P1: FCH PB091E-41 January 10, 2002 21:18 532 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS DTF RTF DTF RTF 10 −1 10 0 10 1 10 2 10 −6 10 −4 −4 −2 0 2 4 10 0 10 −2 10 2 Frequency (Hz) 10 −1 10 0 10 1 10 2 Frequency (Hz) Magnitude Phase (rad) (a) DTF RTF DTF RTF 10 −1 10 0 10 1 10 2 10 −6 10 −4 −4 −2 0 2 4 10 0 10 −2 10 2 Frequency (Hz) 10 −1 10 0 10 1 10 2 Frequency (Hz) Magnitude Phase (rad) (b) DTF RTF DTF RTF 10 −1 10 0 10 1 10 2 10 −6 10 −4 −4 −2 0 2 4 10 0 10 −2 10 2 Frequency (Hz) 10 −1 10 0 10 1 10 2 Frequency (Hz) Magnitude Phase (rad) (c) DTF RTF DTF RTF 10 −1 10 0 10 1 10 2 10 −6 10 −4 −3.5 −3 −2.5 −2 −1.5 −1 0 −0.5 10 0 10 −2 10 2 Frequency (Hz) 10 −1 10 0 10 1 10 2 Frequency (Hz) MagnitudePhase (rad) (d) DTF RTF DTF RTF 10 −1 10 0 10 1 10 2 10 −6 10 −4 −4 −2 0 2 4 10 0 10 −2 10 2 Frequency (Hz) 10 −1 10 0 10 1 10 2 Frequency (Hz) Magnitude Phase (rad) (e) DTF RTF DTF RTF 10 −1 10 0 10 1 10 2 10 −10 −4 −2 0 2 4 10 0 10 −5 10 5 Frequency (Hz) 10 −1 10 0 10 1 10 2 Frequency (Hz) Magnitude Phase (rad) (f) Figure 16. DTF and RTF response of free-free beam structure: (a) v 0 /F 3 ; (b) v 1 /F 3 ; (c) v 2 /F 3 ; (d) v 3 /F 3 ; (e) v 4 /F 3 ; (f) v 5 /F 3 . P1: FCH PB091E-41 January 10, 2002 21:18 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS 533 Table 2. Properties of the Beam Structure Element Number 1 2 3 4 5 EI 4 × (1 + 0.0001i)1× (1 + 0.0001i)10×(1 +0.0001i)2×(1 +0.0001i)3×(1 +0.0001i) ρ A 21531 Length 2 3 3 1 1 Consider the fixed-free beam shown in Fig. 17. The left end of the beam is fixed, and the right end is free. Energy is input into this beam by F 3 = 0. The conventional rever- berated transfer function matrix of this five-element beam can be obtained experimentally. The inverse of the RTF is the dynamic stiffness matrix of the beam. RTF −1 =           K 1 4 + K 2 1 K 2 2 0 K 2 3 K 2 4 + K 3 1 K 3 2 K 3 3 K 3 4 + K 4 1 K 4 2 0 K 4 3 K 4 4 + K 5 1 K 5 2 K 5 3 K 5 4           . (40) As previously shown for the free-free beam, the vertical displacement and angular displacement of each node are fed back into local controllers to generate control forces that caneliminate wave reflection at each node. Controllers can be attached at all nodes except for the node at the fixed end of the fixed-free beam. The DTF responses with respect to input F 3 can be obtained from the following expression:                                         v 1 F 3 θ 1 F 3 v 2 F 3 θ 2 F 3 v 3 F 3 θ 3 F 3 v 4 F 3 θ 4 F 3 v 5 F 3 θ 5 F 3                                         DTF =       RTF −1 +       −K 1 4 + G 2l G 3l − G 2l 0 0 2×2 0 G 4r − G 5r G 5r             −1                                  0 0 0 0 1 0 0 0 0 0                                  , (41) where K 1 4 can be obtained because the physical properties of the first element are known. The RTF and DTF res- ponses are shown in Fig. 18 Comparing the expressions in Eqs. (39) and (41), the dif- ference between the free-free beam and the fixed-free beam 01 2 3 F 3 45 EI 1 ρA 1 EI 2 ρA 2 EI 3 ρA 3 EI 4 ρA 4 EI 5 ρA 5 Figure 17. Fixed free beam that has five elements. dereverberated responses depends on the way the bound- ary conditions are treated. A controller is needed for the free-free beam to extend the free end to infinity. However, for the fixed-free beam, element 1 is ignored, and element 2 needs to be extended to the left to infinity. To achieve this, K 1 4 is the affecting part from element 1, and G 2l is from element 2. DAMAGE DETECTION APPROACH BASED ON DTF RESPONSE Detecting the Presence and Location of Damage In the previous section, collocated noncausal controllers were developed to obtain the DTF responses for discrete spring-mass structural elements and spectral rod and beam finite elements. In this section, a methodology for detecting damage in one-dimensional structures is devel- oped based on the characteristic changes in the undamaged DTF responses. For an undamaged structure, collocated P1: FCH PB091E-41 January 10, 2002 21:18 534 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS 10 −1 10 0 10 1 10 2 10 −1 10 0 10 1 10 2 10 −10 10 −5 10 0 −4 −2 2 4 0 10 5 Frequency (Hz) Frequency (Hz) Magnitude Phase angle (a) RTF DTF RTF DTF 10 −1 10 0 10 1 10 2 10 −1 10 0 10 1 10 2 10 −10 10 −5 10 0 −4 −2 2 4 0 10 5 Frequency (Hz) Frequency (Hz) Magnitude Phase angle (b) RTF DTF RTF DTF 10 −1 10 0 10 1 10 2 10 −1 10 0 10 1 10 2 10 −6 10 −4 10 −2 10 0 −2 −3 −2.5 −3.5 −1.5 -0.5 0 -1 10 2 Frequency (Hz) Frequency (Hz) MagnitudePhase angle (c) RTF DTF RTF DTF 10 −1 10 0 10 1 10 2 10 −1 10 0 10 1 10 2 10 −10 10 −5 10 0 −4 −2 2 4 0 10 5 Frequency (Hz) Frequency (Hz) Magnitude Phase angle (d) RTF DTF RTF DTF 10 −1 10 0 10 1 10 2 10 −1 10 0 10 1 10 2 10 −5 10 0 −4 −2 2 4 0 10 5 Frequency (Hz) Frequency (Hz) Magnitude Phase angle (e) RTF DTF RTF DTF Figure 18. DTF and RTF response of fixed-free beam structure: (a) v 1 /F 3 ; (b) v 2 /F 3 ; (c) v 3 /F 3 ; (d) v 4 /F 3 ; (e) v 5 /F 3 . P1: FCH PB091E-41 January 10, 2002 21:18 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS 535 F Structure Sensor Sensor Sensor Sensor RTF Presence Location Type Extent Element # PDI Mag Damaged @ undamaged Phase damage index (PDI) Phase DTF System ID M,K,etc 0 G RTF Response Collocated non-causal controllers DTF DTF = (RTF −1 +G) −1 System Dynamics Responses Responses Figure 19. Damage detection methodology based on the DTF response. noncausal controllers can be developed based on an identi- fied model of the structural system. Implementing these noncausal controllers off-line in a computer will yield the DTF response from each actuator to each sensor. To infer the presence of damage, these undamaged controllers can be repeatedly applied to the identified RTF responses of a structural system. Thus, as damage appears in the struc- ture, its presence will be revealed in the magnitude and phase plots of the DTF responses. Therefore, by tracking relative magnitude and phase errors between the undam- aged and damaged structure’s DTF response, it is possible to infer both the presence and location of damage. Fur- thermore, to determine damage in a particular structural element, one needs only to track the transmission of in- cident energy through the structure. This can be done by computing the ratio of the DTF responses for sequential degrees of freedom. This damage detection methodology is illustrated in Fig. 19. Using examples of a free-free and fixed-free beam, the concept of a phase damage index is introduced following to illustrate how the presence, location, type, and amount of damage can be determined. This approach is later demon- strated on a model of a civil building structure. Free-Free Beam. Consider the free-free beam structure previously shown in Fig. 14. Energy is input at node 3 where external force F 3 = 0. Figure 20 shows the DTF re- sponses for the following transmission ratios of sequen- tial degrees of freedom (v 1 /v 0 , v 2 /v 1 , v 3 /v 2 ). The solid line refers to the undamaged case, and the dashed line refers to the damaged case. Each part of the figure dis- plays the respective magnitude and phase response of the transmission ratio. It is assumed that damage in the form of a loss of stiffness is simulated in element number 3. Careful inspection of Fig. 20 allows two observations: First, in the case of stiffness damage, the phase of the DTF re- sponse will wrap around −π earlier than in the undam- aged case. Secondly, there is not a significant change in the DTF magnitude response due to damage in the structure. To explain these observations in more detail, one has to consider the effect of damage on the structure’swave propagation response. When a leftward propagating wave passes through a damaged element such as element num- ber 3, the effect of a stiffness loss is to increase the wave number and slow down the propagation of the transmit- ted wave. This introduces an additional phase lag in the DTF response. Undamaged structural elements have no effect on propagating the transmitted wave. In terms of the magnitude response, the noncausal controllers are de- signed to prevent wave reflection at the interfaces between structural elements. However, when damage occurs, wave reflection is not prevented because the controllers can no longer provide a match terminating boundary condition at each end. Nevertheless, because the simulated damage is small, the magnitude of the DTF displays modest sensi- tivity to damage. Therefore, a phase damage index (PDI) has been developed to locate the element that contains the damage. The PDI is based on the relative phase error from successive DOFs and is given by PDI i =  ω l ≤ ω ≤ ω u     phase  v i v i−1 (ω)  undamaged DTF  −phase  v i v i−1 (ω)  damaged DTF      , (42) where i = 1, 2, ,5 and refers to the ith element. ω l , ω u are the lower and upper bounds of the frequency range of interest. Computing the value of the PDI for each struc- tural element indicates which element is damaged. This is evident from the PDI computed for the free-free beam ex- ample described earlier in this section. Note that the PDI is significantly larger for element number 3 in which simu- lated stiffnessdamage was assumed(see Fig.20). Figure21 displays the PDI for assumed stiffness damage in each of the five structural elements. Note that in all cases, the PDI that has the largest absolute value indicates, the structural element in the free-free beam example that is damaged. P1: FCH PB091E-41 January 10, 2002 21:18 536 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS DTF response v 1 /v 0 Magnitude Frequency (Hz) Undamaged Damaged Undamaged Damaged −4 −2 0 2 4 Phase (rad) Frequency (Hz) 10 −1 10 −1 10 −1 10 0 10 0 10 0 10 1 10 1 10 2 10 2 (a) 10 1 10 1 10 1 10 2 10 2 10 0 10 0 10 0 10 −1 10 −1 10 −1 DTF response v 2 /v 1 Magnitude Frequency (Hz) −4 −2 0 2 4 Phase (rad) Frequency (Hz) Undamaged Damaged Undamaged Damaged (b) DTF response v 3 /v 2 Magnitude Frequency (Hz) Undamaged Damaged −4 −2 0 2 4 Phase (rad) Frequency (Hz) Undamaged Damaged 10 1 10 1 10 1 10 2 10 2 10 0 10 0 10 0 10 −1 10 −1 10 −1 (c) DTF response v 4 /v 3 Magnitude Frequency (Hz) Undamaged Damaged −4 −2 0 2 4 Phase (rad) Frequency (Hz) 10 0 10 0 10 0 10 1 10 1 10 2 10 2 10 −1 10 −1 10 −1 Undamaged Damaged (d) DTF response v 5 /v 4 Magnitude Frequency (Hz) Undamaged Damaged Undamaged Damaged −4 −2 0 2 4 Phase (rad) Frequency (Hz) 10 1 10 1 10 1 10 2 10 2 10 0 10 0 10 0 10 −1 10 −1 10 −1 (e) Element number Amount of damage index for elements of interest Element 3 is damaged (5% loss in stiffness) 1 2 3 4 5 0 10 20 30 40 50 60 70 80 90 (f) Figure 20. Element 3 is damaged in terms of 5% loss of stiffness. (a) DTF response v 1 /v 0 ; (b) DTF response v 2 /v 1 ; (c) DTF response v 3 /v 2 ; (d) DTF response v 4 /v 3 ; (e) DTF response v 5 /v 4 ; (f) damage index vs. element number. P1: FCH PB091E-41 January 10, 2002 21:18 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS 537 Element number Amount of damage index for elements of interest Element 1 is damaged (5% loss in stiffness) 0 10 20 30 40 50 60 1 2 3 4 5 (a) Element 2 is damaged (5% loss in stiffness) 12 3 4 5 0 20 40 60 80 100 120 Element number Amount of damage index for elements of interest (b) 1 2 3 4 5 0 5 10 15 20 25 30 35 Element number Amount of damage index for elements of interest Element 4 is damaged (5% loss in stiffness) (c) 1 2 3 4 5 0 2 4 6 8 10 12 14 16 Element number Amount of damage index for elements of interest Element 5 is damaged (5% loss in stiffness) (d) Figure 21. Damage index vs. element number. (a) Element 1 is damaged (5% stiffness loss). (b) Element 2 is damaged (5% stiffness loss). (c) Element 4 is damaged (5% stiffness loss). (d) Element 5 is damaged (5% stiffness loss). Fixed-Free Beam. For a fixed-free beam,  v 0 θ 0  =  0 0  . Only i = 2, 3, 4, 5 can be substituted in Eq. (42), and four damage indices, PDI 2 , PDI 3 , PDI 4 , PDI 5 , can be formu- lated. They are associated with elements 2,3,4, and 5, re- spectively. Five damage cases are simulated. Figure 22a shows damage indexes versus element number when ele- ment 1 is damaged. Remember that no damage index is formulated for element 1. However, from Fig. 22a, the in- dex for element 2, PDI 2 , is large. This implies that damage occured in element 1, demonstrated by the damage index of element 2. Of course, from Fig. 22b, damage in element 2 leads to a blowup of PDI 2 , too. Therefore, if the damage index of element 2, PDI 2 , exhibits some large value, both element 1 and element 2 need to be checked to see whether they are damaged. The damage indexes of elements 3,4, and 5 can locate the damage of the associated elements as shown in Fig. 22c–e. Damage Type and Extent Most structural damage can be categorized as mass or stiff- ness related. In stiffness loss, the phase curve of the DTF response of a damaged structure tends to wrap at π or −π earlier than in the undamaged case. In mass loss, the phase of the DTF response of a damaged structure tends to wrap later than in the undamaged case. This pattern in the phase behavior of stiffness or mass damage is pri- marily a result of the way wave number variations affect the transmission properties of an incident wave through a structural element. Thus, by simply examining the phase behavior of undamaged and damaged DTF responses, it is possible to ascertain the type of damage, albeit stiff- ness or mass. In addition, damping effects can also be identified. Although the type of damage in a structural system can be determined from the phase response of the DTF, P1: FCH PB091E-41 January 10, 2002 21:18 538 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS Element number Amount of damage index for interested elements Element 1 is damaged (5% loss in stiffness) 2 3 4 5 0 20 40 60 80 100 120 140 180 160 200 (a) Element number Amount of damage index for interested elements Element 2 is damaged (5% loss in stiffness) 2 3 4 5 0 20 40 60 80 100 120 (b) 432 Element number Amount of damage index for interested elements Element 3 is damaged (5% loss in stiffness) 0 10 20 30 40 50 60 70 80 90 5 (c) Element number Amount of damage index for interested elements Element 4 is damaged (5% loss in stiffness) 0 5 10 15 20 25 30 35 2 3 4 5 (d) Element number Amount of damage index for interested elements Element 5 is damaged (5% loss in stiffness) 0 2 4 6 8 10 12 14 16 2 3 4 5 (e) Figure 22. Damage indices vs. element number. (a) Element 1 is damaged (5% stiffness loss). (b) Element 2 is damaged (5% stif- fness loss). (c) Element 3 is damaged (5% stiffness loss). (d) Element 4 is damaged (5% stiffness loss). (e) Element 5 is damaged (5% stiffness loss). P1: FCH PB091E-41 January 10, 2002 21:18 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS 539 0 0 100 200 300 400 Damage index of element 5 500 600 700 800 10 20 30 40 50 Percent of stiffness loss at element 5 Damage index vs. damage extent 60 70 80 90 100 Figure 23. Fixed-free beam; the fifth element is damaged. determining the extent of damage in a structural element is a bit more dificult. The phase damage index (PDI) de- scribed before represents a qualitative measure of damage in a structure. However, by conducting a numerical study of the structural model identified, one can simulate struc- tural damage in each structural element and correlate the PDI with percentage damage. Thus, a lookup table or curve-fitted database can be developed for each structural element to provide an estimate of the amount of damage in that element. Figure 23 displays such a correlation for the (a) x g m 1 x 1 m 2 x 2 m 3 x 3 k 1 /2 k 2 /2 k 3/ 2 k 1 /2 k 2 /2 k 3 /2 (b) Figure 24. (a) Framed building structure. (b) Discrete three-DOF model. fixed-free beam example developed earlier. Note that the PDI increases as structural damage increases in the struc- tural element of interest. Similar trends exist for other elements. DAMAGE DETECTION IN A BUILDING STRUCTURE USING DTF Consider the discrete three-DOF model of the three-story framed building structure displayed in Fig. 24. The prop- erties of the structure are given in Table 3. The equation of motion for the building structure is M ¨ X + KX = q (43) or    m 1 00 0 m 2 0 00m 3         ¨x 1 ¨x 2 ¨x 3      +   k 1 + k 2 −k 2 0 −k 2 k 2 + k 3 −k 3 0 −k 3 k 3        x 1 x 2 x 3      =      k 1 x g 0 0      . (44) Transformed into the frequency domain and assuming that x = Xe iωt , Eq. (44) becomes [−Mω 2 + K ]X = Q (45) P1: FCH PB091E-41 January 10, 2002 21:18 540 HEALTH MONITORING (STRUCTURAL) USING WAVE DYNAMICS Table 3. Properties of the structure Mass m 1 = 16.7676 m 2 = 16.7676 m 3 = 16.7676 unit: kg Stiffness k 1 = 9158.7 k 2 = 13084 k 3 = 36635 unit: N/m m 1 m 1 k 1 k 2 k 3 k 3 m 3 m 3 k 2 k 1 m 2 m 2 Element 1 Element 2 Element 3 =++ Figure 25. Summing asymmetrical spring-mass elements to con- struct a building structure. or   k 1 + k 2 − m 1 ω 2 −k 2 0 −k 2 k 2 + k 3 − m 2 ω 2 −k 3 0 −k 3 k 3 − m 3 ω 2      X 1 X 2 X 3    =    −k 1 ω 2 X g 0 0    . (46) From Eq. (46), note that energy is input into the build- ing structure from ground motion by F 1 =−k 1 ω 2 X g (ω)or k 1 ¨ X g (ω). A model of the three-DOF three-story building struc- ture in Fig. 24 can be constructed by simply adding three asymmetrical spring-mass elements (see Fig. 25). The DTF of the whole building structure can be obtained by first adding a virtual controller to the right end of element 1, then adding elements 2 and 3 that have similar virtual controllers attached to both ends. This process is shown in Fig. 26. The control architecture is shown in Fig 27. The controller gains G 1r , G 2l , G 2r , G 3l , G 3r are obtained by us- ing Eq. (8) and substituting different k, m, and µ values for the different structural elements. The RTF and DTF are shown in Fig. 28. Considering the structure shown in Fig. 26, energy is input from the left end of the structure. After passing ele- ment 1, controller G 1r will prevent the energy from being reflected back into element 1. A portion of the energy will continue transmitting into element 2. At the end of ele- ment 2, controller G 2r acts like controller G 1r to ensure that no energy is reflected back. The same effect willbe achieved by controller G r3 . The result is that wave reflection does Figure 26. Applying controllers to obtain the DTF of a building struc- ture. DTF of Element 1 Element 2 Element 3 K 1 K 1 m 1 m 2 m 2 m 3 m 3 m 1 K 2 K 2 K 3 K 3 + = G 1r G 2r + G 2r G 3r G 3r not occur across the structure. If damage is present in a structural element, the idealized local controllers can not prevent reflection. Damage in civil structures is usually categorized as stiffness loss. The following damage case studies simulate one-quarter damage loss for each stiffness of the three- DOF three-story building. This study attempts to show the characteristic change in the DTFs before and after damage. Further, a methodology is developed to locate damage. Six sets of figures are displayed for each case study, and each figure has two parts: the magnitude and phase of a parti- cular DTF response. Emphasis is placed on examining the incident path of energy traveling through the structure. As the energy is transmitted across a damaged structural element, either the magnitude or the speed of wave pro- pagation will change compared to the undamaged element. This change in local structural properties is revealed in the magnitude and phase plot of the DTF response. Case Study: One-Quarter Stiffness Loss for k 1 Considering k 1 damaged in terms of stiffness loss, the energy is still input from the left end of the structure. Re- member that G 1r is determined by m 1 and k 1 . The controller gains G 1r can prevent wave reflection perfectly at ele- ment 1. But for the damaged element, we do not know how much change there is in k 1 . If we still use G 1r as a virtual controller for the damaged element, wave reflections will occur at the end of element 1. The DTF response of ¨x 1 / ¨x g will change before and after damage, as is illustrated in Fig. 29 a. Strictly speaking, ¨x 1 / ¨x g is no longer the DTF after damage because a wave is reflected back into ele- ment 1. But that is not the case for elements 2 and 3. Controller gains G 2r and G 3r will do their jobs perfectly. This leads to identical DTF responses of ¨x 2 / ¨x 1 , ¨x 3 / ¨x 1 , and ¨x 3 / ¨x 2 , illustrated in Fig. 29d–f before and after damage. Now, consider Fig. 29a. In the low-frequency range, the DTF phase of an undamaged element is smaller than that of an damaged element. This means that the wave number of an undamaged element is smaller than that of an dam- aged element. The reason for the increased wave number is either stiffness loss and/or mass increase. Mass increase is unlikely in the event of damage in a civil structure. Thus, the cause of damage is assumed to be stiffness. [...].. .10 2 10 0 (b) 10 2 DTF RTF Magnitude (a) Magnitude Figure 27 Control architecture 10 −2 10 −4 10 0 10 1 D R 10 0 10 −2 10 −4 10 −6 10 0 10 2 10 1 Frequency (Hz) 10 1 Phase (rad) DTF RTF 10 2 0 1 −2 −3 −4 −5 −6 7 10 0 D R 10 1 Hz Hz (c) 10 2 Magnitude 0 −0.5 1 1. 5 −2 −2.5 −3 −3.5 10 0 DTF RTF 10 0 10 −2 10 −4 10 −6 10 0 10 1 Frequency (Hz) 10 2 0 Phase (rad) Phase (rad) Frequency (Hz) −2 DTF RTF −4 −6 −8 10 10 0 10 1 Hz 10 2... (Hz) 10 0 Undam 1/ 4 loss 10 1 10−2 10 −3 0 10 10 2 Undamaged 1/ 4 loss for k1 10 1 Frequency (Hz) 10 1 Magnitude −4 (e) Magnitude Undamaged 1/ 4 loss for k1 Phase (rad) Phase (rad) −2 Undama 1/ 4 loss 10 0 Frequency (Hz) 10 10 0 10 1 Frequency (Hz) 10 −5 10 0 −5 −6 7 10 0 (d) 10 1 (c) 10 0 Magnitude 10 2 Magnitude Pha −2 −2.5 −3 −3.5 0 10 0 −0.5 1 1. 5 −2 −2.5 −3 −3.5 10 0 10 1 Frequency (Hz) Undam 1/ 4 loss 10 1 Frequency... (a) x1 /xg ; (b) x2 /xg ; (c) x3 /xg ; (d) x2 /x1 ; (e) x3 /x1 ; (f ) x3 /x2 ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ −4 10 0 10 1 Frequency (Hz) 10 2 (d) 10 −5 10 1 Frequency (Hz) 10 0 −8 10 2 10 −5 0 1 −2 −3 −4 −5 −6 7 10 0 Frequency (Hz) (f) Undam 1/ 4 loss 10 1 10 1 10 0 10 2 Undam 1/ 4 loss 10 1 10−2 10 −3 0 10 10 2 Undamaged 1/ 4 loss for k2 10 1 0 −0.5 1 1. 5 −2 −2.5 −3 −3.5 10 0 10 1 Frequency (Hz) Frequency (Hz) Undamaged 1/ 4... 28 RTF and DTF response of a building structure: (a) x1 /xg ; (b) x2 /xg ; (c) x3 /xg ¨ ¨ ¨ ¨ ¨ ¨ Ph 10 1 Frequency (Hz) Undamaged 1/ 4 loss for k1 10 1 10−2 10 2 10 −3 0 10 10 2 10 1 0 −0.5 1 1. 5 −2 −2.5 −3 −3.5 10 0 −0 Phase (rad) −6 −8 10 1 Frequency (Hz) (f) 10 0 Undamaged 1/ 4 loss for k1 10 −5 Phase (rad) 10 0 0 1 −2 −3 −4 −5 −6 7 10 0 10 1 Frequency (Hz) 10 1 Frequency (Hz) 10 2 Undama 1/ 4 loss 10 1 Frequency... x1 /xg ; (b) x2 /xg ; (c) x3 /xg ; (d) x2 /x1 ; (e) x3 /x1 ; (f ) x3 /x2 ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ −4 10 0 10 1 Frequency (Hz) Magnitude Ph −6 −8 10 0 Undamaged 1/ 4 loss for k3 10 −5 10 1 Frequency (Hz) 10 1 10−2 10 −3 0 10 10 2 −4 Undamaged 1/ 4 loss for k3 −6 −8 10 1 Frequency (Hz) (f) 10 0 10 −5 10 0 0 1 −2 −3 −4 −5 −6 7 10 0 −3 −4 10 0 10 2 Undamaged 1/ 4 loss for k3 10 1 Frequency (Hz) 10 0 10 1 10−2 10 −3 0 10 ... loss for k2 10 1 Frequency (Hz) 10 −2 Magnitude Magnitude 10 1 Frequency (Hz) 10 0 10 0 Phase (rad) Phase (rad) −6 Phase (rad) Phase (rad) −4 10 10 0 (e) Undamaged 1/ 4 loss for k2 Undama 1/ 4 loss 10 1 10−3 0 10 10 2 0 −2 10 1 10 1 Magnitude Undamaged 1/ 4 loss for k2 10 0 −5 −6 7 10 0 Frequency (Hz) 10 0 (c) Magnitude Ph Ph −3 0 −0.5 1 1. 5 −2 −2.5 −3 −3.5 10 0 10 1 Frequency (Hz) Undam 1/ 4 loss 10 1 Frequency... 10 2 0 −0.5 1 1. 5 −2 −2.5 −3 −3.5 10 0 Undamaged 1/ 4 loss for k3 10 1 Frequency (Hz) 10 1 10 2 10 1 Frequency (Hz) Undama 1/ 4 loss −2 Magnitude Magnitude Phase (rad) 1 Phase (rad) Phase (rad) −2 10 1 Frequency (Hz) 0 10 10 0 Phase (rad) Undama 1/ 4 loss 10 0 0 (e) 10 1 Frequency (Hz) (d) 10 1 (c) 10 0 10 0 −4 10 2 Magnitude Ph 1/ 4 loss for k3 −2 Undamaged 1/ 4 loss for k3 10 1 Frequency (Hz) Undama 1/ 4 loss f 10 1... 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BIBLIOGRAPHY 1 S.W Doebling, C.R Farrar, and M.B Prime, and D.W Shevitz, Los Alamos National Laboratory Report No LA -1 3 070 -MS, Los Alamos, NM, 19 95 2 C.R Farrar and D.A Jauregui, Smart Mater Struc 7, (5): 70 4– 71 9 (19 98) 3 C.R Farrar and D.A Jauregui, Smart Mater Struct 7, (5): 72 0 7 31 (19 98) 4 F.K Chang, Proc 2nd Int Workshop Struct Health Monitoring, Stanford University, Stanford, CA, Sept 8 10 , 19 99 5 . (Hz) Undamaged Damaged Undamaged Damaged −4 −2 0 2 4 Phase (rad) Frequency (Hz) 10 1 10 1 10 1 10 0 10 0 10 0 10 1 10 1 10 2 10 2 (a) 10 1 10 1 10 1 10 2 10 2 10 0 10 0 10 0 10 1 10 1 10 1 DTF. angle (c) RTF DTF RTF DTF 10 1 10 0 10 1 10 2 10 1 10 0 10 1 10 2 10 10 10 −5 10 0 −4 −2 2 4 0 10 5 Frequency (Hz) Frequency (Hz) Magnitude Phase angle (d) RTF DTF RTF DTF 10 1 10 0 10 1 10 2 10 1 10 0 10 1 10 2 10 −5 10 0 −4 −2 2 4 0 10 5 Frequency. (Hz) 10 1 10 1 10 1 10 2 10 2 10 0 10 0 10 0 10 1 10 1 10 1 (e) Element number Amount of damage index for elements of interest Element 3 is damaged (5% loss in stiffness) 1 2 3 4 5 0 10 20 30 40 50 60 70 80 90 (f) Figure