Shock and Vibration

Volume 2019, Article ID 1362954, 12 pages

https://doi.org/10.1155/2019/1362954

## One-Step FE Model Updating Using Local Correspondence and Mode Shape Orthogonality

^{1}Brincker and Georgakis ApS, Department of Engineering, Inge Lehmanns Gade 10, 8000 Aarhus, Denmark^{2}Technical University of Denmark, Department of Civil Engineering, Brovej, Building 118, DK-2800 Kongens Lyngby, Denmark^{3}Aarhus University School of Engineering, Department of Engineering, Inge Lehmanns Gade 10, 8000 Aarhus, Denmark^{4}University of Oviedo, Department of Construction and Manufacturing Engineering, C/ Pedro Puig Adam, s/n, 33204 Gijón, Spain

Correspondence should be addressed to Sandro D. R. Amador; kd.utd.gyb@oids

Received 14 August 2018; Accepted 31 October 2018; Published 2 January 2019

Academic Editor: Filippo Ubertini

Copyright © 2019 Martin Ø. Ø. Jull et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, it is described how the matrix mixing model updating technique can be combined with the local correspondence (LC) mode shape expansion algorithm, to give a new finite element (FE) model updating method. The matrix mixing method uses that the inverse mass and stiffness matrices can be expressed as a linear combination of outer products of FE mode shape vectors, where the low-frequency part of these sums are substituted with expanded test modes. The approach is meant to update FE models in one-step and is exact, except for the following two approximations: the mode shape smoothing and the mass scaling of the expanded experimental mode shapes. A simulation study illustrates the errors from the two approximations and shows the ability of the technique to improve the modal assurance criterion (MAC) values so that they get very close to unity. Finally, the performance of the proposed updating method is assessed by means of an application example in which the FE model is updated based on the test modes of a real structure.

#### 1. Introduction

Finite element models are numerical idealizations of real structures used in structural design or to predict the structural response under operational conditions. The accuracy of these models is highly dependent on how precise the localized and distributed imperfections are accounted for in the model. For example, joints and boundary conditions tend to be inaccurately modelled by standard components embedded in the FE software that may be different from the ones in the actual structure. Many other examples of inaccurate modelling can be found, and, in general, we accept that the discrepancies between the model and the modelled structure are not significant.

We do expect, however, that the model can simulate the structural behaviour of the modelled structure in every respect that matters. When the structure is built, we can improve the model by adjusting the model parameters to match a subset of measured responses of the structure. This can be done by using the dynamic response of the structure [1]. A review of the existing updating techniques prior to 1993 is found in [2]. Nowadays, a commonly used updating technique is a sensitivity-based method explained in [3], in which the FE model spatial matrices are both parametrically and iteratively updated to match the experimental modal parameters estimated from vibration tests.

In this paper, a novel approach is proposed to update the FE model based on the test modal properties coming from an operational modal analysis (OMA) test [4]. Contrary to sensitivity-based updating techniques, the main advantage of this approach relies on the fact that the spatial matrices of the FE model are updated in one step, hence the name “one-step approach” was framed. The idea behind this approach is basically to replace the modal properties estimated with the FE model by their experimental counterparts so that a good correlation between the updated FE and the test modal parameters is obtained.

In this proposed approach, the method of matrix mixing [5, 6] is used in combination with the local correspondence (LC) expansion technique [7]. The idea is to expand experimental mode shapes in a limited number of degrees of freedom (DOFs) to an entire structure using modes shapes from a FE model. Since the expansion procedure is a crucial task carried out by the one-step approach, it is also herein extensively and detailed described. In order to update the FE model in one step, the technique takes advantage of the matrix mixing method in which the inverse mass and stiffness matrices are written as a sum of outer products of mode shapes. This updating procedure is also discussed in [8, 9]. By making use of such method, the FE model modes in these sums can be replaced by the corresponding expanded test modes, yielding the updated inverse mass and stiffness matrices.

Since the inverse stiffness matrix is of interest in its own right in structural health monitoring (SHM) as shown in [10, 11], the accurate updating of will also be of interest in SHM. The one-step algorithm presented in this paper is a perturbation technique since it relies on the LC principle [7]. The LC principle is a first-order perturbation technique based on the sensitivity equations created by Fox and Kapoor [12]. This means that one-step requires normal model updating to be done before the algorithm can be applied. The idea is that, after the application of the one-step updating approach, the updated FE model completely recreates all natural frequencies and mode shapes found from test measurements. Since no explicit mass scaling is done in the technique, errors in the modal mass are not adjusted by the technique.

Thus, the one-step technique ensures that the modal properties of the updated FE model match the ones estimated from measurements. The technique is meant to be applied only after classical FE model updating takes place, i.e., after the FE model is updated by making use of the analyst’s engineering knowledge and/or of a classic sensitivity-based FE updating technique. This is to assure that the FE model being updated by the one-step approach is physically equivalent to the tested structured and that the discrepancies between model and real structure are caused by perturbations distributed over the FE model. Once these assumptions are fulfilled, the one-step approach can be applied to remove the existing discrepancies and bring the FE model closer to the experimental results.

It is worth also highlighting that, similarly to any sensitive-based FE model updating technique, the one-step FE updating approach herein proposed is suitable for cases where the structural system being updated can be modelled by a linear FE model with orthogonal mode shape vectors. This implies that the one-step technique cannot be used to update FE models of nonlinear structural systems with nonorthogonal mode shape vectors. This is the case of a fairly amount of constructed systems whose structural dynamic behaviour is dominated by nonlinearities.

The paper can be basically divided into three different parts. In the first part, the derivation and description of the one-step approach is presented. In order to illustrate the efficiency and accuracy of the technique from a practical point of view, four application examples are presented in the second part. Finally, some remarks regarding the results obtained from the application examples are presented in the last part of the paper.

#### 2. The One-Step Updating Approach

In the one-step updating approach, it is considered that the discrepancies between the modelled and the real structure are small. If this condition is satisfied, the LC will enable the mixing technique to directly update the mass and stiffness matrices of the FE model. This can be done by using the expanded test modal vectors, without introducing errors in the matrices. An iterative scaling technique ensures that the test mode shapes are correctly mass scaled relative to the new updated mass matrix. Once the test modal vectors are expanded and mass scaled, they are used to replace the modal properties estimated with the FE in order to improve the correlation between the FE and the test modal parameters. The derivation and the detailed description of the one-step approach are presented in the following subsections.

##### 2.1. Basic Equations

An undamped multiple degree of freedom (MDOF) system in structural dynamics is described by the equation of motion:where and are the mass and stiffness matrices, respectively. is an -dimensional column vector containing the deformations in all DOFs of the system. is a function of time , so we have for every instance in time. The solution of this linear system of equations is thoroughly explained in [4].

The solution is a linear combination of mode shapes multiplied by harmonics at the corresponding eigenfrequencies. We order the mode shapes as the columns of the mode shape matrix , and the squares of the angular frequencies as the nonzero elements of the diagonal matrix , where . The ordering of the frequencies on the diagonal corresponds to the ordering of the columns in the mode shape matrix. With this solution, we obtain the following orthogonality relations:when the mode shapes are mass normalized [4].

##### 2.2. One-Step Updating Equations

Pre- and postmultiplying equations (2) and (3), respectively, by and , and inverting the resulting equations yieldsEquation (4) can be written as a sum of outer products:where is the column in the mode shape matrix . If we split this sum into a sum containing contributions from the first mode shapes, and a sum containing the remaining contributions, we havewhere the second sum has elements.

Letting be the matrix with the first mode shapes as columns and be the matrix with the remaining mode shapes, we can write equation (7) as

Using the same techniques, the stiffness equation (5) can be rewritten aswhere is an diagonal matrix containing the first inverses of the squared angular frequencies and is an diagonal matrix containing the remaining inverses of the squared angular frequencies. Equations (8) and (9) are central in the one-step updating approach.

##### 2.3. Mode Shape Expansion

Using the notation from [4], we know from structural modification theory [13] that if we have a system with mode shape matrix , any perturbed system with mode shape matrix can be written aswhere contains the weights when writing the modes of as a linear combination of modes in . This is always possible for any modification.

Provided that has only a few columns and that the modes in are well described by the first modes of , we can let be an matrix and use a smaller matrix of size .

If we measure the responses of the structure at a set of DOFs, we can represent the mode shape matrix aswith and containing the modal coordinates of the FE mode shapes at the measured and unmeasured DOFs, respectively. We, hereinafter, designate them as active and deleted DOFs and denote them with subscripts and , respectively.

We can write an experimentally identified mode shape column vector aswhere the column vector is the linear combination of modes from that gives the identified mode. This is known from the system equivalent reduction-expansion process (SEREP) [14]. Having identified the mode shape , we solve equation (12) to getwhere is the Moore–Penrose pseudoinverse [15] of matrix .

Inserting into (12) gives the least squares smoothed identified mode shape:

Extending equation (14) to all identified modes giveswhere . Equation (15) that assumes that can be written as a linear combination of the first modes of the full mode shape matrix is a reduced version of equation (10).

Equation (15) can now be used for expanding the measured mode shapes in . This is accomplished by appending the unmeasured or deleted DOFs to the linear combinations and holding constant. This gives

Inserting into (16) giveswhere is the global mapping transformation matrix obtained by means of SEREP expansion [14]. The matrix consists of the expanded test modes. The assumption in equation (16) is that the estimate of based on the active subset is close to for the full set of DOFs. Solving equation (12) requires to have fewer mode shapes than DOFs to make this an overdetermined system. In case of few active DOFs, only a small set of modes can be used. A better way to solve equation (12) is to use the LC approach [7], which gives an alternative way to compute .

##### 2.4. LC Expansion

Based on the sensitivity equations [12], the LC principle [7] expands an experimental mode shape by selecting an optimal subset of the FE mode shapes in as expansion basis. Only FE mode shapes with frequencies close to those of the test mode shapes are considered to make sure that the number of modes is lower than the number of active DOFs. The idea behind this strategy is to obtain an overdetermined set of equations in (12) by selecting a small subset of FE modes that locally correspond to an experimentally estimated mode shape.

The best subset, or cluster, is chosen based on a resampling technique known as leave-one-out cross validation (LOOCV) [16]. The LOOCV technique leaves out one of the mode shape parameters, selects a subspace , of , and measures how well the expansion predicts the missing parameter. In this way, the LC principle chooses the optimal cluster based on optimality of predicting unknown responses. Each of the measured responses takes on the role of being unknown test data, one at a time. Using the LOOCV technique enables LC to do a high-level mode shape smoothing since the noise in the subspace of the optimal cluster is completely removed. This has been shown to be a significant advantage over SEREP.

It is possible to use the LC expansion in a simpler way, by only selecting a subspace of whose frequencies are located around that of the test mode . This subspace, containing a fixed number of modes, should then be used to compute the least squares fit and remove the elements of associated with the smallest contributions to . This is a kind of “noise floor” alternative to the traditional LC principle. The LOOCV technique will give its own optimality, and this is the principle used here.

In the simulation and test cases in this paper, there is no significant change, in terms of results, by choosing a fixed subspace of 5 modes locally around all test modes and solving equation (12) in a least-squares sense using the LC LOOCV technique. An illustration of using a 5-mode subspace in the LC expansion is shown in Figure 1. The subspace of modes used in the LC expansion of each experimental mode consists of a FE modal vector with the same natural frequency of the test modal vector being expanded, of one FE modal vector above this, and by three FE modal vectors below, giving a total of five modes. Once the subspace is defined, the LOOCV technique is then used to identify the modes that actually contribute to the expansion of the test mode shape. The empty circles indicate the modal vectors that do not contribute to expansion and, therefore, are not included in the (actual) subspaces used to expand the test mode shapes. The filled circles, on the contrary, show the modes used in the actual expansion. For test modes 1–3 fewer than 5 modes are shown since this gives a better visual illustration of the narrow-band nature of LC. The modes that are not shown are effectively unused in the expansion.