Damage Detection in Aluminum Cylinders Using Modal Analysis

Transcription

1 Daage Detection in Aluinu Clinders Using Modal Analsis Ivan C. Davis Thesis subitted to the Facult of the Virginia Poltechnic Institute and State Universit in partial fulfillent of the requireents for the degree of Masters of Science in Mechanical Engineering Coittee: Alfred L. Wicks, Chair Stefan Dua Charles Garnett Jul 31, 2002 Blacksburg, VA Kewords: Modal Analsis, Daage Detection, Statistics, Mechanical Engineering Copright 2002, Ivan C. Davis

2 Daage Detection in Aluinu Clinders Using Modal Analsis Ivan C. Davis (ABSTRACT) Man studies have attepted to detect structural daage b eaining differences in the frequenc response functions of a structure before and after daage. In an eperiental setting, this variation can not be attributed solel to the addition of daage. Other sources of variation include testing and structure variation. Eaples of testing variation include the error introduced b odal paraeter etraction, easureent noise, and the ass loading of the acceleroeter. Structure variabilit is due to slight differences in the supposedl identical structures. Diensional tolerancing is one eaple. This stud began with si identical undaaged aluinu clinders, of which three were later daaged to varing etents. The frequenc response functions of the undaaged and daaged clinders were easured. Also, the frequenc response function of the sae undaaged clinder was easured ultiple ties to investigate testing variation. The contributions of testing, clinder, and daage variation to the differences between clinder responses was elucidated b specificall eaining their frequenc response functions in two was: coparing the natural frequencies and directl investigating the entire frequenc response function. The curvature of the frequenc response functions was then used to deterined the presence, location, and severit of the iparted daage. ii

3 Acknowledgent I would like to thank advisor Dr. Alfred Wicks for his support and guidance throughout tie as a Master s student here at Virginia Tech. His upbeat attitude and knack for offering the correct piece of advice at the correct tie were invaluable in keeping e on the track to finishing research in tie. I would also like to thank Dr. Stefan Dua and Mr. Charles Garnett for serving on graduate coittee. Of course o and dad, Ruth Ann and Brad Davis, deserve an enorous deal of credit as well. The stood behind e in all of decisions and never ceased to offer their love and support throughout entire college career. There gifts surpass all that I could have ever epected. And just this once, I will even thank sister for keeping e on toes. I would also like to thank officeate Julian Davis. I a grateful to hi for selflessl offering his assistance for an of the probles that I encountered. His input will not be forgotten. Finall, I a grateful for all of the friendships that I have ade here at Virginia Tech as well as high school. I will alwas reeber the eperiences we shared. iii

4 Contents Abstract ii Acknowledgent iii List of Tables v List of Figures vi Noenclature viii SOURCES OF FRF VARIATION IN DAMAGE DETECTION 1 EFFECTS OF FRF VARIATION SOURCES ON THE NATURAL FREQUENCIES OF CYLINDERS 9 VIABILITY OF FRF CURVATURE FOR VARIOUS LEVELS OF DAMAGE DETECTION 19 Vita 29 iv

5 List of Tables SOURCES OF FRF VARIATION IN DAMAGE DETECTION Table 1: The ean and standard deviation of various coparisons...7 EFFECTS OF FRF VARIATION SOURCES ON THE NATURAL FREQUENCIES OF CYLINDERS Table 1: Linearit constants and correlation coefficients.14 Table 2: Natural frequenc estiates and z ep s for testing variation tests...15 Table 3: Estiates of the undaaged and daaged natural frequencies with 95 percent confidence bounds..16 Table 4: Natural frequenc differences (z ep s) due to testing and clinder variation Table 5: Natural frequenc differences (z ep s) due to testing and daage variation..16 Table 6: Natural frequenc differences (z ep s) due to testing, clinder, and daage variation VIABILITY OF FRF CURVATURE FOR VARIOUS LEVELS OF DAMAGE DETECTION Table 1: Individual curvature differences eans.. 25 Table 2: Mean and standard deviations for curvature differences 26 Table 3: z-values at DOFs near daage...26 v

6 List of Figures SOURCES OF FRF VARIATION IN DAMAGE DETECTION Figure 1: The undaaged test clinder and the three daage cases.. 2 Figure 2: Laout and nubers of the testing points... 3 Figure 3: General test setup 3 Figure 4: T2T coparison of T1 and T2 4 Figure 5: C2C coparison of C1 and C Figure 6: D2U coparison of C4 and D4 (light daage and sae clinder) 5 Figure 7: D2U coparison of C3 and D3 (heav daage and sae clinder)..6 Figure 8: D2U coparison of C3 and D4 (light daage and different clinder)...6 Figure 9: D2U coparison of D4 and D3 (heav daage and different clinder) 6 EFFECTS OF FRF VARIATION SOURCES ON THE NATURAL FREQUENCIES OF CYLINDERS Figure 1: Test clinder..10 Figure 2: Test scheatic showing the clinder hung in the test rig..10 Figure 3: Daage tpes and locations..11 Figure 4: First three odes of driving point FRF.11 Figure 5: Undaaged and daage odes at approiatel 712 Hz Figure 6: A scatter diagra for 3 U1 versus 3 U2 with a regression line..14 Figure 7: Distribution of the natural frequencies for the first ode of U Figure 8: Graphical depiction of Table VIABILITY OF FRF CURVATURE FOR VARIOUS LEVELS OF DAMAGE DETECTION vi

7 Figure 1: Test clinder..20 Figure 2: Test scheatic showing the clinder hung in the test rig..20 Figure 3: Daage tpes and locations..21 Figure 4: Scheatic showing the full population of FRFs for U 3 21 Figure 5: FRF plotted against frequenc and DOF...22 Figure 6: Average curvature differences for the undaaged and daaged clinders..23 Figure 7: Renubered DOFs for the longitudinal direction.23 Figure 8: Individual undaaged and one-hole daaged curvature differences...24 Figure 9: Individual undaaged and two-hole daaged curvature differences Figure 10: Individual undaaged and slot daaged curvature differences.24 Figure 11: Curvature differences for one-hole daage. ( H Figure 12: Curvature differences for two-hole daage. ( H Figure 13: Curvature differences for slot daage. ( H D 3 U 3 D 4 U 4 D 2 U 2 ). 25 ).26 )..26 vii

8 Noenclature SOURCES OF FRF VARIATION IN DAMAGE DETECTION ω FDAC Denotes a specific spectral line Frequenc Doain Assurance Criterion H 1 Frequenc Response Function 1 H 2 Frequenc Response Function 2 N Total nuber of degrees-of-freedo (66) n Denotes a specific degree-of-freedo * Matri transpose EFFECTS OF FRF VARIATION SOURCES ON THE NATURAL FREQUENCIES OF CYLINDERS i f (i) a ε ρ 2 Natural frequenc deviation of ode and clinder (Hz) Natural frequenc deviation of ode and clinder (Hz) Denotes a specific degree-of-freedo Measured natural frequenc of ode at degree-of-freedo i (Hz) Proportionalit constant Residual coponent Correlation coefficient Average natural frequenc for ode and clinder (Hz) s Standard deviation for ode and clinder (Hz) N Total nuber of degrees-of-freedo (66) σ µ H 0 H 1 d 0 Population variance of natural frequenc deviations True natural frequenc ean for ode and clinder (Hz) Null hpothesis Alternative hpothesis Difference in natural frequenc eans viii

9 z ep T 1 -T 2 U 1 -U 6 D 2 -D 4 Test statistic Clinders used to deterine testing variation Undaaged clinders Daaged clinder VIABILITY OF FRF CURVATURE FOR VARIOUS LEVELS OF DAMAGE DETECTION E I ρ M ω i H (,i) Modulus of elasticit Moent of inertia Bea curvature Applied bending oent Denotes a specific spectral line Denotes a specific degree-of-freedo ω Curvature of frequenc response function H ( ω, i) Frequenc response function s Distance between two adjacent degree-of-freedos H ( i) Curvature difference U{ H ( i)} Average undaaged curvature difference D { H ( i)} Average daaged curvature differences for daage case a a a Denotes a specific daage case N Total nuber of degree-of-freedos (66) Average of an individual curvature difference s Standard deviation of an individual curvature difference µ True ean of an individual curvature difference σ z ep z U 1 -U 6 D 2 -D 4 True standard deviation of an individual curvature difference Test statistic z-value Undaaged clinders Daaged clinders i

10 SOURCES OF FRF VARIATION IN DAMAGE DETECTION I.C. Davis and A.L. Wicks Departent of Mechanical Engineering Virginia Poltechnic Institute and State Universit Blacksburg, VA ABSTRACT Modal analsis techniques have been used to detect structural daage for a nuber of ears. Usuall a odal test is perfored on the structure before and after daage has been iparted. The daage is then correlated to the differences in either the FRF or odal responses. However, changes in the responses, be it FRF or odal, due to easureent and clinder variation of the undaaged and daaged clinder are rarel discussed. This stud seeks to assess the variation in FRF responses of clinders due to easureent variabilit, clinder variabilit, and daage variabilit using the Frequenc Doain Assurance Criterion (FDAC). The FDAC is capable of separating the different sources of variabilit in FRFs of clinders before and after daage. INTRODUCTION Nondestructive daage detection has been the subject of uch research over the past few decades. Daage detection can be broken up into four categories. Level I tests for the presence of daage. The presence of daage is tpicall confired onl b witnessing significant changes in the Frequenc Response Functions (FRF) of the structure before and after daage. Level II tests for the presence and location of the daage on the structure. Level II techniques are ore cople than Level I techniques since the change in the FRF or odal propert ust be correlated to a specific spatial position on the clinder. Level III tests for the presence, location, and severit of the daage. For eaple, Level III tests would deterine how deep and long a crack is. Finall level IV tests for the presence, location, and level of the iparted daage as well as predicting the change in phsical properties of the structure due to the daage. For eaple, how will the fatigue life and strength of the structure be reduced as a result of the daage? In this paper, we used Level I techniques. Higher level tests were unnecessar considering the scope of this paper. The application of odal analsis techniques, or ore specificall the change in odal paraeters, is one tool that can be used for daage detection. Researchers have attepted to perfor daage detection using changes in the frequenc response function (FRF) as well as changes in the odal paraeters. ([1] gives a good overview.) The FRF approach deals with variation in the FRFs such as natural frequencies and curvatures, while odal approaches eaine changes in the daping ratios and the ode shapes. In an cases the FRF approach appears to be ore sensitive to daage than the odal approaches. The odal approaches require curve fitting of the FRFs, which tend to introduce a certain aount of uncertaint as 1

11 well as distributing the response across the structure. As a result, daage localization (Level II) using odal approaches is ore difficult. On the other hand, the FRF approach directl uses the easured data. A general research procedure for daage detection has three steps: 1) perfor a odal analsis on the undaaged structure, 2) perfor the sae analsis on a daaged structure, and 3) relate the changes in the undaaged and daaged responses to the iparted daage. One proble with this technique quickl arises. What percent of the changes in the responses arise fro alterations in the testing schee, differences in the structures due to achining, or fro other factors? The purpose of this paper is to investigate soe possible sources of variation between the responses of daaged and undaaged structures. A population of si identical clinders was used for these tests. The clinders had a length of inches, an outside diaeter of 7.125, and a wall thickness of inches. Also, each end was threaded. The clinders were tested to evaluate the aount of test-totest (T2T) variation, clinder-to-clinder (C2C) variation, and daage-to-undaaged (D2U) clinder variation. Previousl, Marwala and Hunt [2] have investigated the feasibilit of identifing daage in a population of clinders. This stud seeks to etend the work done in [2] concerning the sources of variation between daaged and undaaged clinders used for daage detection. Figure 1 shows the test clinder and the three daage cases. Figure 1. The undaaged test clinder and the three daage cases. TEST SETUP AND METHODOLOGY Each clinder test was set-up such that a freefree boundar condition was siulated. The test clinder was hung level in the test rig using tape. Each clinder was set-up in precisel the sae fashion in attept to iniize the effects of huan factors on the response changes. Consequentl, it is believed that changes in the FRFs will not be the result of slight alterations in the test set-up. Instead these changes will be due to inherent properties of the easureent sste, the clinders, and the iparted daage. However the validit of this belief was investigated b testing the T2T variation, which includes huan factors and properties of the data acquisition sste. Each clinder was arked with a testing point atri of 6 rows and 11 equall spaced circuferential points for a total of 66 points. One row was located on the edge of the threads, two aial rows were located on the flanges, and the reaining three rows were located on the span of the clinder. One point located on the thread was peranentl arked 2

12 as the driving point. The driving point is labeled point one and serves as the basis for nubering all other points. Figure 2 shows the laout of the testing points and the point nubers on the clinder. direction of the ipact force. Since this test features a roving ecitation force, ode shape direction would be a serious concern if this clinder was left ass setric. The second approach consisted of a roving acceleroeter and a stationar force applied to the driving point. For this test a inute acceleroeter was used. Since the ipact force stas in the sae direction throughout the duration of the test, ode direction is not a concern here. Therefore, the need for ass asetr is alleviated. B using such a sall acceleroeter, the ass variation is reduced thus generating a ore consistent set of FRFs. Figure 3 shows the general test setup. Figure 2. Laout and nubers of the testing points. The X s correspond to points and each row has 11 points. Onl five points in each row can be shown on the 2-D iage above. Two different approaches were used for testing the clinders. The first consisted of a stationar acceleroeter and a roving ipact force. A large acceleroeter, weighing 0.25 ounces, was located on the clinder s driving point and a load cell equipped haer was used to ecite the clinder at ever test point. An average of four ipacts was used. A heav acceleroeter was used in this test to help ake the clinder ass asetric. This asetr is desired to reduce the repeated roots proble of setric structures. For setric structures, the direction of the ode shape depends on the Figure 3. General test setup. The responses of the various test cases were easured using Siglab and then iported into MEScope, a odal analsis software package. The FRF and the coherence for each easureent was stored. For each test, the sae data input settings were used in Siglab. To reiterate, this paper investigates the aount of test-to-test variation, clinder-to-clinder variation, and daage-to-undaaged clinder variation present in all odal test coparisons. For T2T variation, the entire 3

13 test procedure was perfored five ties on one clinder (T1-T5). For C2C variation, all si undaaged clinders were tested using identical procedures (C1-C6). Finall three clinders were daaged. Two holes were drilled in one clinder (D2), a slot was illed in a different clinder (D3), and a hole was drilled in the third clinder (D4). Each of these daaged clinders was then tested for D2U clinder coparisons. CORRELATION METHOD Alleang and Brown [3] introduced the Modal Assurance Criterion (MAC) to deterine the correlation between two ode vectors. Since that tie, other functions have been developed to serve an different correlative needs ([4] and [5] for eaple). This paper will use a version of the Frequenc Doain Assurance Criterion (FDAC) [5] to deterine the relationship between different tests. In the FDAC introduced in [5], eperiental frequenc data was correlated to analtical (FEA for eaple) frequenc data. In this version of the FDAC two eperiental data sets are copared. More specificall, the FRF values at each degree of freedo (DOF) of one clinder are copared to each DOF of another clinder for each spectral line. The FDAC, as used in this paper, is given b: RESULTS AND DISCUSSION Figure 4 shows a T2T coparison for T1 and T2. The driving point FRF s for both tests are superiposed onto the FDAC displa the position of the odes. As stated previousl, the FDAC is calculated at each spectral line using all 66 DOFs. Since a relativel large aount of DOFs were included, the calculated FDAC is considered a strong estiator of test correlation. Since the sae clinder was tested twice under the sae conditions, T2T coparisons can be considered the baseline scatter for all other tests. The T2T coparison is highl correlated since the FDAC is 0.95 or higher for uch of the frequenc range. As a result, T1 and T2 are highl correlated. All T2T coparisons showed a siilar correlation pattern. FDAC ( ω) = N N * * { H1 ( ω, n) }{ H 2 ( ω, n) } { H 2 ( ω, n) }{ H 1 ( ω, n) } n= 1 N N * * { H1 ( ω, n) }{ H 1 ( ω, n) } { H 2 ( ω, n) }{ H 2 ( ω, n) } n= 1 n= 1 n= 1 where ω is the desired frequenc, H 1 is the first set of FRF data, H 2 is the second set of FRF data, n is a spatial testing point or DOF, N is the total nuber of spatial testing points or DOFs, and * is the cople conjugate. Like the MAC, the FDAC shows perfect correlation when equal to 1 and shows perfect non-correlation when equal to 0. Figure 4. T2T coparison of T1 and T2. The FDAC of Figure 4 displas ore scatter in the upper frequenc ranges. This is epected because the higher frequencies are ore sensitive to changes in the structure, therefore these frequencies will show ore scatter. 4

14 Figure 5 shows CSC coparisons for C1 and C2. As epected, the C2C coparisons show a significant increase in MAC differences copared to the T2T coparisons, especiall in the upper frequenc ranges. The C2C variation is due to structural variations such as clinder achining variations in the clinder due to tolerances. in the upper frequenc region. For instance, there is a natural frequenc shift of alost 14 Hz for the ode at 1820 Hz in the D2U coparison. Clearl, if the inflicted daage is onl slight, then the difference in responses for D2U coparisons is overshadowed b the difference in responses of C2C coparisons. Figure 5. C2C coparison of C1 and C2. Figure 6 shows a D2U coparison for C4 and D4 (clinder 4 before and after daage). The D2U coparison of Figure 6 sees to show less FDAC scatter than the C2C coparison of Figure 5. This observation is precisel opposite to the one we epected. We thought that the difference in the daaged and undaaged clinder responses would be larger than the difference between two undaaged clinder responses. The daage iparted to clinder 4 was sall, onl a hole. Perhaps the daage was not severe enough to affect a significant change in the responses. Figure 7 shows the D2U coparison of C3 and D3 (Clinder 3 was daaged b adding a slot). Coparing Figures 7 and 5, the D2U coparison shows greater scatter then the C2C coparison. Again, the changes are especiall pronounced Figure 6. D2U coparison of C4 and D4 (light daage and sae clinder). Both of the D2U coparison cases described so far have one attribute in coon: the sae clinder was tested before and after daage. In an situations the undaaged response of a structure will not be known a priori. Therefore, we have also perfored D2U coparisons of an undaaged clinder and a different daaged clinder. Two eaple D2U tests of this tpe are shown in Figures 8 and 9. Figure 8 shows a case where slight daage was iparted (single hole) and Figure 9 shows a case where ore severe daage (slot) was iparted. The D2U coparisons of Figures 8 and 9 (different clinders) displa greater aounts of FDAC variation than the undaaged C2C coparison tests (Figure 5). However, different clinder D2U coparisons have two 5

15 sources of FDAC variation: 1) variation due to different clinders, and 2) variation due to daage. Coparing Figure 4 to Figures 5, 6, and 7, it is clear that the T2T variation coponent is inor copared to the C2C and D2U coponents. Therefore, the T2T variation can be considered as negligible. Figure 9. D2U coparison of C4 and D3 (heav daage and different clinder). Figure 7. D2U coparison of C3 and D3 (heav daage and sae clinder). In Table 1, we have attepted to quantif the sources of variation. In coluns 2 and 3 of Table 1, the ean and standard deviation of the FDAC for each coparison case is shown. The ean is an overall indicator of the coparison s correlation. Of course, as the ean approaches one, the coparison approaches perfect correlation. The standard deviation is a global indicator of the FDAC s scatter. For the ost part, the nubers of Table 1 follow the results taken fro Figures 4-9. However, there is one interesting result that should be pointed out. The ean and standard deviation of the C3 vs. D3 coparison are less then the ean and standard deviation of the C4 vs. D3 coparison. We epected the D2U test of different clinders to show less overall correlation and ore scatter than a D2U test of the sae clinder. Figure 8. D2U coparison of C3 and D4 (light daage and different clinder). One possible eplanation is that once the iparted daage is severe enough, the variation in the FRFs due to D2U coparison copletel eclipses the variation due to the clinder difference. Therefore, an variation 6

16 introduced b C2C coparison can not be easured and separated. Perhaps this propert can be utilized in deterination of severit of daage using FRF s. If the clinder s daaged response variation reaches a certain threshold where C2C coparisons are not separable, then the daage iparted is severe. If the C2C variation can be separated fro the overall variation, then the iparted daage is lighter. Further research will be needed to deterine if FRF variation aount is a revealing indicator of structural daage severit. CONCLUSION In this paper we have attepted to deterine the sources of FRF variation for daage detection. A population of clinders was used for the testing. Test-to-test variation, clinder-to-clinder variation, and various trials of daage-to-undaaged variation were easured and copared. T2T variation was found to be negligible when copared to C2C and D2U variation. C2C variation was found to be ore significant then D2U variation if the iparted daage was light. If the iparted daage was heav, D2U variation was found to overshadow C2C variation. One possible easure of daage severit was also introduced. It would be wise to stress that these results are onl true for the population of clinders we tested. If the sources of variation in a different structure need to be quantified, then a siilar testing ethodolog should be perfored for that structure. ACKNOWLEDMENT We would like to thank Charles Garnett and Donna Reedal at NSWC Dahlgren for all of their support in this stud. Table 1. The ean and standard deviation of various coparisons. Variation Source FDAC Mean FDAC Std T2T (T1 vs. T2) C2C (C1 vs. C2) D2U - sae clinder, light daage (C4 vs. D4) D2U - sae clinder, heav daage (C3 vs. D3) D2U - different clinder, light daage (C3 vs. D4) D2U - different clinder, heav daage (C4 vs. D3) REFERENCES [1] Doebling S.W., Farrar C.R., Prie M.B., and Shevitz D.W. Daage identification and health onitoring of structural and echanical sstes fro changes in their vibration characteristics: a literature review. Los Alaos National Laborator Report, LA MS, [2] Marwala T., and Hunt H.E.M. Feasibilit of daage detection using vibration data in a population of clinders. Proceedings of the 18 th International Modal Analsis Conference, pp , [3] Alleang R.J., and Brown D.L. A correlation coefficient for odal vector analsis. Proceedings of the 1 st International Modal Analsis Conference, pp ,

17 [4] Lieven N.A.J., and Ewins D.J. Spatial correlation of ode shapes, the coordinate odal assurance criterion (COMAC). Proceedings of 6 th International Modal Analsis Conference, pp , [5] Avitabile P., and Pechinsk F. Coordinate Orthogonalit Check. Proceedings of the 12 th International Modal Analsis Conference, pp , [6] Fotsch D., and Ewins D.J. Application of the MAC in the frequenc doain. Proceedings of the 18 th International Modal Analsis Conference, pp ,

18 EFFECTS OF FRF VARIATION SOURCES ON THE NATURAL FREQUENCIES OF CYLINDERS I.C. Davis and A.L. Wicks Departent of Mechanical Engineering Virginia Poltechnic Institute and State Universit Blacksburg, VA ABSTRACT Man studies have attepted to detect structural daage using differences in the preand post-daage odal responses. In the eperiental real, these differences can not be attributed solel to the presence of daage. Testing and structural variation will also contribute. Testing variation includes the ass loading of the acceleroeter and error due to odal paraeter etraction. An eaple of structural variation is diensional tolerancing. Since natural frequencies are affected b daage, the effect of these alternative variation sources on the natural frequencies of aluinu clinders will be investigated. The clinders will then be daaged. The effects of daage variation will then be copared to the effects of testing and clinder variation. INTRODUCTION Structural daage detection (DD) using odal analsis offers an benefits. For eaple, it is a non-destructive technique and usuall the structure can be tested in situ. Man odal-related DD schees copare the odal response of a structure before daage to the response after daage. Modal responses include frequenc response functions (FRFs) and odal paraeters such as odeshapes, natural frequencies, or daping ratios. The difference between these two responses is then soehow correlated to daage ([1-4] for eaple). One drawback to this approach is that the differences between the pre- and postdaage responses can not be attributed solel to the addition of daage if the responses were found eperientall. Other sources of response variation are tered inherent variation and include testing and structure variation. The purpose of this paper is to investigate the sources and effects of this inherent variation. Also a previous stud [5] concerning this aterial will be etended. Test variation is coposed of test setup and analsis procedure variation. An acceleroeter is one source of test variation because the ass of the acceleroeter disrupts the ass distribution of the structure. Measureent noise is also considered test setup variation. The ost proinent eaple of analsis procedure variation is odal paraeter etraction. On the other hand, structure variation is caused b slight differences in supposedl identical structures. Diensional tolerancing is one eaple. Inherent variation can also be broken down into ssteatic and rando variation. Ssteatic variation acts on structures in a deterinistic wa. The effects of ssteatic 9

19 variation on one structure are functionall related to the effects of ssteatic variation on another identical structure. Sources of ssteatic variation include the ass-loading of an acceleroeter and slight structural differences. Sources of rando variation include easureent noise and errors introduced b odal paraeter etraction. It is well known that the addition of daage alters a structure s natural frequencies [7]. Therefore we will investigate how the natural frequencies of aluinu clinders are affected b three sources of inherent variation: a roving acceleroeter, structure differences, and odal paraeter etraction. First, we will deterine how inherent variabilit odifies natural frequencies of the clinders on a DOF to DOF basis. Fro this we can ascertain the nature of the inherent variabilit. For instance, the relative contributions of each source to the overall inherent variabilit can be deterined. Second, we will investigate how the inherent variabilit affects natural frequencies on a clinder to clinder basis. Each clinder has 66 DOFs, so a good estiate of the true natural frequencies of a clinder can be calculated b averaging the easured natural frequencies over the entire set of DOFs. The statistical significance of the natural frequenc differences due to onl testing variation and to the su of testing and clinder variation (testing variation is unavoidable) will be tested using standard statistical inference techniques. Third, the daage variabilit will be copared to the inherent variabilit. Three of the clinders were daaged to different etents and estiates of the daaged natural frequencies were deterined. Those natural frequencies will then be copared to natural frequencies altered b inherent variation using the sae statistical techniques. PROCEDURE Each clinder was arked with a testing point atri consisting of 6 rings of 11 equall spaced circuferential points for a total of 66 DOFs (Figure 1). The clinders were then hung with tape in a test rig to siulate free-free boundar conditions (Figure 2). The clinders were ecited using an ipact fro a load-cell equipped haer, and the dnaic response of the clinders was easured with an acceleroeter. All signals were captured using Siglab. Figure 1. Test clinder. DOFs are labeled with an and soe are nubered. Figure 2. Test scheatic showing the clinder hung in the test rig. Since the clinders are both stiffness and ass aisetric, odeshape orientation will be dependent on forcing direction. We decided to use a roving acceleroeter to itigate the effects of this 10

20 dependenc. Since the forcing function has a constant direction, the odes will have constant orientations. The drawback to this approach is that each tie the acceleroeter is oved, the ass distribution of the clinder is altered, which will increase testing variation. Further raifications of an aisetric structure will be discussed in an upcoing section. To get an idea of testing variation, the FRFs of one undaaged clinder was deterined twice, with the tests being labeled T 1 and T 2. Net, the FRFs of the si undaaged clinders were deterined using identical test conditions and were labeled U 1 - U 6. Three of the clinders were then daaged: U 2 b adding two holes, U 3 b adding a slot, and U 4 b adding a single hole (Figure 3), and were labeled D 2, D 3, and D 4, respectivel. Again using the sae procedure, the FRFs of the daaged clinders were easured. Figure 3. Daage tpes and locations. Once the coplete set of FRFs was deterined for all clinder tests, the first three natural frequencies of their driving point FRFs (Figure 4) were deterined using rational fraction polnoial (RFP) curve fitting. The first three natural frequencies were selected for three reasons. First, these odes are well separated fro each other, and thus are free fro ode coupling. Second, these odes displa inial ode bifurcation after the addition of daage. Mode bifurcation is discussed later in this report. Third, noise containates lower frequenc odes to a lesser etent than high frequenc odes. The RFP approach requires the user to specif a frequenc band to curve-fit. To eliinate a possible source of variabilit, identical frequenc bands were specified for each natural frequenc. Figure 4. First three odes of driving point FRF.. Net the average and standard deviation were calculated for each of the three natural frequencies of each clinder. As stated previousl, test variation can be discerned b investigating the differences in natural frequencies between T 1 -T 2. Clinder variation will be elucidated b coparing the natural frequencies of U 1 to the natural frequencies of U 2 through U 6. Daage variation was deterined b coparing the natural frequencies of the sae clinder before and after daage (i.e. U 2 and D 2 ). Finall the cobined effect of all three sources of variabilit was investigated b coparing a daaged clinder with an unrelated undaaged clinder (i.e. U 2 versus D 3 ). 11

21 NATURE OF VARIATION THEORY This section will describe how the inherent variabilit will be investigated on a DOF b DOF basis. First, we will deterine the deviation of the natural frequencies easured at each DOF fro the ean natural frequenc of the clinder: ( i) = f ( i), (1) where is a specific clinder test (U1 U6, etc), is the ode nuber (1, 2, or 3), f (i) refers to the easured natural frequenc at DOF i for clinder, and is the average natural frequenc for ode for clinder. The deviations of one clinder s natural frequencies will have soe degree of linear relationship with another clinder s: ( i) = a[ ( i)] + ε. (2) where and are different clinder tests, a is the proportionalit constant (slope) of the linear relationship and is deterined using a least squares approach, and ε is the rando coponent which accounts for easureent noise and error introduced b odal paraeter etraction. The linear and rando coponents of Equation 2 describe ssteatic and rando variation, respectivel. The assuption of a linear relationship between deviations can be ade because the clinders are linear structures. The natural frequencies are deterined b solving an eigenvalue proble consisting of the ass and stiffness atrices of the clinders. The roving acceleroeter will predoinatel affect the ass atri. Although the ass atrices of different clinders will be different due to clinder variation, the effects of the acceleroeter on the ass atri of different clinders will be identical. Since the clinders are linear, the change in response due to this perturbed ass atri will be repeated in linear fashion fro clinder to clinder. The size of the proportionalit constant a is a easure of the difference in the effect of acceleroeter on different clinders. For instance, assue the wall thicknesses of clinders 1 and 2 are and 0.620, respectivel. The roving acceleroeter a cause larger natural frequenc deviations for clinder 2, since the wall is thinner. In the case of structural variation between two clinders, the ass and stiffness atrices of the clinders will be different. Since the two ass atrices will be perturbed b the acceleroeter in the sae fashion, the solutions of the eigenvalue proble will still be linear. Onl the clinders will respond with different agnitudes to the roving acceleroeter. In other words, a is a easure of clinder variabilit. The aount of linearit between the natural frequenc deviations is indicated b the correlation coefficient ρ 2 : 2 ( ) ( )( ) 2 ρ =. (3) If ρ 2 is one, the natural frequenc deviations are perfectl correlated, or linearl related with zero rando content. If ρ 2 is zero, the relationship between the deviations is entirel rando. EFFECTS OF VARIATION THEORY This section will discuss how the eans of different natural frequencies will be copared. For a particular ode, each clinder will have a saple of N natural frequencies (one at each DOF). The averages and standard deviations of these saples are deterined using 12

22 s = = 1 N N i= 1 N i= 1 f ( i), (4) [ f ( i) ] N 2 (5). hpothesis can not be proven true. Rather, the null hpothesis is accepted when the data gives insufficient reason to refute it. On the other hand if the null hpothesis is rejected, strong support is given to the alternative hpothesis. We would like to provide strong evidence that certain natural frequencies are statisticall different, hence the order of the hpotheses. If the population variance σ 2 of the natural frequencies was known, a (1-α)100 percent confidence interval for the true natural frequenc µ of the clinder would be given b σ σ zα / 2 < µ < + zα / 2. (6) N N The true natural frequenc will fall between the upper and lower bounds of the confidence interval with a confidence of (1-α)100 percent. Equation 6 is valid when the rando saple is norall distributed, and gives a good approiation when N 30. Also when N 30, the saple standard deviation s a be substituted for σ. In this stud α is equal to 5 percent and N = 66, giving the confidence interval as s s 1.96 < µ < (7) The natural frequencies eans of each clinder will be copared using standard statistical hpothesis testing. First a null hpothesis H 0 and an alternative hpothesis H 1 are ade: H 0 : µ µ = d = 0, (8) o H 1 : µ 0. (9) µ The order of these hpotheses is ver iportant. Using this procedure, the null A point estiator of µ µ is the difference in the saple eans. The sapling distribution of is approiatel norall distributed with ean µ = µ and standard deviation µ 2 2 ( σ / n ) ( σ / n ) σ = +. With n = 66 we can again substitute s for σ, and base the decision on the equivalence of the natural frequencies using the test statistic z ep : z ep ( ) d 0 =. 2 2 (10) s / 66+ s / 66 Since this is a two-tailed test, H 0 will be rejected (H 1 accepted) with 95 percent confidence when z ep > z α/2 or if z ep < -z α/2. Again α, or the level of significance, equals 5 percent. The level of significance is the probabilit of rejecting H 0 when H 0 is true. For α equal to 5 percent, z α/2 is equal to ±1.96. It should be noted that all statistics were referenced fro [6]. MODE BIFURCATION Recall that the first three natural frequencies of the driving point FRFs are used in this stud. The third natural frequenc displas soe unusual behavior (Figure 5). The undaaged ode has separated into two daaged odes. Before the addition of daage, the clinders were aisetric. Since the stiffness and ass of the clinder 13

23 are setric in each direction perpendicular to the centerline, odes with siilar odeshape will be located at the sae natural frequenc. This condition is anifested analticall with repeated eigenvalues, and is referred to as a repeated roots solution. Mode shape orientation will be dependent on forcing direction. Fro the figure, it is iediatel clear that rando content is doinating the coparison between 3 U1 and 3 U2 ρ 2 is onl Figure 5. Undaaged and daaged odes at approiatel 712 Hz. When daage is added, the stiffness and ass setr is destroed. The eigenvalues becoe distinct resulting in different natural frequencies, a situation tered ode bifurcation. The bifurcated odes will still have the sae general odeshape but with different orientations (i.e. still displa 3 lobes but 55º circuferentiall out of phase). In this stud, the third and fourth daaged odes (the bifurcated odes) will alwas be copared to the third undaaged ode. NATURE OF VARIATION Recall that first the natural frequencies were copared on a DOF b DOF basis. A scatter diagra with 3 U1 plotted on the abscissa and 3 U2 plotted on the ordinate is shown in Figure 6 with the regression line. Figure 6. A scatter diagra for 3 U1 versus 3 U2 with a regression line.. The proportionalit constant a and correlation coefficient ρ 2 were calculated for coparisons of U 1 to each of the other undaaged clinders (U 2 -U 6 ), and the results are shown in Table 1. The values of a and ρ 2 are ver sall for the first ode. As the order of the odes increase, both a and ρ 2 increase in value. Table 1. Linearit constants and correlation coefficients. Mode 1 Mode 2 Mode 3 a ρ 2 a ρ 2 a ρ 2 U 1 vs. U U 1 vs. U U 1 vs. U U 1 vs. U U 1 vs. U Minute values of a are characteristic of the first ode, which could initiall be construed as prodigious clinder variation. However, the validit of a, as well as the linear assuption, ust be questioned because of the eceedingl sall value of ρ 2. Equation 14

24 2 is doinated b the rando coponent for ode 1, and ode 2 for that atter, aking the value of a eaningless. The correlation coefficients for ode 3 increase draaticall. The linear coponent of these deviations is becoing ore evident. We epect a and ρ 2 to both approach one as the ode nuber increases. When the frequenc of a ode increases, the corresponding odeshape s wavelength decreases while the odeshape s frequenc increases. As a result, DOFs on the structure oscillate with a higher velocit aking the inertial loading of the acceleroeter ore significant. In other words, the effective ass of the acceleroeter will increase. In turn, natural frequenc deviations will becoe ore linearl related. When the relationship between the deviations approaches coplete linearit (ρ 2 > 0.95), clinder variation will be accuratel described b the value of a. In conclusion, the deviation variabilit for the first three odes is doinated b rando variabilit for these three odes. Modal paraeter etraction, and to a uch lesser etent easureent noise, is ore responsible for natural frequenc deviations than variation due to clinder differences or the acceleroeter. EFFECTS OF VARIATION In Figure 7 a histogra of the natural frequencies for the first ode of U 3 is shown. A standard noral distribution provides a good fit for the natural frequenc distribution with 95 percent confidence using the χ 2 goodness-of-fit criterion. A noral distribution also odels the other natural frequenc saple distributions in this stud. Figure 7. Distribution of the natural frequencies for the first ode of U 3. Estiates for the natural frequencies of T 1 and T 2 with 95 percent confidence bounds are shown in Table 2 with the corresponding statistical coparisons. With 95 percent confidence, there is insufficient evidence to refute the null hpothesis that natural frequenc eans subject onl to test variation are statisticall different. (The null hpothesis is rejected if z ep < or if z ep > 1.96.) Therefore it can be concluded that testing variation will not appreciabl alter the natural frequencies. Table 2. Natural frequenc estiates and z ep s for testing variation tests. T 1 (Hz) T 2 (Hz) Mode 1 Mode 2 Mode ± ± ± ± ± ± z ep In Table 3, estiates of the true natural frequencies for the undaaged and daaged clinders are displaed. Note that there are three natural frequencies for the undaaged clinders, but four for the daaged clinders due to ode bifurcation. 15

25 Table 3. Estiates of the undaaged and daaged natural frequencies with 95 percent confidence bounds. Mode 1 (Hz) Mode 2 (Hz) Mode 3 (Hz) Mode 4 (Hz) U ± ± ± N/A U ± ± ± N/A U ± ± ± N/A U ± ± ± N/A U ± ± ± N/A U ± ± ± N/A D ± ± ± ± D ± ± ± ± D ± ± ± ± In Table 4, the differences in natural frequenc eans due to clinder and testing variation are shown. Each of the coparisons are statisticall different with 95 percent confidence. Fro a coparison of Tables 2 and 4, it is clear that clinder variation is ore significant than testing variation. This soewhat contradicts the conclusion of the previous section which found that natural frequenc deviation was doinated b rando (i.e. testing) variation. Clinder variation also causes a shift in the ean natural frequenc which is not accounted for b natural frequenc deviation. Therefore on a clinder to clinder basis, clinder variation affects a larger change in natural frequencies than testing variation. Table 4. Natural frequenc differences (z ep s) due to testing and clinder variation Mode 1 Mode 2 Mode 3 U 1 vs. U U 1 vs. U U 1 vs. U U 1 vs. U U 1 vs. U Natural frequenc differences (z ep s) due to the cobined effects of daage and testing variation are shown in Table 5. Clinder variation results are included for coparisons. With two eceptions, the natural frequencies affected b daage are different with 95 percent confidence. For the first two odes, the effects of clinder variabilit on natural frequencies is greater than the effect of daage variabilit. Since the natural frequencies of the first two odes are ore sensitive to clinder variabilit, these odes are useful as DD indicators. Table 5. Natural frequenc differences (z ep s) due to testing and daage variation. Mode 1 Mode 2 Mode 3 Mode 4 U 2 vs. D U 3 vs. D U 4 vs. D U 2 vs. U N/A U 3 vs. U N/A U 4 vs. U N/A The effect of daage variabilit begins to eceed the effect of clinder variabilit for the third ode. This can be eplained b considering the general properties of ode shapes. Low frequenc odes have long odal wavelengths which are not affected b sall localized daage patterns. As the 16

26 frequenc increases, the wavelength of the odes decrease, allowing daage to significantl interact with the ode, and thus its natural frequenc. In turn the odeshape and the natural frequenc will be altered. As a result daage variabilit will becoe significant for higher order odes. On the other hand, clinder variation tends to be global in nature, and thus all odes are affected b it. In practice, an eperienter will usuall not have pre-daage natural frequenc inforation about a specific clinder being tested for daage. Rather predaage inforation will coe fro a natural frequenc standard deterined fro a replica clinder. However, we have shown that clinder variation affects natural frequencies, especiall in the low frequenc regions. Natural frequenc differences due to testing, clinder, and daage variation are tabulated in Table 6, and shown graphicall in Figure 8. Again natural frequenc differences due to clinder variation are shown for coparison. Table 6. Natural frequenc differences (z ep s) due to testing, clinder, and daage variation. Mode 1 Mode 2 Mode 3 Mode 4 U 1 vs. D U 1 vs. D U 1 vs. D U 1 vs. U U 1 vs. U U 1 vs. U For the first two odes, natural frequencies are being altered alost eclusivel b clinder variation. The z ep s for the natural frequenc differences including daage are ver siilar to z ep s which do not include daage. One interesting result is that in soe cases, the z ep s actuall decrease with the addition of daage variabilit. The addition of daage has actuall reduced the difference in natural frequenc eans. For odes 3 and 4 (bifurcated odes) daage variation is becoing ore evident. Figure 8. Graphical depiction of Table 6. RESULTS Natural frequenc deviations (the difference between the natural frequenc easured at a DOF and the ean natural frequenc of the clinder) are doinated b error introduced b odal paraeter etraction. Clinder variabilit and roving acceleroeter variabilit are barel observed. For the third ode, the effects of the roving acceleroeter becoe evident. For the first two odes, clinder variation affects the ean natural frequencies of clinders ore than daage variation. Daage variation becoes to overtake clinder variation for the third ode. CONCLUSION The purpose of this stud was to investigate how inherent variabilit (coposed of testing and clinder variabilit) and daage variabilit affect the natural frequencies of aluinu clinders. First, we 17

27 deterined the natural frequencies of undaaged and daaged clinders. The variation between the natural frequencies of these clinders was then investigated on a DOF b DOF basis to eplore the nature of inherent variation. The error introduced b curve-fitting was found to doinate these results. Net we investigated how the ean natural frequencies of the clinders were affected b inherent and daage variabilit. The effect of testing variation was found to be inial, while clinder and daage variation pla a uch larger role B using higher order odes, the relative aount of variabilit caused b daage will increase. Therefore daage detection using natural frequencies would be ore feasible using higher order odes. ACKNOWLEDGEMENT We would like to thank Charles Garnett and Donna Reedal at the Naval Surface Warfare Center at Dahlgren for their support throughout this project. Proceedings of the 13 th International Modal Analsis Conference, pp , [4] Marwala T., and Hunt H.E.M. Feasibilit of daage detection using vibration data in a population of clinders. Proceedings of the 18 th International Modal Analsis Conference, pp , [5] Davis I.C. and Wicks A.L. Sources of FRF variation for daage detection. Proceedings of the 20 th International Modal Analsis Conferences, [6] Walpole R.E. and Mers R.H. Probabilit and Statistics for Engineers and Scientists, Macillan Publishing Copan, New York, [7] Willias E.J., Messina A., and Pane B.S. A frequenc-change correlation approach to daage detection. Proceedings of the 15 International Modal Analsis Conference, pp , REFERENCES [1] Willias E.J., Messina A., and Pane B.S. A frequenc-change correlation approach to daage detection. Proceedings of the 15 th International Modal Analsis Conference, pp , [2] Maia N.M.M, Silva J.M.M, and Sapaio R.P.C. Localization of daage using curvature of the frequenc-responsefunctions. Proceedings of the 15 th International Modal Analsis Conference, pp , [3] Stubbs N., Ki J., and Farrar C.R. Field verification of nondestructive daage localization and severit estiation algorith. 18

28 VIABILITY OF FRF CURVATURE FOR VARIOUS LEVELS OF DAMAGE DETECTION I.C. Davis and A.L. Wicks Departent of Mechanical Engineering Virginia Poltechnic Institute and State Universit Blacksburg, VA ABSTRACT Man studies have attepted to detect structural daage using changes in pre- and post-daage odeshapes. In general, the lower odes tend to be insensitive to daage, but it has been shown that daage can appreciabl affect odeshape curvature. One proble with using odeshape curvature is that the odeshapes ust be etracted fro the frequenc response functions (FRFs) which will introduce error. This error could easil ask the inute changes induced b the daage. Therefore, this paper will investigate whether FRF curvatures in the longitudinal and circuferential directions of aluinu clinders are sensitive to daage. This stud will use si undaaged clinders, three of which will be daaged to varing degrees. We will then deterine whether FRF curvatures can resolve the presence, location, and severit of daage using siple statistical procedures. INTRODUCTION Structural daage detection (DD) using odal analsis offers an benefits. For eaple, it is a non-destructive technique and usuall the structure can be tested in situ. There are four categories of DD using a odal approach. Tests for the presence of daage are tered Level I and are tpicall perfored b investigating the changes in the frequenc response functions (FRF) of the structure before and after daage. Tests for Level II DD ai to locate daage. Changes in FRF or odal paraeters ust be correlated to spatial positions on the structure. Deterination of the severit of daage is tered Level III DD. For eaple, Level III tests would deterine how deep and long a crack is. Level IV DD investigates how the properties of the structure are altered due to the daage. For eaple, Level IV DD elucidates how the fatigue life of a structure is reduced due to a crack. We are concerned with Level I, II, and III DD in this stud. Man studies have attepted DD b eaining the differences in the structure s pre- and post-daage odeshapes. In general, odeshapes are fairl insensitive to daage, especiall in the lower frequenc regions where FRF easureent is often applied. Pande et al [1] noted that the odeshape curvature of a bea, ρ, a be ore sensitive to daage according to Equation 1. The curvature of a bea is dependent on the bea s odulus of elasticit, E, and oent of inertia, I, both of which will be odified b the addition of daage. 1 = ρ M EI (1) 19

29 One drawback of a curvature approach using odeshapes is that the odeshapes ust be etracted fro the FRFs which will introduce error. Changes caused b daage are subtle and can easil be asked b this additional error. Therefore, Maia et al [2,3] investigated DD using the curvatures of FRFs. The benefits of such an approach are twofold. First, no errors are introduced b odal identification, since the eperiental data is used directl. Second, FRFs contain ore independent statistical degrees-of-freedo (DOFs) than odeshapes. With an FRF approach, an entire frequenc range of inforation is associated with each DOF, while odeshapes onl reveal odal displaceents at each DOF. Daage affects the odeshape curvature of a bea b directl altering structural properties that define the curvature. The benefit of using FRF curvatures is that derivatives are ver sensitive to changes in a function. The addition of daage will onl induce sall changes in the FRFs, which a not be recognized b direct investigation of FRFs. However, these sall changes will be aplified in the second derivative (curvature) of the FRF. Maia et al investigated the FRF curvatures nuericall [2] and eperientall [3] of a one-diensional bea. In this stud, the FRF curvatures of si undaaged aluinu clinders will be calculated in two directions: longitudinall and circuferentiall. Three of these clinders will then be daaged to different etents, and the daaged FRF curvatures will be calculated in both directions. The viabilit of Level I, II, and III DD using these FRF curvatures will be eained using siple statistical procedures. PROCEDURE Each clinder was arked with a testing point atri consisting of 6 rings of 11 equall spaced circuferential points for a total of 66 DOFs (Figure 1). The clinders were then hung with tape in a test rig to siulate free-free boundar conditions (Figure 2). The clinders were ecited using an ipact fro a load-cell equipped haer, and the dnaic response of the clinders was easured with an acceleroeter. All signals were captured using Siglab. Figure 1. Test clinder. DOFs are labeled with an and soe are nubered. Figure 2. Test scheatic showing the clinder hung in the test rig. Since the clinders are both stiffness and ass aisetric, odeshape orientation will be dependent on forcing direction. We decided to use a roving 20

30 acceleroeter to itigate the effects of this dependenc. Since the forcing function has a constant direction, the odes will have constant orientations. The FRFs of the si undaaged clinders were deterined using identical testing conditions, and labeled U 1 -U 6. Afterwards, three of the clinders were daaged: U 2 b adding two holes, U 3 b adding a slot, and U 4 b adding a single hole (Figure 3), and were labeled D 2, D 3, and D 4, respectivel. Again using the sae procedure, the FRFs of the daaged clinders were found. Note that the locations of daage with respect to the DOFs are shown in Figure 3. Once the entire set of FRFs was obtained, their curvatures were deterined using the procedure described in the net section. Siilarl, 1 2 ( ) = f ( ) f '( ) + f ' ( )( ) + L f 0 (3) 2! 0 0 ' Adding Equations 2 and 3, approiating the Talor epansion to the second derivative, and rearranging, we obtain the central difference approiation for a general function f: f '' ( ) 0 f ( 0 + ) 2 f ( 0 ) ( ) 2 + f ( ) 0. (4) Figure 4 shows a siplified diagra of the entire set of FRFs obtained for U 3. FRF curvatures can be deterined in two directions: along the DOF ais or the frequenc ais. Slices of Figure 4 in both directions are shown in Figure 5. In this stud, the curvature will be calculated along the DOF ais (i.e. at a constant frequenc) to eaine the possibilit of daage localization. Figure 3. Daage tpes and locations. FRF CURVATURE DETERMINATION The FRF curvatures will be approiated using the central difference approiation, which can be derived fro a Talor series epansion. This derivation, starting fro the Talor series epansion, is given below Figure 4. Scheatic showing the full population of FRFs for U ( + ) = f ( ) + f '( ) + f ' ( )( ) + L f 0 (2) 2! 0 0 ' 21

31 where and refer to specific clinder tests (U 1, U 2,, D 2, ). In this stud, the curvature differences will alwas be sued over the entire eperiental frequenc range (0 to 2 khz). Figure 5. FRF plotted against frequenc and DOF. FRFs will be represented b H(ω,i) where ω refers to a spectral line and i refers to a DOF. Substituting H for f in Equation 4, and calculating the derivative along the DOF ais of Figure 4 gives the central difference approiation as H ( ω, i+ 1) 2H ( ω, i) + H( ω, i 1) (, i) = 2 H ω, (5) s where i is the DOF, H (ω,i) is the value of the FRF labeled at DOF i and spectral line ω, H (ω,i) is the second derivative of H (ω,i), and s is the phsical distance between the DOFs. For the longitudinal direction, s was deterined b easuring the longitudinal distance between adjacent DOFs (see Figure 1). For the circuferential direction, s was deterined b finding the circuferential distance between adjacent DOFs (i.e. around the clinder, not straight line distance). This for of the central difference approiation is siilar to the one defined in [2]. The siple difference between FRF curvatures is given b ω H ( i) = H ( ω, i) H ( ω, i), (6) As stated previousl the curvatures will be coputed in both the circuferential and longitudinal directions of the clinder. In the circuferential direction, si sets of curvature data will be deterined corresponding to the si different rings of testing points (Figure 1). In the longitudinal direction, the curvature will be calculated along the length of the clinder. Eleven sets of data will be deterined corresponding to eleven coluns of testing points. Note that longitudinal curvature can not be calculated at DOFs located on the top or botto ring of the clinder since these DOFs are not flanked on both sides b other DOFs. LEVEL I DAMAGE DETECTION In this section we will deterine if FRF curvature is sensitive to the presence of daage. First, the average aount of curvature difference between undaaged clinders was calculated: U{ H ( i)} = 5 6 e= 1 f = e+ 1 ω H U f ( ω, i) H ( ω, i) 15 U e. (7) Basicall, Equation 7 sus the perutation of non-repeated undaaged curvature differences and divides b the total nuber of these differences. Secondl, the average difference between the curvatures of each undaaged clinder and each daaged clinder was deterined: 22

32 D { H a ( i)} = 6 f = 1 ω H D a ( ω, i) H ( ω, i) 6 U f, (8) where a is 2, 3, or 4 corresponding to the clinder with two-hole, slot, or one-hole daage patterns, respectivel. Equation 8 was calculated for each daage level. The values calculated b Equation 7 are average curvature differences which include testing and clinder variation. The values calculated b Equation 8 include testing, clinder, and daage variation. We epect daage variabilit to be ore significant than clinder variation, so the values calculated b Equation 8 should be larger than those calculated b Equation 7. The average undaaged and daaged curvature differences were calculated in both the circuferential and longitudinal direction. The average difference in the undaaged curvatures was then subtracted fro each of the average differences calculated b Equations 7 and 8 to siplif the results (Figure 6). Thus the average undaaged difference appears as a line at zero in Figure 6. the clinder DOFs shown in Figure 1, while the circuferential DOFs do not correspond to Figure 1. The DOFs were renubered for the longitudinal case to siplif FRF curvature calculation in this direction. The renubered DOFs for the longitudinal direction are shown in Figure 7. Also note the apparent periodicit displaed for the longitudinal direction of Figure 6. This is the result of assigning the DOFs at the top and botto of the clinder a longitudinal curvature of zero, as described previousl. Figure 7. Renubered DOFs for the longitudinal direction. According to Figure 6, there sees to be differences between the average undaaged and daaged curvature differences. This difference due to daage ust be greater than the differences between individual undaaged clinders for Level I DD to be viable. B averaging the undaaged curvature differences, knowledge of individual undaaged curvature differences (as calculated b Equation 6) is lost. Therefore individual undaaged and daaged differences will be investigated graphicall in Figures 8, 9, and 10 and statisticall afterwards. Each figure displas an individual undaaged curvature difference as well as different cases of individual daaged curvature difference. Onl curvature differences in the circuferential direction are shown. Figure 6. Average curvature differences for the undaaged and daaged clinders. Note that the abscissa labels of the two plots in Figure 6 are different. The DOFs of the circuferential plot eactl correspond to 23

33 Graphicall, the curvature difference of the slot daage (Figure 10) is noticeabl larger than the undaaged curvature difference. Conversel, the undaaged and daaged differences are siilar in agnitude for the other two daage cases. The statistical significance of the differences between the undaaged and daaged curvature differences can be deterined b eaining the ean and standard deviation of each individual curvature difference: Figure 8. Individual undaaged and one-hole daaged curvature differences = N 1 { H } = N i= 1 H ( i), (9), s = s { H } = N i= 1 ( H ( i) N ) 2 (10) where N is the total nuber of DOFs ( = 66). The true ean of the curvature difference can then be estiated with (1-α)100 percent confidence using Figure 9. Individual undaaged and two-hole daaged curvature differences. σ σ zα / 2 < µ < + zα / 2 (11) N N where σ is the standard deviation of the population of curvature differences. Since N = 66, we a substitute the saple deviation s - for σ. The level of significance α will be set to 5 percent, thus z α/2 equals Figure 10. Individual undaaged and slot daaged curvature differences. The significance of the differences between the averages calculated b Equation 9 will be deterined using standard statistical hpothesis testing. A siplified description of statistical hpothesis testing will be given here, but a ore thorough discussion can be found in [4] and [5]. A test statistic z ep is calculated using the eans and standard deviations of an undaaged and a daaged curvature difference (specified as u and d): 24

34 z ep ( ) u ( ) d =. 2 2 (12) ( s ) / N + ( s ) / N u If z ep < or z ep > 1.96 then the curvature difference averages are statisticall different with a confidence of 95 percent. Estiates of individual curvature difference eans are displaed in Table 1 with 95 percent confidence intervals. Also shown are the z ep s for coparisons between the undaaged curvature difference (denoted with u) and each of the daaged curvature differences (d). For eaple, the z ep reported for D 2,U 5 was calculated b coparing the ean of U 2,U 5 with the ean of D 2,U 5. Table 1. Individual curvature difference eans., µ - z ep u U2,U ± N/A D2,U ± d D3,U ± D4,U ± d ust be present at DOFs spatiall located near daage. The difference between FRF curvatures of the sae clinder before and after daage (as calculated with Equation 6) are shown in Figures 11, 12, and 13. Ste plots are shown for clarit. In each figure, the DOFs nearest to the daage are annotated with squares (Figure 3 shows the circuferential locations of daage patterns). Recall that the DOFs were renubered to siplif calculation of the longitudinal curvatures (Figure 7). Dashed vertical lines separate the circuferential plots into the si regions corresponding to the si rings of testing points on the clinder. To reiterate, the apparent periodicit of the longitudinal plots is due to the fact that DOFs at the top and botto of the clinder were assigned a zero FRF curvature. Note that the longitudinal plot appears to be separated into eleven regions corresponding to the coluns of testing points on the clinder. Since the z ep for each coparison is less than -1.96, the averages of the curvature differences are statisticall significant with a confidence of 95 percent. Therefore differences in FRF curvature are sensitive to the presence of each tpe of daage. The results of Figure 7 further support this conclusion. LEVEL II DAMAGE DETECTION Up to this point, it has been shown that the presence of daage can successfull be detected b statisticall coparing the averages of the undaaged and daaged curvature differences. For successful Level II DD, the curvature difference inforation can not be averaged over the entire DOF range statisticall significant curvature differences Figure 11. Curvature difference for onehole daage. ( H ) D 4 U 4 25

35 Table 2. Mean and standard deviations for curvature differences., s U 4,D U 2,D U 3,D Figure 12. Curvature difference for twohole daage. ( H ) D 2 U 2 With N = 66, the ean and standard deviation of the noral distribution that is approiating the curvature difference distribution can be substituted with and s, respectivel. The probabilit of obtaining a certain curvature difference fro a population is deterined b first calculating a z-value: z = H ( i) s. (13) Figure 13. Curvature difference for slot daage. ( H ) D 3 U 3 Graphicall, the curvature differences at DOFs near the daage do not appear to be significantl different than other DOFs for the one-hole and two-hole daage cases while the slot daage does show coe curvature difference at DOFs near daage. The statistical significance of these curvature differences can be deterined b calculating the probabilit of their occurrence fro a noral distribution defined b the ean and standard deviation of their respective curvature difference populations. The eans and standard deviations for the curvature difference saples were calculated using Equations 9 and 10 and are shown in Table 2. A z-value is sipl the difference between a curvature difference at a specific DOF i and the average curvature difference over all of the DOFs, noralized b the standard deviation of the curvature differences. The z-values for curvature differences at DOFs near daage are shown in Table 3. Table 3. z-values at DOFs near daage. One-hole Two-hole Slot DOF z DOF z DOF z The z-value is a easure of the probabilit of obtaining a certain value fro a stationar statistical sste described b a noral distribution. For eaple, a z-value of 26