- Email: [email protected]

Substructural and progressive structural identification methods C.G. Koh a,∗, B. Hong b, C.Y. Liaw a a

Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b DSO National Laboratories, 20 Science Park Drive, Singapore 118230, Singapore Received 1 November 2002; received in revised form 12 May 2003; accepted 15 May 2003

Abstract While it is possible in principle to determine unknown structural parameters by system identification techniques, a major challenge lies in the numerical difficulty in obtaining reasonably accurate results when the system size is large. Adopting the strategy of “divide-and-conquer” to address this issue, substructural identification and progressive structural identification methods are formulated. The main idea is to divide the structure into substructures such that the number of unknown parameters is within manageable size in each stage of identification. A non-classical approach of genetic algorithms is employed as the search tool for its several advantages including ease of implementation and desirable characteristics of global search. Numerical simulation study is presented, including a fairly large system of 50 degrees of freedom, to illustrate the identification accuracy and efficiency. The methods are tested for known-mass and unknown-mass systems with up to 102 unknown parameters, accounting for the effects of incomplete and noisy measurements. 2003 Elsevier Ltd. All rights reserved. Keywords: Structural dynamics; System identification; Substructure; Genetic algorithms

1. Introduction Engineering analysis can be broadly categorized as direct analysis and inverse analysis. Direct analysis for structural systems aims to predict structural response (output) for given excitation (input) and known system parameters, whereas inverse analysis deals with identification of structural parameters based on given input and output (I/O) information. The latter may be termed as “structural identification” and falls within the domain of system identification. Structural identification can be applied to update the model so as to better predict structural response. It can be used for structural health monitoring and damage assessment in a non-destructive way by tracking changes in relevant structural parameters. For structural control applications, identification of actual parameters is also essential for effective control. From computational point of view, structural identification presents a challenging problem particularly when the system involves a large number of unknown para-

∗

Corresponding author. Tel.: +65-6874-2163; fax: +65-6779-1635. E-mail address: [email protected] (C.G. Koh).

0141-0296/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0141-0296(03)00122-6

meters. Besides accuracy and efficiency, robustness is an important issue for selecting the identification strategy. The following factors are often considered in numerical simulation study to test the identification strategy. 1. The strategy should not require unreasonably good initial guess for convergence. 2. In practice, dynamic response is normally measured by using accelerometers. Error is incurred to obtain velocity and displacement signals by integration. Hence, direct use of acceleration signals is preferred over velocity and displacement signals. 3. While accurate measurements are possible due to advances in sensor technology, noise is still inevitable and affects the identification accuracy. The strategy should thus be tested in the presence of I/O noise. 4. Though more measurements give better results in general, the strategy should not assume complete measurement since this is difficult to achieve in reality. Research interest in this subject area has increased steadily over the years, mainly due to rapid enhancement in computer power and development in new algorithms.

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Structural identification methods can be classified under various categories, such as frequency and time domains, parametric and nonparametric models, and deterministic and stochastic approaches [1–4]. Alternatively, these methods can be categorized into classical and non-classical methods. Classical methods can be dated back to the late 1970s and early 1980s [5–8], and literature review of recent works can be found in Refs. [9–11]. In many studies, however, classical methods have been tested only on relatively small structural systems with typically not more than 10 unknowns. As mentioned, the main challenge is to achieve reasonable accuracy of identified results for larger systems. Due to the ill-conditioned nature of inverse analysis, convergence becomes more difficult as the number of unknown parameters increases. A logical approach is to divide the structural system into smaller substructural systems so that the number of unknowns, and hence the convergence difficulty, is reduced in each identification effort. To this end, Koh et al. [12] formulated a “substructural identification” (SSI) strategy and illustrated its significant improvement in terms of identification accuracy and efficiency. In the context of classical identification methods, subsequent works adopting the substructure concept included Refs. [13–15]. Similar attempts were made to reduce the number of unknowns by an improved condensation method for multi-story frame buildings [16,17]. Classical methods are derived from rigorous mathematical theories and are typically calculus-based, e.g. use of gradients to obtain the next estimate. In contrast, non-classical methods largely depend on computing power for the extensive and hopefully robust search. Forward analysis is employed to evaluate the “fitness” of trial parameters and the next guess is deduced based on some heuristic rules such as neural and evolutionary concepts. The availability of faster computers has, in particular, facilitated research effort for the computationally demanding task as often required by nonclassical methods. These “soft-computing” methods include neural networks and evolutionary algorithms, developed as general search and optimization tools. The associated research work is relatively new in the context of structural identification; some representative research works are described below. Neural networks are computing systems with the ability to learn by extensive training data sets, imitating the way humans process information and make inference. For structural identification, unknown parameters can be recognized from given measurements by selforganization. A key feature of the neural network methods is that they can potentially cope with complex nonlinear system and on-line identification [18]. Wu et al. [19] applied neural networks to identify damage of a simple three-store frame by comparison of parameters of the healthy (undamaged) and damaged structures. Chas-

siako and Masri [20] applied a neural network-based approach for damage detection, using healthy system signals as training data. On the basis of substructural identification by means of neural networks, Yun and Bahng [21] adopted natural frequencies and mode shapes as input patterns, and Yun et al. [22] proposed a joint damage assessment method for framed structures. Very recently, Tsai and Hsu [23] applied the neural network approach to identify damage of reinforced concrete structures. Evolutionary algorithms can be classified into four categories: evolutionary programming, evolutionary strategies, genetic programming and genetic algorithms [24,25]. Based on the principle of “survival of the fittest” and genetic operations, these heuristic algorithms have been developed for diverse optimization and search problems in engineering fields including civil engineering such as construction scheduling and structural optimization. Nevertheless, relatively few studies have been reported on structural identification, and the genetic algorithms (GA) method appears to be a popular choice. First introduced in the 1960s and later embedded into a general framework of adaptation [26,27], the GA method aims to imitate evolution of living things by natural selection. Providing a remarkable balance between exploitation of good candidates and exploration by random chances, this method has been shown to have several advantages over classical methods in the context of structural identification [28]. The advantages include enhancement of global convergence by conducting population-to-population search, no requirement of gradient information, relative ease of implementation, convenient use of any measured response in defining the fitness function, and self-start feature with random initial guess in a specified search range. Besides, it has a high level of concurrency and is thus suitable for distributed computing [29]. In the context of spectral elements, Doyle [30] used the GA approach to identify the location of impact load (one unknown in the example presented). In a similar study, Zhao et al. [31] identified the flaw sizes and locations in a rectangular plate with up to eight unknowns in the numerical examples. Dunn [32] employed the GA approach to update the finite element model (with up to three unknowns) and the fitness function which was defined in terms of the frequency response function. With an associative memory approach, Udwadia and Proskurowski [33] to generate the relevant training vectors so as to improve the GAidentification results and applied it to a simple structure of 5 degrees of freedom (DOFs). Chakraborty et al. [34] showed that the use of GA could provide good initial estimate of parameters of stiffened plates. Nonetheless, the structures dealt with in the above GA-based structural identification studies were rather simple with few unknowns. A rare exception is the numerical example

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of a 50-DOF system, perhaps the largest system thus far in terms of the number of unknowns (52) for structural identification, as presented by Koh et al. [28]. This was made possible by conducting identification task in modal domains with few unknowns each. Just as in the case of substructural identification, the modal GA method can be seen as another divide-and-conquer strategy to improve the numerical convergence by splitting the problem into smaller problems of manageable size. In this paper, the idea of divide-and-conquer is pursued in the framework of GA-based substructural identification. Much as it appears as a simple idea, it should be noted that substructures are not isolated from the remainder of the structure (or adjacent substructures). It is necessary to account for interaction forces at interface where the substructure of concern is “cut” from the rest of the structure. The SSI method proposed by Koh et al. [12] is herein modified using the concept of quasi-static displacement vector to facilitate the handling of interaction force at substructural interfaces. Consequently, the required information is less than the SSI method originally proposed in Ref. [12]. In addition, a variation of the SSI method is developed in this paper. The GA approach is applied to identify structural parameters in stages beginning from a substructure and progressively to bigger substructures until the whole or the necessary part of the structure is identified. Numerical results are presented for up to 50-DOF structures involving all mass and stiffness values unknown.

2. Substructural identification As an illustration of the SSI method, a linear structure as shown in Fig. 1(a) is considered. In order to write the equations of motion for the substructure as shown in Fig. 1(b), the equations of motion for the entire structure can be arranged as follows:

冤

Muu Muf Mfu Mff Mfr

u¨f

Mrf Mrr Mrg

u¨r

Mgr Mgg Mgd Mdg Mdd

⫹

冤

冥冦 冧 冥冦 冧 u¨u

Cuu Cuf

u¨g u¨d

u˙r

Cgr Cgg Cgd

u˙g

Cdg Cdd

u˙d

冤

Kfu Kff Kfr Krf Krr Krg Kgr Kgg Kgd

(1)

冥冦 冧 冦 冧

Kdg Kdd

uu

Pu

uf

Pf

ur

⫽ Pr

ug

Pg

ud

Pd

where subscript ‘r’ denotes internal DOFs of the substructure concerned, subscripts ‘F’ and ‘G’ denote the DOFs of the remaining structure on the two sides marked as F and G in Fig. 1(a). Subscripts ‘f’’ and ‘g’ denote interface DOFs of the substructure with the remaining structure on the two sides F and G, respectively. Let subscript j denote all interface DOFs (i.e. f and g included) for concise presentation. For the substructure considered, the equations of motion may be extracted from the above equation system to yield

再 冎 再 冎 u¨j(t)

u¨r(t)

⫹ [Krj Krr]

u˙f

Crf Crr Crg

⫹

(a) Complete structure; (b) a substructure.

Kuu Kuf

[Mrj Mrr]

u˙u

Cfu Cff Cfr

Fig. 1.

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⫹ [Crj Crr]

uj(t)

ur(t)

再 冎 u˙j(t)

u˙r(t)

(2)

⫽ Pr(t)

Treating interaction effects at the interface ends as “input”, the above equation system can be re-arranged as Mrru¨r(t) ⫹ Crru˙r(t) ⫹ Krrur(t) ⫽ Pr(t)⫺Mrju¨j(t) ⫺Crju˙j(t)⫺Krjuj(t)

(3)

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From identification point of view, there are two versions of the SSI approach depending on whether overlap is allowed between adjacent substructures [12].

In the version of SSI without overlap as illustrated in Fig. 2(a), no members of any two adjacent substructures overlap. The SSI procedure is fairly straightforward, provided that response measurements at interface DOFs are available which are, in fact, treated as input to the substructure of concern. In the SSI formulation by Koh et al. [12], accelerations, velocities and displacements at the interface DOFs are required as evident in the RHS of Eq. (3). Due to practical reason as mentioned earlier, acceleration is preferred over velocity and displacement. To eliminate the requirement of time signals of displacement and velocity, the concept of “quasi-static displacement” vector is adopted. The displacements for internal DOFs are split into quasi-static displacements (usr) and “relative” displacements (u∗r ), i.e. (4)

Quasi-static displacements are obtained by solving Eq. (3) while ignoring the applied force (Pr), inertia effect and damping effect (all time-derivative terms set to zero). Hence, Krrusr ⫽ ⫺Krjuj

(5)

or usr ⫽ ⫺K⫺1 rr Krjuj ⫽ ruj

Mrru¨ ∗r (t) ⫹ Crru˙ r∗(t) ⫹ Krrur∗(t) ⫽ Pr(t)⫺(Mrj

(7)

⫹ Mrrr)u¨ j(t)⫺(Crj ⫹ Crrr)u˙ j(t)

2.1. Substructural identification without overlap

ur(t) ⫽ usr(t) ⫹ u∗r (t)

static condition. Substituting the above equation into Eq. (3) leads to

(6)

where r is called the influence coefficient matrix which relates internal DOFs to interface DOFs under the quasi-

The RHS without Pr term represents forces induced by motion relating to interface DOFs and may be referred to as “interface motion forces” for convenience. Since damping force is usually small compared to inertia force in typical civil engineering structures, the velocitydependent part in the interface motion forces is assumed to be negligible. Hence, Mrru¨ ∗r (t) ⫹ Crru˙ r∗(t) ⫹ Krrur∗(t) ⫽ Pr(t)⫺(Mrj

(8)

⫹ Mrrr)u¨ j(t) Accordingly, only accelerations (no displacements or velocities) at interface DOFs are required to compute the interface motion forces. If there is no excitation within the substructure, Pr simply vanishes and the method can advantageously be used for “output-only” identification (i.e. no force measurement is necessary) for the substructure. The forward analysis as required in the GA approach involves solving the above equations of motion subjected to the excitation and interface motion forces. Note that the fitness function is defined in terms of the measured components of the relative accelerations (u¨ ∗r ). A block diagram of the GA-implementation for the SSI method is presented in Fig. 3. For lumped mass systems, Mrj vanishes and Eq. (8) can be simplified to Mrru¨ ∗r (t) ⫹ Crru˙ r∗(t) ⫹ Krrur∗(t) ⫽ Pr(t)⫺Mrrru¨ j(t)

(9)

In addition, if a substructure includes the free end, such as substructure S1 in Fig. 2(a), the influence coefficients matrix reduces to simply r ⫽ [1 1 % 1]T

(10)

2.2. Substructural identification with overlap For SSI with overlap as depicted in Fig. 2(b), neighbouring substructures are allowed to have common, or overlap, members. The overlap has the following four implications.

Fig. 2.

(a) SSI without overlap; (b) SSI with overlap; (c) PSI.

1. A node may be interface to one substructure but internal to another. For example, for identification of substructure S1 in Fig. 2(b), acceleration measurement at interface node A is required as input. Nevertheless, for S2-identification, node A is an internal node and its measured response is treated as output. 2. At the overlap, interface acceleration required in substructure Si+1 can actually be computed from the previously identified substructure Si. In Fig. 2(b), the

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Fig. 3.

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Block diagram for GA-based SSI method.

upper interface node of S2 is node B at which no acceleration measurement is required. Instead, having identified S1, the acceleration at node A can be computed by solving the equations of motion for S1. Thus, less response measurements are required as compared to SSI without overlap. Nevertheless, error propagation may result since any identification error of S1 is carried forward to S2 in SSI with overlap, whereas the identification of one substructure is independent from others in SSI without overlap. 3. Once identified in a substructure Si, the overlap member can (optionally) be taken as known in the subsequent substructure Si+1 but this gives another source of error propagation. 4. While interface masses are excluded in SSI without overlap, these masses are included in one substructure or another in SSI with overlap. This facilitates the identification of all masses if unknown. Despite the above differences, the GA-implementation procedure is similar to that for the SSI without overlap.

The block diagram as shown in Fig. 3 still applies. The main deviation is that some interface accelerations are not obtained from measurement but from computation in the previous substructure as explained earlier.

3. Progressive structural identification In general, identification results are improved with increasing number of response measurements. In the above-mentioned SSI approach, only measurements within the substructure of concern and at interface ends are used when identifying the substructure parameters. Given a limited number of sensors, the identification results can be enhanced greatly if the measurement program allows shift of measurement sensors according to the substructure under investigation. If this is not possible, an alternative is to make use of as many response measurements as possible by progressively expanding the domain of substructural identification. To this end, a new method called the progressive

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structural identification (PSI) method is proposed as illustrated in Fig. 2(c). The PSI method can be seen as a variation of the SSI method, with the same idea of dividing the system to be identified into several sub-systems to improve the identification performance. The primary difference is that the substructure grows progressively while still keeping the number of unknowns small at each stage. For the structure in Fig. 2(c), the top substructure is selected as the first substructure (S1) for identification. After its unknown structural parameters are identified, an extended substructure is considered, i.e. the second substructure (S2) which includes the previous substructure (S1). The identified parameters of S1 are taken as known, and the measurements in substructure S1 are used again in the fitness evaluation while identifying the parameters of S2. This procedure continues until the whole or required part of the structure is identified. As compared to the SSI method, the main advantage of the PSI method is that it utilizes increasing availability of response measurements without increasing the number of unknowns as the substructure enlarges. The cost incurred is the increase in computational time for the forward analysis involving solving larger systems. This shortcoming may, nevertheless, be offset by faster convergence at each stage due to the availability of more response measurements.

4.1. Example 1—identification of 10-DOF known-mass system The 10-DOF lumped mass system similar to that presented in Fig. 1 is considered, with node numbers beginning from 1 at the lowest node to 10 at the highest node. The focus is here to identify the 10 stiffness parameters though damping constants are also treated as unknown. The exact parameters are: m 1 = 600 kg, m 2 = … = m 5 = 400 kg, m 6 = … = m 10 = 300 kg, k 1 = 700 kN / m, k 2 = k 3 = 650 kN / m, k 4 = k 5 = 600 kN / m, k 6 = k 7 = k 8 = 400 kN / m, and k 9 = k 10 = 300 kN / m. The GA software used is GENESIS [36] which uses binary encoding of chromosomes as opposed to realvalue encoding. For structural identification, the use of binary encoding can be turned into an advantage of controlling the resolution of identified parameters via the number of chromosome bits. In this study, the GA search is carried out in a range from a half to twice of the exact value with 6-bit resolution within the chosen range. The resolution of each parameter is therefore 1 / 26 × (2⫺ 0.5) = 2.3% of the corresponding exact value. The GA parameters used are crossover rate of 0.6 and mutation rate of 0.01 which are commonly adopted. The population size of 50 is found to be sufficient for this example. As mentioned earlier, only acceleration signals (in time domain) are used and the fitness value is, in the CSI method, defined as:

冘冘 M

4. Numerical results and discussion

To check the applicability and performance of the proposed methods, numerical simulation study is carried out on examples with known exact values. First, a 10-DOF structural system is used to compare the SSI methods with and without overlap, and the PSI method. For comparison purpose, the GA method is also applied to the complete structure in a direct manner and is referred here as the “complete structural identification” (CSI) method. Next, a larger system of 50 DOFs with up to 102 unknowns is investigated. Finally, an example of long truss bridge is presented. Damping effect is taken into account by adopting Rayleigh damping. The modal damping ratios are taken as 5%. The associated damping constants are assumed as unknown in the identification. Time domain is adopted and acceleration time history is used as the main response quantify in the identification procedure. To excite high modes for better identification, the input force is random excitation by means of Gaussian white noise and taken as known (measured). The numerically simulated response is obtained for 2 s at time step of 0.002 s by using Newmark’s constant-acceleration method [35].

fe ⫽

L

| u¨ m(i,n)⫺u¨ e(i,n)|2

i ⫽ 1n ⫽ 1

ML

(11)

where subscripts m and e denote measured and estimated quantities, respectively, L is the number of time steps and M is the number of measurement sensors used. For the SSI and PSI methods, the relative accelerations are used in the fitness evaluation and are derived from Eqs. (4) and (6) as follows: u¨ r∗(t) ⫽ u¨ r(t)⫺ru¨ j(t)

(12)

Three excitation forces act on the 3rd, 6th and 9th node. Acceleration response measurements (simulated) are assumed to be available at three nodes, viz. the 2nd, 5th and 10th nodes. Though it is possible to use fewer measurements in some methods than others, the same force and response measurements (in terms of total number, locations and signals used) are used for fair comparison of the four methods studied. 1. CSI: The complete structure is identified as a whole without substructuring. 2. SSI without overlap: The structure is divided into two substructures as shown in Fig. 4. The first substructure begins from node 5 to 10, but the mass at node 5 (where acceleration is used as input) is not included

C.G. Koh et al. / Engineering Structures 25 (2003) 1551–1563

Fig. 4.

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Two substructures used in SSI without overlap.

in the substructure formulation. This substructure is denoted as S1 = [5–10) for convenience. Square bracket indicates that acceleration measurement at interface node 5 is required (as input). Parenthesis indicates that acceleration measurement at node 10 is optional and, if available (as in this case), is treated as output. The second substructure is S2 = [0–5], for which the measurement at interface node 5 is treated as input. Node 0 represents the support (or “ground”) and if there is any support motion, its response measurement is also required as input. For this substructure, acceleration at node 2 is the output. Note that S1 is used for identification of k6 to k10, whereas S2 is for k1 to k5. 3. SSI with overlap: The structure is divided into two substructures but with overlap as shown in Fig. 5. The first substructure is S1=[5–10) with measurement at node 5 as input and measurement at node 10 as output, as in the case of SSI without overlap. But the second substructure is S2=[0–6) for which acceleration measurement at interface node 6 is not available. The required acceleration at node 6 is, instead, computed based on consideration of S1 that has been identified. Accelerations at nodes 2 and 5 are treated as output to S2. It is noted that the measured acceleration at node 5 is treated as input to S1 but output to S2. The overlap element is stiffness element k6. Its value is identified in S1 and taken as known in S2. As in SSI without overlap, S1 is used for identification of k6 to k10 and S2 for k1 to k5. But the identification results are not necessarily the same even with the same set of measurements used, owing to the different numerical procedures particularly with respect to S2. 4. PSI: The first substructure is S1 = [5–10) with one output at the 10th node. As before, S1 is used for

Fig. 5.

Two substructures used in SSI with overlap.

identification of k6 to k10. The identified values will then be treated as known in the subsequent substructure which is the whole structure, i.e. S2 = [0–10), with all available measurements used as output to identify k1 to k5. The total trial numbers chosen are the same. For CSI, the number of trials used is 10,000. For the SSI and PSI methods, the trial number used is 5000 in each of substructures S1 and S2. Due to the stochastic nature of the GA approach, the comparison is performed using several sets of identification results corresponding to different initial populations which are randomly generated within the specified search range. Table 1 presents the mean absolute errors of identified stiffness parameters corresponding to each data set of initial population for all the methods considered. The results consistently show that the divide-and-conquer idea indeed works. The SSI and PSI methods give better identification accuracy than the CSI method, and the improvement is clearly due to the much smaller search domains involved. When the identification is processed on the whole structure by the CSI method, there are 12 parameters to be identified. Since the resolution of each parameter is 1/26, the number of possible combinations is 272⬇4.7 × 1021. In contrast, the number of possible combinations at each substructure for SSI (and PSI) is at most 242⬇4.4 × 1012, an enormous reduction by about nine orders of magnitude.

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Table 1 Mean absolute error (%) of identified stiffness parameters for 10-DOF known-mass system Data set 1 2 3 4 5 Average

CSI 15.6 10.9 14.1 9.3 12.4 12.5

SSI without overlap

SSI with overlap

PSI

4.3 4.3 4.1 3.8 3.6 4.0

4.1 4.1 3.4 4.1 2.9 3.7

3.4 4.1 4.1 4.3 2.9 3.8

4.2. Example 2—identification of 50-DOF known-mass system The previous example serves to illustrate the implementation procedures and relative performance of the SSI and PSI methods. A larger system is now considered by increasing the number of DOFs to 50. The exact parameter values of the 50-DOF system are that all masses are 300 kg except for m1 which is 600 kg, and all stiffness elements are 700 kN/m. For every 10 levels, three random forces are applied at the respective 3rd, 6th and 9th nodes, and four acceleration measurements are taken at the respective 3rd, 5th, 8th and 10th nodes. The GA parameters used are the same as in the previous example. This large system is also used to study the effects of I/O noise by polluting the time signals of excitations and (numerically simulated) responses with noise. The noise level is defined as the ratio of standard deviation of the zero-mean white Gaussian noise to the root-mean-square value of the unpolluted time signals. Five percent I/O noise is considered in this study. It is noted that SSI without overlap generally requires more measurements than SSI with overlap. Thus, for SSI, only the version with overlap will be used here by dividing the structure into 10 substructures: S1 = [45–50), S2 = [40–46), S3 = [35–41), S4 = [30–36), … S10 = [0–6). For PSI, the identification is executed in 10 stages: S1 = [45–50), S2 = [40–50), S3 = [35,50), … S10 = [0,50). Expressed as the ratio of identified stiffness to exact value, the results obtained by the CSI, SSI with overlap and PSI methods are presented in Fig. 6. The plot range is set as the parameter search range (i.e. 0.5–2). For CSI, the mean absolute of identified stiffness parameters is 7.3%. SSI with overlap reduces the mean absolute is to 4.5%, and PSI further to 3.1%. The improvement in identification accuracy also comes with considerable saving in computational time mainly due to smaller systems involved in the forward analysis. On the same personal computer, the saving in computational time is 43% for SSI with overlap and 37% for PSI as compared to CSI.

Fig. 6. Ratio of identified stiffness to exact value for 50-DOF knownmass system.

4.3. Example 3—identification of 10-DOF unknownmass system While many structural identification studies deal with identification of stiffness and/or damping parameters, a greater challenge is to identify unknown mass para-

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meters as well. In this regard, many time-domain identification methods require state-space formulation and are not applicable since the mass matrix has to be known. Frequency-domain methods that depend solely on matching of eigenvalue solutions (i.e. natural frequencies and mode shapes) are not applicable either, because mass and stiffness parameters can change by the same ratio while still giving the same eigen-solutions. In contrast, the proposed identification methods are not based on state-space formulation nor eigenvalue solution matching. They are thus, in principle, able to identify stiffness, damping as well as mass parameters though the numerical difficulty increases greatly as compared to identification of known-mass systems. The previously considered 10-DOF system is studied again but assuming that all mass parameters are also unknown, leading to a total of 22 unknowns (10 mass, 10 stiffness and 2 damping parameters). As the number of unknown parameters now increases to 22 from 12, it is found necessary to increase the number of measurements to achieve the desired identification accuracy. Four measurements (instead of three) are available, viz. at nodes 2, 5, 8 and 10. The same GA parameters are used as before. Only the SSI method with overlap and the PSI methods are studied in this example, and the better method will later be applied to identify the larger system, i.e. the 50-DOF system with a total of 102 unknown parameters. For the SSI method with overlap, two substructures are used, i.e. S1 = [5–10) and S2 = [0–6). Five thousand trials are used for each substructure. For the PSI method, the structure is identified in four stages: S1 = [8–10), S2 = [5–10), S3 = [2–10) and S4 = [0–10). The trial number is 2000 each for S1 and S4, while 3000 is used each for S2 and S3 (which have more unknowns than S1 and S4). In view of greater difficulty to identify unknown mass, stiffness and damping parameters simultaneously and the stochastic nature of the GA method, identification is performed using three different sets of initial populations (randomly generated). For each substructure, the fitness values for the three identification sets are compared and the identification results corresponding to the best fitness value are adopted. As presented in Table 2, the PSI method gives better results of identified stiffness parameters in general with mean absolute of about 7% as compared to that of about 27% for SSI with overlap. Similarly, there is significant improvement in identified mass parameters with mean absolute of about 8% for PSI as compared to about 32% for SSI with overlap. The improvement in identification accuracy is due to, by construction, the increasing availability of response measurements as the identification progresses in stages while keeping the number of unknowns within small and manageable. The utilization of as many response measurements as possible also yields faster convergence

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than SSI. The faster convergence offsets, to some extent, the increase in computational time required for forward analysis due to increasing system size in PSI. For this example, the computational time required for PSI is only 11% more than that for SSI on the same computer. 4.4. Example 4—identification of 50-DOF unknownmass system Based on the above comparison study, the PSI method is selected for the challenging identification problem, i.e. to identify 102 unknowns (50 mass, 50 stiffness and 2 damping parameters) of the 50-DOF system as considered in Example 2. Since the number of unknowns approximately double as compared to the known-mass system, it is necessary to have smaller substructures. Thus, the structure is identified in 25 stages starting from S1 = [48–50) and increasing by 2 nodes in each subsequent stage, i.e. S2 = [46–50), S3 = [44,50), … S25 = [0,50). The number of trials is 2000 at each stage, and the search range is from 0.5 to 1.5 of the exact value. Other GA parameters remain the same as in Example 2. For comparison purpose, the CSI method is also used to identify the same system and the number of trials used is 50,000. Identification is performed using the five sets of initial random population and the best-fitness results are adopted. In terms of identified-to-exact ratios, the results for stiffness parameters are presented in Fig. 7. The PSI method gives generally much better results than the CSI method. The mean absolute error is 5.1% for the PSI method, significantly less than 21.0% for the CSI method. A similar trend is observed for identified mass parameters, as shown in Fig. 8, with the mean absolute error reduced to 5.0% for PSI from 17.8% for CSI. Not only are the results significantly improved by the PSI method, but also there is substantial saving in the computational time by about 40%. To our knowledge, structural identification of such a large system with more than 100 unknowns has not been reported in the literature. 4.5. Example 5—substructural identification of longspan truss structure In the final example, a long-span truss structure is considered to illustrate the applicability of substructural identification for realistic and statically indeterminate structures. The structure comprises 57 truss members, as shown in Fig. 9(a). The exact parameters for all members are Young’s modulus, E = 200 GPa (steel) and crosssectional area, A = 0.0015 m2. A substructure as shown in Fig. 9(b) comprising 11 members is to be identified. For each of these members, the axial rigidity EA is treated as unknown. In addition, the two Rayleigh damping constants for the substructure are assumed to be unknown. Assuming that only this substructure is to be

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Table 2 Identification error (%) of stiffness and mass parameters for 10-DOF unknown-mass system Stiffness k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 Mean absolute error

SSI with overlap +83.3 ⫺19.0 ⫺7.1 +26.2 ⫺12.3 +4.8 +38.1 ⫺4.8 +23.8 +47.6 26.7

PSI ⫺7.1 +14.3 ⫺3.2 0.0 +11.9 ⫺2.4 ⫺14.3 +2.4 +2.4 ⫺11.9 6.99

Fig. 7. Ratio of identified stiffness to exact value for 50-DOF unknown-mass system.

identified out of the large structure considered, the method of SSI without overlap is appropriate. The response measurements comprise all interface accelerations as well as three internal accelerations, i.e. a1, a2 and a3 as shown in Fig. 9(b). Though the full structure is subjected to eight random forces (F1 to F8), only the forces within the substructure, i.e. F3 and F4, are required for the identification. The I/O signals are numerically contaminated with 10% noise. The identified results of EA for all 11 members are presented in Table 3, and the mean absolute error is only about 9% which is good in view of the 10% noise level imposed. Note that the identification is successfully achieved with-

Mass m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 Mean absolute error

SSI with overlap +45.2 ⫺35.8 +50.0 +4.8 ⫺33.3 +73.7 ⫺2.3 ⫺14.3 +47.7 +9.7 31.7

PSI +2.3 +2.5 0.0 ⫺12.0 +16.8 ⫺16.7 ⫺9.7 +12.0 ⫺9.7 0.0 8.17

Fig. 8. Ratio of identified mass to exact value for 50-DOF unknownmass system.

out involving the other parts of the structure and does not depend on the size or complexity of the entire structure.

5. Conclusions In this paper, two divide-and-conquer methods, namely the SSI and PSI methods, are formulated for handling identification of large systems with many unknowns. The identification endeavour is accomplished by dividing a large search domain into smaller and more manageable search domains. The GA approach is adopted as the identification tool for several advantages mentioned previously, notably including the direct use

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Fig. 9.

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Long-span truss structure: (a) full structure; (b) substructure.

Table 3 Substructural identification of truss structure Member number

Exact EA (MN)

Estimated EA (MN)

Error (%)

1 2 3 4 5 6 7 8 9 10 11

300 300 300 300 300 300 300 300 300 300 300

338.27 289.00 326.54 285.19 243.26 315.98 319.50 305.13 251.76 316.57 256.15 Mean absolute error

+12.76 ⫺3.67 +8.85 ⫺4.94 ⫺18.91 +5.33 +6.50 +1.71 ⫺16.08 +5.52 ⫺14.61 8.99

of measured accelerations (without the need to integrate to velocities or displacements). It should be pointed out that the results presented in the paper have, to some extent, accounted for modelling error associated with damping. This is because the damping term in the interface motion force is neglected as explained earlier but the identified results are generally good. Furthermore, the damping constants of the Rayleigh model adopted for the full structure are not necessarily the same as those at the substructure level. Based on this study, the main features of each of the methods considered are summarized below. 1. CSI: By applying GA-based identification to the whole structure, this method is relatively easy to implement but the identification results are often not acceptable for large systems. 2. SSI without overlap: The identification of each substructure is completely independent of other substructures. The identification sequence of substructures is immaterial, and it is thus also possible to identify only part of the structure. If the excitation force is outside the substructure, no force measurement is required leading to an output-only identification approach.

Nevertheless, interface masses are not included in any substructure making it inconvenient for mass identification though not impossible, e.g. by re-defining substructures to include the interface masses yet to be determined. 3. SSI with overlap: Some measurements are used both as interface measurements (input) to one substructure and as internal measurements (output) to another. Thus the number of measurements required can be reduced as compared to SSI without overlap. Nevertheless, error propagation may arise from inaccuracy due to (a) some interface measurements are computed from previously identified substructures and (b) structural parameters of overlap members if assumed as known in the subsequent substructure. The identification of each substructure is, to some extent, dependent on the identification sequence and the results of previous substructures. In principle, identification of all masses is possible since they are included in one substructure or another. But the numerical results for the system with 22 unknowns are not satisfactory, illustrating the need to include more measurements. This can be achieved if measurement sensors can be shifted to concentrate on the substructure under inves-

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tigation. Alternatively, suitable local search operators should be incorporated in the GA method to further improve the identification accuracy [37]. 4. PSI: The main idea is to include as many available measurements as possible while keeping the number of unknowns small at each stage of identification. As a result, the PSI method is found to be the most successful for identification of unknown-mass systems. In the presence of 5% I/O noise, the mean identification error is about 5% in Example 4 which has 102 unknowns including all unknown mass parameters. The computational cost for forward analysis increases as the substructure grows in size. Nevertheless, this is offset by faster convergence and the total computational time is thus not necessarily much longer as compared to SSI.

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Acknowledgements

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The authors gratefully acknowledge the research funds and scholarship provided by DSO National Laboratories (Singapore) and the National University of Singapore. Special thanks go to Dr. Y.W. Chan, Ms L.P. Pey and Mr T.Y. Soh of DSO National Laboratories for their technical assistance.

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