vicinities of the resonance frequencies. Therefore, it seems natural to scale it with respect to the dissipated energy (in one cycle) at these frequencies. The dissipated energy is a quantity that in practice is easy to measure at a given frequency w. In order to illustrate the procedure to apply, consider a linear structure under testing. Let f(w)eiwt and TI~eiwt be the force excitation and the measured displacements, respectively. Let ih, (w )eiwt be the measured displacements at the force locations. f(w), II~ and :ih,(w) are complex quantities; [wmin, Wmax] is the frequency bandwidth of the measurements, comprising N eigenfrequencies. The proposed procedure seeks to identify the forces to be applied to the structure in such a way as to produce the same dissipated energy in the vicinities of the resonances. We should recall at this point that the structures under investigation are assumed to be linear. The procedure to follow thus consists of:
(7)
where II 11 2 is an energetic norm. TJ~ ( s) contains all the accurate quantities: the equilibrium equation and the accurate experimental information. The less accurate experimental data are within the . r second term, ponderated by the coeflic1ent --,where r 1- 1' is a parameter belonging to [0, 1J whose value depends Oll the reliability of the
