Identification of Structural Parameters Based on Inverse Modification Theory

During the service life of a structure its stiffness, mass and damping parameters may be changed due to design modifications or due to a damage event. The effects of such modifications are usually analysed by either supplementing the initial Finite Element model with a model of the modification or, more directly, by calculating the modified structure response using structural modification theory. In practice the parameters of the modification are often unknown, for example when the modification was caused by a crack or by bolting a stiffener to the structure. In such cases a model updating approach can be applied where the parameters of the modified structure model are fitted to the test data of the modified structure. The drawback of this approach comes from the necessity to include not only the additional parameters related to the modification but also parameters related to the unmodified part of the structure since the model of the unmodified structure might include not only parametric errors but also physical modelling errors (e.g. due to oversimplification). Therefore, the model of the unmodified structure is usually fitted to test data of the unmodified structure in advance. In any case the success of the model updating approach is based on the ability of the initial model to represent the physics, i.e. the model should not contain physical model errors but only errors in the parameters of the model [1]. In the present paper an approach based on inverse modification theory is presented which allows the identification of the modification parameters. This approach includes the more classical model updating approach where the parameters of an initial analytical are fitted to measured frequency responses. However, the procedure also allows to use only measured frequency responses of the initial unmodified structure instead of analytical responses, i.e. in this case there is no need for a model of the whole structure, only a model of the modification is necessary. Structural modification theory is used as a tool to calculate the frequency response of a modified structure given the frequency response of the unmodified structure and the location, the type and the magnitude of the modification [4]-[7]. This theory is used here to solve the inverse problem: identify the magnitude of the modification with the frequency responses of the modified and unmodified structure taken either from analysis or test. 1. BASICS OF COMPUTATIONAL MODEL UPDATING Computational updating procedures are aimed at fitting selected model parameters such that the test/analysis deviations are minimised. Using appropriate residuals (containing the test/analysis differences) the following objective function J can be derived: J(p) = ∆z W ∆z + p Wp p → min (1) with: ∆z residual vector containing the test/analysis differences, W, Wp weighting matrices. The minimisation of equation (1) yields the desired correction parameters p = [pi], i=1,2...pe= no. of correction parameters. The second term in equation (1) is used to constrain the parameter variation. The residuals ∆z = zT z(p) (zT: test data vector, z(p): corresponding analytical data vector) usually depend in a non-linear way on the parameters. Thus the minimisation problem is also non-linear and must be solved iteratively. One way is to use the classical sensitivity approach (e.g. references [1] – [3]) where the analytical data vector is linearized by a Taylor series expansion truncated after the first term which leads to: ∆z = ∆z0 G0 ∆p (2) with: p0 design parameter vector at linearization point (index “ 0”), ∆p = p p0 design parameter changes, r 0 = z T – z (p0) test/analysis differences (residual vector) at linearization point, for example, the differences between eigenfrequencies and/or mode shapes and/or frequency responses G0 = ∂z/∂p|p=p0 sensitivity matrix at linearization point. Introducing eq.(2) into eq.(1) and constraining only the parameter changes yields the linear problem (3) which has to be solved for the parameter changes ?p in each iteration step for the actual linearisation point. (G0 T W G0 + Wp) ∆p = G0 T W r0 (3) For Wp = 0 equation (3) represents a standard weighted least squares problem. The above minimisation procedure requires an appropriate parameterisation of the model. The most widely used parameterisation of the finite element (FE) model matrices concerning physical parameters like mass, material or beam cross section parameters is given by: K = KA + ∑ αi Ki , i = 1...nα (4a) M = MA + ∑ βj Mj , j = 1...nβ (4b) D = DA + ∑ γk Dk , k = 1...nγ (4c) with: KA, MA, D initial analytical stiffness, mass and damping matrices, p = [.. αi ... βj .. γk..] vector of unknown modification parameters, Ki , Mj , Dk given substructure matrices defining location and type of parameter uncertainties. This parameterisation allows to modify the mass, stiffness and damping parameters of selected substructures. The solution of eq.(3) yields the parameter changes ∆p which are used to update the parameter vector in the next iteration step by p(p0+∆p) = p(p0) + ∆p (5) 3. CALCULATION OF RESIDUAL VECTOR AND RESPONSE SENSITIVITIES In this paper we report about minimizing the residual vector m i (p ) = − r u u (6) which represents the difference of frequency responses m u of a modified structure due to an excitation force vector Fe applied at one or more of the m modification DOFs and the frequency responses i (p ) u of an unmodified structure to identify the unknown modification parameters pi . The standard procedure to calculate response sensitivities is based on deriving the equation of motion, 2 j − ω + ω = K M D F , after introduction of eqs.(4), with respect to the parameter vector p which yields [ ] [ ] 0 i i um i me e p p p p i i0 i i0 ... (p ) / p ... [... ...] = = = ∂ ∂ = − G u H Z H F (7) sensitivity matrix at linearization point “0” where Hnm (n,m)FRF –matrix n= no. of measured DOFs m= no. of DOFs where modifications are applied Hme (m,e)FRF –matrix e= no. of DOFs where excitation forces are applied