Abstract Natural frequecies, mode shapes ad modal dampig values are the most importat parameters to describe the oise ad vibratio behaviour of a mechaical system. For rotatig machiery, however, the directivity of the propagatio wave of each mode should also be take ito accout. For rotatig systems, this directivity ca be determied by complex modal testig. I this paper, a rollig tire is represeted as a flexible rig model. The limitatio of applicatio of the complex modal testig which requires two directioal measuremets at a certai poit, which is difficult to measure i practice, has bee overcome through a modified complex modal testig which requires oly oe directioal measuremets at ay two poits. The techique is described i detail ad applied to both a umerical example ad to a experimetal data set of a real rotatig tire. Keywords: tire dyamics, rotatig tire aalysis, forward ad backward waves, complex modal testig 1 Itroductio The dyamic characteristics of rollig tires have bee widely studied i literature by lookig at modes ad propagatig waves [1-3]. Next to the atural frequecies ad the dampig values, the directivity of the propagatio wave of each complex mode (forward ad backward wave) is a importat parameter for the uderstadig of the dyamic characteristics of rollig tires. Furthermore, a accurate estimatio of the complex mode shape itself is also of great importace [4]. owever, the experimetal setup eeded to idetify the mode shapes usig covetioal modal testig approaches is quite complex ad expesive as may poits should be measured o the rotatig object [1, 2]. For this reaso, Lee [5-7] itroduced a complex modal testig (CMT) approach for rotatig machiery, which makes it possible to estimate the directivity of each wave with oly two resposes measured at differet poits ad two forces applied at differet locatios. The equatio of motio for the rotatig system is composed of two directios (e.g. for the rotor/bearig system it is described i the Y ad Z directio ad for the flexible rig model it is described i the radial ad circumferetial directio). These two directioal vibratio measuremets at a certai poit are used i the covetioal CMT techique. I curret experimetal setup of a rotatig tire, it is very difficult
to measure the both directioal resposes (radial ad circumferetial directio) at a certai poit. Therefore, the covetioal CMT ad the modified CMT that is proposed i this paper ca be classified by followig explaatios. The modified CMT techique, which uses oly oe directioal measuremet measured at ay two poits (i this research the radial directio is used), is itroduced with the flexible rotatig rig model to get same results to the covetioal CMT. Ad to itroduce i a simplified maer, the covetioal CMT is developed for estimatig the directivity of each wave as well as the degree of aisotropy or asymmetric of rotors by usig directioal frequecy respose fuctio (dfrf). I this techique, the dfrf is idetified with two directioal vibratio measuremets, which compose the equatio of motio of a rotor/bearig system i a Cartesia coordiate, at oe positio. The proposed modified complex modal testig i this study is developed based o a rotatig flexible rig model. I the modified techique, the dfrfs are composed of two vibratioal measuremets i radial directio measured at two differet positios ad these are used to cofirm the directivity of each wave by the differece of magitudes betwee the results i the positive ad egative frequecy regios of the dfrfs I fact there are several limitatios o aalysig dyamic behaviour of a rotatig tire with the flexible rig mode as itroduced i Ref. [8]. For istace, a rollig tire is deformed by vehicle s weight durig drivig coditio [3]. The proposed rig model, however, has always a circular cofiguratio durig rotatig coditio, i additio, tire icludes bedig mode i axial directio but rig model caot take this ito accout [4]. Despite of these limitatios, the reaso of adoptig the flexible rig model is that it is easy to cofirm ad to estimate the rotatioal effects (forward ad backward wave) i the aalytic FRF result. Ad this study does ot focus o ivestigatio of dyamic chages of a rollig tire by loadig ad rotatig coditios but developmet of a method to estimate the rotatioal effects. I additio, it is observed that the estimated rotatioal effects cofirmed by the proposed method are i well agreemet with the cofirmed results i experimetal measuremet withi iterestig frequecy regio. ece, it ca be cocluded that the itroduced flexible rig model is effective to aalyse rotatioal effect of a tire. I the curret paper, the covetioal CMT theory will be briefly itroduced ad two measuremet
coditios will also be itroduced to estimate the directivity of each wave with cofidece for the modified CMT techique. Ad the approach with these coditios will be applied o the case of a rotatig tire. First the approach is applied o the umerical case of a tire rig model [9] ad afterwards o measuremets o a real rotatig tire. 2 Complex Modal Testig [5] 2.1 Directioal frequecy respose fuctio The equatio of motio of a isotropic rotor/bearig system uder complex force excitatio ca be writte as with Mc p Cc pkc p g (1) M M jm, C C jc, K K jk c 1 2 c 1 2 c 1 2 M M M, C C C, K K K 1 yy zz 1 yy zz 1 yy zz M M M, C C C, K K K 2 yz zy 2 yz zy 2 yz zy ad p y jz, g f jf y z I eq. (1), vector p ad g represet the complex respose vector ad the complex force vector which are composed of displacemet vectors ad exteral force vectors i y ad z directio, with y ad z describig the plae perpedicular to the axis of rotatio, ad the size of each vector is N 1. I additio, subscript c represets complex matrices that are composed of complex values. Ad the size of M c, C c ad K c is NN ad the equatio ca be trasformed ito a state space form with Ac w c + Bc w c = F c (2) Mc -Mc p A c =, B c =, w c = ad F c = Mc Cc Kc p g The size of the state matrices A ad B, i eq. (2), is 2N2N ad these are composed of complex c c values ad are o-symmetric matrices. The size of the state respose vector w c ad the state force vector F c is 2 N1. If the eigevalue problem of eq. (2) is solved with the assumptio that F c is
equal to, the the eigevectors ad adjoit eigevectors ca be obtaied, ad the bi-orthogoal property ca be satisfied as follows l l A r cj c ci ji B r cj c ci i ji I eq. (3), the vectors r c ad l c represet eigevectors (right eigevectors) ad adjoit eigevectors (left eigevectors), respectively, ad λ represets the eigevalues of the system, which are expressed by complex values. The real terms of λ imply eergy dissipatio ad the imagiary terms of λ refer to the resoace frequecies of the system. ere, positive (+) resoace frequecies represet propagatig directivity of the wave i forward directio, while egative (-) resoace frequecies represet (3) propagatig i backward directio with respect to the rotatig directio of the system. I eq. (3), δ ij represets the Kroecker delta fuctio, which is defied as 1, whe the idex i = j, ad as otherwise. The superscript idicates the ermitia matrix. The eigevectors ad adjoit eigevectors are composed as φc ψc rci, lcj (4) φc i ψc j ere, the bar represets the complex cojugate. The φ c ad ψ are the modal vector ad adjoit modal vector of the eq. (1) ad the modal vectors ca be used to express the complex displacemet vector p. N 1 if, B i t () t c c c p φ (5) I eq. (5), η c represets the geeral coordiate of the complex displacemet vector p ad the idex i stads for each propagatig directio where, F is the forward ad B is the backward wave compoet of each resultig complex mode. Combiig eq. (5) ad (1), the relatio betwee the complex force ad the complex respose is obtaied i frequecy domai (eq. (6)) ad the elemets of the directioal frequecy respose fuctio (dfrf) are represeted i eq. (7). P( j) ( j) G ( j) (6) N c c ( j) φ ψ (7) 1 if, B j Note that the dfrfs i eq. (7) do ot have a cojugate part (i.e. each wave is represeted by first order i
basis ot secod order basis) which esures that the forward wave, which is represeted by a positive frequecy, is oly observed i the positive frequecy regio ad the backward wave, which is represeted by a egative frequecy, is oly observed i the egative frequecy regio. Therefore, dfrf aalysis of rotatig systems makes it possible to estimate the propagatig directivity of each wave whe the dfrfs are ivestigated i both frequecy regios (positive ad egative). 3 Modified Complex Modal Testig for a flexible rig 3.1 Trasfer fuctio of a rotatig flexible rig i global coordiates I previous works [9-11], a rig model has bee used to aalyse the dyamic characteristics of tires. The rig is represeted i cylidrical coordiates ad two referece systems ca be used to express its dyamic behaviour. Oe is a rotatig referece which is a local coordiate system. The other oe is a fixed referece which is a global coordiate system. I the preseted research work, laser Doppler vibrometer is adopted for gaiig vibratio iformatio of the tire vibratio patter. Therefore, the global coordiate system is used to aalyse the dyamic behaviour. Fig. 1 shows the cofiguratio of the rotatig flexible rig ad its two coordiate systems (global ad local). Fig. 1 Global ad local coordiates of a rotatig flexible rig. The equatio of motio of a rotatig rig cosiderig dampig i global coordiates is as follows [9, 12]
with, t, t, t, t Mu + Cu + Ku = q (8) A M A (9-1) j2 2 C A 2 j2 (9-2) 4 3 EI EA 2 2 2 EI EA 2 A 4 2 j j 2 4 2 R R R R K (9-3) 3 4 EI EA 2 EI EA 2 2 2 j 2 A 4 2 4 2 j R R R R ur qr u, q (9-4) u q I eq. (8), M, C ad K represet the mass, dampig (with gyroscopic terms) ad stiffess matrices, respectively, while, the vectors u ad q represet displacemets ad exteral forces i radial ad circumferetial directios, respectively. The size of each matrix is 2 2 ad the detailed descriptio of these matrices ad vectors are show i eq. (9). I eq. (9), ρ is the material desity, A is the cross sectioal area, λ is the iteral dampig, Ω is the rotatioal speed, E is the Youg s modulus ad I is the momet of iertia. The equatio shows that the mass matrix is diagoal, while, the dampig ad stiffess matrices are skew-symmetric matrices which are composed of complex values. I eq. (9-4), u r ad u Θ represet the radial ad circumferetial displacemet for th mode, receptively, ad these are fuctio of time ad spatial domai. u (, t) U e, u (, t) U e (1) r r j( t) j( t) The equatio of motio (eq. (8)) ca be represeted i state space as follows with Ap + Bp = f (11) j M j -M u() t A = e, B = e, p ( t) = ad f ( t) = M C K u( t) q( t) I the previous sectios, matrices A ad B were oly composed of system properties, however, i this case (rotatig tire rig model), the matrices are composed of complex values ad their size is 4 4 ad they iclude the space term which is show i eq. (1). Therefore, the respose vector p is oly a
fuctio of time. If the vector f is put to zero ad the eigevalue problems are solved, the, both eigevectors ad adjoit eigevectors ca be obtaied; these eigevectors satisfy the bi-orthogoal property. T l j Ari ji (12) T l j Bri i ji i, j 1, 2,..., N ere, the superscript T represets the traspose matrix. Ad the eigevectors ad the adjoit eigevectors are made up of the combiatio of modal vectors. r e i j φ j ψ, l j e φ i ψ j The space term ( e ) is icluded i the eigevectors ad the adjoit eigevectors because the term is multiplied to the matrices A ad B as a scalar. Therefore, the vectors r ad l are depedet of space whereas the modal vectors φ ad the adjoit modal vectors Ψ are the uique properties of the system, which are space idepedet. ere, the modal vectors are composed of radial ad circumferetial directioal elemets (φ r, φ Θ ad Ψ r, Ψ Θ). I order to get the FRF, the state vector p ca be represeted as follows p = R η (14) ere, R idicates the matrix of eigevectors ad η idicates the vector of geeral coordiates. Substitutig eq. (14) ito eq. (11) ad multiplyig L (the matrix of adjoit eigevectors) to the left had of both sides, oe ca obtai j (13) si-λ η = L f (15) I eq. (15) s idicates Laplace domai, I idicates the idetity matrix of size 44, λ is a diagoal matrix composed of eigevalues. Pre-multiplyig the iverse of si-λ to both sides of the equatio gives a equatio havig vector η aloe o the left had side. Oe ca obtai the result (eq. (16)) after multiplyig matrix R to both sides of the resultig equatio. Eq. (16) ca be expressed i summatio form. 1, s s s p R I - λ L f (16) i N N T r l r l r l p, s f s s N ; if, B s f - 1 if, B s - s - i (172)
Eve though the system matrices for th mode A ad B have size of 44, if oe takes ito accout N umbers of mode, the the size of these system matrices ca be cosidered as 4N4N likewise the rotatig system; therefore, the respose of the rig should be expressed by 4N bases fuctios, as ca be deduced from eq. (17) (2N (-N to N) 2 (forward, backward)). I eq. (11), the displacemet vector u ad the exteral force vector q are located i below half of the respose state vector p ad the force state vector f, respectively. Therefore, ot every elemet i matrix eq. (17) eeds to be cosidered. It is oly ecessary to cosider the lower part of the eigevectors ad the adjoit eigevectors to express vector u. N T Bφ B ψ φ ψ u(, s) q( s) 1 if, B s - s - (18) j s f where B e Because both the eigevectors ad adjoit eigevectors iclude the space iformatio (θ) as show i i eq. (13), it is also icluded i eq. (18). ere, the locatio of measuremet is idicated by subscript s ad the locatio of the force by subscript f. The scalar B stads for the phase delay which is determied by the locatios of the measuremet ad the force. Oe ca cofirm that the priciple of reciprocity [13] does ot hold i the case of flexible rotatig rig sice the phase i the scalar B is chaged whe the two locatios (measuremet ad force) are switched. The equatio ca be trasformed ito the frequecy domai by substitutig s by jω ad represetig the result i matrix form. r (, j) rr (, j) r (, j) r ( j) (, j) (, j) ( ) or U Q U Q (, j) r (, j) (, j) (19) U Q ( j) Eq. (19) shows the frequecy respose matrix (FRM) i which every FRF related to each directio is show. The elemet for the radial displacemet ad the radial force i the FRM is i N T Bφr ψr Bφr ψ r rr (, j) (2) 1 if, B j j From eq. (18) ad (2), the FRF of a rotatig rig follows with the followig relatio ( j) ( j), i, k r, (21) ik ik I other words, if a FRF is ivestigated i both positive ad egative frequecy regio, the magitude i the egative frequecy is just the replica of the result i the positive frequecy; therefore, it is
impossible to estimate the iformatio of the propagatig directivity of each wave with a geeral FRF similar to the rotatig system. The CMT [5] was itroduced for the lumped elemet system (rotor/bearig). I this system it is possible to measure the two directioal (y, z) resposes ad forces i a Cartesia coordiate to comprise the complex respose ad force. owever, i order to apply the CMT to a flexible rig model, it is required that the radial ad circumferetial displacemet ad force are directioal elemets i the equatio of motio. I real experimetal cofiguratios of a rollig tire, the circumferetial displacemet ad force are impossible or at least very difficult to measure with curret techology (a detailed explaatio of the experimetal cofiguratio is give i sectio 5). Therefore, i this research, two vibratio measuremets oly i radial directio, which are measured at differet locatios, ad the radial forces applied at that time (the force always applied at the same positio) are used to comprise the complex respose ad the complex force, p u ju g f j f (22) 1 2, r r ere, u 1 ad u 2 are the radial displacemets measured at differet positio (i.e. Θ s = Θ 1 ad Θ 2), uder the assumptio that the force is applied with the same magitude at the same positio i every experimet, f r idicates the applied radial force. Eq. (22) ca be used to obtai the followig results. P 2 ( j) 1 2 j( 1 2) G P 2 ˆ ( j) 1 2 j( 1 2) Gˆ Pˆ 2 ˆ ( j) 1 2 j( 1 2) G Pˆ 2 ( j) 1 2 j( 1 2) Gˆ U U where, (23) 1 2 1 2 Fr Fr I eq. (23), P ad G are trasformatios of p ad g ito the frequecy domai ad ˆP ad Ĝ are trasformatios of the complex cojugate of p ad g ito the frequecy domai. 1 ad 2 idicate the FRFs i terms of the locatio of measuremet at Θ 1 ad Θ 2. I other words, it is cofirmed that the two sided dfrfs show differet magitudes betwee the egative frequecy result ad the positive frequecy result ad it ca be also expected that the directivity of each wave ca be estimated from this. Detailed uderstadig for the dfrfs i eq. (23) is coducted i the ext sectio.
3.2 Directioal frequecy respose fuctio Itroduced i the previous sectio, the directioal FRFs obtaied by the radial displacemet measured at two positios ad radial force applied at oe positio caot be aalysed by the equatio of motio i the same way as was doe i sectio 2.1 because the equatio of motio of a rotatig rig is coupled (radial ad circumferetial directio) due to Coriolis effects whereas the values i eq. (23) oly cosist of radial iformatio. Therefore, i this research, each magitude of a directioal FRF at a resoace frequecy (jω ) ad at the opposite frequecy (-jω ) ivestigated. Through this, the propagatig directivity of the each wave is estimated. I eq. (23), ca be expressed as ( j) 1 2 j( 1 2) / 2 2 K(1 j) (1 j) / 2 1( j) (24) where K( j) a j b 2( j) ere, K idicates the ratio of 1 ad 2. The result ca be assumed as a complex value cosistig of real costats a ad b ad with also K( j) K( j) (25) Substitutig K = a + jb ito eq. (24) yields ( j) 2 ( a b 1) j( a b 1) / 2 2W where W A ad A ( a b 1) ( a b 1) 2 2 From eq. (26), it is cofirmed that the directioal FRF ca be expressed as the product of 2 ad W ω (26) ad the magitude of W ω is represeted by A. I other words, it is cocluded that the magitude of at the positive frequecy ca be expressed by the product of the magitude of 2 ad A. The magitude of at the egative frequecy is ( j) 2 K(1 j) (1 j) / 2 (27) It ca be expressed as i eq. (27) i which 2 ad K are represeted by the complex cojugate of the positive frequecy results of those i the egative frequecy. Substitutig the cojugate of K ito the equatio yields
( ) j 2 ( a b 1) j( a b 1) / 2 2 W where W From eq. (28), the directioal FRF at the egative frequecy is represeted as the product of 2 ad A (28) W -ω ad the magitude of W -ω is represeted by A. From comparig eq. (26) ad eq. (28), it follows that the magitude of 2 ad 2 does ot affect the magitude of the directioal FRF betwee the positive ad the egative frequecy sice the relatio of the cojugate gives the same magitude. Therefore, the compariso ca be achieved by the differece i magitude of each W at the positive ad the egative frequecy. A A 8b (29) Eq. (29) shows that the magitude of the directioal FRF at the positive ad the egative frequecy is govered by the imagiary term of K. If the imagiary term has a egative value (i.e. b<), the it ca be cocluded that ( j) ( j) (3) The forward ad backward waves of a rotatig rig are represeted by the egative ad the positive frequecies [1, 14]. Therefore, if the imagiary value of the K satisfies the coditio (b<), it implies that the forward wave has a larger magitude i the egative frequecy regio, while the backward mode has a larger magitude i the positive frequecy regio. This fact is used to estimate the directivity of each wave with directioal FRFs. The FRFs for radial directio i eq. (2) are re-expressed for a deeper uderstadig of K. rr N js f js f T N js f js f e e Q e Q e ( j) 1 i F, B j j - 1 if, B j - j s f j s f j - j j j j Q e j j Q e j j N - 1 if, B j where Q e ad j (31) Eq. (31) is the represetatio which is replaced from the eq. (2) expressed by two umbers of first order basis to oe umber of secod order basis ad the mode shape iformatio is also replaced with the Euler s form represeted by combiatio of amplitude Q ad phase Θ from the product of φ ad i i i
Ψ. The eigevalue of a rotatig rig λ is represeted by the combiatio of a real value σ ad a imagiary value ω, where the real value stads for the eergy dissipatio ad the imagiary value stads for the resoace frequecy. Expressig K at a resoace frequecy (ω ) usig eq. (31) yields j j e j j e j j K( j ) 1 f 1 f - j 2 f j 2 f e j j e j - j 2 2 j2 1 f j21 f e j e j2 2 f j22 f e j e e e ' j21 ' j2 2 j2 where e j2 Eq. (32) shows the ratio of FRFs at two locatios, which results ito the ratio of the residues of each mode, assumig that each mode is well separated from each other, which is the case at the resoace frequecies. Represetig the result i a trigoometric form to idetify the imagiary value yields ' j21 e ' j 2 ' ' cos2 1 si 2 1 ' ' cos2 2 j si 2 2 A A B B j A B A B 1 2 1 2 2 1 1 2 2 2 A2 B2 ' ' 1 j 1 2 2 cos 2 si 2 K( j ) 2 ' ' e cos 2 j si 2 j A jb A jb ' where i i f, i 1, 2 1 1 2 2 ere, A ad B represet the real ad imagiary terms comprisig the deomiator ad the umerator of K. Idex i represets the measuremet positio 1 ad 2. Cosiderig oly the imagiary term i eq. (33), the sig of the imagiary term is govered by the umerator sice the deomiator has always a positive value. This umerator of the imagiary part ca be arraged by applyig the trigoometric idetity. A B A B si 2 2 si cos (34) 2 2 1 1 2 1 2 1 2 1 2 I the expressio of eq. (34), it is expected that the magitude of (32) (33) 2 is much larger tha by eq. (32) (i the experimetal results of a rotatig tire, it is observed that the magitude of 2 is
ormally te times bigger tha the magitude of ). Therefore, it ca be assumed that the result of eq. (34) will be domiated by the first term of the equatio. Uder the assumptio that the positio of the secod ode is ahead of the first ode (i.e. Θ 2> Θ 1), the phase i the first term should be located at -18 ~ to have a egative value, hece, the followig relatio ca be established. 18 2 1 (35) 2 Eq. (35) shows that the locatios of the two measuremets ca be decided based o the rage of the iterested mode umber. For example, if oe has iterest up to 9 th mode, the two locatios should be located withi 1. For the case of takig ito accout the secod term of the equatio, the result ca be chaged. The cosie part of the secod term is assumed to have uit value (i.e. the maximum value of cosie) sice the phase i the cosie caot be estimated by experimets due to 1 ad 2. The, the two measuremet positios should satisfy the followig relatio to have a egative value of eq. (34) by cosiderig the sie i the secod term. 9 2 1 (36) 2 The relatio i eq. (35) ca be assumed as a ecessary coditio for applyig the modified CMT techique ad eq. (36) ca be assumed as the sufficiet coditio. If eq. (36) is satisfied, the directioal FRF ( j) shows larger value of the residue i the egative frequecy for the forward wave ad i the positive frequecy for the backward wave. By usig this fact, the directivity of each wave ca be estimated. I a similar way as ( j), other directioal FRFs ca be idetified of characteristics i the two sided frequecy domai. Checkig the results, ( j) ad ˆ ˆ ( j) show a similar ˆ tred to ( j), while ( j) ad ˆ ( j) show opposite treds to ˆ ˆ ( j). These results ca be idetified i the ext sectio (simulatio). 4 Simulatio I this sectio, the modified CMT techique itroduced i the previous sectio will be verified by umerical simulatio by the aalytic formulae (eq. (2)). The rig model which is used i the simulatio
is composed of 6 evely distributed odes from to 36 with a icremet of 6 ad it rotates i couter clockwise directio with a speed of 8z. The applyig force is located at the bottom of the rig i global coordiates (Fig. 2). Fig. 2 Cofiguratio of a rotatig rig for simulatio I the simulatio, the displacemet respose is obtaied by eq. (18). The parameters used i eq. (9) are summarized i table 1. The simulatio is coducted i two cases i terms of the locatio of measuremet. I the first case, it is assumed that the measuremets are located at (Θ 1) ad 6 (Θ 2). I the other case, the measuremets are located at (Θ 1) ad 3 (Θ 2). The first case satisfies both coditios, while the secod case does ot satisfy both coditios imposed by eq. (35) ad eq. (36) because the results are oly idetified up to the 7 th mode for the both cases. Table 1 Parameters for the rig model used i the preseted simulatio Youg's modulus E = 4.76 e9 N/m 2 Desity ρ = 145kg/m 3 Thickess h = 14.1e -3 m Radius R =.3m Width b =.35m Dampig ratio 5%
I the first simulatio (for the case of ad 6 ), it is assumed that oly the 3 rd mode exists ad the resposes of the applied force are obtaied at two positios. These two resposes ad oe force are used to obtai the FRFs ( 1 ad 2) ad the averaged result of the FRFs (Fig. 3 (a)). These data are also used to obtai directioal FRFs (Fig. 3 (b)). Displacemet (m).12.1.8.6.4.2 Geeral FRF (rr) -(1 + 2)/2- -1-8 -6-4 -2 2 4 6 8 1 Frequecy (z) Displacemet (m) Displacemet (m) Displacemet (m) Displacemet (m).4.2 Directioal FRF (dfrf) -1-8 -6-4 -2 2 4 6 8 1 Frequecy (z).4.2-1 -8-6 -4-2 2 4 6 8 1 Frequecy (z).4.2 p*g* -1-8 -6-4 -2 2 4 6 8 1 Frequecy (z).4.2 p*g * -1-8 -6-4 -2 2 4 6 8 1 Frequecy (z) (a) Fig. 3 (a) Averaged FRF for 3 rd mode at ( 1) ad 6 ( 2) (b) from top to bottom ( ( j), (b) ( j), ˆ ( j) ad ˆ ( j) ) The results of the resoace frequecies of the 3 rd mode idetified by eq. (12) are -56.91z for the forward wave ad 18.53z for the backward wave. The geeral FRF caot be used to estimate the directivity of each mode sice it has idetical magitudes i the positive ad the egative frequecy regios as show i Fig. 3 (a) (i.e. the magitudes are.727 at +18.53z ad -18.53z ad.159 at +56.9z ad -56.9z). Fig. 3 (b) shows the directioal FRFs i the sequece of ( j), ˆ ˆ ( j), ( j) ad ˆ ( j) goig from top to bottom. As predicted i the previous sectio, it is ˆ observed i the secod plot ( ( j) ) that the peak at the forward wave resoace frequecy (- 56.91z) shows larger magitude (.24) tha the magitude (.175) at the positive frequecy (56.91z)
peak, while the peak at the backward resoace frequecy (18.53z) shows lager magitude (.165) tha the magitude (.12) at the egative frequecy (-18.53z) peak. It is also observed that the plot of ( j) shows the same treds as ˆ ˆ ( j) but that the plots of ˆ ( j) ad ( j) ˆ show opposite treds to ( j). Through this figure, the predicted treds i the previous sectio are verified. Similar simulatios as show i the previous result ( ( j) i Fig. 3 (b)) have bee coducted for both coditios ( to 6 ad to 3 ) but with varyig the assumed mode from 4 th to 7 th to compare the variatio of the results with respect to measuremet positios (Fig. 4). Displacemet (m) Displacemet (m) Displacemet (m) Displacemet (m).2.1 Directioal FRF (dfrf) -p*g*- 4th mode -3-2 -1 1 2 3 Frequecy (z).2.1 5th mode -3-2 -1 1 2 3 Frequecy (z).2.1 6th mode -3-2 -1 1 2 3 Frequecy (z).2.1 7th mode -3-2 -1 1 2 3 Frequecy (z) Displacemet (m) Displacemet (m) Displacemet (m) Displacemet (m).2.1 Directioal FRF (dfrf) -p*g*- 4th mode -3-2 -1 1 2 3 Frequecy (z).2.1-3 -2-1 1 2 3 Frequecy (z).1.5 5th mode -3-2 -1 1 2 3 Frequecy (z).2.1 6th mode 7th mode -3-2 -1 1 2 3 Frequecy (z) (a) (b) Fig. 4 directioal FRF simulatio results ( ( j) ) for 4 th to 7 th mode from top to bottom (a) measured at (Θ 1) ad 6 (Θ 2) (b) measured at (Θ 1) ad 3 (Θ 2) The resoace frequecies (eq. (12)) -95.16 z, -141.14 z, -195.1 z ad -257.1 z are for the forward waves ad the frequecies 38.72 z, 67.34 z, 14.33 z ad 149.64 z are for the backward waves. Based o these results, ivestigatig the plots of the first simulatio (Fig. 3 (b)), it shows that every forward wave has larger magitude at egative frequecy tha the result i the positive regio, while every backward wave always has a larger magitude result at the positive frequecy. For the
secod simulatio case (Fig. 4 (b)), however, the same coclusios as draw from Fig. 4 (a) are cocluded for the 4 th ad 5 th mode, but for the 6 th mode it shows similar magitude results i both frequecy regios ad for the 7 th mode the larger value appears i the egative frequecy regio for the backward wave ad i the positive frequecy regio for the forward wave. From these results, it ca be cocluded that the rage of the estimated mode umber by the directioal FRF is restricted by the iterval of two measuremet positios (verificatio of eq. (35) ad eq. (36)). 5 Experimets I order to validate the modified CMT itroduced i this study, 45 locatios o a rollig tire are measured, excited by a broadbad excitatio. Two of these 45 measuremet locatios are selected to apply the techique ad the obtaied results are compared to the results which are obtaied by determiig the mode shapes from all the measuremets at all 45 odes. 5.1 Experimetal cofiguratio for a rollig tire The tire rolls o a steel drum ad is excited by a metal strip (so-called cleat) which is fixed to the drum surface. A Laser Doppler Vibrometer (LDV) is used to measure the vibratio velocity i the radial directio. As show i Fig. 5, a array of 45 mirrors is used to direct the laser beam i the radial directio towards the tire surface. Fig. 5. Cofiguratio for the vibratio measuremet of a rollig tire.
Sice poits close to the tire footprit are ot accessible, the measuremets start at about 48 clockwise (at 222 ) from the footprit/cleat cotact (referece at 27 ) ad ed at 48 couter-clockwise (at 318 ) from the referece. The 45 measuremet positios are evely distributed every 6 from the start poit ad the measuremet directio is clockwise. The respose at each fixed locatio has bee measured for 1 secods. The umber of cleat excitatios ad revolutios of the tire durig this time period are determied by the rotatioal speed of the tire. Tachometers are used to cout the umber of cleat excitatios (i.e. the umber of revolutios of the drum) ad the umber of revolutios of the tire durig 1 secods. The time respose due to every cleat passage is used i a time averagig process. At higher rotatio speeds, the umber of averages icreases. owever, the duratio of the time respose decreases sice the time betwee two cosecutive cleat passages become smaller. This causes a poorer resolutio i the frequecy domai. The tire spidle force sigal is used to sychroize the vibratio respose at differet poits alog the tire circumferece. The excitatio force at the tire footprit due to rollig over the cleat is ot kow. The measuremet coditios are listed i table 2. Table 2 Measuremet coditios for the rollig tire Measuremet coditios Tire rollig speed [km/h] 1 Tire dimesios 25/55 R16 Iflatio Pressure [bar] 2.2 Load [N] 7 Applied Excitatio 3x25 mm cleat Samplig Frequecy [kz] 64 Fig. 6 shows the results of the measuremet. Fig. 6 (a) shows the measured results of four quatities (tire vibratio velocity, vertical spidle force, drum tachometer ad tire tachometer) at fixed poit 34. All results show are ormalized by the maximum value of each quatity. Fig. 6 (b) shows the averaged velocity calculated out of the respose of 41 cleat passages. The duratio of a sigle drum revolutio is estimated to be.213 secods. Fig. 6 (c) shows all averaged ad sychroized velocity results ad (d) shows all averaged ad sychroized force results.
1 Normalized measuremet data 4 Averaged Measuremet at 34th Node Tacho Drum.8 Tacho Tire Laser Fz 3.6 2.4 1 Velocity (Voltage).2 -.2 Velocity (Voltage) -1-2 -.4-3 -.6 -.8-4 -1 1 2 3 4 5 6 7 8 9 1 11 Time (Sec.) -5.2.4.6.8.1.12.14.16.18.2 Time (Sec.) (a) (b) 8 45 Number of Measuremets (Averaged & Sychroized) 45 Number of Measuremets (Averaged & Sychroized) 4 x 14 6 3 4 2 Velocity (Voltage) 2-2 Force (Voltage) 1-1 -4-2 -6-3 -8.2.4.6.8.1.12.14.16.18.2 Time (Sec.) -4.2.4.6.8.1.12.14.16.18.2 Time (Sec.) (c) (d) Fig. 6 Example of measuremet data (a) 4 measured quatities (Tacho drum, tacho tire, laser ad spidle force) at fixed poit 34, (b) averaged velocity measuremet at fixed poit 34 over 1 secods, (c) all 45 averaged ad sychroized velocity resposes, (d) all 45 averaged ad sychroized vertical spidle force measuremets. Comparig the spidle force results of the 45 measuremets idicate that the cleat excitatio has a good repeatability ad test coditios stay costat durig the measuremet of the vibratio respose at the differet fixed poits. I additio, the compariso also shows that the differet resposes are well sychroized. The frequecy of the mai spidle force oscillatio correspods to the first vertical tire resoace frequecy [2]. The frequecy cotets of the vibratio velocity at differet poits are foud to be differet from each other. All tire resoaces have a differet cotributio to the total respose i
differet poits. From these 45 results which are obtaied by post processig, the dyamic characteristics of the tire ca be aalysed from the 45 averaged ad sychroized resposes. 5.2 Results For the rollig tire, the excitatio force as a result of rollig over the cleat is ot kow. Therefore, a geeral modal aalysis method, which uses frequecy respose fuctio, caot be applied. Istead, a Operatioal Modal Aalysis (OMA) method should be applied which oly requires resposes to estimate the modal parameters of the structure. Fig. 7 shows the measured power spectrum (solid lie) of the vibratio velocity for a tire rollig at 1km/h. The modal parameters of the rollig tire are estimated by usig oly respose power spectra sice the applied excitatio force caot be measured. The excitatio force due to rollig over the 3x25 mm cleat has a relatively flat excitatio spectrum. The PolyMAX modal parameter estimatio method [15, 16] is applied to the power spectra of the resposes because the frequecy respose fuctios of the rotatig system have a similar form (eq. 2) as compared to the statioary system. Thus, the estimated modal parameters ca be used to recostruct the frequecy respose. Fig. 7 shows a good agreemet betwee the measured ad estimated power spectra. This agreemet proves that the estimated modal parameter has fully described the measured dyamic behaviour. All estimated modal parameters (resoace frequecy, dampig ad mode shape) are show i Fig. 8.
6 5 Measured Power Spectrum Estimated Power Spectrum db (mm 2 /s 2 -Power of Velocity-) 4 3 2 1-1 -2 5 1 15 2 25 3 Frequecy (z) Fig. 7 Measuremet (solid lie) ad estimated results (dotted lie); Respose power spectrum ad its estimatio of a rollig tire (1 km/h) Fig. 8 Mode shapes for the rollig tire at 1km/h.
From Fig. 8, it is cofirmed that several modes are ot observed because the frequecy cotet of the excitatio force is depedet o the rotatio speed ad the cleat geometry for a rollig tire [1, 17]. More modes at higher frequecies will be idetified whe the rollig speed icreases. This idicates that the excitatio force levels at higher frequecies are higher as the rollig speed icreases. As show i Fig. 8, up to the 9 th mode has bee observed i the experimetal results. Therefore, the satisfyig locatio iterval betwee two sesor positios i eq. (36) is less tha 1. Ay two odes ca be selected i this regard, so here two cases are cosidered, oe is the 34 th ode (Θ 1 = 54 ) ad 33 th ode (Θ 2 = 6 ) ad aother oe is 19 th ode (Θ 1 = 144 ) ad 18 th ode (Θ 2 = 15 ), amog the odes satisfyig this coditio are selected to obtai the directioal FRF. The applied force f r is assumed as a impulse force. The obtaied directioal FRF ( ) is show i Fig. 9. 2.5 x 14 p*g* 5B 2 Velocity (Voltage) 1.5 1 6F 5F 4F 3F 1B 2B 4B 6B 7B 8B 9B.5-3 -2-1 1 2 3 Frequecy (z) (a)
1.8 2 x 14 p*g* 2B 5B 8B 1.6 1B 4B 1.4 6B 9B Velocity (Voltage) 1.2 1.8 6F 5F 4F 3F 7B.6.4.2-3 -2-1 1 2 3 Frequecy (z) (b) Fig. 9 directioal FRF ( (Θ 1) ad 15 (Θ 2) ) results of a rollig tire (a) measured at 54 (Θ 1) ad 6 (Θ 2) (b) measured at 144 As show i Fig. 9 (a), it is observed that for the frequecy ragig from -2z~2z, the most of magitude has larger values i the positive frequecy regio, while, it shows the opposite results for higher frequecy values (i.e. -33z~-2z ad 2z~33z). This tred show i Fig. 9 (a) is also show i Fig. 9 (b), eve though it has differet magitude. Ad all resoace frequecies for each directioal wave show i Fig. 8, which are estimated by PolyMAX, are also show i Fig. 9 with wave umber ad directioal iformatio (e.g. F ad B). All results show i Fig. 8 (resoace frequecy, dampig ad directivity of wave) are listed i table 3.
Table 3 * idicates a o-idetified mode The estimated residues i the egative ad positive frequecies are listed i table 3 with correspodig egative ad positive resoace frequecies. There is a small differece betwee the results, i which oe result is estimated by traditioal modal estimatio method (PolyMAX) ad aother oe is estimated by the directioal FRF. For the 5 th forward wave case, for istace, the resoace frequecy idicated by the modified CMT techique is -263.47z, because the residue i the egative frequecy is larger tha the positive frequecy, it has aroud 8z magitude differece with the frequecy result which is estimated by PolyMAX. PolyMAX used 45 measuremets while the directioal FRF used oly 2 measuremets to estimate each result. The differece i the umber of averages results i small differece betwee two results. From this table, oe ca cofirm that the results are exactly the same as those predicted i the previous sectio i that the magitude of the residue has a larger value i the positive frequecy for the backward waves (otified by B i the table) ad the egative frequecy for the forward waves (F). From this fact it ca be cocluded that the modified CMT techique makes it possible to estimate the directivity of each wave.
6 Coclusio There are two differet directioal waves i the rotatig systems due to the Coriolis effect. Geerally, directioal waves are examied by usig mode shapes which are obtaied from measuremets o a large umber of positios. The complex modal testig techique, however, ca give the directioal iformatio with oly two positio measuremets. I this research, theoretically ad experimetally, the difficulties of applyig the complex modal testig to rollig tire aalysis are itroduced. I order to apply the techique with cofidece, two coditios which should be satisfied i the experimet are itroduced ad these are explaied ad verified by simulatios. I additio, the techique is applied to experimetal measuremets where by usig oly 2 measuremet poits, the same coclusios are obtaied as i usig a full data set of 45 measuremet poits. 7 Ackowledgemet The authors would like to thak the EU Seveth Framework Program (FP7/21) for its support uder the TIRE-DYN project (grat agreemet o 251211). The first author received fiacig support from Dasa project ad LMS Iteratioal for the research stay i Leuve. e also thaks prof. Chog-Wo Lee gratefully for his techical advice i this research. Ad this work was partially supported by a Natioal Research Foudatio of Korea (NRF) grat fuded by the Korea govermet (214R1A2A1A15264). Referece [1] P. Kidt, Structure-bore tyre/road oise due to road surface discotiuities, Ph.D. thesis, Leuve, 29. [2] P. Kidt, D. Berckmas, F. D. Coick, P. Sas ad W. Desmet, "Experimetal aalysis of the structure-bore tyre/road oise due to road discotiuistes," Mechaical Systems ad Sigal Processig, vol. 23, pp. 2557-2574, 29.
[3] I. Lopez, R. E. Blom, N. B. Rooze ad. Nijmeijer, "Modellig vibratios o deformed rollig tyres - a modal approach," Joural of Soud ad Vibratio, vol. 37, pp. 481-494, 27. [4] Y. -J. Kim ad J. S. Bolto, "Effects of rotatio o the dyamics of a circuar cylidrical shell with applicatio to tire vibratio," Joural of Soud ad Vibratio, vol. 275, pp. 65-621, 24. [5] C.-W. Lee, "A complex modal testig theory for rotatig machiery," Mechaical Systems ad Sigal Processig, vol. 5, o. (2), pp. 119-137, 1991. [6] S. Gog, A study of I-Plae Dyamics of Tires, Ph.D. Thesis, TU Delft, 1993. [7] L. Kug ad W. Soedel, "Free vibratio of a peumatic tire-wheel uit usig a rig o a elastic foudatio ad a fiite elemet model," Joural of Soud ad Vibratio, vol. 17, o. 2, pp. 181-194, 1986. [8] P. Kidt, P. Sas ad W. Desmet, "Developmet ad validatio of a three-dimesioal rig-based structural tyre model," Joural of Soud ad Vibratio, vol. 326, pp. 852-869, 29. [9] W. Soedel, Vibratio of shells ad plates, third editio, revised ad expaded, New York: Marcel Dekker, Ic., 24. [1] D. J. Ewis, Modal Testig : Theory, Practice ad Applicatio, Secod Editio ed., Research Studies Press Ltd., 2. [11] C. G. Díaz, S. Vercamme, J. Middelberg, P. Kidt, C. Thiry ad J. Leysses, "Numerical predictio of the dyamic behaviour of rollig tyres," i ISMA212, Lueve. [12] W. eyle, S. Lammes ad P. Sas, Modal Aalysis Theory ad Testig, K. U. Leuve - PMA, 1998. [13] L. ermas ad. V. d. Auweraer, "Modal testig ad aalysis of structures uder operatioal coditios: Idustrial applicatios," Mechaical Systems ad Sigal Processig, vol. 13, o. (2), pp. 193-216, 1999. [14] P. Zegelaar, The dyamic respose of tyres to brake torque variatios ad road udeveesses, Ph.D. Thesis, TU Delft, 1998.
[15] C.-W. Lee ad Y.-D. Joh, "Theory of excitatio methods ad estimatio of frequecy respose fuctios i complex modal testig of rotatig machicery," Mechaical Systems ad Sigal Processig, vol. 7, o. 1, pp. 57-74, 1993. [16] C.-W. Lee ad C.-Y. Joh, "Developmet of the use of directioal frequecy respose fuctios for the diagosis of aisotropy ad asymmetry i rotatig machiery: theory," Mechaical Systems ad Sigal Processig, vol. 8, o. 6, pp. 665-678, 1994. [17] J. Lee, S. Wag, P. Kidt, B. Pluymers ad W. Desmet, "Dampig aalysis with respect to rollig speed by aalytic solutio of a flexible rig model ad its frequecy respose fuctio derivatio by modal summatio method," Iteratio Joural of Applied Mechaics, vol. 6, o. 5, 214 DOI: 1.1142/S17588251145549.