Identification of damage location using Operational Deflection Shape (ODS)

Identification of damage location using Operational Deflection Shape (ODS) D. Di Maio, C. Zang, D. J. Ewins Mr. Dario Di Maio, Mechanical Engineering Bldg. Imperial College London, South Kensington Campus. SW7 2AZ, London Dr Chaoping Zang, Mechanical Engineering Bldg. Imperial College London, South Kensington Campus. SW7 2AZ, London Prof. David J. Ewins, Mechanical Engineering Bldg. Imperial College London, South Kensington Campus. SW7 2AZ, London ABSTRACT The ability to predict and to measure the ODS of a vibrating structure suggests its use to increase the potential for structural damage detection, localization and severity assessment with high spatial density. Previous research based on simulation and testing of some simple test cases showed the effect of excitation locations on structural damage detection. The measurement of the ODS, at specific excitation location, of healthy and unhealthy structures showed a substantial difference in the deflection shape of the structures. Following the positive results of the simulations and measurements of ODS between undamaged and damaged strucuture, it can be possible to start a theoretical identification of the damage location. Results of the damage location identification will presented and discussed. NOMENCLATURE M K D F α(ω) Φ η r ω r ω {} ε s p xy, ( ) Mass matrix Stiffness matrix Hysteretic proportional damping Force Frequency Response Function Eigenvector Damping loss factor of r th mode Natural frequency of r th mode Vibration frequency Perturbation function Coefficient Generic node position

1. Introduction When we started to study the effect of simulated structural damage on two dimensional structures, we focused our studies on the use of the Operational Deflection Shapes (ODS) because: (i) ODS is a summation of all the mode shapes of the structure and (ii) the ODS can be a result of vibration measurement without any substantial post processing such as modal analysis. Our first approach to that problem was to study the effect of the excitation location and the excitation frequency on the calculation of an undamaged structure ODS and that of a damaged one when both those excitation parameters were varied. A square plate made of steel was the test piece designed to be used to run the ODS simulations. First results showed that the ODS of the damaged structure can significantly change from the undamaged one for some excitation frequencies and for some excitation locations. The interesting results of the simulations forced our initial guess to validate them against reference data such as measured ODSs. Having produced several simulations to identify the best excitation frequencies and excitation positions, we used those indications to carry out measurements on a real test piece, both in undamaged and damaged conditions. The damaged identification is an invasive issue for the structure because it has to be either cracked or machined. To avoid any irreparable damage we decided to use magnets to simulate the damaged and to physically produce it on the metal structure. The simulations of ODS changes between healthy and unhealthy models were confirmed by the results of measured ODSs. Those early results were not enough to locate the damage position. However the path started using ODS methodology was interesting. In this paper, we will discuss the theoretical description of the ODS for the purpose of damage detection and its use to produce an algorithm to identify the damage position. FE model simulations, using a rectangular plate as an example, demonstrate the results of the algorithm to locate the simulated damage on the models. The anomalies of ODS of the damaged structures with respect to the healthy structure can be detected in terms of the geometric features. Results show the feasibility of structural damage detection and localization using an ODS based technique. 2. Operational Deflection Shape theory The first part is focused on the method used to produce ODS from modal data obtained by FEM model. The second part of this section is focused on the description of the algorithm to calculate the damage location making use of the ODS. 2.1 ODS theory If we consider the general equation of forced vibration for the particular case of harmonic excitation and response of a MDOF system with proportional hysteretic damping: 2 iωt iωt ([ K] i[ D] ω [ M ]){ X} e = { F} e + (1) and the hysteretic damping matrix [D] is proportional, typically: [ D] = β [ K] + γ[ M ] (2) the solution of the general equation is: { X} = 2 1 ([ K] + i[ D] ω [ M ) { F} = [ α( ω)]{ F} (3) but this equation is not helpful for numerical application therefore a more explicit form of this solution may be derived: { X} = { φ} { F}{ φ} N T r 2 2 r= 1 ωr ω + iη r 2 rωr (4)

where γ η r β + ω = is damping loss factor for that mode. Equation (4) may be used to study ODS in terms of 2 r mode shapes, force vectors, the excitation frequencies and the damping factors. Equation (4) was then coded using the Labview software platform to perform simulations of ODS sweeping the excitation frequency and the excitation point location. 2.2 Identification algorithm We theoretically simulated and practically measured the change in ODS when a damage is either simulated or physically introduced to the structure. The use of the ODS involves a very important feature consisting in the number of ODSs we can either simulate or measure because it depends by the excitation location and excitation frequency. We are not forced to focus our search around the mode shapes only but this study can be wider and any way connected to the mode shapes being the ODS a summation of all. Being able to simulate an undamaged ODS as well as a damaged one, it was important to link them by a parameter or parameters as expressed by equation: { X } = { X } + { ε}* s D UD (5) where { X D} is the damaged ODS, { XUD} is the undamaged ODS, { ε} is the perturbation function and s is a coefficient. Equation (5 can be an initial link between the damaged and undamaged ODSs. However used as described, it is not very useful and also a precise definition of the coefficient, s, was not assessed. The first order derivation of the ODS was then used to outline any discontinuity in the deflection shape produced by the damage and it is expressed by the equation: N T { XD} { φd} r{ F}{ φ } D r = 2 2 2 ( p( x, y) ) ( p( x, y) ) r= 1 ωrd, ω + iηrω (6) rd, (, ) p xy being a generic location. Several simulations were performed using the algorithm presented in equation 6 and the results were satisfactory when the simulated damage was significant while for small damages it was neither easy nor evident to detect. Interesting results were produced when the parameter linking the damaged ODS and the undamaged one was defined as expressed in equation: ( ) ( ) T T { X } { X } N N { φ } { F}{ φ } { φ } { F}{ φ } ε = = D UD D r D r UD r UD r 2 2 2 2 2 2 ( p x, y ) ( p( x, y) ) r 1ωrD, ω iηrω rd, ( p( x, y) ) (7) = + r= 1ωrUD, ω + iηrω rud, the new definition of the perturbation function, ε ε( xyz,, ) =, producing very interesting results. Using this algorithm it was possible to locate the damage even when it was designed to be very light, meaning that the deepness of the cut was very small. ODSs can be recalculated, using equation 7, along two dimensions, (X, Y). In this paper calculations of the damage position were performed along one dimension only as expressed by equation: ( ) ( ) T T { X } { X } N N { φ } { F}{ φ } { φ } { F}{ φ } ε = = D UD D r D r UD r UD r 2 2 2 2 2 2 ( p x ) ( p( x) ) r 1ωr, D ω iηω r r, D ( p( x) ) (8) = + r= 1ωr, UD ω + iηω r r, UD

3. Damage detection ODS simulation using a rectangular plate 3.1 Preliminaries A finite-element model of a square plate of dimensions 0.5m x 0.5m x 0.01m was produced using Shell elements type to generate a mesh of 2600 nodes spaced on a grid of 0.01m. Material properties of the mild steel were used for the FEM model hence: material density was set at 8000 Kg/m 3 and Young s modulus was set at 217.2 GPa. The model described was largely used until was decided to avoid symmetrical structure because the double modes which inevitably split the natural frequency whenever the structure become detuned due to a damage for example. The Fe model stayed formally the same except to remove two rows of element to make the structure rectangular. The new plate has now dimension 0.5m x 0.48m x 0.01m for a total number of nodes of 2499. The first 20 mode shapes were calculated under Free-Free conditions both for the undamaged structure and for the damaged ones. Figure 1 shows the four locations chosen to simulate the damaged plate. Figure 3a shows a plate damaged at two locations, the damage depicted in red is simulated with 6 elements, 0.01m x 0.06m, whose thickness is 0.007m thus to obtain a damage of 30% while a deeper damage, in purple, was simulated reducing the thickness of the 4 elements, 0.01m x 0.04m, up to 0.003m producing a cut of 70%. Figure 3b shows the same plate where the cut was simulated reducing the thickness of the 4 elements, 0.01m x 0.04m, of 10% and 50% producing a cut deep 0.001m and 0.0005m, respectively. Figure 3c shows a plate whose damage is now produced diagonally reducing the thickness of the 3 elements of 10% and 50% while the last figure 3d shows the plate with a very small cut made of 2 elements whose thickness was also for this case reduced of 10% and 50%. a) b) c) d) Figure 1 Damage locations

All eigenvalues of undamaged and damaged plate are reported in Table. The table is divided to report the undamaged eigenvalues and the damaged ones labelled Damage (a); Damage (b); Damage (c); Damage (d) with the degree of deepness of the cut. Undamaged Damage (a) Damage (b) Damage (c) Damage (d) 70% & 30% 10% 50% 10% 50% 10% 50% Mode Eigenvalues [Hz] Eigenvalues [Hz] Eigenvalues [Hz] Eigenvalues [Hz] Eigenvalues [Hz] 1 140.83 140.59 140.79 140.68 140.82 140.79 140.81 140.75 2 203.34 203.09 203.27 203.08 203.32 203.26 203.30 203.19 3 255.75 254.47 255.66 255.12 255.73 255.67 255.68 255.37 4 358.67 358.19 358.63 358.51 358.62 358.45 358.65 358.60 5 369.55 368.95 369.47 369.28 369.49 369.34 369.53 369.47 6 613.74 613.28 613.72 613.69 613.56 613.05 613.65 613.37 7 665.14 660.56 664.82 662.33 665.02 664.65 665.14 665.13 8 666.36 665.44 666.33 666.24 666.18 665.66 666.32 666.08 9 719.90 719.21 719.84 719.68 719.77 719.43 719.90 719.90 10 812.83 810.90 812.81 812.76 812.63 811.94 812.78 812.66 11 1091.70 1089.90 1091.50 1090.90 1091.60 1091.40 1091.60 1091.00 12 1115.20 1111.10 1114.80 1111.80 1115.00 1114.30 1115.20 1115.00 13 1195.50 1194.80 1195.20 1194.20 1195.20 1194.50 1195.50 1195.50 14 1311.70 1310.40 1311.40 1309.70 1311.40 1310.20 1311.50 1310.50 15 1331.60 1330.10 1331.30 1330.40 1331.30 1330.30 1331.50 1331.30 16 1421.20 1419.90 1421.10 1420.90 1420.80 1419.80 1421.10 1420.70 17 1599.50 1596.30 1599.40 1599.10 1599.30 1598.80 1599.40 1599.40 18 1676.40 1673.30 1676.20 1674.50 1676.30 1676.00 1676.30 1676.10 19 1777.30 1774.90 1776.90 1773.70 1777.20 1777.00 1777.20 1776.50 20 1992.10 1988.70 1992.00 1991.70 1992.00 1991.60 1991.80 1991.00 Table 1 3.3 Damage detection using ODS Having produced several FE models with different damage positions and with different degrees of deepness of the cut we were able to run a code to calculate any ODSs both of the healthy and unhealthy model making use of the equation (4). We run modal analysis for all the FE models to obtain the eigenvectors, eigenvalues and coordinates. All the data saved were used to calculate any ODS in the frequency range of the calculated modal solutions using equation (4), the damping loss factors required by the equation were measured and reported in Table 2. Mode Loss factor Mode Loss factor Mode Loss factor Mode Loss factor 1 1.60E-03 6 1.20E-03 11 7.60E-03 16 2.80E-03 2 4.00E-04 7 1.40E-03 12 3.20E-03 17 1.40E-03 3 2.00E-03 8 2.00E-03 13 8.00E-04 18 1.40E-03 4 6.70E-03 9 3.00E-04 14 2.90E-03 19 3.60E-03 5 6.60E-03 10 1.12E-02 15 2.10E-03 20 4.00E-03 Table 2 The excitation location was kept fixed for all ODS simulations that was set at node number 1 as shown in Figure 2 and the level of excitation force was kept constant for all the simulations.

Node 2448 Node 2499 Node 1 Node 51 Figure 2 Nodes numbering The code written in Labview used equation (4) to calculate the ODSs and equation (8) to recalculate the ODS. The code could output simultaneously both the undamaged and damaged ODSs to one plot and the damage location to another plot. That option was useful to compare the two ODSs, being overlapped one to each other and to asses whether the damaged ODS simulated was sensitive to the damage position obviously looking at the plot due for that task. The next paragraph will report the results of some ODS simulations for each damage location and deepness of the cut. Some plots showing the damage position had to be rotated to give a better view of the damage position otherwise unclear to see. 3.3.1 Case (a): two damages localized at the centre and at the side of the structure Figure 3 shows the rectangular plate, where two damages were simulated reducing the element thickness of 30% for the vertical one and 70% for the horizontal one from the original element thickness. Figure 3 Vertical and Horizontal damages, 30% and 70% respectively Figure 4 shows on the right hand plot two overlapped ODS simulated at 130 Hz, being 140.83 Hz and 140.59 Hz the resonances of the undamaged and damaged models respectively. The damages do not change significantly the ODS of the models at that excitation frequency however looking at the left hand plot where the ODSs were replot using equation (8) it is very clear to identify the damages position. Figure 5 shows on the right hand plot two ODSs simulated at excitation frequency very close to the resonance frequencies of the two FE models. The damages are now affecting clearly the ODSs which are showing a change but the damage location plot on the left hand side does not show any presence of damages.

Undamaged(coloured) and damaged ODS(black) Figure 4 ODS simulations at 130HZ Undamaged(coloured) and damaged ODS(black) Figure 5 ODS simulations at 140.86Hz In the last example of this test case we provide simulations of ODSs whose excitation frequency is near the 17 th mode shape. Figure 6 shows undamaged and damaged ODS exited at 1525 Hz on the right hand plot while in the left hand plot is shown the damage position plot. It seems that when the excitation frequency is between two resonances the equation 8 produces better results and this is due the small contribution of the mode shapes which they are very sensitive to the damage position. When the excitation frequency is set close to the resonance such as 1600 Hz, being the resonance of undamaged and damaged ODS 15999.5 Hz and 1596.3 Hz respectively, the two ODSs do not match perfectly as it happened for the cases showed before however looking at the left hand plot it is still possible to see the damage position, more clearly the damaged done at the centre which cut is deeper.

Undamaged(coloured) and damaged ODS(black) Figure 6 ODS simulations at 1525Hz Undamaged(coloured) and damaged ODS(black) Figure 7 ODS simulations at 1600Hz 3.3.2 Case 2: damaged localized at the centre The second case presents a FE model where the cut was done near the centre of the structure. We decided to produce two degree of deepness of the cut such as 10% and 50% of the original thickness of the element which was 0.01m. Figure 8 shows clearly where the damage was simulated. We present one example of damage location for low excitation frequency which was chosen away from the two adjacent resonances and one a higher excitation frequency, this time closer to one resonance.

Figure 8 Horizontal damage Figure 9 shows an example of ODSs simulated at 500 Hz, being the closest resonances at 369.28 and 613.73, thus to be quite away from both of them. The simulation was performed using the model whose damage was set at 90% of its original element thickness therefore only 0.001m was removed from the 4 elements. Despite the ODS amplitude displacement at this excitation frequency cannot be very high, for obvious reasons, the left hand plot shows clearly the damage position. Undamaged(coloured) and damaged ODS(black) Figure 9 ODS simulation at 500Hz 3.3.3 Case 3: damage localized at the corner of the structure Figure 10 shows the FE model of the plate when the damage was simulated diagonally to the structure and also for this case we decided to produce two cuts of 10% and 50% of the original element thickness.

Figure 10 Diagonal damage This time we present ODSs with a model in which the damage deepness is 10%, 0.001m, of the original element thickness. The simulated is performed at excitation frequency of 250 Hz, being close to the 3 rd mode shape resonance, the ODS has a shape of mushroom and the nodal line is crossing more or less the position of the damage as shown in Figure 11 right hand plot. Despite the amplitude of vibration around of the nodal line is very small the damage plot shows clearly the area where the cut is produced as shown Figure 11 left hand plot. Undamaged(coloured) and damaged ODS(black) Figure 11 ODS simulations at 250 Hz; damage 10% 3.3.4 Case 4: small damage localized at the centre of the structure The last case proposed it is the model for which the damage was simulated nearly at the centre of the plate and also for this FE model the cut was made of 10% and 50% of the original element thickness. Figure 12 shows the plate and the position of the damage.

Figure 12 Small horizontal damage For the last case we present a series of simulations of ODSs in the frequency range of the 6 th, 7 th, and 8 th mode shapes of the FE model in which the cut is the 10% of the original element thickness. Figure 13 shows the ODSs at 613 Hz on the right hand plot while on the left hand plot it is possible to see the damage position outlined by a dotted circle, to have a better visualization a zoomed picture has been inserted. The damage is small but the recalculation of the ODSs at this excitation frequency still helps to locate the distortion in the plot. When the excitation frequency is set at 655 Hz near to the 7 th resonance the position of the damage is even clearer as shown in Figure 14 on the left hand plot while on the right hand one the simulated ODSs are shown. Undamaged(coloured) and damaged ODS(black) Figure 13 ODS simulations at 613 Hz Zoom of the dotted circle

Undamaged(coloured) and damaged ODS(black) Figure 14 ODS simulations at 655Hz When the excitation frequency is moved close to the 8 th resonance and set at 666.5 Hz the recalculation of the ODSs using equation 8 cannot help to locate the damage as shown in the left hand plot in Figure 15, the right hand plot shows the simulated ODSs of undamaged and damaged model which are not very well overlapped because the cut. Last series of plots, shown in Figure 16, are simulated setting the excitation frequency at 680 Hz between the 8 th and 9 th resonances. Again, simulating the ODSs of healthy and unhealthy models away from their natural mode shapes the use of equation 8 to recalculate the ODSs produces very good results in locating the damage as shown on the right hand plot of Figure 16. Undamaged(coloured) and damaged ODS(black) Figure 15 ODS simulations at 666.5Hz

Undamaged(coloured) and damaged ODS(black) Figure 16 ODS simulations at 680Hz Conclusions We have presented in this paper the idea of using ODS data in the damage identification. By definition of ODS it is a summation of all mode shapes, being impossible to be calculated all, the equation (4) make use of the ones calculated in a predefined frequency range. There fore, the ODS offers a great advantage because our study and, later our measurement test, are not focused on the mode shapes only. We presented a possible way to recalculate the ODSs, undamaged and damaged one, which, finally, provided very interesting results in locating the simulated damage. All ODS simulations are performed noise free and, obviously, this is a great advantage which in the real measurements we cannot have. However, additional studies must be done on this path to asses the maximum level of noise admissible in a real measurement test. References [1] D. Di Maio, C. Zang, D. J. Ewins, Effect of vibration excitation locations on structural damage detection using the CSLDV technique: simulation and testing, AIVELA conference, Ancona (IT), 2006 [2] D. Di Maio, C. Zang, D. J. Ewins, Effect of vibration excitation location on structural damage detection using the CSLDV technique, XXIV IMAC conference, St. Louis 2006 [3] K.Waldron, A. Ghoshal, M.J. Schulz, M.J. Sundaresan, F. Ferguson, P.F. Pai, J.H. Chung, Damage detection using finite element and laser operational deflection shapes, Finite Elements in Analysis and Design 38 (2002) 193-226 [4] D.J.Ewins, Modal testing theory and practice and application, RSP editor [5] A. Khan, A. Stanbridge, J. Kim, D.J. Ewins, Detecting damage in vibrating structures with a scanning LDV, AIVELA conference, Ancona 1998