TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM IDENTIFICATION USING THE HILBERT TRANSFORM

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1 Proeeings of ASME VIB 5: th Biennil Conferene on Mehnil Virtion n Noise Septemer 4-8, 5 Long Beh, CA, USA DETC TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM IDENTIFICATION USING THE HILBERT TRANSFORM Mihel Felmn Fulty of Mehnil Engineering Tehnion Isrel Institute of Tehnology Hif,, Isrel ABSTRACT The ojetive of the pper is to explin moern Hilert trnsform metho for nlysis n ientifition of mehnil non-liner virtion strutures in the se of qusiperioi signls. This speil kin of perioiity rises in experimentl virtion signls. The metho is se on the Hilert trnsform of input n output signls in time omin to extrt the instntneous ynmi struture hrteristis. The pper fouses on the ynmi nlysis n ientifition of three groups of ynmis systems: o Fore virtions of liner n non-liner SDOF systems exite with qusiperioi fore signl. o Comine fore virtions of qusiperioi time vrying liner n non-liner SDOF systems exite with hrmoni signl. o Comine self-exite n fore virtions of non-liner SDOF systems exite with hrmoni signl. The stuy fouses on signl proessing tehniques for nonliner system investigtion, whih enle us to estimte instntneous system ynmi prmeters (nturl frequenies, mping hrteristis n their epenenies on virtion mplitue n frequeny) for ifferent kins of system exittion. Keywors: non-liner system, Hilert trnsform, ientifition, envelope, instntneous frequeny. INTRODUCTION Reent work in the re of time omin representtions of virtion, suh s the Hilert Trnsform (HT) [, ], shows gret promise for pplitions in ynmi system ientifition. The propose methos, FREEVIB n FORCEVIB, for ientifying instntneous mol prmeters (nturl frequenies, mping hrteristis n their epenenies on virtion mplitue n frequeny) prove to e very simple n effetive. Of prtiulr interest n importne is the use of the HT to interpret qusiperioi non-liner systems motion. In the simplest se, qusiperioi signl is signl tht onsists of sum of two sinusoil signls with onstnt frequenies. The tritionl Fourier trnsform enles estimting only two seprte frequeny points of the Frequeny Response Funtion (FRF). The HT of the sme signl ontrstingly llows estimtion of the FRF for wie ontinuous frequeny rnge. Qusiperioi motion The lss of the so-lle lmost perioi motions is prtiulr su-lss of reurrent trjetories whih is of interest in non-liner ynmis. The remrkle feture whih revels the origin of these trjetories is tht eh omponent of n lmost perioi motion is n lmost perioi funtion with well stuie nlytil properties. An lmost perioi funtion is uniquely efine "in verge'' y trigonometri Fourier series i nt f() t e λ, where λ n re rel numers. If ll λ n re n= n liner omintions (with integer oeffiients) of finite numer of rtionlly inepenent elements from sis of frequenies, then we hve prtiulr se of lmost perioi funtions, nmely qusiperioi funtions. A qusiperioi signl, in this ontext, is signl tht onsists of sum of given numer of sinusoil signls with known frequenies n unknown, time vrying mplitues n phse. This kin of qusi-perioiity rises in virtion signls. In mny prtil situtions, it is esirle tht prmeters of qusiperioi signl e estimte in rel time. A ontinuous estimtion of these prmeters n e use, for exmple, for mesuring, monitoring, or ignostis purposes. Instntneous hrteristis of qusiperioi virtion. Two-omponent signl representtion Consier virtion signl ompose of two qusi-hrmonis, eh with slow vrile mplitue n frequeny in the time omin. In this se, the signl n e moele s weighte sum of monoomponent signls, eh with its own instntneous frequeny n mplitue funtion: tht is, Copyright #### y ASME

2 t t i ωt i ω t Ft () = e + e with,, ω n ω eing unknown funtions in the time omin. The envelope n the instntneous frequeny of the oule-omponent virtion signl ω () t re: ( ) ω ω ( ) t () = + + os ( ) t ( ω ω) + os ( ω ω) t ω() t ω = + () t Eq.() shows tht the signl envelope t () onsists of two ifferent prts, tht is, slow vrying prt inluing the sum of the omponent mplitues squre + n rpily vrying (osillting) prt, the multiplition of these mplitues with funtion os of the reltive phse ngle etween two omponents. The instntneous frequeny of the two onstnt frequeny tones onsiere in Eq.() is generlly time-vrying n exhiits symmetril evitions out the frequenyω. For the two tones, not only re there time-vrying evitions in the instntneous frequeny, ut these evitions lwys fore the instntneous frequeny eyon the frequeny rnge of the signl omponents. It ppers, then, tht in generl, the instntneous frequeny of signl n the verge frequeny t eh time of the signl re ifferent quntities. The time vrying ifferene etween the two initil onstnt frequenies n the instntneous frequeny of the ompose signl n e interprete s frequeny moultion within the signl. The evelope HT tehnique oul exploit this moultion property of the input exittion signl for mol nlysis of ynmi systems. Non-liner visous mping n frequeny, or istriute liner frequenies n ssoite mol mping, re oth me possile, in omputtionlly fst mnner using HT ientifition. The Hilert trnsfer ientifition The moern HT ientifition metho, s non-prmetri metho, is reommene for instntneous mol prmeters ientifition in time omin, inluing etermintion of the system kone, mping urves, n stti fore hrteristis. The HT metho is suggeste for ientifition of SDOF liner n non-liner systems uner free or fore virtion onitions. A seon-orer onservtive system with non-liner restoring fore kx ( ) n solution xt () = Aosωt n e represente in generl power series form x+ k( x) = () 4 kx ( ) = ( α+ αx+ α5x+...) x Applying the multiplition property of the HT for overlpping funtions [, ] to Eq.(), we otin new form of time vrying eqution of motion x+ jδ() t x+ ω () t x= () where ω () t is the fst vrying nturl frequeny funtion n δ () t is the fst vrying fititious mping funtion. If we onsier only the men vlue of the vrying nturl frequeny funtion squre, we get n importnt result () T 5 4 T t t A 5A 4 8 ω = ω () = α + α + α +... (4) whih proves tht the verge nturl frequeny hs, orret to the polynomil onstnt oeffiients, the sme expression s the initil non-liner restoring fore kx ( ) Eq.(). This generl result mens tht the estimte verge nturl frequeny n hene the system skeleton urve (kone) A( ω ) inlues the min informtion out the initil non-liner elstis hrteristis n n e use for non-liner system ientifition. Clerly, fter the verging we get only the first term of the motion, so the HT ientifition metho restores the initil non-liner fores pproximtely orret to the time vrying first term of motion. In the first stge of the propose ientifition tehnique, the envelope A() t n the instntneous frequeny ω () t re extrte from the virtion n exittion signls on the se of the HT signl proessing. In the next stge, the instntneous unmpe nturl frequeny n the instntneous mping oeffiient of the teste system re lulte oring to formuls [, ]: A A A ω A ω ω() t = ω + + ; h () t =, where A A Aω A ω A() t n ω () t re the envelope of the instntneous frequeny of the virtion. The mss prmeter of eh virtion moel uner ientifition is priori tken s equl to. The otine instntneous funtions re low-pss filtere to get the first term of virtion motion. In the finl stge, the non-liner "frequeny response funtion" [ ] is onstrute together with oth the elsti n the mping stti fore hrteristis, whih re lulte oring to the eomposition tehnique [ ]: ω() tat (), x > h() tx (), t x > kx ( ) ; hxx ( ) ω() tat (), x< h() t x (), t x < It is onvenient to represent the result of the HT ientifition in stnr form whih inlues the skeleton urve with the FRF n lso the stti fore hrteristis of the ynmis system. Virtion moeling n numeril results This pper onentrtes on ynmi nlysis n ientifition of three groups of ynmis systems: () fore virtions of liner n non-liner SDOF systems exite with qusiperioi fore signl; () omine fore virtions of qusiperioi time vrying liner n non-liner SDOF systems exite with hrmoni signl; n () omine self-exite n fore virtions of non-liner SDOF systems exite with hrmoni signl. For the given ifferent equtions of virtion motion, there is omine nlytil expression Eq.(5) for ll the numeril for liner n non-liner moelling vrints: x+ γx + δ x x + µ x ( x ) + kx+ αx = F( t) k = ω (+ βos ωβt) (5) Ft ( ) = Aosπ ft+ Aosπ ft+ Aosπ ft Copyright #### y ASME

3 Consequently, we n onstrut virtion motion y forming omintion of the following prmeters: γ - the liner visous frition oeffiient, δ - the non-liner "qurti" frition oeffiient, µ - the frition oeffiient of the vn-er- Pol eqution, k - the stti elsti fore oeffiient, ω - the liner unmpe nturl frequeny squre, α - the ui oeffiient of the Duffing eqution, β - the mplitue moultion oeffiient of the elsti fore oeffiient, ω β - the frequeny moultion oeffiient of the elsti fore oeffiient, A i - the mplitue of the of exittion, f i - the mplitue of the of exittion. Their orresponing numeri vlues re given in Tle. System Prmeters Dmping Elstiity Qusiperioi Sweep γ Moel prmeter Tle Moel δ.. µ... ω α.4.4 β ω β.. A e - f A f A. f e -5 e -5 The simultions of ll ifferentil equtions of motion re performe with SIMULINK (MATLAB) with the permnent step vlue ODE4 (Runge-Kutt) solver. Moel Liner system In the first test se (Tle ), we use two hrmonis to generte "eting" virtion regime. The frequeny of the first hrmonis f =.6 Hz is lose to the liner system resonne frequeny ω /.59 π Hz, n the frequeny of the seon hrmonis f =.55 Hz is hosen to hve three full perio etings uring the totl time of reore virtion T f f (Figure ). Nturlly, the liner system oes not hnge the typil "two hrmonis" wve n the spetrum form of the output (isplement) signl reltive to the input (exittion) signl (Figure ). Intermeite results of the HT ientifition, suh s the time segment of the isplement with its envelope together with the instntneous frequenies, re given in Figure. The instntneous frequeny of the isplement (Figure,, she line) vries in time roun the onstnt vlue of the estimte instntneous unmpe nturl frequeny (ol line). Displement Power Spetrum Mgnitue Solution (Displement) Frequeny, Hz Power Spetrum Mgnitue 4... Frequeny, Hz Figure. Moel - "Liner system": ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. Amplitue Frequeny, Hz Displement n Envelope Instntneous Frequenies Unmpe nturl frequeny Figure. Moel Liner system: ) isplement n envelope; ) instntneous frequenies. This frequeny moultion performne llows reonstrution of the teste ynmi struture in wie frequeny rnge. The otine finl results of the HT ientifition, inluing the skeleton urve, the FRF, the elsti stti fore hrteristis, the mping urve, n the frition fore, re shown in Figure with the ol line. In the sme figure, the initil hrteristis from Tle, nmely the liner skeleton line k ω = =.59 Hz, the liner elsti stti fore π π π Copyright #### y ASME

4 γ kx = x, the liner mping urve =., n the liner frition fore γ x =.x, re shown with she line. However, the ifferene etween the initil liner n the estimte hrteristis is less thn.%, so it n not e istinguishe in Figure. Skeleton urve n FRF Dmping urve Displement Amplitue Fore Frequey, Hz Elsti Stti Fore Displement Fore Veloity Amplitue Dmping oeffiient.5.5 Frition Fore 5 5 Veloity Figure. Moel Liner system. Ientifition results: ) skeleton urve n FRF; ) elsti stti fore; ) mping urve; frition fore hrteristis. Moel "Non-Liner" system Let us now exmine the ientifition results for the non-liner moel, whih ontins two ifferent types of non-linerity: non-liner "qurti" mping n lso non-liner ui elsti omponent inherent to the Duffing eqution (Tle ). Agin, fore virtion regime is proue y the sme qusiperioi fore input signl. The existene of non-linerity Power Spetrum Mgnitue Displement Solution (Displement) Frequeny, Hz Power Spetrum Mgnitue Frequeny, Hz Figure 4. Moel - "Non-liner" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. n e notie immeitely from the time t (Figure 4,, ), whose input n output wve shpes iffer, n lso from the output spetrum (Figure 4, ), whih hs high frequeny multiple hrmonis. The HT ientifition results re given in Figures 5 n 6. The omprison etween the estimte (ol line) n the initil (she line) skeleton urve, tken s the first term of Eq.(4), α+ αa ω ( A) 4 + A f( A) = = =, shows π π π Displement n Envelope Amplitue Frequeny, Hz Instntneous Frequenies Nturl unmpe frequeny Figure 5. Moel - "Non-liner" system: ) isplement n envelope; ) instntneous frequenies. Displement Amplitue Fore Skeleton urve n FRF.5..5 Frequey, Hz Elsti Stti Fore Displement Veloity Amplitue Fore Dmping urve Dmping oeffiient Frition Fore Veloity Figure 6. Moel - "Non-liner" system. Ientifition results: ) skeleton urve n FRF; ) elsti stti fore; ) mping urve; frition fore hrteristis. tht these skeleton urves re in goo lose greement. The ientifie (ol line) n the initil (she line) stti fore hrteristis kx + α x = +.4x (Figure 6, ) re very lose together. The ientifie (ol line) n the initil (she line) 4 Copyright #### y ASME

5 frition fore hrteristis γx + δx x =.x +. x x re lso lose to eh other (Figure 6, ). This moel illustrtes tht HT ientifition mkes it possile to restore non-liner hrteristis even in the se of omine non-linerity in oth the elstis n the frition prts of the eqution of motion. Note tht the ientifie stti fore hrteristis hve smll evition from the initil hrteristis to the "liner" iretion. In other wors, they re slightly less non-liner thn the initil fore hrteristis. This mens tht the propose HT ientifition restores only the min first term of the motion. Moel "Prmetri" system As next test, let us onsier struture tht esries slow moulte elsti fore k = ω ( + β os ω t β ) = +.5os.t of the trivil ynmis system uner externl Displement Power Spetrum Mgnitue Frequeny, Hz Power Spetrum Mgnitue Solution (Displement)... Frequeny, Hz Figure 7. Moel - "Prmetri" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. Displement n Envelope Amplitue Frequeny, Hz Instntneous Frequenies Figure 8. Moel - "Prmetri" system: ) isplement n envelope; ) instntneous frequenies. hrmonis exittion (Tle ). The generte virtion hs rther omplite form over time n lso in the frequeny omin (Figure 7). But HT ientifition restores this moultion in lose etil. Thus, Figure 8, shows tht the ientifie instntneous unmpe nturl frequeny of the virtion (ol line) ompletely oinies with the vrying +.5os.t initil elsti fore moultion funtion f = π Veloity Amplitue 5 5 Dmping urve.... Dmping oeffiient Fore Frition Fore Veloity Figure 9. Moel - "Prmetri" system: Ientifition results: ) mping urve; ) frition fore hrteristis. (she line).the otine mping hrteristis (Figure 9,,, ol line) emonstrte the liner type of frition fore, whih is goo mth for the initil liner type of the "prmetri" γ moel (she line) with =. n γ x =.x. This exmple emonstrtes tht the HT metho enles ientifition of slow moultion of the system prmeters, even in the se of the simplest monohrmoni exittion. Power Spetrum Mgnitue Displement Solution (Displement)... Frequeny, Hz Power Spetrum Mgnitue Frequeny, Hz Figure. Moel 4 - "Prmetri + non-liner + sweep" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. 5 Copyright #### y ASME

6 Moel 4 "Prmetri + non-liner + sweep" system The next system s well s the previous one hs slow moulte elsti fore, ut in ition to the liner visous, it lso hs non-liner qurti frition memer. The teste moel hs ifferent externl fore exittion with sweeping inresing frequeny inste of single hrmonis (Tle ). The generte virtion in time n lso in the frequeny omin (Figure ) tkes rther omplite form. The results of HT ientifition re shown in Figures n. Displement n Envelope Frequeny, Hz Amplitue Instntneous Frequenies Figure. Moel 4 - "Prmetri + non-liner + sweep" system: ) isplement n envelope; ) instntneous frequenies. The ientifie instntneous unmpe nturl frequeny (ol line) ompletely oinies with the initil elsti fore +.5os.t moultion f = (Figure,, she line). π Dmping urve Frition Fore Veloity Amplitue Fore.5 The otine non-liner frition fore hrteristis (ol line) re in lose greement with the initil non-liner type of frition fore (she line) γx + δx x =.x +. x x (Figure, ). Moel 5 "Prmetri instility" system The stti elsti fore for this se is now moulte with high moultion frequeny equl to the onstnt nturl frequeny of the system ω = ω β = (Tle ). The mplitue of the solution of the system uner externl hrmoni exittion inreses infinitely (Figure, ). This ehvior illustrtes the prmetri instility of the teste system. Power Spetrum Mgnitue Displement 4 5 x Frequeny, Hz Power Spetrum Mgnitue x 6 4 Solution (Displement)... Frequeny, Hz Figure. Moel 5 - "Prmetri instility" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. Displement Amplitue Skeleton urve n FRF Frequey, Hz x 4 Elsti Stti Fore Veloity Amplitue Dmping urve.... Dmping oeffiient 5 Frition Fore..5 Fore Fore Dmping oeffiient Veloity Figure. Moel 4 - "Prmetri + non-liner + sweep" system: Ientifition results: ) mping urve; ) frition fore hrteristis. Displement x 4 5 Veloity x 4 Figure 4. Moel 5 - "Prmetri instility" system: ) skeleton urve n FRF; ) elsti stti fore; ) mping urve; frition fore hrteristis. In this se, the input hrmonis exittion hs prtilly no influene on the oserve unstle osilltion. Therefore, we 6 Copyright #### y ASME

7 will use the HT ientifition FREVIB metho tht nlyses only the virtion output of the struture. The HT ientifition metho restores only the orret skeleton urve n the elsti fore hrteristis (Figure 4,, ). It is ler tht in this se, the HT inste of the initil liner mping restores negtive inrement n the orresponing negtive (in the opposite iretion) frition hrteristis (Figure 4,, ). In some speil ses, the otine inrement n e use for qulity nlysis of the instility growth rte of suh unstle virtion solutions. Moel 6 "Self exite + non-liner" system The next test omines the non-liner frition prt ommon to the vn-er-pol osilltor n the non-liner ui elsti fore prt typil to the Duffing eqution. Moel 6 eing teste hs very low level hrmoni input exittion (Tle ). As expete, the teste moel 6 isplys known self-exite regime of non-liner virtion in time (Figure 5, ). x Power Spetrum Mgnitue Displement Frequeny, Hz Power Spetrum Mgnitue 4 6 Solution (Displement) Frequeny, Hz Figure 5. Moel 6 - "Self exite + non-liner" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. The orresponing spetrum of the self-exite virtion tht shows high frequeny multiple hrmonis onfirms the existene of the non-liner elsti prt (Figure 5, ). Agin, the oserve self-exite osilltion regime prtilly oes not epen on the input exittion signl. The HT ientifition metho FREEVIB use here restores in full etils oth the non-liner frition n the non-liner elstis prts. Thus, the ientifie (ol line) n the initil (she line) skeleton urve, s the first term of Eq.(4), prtilly oinie (Figure 7, ). Similrly, the ientifie (ol line) n the initil (she line) stti fore hrteristis kx + α x = +.4x prtilly oinie s well (Figure 7, ). The ientifie non-liner frition fore hrteristis (ol line) re in lose greement with the initil non-liner type of the frition fore (she line) µ xx ( ) =. xx ( ) (Figure 7, ). Amplitue Frequeny, Hz Displement n Envelope Instntneous Frequenies Figure 6. Moel 6 - "Self exite + non-liner" system: ) isplement n envelope; ) instntneous frequenies. Displement Amplitue Fore Skeleton urve n FRF Frequey, Hz Elsti Stti Fore.5.5 Displement Fore Veloity Amplitue Dmping urve Dmping oeffiient.4.. Frition Fore.4 Veloity Figure 7. Moel 6 - "Self exite + non-liner" system: ) skeleton urve; ) elsti stti fore; ) mping urve; ) frition fore hrteristis. Moel 7 Self + fore exite system Let us now exmine the next system with the omine nonliner frition prt ommon to the vn-er-pol osilltor n high level qusiperioi exittion (Tle ). The hosen fore virtion regime eomes ominnt, n the virtion shpe shown in Figure 8,, is similr in pperne to the fore virtion of the non-liner system in Figure 4,,. The HT, however, etets the tul properties of the teste system. The ientifie skeleton urve n the elsti stti fore hrteristis (ol line) tke trivil liner form orresponing to the initil liner elstis prt (she line) of the teste system (Figure,, ). The ientifie non-liner frition fore hrteristis (ol line) re in lose greement 7 Copyright #### y ASME

8 with the initil non-liner type of the frition fore (she line) µ xx ( ) =. xx ( ) (Figure, ). Displement Power Spetrum Mgnitue Frequeny, Hz Power Spetrum Mgnitue Solution (Displement)... Frequeny, Hz Figure 8. Moel 7 - "Self + fore exite" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. Frequeny, Hz Amplitue Displement n Envelope Instntneous Frequenies.6 Nturl unmpe frequeny Figure 9. Moel 7 - "Self + fore exite" system: ) isplement n envelope; ) instntneous frequenies. Moel 8 Self exite + sweep system The system uner onsiertion repets the previous system with the non-liner frition prt ommon to the vn-er-pol osilltor, ut now it involves sweeping frequeny fore exittion (Tle ). The otine wve shpe in time n lso Displement Amplitue Fore.5.5 Skeleton urve n FRF Frequey, Hz Elsti Stti Fore Displement Veloity Amplitue Fore.5.5 Dmping urve.... Dmping oeffiient Frition Fore Veloity Figure. Moel 7 - "Self + fore exite" system: ) skeleton urve; ) elsti stti fore; ) mping urve; ) frition fore hrteristis. the orresponing spetrum shpe emonstrte the typil resonne performne of the struture (Figure ). Agin the hrteristis ientifie on the se of the HT skeleton urve n the elsti stti fore hrteristis (ol line) tke trivil liner form tht orrespons to the initil liner elstis prt (she line) of the teste system (Figure,, ). The ientifie non-liner frition fore hrteristis (ol line) re in lose greement with the initil non-liner type of the frition fore (she line) µ xx ( ) =. xx ( ) (Figure, ). Displement Power Spetrum Mgnitue x Solution (Displement) Frequeny, Hz Power Spetrum Mgnitue Frequeny, Hz Figure. Moel 8 - "Self exite + sweep" system: ) exittion, ) isplement, ) exittion spetrum, ) isplement spetrum. 8 Copyright #### y ASME

9 Frequeny, Hz Amplitue.5..5 Displement n Envelope Instntneous Frequenies Nturl unmpe frequeny Figure. Moel 8 - "Self exite + sweep" system: ) isplement n envelope; ) instntneous frequenies. Displement Amplitue Fore Skeleton urve n FRF Frequey, Hz Elsti Stti Fore Displement Veloity Amplitue.5.5 Dmping urve... Dmping oeffiient.5.5 Frition Fore Veloity Figure. Moel 8 - "Self exite + sweep" system: ) skeleton urve n FRF; ) elsti stti fore; ) mping urve; ) frition fore hrteristis. Fore Conlusions This pper hs onsiere the etils n otine results of estimting the prmeters of qusiperioi systems onsisting of sum of given numer of hrmonis signls. The HT ientifition methos use, FREEVIB n FORCEVIB, enle etile reonstrution n seprtion of the tul omine non-liner elsti n the frition fore of the eqution of motion of n SDOF system. The propose HT tehnique llows ientifition of liner, non-liner, n moulte prmeters uner ifferent kins of exittion inluing the qusiperioi input signl with only two hrmonis omponents. HT ientifition llows reonstrution of the teste ynmi struture in wie ontinuous frequeny rnge roun the resonne, wheres the tritionl Fourier trnsform enles estimtion only of the two orresponing isrete frequeny points on the FRF. A further ontinuous rel-time estimtion of the initil prmeters of non-liner ynmil systems n e use, for exmple, for mesurement, monitoring n ignostis purposes. ACKNOWLEDGMENTS I woul like to thnk Dr. Oe Gottlie for his help in hoosing the qusiperioi virtion moels n their prmeters. REFERENCES. M. Felmn, "Non-liner system virtion nlysis using Hilert trnsform -- II. Fore virtion nlysis metho "FORCEVIB"". Mehnil Systems n Signl Proessing, 994, 8(), pp M. Felmn, "Non-Liner Free Virtion Ientifition vi the Hilert Trnsform," Journl of Soun n Virtion. v 8, n, Deemer 4, 997, pp M. Felmn, "Hilert Trnsforms", Enylopei of Virtion, Aemi Press, pp ,. 4. M. Felmn. "Mtl progrms for the HT ientifition", Otoer, from Tehnion. We site: 9 Copyright #### y ASME