A Comparison of Dynamic Tyre Models for Vehicle Shimmy Stability Analysis

Transcription

1 A Comprison of Dynmi Tyre Models for Vehile Shimmy Stility Anlysis J.W.L.H. Ms DCT MS Thesis Supervisors: Prof. Dr. H. Nijmeijer (TU/e) Dr. Ir. I.J.M. Besselink (TU/e) Ir. S.G.J. de Cok (DAF Truks) Ir. R.M.J. Lieregts (DAF Truks) Mster Thesis Committee: Prof. Dr. H. Nijmeijer (TU/e) Dr. Ir. I.J.M. Besselink (TU/e) Dr. Ir. T. Hofmn (TU/e) Ir. R.M.J. Lieregts (DAF Truks) Prof. Dr. Ir. H.B. Pejk Eindhoven University of Tehnology Deprtment of Mehnil Engineering Dynmis nd Control Group Eindhoven, Otoer 009

2 Astrt In this thesis, omprison is mde etween different dynmi tyre models for the nlysis of shimmy instility. First, three different tyre models whih inlude the stright tngent tyre model, the Von Shlippe tyre model nd the rigid ring tyre model re disussed. The equtions of motion for these tyre models re derived nd re vlidted using mesurements. For this vlidtion, oth step responses t low forwrd veloity re used s well s frequeny responses t high forwrd veloities. The tyre models re then oupled to si front xle model nd the resulting equtions of motion re derived. For the system with the reltively simple stright tngent tyre model, nlytil expressions for the stility oundries re derived s funtion of system prmeters. Finlly, numeril nlysis of the eigenvlues nd eigenvetors of the system with the different tyre models is rried out. 1

3 Tle of Contents List of Symols Introdution Bkground nd motivtion Prolem sttement Outline of the thesis Tyre Modelling Survey of literture on dynmi tyre modelling Von Shlippe Stright tngent Kluiters Rigid ring Stte spe representtion nd rim interfe Tyre Model Comprison nd Vlidtion Prmeter vlues Step responses Frequeny responses Conlusions Bsi Front Axle Survey of literture on shimmy stility nlysis Font xle model Anlytil omputtion of the stility oundries Prmeter influene on the nlytil expressions of the stility oundries Numeril omputtion of the stility oundries Stility oundry sensitivity to prmeters djustments Conlusions Truk Front Axle Prmeter vlues Eigenvlue nd eigenvetor nlysis Conlusions Conlusions nd Reommendtions Conlusions Reommendtions Referenes A. Pdé versus Tylor Approximtion B. Equtions of Motion for the Rigid Ring Tyre Model C. Optimiztion of Prmeters for the Rigid Ring Tyre Model D. System Mtrix of the Front Axle with the Stright Tngent Tyre Model E. System Mtrix of the Front Axle with the Rigid Ring Tyre Model... 91

4 List of Symols i C Fϕ C Mϕ C Fα C Mα F y H i H y,x (s) H CM I k m M x M z n N q q i Q i r i R R e s s t t t p T U v 1 v V w 1, w x y y 1 y z z hlf of the tyre ontt length oeffiient of the hrteristi eqution stiffness turn slip stiffness for the lterl fore turn slip stiffness for the self ligning moment ornering stiffness self ligning stiffness tyre lterl fore i-th Hurwitz determinnt trnsfer funtion: input x, output y ngulr momentum with respet to the entre of mss moment of inerti dmping mss overturning moment tyre self ligning moment steering xis offset with respet to the wheel entre distne etween entre of grvity nd swivel xis, N = q + n entre of grvity offset with respet to the wheel entre i-th generlised oordinte i-th generlised fore i-th stility oundry expression tyre rdius effetive rolling rdius Lple vrile trvelled distne time tyre pneumti tril kineti energy potentil energy defletion of the tyre string t the leding ontt point defletion of the tyre string t the triling ontt point forwrd veloity weighting ftors longitudinl diretion lterl position, lterl diretion lterl position of the tyre string t the leding ontt point lterl position of the tyre string t the triling ontt point intermedite vrile vertil diretion 3

5 α α' γ δq i δw ε η λ ξ σ τ ϕ ϕ' ψ ω Ω tyre side slip ngle tyre deformtion ngle mer ngle virtul hnge in generlised oordintes (Lgrnge equtions) virtul work (Lgrnge equtions) ster ngle mplitude rtio root of the hrteristi eqution reltive phse ngle tyre relxtion length time dely turn slip trnsient turn slip yw ngle ngulr veloity ngulr veloity of the wheel A 1 trnsformtion mtrix etween frme 1 nd A system mtrix (stte spe) B input mtrix (stte spe) C output mtrix (stte spe) C stiffness mtrix D feedthrough mtrix (stte spe) F mtrix relted to the tyre fores I unity mtrix I CM inerti tensor with respet to the entre of mss K dmping mtrix M mss mtrix q vetor ontining generlised oordintes Q vetor ontining generlised fores u input vetor W v, W p mtries relted to the tyre deformtion ngle x stte vetor y output vetor Z 1, Z, Z 3 omponent mtries xɺ xɺɺ x x A -1 A T first time derivtive of x seond time derivtive of x vetor x mgnitude of x inverse of mtrix A trnspose of mtrix A 4

6 Susripts nd supersripts r rr st t vs w x y z α γ ψ ϕ ω xle elt ontt pth rim rigid ring stright tngent totl (xle nd wheel) Von Shlippe wheel x-diretion y-diretion z-diretion side slip rottionl omponent round the x-xis rottionl omponent round the z-xis turn slip rottionl omponent round the y-xis 5

7 1. Introdution 1.1 Bkground nd motivtion Osionlly, vehiles my show n unstle osilltion of the steered wheels. This phenomenon n our on motoryles, utomoiles nd irplnes. This osilltion is referred to s shimmy nd is used y vriety of flexiilities in the design of the suspension of the vehile. This instility is not only n unomfortle phenomenon for oupnts, ut n use more dngerous effets suh s loss of ontrol, exessive tyre wer nd even filure of mehnil omponents. Pejk [Pejk; 1966, Pejk; 004] points out tht there re two types of shimmy instility, eh used y different ftors. One of the first investigtors who developed theory for the shimmy motion of utomoiles is Fromm [Beker, Fromm & Mruhn; 1931]. He investigtes the first type of shimmy where the min ftor using this instility is the gyrosopi oupling etween the ngulr motions of the wheel out the longitudinl nd steering xis. The min ftors using the seond type of shimmy instility re the lterl omplines in tyres nd suspension. This type of instility is disussed in this thesis. At DAF Truks N.V. in Eindhoven, the shimmy phenomenon is studied. However, nlysis methods nd models re required to ssess the sensitivity of ertin design modifitions with respet to stility. The shimmy sensitivity is generlly ddressed y the nturl dmping or frition round the steering xis. It is, however, desirle to design front xle tht is stle regrdless of the dmping in the system. 1. Prolem sttement This projet fouses on the shimmy stility of steered front xle tyres. The prolem presented in this thesis n e summrised s follows: Provide detiled prmeter study regrding shimmy stility in si front xle model. The front xle model is split into tyre model nd suspension model to nlyse nd vlidte the dynmi ehviour of the seprted tyre models. 1.3 Outline of the thesis As stted in the prolem sttement, distintion is mde etween the tyre model nd suspension model. A shemti representtion of this nd of the nomenlture used is displyed in Figure 1.1. Figure 1.1. Shemti representtion nd nomenlture of the two prts of the front xle. 6

8 The initil fous in this thesis is on the tyre modelling. The tyre models disussed here hve the restrition tht they re only ple of showing liner tyre ehviour, thus limited to smll side slip ngles. At first, the simple stright tngent tyre model developed y Pejk [Pejk; 1966] is used, ontinuing with Kluiters pproh to the tyre model developed y Von Shlippe [Von Shlippe & Dietrih; 1954] nd finlly ends with the reently developed rigid ring tyre model, sed on the work of Murie [Murie; 000] nd Zegelr [Zegelr; 1998]. Chpter dels with the derivtion of these tyre models. In Chpter 3, the dynmi ehviour of these tyre models is nlysed using oth step responses t low forwrd veloity nd frequeny responses t higher forwrd veloities. The frequeny responses re nlysed t three different forwrd veloities of 5, 59 nd 9 km/h. Both step nd frequeny responses onsist of three different inputs whih re pure yw input, side slip input nd turn slip input. Susequently, the responses re vlidted using mesurements on pssenger r tyre where ville. The fous in the seond prt is on the stility nlysis of si front xle model with the tyre models used in the first prt of this thesis. In the first prt of Chpter 4, equtions of motion re derived for the suspension model with two degrees of freedom nd re oupled to the tyre models. With the equtions of motion eing derived, the fous in remining prt of Chpter 4 is on the stility oundries. First, nlytil expressions for the stility oundries re derived for the front xle with the stright tngent tyre model. Seond, numeril omputtion of the stility oundries is mde using ll three models for prmeter vlues of pssenger r. Chpter 4 ends with n overview of the sensitivity of the stility of the front xle with ll three models to the most influentil prmeter hnges. In Chpter 5, the sme front xle model is used with prmeter vlues of truk for oth the tyre nd the suspension. The nlysis in this hpter is sed on n eigenvlue nd eigenvetor nlysis of the front xle model with ll three tyre models. The dmping nd eigenfrequenies of the two modes representing the two degrees of freedom of the suspension re lulted s well s the mplitude rtio nd the reltive phse ngle of the motions of these two degrees of freedom. Finlly, onlusions re drwn nd reommendtions for future reserh re given in Chpter 6. 7

9 . Tyre Modelling In this hpter, the equtions of motion of three dynmi tyre models re derived. These tyre models re the stright tngent tyre model, the Von Shlippe tyre model nd the rigid ring tyre model. These tyre models re nlysed nd vlidted using step nd frequeny responses further on in this thesis. The tyre models presented in this hpter hve the restrition tht they n only show liner tyre ehviour in stright line motion nd re restrited to lterl dynmis. Setion.1 strts with short survey of literture, followed y the explntion of the Von Shlippe tyre model in Setion.. In Setions.3 nd.4, two of the derivtives of the Von Shlippe tyre model re disussed. These re the stright tngent tyre model nd Kluiters pproh respetively. In Setion.5, the rigid ring tyre model is disussed. Finlly, the models re trnsformed into stte spe representtion in Setion.6..1 Survey of literture on dynmi tyre modelling Over the yers, mny tyre models hve een developed. The first models tht re used in the nlysis of shimmy re sed on n pproh where the rod-tyre interfe is redued to single ontt point. However, in 194 Von Shlippe [Von Shlippe; 1954] introdued the onept of strethed string with finite ontt length. Mny tyre models sed on this onept hve sine then een developed nd some of them re disussed in this hpter. Besselink [Besselink; 000] mde detiled overview of severl of the derivtives of the strethed string pproh on whih the mjority of this setion is sed. More reently, the dynmi ehviour of the tyre hs een studied nd hs resulted in rigid ring pproh. This rigid ring pproh is sed on the work of Murie [Murie; 000] nd Zegelr [Zegelr; 1998]. The stright tngent tyre model developed y Pejk [Pejk; 1966] is simple liner pproximtion of the strethed string onept. It uses only the defletion in the leding ontt point of the tyre to lulte the lterl fore nd self ligning moment generted y the tyre. Due to the simpliity of this model, Besselink [Besselink; 000] proves tht this model eomes less urte t low forwrd veloities. More elorte models hve, mongst others, een developed in [Segel; 1966], [Von Shlippe; 1954], [Rogers; 197], [Pejk; 1966] nd [Smiley; 1957]. Unlike the stright tngent pproh, these models onsider oth the defletions of leding nd triling ontt points of the tyre. The strethed string pproh ssumes tht no sliding ours in the ontt re nd s result, the triling ontt point follows the sme pth s the leding ontt point with time dely. Due to this time dely, n exponentil funtion ours in the nlytil expressions for the tyre trnsfer funtions of the strethed string pproh. This n e seen from the expressions of Segel [Segel; 1966] nd Von Shlippe [Von Shlippe; 1954]. Segel inludes the defletion of the entire string nd this is therefore ext in his expressions. However, Pejk [Pejk; 004] points out tht Segel forgets to inlude the rdil fores ting on the irulr shpe of the string due to the string tension. Beuse of the lterl defletion of the string, these rdil fores use moment round the vertil xis. Von Shlippe pproximtes the ontt line y forming stright onnetion etween the ext leding nd triling ontt point. Getting rid of the exponentil funtions in order to otin set of differentil equtions with onstnt oeffiients, severl pproximtions hve een mde. These pproximtions redue the omplexity of the mthemtis involved. For exmple, Smiley [Smiley; 1957] uses Tylor series to pproximte the lterl position of the entre of the ontt pth. Severl derivtives of oth Segel s nd Von Shlippe s expressions hve een mde s well. These derivtives use Tylor expnsions nd re mde y Rogers [Rogers; 197] nd Pejk [Pejk; 1966]. Rogers uses Tylor series expnsion to pproximte the defletion of the triling ontt point with 8

10 respet to the leding ontt point nd this is therefore n pproximtion to the Von Shlippe tyre model. Pejk develops the exponentil funtion ourring in the trnsfer funtions of Segel in Tylor series of seond order (proli) nd first order (stright tngent). Insted of the Tylor pproximtions, Kluiters reples the time dely in the Von Shlippe pproximtion etween leding nd triling ontt point with Pdé filter [Kluiters; 1969]. More reently, the rigid ring tyre model hs een developed y Zegelr [Zegelr; 1998] nd Murie [Murie; 000]. Unlike the strethed string pproh, it inludes the dynmis of the elt nd ontt pth nd is therefore ple of showing for exmple gyrosopi effets nd resonnes of the elt.. Von Shlippe In the strethed string pproh [Von Shlippe; 1954], the tyre is onsidered s mssless string of infinite length under onstnt pre-tension fore nd it is uniformly supported elstilly in the lterl diretion, see Figure.1. Figure.1. Strethed string model. Here, represents hlf of the ontt re nd σ represents the relxtion length. The lterl stiffness per unit length etween the string nd the wheel plne is represented y. In [Pejk; 004], the oundry ondition is used tht through the rolling proess of the tyre, the string forms ontinuously vrying slope round the leding ontt point. At the rer however, the sene of ending stiffness my use disontinuity in slope. This oundry ondition yields the following differentil eqution for the string defletion t the leding ontt point v 1 with respet to the trvelled distne s t [Besselink; 000, Pejk; 004]: dv1 1 + v1 = α + ϕ (.1) ds σ t Here, α represents the tyre side slip ngle nd ϕ represents turn slip nd they re given y: α dy = ψ (.) dst dψ ds ϕ = (.3) t where y nd ψ represent the lterl position nd the yw ngle of the ontt pth respetively. A more detiled desription of turn slip is given further on in this hpter. Figure. shows shemti representtion of the sitution. In this figure, v 1 nd v re the string defletions of the 9

11 leding nd triling ontt point respetively nd y 1 nd y re the lterl positions of the leding nd triling ontt point respetively. Figure.. Shemti representtion of the string defletion. When the forwrd veloity V is ssumed to e onstnt, it my e ssumed tht s t = Vt, where t is the time. As result, (.1) n e rewritten s: V vɺ 1 + v1 = Vψ yɺ ψɺ (.4) σ From Figure. it n e seen tht the lterl position of the leding ontt point of the string y 1 is given y: y = y + ψ + v (.5) 1 1 By differentiting (.5) with respet to time ( is onstnt) nd sustituting the otined eqution in (.4), the string defletion n e eliminted: σ yɺ 1 + y1 = y + ( σ + ) ψ (.6) V Furthermore, Von Shlippe mde the ssumption tht no sliding ours with respet to the rod in the ontt region. Therefore, the triling ontt point follows the sme pth s the leding ontt point with time dely τ. The lterl position of the tyre string t the triling ontt point y n e desried y: ( ) ( ) y t = y t τ (.7) 1 This time dely is equl to the time etween the leding ontt point nd the triling ontt point following the sme pth nd n e desried with: τ = (.8) V 10

12 Von Shlippe ssumes the ontt line to e stright line etween y 1 nd y. This is shemtilly displyed in Figure.3 where the interprettion of the stright tngent tyre model, disussed in the next setion, is lso displyed. Figure.3. Physil interprettions of the Von Shlippe tyre model nd stright tngent tyre model. Consequently, the lterl fore F y nd self ligning moment M z for the Von Shlippe tyre model re given y: v1 + v y1 + y Fy = ( σ + ) = ( σ + ) y 1 v1 v 1 y1 y M z = σ ( σ ) σ ( σ ) ψ + + = (.9) The tyre ornering stiffness C Fα nd the tyre self ligning stiffness C Mα, ssuming stright line etween y 1 nd y, re given y: ( σ ) CFα = + 1 CMα = σ ( σ + ) + 3 (.10) The ornering stiffness nd self ligning stiffness re tyre prmeters tht re given y: C C Fα Mα Fy = α α = 0 M z = α α = 0 (.11) Given the sign onvention displyed in Figure.4, the self ligning stiffness is negtive prmeter. Finlly, omining (.9) nd (.10) yields the following expression for the lterl fore nd self ligning moment: 11

13 F C y + y y σ + y y = C ψ Fα 1 y = M 1 z Mα (.1) The sign onvention used throughout this thesis is displyed in Figure.4. In this figure, x, y nd z represent the longitudinl, lterl nd vertil diretion respetively. Figure.4. Sign onvention used in this thesis with top view (left) nd rer view (right)..3 Stright tngent As n e seen from Figure.3, the ontt line of the stright tngent tyre model is solely governed y the defletion v 1 t the leding ontt point. It uses the tyre deformtion ngle α' to ompute the fores generted y the tyre. This deformtion ngle α' is given y: v 1 α = (.13) σ Comining (.13) with (.4) yields the expression for the tyre deformtion ngle: σαɺ + Vα = Vψ yɺ ψɺ F C α y = Fα z = CMαα M (.14) The tyre self ligning moment M z for the stright tngent tyre model n e repled y giving the tyre lterl fore F y n offset t p in longitudinl diretion with respet to the wheel entre. This offset is referred to s the pneumti tril of tyre nd n e lulted y: t C M Mα z p = = (.15) CFα Fy 1

14 .4 Kluiters Kluiters uses the Von Shlippe pproh nd reples the time dely etween leding nd triling ontt point with Pdé filter [Kluiters; 1969]. This tyre model is referred to s the Von Shlippe tyre model in the reminder of this thesis. Kluiters oserves tht Pdé filter hs etter onvergene properties when pproximting the time dely ompred to the Tylor expnsion s n e seen from Appendix A. He points out tht Pdé filter of order is suffiient nd inresing the order of the filter does not hnge the results signifintly. The trnsfer funtion of the seond order Pdé filter tht he uses n e desried y: H ( s) s 1 s 1 + V 3 V = s 1 s 1+ + V 3 V y, y1 (.16) where s is the Lple vrile. Beuse (.16) defines the reltionship etween y 1 nd y, the tyre fores n e omputed using n intermedite vrile z. This vrile hs no physil mening ut hs to e introdued in order to e le to lulte the tyre fores. This yields the following set of equtions: s 1 s y1 = 1+ + z V 3 V s 1 s y = 1 + z V 3 V (.17) Sustituting (.17) into (.6) nd (.1) yields the following expressions desriing the tyre model: σ s 1 s s 1 s 1+ + zɺ z = y + ( σ + ) ψ V V 3 V V 3 V C Fα 1 s Fy = 1+ z y σ + 3 V s M z = CMα z ψ V After rerrnging (.18), the equtions n e trnsformed into: (.18) 13

15 1 σ σ + 3 σ + ɺɺɺ z + ɺɺ z + zɺ + z = y 3 + ( σ + ) ψ 3V V V C Fα 1 Fy = z + ɺɺ z y σ + 3 V 1 M z = CMα zɺ ψ V (.19) As is pointed out erlier, the pneumti tril (.15) is equl to minus the tyre self ligning moment divided y the tyre lterl fore. As n e seen from (.19), unlike the stright tngent tyre model, the pneumti tril for this tyre model is not onstnt, ut dynmi vrile..5 Rigid ring The rigid ring tyre model, sed on the work of Murie [Murie; 000] nd Zegelr [Zegelr; 1998], onsists of three msses whih orrespond to rim, elt nd ontt pth. The elt is elstilly suspended with respet to the rim whih represents the flexile rss. The out-ofplne reltive motion etween the rim nd elt hs three degrees of freedom whih re lterl trnsltion nd rottion out the vertil xis (yw) nd longitudinl xis (roll/mer). The interfe etween the elt nd the rod surfe is modelled with residul stiffnesses nd the tul slip model. The residul stiffnesses, whih onsist of yw nd lterl degree of freedom, represent the lol deformtions of the tyre ontt pth. The residul stiffnesses hve to e inluded to otin orret stti tyre deformtions in the orresponding diretions. The ontt pth mss hs to e inluded to ensure tht no numeril prolems our. Figure.5 shows shemti representtion of the tyre model. In this figure, ψ represents the yw ngle, γ represents the mer ngle nd Ω represents the ngulr veloity of the wheel. Figure.5. Shemti representtion of the rigid ring tyre model. 14

16 Rim nd fixed prt of the tyre The rim hs three degrees of freedom (Figure.6). These degrees of freedom n e desried with the following equtions of motion: ( ɺ ɺ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) mɺɺ r yr = ky y yr + y y yr + Fyr I ɺɺ rψψ r = IrωΩ ɺ γ r + k ψ ψɺ ψɺ r + k ψ Ω γ γ r + ψ ψ ψ r + M zr I ɺɺ rγγ r = Ir ωω ψɺ r + k ɺ γ γ ɺ γ r k γ Ω ψ ψ r + γ γ γ r + M xr (.0) In this eqution, m r, I rγ, I rψ nd I rω represent the mss of the rim nd the moments of inerti of the rim round the longitudinl xis, vertil xis nd lterl xis respetively. In these inerti prmeters, prt of the tyre is inluded tht moves long with the rim. Rules of thum re pointed out in [Pejk; 004] whih stte tht 5% of the tyre mss is fixed to the rim nd 75% of the tyre mss represents the elt. For the moments of inerti of the tyre, 15% nd 85% re used for the fixed prt nd the elt respetively. In the reminder of this thesis, the susript r is used for the inerti prmeters of the rim inluding the fixed prt of the tyre. The degrees of freedom of the rim indited with y r, ψ r nd γ r re the lterl position, yw ngle nd mer ngle respetively nd y, ψ nd γ re the lterl position, yw ngle nd mer ngle of the elt respetively. Finlly, k y nd y represent the lterl dmping nd stiffness etween the rim nd elt respetively, k γ nd γ represent the mer dmping nd stiffness etween the rim nd elt respetively nd k ψ nd ψ represent the yw dmping nd stiffness etween the rim nd elt respetively. Beuse the motions of the rim re defined s inputs in this hpter, these dynmis re only used for the lultion of the output fore nd moments lter on this hpter, whih re mesured on the mounting point of the rim. F yr, M zr nd M xr re the lterl fore, self ligning moment nd overturning moment ting on the rim respetively. The output fore nd moments tht re nlysed further on in this thesis re the fore nd moments ting on the mounting point (xle) of the rim, represented y F y, M z nd M x, nd re therefore equl to minus the fore nd moments ting on the rim: F M M yr zr xr = Fy = M z M x = (.1) Belt nd ontt pth The differentil equtions tht desrie the three degrees of freedom of the elt re derived in Appendix B nd re given y: 15

17 ( ) ( ɺ γ ) y ( y yr ) + y ( y y Rγ ) ( ) ( ) k ψ Ω( γ γ r ) ψ ( ψ ψ r ) + ψ ( ψ ψ ) ( ) ( ɺ ɺ ) + k ( ψ ψ ) ( γ γ ) R ( y y Rγ ) mɺɺ y = k yɺ yɺ + k yɺ yɺ R y r y I ɺɺ ψ = I Ωɺ γ k ψɺ ψɺ + k ψɺ ψɺ ψ ω ψ r ψ I ɺɺ γ γ = I ωωψɺ k ɺ γ γ ɺ γ r + Rky y y Rɺ γ γ Ω r γ r + y (.) Here, m, I γ, I ψ nd I ω represent the mss of the elt nd the moments of inerti of the elt round the longitudinl xis, vertil xis nd lterl xis respetively. k y nd k ψ represents the lterl nd yw dmping etween the elt nd the ontt pth respetively nd y nd ψ represent the lterl nd yw stiffness etween the elt nd the ontt pth respetively. Finlly, R is the tyre rdius. Ω is the ngulr veloity of the wheel nd is given y Ω = V/R e, where V is the forwrd veloity nd R e is the effetive rolling rdius. Figure.6. Rer view (left) nd top view (right) of the degrees of freedom of the rigid ring tyre model. The ontt pth hs two degrees of freedom with respet to the elt whih re lterl trnsltion nd rottion round the vertil (yw) xis. These degrees of freedom n e desried with the following differentil equtions: ( ɺ ɺ ɺ γ ) ( γ ) ( ɺ ɺ ) ( ) mɺɺ y = ky y y R y y y R + Fy I ɺɺ ψψ = k ψ ψ ψ ψ ψ ψ + M z (.3) Here, m nd I ψ represent the mss of the ontt pth nd the moment of inerti of the ontt pth round the vertil xis respetively. The six degrees of freedom desried in (.) nd (.3) long with the degrees of freedom of the rim re displyed in Figure.6. 16

18 Slip model The lterl fore nd self ligning moment originting from reltive movement of the ontt pth with respet to the rod onsists of two ontriutions: one due to side slip nd one due to turn slip. The ontriution of the side slip is lulted using the deformtion ngle α' ording to first-order eqution tht tkes the relxtion effet into ount. It is similr to the eqution desriing the deformtion ngle of the stright tngent tyre model (.14) without the turn slip omponent: σ ɺ α + Vα = α = Vψ yɺ (.4) Here, σ indites the relxtion length of the ontt pth whih is equl to hlf of the ontt length. Furthermore, mesurements show n initil slope of the self ligning moment equl to zero upon pplying step in side slip ngle [Higuhi; 1997]. This step response is shemtilly displyed in Figure.7. Note tht the trvelled distne in this figure is defined s s t = Vt. Pejk [Pejk; 004] desries similr response for the lterl fore for step in turn slip. He suggests tht this initil slope n e hieved y using the first-order eqution with σ s relxtion length (Figure.7), nd sutrting response urve tht strts with the sme slope ut dies out fter hving rehed its pek (Figure.7). This ltter response urve n e otined y tking the differene of two responses, eh leding to the sme level ut strting t different slopes. These different slopes n e hieved y two first-order equtions with relxtion lengths equl to σ (Figure.7) nd σ / (Figure.7d). Figure.7. Shemti representtions of step responses in self ligning moment to side slip. Consequently, n dditionl first order differentil eqution is needed for the lultion of the self ligning moment: σ ɺ α + Vα = α = Vψ yɺ (.5) t t t where σ t is given y: σ = σ / (.6) t Summrising this extr effet results in the following eqution for the deformtion ngle for the self ligning moment α' M : α = α α (.7) M where α' nd α' t re given in (.4) nd (.5) respetively. The trnsfer funtion etween the deformtion ngle for the self ligning moment α' M nd the side slip ngle α eomes: t 17

19 1 1 Hα, ( ), ( ), ( ) M α s = Hα α s Hαt α s = = σ σ σ 3 s 1 s 1 σ + + s + s + 1 V V V V (.8) Figure.8 shows the response of the deformtion ngle to step in side slip using the trnsfer funtion desried in (.8). The forwrd veloity is equl to 5 m/s whih is typil veloity for step response mesurements mde on the flt plnk tyre tester. As expeted, the initil slope of the deformtion ngle is 0. Beuse the initil slope of the lterl fore in this step response is not equl to zero, the effet disussed ove is not pplied to the lterl tyre fore nd s result, only the deformtion ngle desried in (.4) is used for the omputtion of the lterl fore. This results in the following expressions for the lterl fore F yα nd self ligning moment M zα regrding side slip: F M yα = CFαα = t C α zα p Fα M (.9) where C Fα is the ornering stiffness for the ontt pth nd α' nd α' M re desried in (.4) nd (.7) respetively. Figure.8. Response of the deformtion ngle for the self ligning moment to step in side slip (V = 5 m/s). The seond ontriution to the lterl fore nd self ligning moment is mde y turn slip. This typilly ours when the tyre is trvelling in irulr pth. A shemti representtion of this sitution is given in Figure.9. It n lso e seen from this figure tht the veloity vetor is tngent to the trvelled pth nd the side slip ngle is equl to zero in this sitution onsequently. Figure.9. Shemti representtion of typil sitution where turn slip ours. 18

20 The turn slip ϕ is given y: ψ V ϕ = ɺ (.30) Some tyre models hve n inurte response to step in turn slip euse the irulr pth uses the side slip ngle desried in (.) to remin 0. Turn slip dynmis n e desried using four first-order equtions s hs een pointed out y Pejk [Pejk; 004]. Mesurements show n initil slope of the lterl fore equl to zero in step response in turn slip. Similr s is done for the self ligning moment for step in side slip, this n e hieved y using two first-order equtions with σ nd σ / s relxtion lengths: σ ɺ ϕ + Vϕ = Vϕ = ψɺ σ ɺ F ϕ F + Vϕ F = Vϕ = ψɺ (.31) where σ F is given y: σ F = σ / (.3) Here, ϕ' nd ϕ' F re the two omponents desriing the trnsient turn slip for the lterl fore. The trnsient turn slip for the lterl fore ϕ F used for the lultion of the lterl fore equls: ϕ = ϕ ϕ (.33) F F where ϕ' nd ϕ' F re given in (.31). The tyre self ligning moment response to step in turn slip is urve tht fter hving rehed pek tends to lower sttionry vlue, whih is shemtilly displyed in Figure.10. This urve n e hieved y using the first-order eqution with σ s relxtion length (Figure.7), nd dding response urve tht dies out fter hving rehed its pek (Figure.10). This ltter response urve n e otined y tking the differene of two responses, eh leding to the sme level ut strting t different slopes. Vlues for the relxtion lengths re tken s σ / (Figure.10) nd σ /1.5 (Figure.10d) nd re otined from [Pejk; 004]. Figure.10. Shemti representtions of step responses in self ligning moment to turn slip. The two dditionl equtions tht desrie the turn slip influening the self ligning moment re: σ ɺ ϕ ɺ ϕ ϕ ψɺ ϕ1 1 + V 1 = V = ɺ ϕ Vɺ V ɺ σ ϕ + ϕ = ϕ = ψ (.34) 19

21 where σ ϕ1 nd σ ϕ re given y: σ σ ϕ1 ϕ = σ / = σ /1.5 (.35) The trnsient turn slip for the self ligning moment ϕ' M used for the lultion of the self ligning moment equls: M ( ) ϕ = wϕ + w ϕ ϕ (.36) 1 1 where ϕ', ϕ' 1 nd ϕ' re desried in (.31) nd (.34). In (.36), w 1 nd w re ftors tht govern the hrteristis of the tyre moment response to turn slip. From physil point of view, w 1 influenes the effet of tred width nd w influenes the mgnitude of the overshoot visile in Figure.10. Vlues for these ftors re tken equl to 1 nd 4 respetively nd they re otined from [Pejk; 004]. The trnsfer funtion etween trnsient turn slip for the self ligning moment ϕ' M nd the turn slip ϕ eomes: Hϕ, ( ), ( ) 4( ( ) ( )) M ϕ s = H ϕ ϕ s + Hϕ 1, ϕ s Hϕ, ϕ s = + σ σ σ s + 1 s + 1 s + 1 V V 1.5V σ 11σ s + s + 1 = V 6V 3 σ 3 1.5σ 13σ s + s V V 6V (.37) Figure.11 shows the response of the trnsient turn slip for the self ligning moment to step in turn slip using the trnsfer funtion desried in (.37). Also inluded re the seprte omponents of whih the totl trnsfer funtion is omposed of. The prmeter vlues re hosen to orrespond to the vlues used in Chpter 3. Figure.11. Response of the deformtion ngle for the self ligning moment to step in side slip (V = 5 m/s). 0

22 As n e seen from Figure.11, the vlue of 4 for w is not suffiient to hve n overshoot in the response urve for ϕ' M. However, s n e seen from the next hpter, the dynmis etween the rim nd the ontt pth use the response of the self ligning moment t the wheel entre to hve n overshoot similr to the shemti representtion in Figure.10. It n e noted tht the response urve of ϕ' M is similr to first-order response urve with smll relxtion length nd the question rises if the four differentil equtions desriing the turn slip response of the self ligning moment re neessry. This is however eyond the sope of this reserh. Assuming the turn slip remins reltively smll, this results in the following expression for the lterl fore F yϕ nd self ligning moment M zϕ regrding turn slip: F M = C ϕ = C ϕ yϕ Fϕ F zϕ Mϕ M (.38) where C Fϕ nd C Mϕ re the turn slip stiffnesses for the lterl fore nd self ligning moment respetively nd ϕ' F nd ϕ' M re desried in (.33) nd (.36) respetively. Beuse the moment generted y turn slip M zϕ uses smll differene in ngle etween the rim nd elt nd therefore lso smll side slip ngle nd side fore, the turn slip stiffnesses nnot e otined from the stedy-stte vlues diretly. Insted, liner simultions re mde to otin the sme stedy-stte vlues s the mesurements further on in this thesis. Finlly, the totl lterl fore nd self ligning moment inluding oth side slip nd turn slip eomes: Fy = Fy α + Fy ϕ M z = M zα + M zϕ (.39) Figure.1. Shemti representtion of the slip model of the rigid ring tyre model. 1

23 A shemti representtion of the different omponents of the slip model desried in this setion is displyed in Figure.1. In this setion, there is differene in prmeters tht re used for the relxtion length nd ornering stiffness for the rigid ring tyre model with respet to the other two tyre models. This is result of the dynmis involved etween rim nd ontt pth. First, n djustment hs to e mde to the ornering stiffness. The yw stiffnesses in the rigid ring tyre model etween rim nd ontt pth uses differene in the yw ngle pplied to the rim nd the yw ngle of the ontt pth. Beuse this is not the se for the stright tngent tyre model nd the Von Shlippe tyre model, the slip ngle of the ontt pth is different for the tyre models upon pplying the sme yw ngle to the rim. The differene in side slip ngle δα etween the stright tngent tyre model nd the rigid ring tyre model n e desried s: α α δα α M z rr = st = st (.40) ψ, totl Here, the susripts rr nd st refer to the rigid ring tyre model nd stright tngent tyre model respetively. The totl yw stiffness etween rim nd ontt pth ψ,totl is given y: ψ, totl = ψ ψ (.41) To otin the sme stedy-stte hrteristis in spite of the different side slip ngles, the prmeter vlue of the ornering stiffness hs to e djusted ording to the differene in side slip ngle δα [Pejk; 004]: C Fα C = CFα 1 t Fα p ψ, totl 1 (.4) Here, C Fα nd C Fα indite the ornering stiffness of the rigid ring tyre model nd the stright tngent tyre model respetively. A similr djustment hs to e mde for the relxtion length of the tyre. For the stright tngent tyre model nd the Von Shlippe tyre model, the relxtion effet of the tyre is used only y the prmeter vlue of the relxtion length. The rigid ring tyre model hs n dditionl relxtion effet used y the lterl stiffness etween rim nd ontt pth. The relxtion length of the rigid ring tyre model σ is equl to hlf of the ontt length. Pejk [Pejk; 004] sttes tht this differene in relxtion lengths n e desried y: C C σ = σ + = σ + C t C t ψ, totl Fα ψ, totl Fα ψ, totl + Fα p y, totl ψ, totl + Fα p y, totl (.43) Here, σ nd σ re the relxtion length of the rigid ring tyre model nd stright tngent tyre model respetively nd y,totl is the totl lterl stiffness etween rim nd ontt pth nd n e lulted y:

24 1 = 1 1 R + + γ y, totl y y (.44).6 Stte spe representtion nd rim interfe In order to ompre the models to mesurements, the systems n e trnsformed into stte spe representtion: ( t) ( t) ( t) ( t) ( t) ( t) xɺ = Ax + Bu y = Cx + Du (.45) Here, x(t) represents the stte vetor, y(t) the output vetor nd u(t) the input vetor. The mtries A, B, C nd D represent the system mtries. Beuse the stright tngent tyre model nd the Von Shlippe tyre model ssume the wheel to e rigid mss, the sttes of the ontt pth indited with the susript re given y: y = yr + Rγ r ψ = ψ r (.46) γ = γ r These formuls n e sustituted in (.14) nd (.19) so tht y, ψ nd γ do not hve to e inluded in the stte vetor of the stright tngent tyre model nd the Von Shlippe tyre model. In the next hpter, the tyre models re nlysed nd vlidted using mesurements on tyre. In these mesurements rried out y Higuhi [Higuhi; 1997] nd Murie [Murie; 000], the input onsists of pplied motions to the rim nd the output is mesured t the mounting point of the wheel t the wheel entre. To imitte this sitution, the fore nd moment ting on the mounting point of the rim re lulted in the simultions rther thn the tul tyre fore nd moment in the ontt re. Moreover, lulting the outputs t the wheel entre inludes the inerti effets of the wheel in the nlysis of the fore nd moments ting on the mounting point. Consequently, the output fore nd moments in the simultions re given y: y Fy mɺɺ w yr + CFαα = M = I ɺɺ γ + I Ω ψɺ + C α R M I ɺɺ ψ I Ωɺ γ C α t st x wγ r wω r Fα z wψ r wω r Fα p (.47) y CF CF C α α Fα mɺɺ w yr yr + ɺɺ z + z σ + 3( σ + ) V σ + F C R C R C R M I I y z z σ 3( σ ) V σ M z CMα I ɺɺ wψψ r Iw ωω ɺ γ r + CMαψ r zɺ V y Fα Fα Fα vs = x = ɺɺ wγ γ r + wωωψɺ r r + ɺɺ + (.48) 3

25 y ɺɺ + ( ɺ ɺ ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) Fy mr yr ky y yr y y yr = M = I ɺɺ γ + k ɺ γ ɺ γ + I Ω ψɺ + γ γ Ωk ψ ψ M I ɺɺ ψ I Ω ɺ γ + k ψɺ ψɺ + Ωk γ γ + ψ ψ rr x rγ r γ r rω r γ r γ r z rψ r rω r ψ r ψ r ψ r (.49) Here, the susripts st, vs nd rr refer to stright tngent, Von Shlippe nd rigid ring respetively. The inerti prmeters m w nd I wψ re equl to the mss nd moment of inerti of the wheel respetively. They re equl to the inerti vlues of the rim plus the tyre. A distintion is mde etween the outputs of the step nd frequeny response simultions. This is done euse the step response mesurements [Higuhi; 1997] hve een mde with initil onditions insted of step motion pplied to the rim. Consequently, the terms in (.47), (.48) nd (.49) depending on derivtives of y r, ψ r nd γ r re negleted for the step responses. The inputs in the step response mesurements onsist of side slip input dy r /dt, pure yw input ψ r nd turn slip input. The turn slip input is omintion of the side slip nd pure yw input, suh tht the side slip ngle desried in (.) equls 0. Beuse the equtions of motion of the tyre models ontin derivtives of oth the side slip input nd the pure yw input from the mesurements, the inputs for the simultions re tken s the derivtives of the mesurement inputs. Consequently, n impulse is pplied to the simultion inputs to represent the step responses from the mesurements. The input u is given y: ( ψ ψ ) u ( t) = Vy ɺɺ ɺ Vy ɺɺ ɺ (.50) impulse r r r r Here, the first input is used for the side slip input. This input is negtive nd multiplied with the forwrd veloity to otin the sme side slip ngle s for the pure yw input. Applying n impulse to this input represents the step in side slip ngle in the mesurements. The seond input is used for the pure yw input. Applying n impulse to this input represents the step in pure yw ngle in the mesurements. Finlly, the third input is used for the turn slip input. Applying n impulse to this input represents n impulse in turn slip. Consequently, this ltter urve hs to e integrted over the trvelled distne to otin step in turn slip. In the frequeny response mesurements from [Murie; 000], the inerti terms for the rim nnot e negleted. Therefore, the seond derivtives of oth the lterl position nd the yw ngle hve to e inluded in the inputs. The inputs for the frequeny response of the side slip nd yw input re therefore given y: u ( Vy ψ ) T T ( t) = ɺɺ ɺɺ (.51) frequeny r r Beuse the inputs re defined s the seond derivtives of the lterl position nd yw ngle, the inputs hve to e integrted to otin the proper trnsfer funtions nd e le to mke omprison with mesurements. This n e hieved y multiplying the trnsfer funtion for side slip with s nd for pure yw with s. The third input tht is used in this hpter is the turn slip input. As is pointed out lredy, it is omintion of the side slip nd yw input, suh tht the side slip ngle desried in (.) equls 0. The reltion etween the turn slip input nd the omintion of side slip nd yw input is defined s [Besselink; 000]: 4

26 ( ) V H F ( ) ( ) ( ) y, ϕ s = H Fy, α s H Fy, ψ s s V H M, ( s) ( H, ( ), ( )) z ϕ = M s H z α M s z ψ s (.5) Here, H indites the trnsfer funtion nd the susripts ϕ, α nd ψ refer to turn slip, side slip nd pure yw respetively. Beuse of the different numer of degrees of freedom, eh tyre model hs its own stte vetor. The stte vetors re therefore given y: x x x T ( yɺ ɺ γ ψɺ y γ ψ α ) ( yɺ ɺ γ ψɺ y γ ψ ɺɺ z zɺ z) ( ɺ ɺ ɺ ɺ = T = = y γ ψ y γ ψ y ɺ γ ψɺ y γ ψ... T yɺ ψɺ y ψ α α t ϕ ϕ F ϕ 1 ϕ ) st r r r r r r vs r r r r r r rr r r r r r r (.53) Note tht for the step responses, dψ r /dt is defined s n input. As result, this stte is redundnt for the step responses. 5

27 3. Tyre Model Comprison nd Vlidtion The three tyre models from Chpter n e ompred with eh other nd with mesurements on tyres. This is done y using step responses nd frequeny responses. The step response mesurements re rried out y Higuhi [Higuhi; 1997] nd re ville for three inputs. These inputs inlude step in side slip ngle, step in yw ngle nd step in turn slip. The frequeny response mesurements re rried out y Murie [Murie; 000] nd re only ville for the pure yw input. In Setion 3.1, the tyre model prmeter vlues re given. Step responses nd frequeny responses re given in Setions 3. nd 3.3 respetively. Finlly, Setion 3.4 gives the min onlusions tht n e drwn from this hpter. 3.1 Prmeter vlues The prmeters tht re used re displyed in Tle 3.1 nd re sed on prmeters used in [Murie; 000]. However, the prmeters tht re used y Murie re sed on model without turn slip nd re therefore inurte for the rigid ring tyre model desried in this thesis. Therefore, n optimiztion of prmeters is rried out s desried in Appendix C. This is done using the fminon funtion of the optimiztion toolox of Mtl nd is sed on n ojetive funtion tht desries the reltive error etween the frequeny responses of the mesurements nd the model. The prmeters used in this hpter re pplile for pssenger r tyre. This is done euse, unlike for truk tyres, mesurements on pssenger r tyres re ville. Consequently, the experimentl results n e used to otin model prmeters tht represent the mesurements urtely. Tle 3.1. Tyre model prmeters for pssenger r tyre. Prmeter Vlue Unit Prmeter Vlue Unit 4.88e- m I ψ 1.00e- kgm y 6.39e5 N/m k y 7.68e1 Ns/m γ, ψ.0e4 Nm/rd k γ, k ψ 4.00 Nms/rd y 8.00e5 N/m k y 1.00e Ns/m ψ 1.00e4 Nm/rd k ψ 3.00 Nms/rd C Fφ 8.80e Ns/rd m 7.64 kg C Mφ 1.77e Nms/rd m 1.00 kg C Fα 6.70e4 N/rd m r 1.66 kg I γ, I ψ 3.1e-1 kgm m w 9.30 kg I ω 6.61e-1 kgm R.80e-1 m I rγ, I rψ 6.98e- kgm R e 3.00e-1 m I rω 7.50e- kgm t p 3.00e- m I wγ, I wψ 3.91e-1 kgm σ 4.88e- m I wω 7.36e-1 kgm The turn slip stiffnesses C Fφ nd C Mφ for respetively the lterl fore nd self ligning moment re otined from step response mesurements [Higuhi; 1997]. The orretion of the ornering stiffness of the rigid ring tyre model (.4), results in vlue for C Fα equl to N/rd. The totl relxtion length σ (.43) used for the stright tngent tyre model nd the Von Shlippe tyre model is equl to m. Note tht in Tle 3.1, the totl inerti of the wheel is indited with 6

28 prmeters with the susript w. In this hpter however, these vlues do not inlude the inerti prmeters for the rim, euse the experimentlly otined response funtions presented in this hpter re orreted for these rim inerti effets. Consequently, the inerti prmeters of the wheel, indited with susript w, only inlude the inerti of the tyre nd the inerti prmeters of the rim indited with susript r only inlude the inerti of the prt of the tyre tht moves long with the rim. 3. Step responses Step responses n e lulted with the tyre models in stte spe formt (Setion.6) using the lsim ommnd in Mtl. In this setion, the simulted step responses re ompred to rel mesurements nd differenes re identified. Prmeters re tken s in Tle 3.1. The forwrd veloity is tken equl to 5 m/s whih is similr to the forwrd veloity used in the mesurements. The steps re pplied when the trvelled distne equls 0. As is lredy pointed out in Setion.6, the derivtives of the motions of the rim re negleted here. Figure 3.1. Step response of the lterl fore nd self ligning moment to side slip. 7

29 Figure 3.1 shows the step responses to step of 1 degree in side slip. Zoomed in plots re provided where differenes re too smll to e distinguished in the regulr plots. The models show very smll differenes. The lrgest differene n e found in the initil slope of the self ligning moment equl to zero for the Von Shlippe tyre model nd the rigid ring tyre model, whih n lso e seen from the mesurements. The stright tngent tyre model does not show this response nd hs therefore smll disrepny. Figure 3. shows the string model in step in side slip ngle for the initil sitution when the step is pplied t the right-hnd side nd the stedy-stte sitution on the left-hnd side. The pth of the ontt points hs smll dely with respet to the pth of the wheel entre nd uses the zero initil slope of the self ligning moment. Figure 3.. The string model in step in side slip ngle for the initil sitution (right) nd stedystte sitution (left). Figure 3.3 shows the step responses to step of 1 degree in yw ngle. Differenes etween the tyre models re lrger ompred to the step in side slip. Espeilly t the initil stge, differenes n e seen from oth the mgnitude nd the sign of the tyre fores. For the lterl fore, the stright tngent tyre model hs smll negtive initil vlue whih nnot e seen from the mesurements. The Von Shlippe tyre model shows smll dely in lterl fore uild-up. For the self ligning moment, the stright tngent tyre model hs positive initil vlue, wheres the Von Shlippe tyre model nd the rigid ring tyre model hve negtive initil vlue similr to the mesurements. However, the initil pek vlue of the Von Shlippe tyre model is equl to its stedy-stte vlue nd the pek vlue is therefore too smll. The rigid ring tyre model shows n initil pek tht is too lrge ompred to the mesurements. 8

30 Figure 3.3. Step response of the lterl fore nd self ligning moment to yw. 9

31 Figure 3.4. Step response of the lterl fore nd self ligning moment to turn slip. Finlly, Figure 3.4 shows the step responses to step in turn slip. This response shows muh lrger differenes thn the other step responses. The initil slope of the lterl fore is equl to zero for the Von Shlippe tyre model nd the rigid ring tyre model, whih is not the se for the stright tngent tyre model. Differenes n lso e found in the sttionry vlues of the lterl fore. For the self ligning moment, the lrgest differenes n e found in the sttionry vlues. The stright tngent tyre model hs negtive vlue, wheres the Von Shlippe tyre model nd the rigid ring tyre model hve positive vlues s is the se for the mesurements. However, the Von Shlippe tyre model hs zero stedy-stte response to turn slip. A shemti representtion of the stedy-stte sitution for step in turn slip is displyed in Figure 3.5 for the stright tngent tyre model nd the Von Shlippe tyre model. This figure lrifies wht hppens in the stedystte sitution: the lterl fore of the stright tngent tyre model hs lrger positive vlue thn the Von Shlippe tyre model nd the self ligning moments for the stright tngent tyre model nd the Von Shlippe tyre model re negtive nd zero respetively. 30

32 Figure 3.5. Shemti representtion of the stright tngent tyre model nd the Von Shlippe tyre model in step in turn slip. 3.3 Frequeny responses Frequeny response mesurements re rried out y Murie [Murie; 000] nd onsist of pure yw input motion on the rim. The frequeny responses for side slip nd turn slip re not ville nd re therefore only ompred etween the tyre models. For the input in the mesurements, white noise input signl is pplied to the rim with 0 64 Hz ndwidth. Figure 3.6 shows the Bode plots for the lterl fore nd self ligning moment for the three tyre models with side slip input. Severl effets n e reognised in the Bode plots. The tyre models re similr to eh other for low frequenies. For stedy-stte sitution, the mgnitude of the Bode plot etween the side fore nd side slip input is equl to the ornering stiffness. The phse in this sitution is equl to zero, mening tht positive slip ngle results in positive tyre lterl fore. As the frequeny inreses, the relxtion effet eomes visile. This results in deresing mgnitude nd inresing phse lg. All three tyre models re ple of showing these effets. Aove pproximtely 15 Hz, the mgnitude of the lterl fore inreses euse of the mss of the tyre. As the frequeny of exittion is inresing, the neessry fore to exite the wheel in lterl diretion inreses s well. The differenes etween tyre models eome lrger towrds the resonne frequenies of the tyre elt. These re visile y the peks in mgnitude nd the two peks visile represent the yw nd mer degrees of freedom of the elt. Beuse of the gyrosopi oupling of these degrees of freedom, there is no ler yw nd mer mode however. The stright tngent tyre model nd the Von Shlippe tyre model re not ple of showing these resonne peks. The self ligning moment shows similr effets s the lterl fore, ut there re some differenes however. The stedy-stte mgnitude for the tyre models is equl to the self ligning stiffness insted of the ornering stiffness. The phse in this sitution is 180 degrees euse positive side slip ngle results in negtive tyre self ligning moment. For inresing frequenies, the stright tngent tyre model nd the Von Shlippe tyre model disply the sme relxtion effet s in the lterl fore Bode plot. This results in deresing mgnitude nd inresing phse lg. The inrese in phse lg of the rigid ring tyre model is ompensted y the phse leding nture of the gyrosopi effets of the elt. The stright tngent tyre model nd the Von Shlippe tyre model re not ple of showing this phse led euse gyrosopi effets re not present for these tyre models. 31

33 Figure 3.6. Bode plots for the lterl fore nd self ligning moment for side slip (V = 9 km/h). Figures 3.7 through 3.9 show the Bode plots for the lterl fore nd self ligning moment with pure yw input t different forwrd veloities. For this input, mesurements re ville tht re rried out y Murie [Murie; 000]. Similr effets for pure yw input tht n lso e oserved in side slip input, inlude the stedy-stte mgnitude, the relxtion effet nd the resonne frequenies of the rigid ring tyre model. The gyrosopi effets tht use phse led for the rigid ring tyre model with respet to the other tyre models n gin e found in the self ligning moment t frequenies just elow the resonne frequenies. A differene etween the side slip nd pure yw input n e found in the inerti effets whih use n inrese in mgnitude t higher frequenies. For the pure yw input, this inrese n e found for the self ligning moment insted of the lterl fore euse of the moment of inerti of the tyre. Another differene is the dip in mgnitude of the lterl fore for the Von Shlippe tyre model. This dip ours when the wvelength of the trvelled pth equls the ontt length of the string model. Beuse extly one wvelength fits in the ontt re in this point, the resulting lterl fore equls zero nd dip ours in the mgnitude of the Bode plot. Beuse the wvelength of the 3

34 trvelled pth is dependent on the forwrd veloity, this dip is lso dependent on the forwrd veloity. Therefore, this dip disppers from Figure 3.9 euse it exeeds the frequeny rnge onsidered. As n e seen from Figures 3.7 through 3.9, the veloity hs severl effets on the plots. First, it n e noted tht the relxtion effet shifts to higher frequenies for higher veloities. For the rigid ring tyre model, the shifting of the relxtion effet to higher frequenies uses the relxtion effet to eome less visile in the Bode plots euse of the resonne frequenies of the rigid ring tyre model. The derese in mgnitude euse of the relxtion effet is ompensted y the inrese in mgnitude euse of the resonne frequenies. For the other tyre models, the relxtion effet does not eome smller ut shifts to higher frequenies. Seondly, the two lowest resonne frequenies for the rigid ring tyre model represented y the yw nd mer modes tend to shift from eh other for n inresing veloity. This is euse of the gyrosopi effets in the elt. Figure 3.7. Bode plots for the lterl fore nd self ligning moment for pure yw (V = 5 km/h). 33

35 Figure 3.8. Bode plots for the lterl fore nd self ligning moment for pure yw (V = 59 km/h). 34

36 Figure 3.9. Bode plots for the lterl fore nd self ligning moment for pure yw (V = 9 km/h). When ompring the models with the mesurements, it is ovious tht the rigid ring tyre model hs the most urte frequeny response. Effets suh s resonne frequenies nd gyrosopi effets re not present for the stright tngent tyre model nd the Von Shlippe tyre model. However, the rigid ring tyre model hs some disrepnies s well. One of these disrepnies n e found etween 10 nd 30 Hz for the lterl fore t low forwrd veloities. The lterl fore shows n inrese in phse in this region whih n not e seen from the mesurements. Furthermore, the dip in mgnitude for the self ligning moment visile for ll veloities nd the mgnitude t the resonne frequenies show differenes s well. The third input tht is disussed in this thesis is turn slip. Figure 3.10 shows the Bode plots for the lterl fore nd self ligning moment with turn slip input t forwrd veloity of 9 km/h. For stedy-stte sitution, the sme effets n e seen s the stedy-stte vlues for the step response. The mgnitudes in lterl fore re slightly different ut hve the sme sign. As result, the phse in this sitution is equl for ll tyre models. For the self ligning moment however, the stright tngent tyre model shows negtive stedy-stte vlue in the step 35

37 responses. This n lso e seen from the Bode plots where there is phse differene of 180 degrees etween the stright tngent tyre model nd the rigid ring tyre model for n infinitely low frequeny. The zero response in self ligning moment of the Von Shlippe tyre model to step in turn slip n lso e seen from the Bode plot: the mgnitude goes to 0 for n infinitely low frequeny. For inresing frequenies, the relxtion effet, gyrosopi effet nd the resonne frequenies n e reognised in the lterl fore ode plot. For the self ligning moment, the gyrosopi effet, the inrese in mgnitude euse of the moment of inerti of the tyre nd the resonne frequenies n e reognised. Figure Bode plots for the lterl fore nd self ligning moment for turn slip (V = 9 km/h). Until now, the omprison etween models nd mesurements re mde using the lterl fore nd self ligning moment. However, the mesurements tht hve een mde for the frequeny response to pure yw lso inlude the overturning moment round the x-xis tht ts on the rim. This moment is less relevnt for the sope of this thesis euse the rim does not hve degree of freedom round the x-xis when tthed to the suspension model for the systems studied. This is 36

38 explined in the next hpter. However, it is n dditionl vlidtion output, so the mesurements re ompred to the seond output M x defined in (.47) of the three tyre models. The results re shown in Figure 3.11 for three different forwrd veloities. These Bode plots show effets tht n e found in the other outputs s well. These inlude the relxtion effet, phse led euse of gyrosopi effets round 10 Hz nd the resonne frequenies. However, the min disrepnies s re pointed out erlier re the dip in mgnitude etween pproximtely 0 nd 30 Hz nd the mgnitude t resonne frequenies. Figure Bode plots for the overturning moment for pure yw. 3.4 Conlusions The tyre models in this hpter re vlidted with mesurements. First, this vlidtion is mde regrding step responses. Overll, the rigid ring tyre model hs the most urte response for the step responses, lthough the relxtion effet for ll three tyres ppers to e too smll. However, 37

39 it hs to e noted tht the tyre prmeters re optimised using the frequeny response funtions. Beuse different pssenger r tyres hve een used for the frequeny- nd step response mesurements, the disrepnies of the tyre models in the step responses my e used y inurte prmeters. The results for the step responses n e summrised s is done in Tle 3.. In this tle, n error funtion is dded tht indites the reltive error etween models nd mesurements. The error funtion for the lterl fore nd self ligning moment re respetively given y: Error F y ( ) ( ) Fy,model n Fy,mesurement n = 340 n= 1 Fy,mesurement ( n) mx ( Fy,mesurement ) 1 M ( n) M ( n) = ( ) + ( ) 340 z,model z,mesurement Error M z 340 n= 1 M z,mesurement n 0.05mx M z,mesurement (3.1) Here, n is the numer of mesurement points. This error funtion omputes the reltive error etween model nd mesurement. However, there is n extr term in de denomintor of the funtion whih is equl to 5% of the mximum vlue of the mesurement. This is done in order to void hving very lrge error vlue when dividing y very smll vlue in the initil stge of the step responses. The error vlues show wht n e seen from the step response plots: the lrgest errors n e found for the stright tngent tyre model nd the Von Shlippe tyre model in the turn slip responses. Tle 3.. Results step response. Input Output Tyre model Error Remrks Side slip F y Stright tngent Corret response Von Shlippe Corret response Rigid ring Corret response M z Stright tngent Initil slope is not equl to zero Von Shlippe Corret response Rigid ring Corret response Pure yw F y Stright tngent Negtive initil vlue Von Shlippe Smll dely in fore uild-up Rigid ring Corret response M z Stright tngent Positive initil vlue Von Shlippe Negtive initil vlue is too smll Rigid ring Negtive initil vlue is too lrge Turn slip F y Stright tngent Initil slope is not equl to zero Stedy-stte vlue is too lrge Von Shlippe Stedy-stte vlue is too lrge Rigid ring Relxtion effet is too smll M z Stright tngent Negtive stedy-stte vlue Von Shlippe Stedy-stte vlue is equl to zero Rigid ring Overshoot is too lrge The seond vlidtion tht is mde in this hpter inludes frequeny responses. As well s for the step responses, the rigid ring tyre model is shown to e the most urte tyre model. The results for the frequeny responses n e summrised s is done in Tle

40 Tle 3.3. Results frequeny response. Input Output Remrks Side slip F y Stedy-stte vlues re equl to the ornering stiffness C Fα for ll tyre models Relxtion effet is visile t frequenies ove pproximtely Hz for ll tyre models Inrese in mgnitude euse of the mss of the wheel is visile t frequenies ove pproximtely 10 Hz for ll tyre models Only the rigid ring tyre model shows resonne peks of the elt etween pproximtely 30 Hz nd 60 Hz M z Stedy-stte vlues re equl to the self ligning stiffness C Mα for ll tyre models Relxtion effet is visile t frequenies ove pproximtely Hz for the stright tngent tyre model nd the Von Shlippe tyre model The relxtion effet for the rigid ring tyre model is ompensted y the inrese in mgnitude nd phse euse of the gyrosopi effets of the elt Only the rigid ring tyre model shows resonne peks of the elt etween pproximtely 30 Hz nd 60 Hz Pure yw F y Stedy-stte vlues re equl to the ornering stiffness C Fα for ll tyre models Relxtion effet is visile t frequenies ove pproximtely Hz for the ll tyre models The Von Shlippe tyre model shows dip in mgnitude where extly one wvelength fits in the ontt re Only the rigid ring tyre model shows resonne peks of the elt etween pproximtely 30 Hz nd 60 Hz M z Stedy-stte vlues re equl to the self ligning stiffness C Mα for ll tyre models Relxtion effet is visile t frequenies ove pproximtely Hz for the stright tngent tyre model nd the Von Shlippe tyre model The relxtion effet for the rigid ring tyre model is ompensted y the inrese in mgnitude nd phse euse of the gyrosopi effets of the elt Inrese in mgnitude euse of the moment of inerti of the wheel is visile t frequenies ove pproximtely 10 Hz for ll tyre models Only the rigid ring tyre model shows resonne peks of the elt etween pproximtely 30 Hz nd 60 Hz Turn slip F y Smll differenes exist etween tyre models for stedy-stte vlues Only the rigid ring tyre model shows resonne peks of the elt etween pproximtely 30 Hz nd 60 Hz M z Lrge differenes exist etween tyre models in oth mgnitude nd phse t stedy-stte vlues Only the rigid ring tyre model shows resonne peks of the elt etween pproximtely 30 Hz nd 60 Hz 39

41 As well s for the step responses, n error vlue n e lulted for the frequeny responses. For the lterl fore nd self ligning moment, the error funtions re respetively given y: Mgnitude error F Mgnitude error M ( ) y,mesurement ( ) ( n) 47 1 Fy,model n F n y = 47 n= 1 Fy,mesurement 47 1 Phse error Fy = Fy,model ( n) Fy,mesurement ( n) 47 n= M z,model ( n) M z,mesurement ( n) z = 47 n= 1 M z,mesurement ( n ) 47 1 Phse error M z = M z,model ( n) M z,mesurement ( n) 47 n= 1 (3.) Here, n is the numer of mesurement points. For the mgnitude, the error is defined s reltive error etween the model nd the mesurement, euse of the lrge rnge of the mgnitude vlues. The phse error is defined s the solute vlue of the phse differene. Here, the phse differene in degrees is wrpped to the intervl [-180, 180]. Results for these error funtions re given in Tle 3.4. The error vlues show wht n e seen from the frequeny response plots: the rigid ring tyre model hs the smllest errors with respet to the other two tyre models. Tle 3.4 Results for the error funtion for the frequeny response with yw input. Output Mgnitude /phse Tyre model Error V=5 km/h Error V=59 km/h Error V=9 km/h F y Mgnitude Stright tngent Von Shlippe Rigid ring Phse Stright tngent Von Shlippe Rigid ring M z Mgnitude Stright tngent Von Shlippe Rigid ring Phse Stright tngent Von Shlippe Rigid ring

42 4. Bsi Front Axle After the tyre models re nlysed nd vlidted in the previous hpter, the step towrds the front xle model is mde in this hpter. This hpter fouses on the derivtion of the equtions of motion of si suspension model tthed to tyre models s disussed in the previous hpters. In Setion 4.1, literture survey on shimmy stility nlysis is given. The equtions of motion of the front xle model inluding the three tyre models re derived in Setion 4.. Next, Setion 4.3 fouses on the shimmy stility of the front xle for pssenger r. Beuse of the simpliity of the stright tngent tyre model, this model is used to derive nlytil expressions for the stility oundries s funtion of prmeter vlues. In Setion 4.4, short overview of the effets of prmeter djustments re given with respet to the nlytil expressions for the stility oundries derived in Setion 4.3. Next, ll three tyre models re ompred to eh other y numeril eigenvlue nlysis in Setion 4.5 nd sensitivity nlysis of the stility oundries of ll three tyre models to prmeter hnges is mde in Setion 4.6. Finlly, short overview of the min onlusions is given in Setion Survey of literture on shimmy stility nlysis As pointed out in [Prithrd; 1999], the first fundmentl ontriutions towrd understnding the shimmy phenomenon emerged from the utomoile industry in Frne round 190. Broulhiet [Broulheit; 195] pulished his work on the effet of tyre mehnis on shimmy in 195. He first desried the onept of side slip nd suggested tht the energy for the self-sustinle shimmy osilltions omes from the tyre mehnis vi the side slip. Fromm [Beker, Fromm & Mruhn; 1931] lso reognised the importnt role tht lterl slip of tyres plys in utomoile motions. Initilly, theories tht explin shimmy instility prolems hve een sed on liner models. These models onsisted typilly of wheel with n elsti tyre ple of swivelling out king-pin tht moves long stright line. Amongst others, these models hve een developed nd theoretilly investigted in [Kntrowitz; 1937], [Von Shlippe & Dietrih; 1954], [Morelnd; 1951] nd [Smiley; 1957]. Pejk [Pejk; 1966] is one of the first investigtors to inlude non-liner effets in his shimmy stility nlysis. He inluded non-liner effets in the tyre nd in the suspension. These non-liner effets inlude degressive tyre hrteristis, dry frition in the king-pin erings nd rottionl lerne in the wheel erings. Furthermore, model for non-linerities in yw stiffness, whih inlude frition fores nd free-ply, hs een developed y Blk [Blk; 1976] nd lter on y Bumnn [Bumnn; 1991] nd Li [Li; 1993]. Blk lso points out tht it is dvisle to inlude the movements of the supporting struture in the shimmy nlysis. More reently, the shimmy phenomenon hs een studied y severl reserhers using experimentl, nlytil nd numeril tehniques in [Krher; 1996], [Glser & Hryko; 1996] nd [Besselink; 000]. Besselink fouses his reserh on the shimmy stility of twinwheeled irrft min lnding gers nd uses severl tyre models in his nlysis. Nonliner nlysis is used not only to study the qulittive ehviour of the Hopf ifurtion ut lso to nlyze the system eyond the Hopf ifurtion in [Thot, Kruskopf & Lowenerg; 008]. Also, Somieski [Somieski; 1997] studied shimmy s non-liner dynmis phenomenon for nonliner set of ODEs. Here, time domin nlysis showed se of superritil Hopf ifurtion leding to stle limit yle pst the ifurtion point. 41

43 4. Font xle model In order to derive equtions of motion for the front xle model, some ssumptions hve to e mde: The front xle onsists of one ody whih hs one lterl degree of freedom. The mss of the xle ontins the mss of the xle itself inluding moving prts of the suspension. The inerti prmeters of the rim re inluding dditionl omponents suh s the rkes, hu nd steering rod. Beuse of the ssumed symmetry of the front xle, only hlf of the front xle of the truk is onsidered to redue the omplexity of the model. The rim hs one rottionl degree of freedom round the steering xis with respet to the xle. The forwrd veloity V does not vry in time. The system prmeters re onstnt. No longitudinl fores t on the tyres Vertil fores ting on the tyres re negleted euse of the smll ster- nd steering ngle involved. The omponent of the vertil fore perpendiulr to the steering xis F zψ is given y (see lso Figure 4.1): F zψ = F sinε (4.1) z Here, F z is the vertil fore ting on the tyre euse of the mss of the vehile nd ε is the ster ngle. Assuming the steering ngle remins smll, the resulting moment round the steering xis M zψ used y this omponent is given y: ( ) M ψ = F sinε R tnε + n ψ osε (4.) z z Here, R is the tyre loded rdius, n is the steering xis offset with respet to the wheel entre nd ψ is the steering ngle. As n e seen from (4.), the moment round the steering xis used y the vertil fore remins smll when the ster ngle ε remins smll. Moreover, the tyre models presented in this thesis re urte for smll side slip ngle nd therefore lso for smll steering ngle ssuming stright-line motion. Consequently, the dditionl term for the self ligning moment desried in (4.) is negleted. The front xle model developed in this hpter is shemtilly displyed in Figure 4.1. In this model, two degrees of freedom of the suspension re shown whih re lterl trnsltion y nd rottion round the steering xis ψ. Both degrees of freedom hve spring nd dmper ting on them, indited with y nd k y respetively for the lterl trnsltion nd ψ nd k ψ respetively for the rottion round the steering xis. The xle itself hs mss m whih is equl to hlf of the totl xle mss. The steering xis is rotted in the plne of the wheel with ster ngle ε nd hs n offset n with respet to the wheel entre. The entre of grvity of the wheel hs n offset q with respet to the wheel entre nd hs mss m w nd moment of inerti round the steering xis I wψ. Finlly, the tyre loded rdius is indited with R. 4

44 Figure 4.1. Side view (left) nd top view (right) of the model. The steering xis is rotted y ster ngle ε in the plne of the wheel with respet to the vertil xis. As result, the steering ngle hs omponent in oth the yw nd mer ngle of the rim. For smll steering ngles, this n e desried with: yr = y nψ osε ψ r = ψ osε γ r = ψ sinε (4.3) Here, symols with the susript r re defined in the entre of the rim. The derivtion of fore nd moment equilirium n e mde using the Lgrnge method [De Krker; 001]. For this method, the kineti energy T nd potentil energy U of the system s well s the generlised fores Q re needed. The Lngrngin eqution tht is used for the derivtion of the equtions of motion n e desried y: d T T U + = Q i dt qɺ q q i i i (4.4) where q i indites the i-th generlised oordinte: q q y 1 = = q ψ (4.5) The generlised fores Q i n e determined y the eqution: = iδ i (4.6) i= 1 δw Q q where δw indites the virtul work done y the generlised fores nd δq i indites vertil hnge in the generlised oordinte. The potentil energy nd the generlised fores re dependent on the used tyre model nd re lulted in the next setions. 43

45 Stright tngent The system with two degrees of freedom desried in the previous setion n e nlysed in omintion with the simple stright tngent tyre model. The slip model for the stright tngent tyre model is derived in Chpter nd n e desried y (.14). However, euse of the geometry of the system, (.14) hs to e rewritten in terms of the generlised oordintes given in (4.5). The stright tngent tyre model ssumes the wheel to e rigid mss nd euse the prmeters re ssumed to e onstnt, the following set of equtions holds: y = y Rψ sinε nψ osε yɺ = yɺ Rψɺ sinε nψɺ osε ψ = ψ osε ψɺ = ψɺ osε (4.7) Here, symols with the susript re defined in the ontt entre. Comining (.14) nd (4.7) yields: ( ) σαɺ ' + Vα ' = Vψ osε R tnε n ψɺ osε yɺ Fy = CFαα M z = CMαα (4.8) For the remining equtions of motion, the kineti energy T, the potentil energy U nd the generlised fores Q i re needed. The expression for the kineti energy for the two degrees of freedom of the front xle with the stright tngent tyre model reds s: 1 1 ( ) 1 T = m os yɺ + mw yɺ Nψɺ ε + Iw ψ ψɺ (4.9) In this eqution, N is the distne etween entre of grvity nd swivel xis nd n e lulted y: N = q + n (4.10) where q nd n re the entre of grvity offset nd the steering xis offset respetively. The potentil energy U of the system with the stright tngent tyre model n e desried y: 1 1 U = y y + ψ ψ (4.11) The virtul work δw in the system is done y the externl tyre fores nd dmping fores. The lterl tyre fore ts on the tyre with pneumti tril. Figure 4. shows this effet. 44

46 Figure 4.. Relevnt lengths nd ngles for equtions of motion. The virtul work done y the generlised fores n e desried y: ( ( tn ) os ) δw = F δ y R ε + n + t δψ ε k yɺ δ y k ψ ψɺ δψ (4.1) y p y In this eqution, δy is the virtul lterl displement of the xle nd δψ is the virtul ngulr displement of the wheel round the steering xis. Comining (4.1) nd (4.6) yields expressions for the generlised fores: Q1 = Fy ky yɺ Q = Fy ( R tnε + n + t p ) osε k ψ ψɺ (4.13) Solving the Lgrnge eqution for the first vrile (y ) nd using the expression for F y from (4.8) yields fore equilirium in the y-diretion: ( ) ɺɺ os m + m ɺɺ y m Nψ ε + y = C α k yɺ (4.14) w w y f α y Solving the Lgrnge eqution for the seond vrile (ψ ) nd using the expression for M z from (4.8) nd (.15) yields moment equilirium round the swivel xis: ( ) ɺɺ ( ) m Ny ɺɺ osε + m N os ε + I ψ + ψ = C α R tnε + n + t osε k ψɺ (4.15) w w wψ ψ f α p ψ To nlyse the stility of the front xle with the stright tngent tyre model, the equtions of motion hve to e trnsformed into stte spe form. In order to do this, the system n e trnsformed into seond-order system in mtrix form. This form n e desried s: Mqɺɺ + Kqɺ + Cq = Fα (4.16) 45

47 The x mss mtrix M reds s: mt mwn osε M = mwn osε m w N os ε + Iw ψ (4.17) Here, m t is the totl mss of the system nd n e lulted y: The x dmping mtrix K reds s: mt = m + mw (4.18) ky 0 K = 0 k ψ (4.19) The x stiffness mtrix C reds s: y 0 C = 0 ψ (4.0) The x1 mtrix F is relted to the tyre fores nd reds s: CFα F = CFα ( R tnε + n + t p ) osε (4.1) The stte-spe form ( x = Ax) ɺ of the system n now e derived. Comining the eqution desriing the slip model (4.8) nd the seond-order system in mtrix form (4.16) yields: q 0 I 0 q d dt qɺ = M C M K M F qɺ (4.) V α ' σ α ' Wp Wv where I is x unity mtrix nd W v nd W p red s: W W v V Rsinε nosε osε ( σ σ ) V osε ( 0 ) = + p = σ (4.3) The system mtrix A is given in Appendix D. Von Shlippe As well s the stright tngent tyre model, the Von Shlippe tyre model ssumes the wheel to e rigid mss. As is done for the system with the stright tngent tyre model, the slip model hs to e rewritten in terms of the generlised oordintes (4.5). Sustituting (4.7) in (.19) yields: 46

48 1 σ σ + 3 σ + ɺɺɺ z + ɺɺ z + zɺ + z = y 3 + ( σ + R tnε n) ψ osε 3V V V C Fα 1 Fy = z + ɺɺ z y + ( R tnε + n) ψ osε σ + 3 V 1 M z = CMα zɺ ψ osε V (4.4) The sme expressions for the kineti energy (4.10) nd potentil energy (4.11) n e used for the system with the Von Shlippe tyre model. However, s n e seen from (4.4), the self ligning moment n no longer e lulted y giving the lterl tyre fore onstnt pneumti tril. As result, the generlised fores nd therefore the virtul work of the system hnge due to the different tyre model. The virtul work done y the externl tyre fores nd frition n e desried y: ( ( ) ) δw = Fy δ y R tnε + n δψ osε + M zδψ osε ky yɺ δ y k ψ ψɺ δψ (4.5) Comining (4.5) nd (4.6) yields expressions for the generlised fores: Q1 = Fy ky yɺ Q = Fy ( R tnε + n) osε + M z osε k ψ ψɺ (4.6) Sustituting (4.9), (4.11) nd (4.6) in the Lgrnge eqution (4.4) nd solving for the first vrile (y ) yields fore equilirium in the y-diretion: ( ) ɺɺ ψ os m + m ɺɺ y m N ε + y = F k yɺ (4.7) w w y y y Sustituting the expression for the lterl tyre fore F y from eqution (4.4) into (4.7) nd rerrnging yields n eqution tht n e trnsformed into the stte-spe form: CFα m + m ɺɺ w y mwnɺɺ ψ osε + ky yɺ + y + y σ + CFα CFα CFα ( R tnε + n) ψ osε = ɺɺ z + z σ + 3( σ + ) V ( σ + ) ( ) (4.8) Solving the Lgrnge eqution for the seond vrile (ψ ) yields moment equilirium round the swivel xis: ( ) ɺɺ ( ) m Ny ɺɺ osε + m N + I ψ os ε + ψ = F R tnε + n osε + M osε k ψɺ (4.9) w w wψ ψ y z ψ Sustituting the expression for the lterl tyre fore F y nd the self ligning moment M z from eqution (4.4) into (4.9) nd rerrnging yields n eqution tht n e trnsformed into the stte-spe form: 47

49 C Fα mwny ɺɺ osε + ( mwn + I ) ɺɺ wψ ψ os ε + k ψψɺ ( R tnε + n) y osε σ + CFα + ψ + ( R tnε + n) os ε CMα os ε ψ σ = (4.30) + CFα CMα C Fα ( R tnε + n) ɺɺ z osε zɺ osε ( ) ( ) ( R tnε + n ) z osε 3 σ + V V σ + To nlyse the stility of the front xle with the Von Shlippe tyre model, the equtions of motion hve to e trnsformed into stte spe form. As is done for the system with the stright tngent tyre model, the seond-order system in mtrix form n e used for this trnsformtion. Beuse only the slip model hnges with respet to the system with the stright tngent tyre model, the mss mtrix M (4.17) nd dmping mtrix K (4.19) re unhnged. The stiffness mtrix C hnges however due to the dependeny of the tyre fores on y nd ψ. The x stiffness mtrix C eomes: ( tnε ) CFα CFα y + R + n osε σ + σ + C = CFα CFα ( ) ( ) σ + R tnε + n osε ψ + σ + R tnε + n os ε CMα os ε (4.31) Beuse Kluiters uses third order eqution tht desries the slip model, the x3 mtrix F eomes: CFα CFα 0 3( σ + ) V ( σ + ) F = CFα CMα osε CFα ( R tnε + n) osε ( ) ( ) ( R tnε + n) osε 3 σ + V V σ + (4.3) The stte-spe form ( xɺ = Ax) of the system n now e derived. Comining the eqution desriing the slip model (4.4) nd the seond-order system in mtrix form (4.16) yields: q 0 I 0 q d qɺ M C M K M F qɺ z = 0 0 Z z 1 dt zɺ 0 0 Z zɺ z z ɺɺ Wp 0 Z3 ɺɺ where I is x unity mtrix nd W p, Z 1, Z nd Z 3 red s: W Z Z Z 3 3 3V 3V ( σ + R tn ε + n) osε p = ( σ σ ) 1 3 ( 0 1 0) ( 0 0 1) 3 3V 3( σ + ) V V ( 3σ + ) ( σ σ σ ) = = = (4.33) (4.34) 48

50 Rigid ring For the equtions of motion for the rigid ring tyre model itself, the equtions of motion derived in Setion.5 n e used. As well s for the other two tyre models, these equtions hve to e rewritten in terms of the generlised oordintes (4.5). Sustituting (4.3) in the equtions derived for the elt (.) yields: ( ψɺ osε ) ( ɺ γ ) y ( y y + nψ osε ) + y ( y y Rγ ) ( os ) ( ) k ψ Ω ( γ + ψ sinε ) ψ ( ψ ψ osε ) + ψ ( ψ ψ ) ( sinε ) + Rky ( yɺ yɺ Rɺ γ ) + k ( ψ ψ osε ) ( γ ψ sinε ) R ( y y Rγ ) mɺɺ y = k yɺ yɺ + n + k yɺ yɺ R y y I ɺɺ ψ = I Ωɺ γ k ψɺ ψɺ ε + k ψɺ ψɺ ψ ω ψ ψ I ɺɺ γ γ = I ωωψɺ k ɺ γ γ + ψɺ γ Ω γ + + y (4.35) Equtions for the ontt pth (.3) s well s the equtions for the slip model (.4) through (.39) remin unhnged. The rigid ring tyre model ssumes tht the wheel onsists of more thn one rigid ody. Between these odies, dditionl degrees of freedom exist. A shemti representtion of this tyre model tthed to the front xle model is displyed in Figure 4.3. In this figure, the offsets n nd q re not displyed to improve the redility. Beuse the equtions of motion of the elt nd ontt pth s well s the slip model re derived lredy in Chpter nd Appendix B, only the equtions of motion of the two degrees of freedom of the xle re onsidered here. Figure 4.3. The rigid ring tyre model tthed to suspension model. When ompred to the systems with the other two tyre models, similr eqution for the kineti energy (4.9) n e used. However, insted of the inerti prmeters of the wheel m w nd I wψ, the inerti prmeters of the rim m r nd I rψ hve to e used for the two degrees of freedom of the front xle: 49

51 1 1 ( ) 1 T = m os yɺ + mr yɺ Nψɺ ε + Ir ψ ψɺ (4.36) The potentil energy U hnges euse the tyre fores do not t on the rim itself nymore. Insted, dditionl spring fores t on the rim: U = y y + y ( y nψ osε y ) + ψψ ψ ( ψ osε ψ ) + γ ( ψ sinε γ ) (4.37) The virtul work δw in the xle is done y the dmping fores: ɺ ( ɺ ɺ ɺ os ) ( ɺ ) ψ ( ) ( ) k ( ) δw = ky yɺ δ y k ψψ δψ + ky y y + nψ ε δ y + kynosε yɺ yɺ + nψ osε δψ + k ψ osε γ sinε ψ δψ (4.38) + k ψ Ω γ + ψ sinε δψ osε + γ Ω ψ ψ osε δψ sinε Comining (4.38) nd (4.6) yields expressions for the generlised fores: ( ɺ ɺ ψɺ osε ) os ( os ) ψ ( os sin ) ( γ ψ ε ) ε k ( ψ ψ ε ) ε Q1 = ky yɺ + ky y y + N Q = k ψψɺ kyn ε yɺ yɺ + nψɺ ε + k ψɺ ε ɺ γ ε ψɺ (4.39) + k ψ Ω + sin os + γ Ω os sin Sustituting (4.9), (4.37) nd (4.39) in the Lgrnge eqution (4.4) nd solving for the first vrile (y ) yields fore equilirium in the y-diretion: ( ɺɺ ψ osε ) ( ψ osε ) ɺ ( ɺ ɺ ψɺ osε ) mɺɺ y + m ɺɺ r y N + yy + y y y n = ky y + ky y y + n (4.40) Solving the Lgrnge eqution for the seond vrile (ψ ) yields moment equilirium round the swivel xis. Furthermore, if the rottionl stiffness onstnts ψ nd γ re tken equl to eh other s well s the rottionl dmping onstnts k ψ nd k γ, the moment equilirium n e simplified to: ( ɺɺ osε + ɺɺ ψ os ε ) + ɺɺ ψψ + osε ( ψ osε + ) mr Ny N Ir yn n y y + ψψ + ψ ( ψ ψ osε + γ sinε ) = k ψψɺ kynosε ( yɺ yɺ + nψɺ osε ) (4.41) + k ψ ( ψɺ osε ɺ γ sinε ψɺ ) + k ψ Ω ( ψ sinε + γ osε ) To nlyse the stility of the front xle with the rigid ring tyre model, the equtions of motion hve to e trnsformed into stte spe form. This stte-spe form ( xɺ = Ax) hs eome lrger nd is therefore given in Appendix E. 50

52 4.3 Anlytil omputtion of the stility oundries In order to find nlyti expressions for the oundries etween stility nd instility, the Hurwitz riterion [Hgedorn; 1988, Pejk; 1966] n e pplied to the hrteristi eqution of the system. The dmping oeffiients k y nd k ψ s well s the steering xis offset n nd entre of grvity offset q re negleted. The hrteristi eqution of the system desried y (4.) reds s: where: = (4.4) λ 1λ λ 3λ 4λ = 1 V 1 = σ CFα os ε ( R tnε + t p )( R tnε ) C F α ψ y = Iw ψσ mtσ Iw m ψ t CFαV os ε ( R tnε + t p ) yv V ψ 3 = + + Iw ψσ mtσ Iw ψσ CFαy os ε ( R tnε )( R tnε + t p ) ψ ( CFα + yσ ) 4 = + m I σ m I σ 5 t wψ t wψ ( tnε + ) CFαyV os ε R t p y ψv = + m I σ m I σ t wψ t wψ (4.43) The Hurwitz riterion n e pplied to the hrteristi eqution of the system in order to determine stility. The system is symptotilly stle if ll of the following onditions re met ( 0 = 1): V 1 > 0 : > 0 σ 5 > 0 : ψ > CFα os ε ( R tnε + t p ) H > 0 : Iw ψ > mt os ε ( R tnε + σ + )( R tnε + t p ) t 1 p ψ > r1 for ε < tn R H3 > 0 : t 1 p ψ < r1 for ε > tn R t 1 p ψ < min ( r, r3 ), ψ > mx ( r, r3 ) for ε < tn R H 4 > 0 : t 1 p min ( r, r3 ) ψ mx ( r, r3 ) for ε tn < < > R (4.44) 51

53 where r 1, r nd r 3 re given y: Iw y CF I ψ α wψ r1 = + C os Fα ε ( R tnε + t p ) m os ( tn )( tn ) mt ( R tn t ε R ε + σ + R ε + t ε + σ + p ) I r (4.45) y wψ = mt C I I ( tnε p ) Fα wψ y wψ r3 = + CFα os ε R + t mt ( R tnε σ ) mt + + The first ondition of the Hurwitz riteri is trivilly fulfilled: the forwrd veloity nd relxtion length re oth positive. Furthermore, the ondition for H 3 is lwys overed y omintion of the onditions for H 4 nd 5. The stility riteri re shemtilly displyed s funtion of the yw stiffness nd ster ngle in Figure 4.4. The hoie of these prmeters is mde euse this plot shows ll stility oundries exept for H. However, this riterion is shown to e redundnt lter on in this setion. The shded grey re indites negtive yw stiffness whih is physilly impossile sitution. Figure 4.4. Shemti representtion of the stility riteri. As n e seen from Figure 4.4, there re two hrteristi vlues of the ster ngle whih re indited y the lk dots numered with I nd II. The first hrteristi vlue for the ster ngle, indited with I, desries the point in the given ster ngle rnge where the stility oundry desried y 5 in (4.44) nd the horizontl line desriing yw stiffness equl to zero interset. Computing the intersetion of the ltter two lines results in the following ster ngle vlue: 5

54 1 p + t p = ε = R tnε 0 tn t R (4.46) This point n lso e seen from the Hurwitz riteri H 3 nd H 4. This vlue for the ster ngle desries the sitution where the lterl tyre fore ts extly on the xis of rottion of the wheel. Consequently, this is the point where the self ligning moment swithes its sign. The seond hrteristi vlue indites the mximum vlue for the ster ngle for stle system in the given ster ngle rnge. This vlue n e found y solving the eqution: This eqution n e redued to: yi wψ CF Iw yi w m = α ψ ψ α ε ε m R + m + ( tnε + σ + ) t t t ( tn p ) CF os R t (4.47) ( )( ) I = m R + + R + t (4.48) wψ t os ε tnε σ tnε p The solution is hosen s n expression for the moment of inerti rther thn the ster ngle to redue omplexity. When ompring (4.48) with H of the Hurwitz riteri, it ppers tht the expressions re equl to eh other. Consequently, H implies tht the system is unstle for ster ngle lrger thn the vlue indited with II in Figure 4.4. Therefore, H is overed y H 4 nd does not impose n dditionl stility restrition. 4.4 Prmeter influene on the nlytil expressions of the stility oundries The vertil line indited with I (4.46) seprtes the three stle res in Figure 4.4: there re two stle res on the left-hnd side of this line nd one on the right-hnd side of this line. Beuse oth the pneumti tril t p nd the rdius of the tyre R re lwys positive, this line is lwys desried y negtive vlue. As result, there is only one stle re in the positive ster ngle domin. In prtie, the ster ngle is usully positive in order to improve the stright line stility of the system nd therefore the stle re on the right-hnd side of the vertil line is the most interesting re in prtil point of view nd is therefore onsidered in this setion. The stle re on the right-hnd side of I is ound y three lines whih n e desried y (4.46) nd y r nd r 3. These three oundries n lso e seen from Figure 4.5. However, s pointed out lredy, the vertil line desried y (4.46) is lwys in the negtive ster ngle domin nd is therefore negleted. Consequently, only r nd r 3 re onsidered here. 53

55 Figure 4.5. Shemti overview of the stility oundries. First, the oundry on the right-hnd side of the stle re is the point where r nd r 3 interset. Beuse the ster ngle is usully positive, this point should e shifted towrds lrge positive ster ngle where (4.48) hs to e fulfilled. This n e hieved y deresing the mss of oth the wheel m w nd the xle m. Other hnges tht hve similr effet re deresing the relxtion length σ, hlf of the ontt length nd the pneumti tril t p nd inresing the moment of inerti I wψ. Adjusting the rdius of the tyre R hs different effet. When the mximum stle ster ngle vlue desried y (4.48) hs negtive vlue, deresing R results in derese of the mximum stle ster ngle vlue. For positive vlue, deresing R results in n inrese of the mximum stle ster ngle vlue. Seondly, the differene etween r nd r 3 should e s lrge s possile in order to hve stle system over lrge yw stiffness rnge. This differene n e desried y: Iw ψ r3 r = CFα ( R tnε + t p ) os ε mt ( R tnε + σ + ) (4.49) As n e seen from (4.49), the differene n e inresed y deresing the mss of oth the wheel m w nd the xle m, the relxtion length σ, hlf of the ontt length nd the pneumti tril t p nd inresing the moment of inerti I wψ. This is in ordne with the hnges tht hve to e mde to inrese the mximum stle ster ngle. The remining two prmeters C Fα nd R hve different influene on the stility: they oth ffet the grdient of r 3. This influene is shown in Figure 4.6. In this figure, the solid lines disply the originl stility oundries nd the dshed lines disply the oundries with inresing C Fα nd deresing R. Inresing C Fα results in n inresed negtive grdient of r 3. Beuse the vlue for the ster ngle indited with II in Figure 4.4 is not dependent on C Fα, r 3 rottes round this point. Consequently, n inrese of C Fα is desirle for stility onsidertions. Seondly, deresing R results in deresed negtive grdient of r 3. This my seem undesirle for n inrese of the stle re, ut it seems tht r 3 rottes round the point where it intersets with the vertil line where ε is equl to zero. 54

56 Consequently, deresing R results in n inrese of the differene desried with (4.49) for positive ster ngle nd derese of the differene in (4.49) for negtive ster ngle vlue. The effets of vrition of R nd C Fα re shemtilly displyed in Figure 4.6. However, it hs to e tken into ount tht these re vritions mde to only one prmeter. In relity, djusting one prmeter often results in the djustment of severl prmeters euse they n e physilly linked. A finl remrk n e mde out the lterl stiffness y. This prmeter does not influene the differene etween r nd r 3 nor does it influene the mximum stle ster ngle. However, when inresing y, r nd r 3 oth shift upwrds eqully. Figure 4.6. Effet on stility oundries when hnging C Fα (left) nd R (right). 4.5 Numeril omputtion of the stility oundries In this setion, omprison is mde etween the influenes of the tyre models on the stility regions. Due to the inresed omplexity of the Von Shlippe tyre model nd the rigid ring tyre model with respet to the stright tngent tyre model, numeril omputtion of the eigenvlues of the system mtrix is mde rther thn deriving nlytil expressions of the stility oundries s is done in the previous setion. Tle 4.1. Prmeter vlues for the front xle of pssenger r. Prmeter Vlue Unit Desription y 1.40e6 N/m Lterl stiffness of the front xle ψ 1.91e4 Nm/rd Rottionl stiffness round the steering xis I rψ 3.75e-1 kgm Moment of inerti of the rim inluding prt of the tyre tht is fixed to the rim round the x- nd z-xis I rω 7.5e-1 kgm Moment of inerti of the rim inluding prt of the tyre tht is fixed to the rim round the y-xis k y 0 Ns/m Lterl dmping of the front xle k ψ 0 Nms/rd Rottionl dmping round the steering xis m.00e1 kg Hlf of the mss of the front xle m r 1.50e1 kg Mss of the rim inluding prt of the tyre tht is fixed to the rim n 0 m Steering xis offset with respet to the wheel entre q 0 m Centre of grvity offset with respet to the wheel entre ε 6.46 Cster ngle Prmeters tht re used in this setion re representtive for steering system of pssenger r. The prmeter vlues for tyre hve een disussed lredy nd re displyed in Tle 3.1. However, euse other omponents suh s the rking system nd hu re inluded in this hpter, inerti prmeters m r, I rψ nd I rω re inresed to ount for these omponents. Prmeters tht re used for the steering system re tken from multiody model developed t the University of Tehnology Eindhoven nd re displyed in Tle 4.1. In order to redue the 55

57 omplexity of the stility nlysis in this hpter, the dmping oeffiients k y nd k ψ nd the two offsets q nd n re tken equl to zero. The prmeters re tken s given in Tle 3.1 nd Tle 4.1, unless indited otherwise. Figure 4.7 shows ontour plot of the mximum vlue of the rel prts of the eigenvlues, nd therefore the level of stility, for vrying yw stiffness nd ster ngle. As n e seen from this figure, there re three stle res nd three unstle res. Furthermore, the stility oundries derived in the previous setion n e seen from this figure. The lrgest level of instility ours for lrge negtive ster ngle in omintion with low yw stiffness. In this re, the system is monotonilly unstle. In this sitution, the eigenvlues hve positive rel prt nd no imginry prt. This mens tht the sttes of the system go to infinity without ny form of virtion: the wheel simply steers to one side. Furthermore, positive ster ngle uses the system to e ompletely unstle with n exeption for smll re t mediore vlue of the yw stiffness. Finlly, the third unstle re is loted t negtive ster ngle in omintion with yw stiffness of pproximtely Nm/rd. Figure 4.7. Contour plot of the mximum rel prt of the eigenvlues with vrying ster ngle nd yw stiffness for the system with the stright tngent tyre model (V = 5 m/s). Unlike the system with the stright tngent tyre model, the stility oundries of the system with the Von Shlippe tyre model re dependent on the forwrd veloity. This is euse the Von Shlippe tyre model inludes the triling ontt point of the tyre in the lultions of the tyre fores. As lredy mentioned, the triling ontt point hs veloity dependent dely with respet to the leding ontt point. Figure 4.8 shows the stility results for the front xle model with the Von Shlippe tyre model s funtion of ster ngle nd yw stiffness. Beuse the stility oundries re dependent on the forwrd veloity, vrious forwrd veloities re used. 56

58 The grey res indite n unstle system. At low forwrd veloities, the stle regions hnge drstilly when hnging the forwrd veloity nd there re very few onfigurtions tht re stle for every forwrd veloity. For higher forwrd veloities, the results eome more similr to the system with the stright tngent tyre model (Figure 4.7). Although the stility regions re dependent on the forwrd veloity, there re two stility oundries tht do not hnge when the forwrd veloity hnges. These re lso equl to the orresponding stility oundries of the system with the stright tngent tyre model whih re the stility oundries inditing the monotonilly unstle re in the lower left-hnd orner nd the oundry indited with r 3 in Figure 4.5. Figure 4.8. Unstle res (grey) nd stle res (white) for vrious forwrd veloities for the system with the Von Shlippe tyre model. Figure 4.9 shows the stility results for the system with the rigid ring tyre model s funtion of ster ngle nd yw stiffness for vrious forwrd veloities. In this figure, the grey res indite n unstle system. The system with the rigid ring tyre model shows very few similrities with the systems with the other two tyre models. The monotonilly unstle re in the lower left hnd orner is lso present for the system with the rigid ring tyre model s well s the unstle re in the lower right-hnd orner. As n e expeted for the system with the rigid ring tyre model, the system hs lrger stle re ompred to the systems with the other two tyre models. This is used y the turn slip effet nd the dmping fores etween the rim nd elt. Beuse of the turn slip effet, the system is stle lmost everywhere t low forwrd veloities, exept for the monotonilly unstle re. It is interesting to see is tht t extremely high forwrd veloities, the unstle re in the lower right-hnd orner disppers gin. 57

59 Figure 4.9. Unstle res (grey) nd stle res (white) for vrious forwrd veloities for the system with the rigid ring tyre model. 4.6 Stility oundry sensitivity to prmeters djustments In this setion, the dependeny of the stility oundries on the most influentil prmeters is investigted. So fr, the dmping oeffiients k y nd k ψ nd the offsets q nd n re negleted in this hpter. However, it turns out tht these prmeters hve lrge influene on the stility oundries. Figures 4.10 nd 4.11 show the influene of the prmeters k y nd k ψ with respet to the stility oundries. The vlues for these prmeters re sed on 3% dmping of the orresponding eigenmode when the tyre is not touhing the rod. The vlues re tken equl to 460 Ns/m for k y nd 6.8 Nsm/rd for k ψ. The forwrd veloity is tken equl to 5 m/s throughout this entire setion. In Figures 4.10 nd 4.11, the grey res indite the unstle res where the dmping is tken equl to zero nd the shded res with the dshed stility oundries indite the unstle res with djusted dmping. From Figures 4.10 nd 4.11 it n e seen tht introduing dmping in the system uses the unstle res to eome smller. This effet is lrger for djusting the rottionl dmping insted of the lterl dmping. Moreover, the effet is lso lrger for the systems with the stright tngent tyre model nd the Von Shlippe tyre model ompred to the system with the rigid ring tyre model. However, the stility oundry inditing the monotonilly unstle re is not ffeted y djusting the dmping onstnts euse no virtions re involved in this sitution. 58

60 Figure Unstle res for k y = 0 (grey) nd k y = 460 (shded) for the system with the three tyre models t 5 m/s. Figure Unstle res for k ψ = 0 (grey) nd k ψ = 6.8 (shded) for the system with the three tyre models t 5 m/s. Figures 4.1 nd 4.13 show the influene of the offsets n nd q with respet to the stility oundries. The prmeter vlues re tken equl to 3.11 m nd 3.00 m for n nd q respetively. The vlue for n is sed on multiody model developed t the University of Tehnology Eindhoven nd the vlue for q is sed on the ssumption tht the rke lliper is positioned t the rer side of the rke disk. The grey res indite n unstle system for the offsets equl to zero nd the shded res with dshed stility oundries indite n unstle system for djusted offsets. As n e seen from Figures 4.1 nd 4.13, oth djustments use the unstle res to eome lrger for ll three models. The monotonilly unstle re in Figure 4.1 eomes smller. For positive ster ngle however, positive vlues for the offsets n nd q re desirle for lrge vlue of the yw stiffness nd undesirle for low vlues of the yw stiffness. Figure 4.1. Unstle res for n = 0 (grey) nd n = (shded) for the system with the three tyre models t 5 m/s. 59

61 Figure Unstle res for q = 0 (grey) nd q = 0.03 (shded) for the system with the three tyre models t 5 m/s. As well s for the suspension prmeters tht re nlysed so fr, the sensitivity of the stility oundries to the tyre prmeters n e investigted. As n e seen from previous hpters, there re numerous tyre prmeters tht n e vried. However, most of these prmeters only influene the stility oundries slightly when relisti djustments re mde. It seems tht the prmeters tht influene the stility res the most re the prmeters used for the lultion of tyre fores. These re the pneumti tril t p, the ornering stiffness C Fα nd the turn slip stiffness for the self ligning moment C Mϕ. The prmeters used in the lultion of the tyre fores re lso the prmeters tht re most likely to hnge during lod hnges on the tyre for exmple while driving round. Other prmeters suh s the ontt length nd relxtion length σ re only influening the stility oundries of the system with the stright tngent tyre model nd the Von Shlippe tyre model rther thn the system with the rigid ring tyre model. Beuse the rigid ring tyre model is shown to e the most urte tyre model, these prmeters re left out in the sensitivity nlysis in this setion. Figure Unstle res for C Fα = (grey) nd C Fα = (shded) for the system with the three tyre models t 5 m/s. Figure Unstle res for t p = 0.03 (grey) nd t p = 0.06 (shded) for the system with the three tyre models t 5 m/s. 60