A Bayesian model updating with incomplete complex modal data

Mechanical Systems and Signal Processing 136 (2020) 106524

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

A Bayesian model updating with incomplete complex modal data Ayan Das, Nirmalendu Debnath ⇑ Department of Civil Engineering, National Institute of Technology, Silchar, Assam, India

a r t i c l e

i n f o

Article history: Received 29 May 2019 Received in revised form 10 October 2019 Accepted 12 November 2019

Keywords: Finite element method Bayesian approach Complex modal data Damping Maximum a posterior (MAP)

a b s t r a c t Finite element (FE) model updating using measured dynamic data has gained much attention in view of its contribution towards important areas like model-correction, structural health monitoring. Usually model updating is observed to be performed using measured modal data like modal frequencies and mode shapes – indeed very few works are observed to provide scope for using damping measurements in FE model updating. The present work proposes a Bayesian probabilistic FE model updating using multiple sets of complex modal data (viz. complex modal frequencies and complex mode shapes) and thereby enabling the scope for incorporating damping measurements in updating. Moreover there remains the opportunity for using incomplete complex modal data and avoiding the requirement of mode matching. In this paper, a computationally efficient model updating approach based on maximum a posteriori (MAP) is formulated where all the mass, stiffness and damping parameters are updated in a sequential iterative manner. Detailed formulations are obtained considering damping to be contributed by classical damping (Rayleigh damping) and non-classical viscous damping. Besides, formulations for uncertainty estimation and probabilistic damage detection are also developed. The proposed approach is validated using two numerical examples while utilizing incomplete complex modal data. Performance in updating is evaluated for both undamaged and damaged cases. The proposed approach is observed to be successful for updating of FE model as well as for detection of changes/damages in probabilistic manner, while being computationally efficient as well. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Numerical or mathematical models of a real system are widely used to predict or simulate the actual behaviour of the system in most of the areas of science and engineering. In civil and structural engineering, these numerical models created mainly by finite element (FE) modelling are utilized in structural analysis, prediction of structural response due to different types of dynamic loadings like earthquake, wind etc. and structural health monitoring (SHM). In spite of most sophisticated FE procedures available, in practical or real life applications, discrepancies are often observed between the FE model predicted results and actual results. These errors or discrepancies as elaborately discussed in [1,2] often arise mainly due to (a) errors due to idealisation of the behaviour of the actual physical system in creating the FE model (b) errors due to inefficient discretization of the FE model thereby affecting the convergence of the modal data in the frequency of interest (c) ⇑ Corresponding author. E-mail address: [email protected] (N. Debnath). https://doi.org/10.1016/j.ymssp.2019.106524 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

2

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

errors due to assumptions of inaccurate values of model parameters (e.g., Young’s modulus, density, spring stiffness etc.) (d) errors due to uncertainties in measurement of modal data. This has led to the development of FE model updating techniques which minimizes the gap between the measured and FE model predicted results based on the measured/experimental dynamic characteristics of the corresponding real structural-system. A typical FE model updating technique works on the principle of calibration of the uncertain model parameters till the FE model successfully predicts the behaviour of the real structure. Various works have been performed in the field of FE model updating over a last few years. In this context, a book by Friswell and Mottershead [3] can be mentioned as one of the earliest and important works, where model updating techniques are elaborately discussed. Besides, works by Mottershead and Friswell [4,5] are some of the pioneering works in FE model updating which are worth mentioning. Among the earliest FE model updating techniques, direct methods of model updating are the most important which directly update the mass and stiffness matrices. These methods were primarily were developed by the works of various researchers e.g. Baruch [6–8], Baruch and Bar-Itzhack [9], Berman and Nagy [10], Caesar [11,12], Wei [13], Friswell et al. [14], Datta [15], Carvalho et al. [16]. Later, iterative methods of model updating are developed which overcome the difficulties of direct methods of model updating. These iterative methods update the model parameters in an iterative manner till the FE model predicts the results of the actual structure with acceptable accuracy. Within the class of iterative techniques, the sensitivity technique [1] is usually observed to be considered as quite significant. Min et al. [17] proposed a sensitivity based FE model updating method for damped beam-structure. Grip et al. [18] performed sensitivity-based model updating of a reinforced concrete plate by using regularization. Yang and Chen [19] presented a comparative study on direct versus iterative techniques for FE model updating. Few more popular works may be mentioned in this regard like: model updating using frequency domain data using reduced order models by Friswell and Penny [20], model updating using geometric parameters by Mottershead et al. [21], model updating of large structures using incomplete modal data by Yuen [22], model updating based on frequency response function using Kriging model by Wang et al. [23]. The present work comes under the class of Bayesian probabilistic model updating techniques based on maximum a posteriori (MAP), where optimal values of model parameters are estimated with quantified measure of uncertainty in the form of their respective standard deviation. Various works of Bayesian updating are worth mentioning [24–26] where a probability density function (PDF) is utilized in quantifying all types of errors resulting from uncertain model parameters. A significant development in Bayesian FE model updating (based on incomplete modal data) was evolved having an important advantage like non-requirement of matching of mode shapes between the measured and analytical models (Yuen [27– 30]). In a relatively recent work, Mustafa et al. [31] presented FE model updating of an existing large bridge-structure in Bayesian probabilistic framework using incomplete modal data. Yan and Katafygiotis [32] presented a fast Bayesian methodology for FE model updating using modal data (modal frequencies and mode shapes) where those modal data were considered to be identified from multiple setups. Moreover, Das and Debnath [33] proposed FE model updating in Bayesian framework considering the strictly positive nature of structural parameters with the help of combined normal and lognormal distributions. Other works like hierarchical model updating in Bayesian framework by Behmanesh et al. [34], Bayesian model updating for structural health monitoring by Rocchetta et al. [35], stochastic model updating in Bayesian framework by Wan and Ren [36] are worth mentioning which have utilized Bayesian model updating in recent times. On the other hand, updating and damage-detection problems in Bayesian framework are also observed to be performed using sampling based techniques like: FE model updating and reliability analysis of building frame by Beck and Au [37] in an adaptive Markov Chain Monte Carlo (MCMC) framework, FE model updating by Ching and Chen [38] in a transitional MCMC framework, structural health monitoring of building structure by Ching et al. [39] using Gibbs sampling. Bansal [40] extended the work by updating a structure using Gibbs sampling where modal data are obtained from multiple set ups. Very few works have been observed where complex modal data are utilized in updating the structure in Bayesian framework. In this context, work by Cheung and Bansal [41] is worth mentioning where the authors utilize Gibbs sampling [42] technique for model updating, originally proposed by Ching et al. [39], in successfully updating a structure utilizing complex modal data. The complex modal data include the complex eigenvalues, damping ratios and partial complex mode shape of few of the dominant modes. The existing updating approach (based on Gibbs sampling) by Cheung and Bansal [41] using incomplete complex modal data (also without need of mode-matching) indeed demands high computational cost – moreover efficiency of this approach depends on the initial assumed values of the uncertain parameters and the standard deviation corresponding to their prior PDF. In view of this, there exists an interest for formulating a possible computationally efficient framework for FE model updating using incomplete complex modal data and without need for mode-matching. The present work involves a sequential optimization process having more computational efficiency based on MAP, where the system mode shapes and structural parameters are updated in an iterative manner. Detailed formulations are carried out for model updating, uncertainty-estimation and probabilistic detection of changes/damages of structural parameters. The performance of the proposed approach is validated based on two examples (spring-mass-damper system and ASCE benchmark structure) and compared with Gibbs sampling technique and Sensitivity method observing acceptable performance of the proposed approach. Moreover, the performance of the proposed approach for damage-detection (along with computational efficiency) is investigated using similar numerical examples with multiple damage cases demonstrating satisfactory performance.

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

3

2. Theoretical background of Bayesian model updating using complex modal data n o ^ ðm;sÞ : m ¼ 1; 2; ::: ; N m ; s ¼ 1; 2; ::: ; N s be the experimentally obtained modal data where k ^ðm;sÞ ; ^ ^ðm;sÞ nðm;sÞ ; W Let D ¼ k ^ ðm;sÞ indicate experimental eigenvalue and eigenvector for mth mode of sth modal data set from a linear dynamic system and W ^ ðm;sÞ is ðN 0  1Þ indicating N 0 degrees of freedom measured respectively. Besides ^ nðm;sÞ indicates damping ratio and size of W (DOFs) out of total N d DOFs of the analytical/numerical model. Besides, sizes (no. of rows  no. of columns) of each of massmatrix (M), stiffness matrix (K) and damping matrix (D) can be expressed as: ðN d  N d Þ. It may be mentioned that N m is the number of measured modes and N s is the number of modal data sets. 2.1. Parameterization of system matrices Parameterization is one key step in this proposed approach. Such parameterization of mass matrix ðMðh1 ÞÞ, stiffness matrix ðKðh2 ÞÞ and damping matrix Cðh1 ; h2 ; h3 ; a0 ; a1 Þ are considered as shown in Eqs. (1) and (2) respectively. N h1

Mðh1 Þ ¼ M0 þ R ðh1 Þl  Ml l¼1

Nh2

Kðh2 Þ ¼ K0 þ R ðh2 Þl  Kl l¼1

ð1Þ

ð2Þ

In Eq. (1), Ml is the subsystem matrix for lth mass-parameter and M0 is the non-parameterized component of mass matrix. Similarly, in Eq. (2), Kl is the subsystem matrix for lth stiffness-parameter and K0 is the non-parameterized component of stiffness matrix. Here, h1 and h2 represent mass and stiffness parameter vector having N h1 and N h2 number of uncertain parameters respectively. Similar expression can be obtained for damping matrix where both classical (Rayleigh damping) and non-classical damping (e.g. viscous damping) are considered and is expressed in Eq. (3). N h3

Cðh1 ; h2 ; h3 ; a0 ; a1 Þ ¼ C0 þ R ðh3 Þl  Cl þ a0 Mðh1 Þ þ a1 Kðh2 Þ ¼ Cn þ a0 Mðh1 Þ þ a1 Kðh2 Þ l¼1

ð3Þ

P h3 where, Cn ¼ C0 þ Nl¼1 Cl ðh3 Þl and h3 represents damping matrix and damping parameter vector contributed by the non-classical damping having N h3 number of uncertain parameters respectively. It is easy to understand that Cl is the sub-system matrix for lth damping parameter and C0 is the non-parameterized component of damping matrix due to non-classical damping. Besides, a ¼ ½a0 ; a1 T are the Rayleigh damping coefficients expressed as in Eqs. (4) and (5).

a0 ¼

 2wi wj  wj ni  wi nj w2j  w2j

ð4Þ

a1 ¼

  2 wj nj  wi ni w2j  w2j

ð5Þ

    where wi ; wj and ni ; nj are the modal frequencies and damping ratios of the ith and jth mode respectively of the model corresponding to mass matrix Mðh1 Þ and stiffness matrix Kðh2 Þ. It is interesting to note that the Rayleigh damping coefficients a ¼ ½a0 ; a1 T are not constant and these are to be treated as uncertain parameters in addition to the structural parameters. Hence, the sub-system damping matrices with respect to the Rayleigh coefficients a ¼ ½a0 ; a1 T are the mass matrix and @C @C ¼ M and @a ¼ K. This information is utilized in derivation of the formulations required for stiffness matrix respectively i.e. @a 0 1 model updating in the proposed framework in the upcoming sections. 2.2. Formulation of the proposed Bayesian approach In Bayesian approach, the posterior probability is obtained from the prior probability incorporating likelihood function of the observed data. Such Bayesian probability associated with continuous valued parameters may be expressed using Bayes’ theorem [43] as in Eq. (6).

pðhjDÞ ¼

pðDjhÞpðhÞ pðDÞ

ð6Þ

where, h and D, represent the parameter-vector and observed data-vector respectively. Besides, pðhjDÞ is the posterior probability density function (PDF) of the parameter-vector given the observed data-vector, pðhÞ is the prior PDF of the parametervector, pðDjhÞ is the likelihood function and pðDÞ is the normalizing constant. The basic Bayes’ formula given in Eq. (6), in the context of FE model updating utilizing complex modal data can be expressed in further details as in Eq. (7).

4

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

  ^ WjU; ^   p k; h pðU; hÞ ^ W ^ ¼   p U; hjk; ^ W ^ p k;

ð7Þ

Here, h ¼ ½h1 ; h2 ; h3 T represent the structural updating parameter vector. In this context, the system complex mode shapes [41] is used as optimization parameters in addition to the structural parameters. System mode shapes are denoted here as  T  T  T T U ¼ Uð1Þ ; Uð2Þ ; ::: ; UðNm Þ , where size of rth system mode shape UðrÞ ðr ¼ 1; 2; ::: ; N m Þ is ð2N d  1Þ. In the present   ReðWm Þ work, the system mode shape is expressed in a truncated vector form as UðmÞ ¼ where Wm is the complex mode ImðWm Þ shape explained later in a similar fashion as taken in [41]. Such consideration of system mode shapes as optimization parameter plays a pivotal role to remove the requirement of mode matching [29]. Indeed mode-matching requires high computation time due to solution of eigen equation in each iteration. 2.2.1. Formulation of prior PDF The components of the structural parameter vector h are considered to be independent of each other and hence the prior PDF of the structural parameter vector h can be considered to be product of the individual PDFs of the component vector and can be expressed as in Eq. (8).

pðhÞ ¼ pðh1 Þpðh2 Þpðh3 Þpða0 Þpða1 Þ

ð8Þ

The prior PDF of all the structural parameter vectors mentioned in Eq. (8) are assumed to follow multivariate Gaussian distribution having mean values equal to the nominal values and covariance matrices with diagonal values having large variances. The prior PDFs of the individual components can be expressed as in Eqs. (9)-(13).

  T  1 g pðh1 Þ ¼ ð2pÞNh1 =2 jRh1 j1=2 exp  h1  hg1 R1 h  h 1 h1 1 2

ð9Þ

  T  1 g h  h pðh2 Þ ¼ ð2pÞNh2 =2 jRh2 j1=2 exp  h2  hg2 R1 2 h2 2 2

ð10Þ

  T  1 g h  h pðh3 Þ ¼ ð2pÞNh3 =2 jRh3 j1=2 exp  h3  hg3 R1 3 h3 3 2

ð11Þ

  2 1  pða0 Þ ¼ ð2pÞ1=2 ðra0 Þ1 exp  2 a0  ag0 2ra0

ð12Þ

  2 1  pða1 Þ ¼ ð2pÞ1=2 ðra1 Þ1 exp  2 a1  ag1 2ra1

ð13Þ

  Here, hg1 ; hg2 ; hg3 and ðRh1 ; Rh2 ; Rh3 Þ are the nominal values and covariance matrices of the mass, stiffness and dampingparameter vectors respectively. It may be mentioned that these covariance matrices are diagonal matrices with large vari  ances. On the other hand, ag0 ; ag1 and ðra0 ; ra1 Þ are the nominal values and standard deviation of the Rayleigh coefficients a ¼ ½a0 ; a1 T respectively. The nominal values of the structural parameters depend on the problem type and choice of updating parameters but the nominal values of the Rayleigh coefficients can be calculated using Eqs. (4) and (5) using nominal mass and stiffness parameters. Besides, the prior PDF of the system mode shape is assumed to follow uniform distribution. 2.2.2. Formulation of likelihood function Considering a 2nd order linear dynamical system, the system matrix in a typical state space equation as described in Eqs. (14) and (15) with p inputs and n state variables, can be expressed as in Eq. (16).

X_ ¼ AXðt Þ þ BUðt Þ

ð14Þ

Xð0Þ ¼ X0

ð15Þ

 A¼

0 1

I 1

M K M C

 ð16Þ

Here, XðtÞ consisting of n state variables at time t and Uðt Þ consisting of p state variables denote the state vector and excitation vector at time t. It may be mentioned that the number of state variables is equal to twice the number of total DOFs of the structure i.e. n ¼ 2N d . By solving eigen problem of the system matrix A, shown in Eq. (16), complex eigenvalues km and eigenvectors Wm can be obtained for ðm ¼ 1; 2; ::: ; N m Þ which satisfy the relation as shown in Eq. (17).

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

AWm ¼ km Wm where,

5

ð17Þ



 Wm is the complex eigenvector and km is the complex eigenvalue of mth mode and is given by the expression km Wm shown in Eq. (18). Wm ¼

km ¼ nm wm þ iwm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  n2m

ð18Þ

where wm indicates modal frequency of the mth mode. It may be mentioned that the complex eigenvalues and eigenvectors occur in conjugate pairs and the following relation can be obtained by expanding the relation shown in Eq. (19).



 k2m Mðh1 Þ þ km Cðh1 ; h2 ; h3 ; a0 ; a1 Þ þ Kðh2 Þ Wm ¼ 0

ð19Þ 



^ W ^ consists of two independent compoThe likelihood function of the unknown parameters ðU; hÞ given the outcome k;       ^ ^ ^ nents p kjU; h and p WjU . In order to calculate the likelihood function p kjU; h , eigen-equation error em;s for mth mode of the sth modal data set is considered by putting the measured complex eigenvalues ^ kðm;sÞ in Eq. (19) and is expressed in Eq.   ^ (20). On the other hand, in order to calculate the likelihood function p WjU , an error function em;s for mth mode of the sth modal data set is considered as shown in Eq. (21), which measures the discrepancies between the measured complex eigenvector and system complex mode shape corresponding to N 0 measured DOFs.

 2  em;s ¼ ^kðm;sÞ Mðh1 Þ þ ^kðm;sÞ Cðh1 ; h2 ; h3 ; a0 ; a1 Þ þ Kðh2 Þ Wm

ð20Þ

^ ðm;sÞ  LWm em;s ¼ W

ð21Þ

Here, the matrix L (of size N 0  N d ) is the selection matrix that maps the measured DOFs from the full scale mode shape. It can be understood that the error vectors in Eqs. (20) and (21) are complex in nature. It is assumed that the real and imaginary parts of these two error functions em;s and em;s are independent of each other. The PDF of a complex eigenvector Z ¼ fZ 1 ; Z 2 ; ::: ; Z n g is considered to be joint density of the real and imaginary parts of the complex random vector Z (as discussed in [44]) and is shown as in Eq. (22) at q ¼ fq1 ; q2 ; ::: ; qn g.

f Z ðqÞ ¼

@2 @2 ::::::: F Z ðqÞ @x1 @y1 @xn @yn

ð22Þ

Here, F Z ðqÞ is the cumulative distribution function of Z at q as shown in Eq. (23), where real and imaginary components of qk are denoted as: Reðqk Þ ¼ xk , Imðqk Þ ¼ yk .

F Z ðqÞ ¼ pðReðZ1 Þ  x1 ; ImðZ1 Þ  y1 ; ::: ; ReðZn Þ  xn ; ImðZn Þ  yn Þ

ð23Þ

In view of this, separation of the real and imaginary parts of the error functions em;s and lowing relations can be observed as shown in Eqs. (24)-(27).

em;s is necessary and hence fol-

   2 ^k ^ ðm;sÞ Mðh1 Þ þ kðm;sÞ Cðh1 ; h2 ; h3 ; a0 ; a1 Þ þ Kðh2 Þ Wm

ð24Þ

   2 Imðem;s Þ ¼ Im ^kðm;sÞ Mðh1 Þ þ ^kðm;sÞ Cðh1 ; h2 ; h3 ; a0 ; a1 Þ þ Kðh2 Þ Wm

ð25Þ

  ^ ðm;sÞ  LWm Reðem;s Þ ¼ Re W

ð26Þ

  ^ ðm;sÞ  LWm Imðem;s Þ ¼ Im W

ð27Þ

Reðem;s Þ ¼ Re

The PDFs of error vectors shown in Eqs. (24) and (25) are assumed to be Gaussian based on maximum entropy principle [45,46], with zero means and covariance matrices as diagonal matrices having equal variances along the diagonal. On the other hand, PDFs of error vectors shown in Eqs. (26) and (27) are also assumed to be Gaussian (also based on maximum ðm;sÞ

ðm;sÞ

entropy principle) with zero means and covariance matrices RRe and RIm respectively. It may be mentioned that such separation of real and imaginary parts of these error functions and such assigning of Gaussian distribution is found to be adopted in [41]. The expression for the PDFs are shown in Eqs. (28)–(31).

 2

ðm;sÞ Reðem;s Þ  N 0Nd 1 ; rRe IN d

ð28Þ

6

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

 2

ðm;sÞ Imðem;s Þ  N 0Nd 1 ; rIm INd

ð29Þ

  ðm;sÞ Reðem;s Þ  N 0N0 1 ; RRe

ð30Þ

  ðm;sÞ Imðem;s Þ  N 0N0 1 ; RIm

ð31Þ

where, N indicates Gaussian distribution, and 0 indicates zero vector and I denotes an identity matrix. It may be mentioned that equal standard deviation values are considered for the PDF of the error functions shown in Eqs. (24) and (25) for all     ðm;sÞ ðm;sÞ modes and data sets i.e. rRe ¼ r1 and rIm ¼ r2 for m = 1, 2, . . . , Nm and for s = 1, 2, . . . , Ns . Now as mentioned earlier, assuming independent nature of real and imaginary parts of these error functions, the PDF of these error functions can be considered to be a product of the PDFs of their real and imaginary parts. The likelihood function thus can be expressed as in Eq. (32)

 Y Nm Y Ns Nm       Y 0:5  1 ^ WjU; ^ ^ ^ p k; h ¼ p kjU; h p WjU ¼ ð2pÞ0:5Nd r21 INd  exp  2 k Reðem;s Þ k2 2r1 m¼1 s¼1 m¼1  Y Ns Nm Y     1 0:5  ð2pÞ0:5Nd r22 INd  exp  2 k Imðem;s Þ k2 2r2 s¼1 m¼1  Y Ns Nm 0:5   1 Y 1  ðm;sÞ  ðm;sÞ ð2pÞ0:5N0 RRe  ðReðem;s ÞÞ  exp  ðReðem;s ÞÞT RRe 2 s¼1 m¼1   Ns 0:5    Y 1 1  ðm;sÞ  ðm;sÞ  ð2pÞ0:5N0 RIm  exp  ðImðem;s ÞÞT RIm ðImðem;s ÞÞ 2 s¼1

ð32Þ

2.2.3. Formulation of objective function Finally, the posterior PDF can be obtained using the relation shown in Eq. (7). The most probable values of the unknown parameters can be found by maximizing the posterior PDF. The objective function is here obtained by taking the negative logarithm of the posterior PDF (without including the constant that does not depend on the uncertain parameters). This objective function is minimized instead of maximizing the posterior PDF to perform the optimization. This objective function is represented as in Eq. (33).

F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ ¼

T  T  T  1 1 1 g g g h1  hg1 R1 h2  hg2 R1 h3  hg3 R1 h1 h1  h1 þ h2 h2  h2 þ h3 h3  h3 2 2 2 Ns Nm Nm X 2 2 1  1  1 X 1 X k Reðem;s Þ k2 þ 2 þ 2 a0  ag0 þ 2 a1  ag1 þ 2 2ra0 2ra1 2r1 m¼1 s¼1 2r2 m¼1 Ns X



k Imðem;s Þ k2 þ

s¼1

Nm X Ns   T  1    1X ðm;sÞ ^ ðm;sÞ  LWm ^ ðm;sÞ  LWm þ 1 Re W RRe Re W 2 m¼1 s¼1 2

Nm X Ns   T  1    X ^ ðm;sÞ  LWm ^ ðm;sÞ  LWm  Im W RðImm;sÞ Im W

ð33Þ

m¼1 s¼1

2.2.4. Optimization framework The system mode shapes and structural parameters are updated by minimizing the objective function shown in Eq. (33). In this minimization procedure, derivatives of the objective function with respect to various vectors (evaluated using concept of matrix-calculus [47]) are considered as equal to zero. At first, it may be mentioned that the symbol (*) is used to represent recently optimal/updated value. The minimization of the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to U is @F ¼ 0. obtained by differentiating the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to U and equating it to zero i.e. @U The equation obtained using this procedure is shown in the form of Eq. (34).

r2 1

Ns X

BTs1 Bs1 U þ r2 2

s¼1



Nm X Ns X m¼1 s¼1

Ns X

BTs2 Bs2 U þ

s¼1



ðm;sÞ

HT1 RRe

1

Nm X Ns X

Nm X Ns  1  1 X ðm;sÞ ðm;sÞ HT1 RRe H1 U þ HT2 RIm H2 U

m¼1 s¼1





^ ðm;sÞ  Re W

Nm X Ns X m¼1 s¼1

m¼1 s¼1



ðm;sÞ

HT2 RIm

1



 ^ ðm;sÞ ¼ 0 Im W

ð34Þ

7

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

By simplifying Eq. (34) we get the optimal mode shape vector U as shown in Eq. (35).

"

r



2 1

U ¼

" 

Ns X

BTs1 Bs1

þr

2 2

s¼1

Ns X

BTs2 Bs2

s¼1

Nm X Ns X



ðm;sÞ

HT1 RRe

1

þ

Nm X Ns X

2

6 6 6 Bs1 ¼ 6 6 6 4

6 6 6 Bs2 ¼ 6 6 4

R

ðm;sÞ Re

1

H1 þ

Nm X Ns X

HT2

 1 ðm;sÞ RIm H2

m¼1 s¼1

ð35Þ

m¼1 s¼1

    Re ws1 Im ws1   .. . Re ws2 .. .





  Im ws2 .. .

..



.





0 .. . .. .

0N0 Nd

H2 ¼ 0N0 Nd

L

3

    Re wsNm Im wsNm

    3   0 Im ws1 Re ws1 7     .. .. 7 . . Im ws2 Re ws2 7 7 .. .. .. .. 7 5 . . . .       0   Im wsNm Re wsNm N

H1 ¼ L

#1

# Nm X Ns   X  1   ðm;sÞ T ^ ^ Re Wðm;sÞ þ H2 RIm Im Wðm;sÞ

0 2



m¼1 s¼1

m¼1 s¼1

where,

HT1

7 7 7 7 7 7 5

ð36Þ

N d N m 2Nd N m

ð37Þ d N m 2N d N m

N 0 2Nd

ð38Þ

N 0 2Nd

ð39Þ

 2  wsm ¼ ^kðm;sÞ M þ ^kðm;sÞ C þ K

ð40Þ

It may be mentioned that 0N0 Nd shown in Eqs. (38) and (39) is a zero matrix of size N 0  N d . @F ¼ 0, we Now, by differentiating the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to h1 and equating to zero i.e. @h 1 get Eq. (41).

r2 1

Ns X

As1 h1 þ r2 2

s¼1

Ns X

Ns X

1 g 2 As2 h1 þ R1 h1 h1  Rh1 h1 þ r1

s¼1

As3 þ r2 2

s¼1

Ns X

As4 ¼ 0

ð41Þ

s¼1

By simplifying Eq. (41) we get the optimal mode shape vector h1 as shown in Eq. (42).

"

h1

¼

r

2 1

Ns X s¼1

As1 þ r

2 2

Ns X

#1 "

As2 þ R

1 h1

R

1 g h1 h 1

s¼1

r

2 1

Ns X s¼1

As3  r

2 2

Ns X

#

As4

ð42Þ

s¼1

where,

  As1 ¼ GTh1P1 Gh1P1 þ GTh1P2 Gh1P2  2GTh1P1 Gh1P2

ð43Þ

  As2 ¼ GTh1X1 Gh1X1 þ GTh1X2 Gh1X2 þ 2GTh1X1 Gh1X2

ð44Þ

 3 2  T Gh1P1 G01P1 þ GTh1P1 Gh2Q 1 h2 þ GTh1P1 G02Q 1 þ GTh1P1 Gh3R1 h3 þ GTh1P1 G03R1 þ 6   7 7 6 6 GTh1P2 G01P2 þ GTh1P2 Gh2Q 2 h2 þ GTh1P2 G02Q 2 þ GTh1P2 Gh3R2 h3 þ GTh1P2 G03R2  7 7 6 As3 ¼ 6 !7 7 6 GT G T T T T 5 4 h1P1 01P2 þ Gh1P2 G01P1 þ Gh1P1 Gh2Q2 h2 þ Gh1P1 G02Q2 þ Gh1P1 Gh3R2 h3 þ

ð45Þ

GTh1P1 G03R2 þ GTh1P2 Gh2Q 1 h2 þ GTh1P2 G02Q 1 þ GTh1P2 Gh3R1 h3 þ GTh1P2 G03R1  3 2  T Gh1X1 G01X1 þ GTh1X1 Gh2Y1 h2 þ GTh1X1 G02Y1 þ GTh1X1 Gh3Z1 h3 þ GTh1X1 G03Z1 þ 6   7 7 6 6 GT G01X2 þ GT Gh2Y2 h2 þ GT G02Y2 þ GT Gh3Z2 h3 þ GT G03Z2 þ 7 h1X2 h1X2 h1X2 h1X2 h1X2 7 As4 ¼ 6 6 !7 7 6 GT G T T T T þ G G þ G G h þ G G þ G G h þ 5 4 h1X1 01X2 h1X2 01X1 h1X1 h2Y2 2 h1X1 02Y2 h1X1 h3Z2 3 GTh1X1 G03Z2 þ GTh1X2 Gh2Y1 h2 þ GTh1X2 G02Y1 þ GTh1X2 Gh3Z1 h3 þ GTh1X2 G03Z1

ð46Þ

8

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524 @F By differentiating the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to h2 and equating to zero i.e. @h ¼ 0, we get the 2

equation as shown in Eq. (47).

r2 1

Ns X

As5 h2 þ r2 2

s¼1

Ns X

Ns X

1 g 2 As6 h2 þ R1 h2 h2  Rh2 h2 þ r1

s¼1

As7 þ r2 2

s¼1

Ns X

As8 ¼ 0

ð47Þ

s¼1

By simplifying Eq. (47) we get the optimal mode shape vector h2 as shown in Eq. (48).

"

h2

¼

r

2 1

Ns X

As5 þ r

2 2

s¼1

Ns X

#1 "

As6 þ R

1 h2

R

1 g h2 h 2

r

2 1

s¼1

Ns X

As7  r

2 2

s¼1

Ns X

#

ð48Þ

As8

s¼1

where,

  As5 ¼ GTh2Q1 Gh2Q 1 þ GTh2Q 2 Gh2Q 2  2GTh2Q 1 Gh2Q2

ð49Þ

  As6 ¼ GTh2Y1 Gh2Y1 þ GTh2Y2 Gh2Y2 þ 2GTh2Y1 Gh2Y2

ð50Þ

 3 2  T Gh2Q1 G02Q 1 þ GTh2Q1 Gh1P1 h1 þ GTh2Q1 G01P1 þ GTh2Q 1 Gh3R1 h3 þ GTh2Q 1 G03R1 þ 6   7 7 6 6 GTh2Q2 G02Q 2 þ GTh2Q2 Gh1P2 h1 þ GTh2Q2 G01P2 þ GTh2Q 2 Gh3R2 h3 þ GTh2Q 2 G03R2  7 7 As7 ¼ 6 6 !7 T T T T 7 6 GT G 5 4 h2Q2 h1P1 h1 þ Gh2Q2 G01P1 þ Gh2Q1 Gh1P2 h1 þ Gh2Q1 G01P2 þ Gh2Q 1 G02Q 2 þ

ð51Þ

GTh2Q 2 G02Q 1 þ GTh2Q 1 Gh3R2 h3 þ GTh2Q 1 G03R2 þ GTh2Q 2 Gh3R1 h3 þ GTh2Q 2 G03R1  3 2  GTh2Y1 G02Y1 þ GTh2Y1 Gh1X1 h1 þ GTh2Y1 G01X1 þ GTh2Y1 Gh3Z1 h3 þ GTh2Y1 G03Z1 þ 6   7 7 6 6 GTh2Y2 G02Y2 þ GTh2Y2 Gh1X2 h1 þ GTh2Y2 G01X2 þ GTh2Y2 Gh3Z2 h3 þ GTh2Y2 G03Z2 þ 7 7 As8 ¼ 6 !7 6 T T T T 7 6 GT G þ G G þ G G h þ G G þ G G h þ 5 4 h2Y1 02Y2 h2Y2 02Y1 h2Y2 h1X1 1 h2Y2 01X1 h2Y2 h3Z1 3 T T T T T Gh2Y2 G03Z1 þ Gh2Y1 Gh1X2 h1 þ Gh2Y1 G01X2 þ Gh2Y1 Gh3Z2 h3 þ Gh2Y1 G03Z2

ð52Þ

@F By differentiating the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to h3 and equating to zero i.e. @h ¼ 0, we get the 3

equation as shown in Eq. (53).

r2 1

Ns X

As9 h3 þ r2 2

s¼1

Ns X

1 g 2 As10 h3 þ R1 h3 h3  Rh3 h3 þ r1

s¼1

Ns X

As11 þ r2 2

s¼1

Ns X

As12 ¼ 0

ð53Þ

s¼1

By simplifying Eq. (53) we get the optimal mode shape vector h3 as shown in Eq. (54).

"

h3 ¼

r2 1

Ns X s¼1

As9 þ r2 2

Ns X

#1 "

As10 þ R1 h3

g 2 R1 h3 h 3  r 1

s¼1

Ns X s¼1

As11  r2 2

Ns X

#

As12

ð54Þ

s¼1

where,

  As9 ¼ GTh3R1 Gh3R1 þ GTh3R2 Gh3R2  2GTh3R1 Gh3R2

ð55Þ

  As10 ¼ GTh3Z1 Gh3Z1 þ GTh3Z2 Gh3Z2 þ 2GTh3Z2 Gh3Z1

ð56Þ

2

As11

As12

6 6 6 ¼6 6 6 4

 3 GTh3R1 G03R1 þ GTh3R1 Gh1P1 h1 þ GTh3R1 G01P1 þ GTh3R1 Gh2Q 1 h2 þ GTh3R1 G02Q 1 þ   7 7 GTh3R2 G03R2 þ GTh3R2 Gh2Q 2 h2 þ GTh3R2 G02Q2 þ GTh3R2 Gh1P2 h1 þ GTh3R2 G01P2  7 7 !7 T T T T T Gh3R1 G03R2 þ Gh3R2 G03R1 þ Gh3R2 Gh1P1 h1 þ Gh3R2 G01P1 þ Gh3R1 Gh1P2 h1 þ 7 5 GTh3R1 G01P2 þ GTh3R2 Gh2Q 1 h2 þ GTh3R2 G02Q 1 þ GTh3R1 Gh2Q 2 h2 þ GTh3R1 G02Q 2

 3 2  T Gh3Z1 G03Z1 þ GTh3Z1 Gh2Y1 h2 þ GTh3Z1 G02Y1 þ GTh3Z1 Gh1X1 h1 þ GTh3Z1 G01X1 þ 6   7 7 6 6 GT G03Z2 þ GT Gh2Y2 h2 þ GT G02Y2 þ GT Gh1X2 h1 þ GT G01X2 þ 7 h3Z2 h3Z2 h3Z2 h3Z2 h3Z2 7 ¼6 6 !7 7 6 GT G T T T T þ G G þ G G h þ G G þ G G h þ 5 4 03Z2 03Z1 h1X1 1 01X1 h1X2 1 h3Z1 h3Z2 h3Z2 h3Z2 h3Z1 GTh3Z1 G01X2 þ GTh3Z1 Gh2Y2 h2 þ GTh3Z1 G02Y2 þ GTh3Z2 Gh2Y1 h2 þ GTh3Z2 G02Y1

ð57Þ

ð58Þ

9

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

   2     2  ðm;sÞ ðm;sÞ2 ðm;sÞ ðm;sÞ2 kR , Re ^ kR kI , Im ^ kI Denoting few of the symbols as: Re ^ kðm;sÞ ¼ ^ kðm;sÞ ¼ ^ , Im ^ kðm;sÞ ¼ ^ kðm;sÞ ¼ ^ and " #  UðmÞ ¼

WRðmÞ  WIðmÞ

, the unknown symbols shown in Eqs. (43–46), (49–52) and (55–58) can be expressed as follows:

2

Gh1P1

Gh2Q1

2Nd 1



  ^kð1;sÞ2 þ a ^kð1;sÞ M1 Wð1Þ 0 R R R  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kR M1 WR R

6 6 6 6 ¼6 6 .. 6 . 4   ^kðNm ;sÞ2 þ a ^kðNm ;sÞ M1 WðNm Þ 0 R R R  2  ð1;sÞ ð1Þ 1 þ a1 ^kR K1 WR 6   6  6 1 þ a ^kð2;sÞ K1 Wð2Þ 1 R R 6 ¼6 6 .. 6 .  4 ðN m ;sÞ ðN Þ K1 WR m 1 þ a1 ^kR 2

Gh3R1

^kð1;sÞ C1 Wð1Þ R 6 R  6 ^ð2;sÞ 6 kR C1 Wð2Þ R ¼6 6 .. 6 . 4 ðN m ;sÞ ðN Þ ^k C1 WR m R 2

Gh1P2



^kð2;sÞ C2 Wð2Þ R R .. .



^kðNm ;sÞ C2 WðNm Þ R R



  ^kð1;sÞ2 þ a ^kð1;sÞ M1 Wð1Þ 0 I I I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI M1 WI I

 ð1;sÞ  ð1Þ K1 WI a1 ^kI  ð2;sÞ  ð2Þ a1 ^kI K1 WI

ð1;sÞ

ð1Þ



 .. .

^kð2;sÞ CN Wð2Þ R R h3 .. .



ðNm ;sÞ ðN Þ    ^kR CNh3 WR m

.. .  ðNm ;sÞ  ðN Þ a1 ^kI K2 WI m

^kð1;sÞ C2 Wð1Þ I I



^kð2;sÞ C2 Wð2Þ I I .. .



^kðNm ;sÞ C2 WðNm Þ I I



  ^kð1;sÞ2 þ a ^kð1;sÞ M1 Wð1Þ 0 I I R  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI M1 WR I

6 6 6 6 ¼6 6 .. 6 . 4  ðNm ;sÞ2 ðN m ;sÞ ðN Þ ^k M1 WR m þ a0 ^kI I







 .. .

^kð2;sÞ CN Wð2Þ I I h3 .. .



ðNm ;sÞ ðN Þ    ^kI CNh3 WI m

Nd Nm N h2

.. .

.. .   ðNm ;sÞ2 ðNm ;sÞ ðN Þ ^ ^k k  þ a MNh1 WI m 0 I I

3 7 7 7 7 7 7 7 5 Nd N m N h1

3

.. .. .  . ðN m ;sÞ ðN Þ  a1 ^kI KNh2 WI m 

7 7 7 7 7 7 7 5

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kI MNh1 WI I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI MNh1 WI I



 ð1;sÞ ð1Þ a1 ^kI KNh2 WI  ð2;sÞ  ð2Þ a1 ^kI KNh2 WI

^kð1;sÞ CN Wð1Þ I I h3

Nd N m N h1

3

N d N m N h3





7 7 7 7 7 7 7 5

7 7 7 7 7 7 5

.. .  ðNm ;sÞ2  ðN m ;sÞ ðN Þ ^ ^k k þ a M2 WI m 0 I I



.. .   ðNm ;sÞ2 ðN ;sÞ ðN Þ ^ m ^k k  þ a MNh1 WR m 0 R R

3

3

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kI M2 WI I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI M2 WI I



.. .

.. .. .  . ðNm ;sÞ ðN Þ  1 þ a1 ^kR KNh2 WR m

^kð1;sÞ CN Wð1Þ R R h3

 ð1;sÞ ð1Þ a1 ^kI K2 WI  ð2;sÞ  ð2Þ a1 ^kI K2 WI



 ð1;sÞ ð1Þ 1 þ a1 ^kR KNh2 WR   ð2;sÞ ð2Þ 1 þ a1 ^kR KNh2 WR







 ð1;sÞ2  ð1;sÞ ð1Þ ^k MNh1 WR þ a0 ^kR R  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kR MNh1 WR R







.. .   ðN m ;sÞ ðN Þ 1 þ a1 ^kR K2 WR m



^k C1 W I 6 I  6 ^ð2;sÞ 6 kI C1 Wð2Þ I ¼6 6 .. 6 . 4  ^kðNm ;sÞ C1 WðNm Þ I I 2

Gh1X1

  ð1;sÞ ð1Þ 1 þ a1 ^kR K2 WR   ð2;sÞ ð2Þ 1 þ a1 ^kR K2 WR

^kð1;sÞ C2 Wð1Þ R R

6 6 6 6 ¼6 6 .. 6 . 4  ðN m ;sÞ ðN Þ K1 WI m a1 ^kI 2

Gh3R2



.. .  ðNm ;sÞ2  ðN ;sÞ ðN Þ ^ m ^k k þ a M2 WR m 0 R R

6 6 6 6 ¼6 6 .. 6 . 4  ðNm ;sÞ2 ðN m ;sÞ ðN Þ ^ ^k k þ a M1 WI m 0 I I 2

Gh2Q2

 ð1;sÞ2  ð1;sÞ ð1Þ ^k M2 WR þ a0 ^kR R  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kR M2 WR R

7 7 7 7 7 7 7 5 Nd Nm N h2

3



7 7 7 7 7 7 5 N d N m N h3

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kI M2 WR I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI M2 WR I .. .  ðNm ;sÞ2  ðN m ;sÞ ðN Þ ^k M2 WR m þ a0 ^kI I

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kI MNh1 WR I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI MNh1 WR I

  .. . 



ðN m ;sÞ2

^k I

.. .

ðNm ;sÞ

þ a0 ^kI

 ðN Þ MNh1 WR m

3 7 7 7 7 7 7 7 5 N d N m N h1

10

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Gh2Y1

2  ð1;sÞ  ð1Þ K1 WR a1 ^kI 6   6  6 a ^kð2;sÞ K1 Wð2Þ 1 I R 6 ¼6 6 .. 6 . 4  ðN m ;sÞ ðN Þ ^ K1 WR m a1 kI 2

Gh3Z1

^kð1;sÞ C1 Wð1Þ R 6 I  6 ^ð2;sÞ 6 kI C1 Wð2Þ R ¼6 6 .. 6 . 4 ðN m ;sÞ ðN Þ ^k C1 WR m I 2

Gh1X2

Gh2Y2



^kð1;sÞ C2 Wð1Þ I R



^kð2;sÞ C2 Wð2Þ I R .. .





ð1Þ

K2 WR

ð1;sÞ

ð1Þ

ð2Þ









^kð2;sÞ C2 Wð2Þ R I .. .



ð2;sÞ



ð2Þ

KNh2 WR

.. .

..  ðNm ;sÞ . ðN Þ KNh2 WR m  a1 ^kI



^kð1;sÞ CN Wð1Þ I R h3



 .. .

^kð2;sÞ CN Wð2Þ I R h3 .. .



ðNm ;sÞ ðN Þ    ^kI CNh3 WR m



N d N m Nh3



 .. . 



ðNm ;sÞ2

^k R

.. .  ðNm ;sÞ ðN Þ  1 þ a1 ^kR KNh2 WI m 

 .. .

^kð2;sÞ CN Wð2Þ R I h3 .. .



ðNm ;sÞ ðN Þ    ^kR CNh3 WI m

3



7 7 7 7 7 7 5 N d N m Nh3

2 

3

G02Q 1

Nd N m 1

 ð1;sÞ ð1Þ 1 þ a1 ^kR K0 WR   ð2;sÞ ð2Þ 1 þ a1 ^kR K0 WR

 ð1;sÞ2  ð1;sÞ ð1Þ ^k M0 WI þ a0 ^kI I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI M0 WI I

2

G03R2

3

7 6 7 6 7 6 7 6 ¼6 7 7 6 . 7 6 .. 5 4  ðN ;sÞ ðNm Þ ^ m K0 WR 1 þ a1 kR 3

^kð1;sÞ C0 Wð1Þ I I



3

7 6  7 6 ^ð2;sÞ 7 6 kI C0 Wð2Þ I 7 ¼6 7 6 .. 7 6 . 5 4  ðN m ;sÞ ðN Þ m ^k C W 0 I I Nd Nm 1

Nd N m 1

Nd Nm 1

.. .

ðN m ;sÞ

þ a0 ^kR

.. .

^kð1;sÞ CN Wð1Þ R I h3

3

Nd Nm 1

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kR MNh1 WI R  ð2;sÞ2  ð2;sÞ ð2Þ ^k MNh1 WI þ a0 ^kR R



  ð1;sÞ ð1Þ 1 þ a1 ^kR KNh2 WI   ð2;sÞ ð2Þ 1 þ a1 ^kR KNh2 WI





2

7 6 7 6 7 6 7 6 ¼6 7 7 6 . 7 6 .. 5 4  ðN ;sÞ ðNm Þ ^ m K0 WI a1 kI

N d N m N h2

7 7 7 7 7 7 5

7 6 7 6 7 6  7 6 ^ð2;sÞ 7 6 7 6 kR C0 Wð2Þ 7 6 R 7 6 ¼6 G01P2 ¼ 6 7 7 7 6 . . 7 6 .. 7 6 .. 5 4 5 4    ðN m ;sÞ  ðN m ;sÞ2 ðNm ;sÞ ðN m Þ ðN Þ ^k m  ^ ^ C0 WR k þ a W M k R 0 I Nd N m 1 0 I I  ð1;sÞ  ð1Þ K0 WI a1 ^kI  ð2;sÞ  ð2Þ K0 WI a1 ^kI

7 7 7 7 7 7 7 5

3

.. .  ðNm ;sÞ2  ðNm ;sÞ ðN Þ ^ ^k k þ a M2 WI m 0 R R



3

ð1Þ

KNh2 WR

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kR M2 WI R  ð2;sÞ2  ð2;sÞ ð2Þ ^k M2 WI þ a0 ^kR R

^kðNm ;sÞ C2 WðNm Þ R I

3



a1 ^kI

.. .   ðN m ;sÞ ðN Þ 1 þ a1 ^kR K2 WI m

^kð1;sÞ C2 Wð1Þ R I

ð1;sÞ

a1 ^kI

  ð1;sÞ ð1Þ 1 þ a1 ^kR K2 WI   ð2;sÞ ð2Þ 1 þ a1 ^kR K2 WI

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kR M0 WR R  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kR M0 WR R

^kð1;sÞ C0 Wð1Þ R R





K2 WR

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kR M1 WI R  ð2;sÞ2  ð2;sÞ ð2Þ ^k M1 WI þ a0 ^kR R

7 6 7 6 7 6 7 6 ¼6 7 7 6 . 7 6 .. 5 4   ^kðNm ;sÞ2 þ a ^kðNm ;sÞ M0 WðNm Þ 0 R R R

2

G02Q2

ð2;sÞ

a1 ^kI

^kðNm ;sÞ C2 WðNm Þ I R

^k C1 W I 6 R  6 ^ð2;sÞ 6 kR C1 Wð2Þ I ¼6 6 .. 6 . 4 ðN m ;sÞ ðN m Þ ^k C 1 WI R

2

G03R1



.. .  ðNm ;sÞ  ðN Þ ^ K2 WR m a1 kI

 2  ð1;sÞ ð1Þ K1 WI 1 þ a1 ^kR 6   6  6 1 þ a ^kð2;sÞ K1 Wð2Þ 1 R I 6 ¼6 6 .. 6 .  4 ðN m ;sÞ ðN Þ K1 WI m 1 þ a1 ^kR

2

G01P1



ð1;sÞ

a1 ^kI

6 6 6 6 ¼6 6 .. 6 . 4  ðN m ;sÞ2 ðNm ;sÞ ðN Þ ^ ^k k þ a M1 WI m 0 R R

2

Gh3Z2





 ðN Þ MNh1 WI m

3 7 7 7 7 7 7 7 5 Nd Nm N h2

3 7 7 7 7 7 7 7 5 N d N m Nh1

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

2

G01X1

3

7 6 7 6 7 6 7 6 ¼6 7 7 6 . 7 6 .. 5 4   ^kðNm ;sÞ2 þ a ^kðNm ;sÞ M0 WðNm Þ 0 I I R 2

G03Z1

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kI M0 WR I  ð2;sÞ2  ð2;sÞ ð2Þ ^k þ a0 ^kI M0 WR I

^kð1;sÞ C0 Wð1Þ I R

Nd Nm 1

3 2  ð1;sÞ  ð1Þ K0 WR a1 ^kI 7 6   7 6  6 a ^kð2;sÞ K0 Wð2Þ 7 1 I R 7 6 ¼6 7 7 6 . 7 6 .. 5 4  ðN ;sÞ ðN m Þ ^ m K0 WR a1 kI

N d N m 1

 ð1;sÞ2  ð1;sÞ ð1Þ ^k þ a0 ^kR M0 WI R  ð2Þ2  ð2;sÞ ð2Þ ^k þ a0 ^kR M0 WI R

2

3



G02Y1

11

3

7 6 7 6 7 6  7 6 ^ð2;sÞ 7 6 7 6 kI C0 Wð2Þ 7 6 R 7 ¼6 G ¼ 7 6 01X2 7 6 7 6 .. . 7 6 7 6 .. . 5 4 5 4    ðN m ;sÞ  ðN m ;sÞ2 ðN m ;sÞ ðNm Þ ðN m Þ ^k  ^ ^ C0 WR þ a0 kR kR M0 WI I Nd Nm 1

N d N m 1

G02Y2

 3 2  ð1;sÞ ð1Þ K0 WI 1 þ a1 ^kR 7 6   7 6  6 1 þ a ^kð2;sÞ K0 Wð2Þ 7 1 R I 7 6 ¼6 7 7 6 . 7 6 .. 5 4  ðN ;sÞ ðN m Þ ^ m K0 WI 1 þ a1 kR

2

G03Z2

N d N m 1

^kð1;sÞ C0 Wð1Þ R I



3

7 6  7 6 ^ð2;sÞ 7 6 kR C0 Wð2Þ I 7 6 ¼6 7 . 7 6 . . 5 4  ðN m ;sÞ ðN Þ m ^k C W 0 I R N d N m 1

@F By differentiating the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to a0 and equating to zero i.e. @a ¼ 0, we get Eq. 0

(59).

r2 1

Ns X

As13 a0 þ r2 2

s¼1

Ns X

2 g 2 As14 a0 þ r2 a0 a0  ra0 a0  r1

s¼1

Ns X

As15 þ r2 2

s¼1

Ns X

As16 ¼ 0

ð59Þ

s¼1

By simplifying Eq. (59) we get the optimal mode shape vector a0 as shown in Eq. (60).

"

r2 1

a0 ¼

Ns X

As13 þ r2 2

s¼1

Ns X

#1 "

As14 þ r2 a0

g 2 r2 a0 a0 þ r1

s¼1

Ns X

As15  r2 2

s¼1

Ns X

#

ð60Þ

As16

s¼1

where,

  As13 ¼ ST1 S1 þ ST2 S2  2ST1 S2

ð61Þ

  As14 ¼ UT1 U1 þ UT2 U2 þ 2UT2 U1

ð62Þ

0

1 PT1 S2 þ Q T1 S2 þ RT1 S2 þ ST1 P2 þ ST1 R2  PT1 S1  Q T1 S1  RT1 S1  PT2 S2  RT2 S2   A ¼@ þa1 ST1 T2 þ TT1 S2  ST1 T1  ST2 T2

As15

1 XT1 U1 þ ZT1 U1 þ XT2 U2 þ YT2 U2 þ ZT2 U2 þ XT2 U1 þ YT2 U1 þ ZT2 U1 þ UT2 X1 þ UT2 Z1 þ   A ¼@ a1 UT1 V1 þ UT2 V2 þ UT2 V1 þ VT2 U1

ð63Þ

0

As16

ð64Þ

@F Lastly, by differentiating the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ with respect to a1 and equating to zero i.e. @a ¼ 0, we 1

get Eq. (65).

r2 1

Ns X

As17 a1 þ r2 2

s¼1

Ns X

2 g 2 As18 a1 þ r2 a1 a1  ra1 a1  r1

s¼1

Ns X

As19 þ r2 2

s¼1

Ns X

As20 ¼ 0

ð65Þ

s¼1

By simplifying Eq. (65) we get the optimal mode shape vector a1 as shown in Eq. (66).

"

a1 ¼

r2 1

Ns X s¼1

where,

As17 þ r2 2

Ns X s¼1

#1 "

As18 þ r2 a1

g 2 r2 a1 a1 þ r1

Ns X s¼1

As19  r2 2

Ns X s¼1

#

As20

ð66Þ

12

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

  As17 ¼ TT1 T1 þ TT2 T2  2TT1 T2

ð67Þ

  As18 ¼ VT1 V1 þ VT2 V2 þ 2VT2 V1

ð68Þ

0

As19

1 PT1 T2 þ Q T1 T2 þ RT1 T2 þ TT1 P2 þ TT1 R2  PT1 T1  Q T1 T1  RT1 T1  PT2 T2  RT2 T2   A ¼@ þa0 ST1 T2 þ TT1 S2  ST1 T1  ST2 T2 0

As20

1 XT1 V1 þ ZT1 V1 þ XT2 V2 þ YT2 V2 þ ZT2 V2 þ XT2 V1 þ YT2 V1 þ ZT2 V1 þ VT2 X1 þ VT2 Z1 þ   A ¼@ a0 UT1 V1 þ UT2 V2 þ UT2 V1 þ VT2 U1

ð69Þ

2

3

^kð1;sÞ2 M Wð1Þ R R



2

ð1Þ

K WR

2

3

^kð1;sÞ C Wð1Þ n R R



ð70Þ

3

7 7 6 6 6  ð2Þ 7 6 ^ð2;sÞ2  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 7 6 KW 7 7 6 kR M WR 6 kR Cn WR R 7 6 7 7 P1 ¼ 6 Q1 ¼ 6 R1 ¼ 6 7 7 7 6 6 .. 7 6 .. .. 7 7 6 6 5 4 . . . 5 5 4 4    ðNm ;sÞ2 ðN ;sÞ ðN Þ  m m ðN Þ ðN Þ ^k ^k K WR M WR m Nd Nm 1 Cn WR m Nd Nm 1 Nd Nm 1 R R 2

^kð1;sÞ M Wð1Þ R R



3

2

^kð1;sÞ K Wð1Þ R R

3



2

^kð1;sÞ2 M Wð1Þ I I



3

7 7 7 6 6 6 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ2  ð2Þ 7 7 7 7 6 kR M WR 6 kR K WR 6 kI M WI 7 7 7 6 6 6 S1 ¼ 6 T1 ¼ 6 P2 ¼ 6 7 7 7 . . . 7 7 7 6 6 6 . . . . . . 5 5 5 4 4 4    ^kðNm ;sÞ M WðNm Þ ^kðNm ;sÞ K WðNm Þ ^kðNm ;sÞ2 M WðNm Þ R R I R R I N d N m 1 N d N m 1 N d N m 1 2

^kð1;sÞ C Wð1Þ n I I



3

2

^kð1;sÞ M Wð1Þ I I

3



2

^kð1;sÞ K Wð1Þ I I

3



7 7 7 6 6 6 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 7 7 7 6 kI Cn WI 6 kI M WI 6 kI K WI 7 7 7 R2 ¼ 6 S2 ¼ 6 T2 ¼ 6 7 7 7 6 6 6 .. .. .. 7 7 7 6 6 6 . . . 5 5 5 4 4 4    ðN m ;sÞ  ðN ;sÞ ðN ;sÞ m m ðN Þ ðN Þ ðN Þ   m m m ^k ^ ^ C W M W K W k k n I I I I I I N d N m 1 Nd Nm 1 Nd Nm 1 3 3 3 2 ð1;sÞ 2 ð1;sÞ    ^kð1;sÞ2 M Wð1Þ ^k C Wð1Þ ^k M Wð1Þ n R R R 7 7 7 6 I 6 I 6 I 6 ^ð2;sÞ2  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 7 7 7 6 kI 6 6 k k M W C W M W I I n R R R 7 7 7 X1 ¼ 6 Z1 ¼ 6 U1 ¼ 6 7 7 7 6 6 6 . . . 7 7 7 6 6 6 . . . . . . 5 5 5 4 4 4    ^kðNm ;sÞ2 M WðNm Þ ^kðNm ;sÞ C WðNm Þ ^kðNm ;sÞ M WðNm Þ n I I I R R R Nd Nm 1 N d N m 1 N d N m 1 2

2

3

2

3

2

3

2  ð1Þ 3 K WI 7 7 6 6 6  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ2  ð2Þ 7 7 6 KW 7 7 6 kI K WR 6 kR M WI I 7 6 7 7 V1 ¼ 6 X2 ¼ 6 Y2 ¼ 6 7 7 7 6 6 . 7 6 . . . 7 7 6 6 . . 5 4 . . . 5 5 4 4     ðN m Þ ^kðNm ;sÞ K WðNm Þ ^ðNm ;sÞ2 M WðNm Þ W K k I N d N m 1 I R R I N d N m 1 Nd Nm 1 2

^kð1;sÞ K Wð1Þ I R

^kð1;sÞ C Wð1Þ n I R





^kð1;sÞ2 M Wð1Þ R I



^kð1;sÞ M Wð1Þ R I



3

2

^kð1;sÞ K Wð1Þ R I



3

7 7 7 6 6 6 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 6 ^ð2;sÞ  ð2Þ 7 7 7 7 6 kR Cn WI 6 kR M WI 6 kR K WI 7 7 7 Z2 ¼ 6 U2 ¼ 6 V2 ¼ 6 7 7 7 6 6 6 .. .. .. 7 7 7 6 6 6 . . . 5 5 5 4 4 4    ðN m ;sÞ  ðN ;sÞ ðN ;sÞ m m ðN Þ ðN Þ ðN Þ ^k ^k ^k Cn WI m Nd Nm 1 M WI m Nd Nm 1 K WI m Nd Nm 1 R R R The optimization is performed in parts (once at a time) iteratively having coupled nature of the (vector/scalar) parameters. The coupled nature can be observed from the analytical expressions of the optimal values of the uncertain parameters        U ; h1 ; h2 ; h3 ; a0 ; a1 where the analytical expression of one parameter is actually function of the remaining uncertain parameters. Initial values of h1 , h2 , h3 , a0 and a1 are assumed as nominal values of the respective parameters, while U is updated at beginning with no requirement for assumption of its initial value. The iterative algorithm adopted here is presented in the following steps:

13

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

(a) Find the updated estimates of the system eigenvectors as U (for Nm numbers of modes) using the expression as in Eq. (35). (b) Find the updated estimates of the mass-parameters as h1 using the expression as in Eq. (42). (c) Find the updated estimates of the stiffness-parameters as h2 using the expression given in Eq. (48). (d) Find the updated estimates of the damping-parameters as h3 using the expression given in Eq. (54). (e) Find the updated estimates of the Rayleigh damping coefficient a0 using Eq. (60). (f) Find the updated estimates of the Rayleigh damping coefficient a1 using Eq. (66). (g) Iterate the steps a, b, c, d, e and f until the parameters demonstrate an acceptable convergence. It may be mentioned that the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ is quadratic with respect to each of the updating parameters ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ while remaining parameters considered as constants. In this context, the objective function can be expressed as simple standard quadratic form [48] as shown in Eq. (71).

1 f ðxÞ ¼ xT b þ xT Hx þ c 2

ð71Þ

Here, x is the uncertain parameter vector, H is the Hessian matrix of the objective function f ðxÞ with respect to x, b is a constant vector and c is a constant scalar. The solution of this problem is thus obtained by differentiating Eq. (71) with respect to x as given by Eq. (72).

rf ðxÞ ¼ b þ Hx

ð72Þ 

By equating Eq. (72) to zero, we can obtain the optimal value of parameter x by solving the linear system of equations as in Eq. (73).

Hx ¼ b

ð73Þ



x can be considered to be global minimizer of f ðxÞ under the condition that H being symmetric and positive definite (thus being non-singular). Positive definiteness of the Hessian matrix may be ensured by putting a check on the condition number [49] of the Hessian matrix in each iteration. Finally, it may be mentioned that this optimization problem is likely to be performed using various unconstrained non-linear optimization methods, like methods based on quadratic Taylor approximation e.g. Newton’s method of optimization [50]. However such method requires computation of derivative vector and Hessian matrix (of the objective function with respect to entire set of parameters) in each iteration leading to higher computational cost for optimization. 3. Uncertainty estimation   The posterior PDF is approximated by a Gaussian distribution centered at the optimal parameters U ; h1 ; h2 ; h3 ; a0 ; a1 as discussed in [27–30] and with the covariance matrix CðU; h1 ; h2 ; h3 ; a0 ; a1 Þ equal to the inverse of the Hessian matrix of the objective function F ðU; h1 ; h2 ; h3 ; a0 ; a1 Þ. The expression for the covariance matrix of the posterior PDF is given in Eq. (74).

2

H11

6 6 6 6 CðU; h1 ; h2 ; h3 ; a0 ; a1 Þ ¼ 6 6 6 6 4

H12

H13

H14

H15

H22

H23

H24

H25

H33

H34 H44

H35 H45 H55

sym

H16

31

H26 7 7 7 H36 7 7 H46 7 7 7 H56 5

ð74Þ

H66

The components of the Hessian matrix are shown as follows:

H11 ¼

r

2 1

Ns X

BTs1 Bs1

þr

2 2

s¼1

H12 ¼

r2 1

Ns X

r2 1

Ns X

Ls1 þ r2 2

H14 ¼

r

Ns X s¼1

Ns X

Ls3 þ r2 2

Ns X

Ls2 ! Ls4

s¼1

Ls5 þ r

2 2

Ns X s¼1

! Ls6

þ

Nm X Ns X m¼1 s¼1

!

s¼1

s¼1

2 1

BTs2 Bs2

s¼1

s¼1

H13 ¼

Ns X

HT1

Nm X Ns  1  1 X ðm;sÞ ðm;sÞ RRe H1 þ HT2 RIm H2 m¼1 s¼1

!

14

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

H15 ¼

r2 1

Ns X

Ls7 þ r2 2

s¼1

H16 ¼

r

2 1

Ns X

H22 ¼

R

! Ls8

s¼1

Ls9 þ r

2 2

s¼1

1 h1

Ns X

Ns X

! Ls10

s¼1

þr

2 1

Ns X

Ls11 þ r

2 2

s¼1

H23 ¼

r2 1

Ns X

H24 ¼

r

Ns X

Ls13 þ r2 2

H25 ¼

r

Ns X

Ls15 þ r

2 2

r2 1

Ns X

Ls17 þ r

2 2

! Ls14

Ns X

! Ls16

Ns X

! Ls18

s¼1

Ls19 þ r2 2

s¼1

H33 ¼

Ns X

s¼1

s¼1

H26 ¼

Ns X

! Ls20

s¼1

2 R1 h2 þ r 1

Ns X

Ls21 þ r2 2

s¼1

H34 ¼

r

2 1

Ns X

H35 ¼

r

Ns X

Ls23 þ r

2 2

r2 1

Ns X

Ls25 þ r

2 2

Ns X

! Ls24

Ns X

! Ls26

s¼1

Ls27 þ r2 2

s¼1

H44 ¼

Ns X

! Ls28

s¼1

2 R1 h3 þ r 1

Ns X

Ls29 þ r2 2

s¼1

H45 ¼

r

2 1

Ns X

H46 ¼

r

Ns X

Ls31 þ r

2 2

Ls33 þ r

2 2

Ns X

! Ls32

Ns X

! Ls34

s¼1

2 r2 a0 þ r1

Ns X

Ls35 þ r2 2

s¼1

H56 ¼

r

2 1

Ns X

H66 ¼

Ls37 þ r

r þr

Ns X

! Ls36

s¼1

2 2

s¼1

2 a1

! Ls30

s¼1

s¼1

H55 ¼

Ns X s¼1

s¼1

2 1

! Ls22

s¼1

s¼1

H36 ¼

Ns X s¼1

s¼1

2 1

Ls12

s¼1

s¼1

2 1

!

s¼1

s¼1

2 1

Ns X

Ns X

! Ls38

s¼1

2 1

Ns X s¼1

Ls39 þ r

2 2

Ns X s¼1

! Ls40

15

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

where,

2 ðLs1 Þlth

col

ðLs1 Þl1

6 ðLs1 Þ l2 6 ¼6 .. 6 4 .

3 7 7 7 7 5

ðLs1 ÞlNm

ðLs1 Þlm

2

ðLs2 Þlth

2N d N m 1

3 2 8h     ðm;sÞ2     ðm;sÞ2 i ðmÞ 9 ðm;sÞ ðm;sÞ ^ ^  > > ^k ^k < Re wsm = k k þ a þ a W  M þ M Re w l l 0 R 0 R sm R R R 7 6 6 h     ðm;sÞ2     ðm;sÞ2 i ðmÞ > 7 ðm;sÞ ðm;sÞ 7 6 : >  ^ ^  ^ ^ ; kI þ a0 kI þ a0 kR Ml þ k R Ml Im wsm WI Re wsm 7 6 68 97 ¼6 7  ðm;sÞ2    ðm;sÞ2    h  i    ðm;sÞ ðm;sÞ 6< ðm Þ ^k =7 þ a0 ^kI þ a0 ^kR Ml þ ^kR Ml Im wsm WR þ > 7 6 >  Re wsm I 7 6  ðm;sÞ2    ðm;sÞ2    i 5 4> h     ðm;sÞ ðm;sÞ ð m Þ > ^ ^  ^ ^k : Im wsm ; k k M þ k M Im w þ a þ a W l l 0 I 0 I sm I I I

col

ðLs2 Þl1 6 ðLs2 Þ 6 l2 ¼6 .. 6 4 .

3 7 7 7 7 5

ðLs2 ÞlNm

ðLs2 Þlm

ðLs3 Þlth

col

ðLs3 Þl1 6 ðLs3 Þ 6 l2 ¼6 . 6 . 4 . ðLs3 ÞlNm

ðLs3 Þlm

2N d N m 1

3 28h     ðm;sÞ2     ðm;sÞ2 i ðmÞ 9 ðm;sÞ ðm;sÞ ^ ^  > > ^k ^k < Im wsm = k k þ a þ a W þ M þ M Im w l l 0 0 sm I I I I R 7 6 6    ðm;sÞ2    h   ðm;sÞ2 i ðmÞ > 7 ðm;sÞ ðm;sÞ 7 6> ^ ^  ^k ^ ; k k þ a þ a W M þ k M Im w 7 6 : Re wsm l l 0 I 0 R sm I R I 6 8 9 7 ¼6 h  7 i  ðm;sÞ2           ðmÞ ðm;sÞ2 ðm;sÞ 6 > ðm Þ ^k = 7 þ a0 ^kI þ a0 ^kR Ml þ ^kR Ml Im wsm WR þ > 7 6 < Re wsm I 7 6  ðm;sÞ2    ðm;sÞ2    i 5 4 >h     ðm;sÞ ðm;sÞ ðmÞ > ^k : Re wsm ; þ a0 ^kR þ a0 ^kR Ml þ ^kR Ml Re wsm WI R 2

3 7 7 7 7 5 2N d N m 1

3 28h     ðm;sÞ      ðm;sÞ i ðmÞ 9 ^  > > ^k < Re wsm = k þ a Re w þ 1 K þ 1 K W  a l l 1 R 1 R sm R 7 6 7 6    h    ðm;sÞ    ðm;sÞ i ðmÞ 7 6> >   ^ ^ : ; Re wsm a1 kI Kl þ a1 kR þ 1 Kl Im wsm WI 7 6 7 6 9 ¼6 8 h  7       i     ðm;sÞ ðm;sÞ 7 6 > ðmÞ     > ^ ^ 6 <  Re wsm a1 kI Kl þ a1 kR þ 1 Kl Im wsm WR þ = 7 7 6     h  i 5 4 >    ðmÞ ðm;sÞ ðm;sÞ > : ; Im wsm a1 ^kI Kl þ a1 ^kI Kl Im wsm WI

2N d 1

3 ðLs4 Þl1 6 ðLs4 Þ 7 6 l2 7 7 ¼6 .. 7 6 5 4 . ðLs4 ÞlNm 2Nd Nm 1 2

ðLs4 Þlth

col

3 9     ðm;sÞ    ðm;sÞ    i ðmÞ > ^k ^k = Im w K K W þ þ a Im w a l l sm 1 1 sm I I R 7 6 7 6    h    ðm;sÞ  i ðmÞ > 7 6 > ^  : Re wsm a1 ^kðm;sÞ ; k K þ 1 K W þ a Im w 7 6 l l 1 sm I R I 7 68 9 ¼6 7   h  i        ðmÞ ðm;sÞ ðm;sÞ 6> =7 Re wsm a1 ^kI Kl þ a1 ^kR þ 1 Kl Im wsm WR þ > 7 6< 7 6 h    ðm;sÞ    i 5 4>     ðm;sÞ   ðmÞ > : Re wsm a1 ^kR þ 1 Kl þ a1 ^kR þ 1 Kl Re wsm WI ; 2

ðLs4 Þlm

8 > <

2N d 1

h

2Nd 1

2N d 1

16

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

2

ðLs5 Þlth

col

ðLs5 Þl1 6 ðLs5 Þ 6 l2 ¼6 .. 6 4 . ðLs5 ÞlNm

2 ðLs6 Þlth

col

ðLs6 Þl1

3 7 7 7 7 5

ðLs5 Þlm

2N d N m 1

3

6 ðLs6 Þ 7 6 l2 7 7 ¼6 ðLs6 Þlm .. 7 6 5 4 . ðLs6 ÞlNm 2Nd Nm 1

2 8h  9 3  ðm;sÞ   ðm;sÞ   i  > < Re wsm ^kR Cl þ ^kR Cl Re wsm WRðmÞ  > = 7 6 6     h  i  ðm;sÞ   ðmÞ > 7 ðm;sÞ 6 > ; 7 7 6 : Re wsm ^kI Cl þ ^kR Cl Im wsm WI 6 97 ¼ 68 h  7  ðm;sÞ   ðm;sÞ   i    6> ðm Þ =7 7 6 <  Re wsm ^kI Cl þ ^kR Cl Im wsm WR þ > 7 6  ðm;sÞ   ðm;sÞ   i 5 4> h     : Im wsm ^kI Cl þ ^kI Cl Im wsm WIðmÞ > ; 28h  93  ðm;sÞ   ðm;sÞ   i  > < Im wsm ^kI Cl þ ^kI Cl Im wsm WRðmÞ þ > = 7 6 7 6  ðm;sÞ   ðm;sÞ   h  i    6> ð m Þ > ;7 7 6 : Re wsm ^kI Cl þ ^kR Cl Im wsm WI 7 6 9 ¼ 68h  7 i      ðm;sÞ   ðmÞ ðm;sÞ 6> =7 7 6 < Re wsm ^kI Cl þ ^kR Cl Im wsm WR þ > 7 6  ðm;sÞ   ðm;sÞ   i 5 4> h     : Re wsm ^kR Cl þ ^kR Cl Re wsm WðmÞ > ; I

2N d 1

2N d 1

2

ðLs7 Þ1 6 ðLs7 Þ 6 2 Ls7 ¼ 6 . 6 . 4 . ðLs7 ÞNm

3 7 7 7 7 5 2Nd Nm 1

2 8h   ðm;sÞ   ðm;sÞ   i   9 3 > < Re wsm a0 ^kR M þ a0 ^kR M Re wsm Re Wm  > = 7 6 7 6  ðm;sÞ   ðm;sÞ   h  i     7 6 > >      ^ ^ : ; Re wsm a0 kI M þ a0 kR M Im wsm Im Wm 7 6 7 6 9 ðLs7 Þm ¼ 6 8 h  7     i      ðm;sÞ ðm;sÞ 7 6>      > ^ ^ 6 <  Re wsm a0 kI M þ a0 kR M Im wsm Re Wm þ = 7 7 6     i 4> h         >5 ðm;sÞ   ^ : Im wsm a0 ^kðm;sÞ ; M þ a0 kI M Im wsm Im Wm I 2

ðLs8 Þ1

6 ðLs8 Þ 6 2 Ls8 ¼ 6 .. 6 4 . ðLs8 ÞNm

3 7 7 7 7 5 2Nd Nm 1

28h   ðm;sÞ   ðm;sÞ   i   9 3 > < Im wsm a0 ^kI M þ a0 ^kI M Im wsm Re Wm þ > = 7 6 7 6  ðm;sÞ   ðm;sÞ   h  i     6> >7    6 : Re wsm a0 ^kI M þ a0 ^kR M Im wsm Im Wm ; 7 7 6 ðLs8 Þm ¼ 6 8 h   ðm;sÞ    ðm;sÞ    i    9 7 6> =7 7 6 < Re wsm a0 ^kI M þ a0 ^kR M Im wsm Re Wm þ > 7 6  ðm;sÞ   ðm;sÞ   i   5 4> h    : Re wsm a0 ^kR M þ a0 ^kR M Re wsm Im Wm > ; 2

ðLs9 Þ1 6 ðLs9 Þ 6 2 Ls9 ¼ 6 .. 6 4 . ðLs9 ÞNm

2N d 1

2N d 1

3 7 7 7 7 5 2Nd Nm 1

2 8h   ðm;sÞ   ðm;sÞ   i   9 3 > < Re wsm a1 ^kR K þ a1 ^kR K Re wsm Re Wm  > = 7 6 7 6 h  i    ðm;sÞ   ðm;sÞ     7 6 > >      ^ ^ 6 : Re wsm a1 kI K þ a1 kR K Im wsm Im Wm ; 7 7 68 9 ðLs9 Þm ¼ 6 h   ðm;sÞ    ðm;sÞ     i    > 7 7 6>  ^ ^ 6 <  Re wsm a1 kI K þ a1 kR K Im wsm Re Wm þ = 7 7 6 h i     4>         >5 ðm;sÞ   ^ : Im wsm a1 ^kðm;sÞ ; K þ a1 kI K Im wsm Im Wm I

2N d 1

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

2

ðLs10 Þ1

6 ðLs10 Þ 6 2 Ls10 ¼ 6 .. 6 4 . ðLs10 ÞNm

3 7 7 7 7 5 2N d N m 1

28h   ðm;sÞ   ðm;sÞ   i   9 3 > < Im wsm a0 ^kI M þ a0 ^kI M Im wsm Re Wm þ > = 7 6 7 6  ðm;sÞ   ðm;sÞ   h  i     7 6> >      ^ ^ : ; Re wsm a1 kI K þ a1 kR K Im wsm Im Wm 7 6 7 6 9 ðLs10 Þm ¼ 6 8 h  7 i          ðm;sÞ ðm;sÞ 7 6 >      > ^ ^ 6 < Re wsm a1 kI K þ a1 kR K Im wsm Re Wm þ = 7 7 6     i 4 > h        > 5 ðm;sÞ   ^ : Re wsm a1 ^kðm;sÞ ; k K K W þ a Re w Im 1 R sm m R h

Ls11 ¼ GTh1P1 Gh1P1 þ GTh1P2 Gh1P2  2GTh1P1 Gh1P2

i

h i Ls12 ¼ GTh1X1 Gh1X1 þ GTh1X2 Gh1X2 þ 2GTh1X2 Gh1X1 Ls13 ¼

h     i GTh1P1 Gh2Q1 þ GTh1P2 Gh2Q 2  GTh1P1 Gh2Q2 þ GTh1P2 Gh2Q 1

Ls14 ¼

h     i GTh1X1 Gh2Y1 þ GTh1X2 Gh2Y2 þ GTh1X1 Gh2Y2 þ GTh1X2 Gh2Y1

Ls15 ¼

h     i GTh1P1 Gh3R1 þ GTh1P2 Gh3R2  GTh1P1 Gh3R2 þ GTh1P2 Gh3R1

Ls16 ¼

h     i GTh1X1 Gh3Z1 þ GTh1X2 Gh3Z2 þ GTh1X1 Gh3Z2 þ GTh1X2 Gh3Z1

h i Ls21 ¼ GTh2Q1 Gh2Q 1 þ GTh2Q 2 Gh2Q 2  2GTh2Q 1 Gh2Q2 h i Ls22 ¼ GTh2Y1 Gh2Y1 þ GTh2Y2 Gh2Y2 þ 2GTh2Y1 Gh2Y2 Ls23 ¼

h     i GTh2Q1 Gh3R1 þ GTh2Q 2 Gh3R2  GTh2Q1 Gh3R2 þ GTh2Q2 Gh3R1

Ls24 ¼

h     i GTh2Y1 Gh3Z1 þ GTh2Y2 Gh3Z2 þ GTh2Y2 Gh3Z1 þ GTh2Y1 Gh3Z2

h i Ls29 ¼ GTh3R1 Gh3R1 þ GTh3R2 Gh3R2  2GTh3R1 Gh3R2 h i Ls30 ¼ GTh3Z1 Gh3Z1 þ GTh3Z2 Gh3Z2 þ 2GTh3Z1 Gh3Z2   Ls35 ¼ ST1 S1 þ ST2 S2  2ST1 S2   Ls36 ¼ UT1 U1 þ UT2 U2 þ 2UT2 U1   Ls37 ¼ ST1 T1 þ ST2 T2  ST1 T2  TT1 S2   Ls38 ¼ UT1 V1 þ UT2 V2 þ UT1 V2 þ VT1 U2   Ls39 ¼ TT1 T1 þ TT2 T2 þ 2TT1 T2   Ls40 ¼ VT1 V1 þ VT2 V2 þ 2VT2 V1

2N d 1

17

18

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Besides, the components ðLs17 ; Ls18 ; Ls19 ; Ls20 ; Ls25 ; Ls26 ; Ls27 ; Ls28 ; Ls31 ; Ls32 ; Ls33 ; Ls34 Þ can be obtained analytically similarly to those obtained earlier using differentiation of the objective function with respect to the required parameters. But the formulation of these expressions is quite tedious and complex and hence only the expression for Ls17 is shown in Appendix A and the remaining expressions can be obtained in a similar approach for concise presentation of the formulation. 4. Probabilistic estimation of changes/damages In order to indicate the level of reduction of an updating parameter, a probability is considered implying that a structural parameter has been reduced by a fraction d compared to its initial state (corresponding to the initial/undamaged state of the structure). This probability is computed using the value of updated parameter and its standard deviation obtained from the covariance matrix estimated in Section 3 using the similar procedure as described in [51]. The probability of reduction in lth structural parameter can be obtained as shown in Eq. (75).

  pr in Pred l ðdÞ ¼ P hl < ð1  dÞhl Z ¼

1

1

    in in p hin dhin P hpr l l l < ð1  dÞhl jhl

2

ð75Þ

3 dÞhin l

hpr l

ð1   6 7 U4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pr 2 5 2  in 2 þ rl ð1  dÞ rl where, Uð Þ is the cumulative distribution function of the standard Gaussian random variable. Besides, hin and hpr are the l l th most probable values of l structural parameter for the initial and possibly reduced cases respectively. Further, rin l and

in rpr and hpr respectively. l are the corresponding standard deviations of hl l

5. Validation of the proposed approach In this section, performance of the proposed approach is validated by comparing it with Gibbs sampling technique (MCMC based technique) [41] and sensitivity based technique [52] by utilizing incomplete complex modal data. Two numerical examples are considered in this regard: (a) spring-mass damper system and (b) ASCE benchmark structure. In case of Sensitivity approach, although found in literature like [52] for model updating utilizing complex modal data, such updating is required to be extended for multiple data sets of complex modal data for the present work. This extension is concisely presented in next subsection followed by comparative studies. It may be noted that the updating structural parameters are considered in terms of fraction of the nominal values of the parameters while the Rayleigh coefficients are considered in terms of its absolute values. 5.1. Sensitivity method for multiple data sets of complex modal data The basic principle of sensitivity technique [1] works by iteratively solving the linear relation shown in Eq. (76).

  zm  zðhÞ zm  zðhi Þ þ Gijh¼hi Dhi ¼ zm  zðhi Þ  Gijh¼hi Dhi

ð76Þ

Here, zm indicates measured modal data, zðhÞ indicates the modal data obtained from analytical model for a given set of parameter set h. It can be observed from Eq. (76) that the error function is expressed as a linear relation involving the residual point ri ¼ zm  zðhi Þ , the sensitivity matrix Gijh¼hi and required change in parameter at ith point. The expression shown in Eq. (76) can be expanded for our current problem consisting of N s sets of complex modal data in the form of complex eigenvalues and eigenvectors, as shown in Eq. (77).

2 6 6 6 6 6 6 6 6 6 6 4

3

2

zk;1  zk;1 ðhi Þ 7 6 .. 7 6 . 7 6 6 7 6  z z zk;Ns  zk;Ns ðhÞ 7 k;N k;N s ðhi Þ s 7¼6 6  z ðhi Þ 1 6 1Nm  zMAC;1 ðhÞ 7 1 MAC;1 Nm 7 6 7 6 .. .. 7 6 5 4 . . 11Nm  zMAC;Ns ðhi Þ 1Nm  zMAC;Ns ðhÞ zk;1  zk;1 ðhÞ .. .

3

2 Gk;1jh¼hi 7 6 .. 7 6 7 6 . 7 6 7 6 G k;N 7 6 s jh¼hi 76 7 6 GMAC;1jh¼hi 7 6 7 6 .. 7 6 . 5 4 GMAC;Ns jh¼hi

3

72 3 7 ð Dh 1 Þ i 7 76 7 76 ðDh2 Þi 7 76 7 76 7 .. 76 7 74  .  5 7 7 DhNq i 5

ð77Þ

Here, the measured modal vector zm is contributed by measured eigenvalues and eigenvectors for all data sets and N q ð¼ N h1 þ N h2 þ N h3 þ 2Þ is the total number of unknown parameters corresponding to the structural parameters and Rayleigh coefficients. Also, 1Nm indicates unit vector of size N m  1. In upper portion of the error vector, the discrepancies are

19

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

measured between measured eigenvalues zk;s and FE model based eigenvalues zk;s ðhÞ (corresponding to sth experimental modal data set) for m = 1, 2, . . . , Nm and s = 1, 2, . . . , Ns. On the other hand, in lower portion of the error vector, discrepancies between unity and modal assurance criteria (MAC) values between the experimental modes and FE model based analytical modes for all measured modes and all data sets are considered. Besides, the parameter vector h consists of mass, stiffness, damping parameters and the Rayleigh coefficients. It may be mentioned that the calculation of the MAC value between rth ^

analytical complex mode shape Wa and qth experimental complex mode shape W is performed using a procedure as menx

tioned in [53] and expressed in Eq. (78).

  ^  2    T fWa gr W  x   q ! MACðr; qÞ ¼    ^ T  ^  T  fWa gr fWa gr W W x

x

q

ð78Þ

q

where, * indicates complex conjugate of a complex vector and j:j indicates modulus of a complex number. On the other hand, zk;s ðhi Þ and zMAC;s ðhi Þ denote analytical eigenvalue and MAC value between experimental and analytical modes at ith parameter point hi and corresponding to sth experimental modal data set. Besides, Gk;sjh¼hi and GMAC;sjh¼hi denote sensitivity matrix of respective eigenvalue and MAC at ith parameter point hi . The symbols zk;s , zk ðhi Þ and zMAC;s ðhi Þ can be expressed as shown in Eqs. (79)–(81).

 zk;s ¼

^

^

T

^

k ; k ; :::; k

ð1;sÞ ð2;sÞ

ð79Þ

ðN m ;sÞ



T zk;s ðhi Þ ¼ kð1;sÞ ðhi Þ; kð2;sÞ ðhi Þ; :::; kðNm ;sÞ ðhi Þ

ð80Þ



T zMAC;s ðhi Þ ¼ MAC ð1;sÞ ðhi Þ; MAC ð2;sÞ ðhi Þ; :::; MAC ðNm ;sÞ ðhi Þ

ð81Þ

Here, kðk;sÞ ðhi Þ denotes kth FE model based eigenvalue for sth experimental modal data set and MAC ðk;sÞ ðhi Þ indicates MAC value between the kth FE model based complex mode and kth experimental mode of the sth modal data set at parameter point hi . The sensitivity matrices Gk;sjh¼hi and GMAC;sjh¼hi are actually Jacobian matrices expressed in Eqs. (82) and (83) respectively.

2

Gk;sjh¼hi

6 6 6 6 ¼6 6 6 4

@kð1;sÞ ðhi Þ @h1

@kð1;sÞ ðhi Þ @h2



@kð1;sÞ ðhi Þ @hNq

@kð2;sÞ ðhi Þ @h1

@kð2;sÞ ðhi Þ @h2



@kð2;sÞ ðhi Þ @hNq

.. .

.. .

@kðNm ;sÞ ðhi Þ @h1

2

GMAC;sjh¼hi

6 6 6 6 ¼6 6 6 4



@kðNm ;sÞ ðhi Þ @h2



.. .

3

@kðNm ;sÞ ðhi Þ @hNq

7 7 7 7 7 7 7 5

ð82Þ

N m Nq

@MAC ð1;sÞ ðhi Þ @h1

@MAC ð1;sÞ ðhi Þ @h2



@MAC ð1;sÞ ðhi Þ @hNq

@MAC ð2;sÞ ðhi Þ @h1

@MAC ð2;sÞ ðhi Þ @h2



@MAC ð2;sÞ ðhi Þ @hNq

.. .

.. .

@MAC ðNm ;sÞ ðhi Þ @h1

@MAC ðNm ;sÞ ðhi Þ @h2

.. .



@MAC ðNm ;sÞ ðhi Þ @hNq



3 7 7 7 7 7 7 7 5

ð83Þ

Nm N q

The components of the sensitivity matrices can be computed using analytical expressions discussed in [52,54,55] and shown in Eqs. (84) and (85).

 

@kðr;sÞ ðhi Þ @M @C @K ¼ WTðr;sÞ k2ðr;sÞ þ kðr;sÞ þ Wðr;sÞ =WTðr;sÞ 2kðr;sÞ M þ C Wðr;sÞ @hp @hp @hp @hp ^ W ðr;sÞ H

@MAC ðr;sÞ ðhi Þ ¼ @hp



@ mðr;sÞ @hp

mHðr;sÞ þ mðr;sÞ

@ mH ðr;sÞ @hp



^ ðr;sÞ mH mðr;sÞ  W ^ mðr;sÞ mH W ^ W ðr;sÞ ðr;sÞ ðr;sÞ ðr;sÞ H



mHðr;sÞ mðr;sÞ

2 

H

^ W ^ W ðr;sÞ ðr;sÞ



ð84Þ

@ mH ðr;sÞ @hp

mðr;sÞ mðr;sÞ þ mHðr;sÞ @@h p

ð85Þ

Here, mðr;sÞ ¼ LWðr;sÞ denotes the analytical mode shape corresponding to measured DOFs of the full shape analytical mode shape Wðr;sÞ for rth mode and sth modal data set. Besides, vector

@ Wðr;sÞ @hp

@ mðr;sÞ @hp

¼L

@ Wðr;sÞ @hp

and the sensitivity vector of analytical complex eigen-

as already discussed in [54] can be expressed as in Eq. (86).

20

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Xd @ Wðr;sÞ ¼ arpr Wðr;sÞ þ arpk Wðk;sÞ @hp k–r 2N

ð86Þ

where,

arpr

  T @M @C 1 Wðr;sÞ 2kðr;sÞ @hp þ @hp Wðr;sÞ ¼   2 WTðr;sÞ 2kðr;sÞ M þ C Wðr;sÞ

ð87Þ h

arpk ¼ 



1





WTðk;sÞ 2kðk;sÞ M þ C Wðk;sÞ

Additionally, the sensitivity vector

i

@M @C @K WTðk;sÞ k2ðr;sÞ @h þ kðr;sÞ @h þ @h Wðr;sÞ p p p

@ Wðr;sÞ @hp

kðr;sÞ  kðk;sÞ



ð88Þ

can also be obtained using Nelson’s method as described in [55]. It may be men-

tioned that the relation in Eq. (77) should be normalized as shown in Eq. (89) in order to avoid ill-conditioning of sensitivity matrix.

2

ðzk;1  zk;1 ðhÞÞ=zk;1 ðh0 Þ .. .

6 6 6 6 6 ðzk;Ns  zk;Ns ðhÞÞ=zk;Ns ðh0 Þ 6 6 ð1N  zMAC;1 ðhÞÞ=zMAC;1 ðh0 Þ m 6 6 .. 6 4 . ð1Nm  zMAC;Ns ðhÞÞ=zMAC;Ns ðh0 Þ

3

2

ðzk;1  zk;1 ðhi ÞÞ=zk;1 ðh0 Þ .. .

3

2

Gk;1jh¼hi zk;1hð0h0 Þ

6 .. 7 6 7 6 . 7 6 6 h0 7 6 G k;N s jh¼hi z 7 6 k;Ns ðh0 Þ 7þ6 7 6 G h0 7 6 MAC;1jh¼hi z MAC;1 ðh0 Þ 7 6 7 6 .. 5 6 . 4  zMAC;Ns ðhi ÞÞ=zMAC;Ns ðh0 Þ h0 GMAC;Ns jh¼hi zMAC;N ðh0 Þ

7 6 7 6 7 6 7 6 7 6 ðzk;Ns  zk;Ns ðhi ÞÞ=zk;Ns ðh0 Þ 7¼6 7 6 ð1N  zMAC;1 ðhi ÞÞ=zMAC;1 ðh0 Þ m 7 6 7 6 .. 7 6 5 4 . ð1Nm

3 72 ðDh Þ 3 1 i 7 7 ðh0 Þ1 76 7 76 ðDh2 Þi 7 76 ðh0 Þ2 7 76 7 76 . 7 76 . 7 76 . 7 74 ðDh Þ 5 Nq 7 7 ðh0 ÞN i q 5

ð89Þ

s

Here, zk;s ðh0 Þ and zMAC;s ðh0 Þ represent the analytical eigenvalues and MAC values corresponding to nominal values of the parameter vector h0 for sth modal data set. Subsequently, after solving Eq. (89) we get the change in parameter vector at ith parameter point Dhi . The parameter value at (i + 1)th point is thus given by hiþ1 ¼ hi þ Dhi . The entire procedure is repeated until converged results are obtained. One disadvantage of the sensitivity method of model updating is the requirement of mode matching between the analytical and measured modes for each iteration which requires repeated solving of eigen equation (such repetitive eigen-solutions are not required in the proposed approach and Gibbs sampling approach). 5.2. Comparative study using spring-mass-damper system Same example on spring-mass damper system (as in [41] facilitating validation/comparison with published results) is considered here. Moreover, similar numerical FE modelling and numerical simulation of target complex modal data are also considered in the same way (spring-mass damper system with 4 DOFs) as in [41] for appropriate comparison (having similar modelling error for all three methods of updating). The schematic diagram of a typical n-DOF spring-mass-damper system in generalized form is shown in Fig. 1. The nominal properties of the structural components and the modal data details are already described in [41] but are shown here for better readability. Each lumped mass value is equal to 1 kg, while values of the spring stiffness are: k1 = k3 = k5 = 7000 N/m and k2 = k4 = 8000 N/m and values of the damping coefficient as: c1 = c3 = c5 = 4.2 Ns/m and c2 = c4 = 3.2 Ns/m. A total 10 sets of modal data is considered for model updating using the proposed approach where each set of modal data consists of first two modal frequencies, damping factors and partial complex mode shapes corresponding to DOFs 1 and 2. Each set of noisy modal data is generated by adding random values generated from zero mean Gaussian distribution with standard deviation equal to 3% of the exact values. It may be mentioned that there are 4 mass parameters corresponding to the lumped mass, 5 stiffness parameters for the spring stiffness and 5 damping parameters corresponding to damping coefficient of each of the viscous dampers. The updated values of the parameters obtained using the proposed approach and the coefficient of variation (COV (%)) values are shown in Table 1 along with the values obtained using the Gibbs sampling approach (as obtained from Table 1 of the work by Cheung and Bansal [41]). Additionally results obtained using the Sensitivity approach explained earlier is also shown in Table 1. It has been

Fig. 1. Spring-mass-damper system.

21

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Table 1 Target and updated structural parameters of the spring-mass damper system along with their COV (%) values obtained using the proposed approach, Gibbs sampling approach and Sensitivity method. Parameters

m1 m2 m3 m4 c1 c2 c3 c4 c5 k1 k2 k3 k4 k5

Target

1 1 1 1 1 1 1 1 1 1 1 1 1 1

Gibbs sampling approach

Sensitivity method

Updated

Proposed approach COV (%)

Updated

COV (%)

Updated

0.996 0.994 1.00 0.995 0.992 0.997 1.00 0.993 1.00 1.00 0.994 1.00 0.995 1.00

0.0038 0.0050 0.0024 0.0021 0.0736 0.2135 0.0510 0.2159 0.0708 0.0027 0.0007 0.0026 0.0032 0.0028

0.99 0.99 1.00 1.00 0.99 0.98 0.99 1.14 1.00 0.99 0.97 0.97 0.94 1.05

1.00 1.00 1.00 1.00 1.64 3.49 3.80 13.50 6.60 2.02 3.47 4.15 15.89 7.05

1.031 1.000 1.000 1.000 1.012 1.000 1.026 1.000 0.978 1.043 1.007 1.018 1.000 0.969

observed that the updated results obtained using the proposed approach are consistent with the target values and also its performance is quite similar to the Gibbs sampling approach and Sensitivity method for some parameters while somewhat better than the two approaches for other parameters. Also, it may be noted that the COV values are much lower in case of the proposed approach in comparison of the Gibbs sampling approach which proves its effectiveness. The results shown in Table 1 for the proposed approach are obtained using 2000 iterations of the proposed algorithm while the updated values obtained using Gibbs sampling approach represent the posterior mean of 10,000 samples generated using the sampling procedure as mentioned in [41] and the results obtained using Sensitivity approach are obtained after 5000 iterations. Also, iteration histories of the updating parameters are shown in Fig. 2 obtained using the proposed approach and it shows that the updating parameters are well converged after about 1500 iterations.

5.3. Comparative study using ASCE benchmark structure Subsequently, validation is performed using a three dimensional two-bay by two-bay four storied steel frame structure which was originally designed for IASC-ASCE phase-I simulated structural health monitoring (SHM) benchmark problem [56]. Similar numerical FE modelling is considered in the same way (using 36 DOFs) as in [41] for appropriate comparison having similar modelling error for all three methods of updating. Although modelling of this structure, numerical simulation of target complex modal data and the choice of updating parameters are considered as same as in [41], these are further presented here concisely for better readability. It has a 2.5 m  2.5 m plan and is 3.6 m tall with braces in each story located on the exterior faces – braces however removed and replaced with viscous dampers with damping coefficient 20 kN s/m (as shown in Fig. 3). In addition to the non-classical damping due to viscous dampers, classical damping with 1% damping ratio for all modes is also considered. The nominal properties of the structural members can be seen from [56]. Mass of each floor is considered to be equally contributed by four connected slabs. Masses of each slab associated with 1st, 2nd, 3rd and 4th floors are considered as 800 kg, 600 kg, 600 kg and 400 kg respectively. Besides, 2 sets of simulated modal data are used, where each set of modal data consists of modal frequencies, partial complex mode shapes corresponding to four translational DOFs at each floor, of the first eight translational modes (four each in  and y directions). Each set of noisy modal data is generated by adding random values generated from zero mean Gaussian distribution with standard deviation equal to 2% of the exact values. It may be mentioned that performance in updating for ASCE benchmark structure problem are found to be quite similar while updated with 2 ensembles or 5 ensembles except considerable difference in computation time. It is to be noted that a 36-DOF model is chosen for updating purpose in a similar approach as chosen in [41]. Each floor has 9 DOFs where 2 DOFs correspond to x-direction and y-direction, 1 DOF corresponds to rotational DOF with respect to z-axis, 3 rotational DOFs each with respect to  and y axis. Such choice of DOFs is chosen to consider x-y plane rigidity and allowing rotation along  and y axis for each floor. In order to compare with Gibbs sampling approach performed in [41], the choice of updating parameters are considered similar to [41]. The mass parameters ðh1 Þ correspond to the slab masses for each floor, thus making a total of 4 mass parameters. The nominal values of the slab masses as calculated from [56] are obtained as: 3200 kg (=4  800 kg), 2400 kg (=4  600 kg), 2400 kg (=4  600 kg) and 1600 kg (=4  400 kg). Besides, the stiffness matrix consists of 3 sub-structure stiffness matrices for each floor, thus making a total of 12 stiffness parameters. For each floor, out of the three sub-structure stiffness matrices, one corresponds to 2 translational and 1 rotational DOF (along z-axis), and one each corresponds to 3 DOFs along  and y axis respectively. The order of the stiffness parameter is as follows: ðh2 Þl ¼ gc;l , ðh2 Þ4þl ¼ gRx;l , and ðh2 Þ8þl ¼ gRy;l for l = 1, 2, 3 and 4 where, gc;l , gRx;l and gRy;l represent 3DOFs at centre (2 translational and 1 rotational), 3 rotational DOFs along x-axis and 3 rotational DOFs along y-axis of the lth floor respectively. Lastly, the damping parameters considered in this exercise are the damping coefficient values of the dampers assuming equal damping

22

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

(a)

(b)

(c) Fig. 2. Iteration histories of (a) mass parameters (b) stiffness parameters and (c) damping parameters of the 4-DOF spring-mass-damper system.

Fig. 3. Diagram of modified ASCE benchmark structure.

23

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Table 2 Target and updated structural parameters of the ASCE benchmark structure along with the COV (%) values obtained using proposed approach, Gibbs sampling approach and Sensitivity method. Parameters

(h1)1 (h1)2 (h1)3 (h1)4 (h2)1 (h2)2 (h2)3 (h2)4 (h2)5 (h2)6 (h2)7 (h2)8 (h2)9 (h2)10 (h2)11 (h2)12 (h3)1 (h3)2 (h3)3 (h3)4 a0/a0n a1/a1n

Target

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Proposed approach

Gibbs sampling approach

Sensitivity method

Updated

Updated COV (%)

Updated

Updated COV (%)

Updated

0.98 0.99 1.02 0.97 0.99 0.99 0.97 0.99 0.97 1.00 0.99 0.98 0.96 0.97 0.99 0.98 0.99 1.01 0.99 0.98 0.98 1.02

1.80E07 4.45E07 5.67E07 3.80E07 1.36E07 2.22E07 2.63E07 5.28E07 6.78E09 7.54E09 2.23E08 9.90E09 1.39E08 2.47E08 1.93E07 5.26E08 5.77E07 7.67E07 1.14E06 1.61E06 4.23E04 1.56E04

0.96 0.97 0.97 0.99 0.94 1.01 0.98 0.96 0.86 1.03 1.01 0.94 0.87 1.01 0.97 0.95 1.01 0.99 0.98 0.99 0.95 1.15

0.76 0.69 0.68 0.73 1.96 2.44 1.58 1.49 7.67 3.44 2.72 2.71 5.00 2.85 2.04 1.93 1.58 0.82 0.84 0.72 5.85 8.63

1.00 0.97 0.98 0.96 0.99 0.99 0.99 1.00 1.00 1.00 0.99 1.00 1.01 0.99 0.95 0.986 0.993 1.000 0.958 1.000 0.94 1.08

coefficient values of viscous dampers for each story, thus making a total of 4 damping parameters. In addition to the nonclassical damping due to viscous dampers, Rayleigh damping is also considered and in view of this Rayleigh coefficients are considered as uncertain parameters. The nominal Rayleigh coefficients are calculated using 1st translational mode in xdirection and 4th translational mode in y-direction of the analytical model. Similar to the first example, the updated values of the parameters obtained using the proposed approach and the coefficient of variation (COV (%)) values are shown in Table 2 along with the values obtained using the Gibbs sampling approach (as obtained from Table 5 of the work by Cheung and Bansal [41]). Also, the results obtained using Sensitivity method are also shown in Table 2. Here also, it is observed that the performance of updating of the structure is quite similar in comparison to the Gibbs sampling approach and Sensitivity approach for some parameters while it is somewhat closer to target values in comparison to the two approaches for some parameters. Also, the COV (%) values of the proposed approach are much lower than that obtained using Gibbs sampling approach similarly to the first example. Besides, the results shown in Table 2 for the proposed approach are obtained using 3000 iterations of the proposed algorithm while the updated values obtained using Gibbs sampling approach represent the posterior mean of 20,000 samples generated using the sampling procedure as mentioned in [41] and the results obtained using Sensitivity method are obtained using 7000 iterations. Besides, the updating parameters are found to be well converged after about 2500 iterations for the proposed approach as observed from iteration histories shown in Fig. 4. It may be mentioned that the Rayleigh coefficients are expressed in terms of fraction of the nominal values in Table 2 and Fig. 4 (d) in order to compare it with the scaled parameter c ¼ ½c0 ; c1 T obtained using Gibbs sampling based approach. 5.4. Damage detection using the proposed approach The procedure for estimation of changes/damages in structural parameters using the proposed approach is already discussed in Section 4. In this section, performance of the proposed approach in damage detection is investigated also using spring-mass damper system and ASCE benchmark structure with multiple damage cases. Damage cases are simulated with changes only in mass, stiffness and damping parameters individually along with combined changes in mass, stiffness and damping parameters. Moreover level/extent of damages are simulated as quite low targeting stringent validation. Besides, it is interesting to study the computational efficiency of the proposed approach. To quantify such computational efficiency, a single computer (make: HP, model: EliteDesk 800 GI SFF) is used having key features like:(a) processor: Intel(R) Core(TM) i7- 4770CPU @ 3.40 GHz (b) RAM:4096 MB (c) hard disk drive: 500 GB and (d) operating system: Windows 8.1 Pro 64-bit. Besides, MATLAB 2013a is used for writing the codes however without applying any parallel-computing support. 5.4.1. Damage detection using spring-mass-damper system The first example considered here is a spring-mass-damper system (Fig. 1) having 6 DOFs (problem with higher size of system matrices than the earlier considered system with 4 DOFs). The nominal values of each lumped mass is equal to 2 kg, while values of each spring stiffness is equal to 10000 N/m and values of each damping coefficient of the viscous damper is equal to 5 Ns/m. In order to observe the effectiveness of the proposed approach in detecting change/damage of the structural parameter, four damage cases are considered as shown in Table 3. For these damage cases, 12 sets of simulated

24

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

(a)

(b)

(c)

(d)

Fig. 4. Iteration histories of (a) mass parameters (b) stiffness parameters (c) damping parameters and (d) scaled Rayleigh coefficients of the ASCE benchmark structure.

Table 3 Damage cases of the 6-DOF spring-mass-damper system. Damage cases

Change of mass

Change of stiffness

Change of damping

I II III IV

10% reduction of m2 – – 10% reduction of m2

– 15% reduction of k3 – 15% reduction of k3

– – 20% reduction of c1 20% reduction of c1

modal data (Ns = 12) are generated where for each of the modal data set, first three modal frequencies and modal damping ratios and partial complex mode shape corresponding to DOFs 1, 3 and 6 are considered. The modal data sets are generated by adding different sets of noise to the exact modal data of the target model. Each set of noisy modal data is generated by adding random values generated from zero mean Gaussian distribution with standard deviation equal to 3% of the exact values. Besides, there are 6 mass parameters corresponding to the lumped mass, 7 stiffness parameters for the spring stiffness and 7 damping parameters corresponding to damping coefficient of each of the viscous dampers. The model updating of the spring-mass-damper system is performed using the formulations as presented in Section 2 for all the damage cases. It may be mentioned here that all the results for this exercise are obtained with 4500 iterations for the proposed approach. It is also to be noted that the performance of model updating is likely to improve with increase in number of identified modes and observed DOFs and vice versa. Performance in updating of eigenvalues (or frequencies) using the proposed approach for the undamaged and all the damage cases is shown in Table 4. It can be clearly observed that the updated frequencies are close to the target frequencies for all the cases. On the other hand, diagonal modal assurance criteria (MAC) values between initial and target mode shapes are also significantly improved (as shown in Table 5 applying Eq. (78)) using the proposed approach for the undamaged and damage cases.

25

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524 Table 4 Target and updated frequencies (Hz) of the 6-DOF system for the undamaged and damage cases. Modes

1 2 3

Undamaged case

Damaged case I

Damage case II

Damage case III

Target

Updated

Target

Updated

Target

Updated

Target

Updated

Damage case IV Target

Updated

5.0085 9.7658 14.0335

5.0085 9.7658 14.0334

5.052 9.9016 14.0723

5.052 9.9015 14.0723

4.985 9.669 13.715

4.985 9.669 13.715

5.0085 9.7658 14.0335

5.0085 9.7658 14.0334

5.0262 9.8174 13.7596

5.0263 9.8174 13.7597

Table 5 MAC values (before and after updating) of the 6-DOF system for the undamaged and damage cases. Modes

1 2 3

Undamaged case

Damaged case I

Damage case II

Damage case III

Damage case IV

Initial

Updated

Initial

Updated

Initial

Updated

Initial

Updated

Initial

Updated

0.9997 0.9994 0.9996

0.9998 0.9995 0.9998

0.9996 0.9995 0.9997

1.0000 0.9999 1.0000

0.9998 0.9971 0.9953

1.0000 0.9998 0.9999

0.9997 0.9994 0.9996

0.9999 0.9999 1.0000

0.9997 0.9978 0.9984

1.0000 0.9999 0.9995

It may be mentioned that the results of the updated mode shapes are not the updated system mode shapes, rather these mode shapes are obtained by eigen analysis of the system matrix (as shown in Eq. (16)) using the updated structural parameters. Also, the results shown in Table 5 for the MAC values are obtained using average of the MAC values obtained for all the modal data sets. Performance of the updating parameters are shown in Tables 6 and 7 along with the coefficient of variation (COV) values for all cases. Here, values of COV are calculated as a ratio of standard deviation and the updated parameters. The proposed approach demonstrate quite good level of performance in updating of the structural parameters and also much lower updated COV values (%) of the updating parameter corresponding to the posterior PDF. Also, the probabilities of change/damage of the structural parameters for damage case IV are obtained using the procedure explained in Section 4 and are shown in Fig. 5 and it can be clearly observed that the extents of change/damage of the structural parameters nearly match with the assumed values for change/damage. Besides to study the computational efficiency of the proposed approach, a measure of accuracy known as modal distance value (MD) is adopted as shown in Eq. (90).

MDðh1 ; h2 ; h3 Þ ¼

Ns X Nm X s¼1 r¼1

2

3 !2 ^f     f a ðh1 ; h2 ; h3 Þ ð r;s Þ ^ ðr;sÞ ; Wa ðh1 ; h2 ; h3 Þ 5 4 þ 1  MAC W ^f

ð90Þ

ðr;sÞ

^ ðr;sÞ stand for experimental frequency and experimental eigenvector of the rth mode for the sth modal data where, ^f ðr;sÞ and W set respectively, f a ðh1 ; h2 Þ represents ath analytical frequency, and Wa ðh1 ; h2 ; h3 Þ represents ath analytical eigenvector. Table 6 Target and updated structural parameters of the 6-DOF system along with their COV values for the undamaged case, damage case I and II. Parameters

m1 m2 m3 m4 m5 m6 k1 k2 k3 k4 k5 k6 k7 c1 c2 c3 c4 c5 c6 c7

Undamaged case

Damage case I

Damage case II

Target

Updated

COV (%)

Target

Updated

COV (%)

Target

Updated

COV (%)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.9995 0.9924 0.9790 0.9748 0.9795 0.9781 0.9993 0.9588 0.9753 0.9832 0.9933 0.9770 0.9802 0.9985 0.9270 0.9824 0.9783 1.0731 0.9850 0.9649

0.00249 0.00055 0.00681 0.00638 0.00345 0.00453 0.00107 0.00194 0.00342 0.00297 0.00239 0.00148 0.00168 0.06438 0.10533 0.04499 0.06875 0.04195 0.10345 0.06670

1 0.9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.9847 0.9040 0.9964 1.0087 1.0038 1.0087 0.9953 0.9935 0.9919 1.0028 1.0010 1.0170 1.0083 0.9965 0.9946 0.9907 1.0034 1.0003 1.0153 1.0083

0.0031 0.0009 0.0029 0.0028 0.0032 0.0031 0.0016 0.0009 0.0017 0.0019 0.0016 0.0015 0.0015 0.0632 0.1020 0.0453 0.0676 0.0440 0.1024 0.0649

1 1 1 1 1 1 1 1 0.85 1 1 1 1 1 1 1 1 1 1 1

1.0973 1.0339 1.0453 1.0273 1.0169 0.9827 1.0657 1.0659 0.8623 1.0182 1.0124 0.9902 1.0072 1.0664 1.0583 1.0646 1.0171 1.0165 0.9905 1.0067

0.0029 0.0031 0.0043 0.0040 0.0031 0.0033 0.0011 0.0014 0.0021 0.0018 0.0016 0.0014 0.0023 0.0609 0.1033 0.0376 0.0668 0.0445 0.1153 0.0638

26

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Table 7 Target and updated structural parameters 6-DOF system along with their COV values for damage case III and IV. Parameters

m1 m2 m3 m4 m5 m6 k1 k2 k3 k4 k5 k6 k7 c1 c2 c3 c4 c5 c6 c7

Damage case III

Damage case IV

Target

Updated

COV (%)

Target

Updated

COV (%)

1 1 1 1 1 1 1 1 1 1 1 1 1 0.8 1 1 1 1 1 1

1.0008 1.0010 0.9988 1.0005 0.9993 0.9996 1.0004 0.9991 0.9996 1.0005 1.0006 1.0003 0.9993 0.8031 1.0012 1.0003 1.0027 1.0015 1.0015 1.0011

0.0024 0.0004 0.0039 0.0009 0.0030 0.0026 0.0011 0.0010 0.0022 0.0018 0.0013 0.0015 0.0008 0.0800 0.1014 0.0440 0.0672 0.0438 0.1016 0.0641

1 0.9 1 1 1 1 1 1 0.85 1 1 1 1 0.8 1 1 1 1 1 1

1.0772 0.9266 1.0390 1.0355 1.0204 0.9922 1.0535 1.0561 0.8779 1.0204 1.0135 1.0079 1.0142 0.8123 1.0484 1.0650 1.0197 1.0138 1.0043 1.0179

0.0025 0.0023 0.0005 0.0033 0.0032 0.0039 0.0013 0.0018 0.0013 0.0018 0.0015 0.0016 0.0019 0.0756 0.1035 0.0387 0.0675 0.0447 0.1163 0.0643

(a)

(b)

(c) Fig. 5. Probability of change in (a) mass-parameters (b) stiffness-parameters and (c) damping-parameters of 6-DOF spring-mass-damper system for damage case IV using the proposed approach.

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

27

Analytical frequencies and analytical eigenvectors are represented as functions of mass-parameters ðh1 Þ, stiffnessparameters ðh2 Þ and damping-parameters ðh3 Þ. Besides, MAC represents modal assurance criteria and is calculated using Eq. (78). Computational efficiency of this proposed approach is studied in terms of evolution of accuracy (modal-distance) against the computation time. The evolution of the modal distance value against computation time is shown in Fig. 6 for all the cases. It may be observed that trend of MD tends to improve with increased time for all the damage cases while the trend is almost same for the undamaged case. Additionally number of iterations are also plotted for all the cases (as shown in Fig. 7) against computation time for better understanding of computational efficiency. 5.4.2. Damage detection using ASCE benchmark structure The ASCE benchmark structure (as shown in Fig. 3 and with similar structural properties as mentioned in Section 5.3) is here modelled using 12 DOFs (Fig. 8) for the purpose of damage detection. Consideration of DOFs in such way is observed to facilitate satisfactory damage detection for building structure [28,29]. The details of DOFs at one floor (lth floor) are shown in Fig. 8, where the local DOFs are considered as two translational and one rotational coordinates as denoted by 1, 2, 3 while global DOFs are considered as three translational coordinates as denoted by 1G, 2G, 3G. Multiple damage cases (as shown in Table 8) are considered for probabilistic damage detection using the proposed approach. In the standard ASCE benchmark problem [56], damages are incurred in the external braces primarily resulting in loss/damage in the external face stiffness. To simulate a similar pattern in damage, damages are incurred in the stiffness of corner columns (as can be seen from damage cases shown in Table 8). For each case, the target modal data is obtained by simulating damage in the 12-DOF model and for each case, 2 sets of simulated modal data (Ns = 2) from the model where for each of the modal data set, first four modal frequencies and modal damping ratios corresponding to four translational modes. Besides, it is assumed that sensors are

Fig. 6. Computational efficiency (modal distance against computation time) of the proposed approach for 6-DOF spring-mass-damper system.

Fig. 7. Computational efficiency (repetition-measures against computation time) of the proposed approach for 6-DOF spring-mass-damper system.

28

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Fig. 8. Floor plan of the ASCE benchmark structure with considered DOF.

Table 8 Details of the damage cases in the ASCE benchmark structure. Damage case

Change of mass

Change of stiffness

Change of damping

I

–

–

II

5% increase of lumped mass at 4th floor –

–

III

–

10% reduction of young’s modulus of all the corner columns of 1st story –

IV

–

V

5% increase of lumped mass at 4th floor 5% increase of lumped mass at 4th floor

VI

10% reduction of young’s modulus of all the corner columns of 1st story 10% reduction of young’s modulus of all the corner columns of 1st story 15% and 10% reduction of young’s modulus of all the corner columns of 1st and 2nd story respectively

5% reduction of damping coefficients viscous dampers of 1st story 5% reduction of damping coefficients viscous dampers of 1st story 5% reduction of damping coefficients viscous dampers of 1st story 5% reduction of damping coefficients viscous dampers of 1st story

of of of of

placed on +y and –y faces (with measurement direction along x-direction) of 1st and 3rd floors, and -x face (with measurement direction along y-direction) of all floors having a total of 8 measured DOFs thereby generating partial complex mode shape. The modal data sets are generated by adding different sets of noise to the exact modal data of the target model in a similar fashion as in previous exercise. Each set of noisy modal data is generated by adding random values generated from zero mean Gaussian distribution with standard deviation equal to 2% of the exact values obtained from the 12-DOF model. The mass and damping parameters considered for this example are similar as explained in Section 5.3. On the other hand, the stiffness parameters ðh2 Þ considered in this exercise for l ¼ 1; 2; 3; 4 (story number) are ðh2 Þl ¼ ðSþx Þl , ðh2 Þ4þl ¼ ðSx Þl ,     ðh2 Þ8þl ¼ Sþy l and ðh2 Þ12þl ¼ Sy l , where ðSþx Þl and ðSx Þl are the outer face stiffness of the structure in (+x) and (–x) faces     respectively while Sþy l and Sy l are the face stiffness of the structure in (+y) and (–y) faces respectively of the lth story. A face of a story is considered as a vertical plane along the plane of frame. The face stiffness associated with inner faces of a   story (ðSx0 Þl and Sy0 l for lth story) are considered as constant, thus not a part of stiffness parameters. With four numbers of faces (having normal vectors of these faces along +x, -x, +y, -y) considered in each floor, a total number of 16 stiffness (face stiffness) parameters are considered. The nominal values of the face stiffness (i.e. face stiffness for the undamaged case) of each story are equally found to be 6.558 MN/m for the (+x) and (-x) faces while 19.457 MN/m for the (+y) and (-y) faces in case of the structure. On the other hand, the face stiffness of the inner/middle face of each story is equally obtained and is found to be exactly equal to 6.558 MN/m and 19.457 MN/m for  and y direction respectively. Besides, the nominal Rayleigh coefficients are calculated using 1st translational mode in x-direction and 4th translational mode in y-direction of the analytical model as mentioned in Section 5.3.

29

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

The model updating of the benchmark structure is performed using the formulations as presented in Section 2. The performance of the proposed approach is showed in this section in a similar style as the 6-DOF system. It may be mentioned here that all the results (as presented in Section 5.4.2) for this exercise are obtained with 5000 iterations for the proposed approach. It is also to be noted that like the previous exercise, performance of model updating is likely to improve with increase in number of identified modes and observed DOFs. Performance of the proposed approach in updating eigenvalues (or frequencies) is shown in Table 9 for undamaged and damage cases I to III and in Table 10 for damage case IV to VI and it shows considerable matching between target and updated frequencies. Also, diagonal modal assurance criteria (MAC) values between initial and target mode shapes are also significantly improved (as shown in Table 11 for undamaged and damage cases I to III and Table 12 for damage cases IV to VI). Similarly, performance in updating the structural parameters and Rayleigh coefficients is shown in Table 13 for undamaged and damage cases I to III and in Table 14 for damage cases IV to VI where it can be clearly observed that the updated values are close to the target values. In addition to this, the COV values of the corresponding updating parameter in the posterior PDF are also shown in Table 15 for all the cases. Also, the probabilities of change/damage of the structural parameters

Table 9 Target and updated frequencies (Hz) of the ASCE benchmark structure for undamaged and damage cases I to III. Modes

1 2 3 4

Undamaged case

Damage case I

Target

Updated

Target

Updated

Target

Damage case II Updated

Target

Damage case III Updated

5.077 8.745 9.740 13.778

5.078 8.745 9.739 13.776

5.042 8.684 9.677 13.698

5.020 8.648 9.677 13.700

5.020 8.648 9.576 13.658

5.019 8.658 9.580 13.660

5.077 8.745 9.740 13.778

5.077 8.749 9.742 13.778

Table 10 Target and updated frequencies (Hz) of the ASCE benchmark structure for damage cases IV to VI. Modes

Damage case IV

1 2 3 4

Damage case V

Damage case VI

Target

Updated

Target

Updated

Target

Updated

5.020 8.648 9.576 13.658

5.022 8.649 9.581 13.659

4.986 8.588 9.515 13.578

4.987 8.588 9.519 13.580

4.921 8.477 9.326 13.504

4.921 8.479 9.327 13.506

Table 11 MAC values (before and after updating) of the ASCE benchmark using the proposed approach for undamaged and damage cases I to III. Modes

1 2 3 4

Undamaged case

Damage case I

Initial

Updated

Initial

Updated

Damage case II Initial

Updated

Damage case III Initial

Updated

0.9999 1.0000 0.9999 0.9998

1.0000 1.0000 1.0000 1.0000

0.9999 1.0000 0.9997 0.9996

1.0000 1.0000 1.0000 1.0000

0.9999 0.9999 0.9998 0.9998

1.0000 1.0000 1.0000 1.0000

0.9999 1.0000 0.9997 0.9996

1.0000 1.0000 1.0000 1.0000

Table 12 MAC values (before and after updating) of the ASCE benchmark using the proposed approach for damage cases IV to VI. Modes

1 2 3 4

Damage case IV

Damage case V

Damage case VI

Initial

Updated

Initial

Updated

Initial

Updated

0.9996 0.9999 0.9997 0.9996

1.0000 1.0000 1.0000 1.0000

1.0000 0.9997 0.9999 0.9998

1.0000 1.0000 1.0000 1.0000

0.9993 0.9997 0.9991 0.9992

1.0000 1.0000 1.0000 1.0000

30

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Table 13 Target and updated structural parameters of the ASCE benchmark structure for undamaged and damage cases I to III. Parameters

(h1)1 (h1)2 (h1)3 (h1)4 (h2)1 (h2)2 (h2)3 (h2)4 (h2)5 (h2)6 (h2)7 (h2)8 (h2)9 (h2)10 (h2)11 (h2)12 (h2)13 (h2)14 (h2)15 (h2)16 (h3)1 (h3)2 (h3)3 (h3)4 a0 a1

Undamaged case

Damage case I

Damage case II

Damage case III

Target

Updated

Target

Updated

Target

Updated

Target

Updated

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.1315 5.67E-04

0.9968 1.0118 0.9718 1.0196 1.0010 0.9866 1.0029 1.0275 0.9998 0.9775 1.0072 1.0308 0.9965 1.0047 0.9805 1.0063 0.9965 1.0047 0.9805 1.0063 1.0006 0.9893 1.0046 1.0205 0.1314 5.66E-04

1 1 1 1.05 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.1300 5.80E-04

0.9973 1.0067 0.9815 1.0630 0.9989 0.9931 1.0012 1.0219 0.9977 0.9876 1.0051 1.0161 0.9974 1.0019 0.9880 1.0088 0.9974 1.0019 0.9880 1.0088 1.0014 0.9975 1.0058 1.0156 0.1298 5.79E-04

1 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.9333 1 1 1 1 1 1 1 0.1300 5.74E-04

1.0043 1.0074 0.9967 1.0121 0.9353 1.0052 1.0163 1.0139 0.9375 1.0171 1.0085 1.0214 0.9398 1.0012 1.0106 1.0166 0.9398 1.0012 1.0106 1.0166 1.0005 1.0037 1.0043 1.0091 0.1301 5.74E-04

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.95 1 1 1 0.1308 5.77E-04

1.0026 0.9937 1.0165 0.9895 1.0001 1.0082 1.0000 0.9853 1.0011 1.0150 0.9965 0.9835 1.0028 0.9974 1.0129 0.9980 1.0028 0.9974 1.0129 0.9980 0.9497 1.0069 0.9980 0.9887 0.1308 5.78E-04

Table 14 Target and updated structural parameters of the ASCE benchmark structure for damage cases IV to VI. Parameters

(h1)1 (h1)2 (h1)3 (h1)4 (h2)1 (h2)2 (h2)3 (h2)4 (h2)5 (h2)6 (h2)7 (h2)8 (h2)9 (h2)10 (h2)11 (h2)12 (h2)13 (h2)14 (h2)15 (h2)16 (h3)1 (h3)2 (h3)3 (h3)4 a0 a1

Damage case IV

Damage case V

Damage case VI

Target

Updated

Target

Updated

Target

Updated

1 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.95 1 1 1 0.1297 5.79E-04

1.0067 1.0069 1.0038 1.0055 0.9437 0.9977 1.0116 1.0070 0.9457 1.0067 1.0055 1.0090 0.9399 1.0052 1.0095 1.0082 0.9399 1.0052 1.0095 1.0082 0.9615 1.0074 1.0098 1.0100 0.1298 5.78E-04

1 1 1 1.05 0.9333 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.9333 1 1 1 0.95 1 1 1 0.1289 5.81E-04

1.0076 0.9987 1.0180 1.0457 0.9407 1.0063 1.0093 0.9967 0.9431 1.0217 1.0010 0.9916 0.9409 1.0004 1.0206 1.0094 0.9409 1.0004 1.0206 1.0094 0.9603 1.0143 1.0073 1.0003 0.1291 5.81E-4

1 1 1 1.05 0.9 0.9333 1 1 0.9 0.9333 1 1 0.9 0.9333 1 1 0.9 0.9333 1 1 0.95 1 1 1 0.1275 5.85E-04

1.0068 1.0061 1.0049 1.0599 0.9061 0.9439 1.0149 1.0162 0.9074 0.9524 1.0100 1.0125 0.9084 0.9373 1.0142 1.0178 0.9084 0.9373 1.0142 1.0178 0.9656 0.9810 1.0089 1.0104 0.1274 5.85E-04

31

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524 Table 15 Updated values of COV (%) of the structural parameters for the ASCE benchmark structure for the undamaged case and damage cases IV to VI. Parameters

Undamaged case

Damage case I

Damage case II

Damage case III

Damage case IV

Damage case V

Damage case VI

(h1)1 (h1)2 (h1)3 (h1)4 (h2)1 (h2)2 (h2)3 (h2)4 (h2)5 (h2)6 (h2)7 (h2)8 (h2)9 (h2)10 (h2)11 (h2)12 (h2)13 (h2)14 (h2)15 (h2)16 (h3)1 (h3)2 (h3)3 (h3)4 a0 a1

4.12E06 8.40E06 7.36E06 5.70E06 5.60E06 1.10E05 7.18E06 6.18E06 1.08E05 1.81E05 1.08E05 7.54E06 9.42E06 1.50E05 2.06E05 2.65E05 1.99E05 1.75E05 3.27E05 4.68E05 1.09E05 2.34E05 9.46E06 1.16E05 9.21E04 7.24E04

4.30E06 8.75E06 7.67E06 5.94E06 5.83E06 1.15E05 7.48E06 6.44E-06 1.12E05 1.89E05 1.12E05 7.86E06 9.81E06 1.56E05 2.15E05 2.77E05 2.08E05 1.83E05 3.41E05 4.88E05 1.14E05 2.44E05 9.85E06 1.21E05 4.73E04 3.12E04

4.38E06 9.53E06 5.73E06 6.19E06 5.33E06 1.06E05 7.85E06 7.00E06 1.15E05 1.88E05 1.12E05 1.31E05 1.35E05 1.81E05 8.44E06 2.32E05 1.84E05 1.48E05 2.70E05 3.90E05 1.17E05 2.39E05 1.02E05 1.28E05 9.26E05 7.29E04

4.18E06 9.87E06 7.34E06 5.73E06 2.89E06 1.51E05 8.17E06 7.58E06 1.05E05 2.03E05 1.16E05 1.06E05 4.40E05 3.30E05 8.98E05 8.63E05 4.61E05 3.36E05 8.72E05 1.12E04 1.20E05 2.36E05 1.00E05 1.25E05 9.02E05 8.12E05

4.32E06 9.57E06 6.01E06 6.16E06 4.93E06 1.14E05 7.87E06 7.08E06 1.14E05 1.92E05 1.13E05 1.28E05 1.41E05 2.06E05 2.87E05 1.60E05 2.35E05 1.22E05 3.55E05 4.99E05 1.22E05 2.36E05 1.02E05 1.28E05 3.22E04 6.82E04

4.44E06 8.93E06 5.32E06 6.43E06 6.19E06 9.29E06 8.15E06 6.62E06 1.17E05 1.84E05 1.14E-05 1.39E05 2.66E05 1.92E05 4.62E05 5.64E05 4.09E06 2.24E05 2.57E05 4.22E05 1.23E05 2.38E05 1.04E05 1.28E05 2.58E04 6.02E04

4.45E06 8.61E06 4.71E06 6.75E06 7.18E06 6.62E06 8.38E06 6.34E06 1.23E05 1.81E05 1.15E05 1.48E05 3.99E05 1.06E05 7.92E05 7.98E05 2.48E05 3.72E05 6.26E05 8.24E05 1.33E05 2.40E05 1.08E05 1.32E05 5.53E04 8.61E04

are obtained using the procedure explained in Section 4 and are shown in Fig. 9 for damage case VI and it can be clearly observed that the extents of change/damage of the structural parameters nearly match with the assumed values for change/damage. Lastly, it is interesting to study the computational efficiency of the proposed approach in a similar style to the first exercise. The evolution of the modal distance value expressed in Eq. (90) against computation time is shown in Fig. 10 for the undamaged and damage cases and it has been observed that the modal distance measure tends to decrease with increasing time for all the cases. Also, number of iterations against computation time (as shown in Fig. 11) are also plotted for both cases for better understanding of computational efficiency for all the cases. 6. Conclusion FE model updating in Bayesian probabilistic framework using incomplete complex modal data is proposed based on MAP, where multiple sets of modal data are considered for model updating. Apart from mass and stiffness parameters, damping parameters are also updated in a computationally efficient sequential iterative manner. Damping is considered for updating in the form of classical (Rayleigh) damping and non-classical viscous damping. Detailed formulations for the updating procedure are developed as presented. Apart from this, probabilistic estimation of changes/damages in structural parameters is also formulated in this proposed methodology. The performance of the proposed approach is compared with Gibbs sampling and Sensitivity method using two numerical examples: (a) spring-mass-damper system and (b) ASCE benchmark structure. Besides, performance of the proposed approach in damage detection is also studied using similar examples with multiple damage cases. Based on this performed work, following concluding remarks are considered: A. There has been an opportunity for updating various damping parameters along with mass and stiffness parameters in Bayesian framework. B. With the scope of using incomplete modal measurements and avoiding computationally costly mode-matching, such MAP based updating of diverse structural parameters can be considered effective. C. The proposed approach is found to achieve convergence in lesser computation time than both the Gibbs sampling technique and Sensitivity method. D. With convergence of the updating parameters at much lower number of iterations, the proposed approach can be considered as a suitable alternative to other updating techniques in Bayesian framework (e.g. Gibbs sampling based updating technique) for updating of mass, stiffness and damping parameters utilizing complex modal data. E. With the support of detailed formulation, the proposed approach can be well considered for probabilistic damage detection for diverse structural parameters including the damping parameters.

32

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

(a)

(b)

(c) Fig. 9. Probability of change in (a) mass-parameters (b) stiffness-parameters and (c) damping-parameters of the ASCE benchmark structure for damage case VI using the proposed approach.

(a)

(b)

Fig. 10. Computational efficiency (modal distance against computation time) of the proposed approach for (a) undamaged case, damage cases I to III and (b) damage cases IV to VI of the ASCE benchmark structure.

33

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

Fig. 11. Computational efficiency (repetition-measures against computation time) of the proposed approach for the ASCE benchmark structure.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix. A: Expression for Ls17 In order to derive the expression for Ls17 , we have to divide few matrices related to Ls17 , into two parts, one part having Rayleigh coefficient as multiplier and other part having not. The separation of the expressions Gh1P1 and G01P1 into two parts are shown in Eqs. (A.1) and (A.2) for clear understanding.

Gh1P1 ¼ Ah1P1 þ a0 ðBh1P1 Þ

ðA:1Þ

G01P1 ¼ A01P1 þ a0 ðB01P1 Þ

ðA:2Þ

where,

Ah1P1

2  ð1;sÞ2  ð1Þ ^k M1 WR R 6   6  6 ^kð2;sÞ2 M1 Wð2Þ R R 6 ¼6 6 .. 6 . 4   ^kðNm ;sÞ2 M1 WðNm Þ R R 2

Bh1P1

A01P1

 ð1;sÞ2  ð1Þ ^k M2 WR R  ð2;sÞ2  ð2Þ ^k M2 WR R

 ð1;sÞ  ð1Þ ^k M2 WR R  ð2;sÞ  ð2Þ ^k M2 WR R

.. .   ^kðNm ;sÞ M1 WðNm Þ R R

..  ðNm ;sÞ  . ðN Þ ^k M2 WR m R

3 2  ð1;sÞ2  ð1Þ ^k M0 WR R 7 6   7 6  6 ^kð2;sÞ2 M0 Wð2Þ 7 R R 7 6 ¼6 7 7 6 . 7 6 .. 5 4   ^kðNm ;sÞ2 M0 WðNm Þ R

R



..  ðNm ;sÞ2 . ðN Þ ^k M2 WR m R

 ð1;sÞ  ð1Þ ^k M1 WR R  ð2;sÞ  ð2Þ ^k M1 WR R

6 6 6 6 ¼6 6 6 4



N d N m 1

B01P1

.. .   

 ð1;sÞ2  ð1Þ ^k MNh1 WR R  ð2;sÞ2  ð2Þ ^k MNh1 WR R 

.. .   ^kðNm ;sÞ2 MN WðNm Þ R R h1

 ð1;sÞ  ð1Þ ^k MNh1 WR R  ð2;sÞ  ð2Þ ^k MNh1 WR R

.. .

..  ðNm ;sÞ  . ðN Þ ^k  MNh1 WR m R

3 2  ð1;sÞ  ð1Þ ^k M0 WR R 7 6   7 6  6 ^kð2;sÞ M0 Wð2Þ 7 R R 7 6 ¼6 7 7 6 . 7 6 .. 5 4   ^kðNm ;sÞ M0 WðNm Þ R

3

R

7 7 7 7 7 7 7 5 Nd Nm N h1

3 7 7 7 7 7 7 7 5 N d N m Nh1

Nd Nm 1

In this way we can divide the expressions of other matrices Gh1P2 , Gh2Q 1 Gh2Q 2 , G01P2 , G02Q 1 and G02Q 2 , required for the formulation of Ls17 and hence its expression can be shown in Eq. (A.3).

34

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

20

Ls17

2a0 BTT1P1 BT1P1 h1 þ 2a0 BTT1P1 B01P1 þ 2BTh1P1 Ah1P1 h1 þ BTh1P1 A01P1 þ ATh1P1 B01P1 þ

1 3

7 6B 6 @ BTh1P1 Ah2Q 1 h2 þ BTh1P1 A02Q 1 þ BTh1P1 Gh3R1 h3 þ BTh1P1 G03R1 þ a1 BTh1P1 Bh2Q 1 h2 þ C Aþ 7 7 6 7 6 7 6 a1 BTh1P1 B02Q 1 7 6 ! 7 6 2a0 BTh1P2 Bh1P2 h1 þ 2a0 BTh1P2 G01S2 þ 2BTh1P2 Ah1P2 h1 þ BTh1P2 A01P2 þ ATh1P2 B01P2 7 6 7 6  T T   7 6  T  T þBh1P2 Gh3R2 h3 þ Bh1P2 G03R2 þ a1 Bh1P2 Bh2Q 2 h2 þ a1 Bh1P2 B02Q 2 7 6 7 6     1 0 ¼6 7 T T T T T   2Bh1P2 Ah1P1 h1 þ Bh1P2 A01P1 þ Ah1P1 B01P2 þ Bh1P2 Bh2Q1 h2 þ Bh1P2 A02Q 1 7 6 7 6B     C C 7 6B T T T T T   C7 6 B þ B Gh3R1 h þ B G03R1 þ 2B Ah1P2 h þ B A01P2 þ A B01P1 C7 6B 3 1 h1P2 h1P2 h1P1 h1P1 h1P2 7 6B    C C7 6B 6 B þ BTh1P1 Gh3R2 h3 þ BTh1P1 G03R2 þ 2a0 2BTh1P1 Bh1P2 h1 þ BTh1P1 B01P2 þ BTh1P2 B01P1 C 7 C7 6B A5 4@     þa1 BTh1P1 Bh2Q 2 h2 þ BTh1P1 B02Q 2 þ a1 BTh1P2 Bh2Q1 h2 þ BTh1P2 B02Q1

ðA:3Þ

Here, Gh3R1 , Gh3R2 , G03R1 and G03R2 have usual meanings as explained earlier in Section 2. In this way, we can find the expression for ðLs18 ; Ls19 ; Ls20 ; Ls25 ; Ls26 ; Ls27 ; Ls28 ; Ls31 ; Ls32 ; Ls33 ; Ls34 Þ. References [1] J.E. Mottershead, M. Link, M.I. Friswell, The sensitivity method in finite element model updating: a tutorial, Mech. Syst. Signal Process. 25 (2011) 2275– 2296. [2] D.J. Ewins, Adjustment or updating of models, Sadhana 25 (2000) 235–245. [3] M.I. Friswell, J.E. Mottershead, Finite element model updating in structural dynamics, Kluwer Academic Publishers, Boston, 1995. [4] J.E. Mottershead, M.I. Friswell, Physical Understanding of Structures by Model Updating, in: Proceedings of the International Conference on Structural Identification, Kassel, Germany, 2001, p. 2001. [5] J.E. Mottershead, M.I. Friswell, Model updating in structural dynamics: a survey, J. Sound Vib. 167 (1993) 347–375. [6] M. Baruch, Optimization procedure to correct stiffness and flexibility matrices using vibration tests, AIAA Journal 16 (1978) 1208–1210. [7] M. Baruch, Methods of Reference Basis for Identification of Linear Dynamic Structures. Proceedings of the 23rd Structures, Structural Dynamics and Material Conference, New Orleans, Louisiana, May 557-563, 1982. [8] M. Baruch, Methods of reference basis for identification of linear dynamic structures, AIAA Journal 22 (4) (1984) 561–564. [9] M. Baruch, I.Y. Bar-Itzhack, Optimal weighted orthogonalization of measured modes, AIAA J. 16 (4) (1978) 346–351. [10] A. Berman, E.J. Nagy, Improvement of large analytical model using test data, AIAA J. 21 (8) (1983) 1168–1173. [11] B. Caesar, Update and identification of dynamic mathematical models, Proceedings of the Fourth International Modal Analysis Conference, Los Angeles, 394-401 1986. [12] B. Caesar, Updating system matrices using modal test data, Proceedings of the Fifth International Modal Analysis Conference, London, England, April 453-459 1987. [13] F.S. Wei, Analytical dynamic model improvement using vibration test data, AIAA J. 28 (1) (1990) 175–177. [14] M.I. Friswell, D.J. Inman, D.F. Pilkey, The direct updating of damping and stiffness matrices, AIAA J. 36 (3) (1998) 491–493. [15] B.N. Datta, Finite element model updating, eigenstructure assignment and eigenvalue embedding techniques for vibrating systems, Mech. Syst. Sig. Process. 16 (1) (2002) 83–96. [16] J. Carvalho, B.N. Datta, A. Gupta, M. Lagadapati, A direct method for model updating with incomplete measured data and without spurious modes, Mech. Syst. Sig. Process. 21 (2007) 2715–2731. [17] C.H. Min, S. Hong, S.Y. Park, D.C. Park, Sensitivity-based finite element model updating with natural frequencies and zero frequencies for damped beam structures, Int. J. Nav. Archit. Ocean Eng. 6 (4) (2014) 904–921. [18] N. Grip, N. Sabourova, Y. Tu, Sensitivity-based model updating for structural damage identification using total variation regularization, Mech. Syst. Sig. Process. 84 (2017) 365–383. [19] Y.B. Yang, Y.J. Chen, Direct versus iterative model updating methods for mass and stiffness matrices, Int. J. Struct. Stab. Dyn. 10 (2010) 165–186. [20] M.I. Friswell, J.E.T. Penny, Updating model parameters from frequency domain data via reduced order models, Mech. Syst. Sig. Process. 4 (5) (1990) 377–391. [21] J.E. Mottershead, M.I. Friswell, G.H.T. Ng, J.A. Brandon, Geometric parameters for finite element model updating of joints and constraints, Mech. Syst. Sig. Process. 10 (1996) 171–182. [22] K.V. Yuen, Updating large models for mechanical systems using incomplete modal measurement, Mech. Syst. Sig. Process. 28 (2012) 297–308. [23] J.T. Wang, C.J. Wang, J.P. Zhao, Frequency response function-based model updating using Kriging model, Mech. Syst. Sig. Process. 87 (2017) 218–228. [24] J.L. Beck, L.S. Katafygiotis, Updating models and their uncertainties. I: Bayesian statistical framework, J. Eng. Mech. 124 (4) (1998) 455–461. [25] M.W. Vanik, J.L. Beck, S.K. Au, Bayesian probabilistic approach to structural health monitoring, J. Eng. Mech. 126 (7) (2000) 738–745. [26] J.L. Beck, K.V. Yuen, Model selection using response measurements: bayesian probabilistic approach, J. Eng. Mech. 130 (2) (2004) 192–203. [27] K.V. Yuen, An Extremely Efficient Finite-Element Model Updating Methodology with Applications to Damage Detection, Proceedings of Enhancement and Promotion of Computational Methods in Engineering and Science X, Sanya, Hainan, China, August 21-23 166-179 2006. [28] K.V. Yuen, J.L. Beck, L.S. Katafygiotis, Efficient model updating and health monitoring methodology using incomplete modal data without mode matching, Struct. Control Health Monit. 13 (2006) 91–107. [29] K.V. Yuen, Bayesian methods for structural dynamics and civil engineering, John Wiley & Sons (Asia) Private Limited, 2010. [30] K.V. Yuen, Recent developments of Bayesian model class selection and applications in civil engineering, Struct. Saf. 32 (5) (2010) 338–346. [31] S. Mustafa, N. Debnath, A. Dutta, Bayesian probabilistic approach for model updating and damage detection for a large truss bridge, Int. J. Steel Struct. (2015). [32] W.J. Yan, L.S. Katafygiotis, A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple steps, Struct. Saf. 52 (2015) 260–271. [33] A. Das, N. Debnath, A Bayesian finite element model updating with combined normal and lognormal probability distributions using modal measurements, Appl. Math. Model. 61 (2018) 457–483. [34] I. Behmanesh, B. Moaveni, G. Lombaert, C. Papadimitriou, Hierarchical Bayesian model updating for structural identification, Mech. Syst. Sig. Process. 64–65 (2015) 360–376. [35] R. Rocchetta, M. Broggi, Q. Huchet, E. Patelli, On-line Bayesian model updating for structural health monitoring, Mech. Syst. Sig. Process. 103 (2018) 174–195. [36] H.-P. Wan, W.-X. Ren, Stochastic model updating utilizing Bayesian approach and Gaussian process model, Mech. Syst. Sig. Process. 70–71 (2016) 245– 268.

A. Das, N. Debnath / Mechanical Systems and Signal Processing 136 (2020) 106524

35

[37] J.L. Beck, S.K. Au, Bayesian updating of structural models and reliability using markov chain monte carlo simulation, J. Eng. Mech. 128 (4) (2002) 380– 391. [38] J. Ching, Y.C. Chen, Transitional markov chain monte carlo method for bayesian model updating, model class selection, and model averaging, J. Eng. Mech. 133 (7) (2007) 816–832. [39] J. Ching, M. Muto, J.L. Beck, Structural model updating and health monitoring with incomplete modal data using gibbs sampler, Comput.-Aided Civ. Infrastruct. Eng. 21 (2006) 242–257. [40] S. Bansal, A new Gibbs sampling based Bayesian model updating approach using modal data from multiple setups, Int. J. Uncertainty Quantif. 5 (4) (2015) 361–374. [41] S.H. Cheung, S. Bansal, A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data, Mech. Syst. Sig. Process. 92 (2017) 156–172. [42] S. Geman, D. Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Trans. Pattern Anal. Mach. Intell. 6 (6) (1984) 721–741. [43] J.M. Bernardo, A.F.M. Smith, Bayesian theory, John Wiley & Sons, England, 2000. [44] J.A. Gubner, Probability and Random Processes for Electrical and Computer Engineers, Cambridge University Press, 2006. [45] E.T. Jaynes, in: R.D. Levine, M. Tribus (Eds.), Where Do We Stand on Maximum Entropy?, 1978, The MIT Press. [46] J.L. Beck, Bayesian system identification based on probability logic, Struct. Control Health Monit. 17 (7) (2010) 825–847. [47] K.B. Petersen, M.S. Pedersen, The matrix cookbook (2012). [48] C.T. Kelley, Iterative Methods for Optimization, Society for Industrial and Applied Mathematics Philadelphia, 1999. [49] D. Belsley, E. Kuh, R. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, John Wiley and Sons, New York, 1980. [50] W. Sun, Y.-X. Yuan, Optimization theory and methods: Nonlinear Programming, Springer, 2006. [51] K.V. Yuen, J.L. Beck, L.S. Katafygiotis, Unified probabilistic approach for model updating and damage detection, J. Appl. Mech., ASME 73 (2006) 555–564. [52] Ø.W. Petersen, O. Øiseth, Sensitivity-based finite element model updating of a pontoon bridge, Eng. Struct. 150 (2017) 573–584. [53] M. Pastor, M. Binda, T. Harcarik, Modal assurance criterion, Procedia Eng. 48 (2012) 543–548. [54] S. Adhikari, Rates of change of eigenvalues and eigenvectors in damped dynamic system, AIAA J. 37 (11) (1999) 1452–1458. [55] S. Adhikari, M.I. Friswell, Calculation of eigensolution derivatives for nonviscously damped systems using nelson’s method, AIAA J. 44 (8) (2006) 1799– 1806. [56] E. Johnson et al, Phase I IASC-ASCE structural health monitoring benchmark problem using simulated data, J. Eng. Mech. 130 (1) (2004) 3–15.