IMAC IX April, Determination of Modal Sensitivity Functions for Location of Structural Faults

Transcription

1 Deerminaion of Modal Sensiiviy Funcions for Locaion of Srucural Fauls by Mar H. Richardson Srucural Measuremen Sysems Milpias, California and M. A. Mannan Royal Insiue of Technology Socholm, Sweden Absrac I is well nown ha he modal properies of a srucure are direcly relaed o is mass, siffness, and damping properies. In fac, he modal parameers (eigenvalues and eigenvecors), are soluions o he differenial equaions of moion, which are wrien in erms of he mass, siffness, and damping. This paper focuses on he deerminaion of he funcional relaionship beween variaions in he mass, siffness, and damping, and variaions in he modal properies of srucures. For small changes, his sensiiviy funcion reduces o a very simple funcion of variaions in he modal frequencies and damping only. This maes i possible o deec, locae, and quanify srucural fauls by monioring frequency and damping only. This finding was previously repored by Subbs e.al. [4], [5], [6]. In his paper he complee sensiiviy funcions for mass, siffness, and damping changes are derived, and he validiy of he siffness sensiiviy for small changes is verified. I is also poined ou ha he Srucural Dynamics Modificaion (SDM) echnique can be used o deermine he addiional erms for he complee sensiiviy formulas. Nomenclaure n = number of degrees-of-freedom (DOFs) of he srucural dynamic model m, modes = number of modes [ M ] = ( n by n) mass marix [ C ] = ( n by n) damping marix [ K ] = ( n by n) siffness marix { x () = acceleraion n-vecor { x () = velociy n-vecor { x () = displacemen n-vecor { f () = exernal force n-vecor m by m modal mass marix [ m ] = ( ) [ c ] = ( m by m) [ ] = ( m by m) modal damping marix modal siffness marix w = damped naural frequency for mode d = damping coefficien for mode W = undamped naural frequency for mode n by m mode shape marix [ U ] = ( ) Inroducion The modes of vibraion of a srucure are srongly influenced by sligh changes in is physical properies or is boundary condiions. This fac is also self eviden if one considers he mahemaical definiion of modes as he eigenvalue soluions of he differenial equaions of moion for a vibraing srucure. These equaions resul from a sraighforward applicaion of Newon's Second Law o he srucure, and define a force balance beween he inerial (mass), dissipaive (damping), and resoring (siffness) forces wihin he srucure, and he exernally applied forces. These equaions are also relaed, via he Fourier ransform, o he Frequency Response Funcion (FRF) form of he dynamic model, which can also be represened in erms of he modal parameers of he srucure. This parameric form of he FRF marix in erms of modal parameers is he foundaion upon which all modern day modal esing is done. Finally, he Impulse Responses of a srucure comprise a hird, compleely equivalen form of he dynamic model, and hey oo can be represened paramerically in erms of modes of vibraion. Experimenally deermined impulse responses are also used in modern day modal es sysems o idenify modal parameers. Figure 1 illusraes his inerdependency beween he physical properies, (disribued mass, siffness, and damping), he FRFs, he impulse responses, and he modal properies of a srucure. I is clear, hen, ha changes in he physical properies or boundary condiions, (boh of which affec he mass, siffness, and damping properies), will cause changes in he measured FRFs or impulse response funcions, and also will change he measured modal properies. Page 1 of 7

2 Wha is a Srucural Faul? Failure of he Srucural Maerial due o Faigue. For example, cracing, breaing, delaminaion. Loosening of Assembled Pars. For example, loose bols, rives, glued joins, or wear-ou of pars. Manufacuring Defecs. For example, flaws voids or hin spos due o casing, molding, or forming operaions. Improper assembly of pars. Theoreical Bacground Figure Figure 1 In previous papers, [1], [], [3], we invesigaed ways in which he curren modal esing echnology could be used for deecing and locaing srucural fauls. A srucural faul could be any one of he occurrences lised in Figure. Using modern day digial esing equipmen, a srucure can be excied in a wide variey of ways, and high resoluion, noise free, linear esimaes of is FRFs can be obained. Modal parameer idenificaion mehods can hen be applied o he FRF measuremens o very accuraely deermine he modal properies of he srucure. The echnology exiss for deecing milliherz changes in modal frequencies and damping, and also for obaining changes in mode shapes, if necessary. In he following developmen, we expand upon he wor of Subbs, e.al. [4], [5], [6]. They used he orhogonaliy condiion for classically damped (or lighly damped) srucures, and derived relaionships beween changes in he mass, siffness, and damping marices and changes in he modal parameers. The resuling equaions can be used direcly o locae and quanify damage (physical change) on a srucure, if he undamaged mass, siffness, and damping plus measured changes in he modal properies are nown. A ey finding of heirs, however, and one which we will verify here also, is ha if he modal shapes don' change appreciably, (his usually holds for small changes in he physical properies), hen damage can be locaed and quanified by using only changes in he modal frequencies and damping, plus he mode shapes of he undamaged srucure. This offers a decided advanage from an implemenaion sandpoin since modal frequency and damping can be easily measured a pracically any poin on a srucure. Modes of vibraion are commonly defined as soluions o he following differenial equaions: [ M ]{ x ( ) + [ C]{ x ( ) + [ K ]{ x( ) = { f ( ) (1) The modal properies are acually soluions o he homogeneous equaions (i.e. where { f ( ) = { 0 ), and are found by a sraighforward eigensoluion process. The coefficien marices ( [ M ],[ C ], and [ K ]) are usually assumed o be real valued and symmeric, and wihou any furher assumpions, complex conjugae pairs of eigenvalues and corresponding eigenvecors can be found from hese equaions. These consiue he so-called complex modes of he srucure. However, if he damping erm is assumed o be negligible compared o he mass and siffness erms in equaion (1), he eigenvalues and eigenvecors exhibi a very srong orhogonaliy propery, which will be exploied here. The above assumpion can be applied o he majoriy of real world srucures, and cerainly o hose which vibrae freely. Anoher way of saing i is ha he damping forces are assumed o be negligible compared o he inerial and resoring forces of he srucure. Such srucures are said o be classically damped, or simply lighly damped. The orhogonaliy propery of he modes almos simulaneously diagonalizes he mass, siffness, and damping marices, and herefore almos uncouples he equaions of moion. The erm almos is used because sric diagonalizaion only occurs if here is no damping ([ C ] = [ 0] ), or if he damping marix is proporional o he mass and siffness marices, a difficul assumpion o verify wih real srucures. Neverheless, when damping is negligible, he following orhogonaliy condiions can be applied o he unmodified srucure (srucure wih no faul): [ ] [ M ][ U ] = [ m ] U () Page of 7

3 [ ] [ C][ U ] = [ c ] U (3) [ ] [ K ][ U ] = [ ] U (4) Two oher relaionships which resul from orhogonaliy are: where: and: W = m 0 =1... modes (5) d = c m =1... modes 0 = w0 d 0 W + 0, d 0 (6) w = frequency and damping of he unmodified srucure Equaions () hrough (6) can also be wrien for he modified srucure (srucure wih a faul): [ U ] [ M + dm ][ U ] = [ m + dm ] (7) ( W W ) + dm W d m = (15) ( d1 d0 ) + dmd1 dc (16) m = Siffness Changes Probably he mos sough afer cause of a srucural faul is a reducion in local siffness, which migh be caused by he formaion of a crac, delaminaion, or a loose fasener. The above equaions can be combined o yield a single relaionship beween changes in he siffness marix [ dk ] and changes in he modal parameers. Mode mass ( m ) is simply a scaling consan and herefore, can be arbirarily se o any value. We can always scale he mode shapes o uniy modal masses, so ha m = 1 and ( m + dm ) = 1, which also implies ha dm = 0 for all modes (). [ U ] [ C + dc][ U ] = [ c + dc ] [ U ] [ K + dk ][ U ] = [ + d ] ( + d ) ( m dm ) 1 = =1... modes W + where: 1 1 = w1 d 1 W + ( c + dc ) ( m dm ) (8) (9) (10) d = + =1... modes (11) Noice ha each of physical and modal properies of he modified srucure is wrien in erms of he same propery of he unmodified srucure, plus an addiive change d erm. Therefore, equaions (7) hrough (11) can be expanded and he unmodified condiions subraced from hem o yield a new se of formulas ha relae changes in he mass, siffness and damping marices o changes in he modal properies: [ U ] [ dm ][ U ] + [ du ] [ M ][ U ] + [ du ] [ M ][ du ] = [ dm ] [ U ] [ dc][ U ] + [ du ] [ C][ U ] + [ du ] [ C][ du ] = [ dc ] [ U ] [ dk][ U ] + [ du ] [ K][ U ] + [ du ] [ K][ du ] = [ d ] () () (14) Page 3 of 7 Combining equaions (14) and (15) and using mode shapes scaled o uni modal masses gives he following siffness sensiiviy equaion: { U [ dk ]{ U + { du [ K ]{ U + { du [ K ]{ du = w w 1... modes 1 0 = (17) This formula expresses changes in he siffness marix [ dk ] as funcions of he unmodified siffness [ K ] and changes in he modal properies, i.e. change in he mode shape { du, plus changes in he modal frequency ( w ) w. Siffness 1 0 changes don' affec modal damping. Noice ha an equaion (17) can be wrien for each mode (), which means ha a se of (m) equaions can be solved for up o (m) siffness changes a a ime. Hence, when he number of unnown local siffness changes exceeds he number of modes for which we have measured changes, some sor of a searching algorihm will be required. This issue is considered in he numerical example o follow. Small changes: If i can be furher assumed ha he faul is sligh enough so ha he mode shapes don' change subsan- 0 ), hen he siffness sensiiviy equa- ially, (i.e. { du = { ion is grealy simplified: { U [ dk ]{ U w w = =1... modes (18) 1 0

4 Local Siffness Changes A siffness change beween wo degrees-of-freedom, say DOF i and DOF j, changes he siffness marix in he following manner: DOF DOF d j d i i i j [ dk ] = DOF j d i j d i j i DOF Equaion (18) can herefore be rewrien in erms of local siffness changes as: sumsum i j ( u + u u u ) d = w 1 i j i j ij 1 j 0 =1...modes (19) This formula only requires he mode shapes for he unmodified srucure plus changes in he frequency of he modes. The modal parameers of he unmodified srucure can be obained eiher by modal esing or finie elemen analysis. A faul which causes a local siffness change can hen be deeced, locaed, and quanified by simply racing he modal frequencies of he srucure, and using equaion (19). Mass Changes Equaions (), (14), and (15) can be combined in he same manner as above o yield a mass sensiiviy equaion: { U [ dm ]{ U + { du [ M ]{ U { du [ K ]{ U + { du [ M ]{ du = + { du [ K ]{ du ( W W ) 1 0 W 1 =1...modes (0) This formula requires boh he mass [ M ] and siffness [ K ] of he unmodified srucure, plus changes in he modal parameers due o he faul. Again, as wih he siffness, for small changes which don' affec he mode shapes subsanially, he mass sensiiviy equaion becomes grealy simplified: { U [ dm ]{ U = ( W W ) W =1...modes (1) Damping Changes Equaions () and (17) can be combined, ogeher wih uni modal mass scaling of he mode shapes, o yield a damping sensiiviy equaion: { U [ dc]{ U + { du [ C]{ U + { du [ C]{ du = ( d d ) 1 0 =1...modes () And, for small changes which don' affec he mode shapes, his equaion simplifies o: { [ dc]{ U ( d d ) U 1 0 A 3-DOF Example = =1... modes () The validiy of siffness sensiiviy, equaion (19), will be esed on he 3-DOF vibraory srucure shown in Figure 3. The mass, siffness, and damping, as well as he modal properies of he srucure are also given in Figure 3. Foureen differen siffness changes were made o he 3-DOF srucure o simulae differen faul condiions. These are shown in Figure 4. These changes involved various combinaions of local siffness changes beween DOFs 1x, x, and 3x. (Since all of he DOFs are in he x-direcion, only he poin numbers are used as subscrips). The modal frequency shifs caused by he simulaed siffness changes are given in Figure 5. These were obained by solving for he modes of he modified srucure wih each siffness change using he Srucural Dynamics Modificaion (SDM) echnique. Since modal daa for hree modes is available, equaion (19) can be wrien for all hree modes and he resuling se of equaions solved for hree siffness changes a a ime. Those resuls are shown in Figure 6. Noice ha 5% and 10% changes of siffness are all saisfacorily prediced, (cases 1,, 4, 5, 7, 8, & 10). The 15% o 5% changes (cases 3, 6, 9, 11, & ) were also correcly locaed, alhough wih a larger amoun of numerical error. Case 14, wih 30% overall reducion in siffness, also shows a significan degree of error. As a second es, we simulaed a more realisic siuaion where eiher all he modes of he srucure have no been measured, or else he number of poenial local siffness changes exceeds he number of modes measured. In his es, we used sensiiviy equaions for only he firs wo modes of he 3-DOF srucure. This mean ha we could only solve for a single siffness change, or wo changes a a ime. The wo-a-a-ime soluions are shown in Figure 7. Noice ha here are a number of unrealisic (posiive) changes, which are double underlined. Page 4 of 7

5 Whenever an unrealisic change occurred, ha local siffness was eliminaed from furher consideraion as a possible faul locaion. The summary of he search hrough he daa in Figure 7 is shown in Figure 8. Noice ha he single changes in cases 1 hrough 9 were all correcly locaed, and ha cases 10 hrough 14 encounered some difficuly. For example, in case 10 he correc answer is found when only d and d are used as unnowns, bu when d and d are used, d is posiive and is herefore rejeced as a possible soluion candidae. Figure 3 Figure 4. Simulaed Faul Cases Siffness change case 1 case case 3 case 4 case 5 case 6 case 7 d d d Siffness change case 8 case 9 case 10 case 11 case case case 14 d d d Figure 5. Modal Frequency Shifs ( f ) 1 f0 Mode case 1 case case 3 case 4 case 5 case 6 case Mode case 8 case 9 case 10 case 11 case case case Page 5 of 7

6 Figure 6. Calculaed Siffness Changes Siffness change case 1 case case 3 case 4 case 5 case 6 case 7 d d d Siffness change case 8 case 9 case 10 case 11 case case case 14 d d d Conclusions We have verified by example ha he orhogonaliy condiions for classically damped srucures can be used o accuraely locae and quanify srucural fauls, by simply using changes in measured modal frequencies. This is indeed a powerful resul, which was previously poined ou by Subbs, e.al. [4], [5], [6]. We derived separae sensiiviy equaions for mass, siffness, and damping changes, and we showed ha uniy modal mass scaling of he mode shapes simplifies hese formulas. We found from he numerical example ha changes of 10% or less in siffness could be accuraely prediced using he small change version of he formulas. This form assumes ha he mode shapes don' change significanly. When a sufficien number of modes are moniored, even changes as large as 5% can sill be correcly locaed. Bu wih a reduced se of modes, which is more applicable o real world problems, he small change equaions sill did an adequae job of locaing changes of 10% or less. When he small change assumpion canno be made, a subsanially greaer amoun of daa is required o use he sensiiviy equaions. No only are he mass, siffness, and damping marices of he unmodified srucure needed, bu he mode shapes of he modified srucure would also have o be measured as well. For specific applicaions, however, his addiional wor may be warraned in order o obain he increased accuracy. Finally, a se of modal parameers for an unmodified (undamaged) srucure and he SDM echnique could be used o deermine all of he unnown mass, siffness, and damping erms in he sensiiviy equaions (17), (0), and (). Or, from a differen perspecive, hese sensiiviy equaions ogeher wih he use of he SDM echnique provide anoher way of esimaing he mass, siffness and damping marices of a srucure from measured modal daa. References [1] Wolff, T. and Richardson, M. Faul Deecion in Srucures from Changes in Their Modal Parameers Proceedings of he 7h Inernaional Modal Analysis Conference, Las Vegas, Nevada, SEM, Behel, CT. [] Richardson, M. and Mannan, M.A. Deecion and Locaion of Srucural Cracs Using FRF Measuremens 8h Inernaional Modal Analysis Conference, Kissimmee Florida, Jan 9 - Feb 1, 1990, SEM, Behel, CT. [3] Richardson, M. and Mannan, M.A. Using Measured Modal Parameers and he Siffness Marix o Deec and Locae Srucural Fauls ICSTAD Proceedings, Jul 9 - Aug 3, 1990, Bangalore, India. [4] Subbs, N., Broome, T.H., and Osegueda, R. Non- Desrucive Consrucion Error Deecion in Large Space Srucures AIAA Journal, Vol. 8, No.1,1990. [5] Subbs, N. and Osegueda, R. Global Non -Desrucive Damage Evaluaion in Solids, Inernaional Journal of Analyical and Experimenal Modal Analysis, Vol. 5, No., April, 1990, pp [6] Subbs, N. and Osegueda, R. Global Damage Deecion in Solids-Experimenal Verificaion, Inernaional Journal of Analyical and Experimenal Modal Analysis, Vol. 5, No., April, 1990, pp Page 6 of 7

7 Figure 7. Calculaed Siffness Changes Using Only Two Modes d ) (unnowns d & Siffness change case 1 case case 3 case 4 case 5 case 6 case 7 d d Siffness change case 8 case 9 case 10 case 11 case case case 14 d d d ) (unnowns d & Siffness change case 1 case case 3 case 4 case 5 case 6 case 7 d d Siffness change case 8 case 9 case 10 case 11 case case case 14 d d d & d ) (unnowns Siffness change case 1 case case 3 case 4 case 5 case 6 case 7 d d Siffness change case 8 case 9 case 10 case 1 l case case case 14 d d Figure 8. Summary of Search Through Two Mode Resuls (all posiive d s se o zero) Siffness change case 1 case case 3 case 4 case 5 case 6 case 7 d d d Siffness change case 8 case 9 case 10* case 11 * case case * case 14* d d d Page 7 of 7