Shock and Vibration

Volume 2019, Article ID 4652328, 12 pages

https://doi.org/10.1155/2019/4652328

## Impact of Damping Models in Damage Identification

^{1}Department of Civil Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil^{2}Department of Mechanical Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

Correspondence should be addressed to Michael Souza; rb.jrfu.ilop@enoel_leahcim

Received 20 November 2018; Revised 25 February 2019; Accepted 4 March 2019; Published 9 May 2019

Academic Editor: Filippo Ubertini

Copyright © 2019 Michael Souza 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

Several damage identification approaches are based on computational models, and their diagnostics depend on the set of modelling hypotheses adopted when building the model itself. Among these hypotheses, the choice of appropriate damping models seems to be one of the key issues. The goal of this paper is to analyze the impact of a set of damping models on the damage identification diagnostics. The damage identification is built on a Bayesian framework, and the measured data are the modal data associated with the first modes of the structure. The exploration of the posterior density of unknown model parameters is performed by means of the Markov chain Monte Carlo method (MCMC) with Delayed Rejection Adaptive Metropolis (DRAM) algorithm. The analyses are based on experimental dynamic response obtained from an aluminum beam instrumented with a set of accelerometers. The presence of damage/anomaly within the system is physically simulated by placing lumped masses over the beam, considering three different masses and two different placing positions. For the set of cases analyzed, it is shown that the proposed approach was able to identify both the position and magnitude of the lumped masses and that the damping models may not provide an increase of knowledge of some unknown parameters when damping rates are lower than 1%.

#### 1. Introduction

Computational models were introduced in science and engineering in order to simulate the behavior of physical systems [1]. A key aspect for these analyses is the reliability of the predictions provided by computational models. In general, it is not straightforward to ensure the compliance with the real system due to the great number of unknowns and uncertainties related to material and physical properties, boundary conditions, load conditions, etc. [2]. Hence, one usually adopts simplifying hypotheses which may detract the quality and accuracy of the predictions provided by the model.

These issues had led to the development of model updating techniques [3] which aims to estimate unknown system parameters conjugating model structures, measured data, and optimization strategies. One specific framework that deserves attention is the Bayesian FE model updating methods which are based on Bayes’ theorem [4]. The Bayesian framework provides means to obtain information about unknown model parameters as well as their uncertainties; moreover, its output can be used for uncertainty quantification analysis, global sensitivity analysis [5], and risk assessments, to cite a few. The number of applications of Bayesian FE model updating in the structural dynamics field has increased in the last decades. The review paper by Huang et al. [6] presents a comprehensive material concerning Bayesian inference in structural system identification on both vibration and wave propagation analyses. Katafygiotis and Beck [7] addressed the problem of updating a structural model and its associated uncertainties. Beck [8] used the Bayes’ theorem to update the relative plausibility of candidate models within a set of model classes considering measured data and an initial plausibility of each model. As for its implementation, it should be highlighted that the Bayesian analyses depend on pseudorandom samples generation techniques and acceptance rejection procedures of candidate model parameters. In this context, Nicholas Metropolis and coworkers presented the main background for statistical inference [9–11]. Finally, theoretical aspects can be found in the reference works in [12–15].

One of the many applications of Bayesian framework on civil and mechanical engineering is for structural health monitoring (SHM) [6, 16]. It is based on the premise that damaged structures will significantly modify their stiffness, mass, or energy dissipation properties, which, in turn, will change the measured dynamic response of the system [16]. In this context, it is imperative to fully understand the role of each hypothesis on the dynamical behavior of the structures under analysis. As for damping modelling, one may cite some important aspects to discuss, such as: Which is the best damping model to be used? Once one has obtained damping characteristics of the healthy structure, should one change the damping model when the structure gets damaged? What if one simply ignores damping models? The answer of these complex questions involves structural and material damping formulation. As for damping and damage, Bovsunovsky and Surace [17] studied the influence of damping levels on the nonlinear dynamic behavior of a component in order to assess crack’s parameters; Chandra et al. [18] presented the state of the art of research on damping in fiber-reinforced composite materials and structures with emphasis on polymer composites. Prediction analyses of an anisotropic-damping matrix of a composite material were published in [19]. Gibson and Plunkett [20] described the analytical and experimental internal damping and elastic stiffness of E-glass fiber-reinforced epoxy beams under flexural vibration, and Ling and Haldar [21] proposed a novel system identification that could consider both viscous and Rayleigh-type proportional damping in dynamic models.

The objective of this paper is to analyze the impact of damping model choices on damage identification. Four damping models commonly used by the structural engineering community were considered. The damage scenarios are physically simulated as system anomalies which in the present work correspond to the placement of rigid blocks over an elastic beam. The identification is phrased as a statistical inverse problem using the Bayesian framework. The exploration of the posterior probability density function (pdf) of the unknown parameters is performed by means of the Markov chain Monte Carlo method (MCMC) with Delayed Rejection Adaptive Metropolis (DRAM) algorithm [22].

#### 2. Theoretical Approach

The governing equation of an DOF dynamic system is generally written as follows:where is the mass matrix of the system, is the vector containing the forces external to the system, is the vector describing the system configuration, and the dot over a variable denotes its time derivative. As for the operator , it is in charge of describing the relation between and the restoration forces, as well as the relation between and the dissipation forces. When the system is linear, the restoration force is described as a function of the stiffness matrix and it is equal to . As for the dissipation forces, among some possibilities, in this paper, they are restricted to the class: (i) which may be properly described as being viscous and (ii) which solely depends on the velocity at the current instant of time *t*, i.e., . These simplifications allow one to rewrite the governing equation (2) [23, 24] as follows:where is the viscous damping matrix. The matrix should be positive definite such that the power produced by the dissipative force is always negative, i.e., for all .

There are basically two ways to define the matrix . The first one appears as the natural result of hypotheses that were adopted when writing the governing equations in the continuum. The second way is simply to adopt an ad hoc choice. As for the ad hoc choices, the ones which allow the diagonalization of the matrix by the undamped mode shape matrix are particularly compelling due to the fact that they are prone to analytical treatment in some sense. One may cite, for example, the choice named proportional damping matrix [24], which is described as follows:where and are user defined. The damping matrix in equation (3) can be diagonalized by the undamped mode shape matrix .

One key aspect here is the fact that it does not matter if one builds the viscous damping matrix by means of constitutive equations starting from the continuum or by an ad hoc choice; the user still has to determine the physical parameters or ad hoc parameters that characterize the matrix . At this point, any choice associated to the matrix will have an impact on the model predictions provided by equation (2) and consequently on any identification strategy based on this model. Moreover, it is also true for any model-based damage identification approach.

The structure analyzed in this paper is an aluminum beam, and it will be presented in detail in Section 4. Nevertheless, in order to present which damping models will be treated here, it is necessary to give some information about the identification problem at this point. The present damage identification is rephrased as the quest for information about system anomalies placed over an elastic beam. The anomaly itself is a rigid block whose model parameterization is given by its location over the beam and by its mass magnitude . Therefore, concerning the identification process based on equation (2), the computational model will be described by a set of known model parameters and by a set of unknown model parameters that will be the components of the unknown vector .

The parameters and are of main interest for the identification in this work and are present in the unknown vector . Nevertheless, it should be emphasized that beyond and , some other parameters may possibly be additional components of . In the present work, these additional components will be dependent on the damping model that is adopted when performing the system identification procedures. The damping models are options that are commonly used by the structural engineering community. Next, four damping model hypotheses and the corresponding unknown vector that is associated with each hypothesis are described.(i)Damping hypothesis 1 (DH1): the system is undamped. Thus, , where *E* is Young’s modulus and and are, respectively, the position and magnitude of structural anomaly.(ii)Damping hypothesis 2 (DH2): the system is governed by Rayleigh’s proportional damping model. Thus, , where *α* and *β* are the damping coefficients of the model shown in equation (3).(iii)Damping hypothesis 3 (DH3): the system is governed by Rayleigh’s proportional damping model shown in equation (3). Nevertheless, in this case, the damping coefficients are considered equal to the ones that were estimated for the healthy/intact structure [25]. Thus, .(iv)Damping hypothesis 4 (DH4): the system is governed by a damping model which provides the same damping ratio for all modes. In this case, we also have .

Two points should be emphasized at this point. The first one is that, in this study, it is considered that Young’s modulus *E* is also a component of the unknown vector . Nevertheless, its range of variation is restricted to a tight region. Therefore, it is used here to compensate for possible uncertainties in its nominal value. The second one is concerned with the damping matrix of the fourth damping hypothesis (DH4). It is built as follows:where is the damping rate chosen by the user, is the identity matrix, and is a diagonal matrix containing the undamped natural frequencies of the system, i.e., . A key aspect in this case is that, the user may model as a random variable chosen from a uniform distribution defined between the minimum experimental value of damping rate and its maximum experimental value, i.e., where *k* represents the *k*-th vibration mode of interest.

Finally, it should be highlighted that the damping models adopted are amenable for structural damping and they are not meant to describe localized damping mechanisms such as the ones one may observe when facing cracks, for example. Specific information concerning cracks and damping could be found in [26].

#### 3. Bayesian Framework

##### 3.1. Classical Approach

In a Bayesian framework, all unknown quantities and measurements are modeled as random variables, where the uncertainties related to each parameter are described by their probability density functions [14]. Henceforward, the following notation is used for this framework: denotes the probability density function (pdf) of a random variable *θ*, defines the joint probability of the random variables *θ* and *r*, denotes the conditional probability of *θ* given *r*, and denotes the pdf of a random vector .

Consider that one may measure a set of output variables from the system and organize them in a column vector . The measurements in are often quantities that are of interest for control, identification, and model calibration such as displacements, strains, forces, and modal data, to cite a few. In general terms, the connection between the model prediction and the corresponding measurements is given by an additive error observation model [1, 14]:where is an operator that provides model predictions for the same quantities that were measured in . As for , it denotes the random vector of the unknown parameters of the system and, in the present formulation, they depend on the damping model adopted along the damage identification process as described in Section 2. Finally, it is assumed that the model described by the operator is accurate and that the discrepancies between measurements and model predictions may be properly described by the random variable which is in charge of describing the measurement errors and whose probability density functions is given by .

As , , and are random variables or vectors, the Bayes rule [14] may be used to connect the information among them as follows:where is the posterior probability density function, is the likelihood function, is the prior pdf, and is the density of measured data. The posterior density is the main objective of the Bayesian inverse problem. The prior pdf contains our current state of knowledge about the uncertainty model parameters .

Assuming that the model parameters and the additive noise vector are mutually independent, the likelihood function is given by [14], in which denotes the pdf of the random variable *r* evaluated at the argument . In other words, this gives the probability of observing measured data , given a set of parameters for the model. As for the density , it is hardly available inasmuch as it would require a large number of experimental tests. Fortunately, it acts as a scaling factor and it is of little importance when using sample-based techniques to explore the posterior [5, 14, 27].

The complete probabilistic approach for model updating structural anomalies is presented in equation (6). Therefore, one can compute various point estimates for the random vector along with *a posteriori* uncertainties for these estimates after obtaining the posterior distribution . There are several point estimates that can be computed such as the ones shown as follows:where represents the expected value, defines the maximum *a posteriori* estimator (MAP), denotes the maximum likelihood estimator (ML), and defines the domain of the random variable, or vector, of interest.

It should be noted that integrals like the ones presented in equation (7) are quite common when dealing with statistical inverse problems [14] and uncertainty quantification analysis [5]. Moreover, they generally do not have analytical solutions because the operator generally involves a nonlinear map between the unknown random vector and the model predictions . Therefore, one usually resorts to Markov chain Monte Carlo methods to extract information from the posterior density.

A MCMC method is any method that allows one to draw samples to simulate a target density . In this work, the DRAM algorithm is used to generate samples from the posterior density .

##### 3.2. DRAM Algorithm

DRAM is an algorithm proposed by Haario et al. [22] that aims at improving the efficiency of the MCMC, using concepts of the Delayed Rejection (DR) and Adaptive Metropolis (AM). As earlier stressed, a MCMC is any method that produces drawn samples (states) from a target probability density function [27]. This method often allows drawing samples such that the actual state, i.e., , depends solely on the previous state . There are several MCMC methods, for example, the Metropolis (*M*) algorithm [11], the Metropolis–Hastings (MH) [10], and the Gibbs sampler (GS) [28], to cite a few.

The basic concepts of the DRAM algorithm are the adaptive Metropolis (AM), which is addressed to adapt the proposal distribution based on the past history of the draws that have already been accepted in the actual Markov Chain, and the Delayed Rejection (DR), which is a strategy to modify the classical MH algorithm to improve its efficiency of the resulting MCMC estimates relative to asymptotic variance ordering. Thus the Delayed Rejection Adaptive Metropolis (DRAM) takes the main characteristics of DR and AM in order to improve the acceptance ratio, which leads to a reduction of the computational cost to properly explore the target density .

In a classic Metropolis–Hastings approach, a new candidate is generated from a proposal distribution and its acceptance depends on the MH-acceptance ratio shown as follows:

Upon rejection, instead of keeping the chain at the state , a second candidate is proposed. A second proposal, , depends not only on the current position of the chain but also on what one has just proposed and rejected . The probability of acceptance depends on the updated MH-acceptance ratio which is ruled by equation (9). The methodology of DR can be expanded for an arbitrary number of stages working on an iterative process. Figure 1 presents a prompt but illustrative sketch of the Delayed Rejection step in the perspective of DRAM algorithm.