MODEL-BASED HEALTH MONITORING OF STRUCTURAL SYSTEMS

MODEL-BASED HEALTH MONITORING OF STRUCTURAL SYSTEMS: PROGRESS, POTENTIAL AND CHALLENGES K. C. Park and Gregory W. Reich

ABSTRACT The present paper reviews two complementary methods for model-based structural damage detection with applications. The theoretical basis of these two methods is a partitioned formulation of the equations of motion for structures and its application to localized structural system identification. The first method is based on the changes in the localized flexibility while the second utilizes invariance properties of the elemental or substructural transmission zeros. The paper offers challenges in the development of practical model-based online structural health monitoring systems. INTRODUCTION This paper is focused on model-based structural damage detection methods, which make use of both a reference and a damaged model of the structure. The model-based damage detection methods can be categorized into methods which utilize a global model versus a localized one. The global model approach utilizes a global reference model, and the damage is determined at a global location on the structure. In the localized approach the model is decomposed into substructures, and the damage location is determined in terms of changes in the substructural behavior. There have been two complementary developments of the localized approach: substructural flexibility [1,2] and substructural transmission zeros [3]. The paper begins with a variational partitioning of the global equations of motion for structures into substructural equations for partitioned structures. This transforms the system dynamics into a localized form. The identification of localized flexibilities from measured data will then be reviewed. The identification of the transmission zeros of partitions of the localized system Center for Aerospace Structures, Department of Aerospace Engineering Sciences, Campus Box 429, Boulder, CO, 80309.

transfer functions will be outlined, whose particular transmission zeros are shown to remain invariant to flexibility changes in that substructure. The use of this transmission zero invariance property will then be used to detect damages in the partitioned substructures. Implementation issues of the foregoing damage detection methods for online health monitoring systems are then sketched out. Finally, challenges to translate the methodology into practical systems are discussed.

GLOBAL FLEXIBILITY-BASED DAMAGE DETECTION The discrete linear equations of motion for a structure with typical sensors and actuators for vibration testing can be expressed as ¨ g + Dg u˙ g + Kg ug = Bg fg Mg u yg = C0 u + C1 u˙ + C2 u ¨

(1)

where ug is the displacement vector of the assembled structure; Mg , Dg and Kg are the assembled mass, damping and stiffness matrices; fg is the applied force; Bg is the excitation location operator; C(0,1,2) are the sensor placement Boolean matrices; yg is the measurement output vector, and the subscript g refers to the global structural system, as opposed to the partitioned substructures which will be introduced later. In the frequency domain, with u˙ g (0) = ug (0) = 0 and Cg = C0 , that is only with displacement outputs, the output yg (t) in (1) can be obtained in ˆ g (ω) as terms of its frequency-variable magnitude y y ˆ g (ω) = Hg (ω)ˆfg (ω), ¯g Hg (ω) = Cg K

−1

(ω)Bg ,

·

¸ · ¸ ˆ g (ω) y yg (t) jωt =e ˆfg (ω) fg (t) ¯ g (ω) = (Kg + jωDg − ω 2 Mg ) K

(2)

The first step in a model-based damage detection technique is to obtain from the measured experimental data the following mass-normalized ‘global’ flexibility matrix Fg : Fg = ΦΛ−1 ΦT ,

ΦT Mg Φ = I,

ΦT Kg Φ = Λ

(3)

In practice, the experimentally identified global flexibility matrix is given by T Fg = Φm Λ−1 m Φm

(4)

where the subscript m denotes the set of measured modes and mode shapes which are substantially smaller than the analytical model size. This has hampered the analyst’s ability to obtain uniquely the changes in the elemental flexibility properties, which in turn can lead to incorrect and/or ambiguous detection of damage level and locations.

DYNAMIC FLEXIBILITY PARTITIONING In order for a model-based damage detection technique to be effective, one should employ structural identification methods that can directly offer or naturally lead to localized substructural characterization using the measured data. This section presents a variational formulation that facilitates localized identification. To this end, partitioning of a global structure into elements or substructures is first carried out. Second, since we are primarily interested in detecting damages, a preferred variable is strains instead of the partitioned displacements. Hence, the relation between the global equations of motion and the partitioned equations of motion in a strain basis is established. In other words, one relates the global frequency response function Hg (ω) to that of strain-based partitioned structures. Employing the partitioned strain-basis equations of motion, two complementary damage detection techniques will be presented: one that is based on changes in the identified substructural flexibility properties [1,2], and the other based the invariance properties of the localized transmission zeros [3]. Their applications will then follow. Partitioned Equations of Motion The global form of the energy functional of a structural dynamic system can be expressed as: 1 ¨ g − fg ) Π(ug ) = ug T ( Kg ug + Dg u˙ g + Mg u 2

Figure 1: Structural Partitioning Process

(5)

Consider the structure shown in Figure 1. From left to right, the figure represents the disassembly or partitioning process of the global structure into its substructural parts. On the left, a generic continuum structure is pictured, complete with natural boundary conditions and an underlying pattern of subdomains. On the right, the structure is pictured after the partitioning and discretization process is complete. Each subdomain is represented by internal and boundary nodes (local nodes), as well as the interface forces or multiplier field which relate each local node to the corresponding global node. Note that, after partitioning, the global nodes which lie on the partition boundaries are co-owned by two or more substructures. Mathematically, partitioning is the disassembly process given by u = Lug ,

L T f = fg

(6)

where L is the Boolean disassembly matrix that relates the global and substructural displacements, and f is the elemental force. In finite element discretization, the global stiffness matrix Kg is formed by the assembly of individual substructural stiffnesses via the assembly operator LT :

T

Kg = L KL,



  K=  



K1 K2 s

K

..

. Kns

    

(7)

where K is the block-diagonal collection of unassembled substructural stiffness matrices Ks . In order to maintain the kinematic compatibility of the global and local displacements, the substructural displacements must satisfy the following: BT λ (u − Lug ) = 0

(8)

where Bλ is a constraint matrix which can be chosen in one of several ways. It can be a Boolean matrix which extracts the boundary nodes of partitioned substructures, or it can be chosen as a nullspace of the disassembly operator L: BT (9) λL = 0 The energy functional of equation (5) can be re-written in terms of the substructural displacement and constraints as 1 T Π(u, λb , ug ) = uT ( Ku + D u˙ + M¨ u − f) + λT b Bλ (u − Lug ) 2

(10)

where M and D are the substructural mass and damping matrices, respectively, and the Lagrange multipliers λb are introduced to enforce the kinematic compatibility constraint represented by equation (8).

Derivation of Strain-to-Displacement Relation Suppose that one has managed to obtain the substructural stiffness matrices, K for both the healthy and damaged structures by employing an inverse algorithm from experimental data. A convenient way to assess damage is to compare their corresponding diagonal terms. While their changes indeed indicate damage [1], it often masks the failure modes and the extent of damage. It has been found that by transforming the substructural flexibility matrices into strain-basis ones, both damage levels and damage modes are manifested. This can be accomplished through a strain-to-displacement relation. To this end, first note that the displacement in a discrete element u can be decomposed [4] into a deformation d and a rigid-body motion r: u = d + r = d + Φα α

(11)

where Φα represents the rigid-body modes of the element and is only dependent on the element geometry and type, and α are the associated generalized coordinates. The conventional displacement-to-strain relationship for discrete elements can be written as ² = Su (12) where ² is the strain vector, and S is a discrete operator that can be determined in a variety of ways, e.g. from the finite element shape functions of the corresponding discrete elements. This equation can be combined with equation (11), which results in the expression of strain in terms of deformation ² = S(d + Φα α) = Sd

(13)

since the rigid-body modes Φα do not incur any strain. The inverse of this equation results in an expression of the elemental deformation in terms of the elemental strain. d = Φ² ², Φ² = S+ (14) Combining this equation with (11) results in the desired relation, that is, expressing the elemental displacements in terms of the elemental strain and rigid-body motion. u = Φ² ² + Φα α (15) Examples of this formulation for simple elements can be found in [5]. Strain-Based FRFs The strain-to-displacement relationship of equation (15) can now be inserted into the partitioned energy functional (10). This results in the energy functional expressed in terms of the elemental strain, rigid-body motion, and the interface forces. n1 ˙ Π(², α, λb ) = (Φ² ² + Φα α)T K(Φ² ² + Φα α) + D(Φ² ²˙ + Φα α) 2 o (16) T ¨ − f + λT + M(Φ² ¨² + Φα α) B (Φ ² + Φ α) ² α b λ

where Bλ has been chosen as in equation (9) so that the dependence on the global displacement ug is removed. The stationary value of the resulting functional yields the partitioned, strain-based equations of motion: "

2 (Mφ d 2 + Dφ d + Kφ ) dt dt T Bλ Φφ

ΦT φ Bλ 0

#½

q λb

¾

=

½

ΦT φf 0

¾

(17)

where Φφ = [ Φ² Φα ] · ¸ M² M²α T Mφ = Φφ MΦφ = MT Mα ²α ½ ¾ ² q= α

¸ K² 0 Kφ = = 0 0 · ¸ D² D²α T Dφ = Φφ DΦφ = T D²α Dα ΦT φ KΦφ

·

(18)

fφ = Φ T φf

Since the measured output vector consists of the strain ² and the rigidbody amplitude α, we eliminate λb from the above equation (17). Mφ q ¨ + Pφ Dφ q˙ + Pφ Kφ q = Pφ fφ T −1 Pφ = I − Φb M−1 b Φb Mφ −1 Mb = ΦT b Mφ Φb ,

(19)

Φb = ΦT φ Bλ

This equation is designated as the strain-based equation of motion for linear structures. The analytical strain-based frequency response functions Hφ (ω) can be determined from the preceding equation by transforming to the freˆ: quency domain and solving for q q ˆ = Hφ (ω)ˆfφ © ª Hφ (ω) = −ω2 Mφ + jωPφ Dφ + Pφ Kφ −1 Pφ

(20)

Observe that Hg (ω), equation (2), relates the global inputs to the global outputs, whereas Hφ (ω), equation (20), describes the relationship between quantities unique to each substructure. The difference between these two FRFs is illustrated in Figure 2. The traditional system identification method, shown on the left, measures inputs and outputs from the global system and determines the FRF relating the two. A localized system identification procedure, shown on the right, measures inputs and outputs from the partitioned system. The FRF and other identified characteristics correspond to this partitioned system. As is described in the figure, the interface forces are automatically included when the local outputs are determined.

Localized FRFs Global FRFs Hg (ω) =

ˆ ∗ (ω)·ˆ y fg (ω) g ˆ∗ ˆ fg (ω)·fg (ω)

H` (ω) =

ˆ ∗ (ω)·ˆ y f` (ω) ` ˆ∗ f` (ω)·ˆf` (ω)

Note: y` becomes a function of the interaction force λ` as shown in eq. (19)

Figure 2: Comparison Between Global and Local FRFs

Relation between the Global and Strain-Based FRFs Recalling equations (15) and (6), the relationship between the substructural variables and the global displacements can be determined in the following way: ½ ¾ ² q= = Φ−1 (21) φ Lug α Additionally, invariance of the external work term of the energy functional gives T T uT ⇒ fg = LT Φ−T (22) g fg = u f = q fφ φ fφ This equation can then be used in conjunction with the global FRF expression of equation (2). T −T Hφ (ω) = Φ−1 φ LHg (ω)L Φφ · ¸ (23) H² (ω) H²α (ω) = T H²α (ω) Hα (ω) The transfer functions in equations (20) and (23) are equal because of inputoutput system invariance. Equation (20) highlights the dependence of the

localized FRF on the local dynamics of the partitioned form, while equation (23) highlights the dependence on the global dynamics. Finally, we note that from equation (20) the dynamics of the system in strain-based form are partitioned into the internal strain energy corresponding to the strain DOF and inertial energy corresponding to the rigid-body motion of the system. In particular, if we are to assume that damage occurs only as a change in the stiffness of the structure, then the only terms of interest are those containing K² . Therefore, we could also determine the partition of the substructural FRFs corresponding to strain energy as H² (ω) = SLHg (ω)LT ST

(24)

It is this form of the strain-based FRFs which will be utilized to determine the location of damage. Further details may be found in [3]. DAMAGE DETECTION BASED ON LOCALIZED FLEXIBILITY The quasi-static limit (ω → 0) of the previous strain-based FRF H² is termed the strain-based flexibility F² . It can be determined directly from the global measured flexibility Fm as F² = SLFm LT ST

(25)

This localized flexibility matrix relates the quasi-static strain output at one location to a conjugate strain-based input force at another location. Observe that it retains the coupling inherent between substructures, and as such is typically fully populated. However, comparison of the flexibility changes along the diagonal of this matrix is sufficient to correctly and uniquely determine the location of damage. An example of this will be presented in a later section. DAMAGE DETECTION BASED ON LOCALIZED TRANSMISSION ZEROS It should be noted that in a typical vibration test setting it is not practical to excite all the modes, nor place sensors sufficient to cover every discrete degree of freedom (DOF) desired. Nevertheless, for the clarity of our discussion, we will assume that the entire displacement vector is measured, and Bg is an identity matrix whose size is the same as ug . Note that from equation (2), a ¯g: matrix element Hg (i, j) can be expressed [6] in terms of the cofactor of K Hg (i, j) =

¯ g (i, j) cof K ¯g , det K

¯ g (i, j) = (−1)i+j det K ¯ j,i cof K g

(26)

¯ j,i is obtained by deleting row j and column i from K ¯ g . Thus, The submatrix K g the poles of Hg and the zeros of an element Hg (i, j) are determined according to: ¯g = 0 • poles of Hg ⇒ the roots of det K (27) ¯ g (i, j) = 0 • zeros of Hg (i, j) ⇒ the roots of cof K

Observe that changes in the stiffness and/or mass properties at node i do not affect the zeros of Hg (i, i). This node-by-node invariance property of the global system zeros has previously been exploited [7] in an attempt to detect damage. The node-by-node zero invariance property, however, does not accurately predict damage for general structures, except for elements at a fixed boundary. First, structural failures occur in arbitrary locations and very rarely coincide with an isolated node. Second, for continuum and truss-like structures, a nodal stiffness is made up of contributions from the elements that are connected to that node. The above observations have motivated the development of a damage detection method based on an element-by-element transmission zero invariance property. Specifically, consider the elemental strain-based dynamic flexibility matrix He² as the partition (l, m) of H² , where l and m correspond to the list of localized strain inputs and outputs on element e. A key property of H²e is that the transmission zeros of this transfer function are not affected by changes in the corresponding elemental stiffness parameters. It is this invariance property that is exploited to detect damages. In this damage detection scheme, the transmission zeros of a particular localized transfer function corresponding to one substructure are determined for the reference and damage cases. The variation between the two sets of transmission zeros are determined numerically, and the relative variation for one substructure is compared to that of other substructures. The substructure with the lowest overall variation over the set of transmission zeros is deemed to be the damage location. The numerical indication of the transmission zero invariance can be found in the following manner. As the transmission zeros in a particular set vary over a range of frequencies, the quantitative measure used to determine variance is defined as a cumulative error over the range of zeros. This factor is called the cumulative transmission zero deviation, represented by DT Z . For the j th zero on element i, DT Z is given as: ¯! ï ¯ z d (1 : j, i) − z n (1 : j, i) ¯ ¯ ¯ DT Z (j, i) = mean ¯ ¯ ¯ ¯ z d (1 : j, i)

(28)

The transmission zero set from the element which has the lowest value of DT Z over the range of calculated zeros is defined as the element where damage occurs. This element has the least variance between nominal and damaged TZ sets.

DAMAGE IDENTIFICATION OF ENGINE MOUNTING LADDER As an example of how these two damage detection schemes can be used in a practical sense, a simulated vibration test of a ladder-like structure was conducted. As shown in Figure 3, the model contains 16 plane beam elements.

Figure 3: 16 Element Beam Ladder with Strain Measurement Locations

Each element has three degrees of freedom per node, or 36 global DOF. The substructural model has 48 DOF, corresponding to, for each element, two bending strains, measured at the Gauss points, and one longitudinal strain, measured at the element mid-point, as is shown in the figure. Both nominal and damaged cases were simulated, with the damage modeled as a reduction in stiffness in element 10. The nominal and damaged system responses were computed due to burstrandom inputs at 6 locations for approximately 25 seconds at 1600 Hz. The input and output data was then re-sampled and filtered at 400 Hz, which should capture the 12 flexible modes below 200 Hz. These filtered inputs and outputs were then used in a system identification procedure to determine the modal frequencies and mode shapes. These results are then used for the two damage algorithms. Figure 4 shows the results by using the substructural flexibility method. The top half of the figure shows the change in flexibility along the diagonal of the measured global flexibility matrix Fm . Note that the largest flexibility change is seen at node D, which corresponds to one end of the damaged element. However, it also corresponds to elements 4 and 5, which are undamaged. This is the problem inherent in using global flexibility changes. The flexibility values at each node contain contributions from several elements, and therefore flexibility changes at a node location are ambiguous as to the actual location of damage. On the bottom half of the figure is the flexibility change for the strainbased substructural flexibility F² . Here, the only significant flexibility change occurs for strain degrees of freedom at element 10, which correctly identifies the damaged element. The next figure, Figure 5, shows that the substructural flexibility method is able to correctly and uniquely determine the damage

RDI Using 12 Identified Modes

3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20 Global DOF

20

25 Strain DOF

25

30

35

14 12 10 8 6 4 2 0

0

5

10

15

30

35

40

45

Figure 4: Damage Detection Based on Flexibility Changes

RDI Using 9 Identified Modes

14 12 10 8 6 4 2 0

0

5

10

15

20

25 Strain DOF

30

35

40

45

35

40

45

RDI Using 6 Identified Modes

8

6

4

2

0

0

5

10

15

20

25 Strain DOF

30

Figure 5: Damage Detection Based on Flexibility Changes with Fewer Modes

100

DTZ

Element 1 Element 2 Element 3 Element 4 Element 5 Element 6 Element 7 Element 8 Element 9 Element 10 Element 11 Element 12 Element 13 Element 14 Element 15 Element 16

10-1

Damaged Element -2

10

1

2

3

4 5 6 7 Cumulative Number of TZ

8

9

10

Figure 6: Strain-Based Cumulative Transmission Zero Deviation

location with as few as 6 measured modes. This is an attractive attribute for a damage location identification methodology. Often in experimental modal analysis it is impossible to determine more than the first several modes to any degree of accuracy. In order to determine the transmission zeros of the localized transfer functions, a representative state-space model corresponding to equation (24) is determined based on the measured modes of the system. The transmission zeros for each substructural transfer function are then computed based on a numerical routine [8]. The cumulative transmission zero variance DT Z (equation (28)) for each of the elements are computed, and subsequently plotted in Figure 6. This figure shows that element 10 has the lowest cumulative variation over the span of calculated transmission zeros. Note that a transmission zero of a transfer function is defined as a frequency at which the contribution from all of the system modes sum to zero. Therefore, if only a truncated measurement set of modes are included in the transfer function measurement, the transmission zero value will be different than its analytical value. Therefore, the transmission zeros corresponding to a damaged element will not be exactly invariant as is predicted by the theory. Only in the ideal case where all of the system dynamics are measured will the transmission zeros be exactly invariant. This effect can be seen in the figure. If all of the system dynamics to infinite frequency had been measured, then

the difference between the transmission zero variation in the damaged element and other elements would have been several orders of magnitude. In this case the difference is not even one order of magnitude. However, the trend clearly indicated in the figure is that the damaged element, element 10, has the lowest TZ variation over the range of zeros in the set. CHALLENGES IN MODEL-BASED ONLINE HEALTH MONITORING The present paper has reviewed two methods for model-based structural damage detection: changes in substructural flexibility and the invariance properties of substructural transmission zeros. In order for the model-based damage detection methods to be adopted eventually for online health monitoring, the following should be addressed. 1. Fast algorithms for system identification are needed, if possible on a realtime basis. It is especially critical if damage control strategies must be invoked in time before damage propagates to cause catastrophic failures. 2. The present paper utilized the global flexibility that is obtained from system identification as a starting point for subsequent analysis. A method for localized structural identification is highly desirable so that not all of the sensor output are processed for online monitoring purposes. 3. Although we have not discussed any non-model based damage detection methods, a robust online health monitoring system would require a hybridization of both non-model and model-based methods. Studies are needed to develop sensible hybrid damage detection methods that are easy to implement and robust. 4. Finally, a major reason for developing health monitoring technology is to lengthen the service periods of existing systems. Unfortunately, nearly all of the existing systems are not instrumented to get the response data. Economical sensor placement and data collection methodologies, both onsite and remote, are needed in order for online health monitoring technologies to have practical benefits for the nation’s existing infrastructure and transportation systems. Acknowledgments It is a pleasure to acknowledge support by Sandia National Laboratories (Contract AP-1461) and a donation from Shimizu Corporation. The second author is supported through the Air Force Palace Knight Program. REFERENCES 1.

Park, K. C., G. W. Reich, and K. F. Alvin. 1997. “Damage Detection Using Localized Flexibilities,” in : Structural Health Monitoring, Current Status and Perspectives, ed. F-K Chang, Technomic Pub., pp. 125-139; to appear in Journal of Intelligent Material Systems and Structures.

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