Fisher information-based optimal input locations for modal identification

Journal of Sound and Vibration 459 (2019) 114833

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Fisher information-based optimal input locations for modal identification Debasish Jana a , Suparno Mukhopadhyay b, ∗ , Samit Ray-Chaudhuri b a b

Civil and Environmental Engineering, Rice University, Houston, TX, TX-77005, USA Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

article info

abstract

Article history: Received 9 December 2018 Revised 15 June 2019 Accepted 27 June 2019 Available online 1 July 2019 Handling Editor: I. Trendafilova

A method is proposed to determine optimal locations of input forces for experimental modal analysis. The input locations are optimal in a sense that the measured vibration responses, under inputs thus located, contain maximum information about unknown modal parameters of interest. These optimal input locations are obtained by maximizing the trace of the associated Fisher Information Matrix (FIM). Analytical expressions are obtained for the different derivatives involved, by expressing acceleration responses in terms of pseudo-modal responses. A dimensionless scaling is introduced to normalize the effect of different types of modal parameters. An explicit relation is also derived between the FIM and the mode shape components at input locations. It is shown that, in certain experimental scenarios, this relation may be used to directly obtain the optimal input locations from only the mode shapes. An extensive series of numerical simulations, including situations of single/multiple inputs and single/multiple modes of interest, is used to illustrate the proposed approach. It is observed that the modal parameters, identified using any suitable modal identification technique, have the lowest estimation uncertainty when the inputs are optimally located. As an application in damage detection, it is shown that modal parameters estimated from experiments with nonoptimally located inputs may lead to incorrect damage localization. The proposed method is also applied to data from experiments performed on a laboratory scale truss model. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Experimental modal analysis Optimal input location Fisher Information Matrix Pseudo-modal response Damage detection Experimental application

1. Introduction Measured vibration responses of a structural system are often used to identify modal parameters of the system. This may be achieved using ambient/operational vibration data (operational modal analysis) or through planned forced vibration experiments (experimental modal analysis), using suitable system identification techniques [1–3]. Such identified modal parameters find a wide variety of applications, for example, in model updating and physical parameter estimation [4–7] in structural damage detection [8–12]. The accuracy of the modal parameters identified using any technique depends on the amount of information that the measured responses contain about these parameters; the lower this information content, the lower would be the accuracy in the identified parameters. Inaccuracies in the identified modal parameters would, in turn, adversely affect the results of any further activity performed using these parameters, viz. model updating or damage detection. Hence, it is essential that one tries to collect data, which contain the maximum information about the modal parameters of interest.

∗ Corresponding author. E-mail address: [email protected] (S. Mukhopadhyay).

https://doi.org/10.1016/j.jsv.2019.06.040 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

2

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

The quality of measured data depends on the types, numbers and locations of sensors installed on the monitored system, which are decided when planning the measurement campaign prior to actual data collection. Given the importance of good quality data in identification, several researchers have worked on the problem of optimal sensor placement (OSP) [13]. In this problem, given the available types and numbers of sensors, the objective is to determine their locations on the system so as to have the maximum information about the parameters to be identified, from the measured data. Some of the earliest methods to solve the OSP problem have been based on the modal kinetic energy [14,15], aiming to maximize parameter observability by placing sensors at locations of maximum kinetic energy. Hence, for identifying a particular mode, the degrees of freedom (DOFs) with the maximum kinetic energy for that mode would be chosen as the optimal sensor locations. Recently, a criterion based on distinguishability of modes has also been used to obtain optimal sensor locations, by minimizing correlation(s) among mode shape vectors corresponding to different modes [16]. Another popular approach, closely related to the modal kinetic energy [17], formulates the OSP problem as a maximization of the information content in measured data, which also correspond to minimization of the uncertainty in the parameter estimates from such data. The optimal sensor locations may be found by maximizing some norm, e.g., the trace or determinant of the Fisher information matrix (FIM) [17–19], which will be the same as minimizing the corresponding norm of the covariance matrix of errors in estimated parameters. The associated optimization problem may be solved using different search techniques [20–22]. In a Bayesian context, the OSP problem for systems excited by seismic input has been formulated as a minimization of a Bayesian (squared error) loss function [23], shown to be equivalent to maximizing the expected value of the trace of the inverse of FIM. It has also been shown that maximizing the determinant of FIM corresponds to minimizing the information entropy [24,25]. More recently, the OSP problem has been formulated in Bayesian decision making frameworks with objectives of minimizing the expected cost of false positive and false negative errors in damage detection [26], or of maximizing the Kullback-Leibler divergence (relative entropy) between the posterior and prior distributions of the parameters of interest [27]. As is evident from the preceding discussions, extensive research has been performed and is still ongoing on the OSP problem. On the other hand, to the best of the authors’ knowledge, there is no study which systematically addresses the complementary problem of optimal input locations in the context of modal identification or structural health monitoring (although the problem has been studied in the context of active vibration control [28–33]). However, in experimental modal analysis, where the system is excited using known input forces, the locations of these applied inputs would also affect the uncertainty in the estimated modal parameters. While in some situations the location(s) of the input(s) may be fixed based on practical considerations, in other situations there may exist a number of feasible locations of the input(s) (with the number of feasible locations exceeding the number of inputs, but not necessarily including all the active DOFs). Hence, in these latter class of situations, determining optimal input locations so as to have parameter estimates with least associated uncertainty becomes important in the experiment design stage. This requirement defines the objective of the study in this paper. The problem is cast here again in an information theoretic framework. The optimal location(s) of the input(s) should be such that the measured responses (by a fixed set of sensors) contain the maximum information about the parameters of interest. A complete set of sensors is assumed here, i.e., all active DOFs of the system are instrumented with sensors. This assumption is only made to distinctly highlight the effect of input locations on the identified parameters, distinguishing them from the effects of different sensor locations that would arise in case of incomplete set of sensors. As would be evident from the formulation, the method developed does not require this assumption, and it can be similarly applied in situations when not all DOFs are measured. The parameters to be identified are one or more sets of modes: modal frequencies, damping ratios and mode shapes. The optimal input locations are determined by maximizing the trace (T-norm) of the FIM associated with the modal parameters. This approach was recently introduced by the authors in a preliminary study for the case of identifying a single mode using a single impulse input [34]. Here, the approach is developed for the general case of identifying single/multiple mode(s) using single/multiple input(s), including non-impulse inputs. To facilitate the computation of the required partial derivatives, modal superposition is used to express the measured responses in terms of modal responses. A dimensionless scaling is also introduced in this paper to account for the different types of modal parameters, viz. frequencies, damping ratios and mode shapes. Further, an explicit relation between the FIM and the mode shapes is investigated which, in certain experimental scenarios, may be used to obtain the optimal input locations directly from the mode shape components at potential input locations, thereby avoiding the complexity and cost of computing the FIM. Since the true parameter values are unknown at the experiment design stage, a strategy is also discussed in this paper to obtain the optimal input locations based on candidate models of the system sampled from a model space. A suite of numerical simulations with a spring-mass system, as well as experimental data from a 2-D laboratory scale truss model are used to illustrate the proposed method. For validating the method, T-norms of the FIMs for different input locations are compared with the uncertainties in modal parameters estimated from experiments with these input locations. The adverse effects of using modal parameters, obtained from experiments with non-optimally located inputs, is also studied in the context of structural damage detection. 2. Optimal input location methodology The equation of motion of an N-DOF linear classically damped system can be written as: Mẍ (t ) + Cẋ (t ) + Kx(t ) = PI u(t )

(1)

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

3

where M, C and K are the mass, damping and stiffness matrices, respectively; and x = {x1 x2 · · · xN }T , ẋ and ẍ are the displacement, velocity and acceleration responses in physical coordinates, respectively, with xi being the displacement of the ith degree of freedom (DOF). Supposing that there are nI number of inputs, u is the nI × 1 vector of these input forces, and PI is a N × nI binary input location matrix. For any input, the corresponding column of PI has all elements zero except the element corresponding to the DOF where the input acts; for example, if the ith input is located at the pth DOF, then the ith column of PI has all elements equal to zero, except the pth element which is unity. The goal of modal identification is to identify the modal parameters of the system from the measured vibration responses of the system. Let the vector 𝚯 = {𝚯𝝎 ; 𝚯𝜻 ; 𝚯𝝓 } denotes the set of modal parameters to be identified, where 𝚯𝝎 , 𝚯𝜻 and 𝚯𝝓 , respectively, contain the frequencies, damping ratios and mode shape components to be identified. Further, let the accelerations at all the N DOFs constitute the measured response.1 Accounting for noisy data, the measured response can then be expressed as: y = ẍ + v, where v denotes the measurement noise. The objective of the optimal input location problem is to determine the locations of the nI inputs such that the measured responses in y, under the so located inputs, would contain the maximum information about the modal parameters in 𝚯. In other words, using y obtained with the optimally located inputs and starting from a nearby guess, the unknown parameters in 𝚯 would be identified with least uncertainty. This results in the conditional estimation problem for which the covariance of the vector parameter estimates satisfies the following relation [35]:

̂ )(𝚯 − 𝚯 ̂ ) T ] ≥ Q −1 ( T D ) E[(𝚯 − 𝚯

(2)

̂ denotes the estimate of 𝚯, and TD denotes the total duration of the measured data. where E denotes the expected value, 𝚯 − 1 Q (TD ) is called the Cramer-Rao Lower Bound, whose inverse is the Fisher Information Matrix Q(TD ). Assuming that the measurement noise in v is composed of zero mean Gaussian white noise sequences, with a covariance matrix proportional to the identity matrix, i.e., E[vi vTj ] = RI, the FIM can be expressed as [19]: ( TD

Q( T D ) =

∫0

𝜕ẍ 𝜕𝚯

)T (

𝜕ẍ 𝜕𝚯

R

) dt

(3)

The matrix Q(TD ) is a quantitative measure of the information about the unknown parameters in 𝚯 contained in the response time histories in ẍ . It is evident that the elements of Q indicate the sensitivity of the acceleration responses to the different parameters to be identified. Optimizing an appropriate norm of Q would yield the minimum possible value of the covariance of estimation error, and thus least uncertainty in the identified parameters [36]. Hence, in the optimal input location problem, the inputs should be located on those DOFs, which result in maximizing an appropriate norm of Q. Since R is constant over time, maximizing the norm of Q(TD ) is equivalent to maximizing the norm of Q(TD ), given as: TD

Q( T D ) =

∫0

(

𝜕 ẍ 𝜕𝚯

)T (

)

𝜕 ẍ dt 𝜕𝚯

(4)

2.1. Optimality criterion As Q(TD ) is a matrix, to concisely quantify the total information content about 𝚯 in y, some appropriate scalar norm of Q(TD ) should be considered. Some common norms used in the maximum information problem are [19]:

• D-optimality: Maximizing the determinant of Q • A-optimality: Minimizing the trace of [Q]−1 • T-optimality: Maximizing the trace of Q Although the linearity of the trace operator makes the T-optimality computationally superior [19], in the optimal sensor location problem, it has been shown that the D-optimality corresponds to minimizing the information entropy, and hence, the uncertainty in the parameter estimates in a Bayesian sense [24,25]. However, using D-optimality (or A-optimality) requires Q to be non-singular. If the number of measurements, say m, is less than the number of parameters to be identified, say n𝚯 , then

the matrix Q, although of dimension n𝚯 × n𝚯 , will be rank-deficient with rank m < n𝚯 , and hence singular. In such cases, neither the D-optimality nor the A-optimality can directly be used [37]. In the problem considered in this paper: with a full set of sensors, the number of measurements is m = N; and, if nM modes are to be identified, the number of parameters to be identified is n𝚯 = nM (N + 2) > m (one modal frequency, one damping ratio, and N mode shape components, per mode). Hence, even for identifying a single mode, the associated FIM will be singular, restricting the direct use of D- or A-optimality. To circumvent this problem, only the T-optimality criterion is used in this paper.

1

Although the formulation in this paper is developed assuming acceleration outputs, it can similarly be used for velocity and displacement outputs.

4

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

2.2. Partial derivatives of response with respect to modal parameters To compute the matrix Q(TD ) in Eq. (4), one needs to first compute the partial derivatives of the acceleration response ẍ with respect to the modal parameters in 𝚯. To facilitate these computations, the equation of motion in Eq. (1) is converted to modal coordinates. Then, for any input up located at the pth DOF, the modal equation of motion for any mode j can be written as:

𝜂̈ p,j + 2𝜁j 𝜔j 𝜂̇ p,j + 𝜔2j 𝜂p,j = 𝜙p,j up

∀j ∈ ,

∀p ∈ 

(5)

where 𝜔j and 𝜁 j are the jth modal frequency and damping ratio, respectively; 𝜙p,j is the pth component, corresponding to the pth DOF of the jth mass normalized mode shape; 𝜂 p,j , 𝜂̇ p,j and 𝜂̈ p,j are the jth modal displacement, velocity and acceleration responses, respectively;  is the set of all DOFs with inputs; and  is the set of all modes to be identified. The modal responses depend on both the input location through 𝜙p,j as well as the input time history up . As the objective is to find the optimal p, it would be beneficial to instead consider the pseudo-modal responses to the input up , which are independent of the input location (𝜙p,j ), and are given by the pseudo-modal equation of motion [38]:

𝜂̈ p,j + 2𝜁j 𝜔j 𝜂̃̇ p,j + 𝜔2j ̃ 𝜂 p ,j = u p

∀j ∈ ,

∀p ∈ 

(6)

Using modal superposition, the acceleration response ¨xi , at any DOF (say the ith), can then be written in terms of the pseudomodal accelerations, 𝜂 ̃̈ j ’s, as: ẍ i =

∑ j∈

(

𝜙i,j

∑

p ∈

)

𝜙 p ,j ̃ 𝜂̈ p,j

=

∑ ∑ j∈ p∈

𝜙i,j 𝜙p,j 𝜂̃̈ p,j

(7)

For any mode j ∈ , the partial derivatives in Eq. (4) are then given by:

∑ 𝜕 𝜂̃̈ (t) 𝜕 ẍ i (t) = 𝜙i,j 𝜙 p ,j p ,j ; 𝜕𝜔j 𝜕𝜔j p ∈

∑ 𝜕 𝜂̃̈ (t) 𝜕 ẍ i (t) = 𝜙i,j 𝜙 p ,j p ,j 𝜕𝜁j 𝜕𝜁j p ∈

(8)

and

𝜙k,j 𝜂̃̈ k,j (t), ⎧2∑ ⎪ 𝜙p,j 𝜂̃̈ p,j (t), 𝜕 ẍ i (t) ⎪ = ⎨ p ∈ 𝜕𝜙k,j ⎪𝜙 𝜂̃̈ (t), i,j k,j ⎪ ⎩0,

∀ k = i, k ∈  ∀ k = i, k ∉  (9)

∀ k ≠ i, k ∈  ∀ k ≠ i, k ∉ 

𝜂̈ p,j (t) and its partial derivatives with respect to 𝜔j and 𝜁 j , ∀j ∈  and p ∈ , the Duhamel’s integral solution of To compute ̃ Eq. (6) is first obtained: 𝜂 p ,j ( t ) = ̃

t

1

𝜔dj ∫0

where, 𝜔dj = 𝜔j

up (𝜏 )e−𝜁j 𝜔j (t−𝜏 ) sin(𝜔dj (t − 𝜏 )) d𝜏

(10)

√

1 − 𝜁j2 . Then, differentiating 𝜂 ̃p,j (t) in Eq. (10) with respect to ‘t’ and using Leibniz integral rule, the pseudo-

modal velocity is written as:

𝜂̃̇ p,j (t) = −𝜁j 𝜔j 𝜂̃p,j (t) +

t

∫0

up (𝜏 )e−𝜁j 𝜔j (t−𝜏 ) cos(𝜔dj (t − 𝜏 )) d𝜏

(11)

Using the expressions from Eqs. (10) and (11) in Eq. (6), a closed form expression of the pseudo-modal acceleration is obtained:

𝜔j (1 − 2𝜁j2 )  1 (t ) − 2𝜁j 𝜔j  2 (t ) 𝜂̃̈ p,j (t) = up (t) − √ 1 − 𝜁j2

(12)

where t

 1 (t ) =

∫0 t

 2 (t ) =

∫0

up (𝜏 )e−𝜁j 𝜔j (t−𝜏 ) sin(𝜔dj (t − 𝜏 )) d𝜏 (13) −𝜁j 𝜔j (t−𝜏 )

up (𝜏 )e

cos(𝜔dj (t − 𝜏 )) d𝜏

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

5

Finally, Eq. (12) is differentiated with respect to 𝜔j and 𝜁 j to give:

( ) ( ) 1 − 2𝜁j2 𝜕 𝜂̃̈ p,j (t) 𝜕  (t ) 𝜕  (t )  1 (t ) + 𝜔j 1 = −√ − 2𝜁j  2 (t ) + 𝜔j 2 𝜕𝜔j 𝜕𝜔j 𝜕𝜔j 1 − 𝜁2 j

𝜕 𝜂̃̈ p,j (t) 𝜔j = √ 𝜕𝜁j 1 − 𝜁j2

[

𝜁j (3 − 2𝜁j2 ) 1 − 𝜁j2

]

(

𝜕  (t ) 𝜕  (t )  1 (t ) − (1 − 2𝜁j2 ) 1 − 2𝜔j  2 (t ) + 𝜁j 2 𝜕𝜁j 𝜕𝜁j

)

(14)

where t 𝜕  1 (t ) = g (𝜏 )d𝜏 ∫0 1 𝜕𝜔j

t 𝜕  2 (t ) =− g (𝜏 )d𝜏 ∫0 2 𝜕𝜔j

𝜔j 𝜕  2 (t ) 𝜕  1 (t ) = √ 2 𝜕𝜁j 𝜕𝜔j 1−𝜁

𝜔j 𝜕  2 (t ) 𝜕  1 (t ) = −√ 2 𝜕𝜁j 𝜕𝜔j 1−𝜁

and, with g(𝜏 ) = up (𝜏 )(t − 𝜏 )e−𝜁j 𝜔j (t−𝜏 ) : g1 (𝜏 ) = g(𝜏 ) g2 (𝜏 ) = g(𝜏 )

(√

(15)

)

1 − 𝜁j2 cos(𝜔dj (t − 𝜏 )) − 𝜁j sin(𝜔dj (t − 𝜏 ))

(√

)

1−

𝜁j2

(16)

sin(𝜔dj (t − 𝜏 )) + 𝜁j cos(𝜔dj (t − 𝜏 ))

Hence, for any given set of input time histories and input locations in , the partial derivatives involved in the matrix Q(TD ) may be computed without the necessity of any finite difference scheme. The steps involved in these computations will be: 1. First evaluating the integrals in Eqs. (13) and (15) using any suitable numerical integration technique; 2. Using the values of these integrals in Eqs. (12) and (14) to evaluate the pseudo-modal accelerations and their derivatives with respect to modal frequencies and damping ratios; and 3. Finally, using the above in Eqs. (8) and (9) to get the partial derivatives of acceleration responses with respect to modal parameters.

2.3. Dimensionless scaling The different elements in the vector of modal parameters to be identified 𝚯 will have different units and, in general, different orders of magnitude. Hence, directly using the sensitivity of ẍ to the different parameters in 𝚯 without any scaling may result in the matrix Q(TD ) being ill-conditioned and/or biased towards certain parameters in 𝚯. One approach to circumvent this problem is to scale the different elements in 𝚯 so that the resulting parameter vector is dimensionless. Such scaling has recently been used in the context of Bayesian uncertainty quantification to make the Hessian dimensionless [39]. A similar scaling is adopted here, through the transformation matrix:

[

𝚪=

]

diag {𝚯∗ }−1

(17)

M1∕2

where 𝚯∗ = {𝚯𝝎 ; 𝚯𝜻 } is a subset of 𝚯 containing only the modal frequencies and damping ratios to be identified. Then, using 𝚪, the vector 𝚯 may be transformed to the dimensionless vector:

𝚵 = 𝚪𝚯

(18)

resulting in the scaled information matrix: Q∗ ( T D ) =

TD

∫0

(

𝜕 ẍ 𝜕𝚵

)T (

)

𝜕 ẍ dt = 𝚪−T Q(TD ) 𝚪−1 𝜕𝚵

(19)

The trace norm of this scaled Q∗ (TD ) would not be biased towards any particular modal parameter(s), and would hence serve as a more consistent measure of the information content in ẍ about all parameters in 𝚯. Note that the scaling used in Ref. [39] considered unit normalized mode shapes, and hence required no scaling for the mode shapes using I instead of M1/2 in the transformation matrix. However, to be consistent with the computations in Section 2.2, here the mode shape components in the unscaled 𝚯 are considered to be mass normalized. This necessitates the use of M1/2 in the matrix 𝚪 to scale the mode shapes; if 𝝓 is mass normalized, then M1/2 𝝓 is unit normalized. The required M1/2 can be 1∕ 2 computed as 𝝍 Tm 𝝀m 𝝍 m , where 𝝍 m and 𝝀m , respectively, denote the eigenvector and eigenvalue matrices of the mass matrix M.

6

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

2.4. Experiment design The determination of the optimal input locations is a part of experiment design, and needs to be done prior to actually performing the experiment on the physical structure. Once determined, the experiment can be performed with the inputs located at their optimal locations, and the measured responses can be used to identify the unknown modal parameters. However, the optimal input locations also explicitly depend on the values of these unknown parameters, as the evaluation of Q∗ (TD ) requires these values in the computation of the partial derivatives (Eqs. (8) and (9)) as well as in constructing the transformation matrix (Eq. (17)). To address this requirement, a nominal model of the system, which exists prior to identification, needs to be used. If this nominal model is “close enough” to the real system, then the optimal input locations obtained using the nominal modal parameters can be used for the experiment on the physical system. However, in a general situation, accounting for model parameter uncertainty, possible structural deterioration etc., the nominal model may not always be assumed to be “close enough” to the real system. In such a case, several possible models around the nominal model, would need to be considered in determining the optimal input locations. If the nominal model is characterized by the structural matrices (M, K ), a model space  can be defined around the nominal model (i.e., (M, K ) ∈  ), such that all models in  belong to the same model class as the nominal model. This model space should be large enough to include all possible models that may describe the real system, including any effects of structural damage. Then, using any sampling technique, N models from  : (Mi , Ki ) ∈  , ∀i ∈ {1, 2, … , N } can be generated, with a larger  requiring a larger N . For each of these candidate models, the optimal input locations can be obtained, resulting in a set of N optimal locations. The most frequently occurring locations in this set may then be selected as the optimal input locations for the test on the real physical system. 3. Relation of mode shape components at input locations with trace of FIM In this section, the trace of Q(TD ) is explicitly expressed in terms of the mode shape components at the input locations, for various experimental scenarios, viz. single/multiple modes to be identified using a single/multiple set(s) of inputs. As would be evident by the end of this section, these expressions are important to illustrate that in certain cases, the optimal locations of the inputs may be determined directly from the relative values of the mode shape components at possible input locations, thereby avoiding the complexity and cost associated with computing the trace of Q(TD ). For the general case of an N-DOF system, with  denoting the set of DOFs with inputs, and  denoting the set of modes to be identified, the trace of Q can be expressed as a sum of three terms as: tr(Q) = 𝕋1 + 𝕋2 + 𝕋3

(20)

The first term 𝕋1 is associated with the modal frequencies and may be written, using Eqs. (4) and (8), as:

𝕋1 =

N ∑ ∑ j∈ i=1

=

N ∑ ∑ j∈ i=1

=

N ∑ ∑ j∈ i=1

TD

(

∫0

𝜙2i,j

𝜕 ẍ i 𝜕𝜔j

TD

(

⎛ ⎜∑ ⎜p ∈ ⎝

𝔸p1 ,p2 ,j =

(

∫0

∑

𝜙2p,j 𝔸p,p,j + 2

where for all p1 , p2 ∈ : TD

dt

𝜕 𝜂̃̈ 𝜙 p ,j p ,j 𝜕𝜔j p ∈

∫0

𝜙2i,j ⎜

)2

𝜕 𝜂̃̈ p1 ,j 𝜕𝜔j

)(

𝜕 𝜂̃̈ p2 ,j 𝜕𝜔j

)2 dt

∑ p 1 ,p 2 ∈  p1 ≠ p2

(21)

⎞ ⎟

𝜙p1 ,j 𝜙p2 ,j 𝔸p1 ,p2 ,j ⎟ ⎟ ⎠

) dt

(22)

Similarly, the term 𝕋2 in Eq. (20), associated with modal damping ratios, may be written as:

𝕋2 =

N ∑ ∑ j∈ i=1

=

N ∑ ∑ j∈ i=1

TD

∫0

𝜙2i,j

(

𝜕 ẍ i 𝜕𝜁j

)2 dt

⎛ ⎞ ∑ ⎜∑ 2 ⎟ 𝜙p,j 𝔹p,p,j + 2 𝜙p1 ,j 𝜙p2 ,j 𝔹p1 ,p2 ,j ⎟ ⎜ ⎜p ∈ ⎟ p 1 ,p 2 ∈  ⎝ ⎠ p1 ≠ p2

(23)

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

where

𝔹p1 ,p2 ,j =

TD

(

∫0

𝜕 𝜂̃̈ p1 ,j 𝜕𝜁j

)(

𝜕 𝜂̃̈ p2 ,j 𝜕𝜁j

7

) dt

(24)

The last term 𝕋3 in Eq. (20) is associated with the mode shape components and can be written, using Eqs. (4) and (9), as: N N ∑ ∑ ∑

𝕋3 =

j∈ k=1 i=1

∑ ∑

=

TD

∫0

𝜕 ẍ i 𝜕𝜙k,j

)2 dt

(

(N − nI + 3)𝜙2p,j +

ℂ p ,j

j∈ p∈

(

N ∑ i=1

(25)

)

𝜙2i,j

where nI equals the total number of inputs, and

ℂ p ,j =

TD

𝜂̃̈ 2p,j dt

∫0

(26)

3.1. Single mode identification with a single input For the special case when only the parameters of a single mode j are to be identified, using only a single input located on the pth DOF, substituting Eqs. (21), (23) and (25) into Eq. (20), the trace of Q reduces to:

( tr(Q) =

𝜙2p,j

(𝔸p,p,j + 𝔹p,p,j )

N ∑ i=1

)

𝜙2i,j

+ ( N + 2 ) ℂ p ,j

+ ℂ p ,j

N ∑ i=1

𝜙2i,j

(27)

Note that, for a single input, by Eq. (6), the values of 𝔸p,p,j , 𝔹p,p,j and ℂp,j would be independent of the input location p for any given mode. Further, for any given mode, the sum

∑N

i=1

𝜙2i,j is also independent of the input location. Hence, in this special case,

the trace of Q is affected by the input location only through the factor 𝜙2p,j in the first term of Eq. (27). This illustrates the intuitive

understanding in modal analysis that, to identify a single mode using a single input, the input needs to be located at the DOF where the mode shape component for the mode to be identified has the maximum absolute value. This location corresponds to the maximum trace of the FIM, and hence, the FIM need not be computed to obtain the optimal input location in such a case. 3.2. Single mode identification with multiple inputs For the case when a single mode j is to be identified, but using multiple (nI > 1) inputs located on the DOFs in , the trace of Q can be written as: tr(Q) =

∑ p ∈

+2

(

𝜙2p,j

(𝔸p,p,j + 𝔹p,p,j )

∑

p 1 ,p 2 ∈  p1 ≠ p2

N ∑ i=1

)

𝜙2i,j

+ (N − nI + 3)ℂp,j

𝜙p1 ,j 𝜙p2 ,j (𝔸p1 ,p2 ,j + 𝔹p1 ,p2 ,j )

N ∑ i=1

𝜙2i,j +

∑ p ∈

ℂ p ,j

N ∑ i=1

(28)

𝜙2i,j

From Eq. (28), no immediate conclusion, as in the case of a single input, can be made. However, if all the nI inputs are the same, i.e., they have the same time histories, then the values of 𝔸p1 ,p2 ,j = 𝔸p,p,j , 𝔹p1 ,p2 ,j = 𝔹p,p,j and ℂp,j would again be independent of the input locations. In such a scenario, the trace of Q can be simplified to: tr(Q) = C1

∑

p ∈

𝜙2p,j + C2

∑

p 1 ,p 2 ∈  p1 ≠ p2

𝜙p1 ,j 𝜙p2 ,j + C3

(29)

where C1 , C2 and C3 are constants independent of the input locations. It is evident from Eq. (29) that the optimal locations of the inputs should be those DOFs that: (a) correspond to higher absolute values of the mode shape components, and (b) are in phase, for the mode to be identified. Hence, in such a scenario, the above two criteria can be used to find the optimal input locations, avoiding the costly computation of the matrix Q. When identifying a single mode, in case the time histories of the multiple inputs are different from one another, the terms in {𝔸p1 ,p2 ,j ; 𝔸p,p,j ; 𝔹p1 ,p2 ,j ; 𝔹p,p,j ; ℂp,j } would no longer be independent of the input locations. Similar would be the case when multiple modes are to be identified using a single/multiple input(s). Hence, in such cases, the optimal input locations may not

8

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

Fig. 1. A 5-DOF spring-mass system.

be directly determined using only the mode shapes; the trace of the matrix Q would need to be computed. Finally, in situations where the optimal input locations can directly be obtained from the mode shape components, the same experiment design strategy of Section 2.4 may be employed to account for model parameter uncertainty, structural deterioration etc. 4. Numerical validation A 5-DOF spring-mass system (representative of a typical 5-story shear building), as shown in Fig. 1, is used in this section to illustrate the various theoretical discussions of the preceding sections. The system has the following properties: all lumped masses mi = 2500 kg, spring stiffness ki = 5 × 106 N/m for all springs, and modal damping ratio of 5% in all modes. The DOFs are numbered such that the DOF associated with any mass mi is the ith DOF. Different examples are considered, viz. optimal input locations when identifying one or multiple modes using a single input, and when identifying a single mode using multiple inputs; an illustration of the experiment design strategy of Section 2.4; and finally, an application in modal parameter based damage detection. 4.1. Optimal location of a single input for identifying a single mode In this example, an impulsive force is assumed to be the input. Such a type of input is often imparted using an impulse hammer and used in the modal characterization of mechanical systems (e.g., machine tools) and other laboratory scale models. With the objective of identifying a specific mode of the system: (a) the trace norms of the matrix Q∗ (Section 2), and (b) the absolute mode shape components (Section 3), are computed for different possible locations of the input. These values, normalized to have the maximum value equal to one, are shown in Fig. 2 for modes 1 to 3 of the system. The trace norms of Q∗ are shown using blue filled circles, and the absolute values of mode shape components are shown using red unfilled squares. To illustrate how the uncertainty in the identified modal parameters change with the input locations, for each of the five input locations, 500 simulations of dynamic tests on the system are performed. Each simulation differs from the others in terms of the measurement noise sequences added to the true acceleration responses of the different DOFs. In any simulation, the added measurement noise sequences are obtained as r% root mean square (RMS) zero mean Gaussian white noise, with the percentage computed with respect to the RMS of the corresponding true response. The values of r are sampled from the uniform distribution  (2, 20), and thus, vary from simulation to simulation. The Eigensystem Realization Algorithm (ERA) [1] is then used to identify the modal parameters of the system from the “noisy” measured responses in each simulation. The estimation errors between the identified and corresponding true natural frequencies and damping ratios are computed. For mode shapes, the Modal Assurance Criterion (MAC) values comparing the identified and corresponding true mode shapes are computed, and the estimation errors are obtained as one minus the MAC values. The standard deviations of the estimation errors (relative to

Fig. 2. Comparison of normalized information content with normalized estimation uncertainty, for different input locations, when individually identifying modes 1 to 3 of the spring-mass system.

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

9

Table 1 Combinations of possible input locations for two inputs.

Combination No.

DOFs with inputs

Combination No.

DOFs with inputs

1 2 3 4 5

1,2 1,3 1,4 1,5 2,3

6 7 8 9 10

2,4 2,5 3,4 3,5 4,5

the true values) are next computed from the estimation errors obtained in all the 500 simulations. Finally, for any particular mode, the sum of the standard deviations of the estimation errors in that modal frequency, damping ratio and mode shape is taken as a scalar measure of the total estimation uncertainty for that mode. These total uncertainties for the different possible input locations, normalized to have the maximum value equal to one, are shown in Fig. 2 using black unfilled diamonds. Note that, when the input is located at DOF 2, mode 3 is not detected in any of the 500 simulations; hence, the uncertainty for this input location-mode combination is not included in Fig. 2. It is evident from Fig. 2 that, for all the three modes considered in this example, the input locations with lower trace norms of Q∗ (i.e., lower information content) correspond to larger estimation uncertainties, and vice versa. Further, for input locations with very low information content, it may not even be possible to identify the corresponding mode, as is the case with mode 3 in this example for input located at DOF 2. It can also be seen that the ranking of the possible input locations based on the trace norms of Q∗ agree with the ranking based on the absolute values of mode shape components in all cases. In this example, DOFs 5, 2, and 1 are found to be the optimal input locations for identifying modes 1, 2, and 3, respectively. As may be expected, the optimal location of the input changes with the mode to be identified. Hence, different modes identified from different tests, each with optimally located input for the corresponding mode, would lead to reduced uncertainty in the identified modes. 4.2. Optimal locations of multiple inputs for identifying a single mode For large structures, one input device may not sufficiently excite distant degrees of freedom, necessitating the use of multiple inputs for identifying the global modes of the system. In this example, such a case is considered with two input devices exciting the spring-mass system using zero mean Gaussian white noise signals. The objective is to find the optimal locations of these two inputs to identify a specific mode of the system. With five possible locations for each input, there are 5 C2 = 10 possible combinations of the input locations, as listed in Table 1. The trace of the Q∗ matrices associated with the combination numbers of Table 1 are plotted in Fig. 3 using blue filled circles, for modes 1 to 3 of the system. To quantify the uncertainty in identified modal parameters, a similar approach as in Section 4.1 is adopted with 500 simulations of dynamic testing for each of the combinations of Table 1. In each simulation, the Markov parameters of the system are first obtained from the least squares problem expressing the “noisy” outputs as convolution of these parameters with the inputs [40]. The identified Markov parameters are then used in the ERA to identify the modal parameters of the system [1]. The sum of the standard deviations of the relative errors in identified frequencies, damping ratios and mode shapes is taken as the scalar measure of estimation uncertainty, and plotted in Fig. 3 using black unfilled diamonds. The values for any quantity plotted in Fig. 3 have been normalized to maximum value one for any mode. It is again evident from this

Fig. 3. Comparison of normalized information content with normalized estimation uncertainty, for different locations of two inputs, when individually identifying modes 1 to 3 of the spring-mass system.

10

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

Fig. 4. Comparison of normalized information content with normalized estimation uncertainty, for different input locations, when simultaneously identifying modes 1 to 3 of the spring-mass system.

figure that the locations of inputs corresponding to lower information content generally result in an increased uncertainty in the identified modal parameters. 4.3. Optimal location of a single input for identifying multiple modes As shown in the preceding examples, the optimal input locations would generally be different for different modes, ideally requiring different tests to optimally identify different modes. However, in most modal testing applications the objective may be to identify a set of modes from a single test. In this example, such an objective is considered where the first three modes of the spring-mass system are to be identified from a single test with an impulsive force as input. The problem is to find the optimal location of the input for simultaneously identifying all the three modes with the minimum estimation uncertainty. Fig. 4 compares the normalized trace (blue filled circles) of the Q∗ matrices associated with the different input locations, with the normalized total uncertainty (black unfilled diamonds) in the modal parameters identified with such input locations. The input location at DOF 2 is excluded from this comparison as mode 3 is not detected with this location of the input (see Section 4.1). The estimation uncertainty is again computed based on 500 simulations for each input location, followed by modal identification using the ERA. As before, the total uncertainty is computed as the sum of the standard deviations of the relative estimation errors in the identified modal parameters, with all the three modes now considered together. It is again evident from the comparison in Fig. 4 that the total estimation uncertainty shows an increasing trend with decrease in the information content quantified by the trace of Q∗ . 4.4. Effect of input location on modal parameter based damage localization To highlight the effect of input location on modal parameter based structural damage localization, a damaged version of the spring-mass system of Fig. 1 is considered, with damage simulated as a 10% reduction in the stiffness of the fourth spring. Damage localization is performed using the normalized cumulative stresses (NCS) obtained with the damage locating vectors [9] computed from the first two identified modes of the healthy and damaged systems. Three testing scenarios with impulsive force inputs are considered: (a) a single test with the input located at DOF 3, which is non-optimal for both the two modes; (b) two different tests with the input in each test located at the DOF optimal for identifying the corresponding mode, i.e., at DOF 5 for identifying mode 1, and at DOF 2 for mode 2; (c) a single test with the input located at DOF 5, which is optimal for the simultaneous identification of both the modes. The responses are corrupted by 20% measurement noise prior to modal identification. For both the healthy and damaged systems, 500 simulations are performed with each simulation differing in terms of the added measurement noise sequences. The mean NCS values are computed from the NCS values obtained in the 500 simulations. Fig. 5 shows the mean NCS values in the different springs obtained in the three testing scenarios; the red bar corresponds to the damaged spring, while the blue bars correspond to the healthy springs. It is observed that, for the nonoptimally located input, the damaged spring could not be detected at all, although the location of the input (DOF 3) is adjacent to the damaged spring (k4 ). On the other hand, in both the testing scenarios with optimally located inputs, the damaged spring is, on average, detected accurately, with the mean NCS value for the damaged spring being distinctly lower than the other springs.

Fig. 5. Effect of input location on damage localization in the spring-mass system.

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

11

Fig. 6. Optimal location of single input for 100 candidate models.

4.5. Examples of experiment design The proposed technique of finding optimal input location(s) needs the prior knowledge of the modal parameters of the system, which are unknown at the experiment design stage. In the examples discussed in Sections 4.1 to 4.4, the true values of these parameters have been used solely for validation purposes: (a) to illustrate the inverse relation between the information content represented by Q∗ and the uncertainty in identified modal parameters; and (b) to highlight the adverse effects of non-optimally located inputs in damage detection. This section demonstrates the use of the experiment design strategy of Section 2.4 for the determination of optimal input location(s) in the realistic scenario of unknown modal parameters. The necessary model space is created based on the assumptions that: (a) the lumped masses of the real system lie in the range of 1750–2750 kg, (b) the stiffness of the springs lie in the range of 1 × 106 to 7 × 106 N/m, and (c) the modal damping ratios lie in the range of 1%–7%. These ranges correspond to −30% to +10% variations in mass, and −80% to +40% variations in stiffness and damping ratios, around their respective true values of 2500 kg, 5 × 106 N/m and 5%. Assuming independent and uniform distributions of the mass, stiffness and damping ratios within the above ranges, 100 candidate models are sampled from the resulting model space through Latin hypercube sampling (using the in-built MATLABⓇ function lhsdesign [41]). These 100 candidate models are then used for determining the optimal input location(s). In case of a single input, the location of the input corresponding to maximum trace of Q∗ is obtained for each of the candidate models. The location occurring as optimal for most of the candidate models may then be selected as the optimal input location. This example is illustrated in Fig. 6 for the case of identifying a single mode. For mode 1, it is evident that DOF 5 occurs as the optimal location for all the 100 models. For modes 2 and 3, while the optimal input locations are respectively DOFs 2 and 1 for most of the candidate models, DOF 5 for mode 2 and DOF 3 for mode 3 also appear as optimal for a significant number of the models. In such a situation, performing multiple experiments may provide more reliable parameter estimates. Note further that, DOFs 5 and 3, which appear as optimal for the second-most number of models in Fig. 6, are also the second-to-optimal DOFs in Fig. 2 for identifying modes 2 and 3, respectively. In case of multiple inputs, two different approaches can be adopted. In the first approach, the search for optimal input locations is performed considering all the inputs simultaneously. For an N-DOF system with nI inputs, this results in a search space of N Cn combinations for the potential locations of the inputs. For each candidate model, the combination correspondI

ing to maximum trace of Q∗ is obtained. The combination occurring as optimal for most of the candidate models then gives the optimal locations of the inputs. Fig. 7 illustrates this approach, for the case of identifying a single mode of the system using 2 inputs. Amongst all the 5 C2 = 10 combinations, as listed in Table 1, combinations 10, 7 and 2 are found to be the optimal input locations for identifying modes 1, 2 and 3, respectively. These combinations agree with the ones found earlier, using the true model parameters as shown in Fig. 3. Also, as in the case of single input, the second-to-optimal combinations in Fig. 3 are found to be the optimal combinations for the second-most number of candidate models in Fig. 7, for both modes 2 and 3. In the second approach for multiple inputs, the search for the optimal input locations is performed considering each input sequentially, analogous to the sequential sensor placement strategies proposed in Ref. [25]. First, the optimal location of one input is determined. Next, keeping this input at its optimal location, the optimal location of the second input is determined from amongst the remaining (N − 1) available DOFs (assuming an N-DOF system). This process is then repeated for all the inputs. At each stage, since the optimal location of only one input is being sought, the approach for a single input is adopted. In this way, the search space reduces from N Cn combinations of input locations to only N + (N − 1) + · · · (N − nI + 1) locations. I

While this may result in a final solution which is sub-optimal, it would significantly reduce the computational expense in case

12

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

Fig. 7. Optimal locations of two inputs for 100 candidate models.

Fig. 8. Optimal location of second input, with known location of first input, for 100 candidate models.

of large systems with multiple inputs.2 Fig. 8 illustrates this approach for the case of 2 inputs, showing the optimal locations of the second input, with the first input optimally located as per Fig. 6. The optimal locations of the second input are found to be DOFs 4, 5 and 3, for identifying modes 1, 2 and 3, respectively. In case of all the three modes, the combined locations of the first and second inputs from Figs. 6 and 8 agree with the combinations obtained by considering the inputs simultaneously in Fig. 7. 5. Experimental application The proposed method of determining optimal input locations is applied on a laboratory scale 2-dimensional (2-D) steel truss, shown in Fig. 9(a). The truss is 0.2 m high and 3 m long, with each top and bottom chord being 0.6 m long. All members are of channel sections with 36 sq-mm cross-sectional area, as shown in Fig. 9(b). All joints are bolted, and both end supports are modeled to resemble hinges as closely as possible (Fig. 9(c)). Bolting is adopted instead of rivetting or welding for ease of: (i) fabrication, and (ii) modification in members to simulate damage scenarios. To reduce out-of-plane vibrations, guided wires are used to restrain the top and bottom chords at their center points. The truss has a total of 19 elements (members) and 9 unrestrained nodes (joints). A schematic diagram of the truss with element and node numbering is shown in Fig. 9(d). For any node i, the associated horizontal and vertical DOFs are numbered as 2i − 1 and 2i, respectively, resulting in a total of 18 active DOFs. To simulate external dead load on the truss, additional masses of 6.917 kg, 6.91 kg, 6.912 kg and 6.864 kg are added to the bottom nodes 6 to 9, respectively, through attached pans (the above masses include the weights of the pans). A slight variation of masses of these blocks is due to the variability in sizes during the fabrication of these blocks. A finite element (FE) model of

2 Although not used in this paper, for large systems one would need to use some discrete optimization technique (like discrete genetic algorithm in Ref. [25]) to find the input locations which maximize the trace of Q∗ . A significant reduction in the search space would result in a significant reduction in the computational cost for such algorithms.

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

13

Fig. 9. (a) Experimental set-up with 2-D truss; (b) schematic of member cross-sections; (c) simulation of hinge support; and (d) schematic of truss with element numbers (plain) and node numbers (circled).

the truss so fabricated is developed. The first three frequencies of the FE model of the truss are found to be 21.94, 60.80 and 113.09 Hz, respectively. Each node of the truss is instrumented in the vertical and horizontal directions using strain gauge based accelerometers from Honeywell (https://www.honeywell.com/). The data is recorded using a data acquisition (DAQ) system from National Instruments (NI) (http://www.ni.com/). Dynamic tests on the truss are performed by hitting each node of the truss with an impulse hammer in the vertical direction (i.e., in the plane of the truss). In each test, nodal acceleration responses are recorded for 12 s with a sampling frequency of 2000 Hz. Although such a high sampling rate is not required considering the first three modes of the system to be measured, this sampling rate is decided by the hardware of the NI DAQ system. The optimal input locations for identifying the first three modes of the truss are determined using the experiment design strategy of Section 2.4. 100 candidate finite element (FE) models of the truss are generated using Latin hypercube sampling from a model space where the cross-sectional areas of the members are independently and uniformly distributed between 10

Fig. 10. Optimal input locations for identifying different modes of the 2-D truss: (a) Mode 1, (b) Mode 2, (c) Mode 3, and (d) All three modes simultaneously.

14

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833 Table 2 Identified modal parameters of 2-D truss from tests with optimally located inputs.

Mode

Mode 1

Mode 2

Mode 3

Input location Frequency (Hz) Damping Ratio (%) MAC

Dof 6 16.24 1.28 0.992

Dof 12 54.35 1.59 0.984

Dof 18 90.10 1.66 0.921

Fig. 11. Comparison of FE mode shapes of 2-D truss with identified mode shapes from tests with optimally located inputs.

sq-mm and 40 sq-mm. This range of variation in the member cross-sections is assumed so that the determined optimal input locations are applicable for the healthy truss, as well as when one/more members of the truss is damaged with reduced crosssection. Fig. 10 shows the results of this experiment design. It can be seen that DOFs 6, 12 and 18 (vertical DOFs corresponding to nodes 3, 6 and 9) are the optimal input locations for identifying modes 1, 2 and 3, respectively, while DOF 18 is also the optimal location for identifying all the three modes simultaneously. Further, reflecting the symmetry of the truss: (a) for modes 2 and 3, the second-to-optimal DOFs are symmetrically placed with respect to the optimal ones, while (b) for mode 1 the second- and third-to-optimal DOFs are symmetrically placed with respect to each other. The first three modes of the truss are now identified from the measured experimental response data using ERA [1]. For each mode, the data from the test with the input applied at the corresponding optimal DOF is used, i.e. mode 1 is identified from the test with input at DOF 6, mode 2 from the test with input at DOF 12, and mode 3 from the test with input at DOF 18. The identified modal frequencies and damping ratios for the three modes are listed in Table 2. It is observed that the identified frequencies are about 10–25% lower than the corresponding frequencies of the FE model. This is possibly owing to the additional flexibility introduced in the supports by the wooden blocks used for levelling of the truss (see Fig. 9(c)) as well as due to modelling assumptions associated with the bolted joints and imperfections in fabrication. However, the identified mode shapes are found to compare reasonably well with the corresponding FE mode shapes, as shown in Fig. 11. The MAC values listed in Table 2 further illustrate the good agreement between the identified and FE mode shapes. The MAC values comparing the identified modes to the corresponding FE modes for different input locations are also listed in Table 3. It is evident that the highest MAC values correspond to the optimal input locations, highlighting the advantage of optimally located inputs in improving the accuracy of modal identification, especially for higher modes. To experimentally illustrate the importance of appropriate input location on structural damage detection, a damaged version of the truss is similarly tested as the healthy truss. Damage in the truss is simulated by replacing element 18 (see Fig. 9(d)) with

Table 3 MAC values for different input locations.

Input Location

MAC Values Mode 1

Mode 2

Mode 3

DOF 2 DOF 4 DOF 6 DOF 8 DOF 10 DOF 12 DOF 14 DOF 16 DOF 18

0.863 0.912 0.992 0.917 0.920 0.914 0.927 0.921 0.917

0.728 0.740 0.663 0.742 0.751 0.984 0.744 0.752 0.976

0.545 0.355 0.586 0.517 0.519 0.917 0.533 0.621 0.921

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

15

Fig. 12. Effect of input location on damage localization in the 2-D truss.

a member having 50% reduced cross-sectional area. The first three modes of the damaged truss are identified from the measured data in the same way as is done for the healthy truss. The first three modal frequencies of the damaged truss, identified from tests with inputs located at the corresponding optimal DOFs, are 16.06, 53.36 and 88.76 Hz, respectively. The reduction in the identified frequencies, as compared to those of the healthy truss, signal the presence of damage in the system. The identified modal parameters of the healthy and damaged trusses are then used, along with the FE mass matrix of the healthy truss, to detect the damaged element using the NCS approach mentioned earlier. As in Section 4.4, three monitoring scenarios are considered with tests on the healthy and damaged trusses conducted as follows: (a) Single tests with input at DOF 10, which is a non-optimal location for all the three modes; (b) Three sets of tests, each set with optimally located input for identifying a particular mode, viz. input at DOF 6 for identifying mode 1, at DOF 12 for mode 2, and at DOF 18 for mode 3; and (c) Single tests with input at DOF 18, which is optimal for simultaneously identifying all the three modes. The NCS values in the different members obtained in the three monitoring scenarios are shown in Fig. 12, with red bars denoting the NCS in the damaged member and blue bars denoting the NCS in the other members. It is again evident that the damaged member is not detected when the input location is non-optimal, even though the input is just adjacent to the damage, acting at a node to which the damaged member connects. On the other hand, for both cases of optimally located inputs, whether for individual modes or for all three modes considered simultaneously, the NCS in the damaged member is distinctly lower than the NCS in the other members, enabling correct detection of the damaged member. It may be noted here that some other members also show low NCS values. This is a common issue with the NCS approach [11], owing to a low NCS value only indicating a potentially damaged member. 6. Conclusion In this paper, a method is developed for finding optimal input locations for identification of modal parameters in experimental modal analysis. The method is based on maximizing the trace of the associated Fisher Information Matrix (FIM). This results in maximizing the information content about the modal parameters in the measured vibration response data. Although acceleration responses are considered in this paper, the method can be similarly applied with displacement or velocity responses. Analytical expressions are obtained for the derivatives involved in constructing the FIM, avoiding the need for any finite difference computations. Different testing situations are considered, viz. identifying one/multiple modes using one/multiple inputs. Since the computation of the FIM requires the modal parameters of the system, which are unavailable prior to the actual test, an experiment design strategy is proposed which determines the optimal input location(s) based on candidate models sampled from a model space defined around a nominal model of the system. This experiment strategy is also verified numerically. Finally, the effect of input location on modal parameter based structural damage detection is studied using numerical examples as well as experiments performed on a laboratory scale 2-D steel truss. Detection of damaged elements is performed using the NCS approach. The major findings of this study can be listed as follows:

• Through numerical examples with a spring-mass system, it is shown that input locations corresponding to higher trace norms of the computed FIMs generally result in lower uncertainty in the identified modal parameters. Hence, tests performed with input(s) placed at optimal location(s) will result in identified modal parameters with the least estimation uncertainty. • It is also observed that, the optimal location of any input changes with the mode to be identified. Hence, different modes identified from different tests, each with optimally located input for the corresponding mode, would lead to reduced uncertainty in the identified modes.

16

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

• Based on the explicit relation between the FIM and the mode shape components at input locations, it is shown that, when identifying a single mode using a single input, the optimal input location is the DOF where the corresponding mode shape has the maximum absolute value – an intuitive understanding in modal analysis. • It is also shown that, when identifying a single mode using multiple inputs having the same time histories, the optimal input locations are the DOFs where the corresponding mode shape have higher absolute values and are in phase. In such situations, the optimal input location(s) can be obtained directly from the mode shape, avoiding the computation of the FIM. However, in cases where multiple modes are to be identified simultaneously, or where a single mode is to be identified using multiple inputs having different time histories, the mode shapes alone will not provide the optimal location(s) of the input(s), necessitating the use of the FIM. • It is shown that tests with non-optimally located inputs can result in the damaged elements being not detected, while tests with optimally located inputs are shown to result in correct detection of the damaged elements. While the ERA is used for identifying the modal parameters in these examples, the proposed method of determining the optimal input location(s) is independent of the identification algorithm to be used, and should be applicable for any suitable modal identification technique.

References [1] J.-N. Juang, R.S. Pappa, An eigensystem realization algorithm for modal parameter identification and model reduction, J. Guid. Control Dyn. 8 (5) (1985) 620–627. [2] B. Peeters, G. De Roeck, Reference-based stochastic subspace identification for output-only modal analysis, Mech. Syst. Signal Process. 13 (6) (1999) 855–878. [3] R. Brincker, L. Zhang, P. Andersen, Modal identification of output-only systems using frequency domain decomposition, Smart Mater. Struct. 10 (3) (2001) 441. [4] M. De Angelis, H. Lu, R. Betti, R.W. Longman, Extracting physical parameters of mechanical models from identified state-space representations, J. Appl. Mech. 69 (5) (2002) 617–625. [5] K.-V. Yuen, J.L. Beck, L.S. Katafygiotis, Efficient model updating and health monitoring methodology using incomplete modal data without mode matching, Struct. Contr. Health Monit. 13 (1) (2006) 91–107. [6] S. Mukhopadhyay, H. Lu, R. Betti, Structural identification with incomplete instrumentation and global identifiability requirements under base excitation, Struct. Contr. Health Monit. 22 (7) (2015) 1024–1047. [7] K. Prajapat, S. Ray-Chaudhuri, Prediction error variances in Bayesian model updating employing data sensitivity, J. Eng. Mech. 142 (12) (2016) 04016096. [8] A. Pandey, M. Biswas, M. Samman, Damage detection from changes in curvature mode shapes, J. Sound Vib. 145 (2) (1991) 321–332. [9] D. Bernal, Load vectors for damage localization, J. Eng. Mech. 128 (1) (2002) 7–14. [10] F. Catbas, M. Gul, J. Burkett, Damage assessment using flexibility and flexibility-based curvature for structural health monitoring, Smart Mater. Struct. 17 (1) (2007) 015024. [11] Y. Gao, B. Spencer Jr., D. Bernal, Experimental verification of the flexibility-based damage locating vector method, J. Eng. Mech. 133 (10) (2007) 1043–1049. [12] L. Balsamo, S. Mukhopadhyay, R. Betti, A statistical framework with stiffness proportional damage sensitive features for structural health monitoring, Smart Struct. Syst. 15 (3) (2015) 699–715. [13] A. Deraemaeker, K. Worden, New Trends in Vibration Based Structural Health Monitoring, vol. 520, Springer Science & Business Media, 2012. [14] G.R. Parker, T.L. Rose, J.J. Brown, Kinetic energy calculation as an aid to instrumentation location in modal testing, in: Proceedings of MSC World Users Conference, 1990. [15] Y.-T. Chung, J.D. Moore, On-orbit sensor placement and system identification of space station with limited instrumentations, in: Proceedings of the International Modal Analysis Conference, 1993, 4141. [16] H. Sun, O. Buyukozturk, Optimal sensor placement in structural health monitoring using discrete optimization, Smart Mater. Struct. 24 (2015) 125034. [17] D.C. Kammer, Sensor placement for on-orbit modal identification and correlation of large space structures, J. Guid. Control Dyn. 14 (2) (1991) 251–259. [18] P. Shah, F. Udwadia, A methodology for optimal sensor locations for identification of dynamic systems, J. Appl. Mech. 45 (1) (1978) 188–196. [19] F.E. Udwadia, Methodology for optimum sensor locations for parameter identification in dynamic systems, J. Eng. Mech. 120 (2) (1994) 368–390. [20] M. Papadopoulos, E. Garcia, Sensor placement methodologies for dynamic testing, AIAA J. 36 (2) (1998) 256–263. [21] L. Yao, W.A. Sethares, D.C. Kammer, Sensor placement for on-orbit modal identification via a genetic algorithm, AIAA J. 31 (10) (1993) 1922–1928. [22] H. Guo, L. Zhang, L. Zhang, J. Zhou, Optimal placement of sensors for structural health monitoring using improved genetic algorithms, Smart Mater. Struct. 13 (3) (2004) 528. [23] E. Heredia-Zavoni, L. Esteva, Optimal instrumentation of uncertain structural systems subject to earthquake ground motions, Earthq. Eng. Struct. Dyn. 27 (4) (1998) 343–362. [24] C. Papadimitriou, J.L. Beck, S.-K. Au, Entropy-based optimal sensor location for structural model updating, J. Vib. Control 6 (5) (2000) 781–800. [25] C. Papadimitriou, Optimal sensor placement methodology for parametric identification of structural systems, J. Sound Vib. 278 (4) (2004) 923–947. [26] E.B. Flynn, M.D. Todd, A Bayesian approach to optimal sensor placement for structural health monitoring with application to active sensing, Mech. Syst. Signal Process. 24 (4) (2010) 891–903. [27] G. Capellari, E. Chatzi, S. Mariani, An optimal sensor placement method for SHM based on Bayesian experimental design and polynomial chaos expansion, in: Proceedings of VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Crete Island, Greece, 2016, pp. 6272–6282. [28] N.J. Kessissoglou, P. Ragnarsson, A. Lfgren, An analytical and experimental comparison of optimal actuator and error sensor location for vibration attenuation, J. Sound Vib. 260 (4) (2003) 671–691. [29] W. Liu, Z. Hou, M.A. Demetriou, A computational scheme for the optimal sensor/actuator placement of flexible structures using spatial h2 measures, Mech. Syst. Signal Process. 20 (4) (2006) 881–895. [30] A.R.M. Rao, K. Sivasubramanian, Optimal placement of actuators for active vibration control of seismic excited tall buildings using a multiple start guided neighbourhood search (msgns) algorithm, J. Sound Vib. 311 (12) (2008) 133–159. [31] T. Nestorovi, M. Trajkov, Optimal actuator and sensor placement based on balanced reduced models, Mech. Syst. Signal Process. 36 (2) (2013) 271–289. [32] Y.-J. Cha, A. Raich, L. Barroso, A. Agrawal, Optimal placement of active control devices and sensors in frame structures using multi-objective genetic algorithms, Struct. Contr. Health Monit. 20 (1) (2013) 16–44. [33] K. Morris, S. Yang, Comparison of actuator placement criteria for control of structures, J. Sound Vib. 353 (2015) 1–18. [34] D. Jana, K. Prajapat, S. Mukhopadhyay, S. Ray-Chaudhuri, Optimal input location for modal identification, in: Procedia Engineering (Vol. 199), Proceedings of the X International Conference on Structural Dynamics (EURODYN), Rome, Italy, 2017, pp. 990–995. [35] N.E. Nahi, Estimation Theory and Applications, Wiley, New York, 1969.

D. Jana et al. / Journal of Sound and Vibration 459 (2019) 114833

17

[36] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Courier Corporation, 2007. [37] Y. Jung, R.S. Ranjithan, G. Mahinthakumar, Subsurface characterization using a D-optimality based pilot point method, J. Hydroinf. 13 (4) (2011) 775–793. [38] S. Mukhopadhyay, H. Lu, R. Betti, Modal parameter based structural identification using inputoutput data: minimal instrumentation and global identifiability issues, Mech. Syst. Signal Process. 45 (2) (2014) 283–301. [39] Y.-C. Zhu, S.-K. Au, Bayesian operational modal analysis with asynchronous data, Part II: posterior uncertainty, Mech. Syst. Signal Process. 98 (2018) 920–935. [40] J.-N. Juang, M. Phan, L.G. Horta, R.W. Longman, Identification of observer/Kalman filter Markov parameters: theory and experiments, J. Guid. Control Dyn. 16 (2) (1993) 320–329. [41] MATLAB, Version 8.3.0.532 (R2014a), The MathWorks Inc., Natick, Massachusetts, 2014.