A nonlinear mathematical model for a bicycle

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1 A noninea mathematica mode fo a bicyce Giovanni Fosai 1, Fancesco Ricci 1 1 Dipatimento di Matematica Appicata "G.Sansone", Univesità di Fienze, Itay E-mai: giovanni.fosai@unifi.it, jb.icci@vigiio.it Keywods: Bicyce, Nonhoonomic constaints, Noninea mode. SUMMARY. This pape concens the mathematica modeing of a bicyce taking account of the pesence of a tai. It is we known that the bicyce is a cassica exampe of a nonhoonomic system. Fist we conside the kinematics of oing of a bicyce and we deive the fu Lagangian fo a bicyce with tai. Then we wite the equation of motion fo a simpified mode with zeo caste ange, foowing the constained Lagangian point of view. 1 INTRODUCTION In the ecent past, a geat inteest has been devoted towads bicyce modeing as it is a mechanica system chaacteized by nonhoonomic constaints. The bicyce is pobaby the most common mode of tanspotation in the wod, next ony to waking, and, stating fom some pioneeing papes at the end of the nineteen centuy, [1, many eseaches have tied to find pope equations to descibe the dynamic of this system. Mainy, it is possibe to distinguish between two diffeent appoaches: the fist obtains the motion equations using the Newton s aws [, whie the second studies the system fom a Lagangian o Hamitonian point of view [3, 4. So fa, the geatest pat of the existing iteatue has been devoted to modes with ots of simpifications, even if these have been capabe to expain the dynamica chaacteistics of the bicyce. Fo exampe, ineaized equations of motion ae commony intoduced in ode to cope moe easiy with the pobem, [. In this pape we conside a mode with seven degees of feedom, in ode to study a system which is quite simia to the ea one. Moeove, this choose aows us to intoduce the nonhoonomic constaints due to the oing motion in a moe natua way. In fact, as the system has seven degees of feedom, intoducing fou constaints we can expess fou coodinates in function of thee. In section 3 we conside the kinematics of ou mode and we study the pitch ange as function of the bicyce paametes. Then we deive the fu Lagangian fo a bicyce with tai and without offset. Finay we wite the equation of motion fo a simpified mode with zeo caste ange, foowing the constained Lagangian point of view. THE BICYCLE MODEL.1 Desciption of the 7-dimensiona configuation space In this mode it is assumed that the bicyce taveses a fat gound, so that the system is subject to some symmeties. As esut, we can choose just seven coodinates in ode to descibe competey the dynamica system. In paticua, we conside the coodinates x, y of the ea whee contact point in the hoizonta pane with espect to the inetia efeence fame X, Y, Z, whee the Z-axis is pependicua to the gound pane in the diection opposite to gavity. Ceay, the thid coodinate of this contact point can be assume aways zeo. Then we conside the yaw ange θ, which is taken about the vetica Z-diection. This ange is zeo when the pane of the ea whee passes though the X-axis. We adopt the ight-hand ue, so the ange is positive fo counte-cockwise otations. The o ange α is the ange that the bicyce s pane of symmety makes with the vetica diection, so, oughy speaking, it is a otation about the x-axis of the oca efeence fame of the ea whee. In this pape we take α α in ode to obtain a positive ange when the bicyce eans to the eft. Futhemoe, we take the steeing ange ψ about the steeing axis, which is tited backwad by the caste ange ε, the atte due to the pesence of the tai. As the o ange, even the steeing ange can assume vaues in the open inteva π, π. We obseve that, consequenty to the pevious choice about the α ange, the anges ψ and α itsef ae positive when the bicyce is tuning to the eft. 1

2 Finay, we intoduce a ea and font whee otation anges χ and χ f in ode to conside the inetia moments and the mass of both whees. So, the configuation space is Q = SE S 1 S 1 S 1 S 1. Regading the nonhoonomic constaints, the ea and font whee contacts with the gound ae constained to have veocities paae to the ine of intesection of thei espective whee panes and the gound pane, but fee to otate about the axis passing though the whee-gound contact and paae to the z-axis of the oca efeence fame. To sum up, we choose seven geneaized coodinates in ode to mode ou system, and then we intoduced two nonhoonomic constaints that can be witten each one as a system of two equation, as it is showed in the foowing sections.. Physica paametes in ou mode In addition to the seven geneaized coodinates, the bicyce mode pesents some time-independent paametes that identify the bicyce itsef, as shown in figue 1. ε H z R B G R f C E A x F d λ Figue 1: Geomety of the bicyce In fact, the shape of the bicyce is competey deteminated by the adii of the two whees R and R f espectivey, the wheebase, the steeing axis ength h, the offset d and the caste ange ε. This one depends on the tai by the eation λ = R f tan ε d cos ε, and it pays an impotant oe in the motion of the bicyce. In fact, even if these ae not time-dependent coodinates of the system, by changing these paametes it is possibe to obtain diffeent motion of the bicyce. In paticua, the caste ange has a emakabe infuence on the motion, and the esuts obtained ae competey diffeent as the ange ε is cose to zeo o to π. It is cea that this ange is negative accoding to the ight-hand ue. At the beginning, we conside a compete bicyce mode, with tooida ties and offset. Then, in ode to cope with moe simpe equations, we assume d = 0 and = 0, whee is the whee cown adius. 3 THE KINEMATICS OF BICYCLE The kinematic study of bicyce is the fist step to obtain the equations of motion fo the system. To find the eation between the geneaized coodinates chosen befoe, it is necessay to wite down the otation matices fo the fou igid body which compose the system. In this pape, we assume aias tansfomation, so it is tuned the coodinate system.

3 3.1 Rea whee Fist we conside the ea whee. Without consideing the pope otation of the ea whee, the otation matix associated to the body is cos θ sin θ 0 R 1 = R x αr z θ = cos α sin θ cos α cos θ sin α, 1 sin α sin θ sin α cos θ cos α whee R xi α i is the otation about axis x i of an ange α i. In fact, the ea whee otation ange χ pays its oe when we wite the angua veocities of the bodies. Hence, if B is the cente of the ea whee, as shown in figue 1, the expession of the vecto B A in the oca efeence fame is B A S = R + cos α k S sin α j S, so it can be expessed in the inetia efeence fame by substituting the oca unit vectos with thei expession espect to the inetia fame, which can be taken fom the ows of the otation matix 1. We obtain B A Σ = R sin α sin θ ı + R sin α cos θ j + R cos α + k, whee the coodinates of A espect to the oigin of the inetia efeence fame ae A O Σ = x ı + y j. In this way, it is possibe to wite down the coodinates of each point of the fou bodies in the oca and the inetia efeence fame. In paticua, we ae inteested to obtain the coodinates of the cente of mass of each body, in ode to expess the kinetic enegy of the system. 3. Rea fame Passing to the ea fame, we intoduce an auxiiay ange µ, which epesents the pitch ange. In fact, it is common knowedge that when the steeing axis is tuned, the cente of the font whee goes down, so that the ea fame tuns down too. The fact is that the pitch ange depends on the othe coodinates, so it wi be expessed in the foowing sections as a function µ = µθ, ψ, α. In ode to obtain this eation, we wite the otation matix of the ea fame with the ange µ R = R y µr 1 = cos µ cos θ + sin µ sin α sin θ cos µ sin θ sin µ sin α cos θ sin µ cos α = cos α sin θ cos α cos θ sin α, sin µ cos θ cos µ sin α sin θ sin µ sin θ + cos µ sin α cos θ cos µ cos α so that the coodinates of any point of the ea fame can be expessed as a ı S = acos µ cos θ + sin µ sin α sin θ ı + acos µ sin θ sin µ sin α cos θ j a sin µ cos α k 3 in the oca and the inetia efeence fame espectivey, whee a [0,. In this way, we can expess the coodinates of the point C of intesection of the wheebase with the steeing axis, using the eation 3 with a =. Next we find anothe way to expess these coodinates. 3.3 Font fame and font whee The otation matix associated with the font fame and the font whee is sighty moe compicated espect to those seen befoe. In fact, it is necessay to mutipy fou otation matices, i.e. a θ-otation, an α-otation, an ε-otation and a ψ-otation. Thus we obtain the matix R 3 = c ψc ε c θ s ε s α s θ s ψ c α s θ c ψ c ε s θ + s ε s α c θ + s ψ c α c θ s ε c ψ c α s ψ s α s ψ s ε s α s θ c ε c θ c ψ c α s θ s ψ c ε s θ + s ε s α c θ + c ψ c α c θ s ε s ψ c α c ψ s α, 4 s ε c θ c ε s α s θ s ε s θ + c ε s α c θ c ε c α which can be used to expess the oca unit vectos espect to the inetia fame. Howeve, it is convenient to intoduce anothe otation matix with thee auxiiay anges, in ode to expess in a diect way the font 3

4 whee nonhoonomic constaints. Let θ, α and µ be the auxiiay anges intoduced fo the font pat, so that we can wite the otation matix cos µ cos θ + sin µ sin α sin θ cos µ sin θ sin µ sin α cos θ sin µ cos α R = cos α sin θ cos α cos θ sin α. 5 sin µ cos θ cos µ sin α sin θ sin µ sin θ + cos µ sin α cos θ cos µ cos α Now it is evident that the diecto cosines of matices 4 and 5 ae the same, theefoe we have nine eations such as sin α = sin ε sin ψ cos α + cos ψ sin α, which wi be appied in the foowing sections. b cos ε = d sen ε = b = d tan ε ε C ε E b π d D Figue : Geomety of the bicyce: the offset Now we want to expess the coodinates of the point C with espect to the font pat of the bicyce. We conside the font whee-gound contact point, whose coodinates espect to the inetia fame ae z, w. Then, efeing to figue, we have E F Σ = R f cos ε sin ε cos ψ cos θ sin ε cos ψ sin α sin θ sin ε sin ψ cos α sin θ+ cos ε sin ε cos θ cos ε sin α sin θ ı+ + R f cos ε sin ε cos ψ sin θ + sin ε cos ψ sin α cos θ + sin ε sin ψ cos α cos θ+ cos ε sin ε sin θ + cos ε sin α cos θ j+ + R f sin ε cos ψ cos α sin ε sin ψ sin α + cos ε cos α k + k, and D E Σ = dsin ε cos ψ sin α sin θ cos ε cos ψ cos θ + sin ψ cos α sin θ ı+ + d sin ε cos ψ sin α cos θ cos ε cos ψ sin θ sin ψ cos α cos θ j+ + dsin ψ sin α sin ε cos ψ cos α k, C D Σ = b sin ε cos θ cos ε sin α sin θ ı+ + b sin ε sin θ + cos ε sin α cos θ j + b cos ε cos α k = = d tan ε sin ε cos θ sin ε sin α sin θ ı+ + d tan ε sin ε sin θ + sin ε sin α cos θ j + d sin ε cos α k. Theefoe, the vecto C O Σ can be expessed in two diffeent ways, as eithe C D + D E + 4

5 E F + F O o C B + B A + A O. Equating these expession yieds z c ε s ε c ψ c θ s ε c ψ s α s θ s ε s ψ c α s θ c ε s ε c θ c ε s α s θ w + R f c ε s ε c ψ s θ s ε c ψ s α c θ + s ε s ψ c α c θ c ε s ε s θ + c ε s α c θ 0 s ε c ψ c α s ε s ψ s α + c ε c α +/R f +d s ε c ψ s α s θ c ε c ψ c θ + s ψ c α s θ s ε c ψ s α c θ c ε c ψ s θ s ψ c α c θ + d tan ε s ε c θ s ε s α s θ tan ε s ε s θ + s ε s α c θ = 6 s ψ s α s ε c ψ c α s ε c α = x sin α sin θ cos µ cos θ + sin α sin µ sin θ y + R sin α cos θ + cos µ sin θ sin α sin µ cos θ, 0 cos α + /R cos α sin µ so that we have thee expessions fo the auxiiay vaiabes z, w and µ. 4 THE PITCH ANGLE AND ITS ANALYSIS As we have aeady seen at the end of the ast section, thee is a eation between the pitch ange µ and the othe geneaized coodinates. In fact, accoding to what we anticipated befoe, hods the eation sin µ = R f sin ε sin ψ tan α sin ε cos ψ cos ε+ + d sin ε cos ψ sin ψ tan α sin ε + R 7. Fist of a, we obseve that the pitch ange is function of the ange α and ψ, but is θ-independent, i.e. it is independent of the diection towads the bicyce is moving senμ Figua 3: Pot of sin µ as function of ψ, with α fom π 4 ed to π 4 back. Ψ In addition to this, by expession 7 we can evauate the pitch ange fo diffeent vaues of the o ange. In paticua, we pot the pitch ange and its cosine and sine, assuming the ea and the font whee adii equa, i.e R f = R = 0.35m, and the othes physica paametes with the foowing vaues: d = 0.5m, ε = 0.1πad and = 1.0m. In figue 3 it is potted sin µ as function of the steeing ange ψ fo diffeent vaues of the o ange: in paticua the atte assumes its vaues in the ange [ π 4, π 4 with step of π 8. It is cea that the sine of the pitch ange is quite sma, even if it can assume sighty bigge vaues as d tends to zeo. Howeve, the eo made by substituting the ange with its sine is acceptabe, as it is shown in figue 4. On the othe hand, cos µ is vey cose to one, as shown in figue 5, so that we can assume cos µ = 1 with a maximum eo of the 0.78%. Accoding to this anaysis, it is possibe to conside µ a sma ange. Howeve, in the foowing sections we continue to conside competey the pitch ange, in ode to obtain the whoe expession of the Lagangian functiona. Athough, in ode to dea with shote expession, we now assume the offset d = 0, the adius of the toi = 0 and the two whees with R f = R. 5 THE FRONT WHEEL CONTACT POINT Befoe studying the angua veocities of the system in ode to wite the kinetic enegies of the fou bodies, we want to spend some wods about the position of the font whee contact point. In fact, as we have aeady done with the pitch ange, it is possibe to expess the coodinates of the font whee contact 5

6 Μ senμ Ψ cosψ Ψ Figue 4: Pot of µ sin µ as function of ψ, with α fom π 4 ed to π 4 back. Figue 5: Pot of cos µ as function of ψ, with α fom π 4 ed to π 4 back. point by the vectoia eation 6. In fact, the fist and the second scaa expession gives z = x R sin α sin θ + cos µ cos θ + sin α sin µ sin θ + dtan ε sin ε cos θ+ + sin ε sin α sin θ sin ε cos ψ sin α sin θ + cos ε cos ψ cos θ sin ψ cos α sin θ+ + R f sin ε cos ψ sin α sin θ cos ε sin ε cos ψ cos θ + cos ε sin α sin θ+ + sin ε sin ψ cos α sin θ + cos ε sin ε cos θ 8 and w = y + R sin α cos θ + cos µ sin θ sin α sin µ cos θ + dtan ε sin ε sin θ+ sin ε sin α cos θ + sin ε cos ψ sin α cos θ + cos ε cos ψ sin θ + sin ψ cos α cos θ+ + R f cos ε sin ε sin θ sin ε cos ψ sin α cos θ cos ε sin ε cos ψ sin θ+ cos ε sin α cos θ sin ε sin ψ cos α cos θ. 9 By these expession we can pot the position of the font whee contact point. In paticua, in figue 6 and 7 we have the pot of such position aong the X-coodinate and the Y -coodinate, espectivey. We assume x = y = 0 fo the ea whee contact point, and aso θ = 0. The othes paametes ae equa to those chosen in the ast section. z w Ψ Ψ Figue 6: Pot of z as function of ψ, with α = π 6. Figue 7: Pot of w as function of ψ, with α = π 6. Wheeas, in figue 8 we have the font whee contact point coodinates, so that it is cea how the point moves whie the steeing ange is changing. 6 THE ANGULAR VELOCITIES We now tun to wite the angua veocities fo the fou bodies of the bicyce. In ode to obtain the ight expessions, we use the foowing eation 6

7 w Figua 8: Position of the font contact point, with α = π 6 and ψ [ π, π. z ω S = d ks + dt ı S d js dt k S j S + ı S 10 d ıs dt j S ks, whee the oca unit vectos ae obtain by the otation matices intoduced befoe. In paticua, as anticipated, the otation matices fo the ea and font whees ae given by pe-mutipying the matices R 1 and R 3 espectivey by a matix R y χ and a matix R y χ a. So, appying eation 10, we have α ω 1S = χ θ α cos µ θ cos α sin µ sin α ω S = µ θ sin α θ cos α θ cos α cos µ α sin µ α cos ε cos ψ + θsin ε cos ψ cos α sin ψ sin α ω 3S = α cos ε sin ψ θsin ε sin ψ cos α + cos ψ sin α θ cos ε cos α + α sin ε + ψ α cos ε cos ψ + θsin ε cos ψ cos α sin ψ sin α ω 4S = α cos ε sin ψ θsin ε sin ψ cos α + cos ψ sin α + χ a, θ cos ε cos α + α sin ε + ψ whee the ea whee is the body 1, the ea fame the body, the font fame the body 3 and the font whee the body 4. 7 THE NONHOLONOMIC CONSTRAINTS The nonhoonomic constaints associated with the ea whee ae expessed by {ẋ cos θ + ẏ sin θ = R χ ẋ sin θ ẏ cos θ = 0 whie those associate with the font one ae expessed by { ż cos θ + ẇ sin θ = R χ a ż sin θ ẇ cos θ = 0 whee ż and ẇ ae obtained deiving expessions 8 and 9, and using the auxiiay ange intoduced with the otation matix 5. In paticua, we have the foowing equaities: sin θ = cos ψ cos α sin θ + sin ψcos ε cos θ sin ε sin α sin θ 1 sin ε sin ψ cos α cos ψ sin α sin ε sin ψ cos ψ sin α cos α, and cos θ = cos ψ cos α cos θ sin ψcos ε sin θ + sin ε sin α cos θ 1 sin ε sin ψ cos α cos ψ sin α sin ε sin ψ cos ψ sin α cos α. As the constaints imposed on the ea whee give a eation on ẋ and ẏ, those associated with the font whee give a eation on θ and χ a, afte having expessed the auxiiay anges as function of the geneaized 7

8 coodinates. In paticua, we obtain { θ = hψ, α, µ, χ, ψ, α, µ gψ, α χ a = fψ, α, µ, χ, ψ, α, µ, θ, whee these functions ae competey noninea. Howeve, we can obseve that in concusion we have fou eations which aow to educe the geneaized veocities fom seven to thee. It is cea that we choose the natua veocities that can be nomay contoed by the ide, i.e. the ea whee veocity, the o ange and the steeing ange vaiations. 8 THE LAGRANGIAN FUNCTIONAL Once obtained the eations necessay to expess the coodinates of the centes of mass of the fou body, we can wite the kinetic enegy of the system and its potentia enegy. Hence, et h be the coodinate of the font fame cente of mass in the oca efeence fame, and et m 1, m, m 3 and m 4 be the masses of the fou bodies. In addition to this, we intoduce the inetia tensos of the bodies in ode to wite the pat of the kinetic enegy due to otation: σb = I I 0 σg = I I I I 4 σe = I I 6 0 σh = I I I I 8 Finay, we can wite the compete Lagangian functiona fo a bicyce with tai and without offset. L = 1 [ Aẋ cos θ + ẏ sin θ + AR α + sin α + B µ + α sin µ + θ sin α sin µ + θ cos µ + m 4h [ θ sin ɛ + θ cos ɛ sin α + α cos ɛ + AR α cos αẋ sin θ ẏ cos θ θ sin αẋ cos θ + ẏ sin θ [ + C µ cos µ sin α + α cos α sin µ θ cos µ + m 4 h θ sin ɛ α cos ɛ cos α ẋ sin θ ẏ cos θ [ + C µ sin µ + θ sin α sin µ m 4h θ cos ɛ sin α ẋ cos θ + ẏ sin θ + D sin µ α θ sin α + µ θ sin α +D α θ cos α cos µ + m 4 hr α cos ɛ θ α sin ɛ cos α + θ cos ɛ sin α B µ θ sin α + α θ cos α sin µ cos µ [ +m 4h θ sin ɛ α cos α sin µ + µ sin α cos µ + cos ɛ sin µ θ µ sin α α θ cos ɛ sin µ sin α + θ α cos ɛ cos α cos µ θ sin ɛ cos µ + m 4 h 1 θ α cos ɛ sin ɛ cos α + α + θ cos αi θ sin α + χ I + 1 α cos µ + θ cos α sin µ I [ µ θ sin α + θ cos α cos µ α sin µ I [ α cos ɛ cos ψ + θ sin ψ sin α θ sin ɛ cos ψ cos α + α sin ɛ + θ cos ɛ cos α + ψ I [ α cos ɛ sin ψ θ sin ɛ sin ψ cos α θ cos ψ sin α + χ a I [ α cos ɛ + θ sin ɛ cos α + θ sin α θ α sin ɛ cos ɛ cos α I [ α sin ɛ + θ ψ cos ɛ cos α + I8 AgR cos α + gc cos α sin µ m 4 gh cos ɛ cos α whee A = m 1 + m + m 3 + m 4, B = m 4 + m 3 + m 4, C = m + m 3 + m 4, D = m R + m 3R + m 4 R. It is cea that the expession given above is quite ong and it is difficut to dea with it, even because we have to wite aso the constained Lagangian, imposing the constaints on the Lagangian function. 9 THE SIMPLIFIED MODEL AND ITS EQUATIONS OF MOTION In ode to obtain the equations of motion of the bicyce, we assume that the caste ange is zeo, i.e ε = 0, and thus aso µ = 0. This simpification aows us to wite the equations of motion. 8

9 Fist of a, we intoduce the foowing change of vaiabes: {ẋ sin θ ẏ cos θ = v ẋ cos θ + ẏ sin θ = v, so that we can ewite the nonhoonomic constaints as v = R χ v = 0 θ = R χ tan ψ. 11 χ a = χ cos ψ Taking account of the simpified Lagangian with ɛ = 0, and imposing the constaints, the constained Lagangian has the foowing fom. L c = 1 [ Av + AR α + v tan ψ sin α + B + m 4 h sin α v tan ψ + v m 4 h AR tanψ sinα + v D + m 4h v α tan ψ cos α + m 4hR + 1 v tan ψ sin α + v I + 1 R α I v tan ψ cos α+ ψ I [ α + v tan ψ sin α I α + v α sin ψ v v tan ψ sin α + 1 α + v tan ψ cos α [ α cos ψ+ v tan ψ sin ψ sin α sin ψ sin α + v I 6 R cos ψ v tan ψi tan ψ cos α + ψ I8 AgR cos α m 4gh cos α. The equations of motion ae obtained by the Lagange equations fo nonhoonomic systems [5: d L c dt ṙ α L c α + L c Aa α s a = L ṡ b Bb αβṙ β, whee the tems due to the matix A foow fom the Ehesmann connection, whie the matix B is epesentative of the cuvatue of the connection. In this case the Ehesmann connection is given by the eation θ 0 tan ψ 0 α v v ṙ = 0 χ a 0 1 R cos ψ 0 ψ and we have the foowing coefficients diffeent fom zeo A 1 = tan ψ A 3 = 1 R cos ψ, so that the ony components of the cuvatue that don t vanish ae B3 1 = 1 cos ψ B3 3 = sin ψ R cos ψ B3 1 1 = cos ψ B3 3 = sin ψ R cos ψ. Finay, afte some boing cacuations, the equations of motion ae the foowing. Fist equation: I 1 [ AR + m 4hR + I 1 + I 3 + I 7 + I 5 cos ψ + I 6 sin ψ α + sin ψ cos ψi 6 I 5 ψ α + v [ D + m 4h cos α + 1 R I6 + sin ψ cos ψ sin cos αi6 I5 ψ+ v tan ψ sin αi 5 + I 8 ψ ψ + [D v + m 4h tan ψ cos α + sin ψ sin αi 5 I 6 + R I6 tan ψ v AR + m 4 h + m 4 hr + I 1 I + I 5 I 7 + I 8 tan ψ sin α cos α 9

10 + v AR v m 4 hr I + I 6 tan ψ cos α R tan ψ sin ψ sin α cos αi 5 v sin ψ sin α cos αi 6 AR + m 4hg sin α = 0. Second equation: v A+ I R + B + I 4 + sin α AR + m 4 h + m 4 hr + I + I 5 sin ψ + I 7 + cos tan ψ α I 1 + I 5 + I 8 + v R tan ψ sin α m 4hR + I I 6 AR + I 6 sin ψ sin I 6 α + R cos ψ v +I 5 v ψ tan ψ cos ψ sin α + α tan ψ sin α cos α ψ tan ψ sin ψ sin α + v AR + m 4 h + m 4 hr + I + I 5 sin ψ + I 7 tan ψ ψ cos ψ B + I4 + v sin α ψ m4hr + I R cos I 6 AR ψ + v R α tan ψ cos α m 4 hr + I I 6 AR + α cos α α cos α tan ψ + v ψ tan ψ cos ψ cos α α tan ψ cos α sin α I 1 + I 5 + I 8 + α sin ψ sin α + α ψ sin ψ cos ψ sin α + α sin I5 I 6 ψ cos α + ψ cos α α ψ sin α tan ψ I 5 + I 8 + I 6 α tan ψ + α R ψ 1 + I 6 tan ψ cos v ψ ψ R cos ψ I 6 v ψ tan ψ sin α + I 6 I 5 Thid equation: I 5 +I 8 D + m 4 h + I 6 v ψ cos ψ sin ψ sin α + α sin ψ cos α sin α α ψ tan ψ sin α I 6 R α ψ tan ψ + v R I 6 ψ tan ψ sin α = 0. ψ+ v tan ψ cos α + v ψ cos α cos ψ v α tan ψ sin α sin α B + I 4 v tan ψ cos ψ sin α + AR m 4h v sin ψ cos ψ m 4hv α cos α cos ψ v I1 cos α cos ψ R tan ψ sin α + v R α sin ψ v sin ψ v sin α sin ψ sin α cos ψ + α sin ψ v sin ψ sin α + v R cos ψ v [ AR + m 4 h + m 4 hr + I 7 v α D sin α cos ψ I + I 5 v tan ψ cos α + ψ cos α cos ψ v cos ψ sin α α cos ψ v tan ψ R cos ψ α cos ψ + v tan ψ sin ψ sin α cos α cos ψ I 5 + I 8 I 6 + v tan ψ cos ψ sin α I5 sin ψ + I 6 cos ψ v + I 5 I 6 I 6 R tan ψ v α v sin α tan ψ sin α I 6 R cos ψ + I v α 6 R tan ψ +I 6 v R tan ψ cos ψ = 0. Refeences [1 Whippe, F.J.W., The stabiity of the motion of a bicyce," Quatey Jouna of Pue and Appied Mathematics, 30, [ Meijaad, J.P., Schwab, A. L., Lineaized equations fo an extended bicyce mode, in III Euopean Confeence on Computationa Mechanics Soids, Stuctues and Couped Pobems in Engineeing C.A Mota Soaes et a., Lisbon, Potuga, 5-9 June 006 [3 Neimak, J.I., Fufaev, N.A., Dynamics of nonhoonomic systems, vo. 33 of Tansations of Mathematica Monogaphs. Povidence, Rhode Isand: AMS, 197. [4 Koon, W.S., Masden J.E., The Hamitonian and Lagangian appoach to the dynamics of nonhoonomic systems", Repots in Mathematica Physics, 40, [5 Boch, A.M., Baiieu, J., Couch, P., Masden, J.E., Nonhoonomic mechanics and conto, Spinge Veag, Wien