Common underlying steering curves for motorcycles in steady turns

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Common underlying steering curves formotorcycles in steady turnsVenkata Mangaraju Karanam a b & Anindya Chatterjee ba Research and Development, TVS Motor Company, Hosur, Indiab Department of Mechanical Engineering, Indian Institute ofScience, Bangalore, IndiaPublished online: 10 Sep 2010.

To cite this article: Venkata Mangaraju Karanam & Anindya Chatterjee (2011) Common underlyingsteering curves for motorcycles in steady turns, Vehicle System Dynamics: International Journal ofVehicle Mechanics and Mobility, 49:6, 931-948, DOI: 10.1080/00423114.2010.483282

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Vehicle System DynamicsVol. 49, No. 6, June 2011, 931–948

Common underlying steering curves for motorcyclesin steady turns

Venkata Mangaraju Karanama,b* and Anindya Chatterjeeb†

aResearch and Development, TVS Motor Company, Hosur, India; bDepartment of MechanicalEngineering, Indian Institute of Science, Bangalore, India

(Received 23 November 2009; final version received 28 March 2010; first published 10 September 2010 )

We study the steady turn behaviours of some light motorcycle models on circular paths, using thecommercial software package ADAMS-Motorcycle. Steering torque and steering angle are obtainedfor several path radii and a range of steady forward speeds. For path radii much greater than motor-cycle wheelbase, and for all motorcycle parameters including tyre parameters held fixed, dimensionalanalysis can predict the asymptotic behaviour of steering torque and angle. In particular, steeringtorque is a function purely of lateral acceleration plus another such function divided by path radius.Of these, the first function is numerically determined, while the second is approximated by an ana-lytically determined constant. Similarly, the steering angle is a function purely of lateral acceleration,plus another such function divided by path radius. Of these, the first is determined numerically whilethe second is determined analytically. Both predictions are verified through ADAMS simulations forvarious tyre and geometric parameters. In summary, steady circular motions of a given motorcyclewith given tyre parameters can be approximately characterised by just one curve for steering torqueand one for steering angle.

Keywords: motorcycle; steering torque; steering angle; steady turn; circular path

1. Introduction

Motorcycle dynamics models are useful for both understanding and design. Many simplerigid body treatments, with suspension and tyre slips not included, have been published (fromWhipple’s early bicycle model [1] to the more recent Meijaard et al. [2] and Basu-Mandalet al. [3]). However, the automotive industry uniformly uses tyre slip models in the interestof capturing more realistic dynamic behaviour, and many books and papers also includesuspensions, tyre slip, and other non-ideal effects like frame flexibility [4–16]. The use ofmulti-body dynamics (MBD) software is practically unavoidable for such studies. AUTOSIMis favoured by Evangelou, Limebeer, Sharp and their co-workers [9,17–19]; Cossalter and hisco-authors use their own internally developed software [5,16,20]. In this paper, in common

*Corresponding author. Email: [email protected]†Present address: Indian Institute of Technology, Kharagpur, India.

ISSN 0042-3114 print/ISSN 1744-5159 online© 2011 Taylor & FrancisDOI: 10.1080/00423114.2010.483282http://www.informaworld.com

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932 V.M. Karanam and A. Chatterjee

with much industrial R&D work, we will use the commercially available software ADAMS-Motorcycle.

The dynamic behaviour of a motorcycle involves many facets, and has been studied in manyways over the years (see [19] and the references therein). In this paper, we study one specificaspect of motorcycle dynamics, namely steady behaviour on long curves. In particular, westudy the steering torque T and the steering angle δ needed to hold a fairly realistic motorcyclemodel on a circular path of large radius. The principal finding of this paper is that, even forrather complicated models, the steering angle and steering torque curves for different pathradii and speeds all approximately coalesce into single motorcycle- and tyre-specific curves.We present both simple theoretical prediction as well as detailed numerical verification ofthese approximately unifying curves. The implication of such unifying curves is that, duringthe design stage, time-consuming numerical simulations can be reduced in number.

While the simplification and approximate unification we offer in this paper are new, wemention that in fact there have been prior studies of steady-turn steering torques in motorcycles[20,21]. There, the approximations proposed were quite different from ours, the emphasis wason analytical expressions (though complicated), and the discussion included issues of actualmotorcycle handling as reflected by various qualitative features of these plots. In contrast,we offer a specific theoretical simplification whose implementation and verification are fullynumerical, and whose match with numerics is excellent: subjective implications for actualhandling are not discussed at all. To summarise, this paper focuses on showing how, for agiven motorcycle, many curves predictably and approximately coalesce into one.

We note here that a key determinant of actual motorcycle behaviour is the detailed natureof tyre slip, the modelling of which is a complex empirical activity that we deliberately de-emphasise in this paper. We present here a theoretical viewpoint which we believe survivesfor any reasonable model for tyre slip. In our numerical work, however, we use the ADAMS-Motorcycle with its inbuilt tyre model as a black box to demonstrate the validity of ourtheory.

We close this introduction with a brief description of the MBD model which we have devel-oped and simulated using the ADAMS-Motorcycle. The model has 11 degrees of freedom:these may be counted as 6 for the rear frame, 1 for the steering pivot, 1 for the swing armpivot (including rear suspension effects), 1 for the front telescopic suspension and 1 for eachwheel’s spin. Since the tyre has compliance and slip, no degrees of freedom are eliminated bythe ground contacts. Tyre slip is modelled using the Magic Tyre Formula, which is built intothe ADAMS-Motorcycle and used here as a black box (our theory is expected to apply equallyto other reasonable tyre models). These and other relevant details of the simulation have beendescribed, for purposes of saving space in this paper, in an eprints archive report [22]. Asdescribed in that same report, the simulations conducted are not a sequence of fixed-speedsimulations, but rather a few long simulations of slowly accelerated motions (0.1 m/s2) alongcircular paths. As seen in Karanam and Chatterjee [22] the change in the torques due to thisacceleration is small; the qualitative change is nil; the time saved is large; the theory of thispaper is unaffected and the range of curves we address is large enough for errors from thissmall-acceleration approximation to have no real effect on our key simplifying principle.

To avoid confusion later we mention here that, in this paper, t uniformly refers to mechanicaltrail and not time.

2. Preliminary studies of steering torque and steering angle

The aim of this brief section is to demonstrate the kind of numerical results we obtain usingADAMS, and to motivate the theory that will follow in subsequent sections. In particular,

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Vehicle System Dynamics 933

we present steering torques and angles of a motorcycle traversing various circles at variousspeeds.

The mechanical parameters of the motorcycle match those used in Karanam and Chatterjee[22], except for four tyre parameters. The set of four tyre parameters (two parameters for eachtyre) has a significant effect on motorcycle handling. A total of seven such sets, spanninga fairly large range of physical behaviours, is given in Table 1. Other parameter variationsand other tyre models may give a yet wider range of physical behaviours, but the presentrange suffices for verifying the straightforward theory that follows. We note in passing thatamong our chosen mechanical parameters is the combined (vehicle plus rider) mass of about184 kg, which is light compared with many motorcycles presently on the international market.However, our theory should apply to heavier motorcycles as well.

Figure 1 shows the results for one set of tyres (Set 1 of Table 1), for different path radiiranging from 30 m to 100 m. Steering torque (T ) is plotted against speed (V ) in the Figure 1(a).We observe that the graphs are non-monotonic [21], and that they have similar shapes; thatsome small oscillations are present in one or two locations (see Karanam and Chatterjee [22]for discussion of these oscillations, and some other rapid oscillations that we filter out withinADAMS). The simulations end at some speed where gross slip begins to set in or ADAMSis otherwise unable to find a solution. In that regime, we have V 2/R in excess of 10 m/s2 forall five curves, which is high for typical riders of light motorcycles on long curves, and henceenough for our purposes.

Table 1. Tyre parameters used for computing steering torques and angles.

Set no. PDY (front) LMUY (front) PDY (rear) LMUY (rear)

1 1.5376 1.20 1.3986 1.202 1.5376 1.05 1.3986 1.053 1.5376 0.60 1.3986 0.604 1.5376 1.20 1.5376 1.205 1.00 1.20 1.00 1.206 0.65 1.20 0.65 1.207 1.5376 0.60 1.5376 0.60

Note: Here, PDY is the PDY1 of the appendix of Karanam and Chatterjee [22], and represents thepeak lateral friction coefficient. The parameter LMUY is a scaling factor for lateral friction forces.The many other parameters of the magic tyre formula remain as in Karanam and Chatterjee [22]. Thechanges in PDY and LMUY incorporate reasonably wide variations in lateral friction.

5 10 15 20 25 30 35−1

−0.5

0

0.5

1

1.5

2

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

R=30 mR=40 mR=50 mR=75 mR=100 m

5 10 15 20 25 30 35−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Speed (m/sec)

Ste

erin

g an

gle

(Deg

.)

R=30 mR=40 mR=50 mR= 75 mR=100 m

(a) (b)

Figure 1. Variation of (a) steering torque and (b) steering angle with vehicle speed along circular paths of differentradii.

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5 10 15 20 25−1

0

1

2

3

4

5

6

Speed (m/sec)

Ste

erin

g to

rque

(N

−m

)

SET 1SET 2SET 3SET 4SET 5SET 6SET 7

5 10 15 20 25−1

−0.5

0

0.5

1

1.5

2

2.5

3

Speed (m/sec)

Ste

erin

g an

gle

(Deg

.)

SET 1SET 2SET 3SET 4SET 5SET 6SET 7

(a) (b)

Figure 2. Variation of (a) steering torque and (b) steering angle with vehicle speed on a 50 m radius path, withdifferent tyre sets.

Next, the motorcycle in steady circular motion is studied using the seven tyre sets of Table 1,for a fixed path radius of 50 m. Steering torque results are shown in Figure 2(a). It is observedthat there is rather dramatic variation among the curves, with some being monotonic, othershaving two slope reversals, and two having rather long near-flat portions. Details of thesevarious steering torque curves are, presumably, connected to the feel of the motorcycle whenridden on long curves. Moreover, as indicated above, the steer torque curves themselvesmay change significantly if details of the tyre slip model are changed. However, as alreadyexplained, such issues are not of direct interest here; rather, we simply take these curves ascorresponding to a reasonable variety of reasonable motorcycle behaviours.

For completeness, steering angle results corresponding to Figures 1(a) and 2(a) are shownin Figures 1(b) and 2(b). The first figure shows the strong effect of path radius, and the secondfigure shows the somewhat smaller effect of changing tyre parameters. We now move on toour theoretical development. In what follows, we will change not just tyre parameters butgeometrical parameters as well, because our straightforward theory applies regardless.

3. Unifying curves for steering torque

We will use dimensional analysis and Taylor series to predict a simplified form for steeringtorque curves, and then use extensive numerical simulations to validate empirically a furthersimplification. The simplification involves calculation of a certain constant, which we will dousing somewhat informal mechanics that provides intuition. Subsequent numerical simulationswill leave no doubt regarding the correctness of the results.

3.1. Dimensional analysis and Taylor series

Let us consider the parameters that define the motorcycle model. There is a wheelbase P , withunits of length. All other parameters that have units purely of length, such as wheel radius,trail, height of centre of mass above the ground, etc., can now be normalised with respect to P

and specified in dimensionless form. There is the total mass M , and using this, all parametersinvolving units of mass, such as masses of wheels, of the front fork, etc., can be specified indimensionless form. The moment of inertia of any rigid body, with six components, can con-sequently be specified using normalised mass and normalised length quantities. Acceleration

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due to gravity (g) is used via√

P/g to non-dimensionalise all motorcycle parameters thatinvolve units of time, including spring and dash-pot constants. There are various geometricalparameters, like caster angle (or fork angle with vertical), which are already dimensionless. Ofthe tyre friction parameters, some are dimensionless, and others can be made dimensionlessusing the normalising length, mass and time as described above. In this way, the motorcyclecan be completely specified in terms of the variables

P (length), M (mass),√

P/g (time) and other parameters denoted by � (all dimensionless).

Now a steering torque T is required to hold a motorcycle moving at a steady speed V on apath of radius R. It is anticipated that

T = F0(P, M,√

P/g, V, R, �),

where the function F0 is to be determined. In the above, it is clear that only V and R dependon the motion, while all other parameters are constant once the motorcycle (including tyreparameters) is specified. It is also clear that the above may be rewritten in terms of some otherfunction F1, as in

T = F1(P, M, g, V, R, �).

Since T has the same units as MgP , it can be written as

T = MgPF2(P, M, g, V, R, �),

with the understanding that F2 must be dimensionless. Holding � as a constant because themotorcycle is specified, we can then write

T = MgPF3(P, M, g, V, R),

with F3 dimensionless. The arguments of F3 may be invertibly transformed to an equivalentset, whence

T = MgPF4

(V 2

gR, M, g, P,

P

R

),

with F4 dimensionless. But among the arguments of F4, only its third argument g involvestime; so, since F4 is dimensionless, it cannot depend explicitly on its third argument, giving

T = MgPF5

(V 2

gR, M, P,

P

R

),

with F5 dimensionless. Now, in F5, only the second argument M involves mass, and so it hasto be dropped; and then only P involves length, and so we must drop that as well. This leaves

T = MgPF6

(V 2

gR,P

R

), (1)

where F6 is dimensionless. Now the non-dimensional lateral acceleration V 2/gR is compa-rable with unity; but the non-dimensional path curvature P/R is small. Assuming that F6 is a

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well-behaved function for small P/R, we can hold V 2/gR constant and expand F6 in a Taylorseries in its second argument. This gives, upon dropping all terms beyond first order,

T ≈ MgPF6

(V 2

gR, 0

)+ MgP

∂

∂(P/R)

{F6

(V 2

gR, 0

)}× P

R.

The above, on holding g fixed, can be rewritten more simply as

T ≈ f0

(V 2

R

)+ R−1f1

(V 2

R

), (2)

where the functions f0 and f1 are, of course, no longer dimensionless.Equation 2 is a theoretical prediction. Note that it assumes F6 is a well-behaved function of

P/R. For example, if F6 was proportional to ln(P/R) for small P/R, then the above predictionwould be false. Numerical results will show that the prediction is useful and accurate. Moreinterestingly, our numerical results will additionally suggest that f1(V

2/R) may be usefullyapproximated simply by f1(0). Thus, we can write Equation (2) as

T ≈ f0

(V 2

R

)+ f1(0)

R, (3)

with f0 a function and f1(0) a constant.Equation (3) is one of the main contributions of this paper. In the next section we approx-

imately determine f1(0) for demonstration, making an additional simplifying assumption.Subsequently, simulation over a range of V for one R is enough to determine f0.

3.2. Determination of f1(0)

Consider a circular motion on a large radius path with a vanishingly low steady speed(V 2/R → 0). Both lean (φ) and steering angle (δ) are small because R is large.

We mention that the equations we finally obtain in this section are not really new. Certainly,the spirit of these near-straight, small-lean, small-steer equations is present in, for example,[2,23]. However, due to the difference in context, as well as for continuity and completeness,we retain a brief derivation of the key equations we need here.

Recall Equation (1). For large radius, P/R is small; and for infinitesimal speed, V 2/R issmall as well. These two independent motion parameters invertibly determine a small leanangle φ and a small steering angle δ. Here we change independent variables to φ and δ.Accordingly, we write

T = MgPH1(φ, δ) = H(φ, δ),

where H1 is dimensionless, and H has units of torque.Noting that the arguments have small values, we write

T = H(0, 0) + φHφ(0, 0) + δHδ(0, 0) + higher order terms, (4)

where H∗ represents a partial derivative with respect to ‘*’. Note that H(0, 0) = 0 because itcorresponds to motion in a straight line, whence, up to first order,

T = φHφ(0, 0) + δHδ(0, 0). (5)

The torque can thus be written as the sum of two terms. One is the small torque due to smalllean without steer1 in a stationary motorcycle (recall that V 2/R → 0), and the other is the

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small torque due to small steer without lean in a stationary motorcycle. Finally, for correctlycomputing the sum of these two terms, the actual proportion of steer to lean must be foundseparately from equilibrium. These calculations, though not original except perhaps in minorstylistic detail, are given next for completeness.

3.2.1. Equilibrium

Consider a motorcycle as shown in Figure 3. The height of the motorcycle’s centre of massfrom the ground is h; its horizontal distance from the rear axle is a. The mechanical trail is t .The toroidal or transverse radii of the tyres are rf (front) and rr (rear), respectively.

The vertical motorcycle, no-lean and no-steer, is in equilibrium. Small lean without steerproduces an overall toppling moment; and so does small steer without lean (as seen above,Taylor series expansion about zero shows these effects can be considered independently atfirst order). The sum of these two toppling moments must be zero for a motorcycle traversinga circular path at an infinitesimal speed.

First consider a small lean without steer (Figure 4). The front tyre contact point is representedby Point A; the centre of mass is represented by Point B and the rear tyre contact pointis represented by Point C. The lateral displacements of all three points are indicated. Theresulting toppling moment about the shifted contact line is given by

Mtoppling,lean = Mgφ

(h − a rf + (P − a) rr

P

). (6)

Next, consider small steer without lean. This case is trickier, so we consider it in steps.

ψ

roll axis

stee

r axi

s

rr rf

Rear tire Front tire

(a) (b)

Figure 3. (a) Motorcycle and (b) tyres; rr �= rf in general.

A

B

C

a

P-a

rr

h

rf φ

φ

φ

Figure 4. Displacements due to small lean φ and no steer.

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First, note that lateral symmetry implies that under small steer and no lean, the wheel contactpoints do not roll longitudinally, up to first order. The rear wheel may be thought of, therefore,as rigidly attached to the rear frame. This rear-wheel-and-frame assembly has a fixed point (theground contact point) and so only three degrees of freedom in the infinitesimal displacementsneed consideration. These degrees of freedom are all infinitesimal rotations about the groundcontact point. First, rotation about the contact line constitutes lean, and is disallowed for thiscalculation. Second, rotation about the wheel axle, due to lateral symmetry, is of second orderand dropped. That just leaves rotation about the vertical axis. We conclude that the rear frameremains in a vertical plane but rotates about a vertical axis passing through the rear wheelground contact point. This motion is accompanied by a steering motion such that the frontwheel contact point remains stationary with respect to the ground. That steering motion causesa net toppling moment, which we will compute.

But now it is clear that the toppling moment may be more simply calculated by holdingthe rear frame completely fixed, and letting the front wheel ground contact slip on the grounddue to the small steer. The lateral motion of the front wheel ground contact point is (Figure 5)given by

tδ cos ψ,

where δ is the steer angle.Here we make a significantly simplifying approximation. The total mass of the front fork

assembly and front wheel is usually small compared with the total mass of the motorcycleplus rigid rider. Moreover, the centre of mass of the front fork plus front wheel subsystemis usually close to the steering axis. For these two reasons, the motion of the centre of massof this subsystem relative to the rear frame is neglected here and in the rest of the analyticaltreatment in this paper. Such motion is, of course, accounted for by the ADAMS Motorcycle.We emphasise that such motion could also in principle be incorporated here (in this sense, ourapproximation is not fundamentally required by the theory). However, recalling our earlierdevelopment, we only wish to find f1(0), and it will be seen in our results below that thissimplification is inconsequential.

Under this simplifying assumption about the front-assembly mass, and using the equivalentmotion described above, the toppling moment due to steer without lean is (Figure 5) given by

Mtoppling,steer = Mga

Ptδ cos ψ. (7)

A

B

C

a

P-a

δ ψ

Figure 5. Displacements due to small steer angle δ, as seen relative to rear frame.

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Static equilibrium requires, by Equations (6) and (7), that

φ = t cos ψ a

hP − arf − (P − a)rrδ. (8)

Note that the relative proportion between φ and δ is determined by the above equation, usingoverall toppling equilibrium of the motorcycle; this proportion must hold even in Equation(5); the partial derivatives Hφ and Hδ remain to be determined.

A useful point to note here is that in the asymptotic limit of small speed that we are consid-ering, there is no tyre slip. The steering angle δ can therefore be found from the kinematics ofsteady no-slip turns. Cossalter [5, Section 1.10] gives

δ = P cos φ

R cos ψ,

which for small φ is

δ = P

R cos ψ. (9)

3.2.2. Evaluation of infinitesimal steering torques

We now return to Equation (5).First consider the static motorcycle with zero steer and small lean. Why is the steering

torque required?By the previous assumption that the centre of mass of the front subsystem is effectively on

the steer axis, the steering torque is due to the ground front-contact force alone. The groundcontact force for the straight motorcycle is vertical, and has magnitude

Nf = Mga

P.

Under small lean and no steer, at zero speed, the contact force remains vertical. This is becausewe imagine locking the handlebar and applying a small moment about the longitudinal axis tosustain the lean. Now a lateral force at the contact point would set up an unbalanced momentabout a vertical axis; while a forward (longitudinal) force at the contact point would causethe wheel to spin. Moreover, Nf remains constant to first order in the lean, due to lateralsymmetry.

We therefore have the situation sketched in Figure 6(a). The wheel plane has tilted out ofthe page by an infinitesimal angle φ. We wish to find the moment of the vertical force Nf aboutthe steering axis. Now for finding this moment (which is a scalar quantity) it is permissibleto imagine tilting the whole assembly back to vertical, as shown in the longitudinal view ofFigure 6(b). We will now compute the steering moment using the unit vectors e1, e2 and e3

implied by Figure 6(b).The component of the tilted contact force Nf along e1 clearly contributes no moment, and

can be dropped. The component along e2 is Nf cos φ sin ψ with a moment arm (up to firstorder) of rfφ, and so the contribution of that component, up to first order, is

−Nfrf sin ψφ,

taken positive along e1. Finally, the component along e3 is −Nf sin φ with a moment arm (upto first order) of t cos ψ , giving a moment contribution up to first order of

Nf t cos ψφ.

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N f

t

e1

e 2

tilted wheel

Nf

e3

e1

(a) (b)

Figure 6. Front wheel with ground contact force: (a) actual and (b) tilted.

Adding, and substituting for Nf , we see that in Equation (5),

φHφ(0, 0) = Mga

P(t cos ψ − rf sin ψ)φ. (10)

We now turn to the case of small steer with no lean. As discussed above, this correspondsto the front wheel contact point moving out of the rear frame symmetry plane by a distancetδ cos ψ . In this case, still using front-frame-fixed unit vectors e1, e2 and e3, we find that thevertical contact force has no component along e3. Its component along e1, being parallel tothe steering axis, has no moment about that axis. That leaves only the component along e2,and the resulting contribution to the moment in Equation (5) is seen to be

δHδ(0, 0) = Mga

Pt sin ψ cos ψδ. (11)

Thus, Equation (5) becomes (from Equations (10) and (11))

T = Mga

P((t cos ψ − rf sin ψ)φ + t sin ψ cos ψδ) . (12)

In the above, φ must be rewritten in terms of δ from the toppling equilibrium conditionEquation (8); and then δ itself written in terms of path radius from Equation (9). The net resultis that steering torque, for an infinitesimally slow circular motion on a large circular path (andunder the assumption that the front-assembly mass centre lies practically on the steering axis,as described earlier), is given by

T = Mgat

R

(a(t cos ψ − rf sin ψ)

hP − arf − (P − a)rr+ sin ψ

). (13)

To interpret the above equation, we return to Equation (2):

T ≈ f0

(V 2

R

)+ R−1f1

(V 2

R

),

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where in the limit of V 2/R going to zero, we have

T ≈ f0(0) + R−1f1(0).

But here f0(0) is the R = ∞ limit with finite V , that is, the steering torque for a motor-cycle travelling at finite speed along a straight line; that is, f0(0) = 0. It follows that, byEquation (13),

f1(0)

R= Mgat

R



).

Finally, inserting f1(0) from the above into Equation (3), we have

T = f0

(V 2

R

)+ Mgat

R



).

We emphasise that the above equation is based on four things: (i) dimensional analysis,(ii) Taylor series expansion for large path radius, (iii) an empirical observation that f1(V

2/R)

may be reasonably replaced with f1(0) and (iv) the approximation that the centre of mass ofthe front-fork-and-wheel assembly may be treated as being on the steering axis. Under thisapproximation, the prediction is that

T − Mgat

R



)≈ f0

(V 2

R

), (14)

that is, the left-hand side, for a variety of speeds and path radii, may be approximated as asingle function of V 2/R. We will now present numerical verification, wherein the torque T isdetermined from simulation in ADAMS Motorcycle.

3.3. Numerical verification

For numerical verification we will consider nine different cases, as shown in Table 2. Of these,the first five cases involve only changes in tyre parameters; the remaining four cases involvechanges in geometrical parameters as well, from those used in [22]. In Cases 1–5, the lateralfriction coefficient ranges from 1.54 to 0.6 and the scaling coefficient ranges from 0.6 to 1.2,covering a wide range of motorcycle tyre behaviour. Geometrical parameter changes includedecreasing the wheel base (Case 6: wheel base from 1300 mm to 1200 mm), decreasing thefork offset from 60 mm to 10 mm (Case 7: will increase trail from 82 mm to 134.4 mm) andincrease of caster angle from 26◦ to 35◦ (Case 8: trail changes from 82 mm to 147 mm andwheel base changes from 1300 mm to 1403 mm). The wheelbase reduction (Case 6) is achieved

Table 2. Parameter value changes used for numerical verification of unifying formulas.

Case no. PDY (front) LMUY (front) PDY (rear) LMUY (rear) Other changes

1 1.5376 1.2 1.3986 1.2 –2 1.5376 1.2 1.5376 1.2 –3 1.0 1.2 1.0 1.2 –4 0.6 1.2 0.6 1.2 –5 1.0 0.6 1.0 0.6 –6 1.0 1.2 1.0 1.2 Wheelbase P = 1200 mm7 1.0 1.2 1.0 1.2 Fork offset = 10 mm8 1.0 1.2 1.0 1.2 ψ = 35◦ and P = 1403 mm9 1.0 1.2 1.0 1.2 ψ = 30◦

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0 20 40−1

−0.5

0

0.5

1

1.5

2

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 10−1

−0.5

0

0.5

1

1.5

2

V2/R

R=30 mR=40 mR=50 mR=75 mR=100 m

0 5 10−1

−0.5

0

0.5

1

1.5

2

V2/R

Figure 7. Case 1: steering torque plots. Left, torque against speed for various R; centre, torque against V 2/R forvarious R and right, torque with f1(0)/R added.

Table 3. Values of f1(0) for the nine cases of Table 2.

Case f1(0)

1, 2, 3, 4 and 5 −35.7776 −37.7837 −62.2508 −82.1309 −36.480

by shifting the entire front fork and front wheel assembly 100 mm towards the rear wheel. Thefront fork offset (Case 7) is decreased by changing the fork offset value resulting in reductionof wheel base which is then compensated by moving the front fork and wheel assembly. InCase 8, the caster angle is changed and in turn changes trail and wheelbase. Case 9 has beenarranged to keep all other things fixed even as the caster angle is changed.

We now present our numerical results for steering torques. See Figure 7, which presentsCase 1 of Table 2. In the figure, the original curves are on the left; in the centre are the samecurves plotted against V 2/R and on the right are the central-figure curves each suitably shiftedby an amount f1(0)/R. The approximate coalescence into a single function of V 2/R is clearlyseen.

For further verification, all the other cases (2–9) were similarly numerically investigated.Results are presented in Appendix 1 and Figure A1. The values of f1(0), from theory, are givenin Table 3. The match with theory is good, considering that f1(V

2/R) has been approximatedusing f1(0). A numerical determination of a suitable function f1(V

2/R) would improve thematch, but that is not attempted here for the sake of simplicity and analytical understanding.

We now consider steering angles.

4. Unifying curve for steering angles

Our theory for the steering angle proceeds a little differently from steering torques.

4.1. Dimensional analysis, large radius and small slip

Consider a motorcycle on a circular path. Now the steering angle δ may be written in termsof some as yet undetermined function G0 as

δ = G0(P, M, g, V, R, �).

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However, we will be interested in small slip, and so we assume that a small parameter governingtendency to slip exists within the set of non-dimensional parameters, �. Let this slip-relatedparameter, which is a property of the machine and not the motion, be called λ; and let theremaining set of non-dimensional parameters be �̃. Then we may write

δ = G1(P, M, g, V, R, λ, �̃),

with G1 dimensionless. By similar arguments as in Section 3.1, we can then write

δ = G2

(V 2

gR,P

R, λ, �̃

),

with G2 dimensionless. Now we hold the motorcycle constant, except that we allow variationsin λ; in other words, we hold �̃ constant. Then

δ = G3

(V 2

gR,P

R, λ

),

with G3 dimensionless. Finally, for large turn radius and small slip, we have

δ = G3

(V 2

gR, 0, 0

)+ P

RG3,P/R

(V 2

gR, 0, 0

)+ λG3,λ

(V 2

gR, 0, 0

)+ higher order terms,

(15)where G3,∗ represents a partial derivative with respect to ‘*’. Interestingly, the first two termson the right-hand side in Equation (15) can be determined from the kinematics of a non-slippingmotorcycle. Cossalter [5, Section 1.10] gives

tan

(P

R

)= tan δ cos ψ

cos φ,

where ψ is the caster angle and φ the lean, given to an excellent approximation for longturns by

φ ≈ arctan

(V 2

gR

), (16)

when V 2/R is not small (recall that in the previous section, during our consideration ofinfinitesimal V 2/R, we did not use this approximation). For large path radius R and smallsteering angle δ, then,

δ = P cos φ

R cos ψ.

Note that the above is a no-slip, purely kinematic result. It shows that, in Equation (15),

G3

(V 2

gR, 0, 0

)= 0

andP

RG3,P/R

(V 2

gR, 0, 0

)= P cos φ

R cos ψ,

with φ given by Equation (16). We now have

δ ≈ P cos φ

R cos ψ+ λG3,λ

(V 2

gR, 0, 0

). (17)

The second term on the right-hand side above involves the small-slip quantity λ. Now recallthat λ is a property of the machine and not the motion, so it is just some constant for a given

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0 20 40−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Speed (m/sec)

Ste

erin

g an

gle

(Deg

.)

0 5 10−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

V2/R0 5 10

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

V2/R

R=30 mR=50 mR=100 m

Figure 8. Case 1: steering angle plots. Left, angle against speed for various R; centre, steering angle against V 2/R

for various R and right, steering angle with −180P cos φ/πR cos ψ added, with φ = arctan(V 2/gR).

motorcycle (and can be reabsorbed in the original non-dimensional parameter set �). Thisfinally gives, for a given motorcycle,

δ − P cos(arctan V 2/gR)

R cos ψ≈ G4

(V 2

R

), (18)

where in the non-dimensional function G4 we have dropped the dependence on g. Thatis, like we did for the steering torque curves, we have found a quantity that is (within ourapproximations) purely a function of V 2/R.

4.2. Numerical verification

For numerical verification of Equation (18), we will consider five cases from Table 2. Theseare Cases 1, 3, 5, 8 and 9; the others show results so similar that we decided to omit themto save space. Of these, Case 1 is shown in Figure 8. In the figure the original steering anglecurves (converted to degrees) are on the left; in the centre are the same curves plotted againstV 2/R and on the right are the central-figure curves each shifted by −180P cos φ/πR cos ψ

(the factor of 180/π is for consistency with the use of degrees to measure angles). The threedifferent steering angle curves coalesce quite well into a single curve, for lateral accelerationup to 10 m/s2 or so. The imperfect match is presumably due to non-negligible slip and the factthat the smallest radius R is 30 m, which may not be large enough. The remaining four casesare shown in Appendix 2. Overall, the match with the theory is good.

5. Conclusions and future work

Detailed simulations of motorcycle dynamics are now common in industry, especially inthe design stage. One intuitively important aspect of motorcycle handling is the behaviourof the machine in steady turns. Usually, for each motorcycle design (including a setof tyre parameters), detailed simulations are conducted for motions at many differentsteady speeds on paths of many different radii. This observation has motivated the presentpaper.

Using dimensional analysis and a large-radius, small-steer and small-slip approximation,we have found simple approximate unifying formulas using which steering torque and steeringangle curves for different radii approximately coalesce into single, motorcycle-specific, curves.

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So, in principle, only one simulation is sufficient for both steering torque and steering angleto find approximately the result of any other radius of interest. The role of simulation has notbeen eliminated by the present work; however, the amount of time to be spent on simulationcan potentially be reduced dramatically if the present approximations are found acceptable.The theory and numerical results presented here can be used not only in future industrialdesign-oriented studies, but will also lead to improved understanding of motorcycle dynamicsas well.

An anonymous reviewer of this paper raised the interesting question of what radius mightbe best for the master curves, from which the torque and angle curves for other radii wouldbe computed in practice. Our results here, being based on asymptotic arguments, treat allsufficiently large radii as equally suitable. Greater clarity on this issue may emerge with futurework. For now, we tentatively suggest that any radius greater than about 30 times the wheelbasemight be used.

Future work may include direct experimental validation of the theory, which would involvedetailed measurements of steering torques and angles, at many steady speeds, on circles ofmany different radii, with many motorcycles. We have, so far, only limited experimental ver-ification of the general utility of the ADAMS model (unpublished data, within TVSM R&D),but no detailed experimental verification of ADAMS for the simulation results given here.In this paper, we have taken the conservative view that ADAMS represents some reasonablephysics, and at least that physics has borne out our theoretical predictions.

Acknowledgements

We thank TVS Motor Co., Ltd, for permission to publish this work, and Jim Papadopoulos and Amrit Sharmafor commenting on an earlier draft. VMK thanks HB Pacejka for insights into tyre models. AC thanks DavidLimebeer for discussions of background issues, and Jim Papadopoulos for many insights, scattered among sev-eral documents on Andy Ruina’s web page at Cornell University, http://ruina.tam.cornell.edu/research/topics/bicycle_mechanics/overview_papers_and_links.htm.

Note

1. The physical interpretation of φHφ(0, 0) is indeed ‘small torque due to small lean without steer’.

Nomenclature

T steering torque

δ steering angle

V speed

g acceleration due to gravity

R radius of the circular path

φ lean or roll angle

ψ caster angle

t mechanical trail

M vehicle mass

P wheel base

h height of the centre of mass of motorcycle

a horizontal distance of the centre of mass from the rear axle

rr toroid radius of the rear tyre

rf toroid radius of the front tyre

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References

[1] F.J.W. Whipple, The stability of the motion of a bicycle, Q. J. Pure Appl. Math. 30 (1899), pp. 312–348.[2] J.P. Meijaard, J.M. Papadopoulos, A. Ruina, and A.L. Schwab, Linearized dynamics equations for the balance

and steer of a bicycle: a benchmark and review, Proc. R. Soc. A: Math. Phys. Eng. Sci. 463 (2084), 2007, pp.1955–1982.

[3] P. Basu-Mandal, A. Chatterjee, and J.M. Papadopoulos, Hands-free circular motions of a benchmark cycle,Proc. R. Soc. A: Math. Phys. Eng. Sci. 463(2084) (2007), pp. 1983–2003.

[4] R.S. Sharp, The stability and control of motorcycles, Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci. 13(5) (1971),pp. 316–329.

[5] V. Cossalter, Motorcycle Dynamics, Race Dynamics, Greendale, MI, 2002.[6] H.B. Pacjeka, Tire and Vehicle Dynamics, Society of Automotive Engineers Inc, Warrendale, PA, 2002.[7] R.S. Sharp and D.J.N. Limebeer, A motorcycle model for stability and control analysis, Multibody Syst. Dyn.

6(2) (2001), pp. 123–142.[8] D.J.N. Limebeer, R.S. Sharp, and S. Evangelou, The stability of motorcycles under acceleration and braking,

Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci. 215(C9) ( 2001), pp. 1095–1109.[9] R.S. Sharp, S. Evangelou, and D.J.N. Limebeer, Advances in the modeling of motorcycle dynamics, Multibody

Syst. Dyn. 12(3) (2004), pp. 251–283.[10] T.R. Kane, Steady turning of single track vehicles, Paper No. 77057, Society of Automoble Engineers (SAE),

1977.[11] T. Nishimi, A. Aoki, and L. Segel, Analysis of straight running stability of motorcycles, Veh. Syst. Dyn. 9(3)

(1980), pp. 181–206.[12] M.K. Verma, R.A. Scott, and T. Katayama, Effect of frame compliance on the lateral dynamics of motorcycles,

Proceedings of the 10th International Technical Conference, Experimental SafetyVehicles, 1985, pp. 1080–1094.[13] T. Katayama and T. Nishimi, Energy flow method for the study of motorcycle wobble mode, Veh. Syst. Dyn.

19(3) (1990), pp. 151–154.[14] V.M. Karanam, S. Ghosh, R.B. Anand, R. Babu, and R. Venkatesan, Vehicle handling comparison of motorcycles

and bebek vehicles, Paper No. 032-0109, Society of Automoble Engineers (SAE), 2007.[15] S.P. Velagapudi, D. Sharma, R.B. Anand, V.M. Karanam, and R. Babu, Effect of seat support structure stiffness

on handling of motorcycles, Paper No. 032-0070, Society of Automoble Engineers (SAE), 2008.[16] V. Cossalter, R. Lot, and F. Maggio, The modal analysis of a motorcycle in straight running and on a curve,

Meccanica 39(1) (2004), pp. 1–16.[17] R.S. Sharp, Stability, control and steering responses of motorcycles,Veh. Syst. Dyn. 35(4–5) (2001), pp. 291–318.[18] S. Evangelou, D.J.N. Limebeer, R.S. Sharp, and M.C. Smith, Mechanical steering compensators for high

performance motorcycles, Trans. Am. Soc. Mech. Eng., J. Appl. Mech. 74 (2007), pp. 332–346.[19] D.J.N. Limebeer and R.S. Sharp, Bicycles, motorcycles and models; special issue on single track vehicle stability

and control, Inst. Elect. Electron. Eng. Control Syst. Mag. 36 (2006), pp. 34–61.[20] V. Cossalter, A. Doria, and R. Lot, Steady turning of two wheeled vehicles, Veh. Syst. Dyn. 31 (1999), pp.

157–181.[21] V. Cossalter, R. Lot, and M. Peretto, Steady turning of motorcycles, Proc. Inst. Mech. Eng. D: J. Automob. Eng.

221 (2007), pp. 1343–1356.[22] V.M. Karanam and A. Chatterjee, Some Procedural Details of Analysis Using ADAMS-Motorcycle. Available at

http://eprints.iisc.ernet.in/17639/[23] J.P. Meijaard and A.L. Schwab, Linearized equations for an extended bicycle model, in III European Con-

ference on Computational Mechanics – Solids, Structures and Coupled Problems in Engineering, C.A.Motasoares et al., eds., Lisbon, Portugal, 5–9 June 2006. Available at http://audiophile.tam.cornell.edu/∼als93/Publications/MeijaardSchwab2006.pdf

Appendix 1. Steering torque curves for eight remaining cases from Table 2

For each case depicted in this section (Figure A1) the first subplot shows the steering torque against speed for variousR held fixed; the second subplot shows the same against V 2/R for various R held fixed and the third subplot showsthe approximation to a single curve obtained by shifting each curve by f1(0)/R. The match with theory is good.

Appendix 2. Steering angle curves for four cases from Table 2

For each case depicted in this section the first subplot shows the steering angle against speed for various R held fixed;the second subplot shows the same against V 2/R for various R held fixed and the third subplot shows the singlecurve obtained by Equation (18). The four remaining cases, not shown here, were very similar. Overall, agreementwith theory is good (Figure A2).

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0 20

0

1

2

3

4

5

6

7

8

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 10

0

1

2

3

4

5

6

7

8

V2/R

R=30 mR=50 mR=100 m

0 5 10

0

1

2

3

4

5

6

7

8

V2/R

Case 8

Ste

erin

g to

rque

(N

m)

0 10 20 30

0

1

2

3

4

5

6

Speed (m/sec)0 5 10

0

1

2

3

4

5

6

V2/R

R=30 mR=50 mR=100 m

0 5 10

0

1

2

3

4

5

6

V2/R

Case 9

0 20-3

-2

-1

0

1

2

3

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 10-3

-2

-1

0

1

2

3

V2/R

R=30 mR=50 mR=100 m

0 5 10-3

-2

-1

0

1

2

3

V2/R

Case 7

0 10 20 30

0

1

2

3

4

5

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 10

0

1

2

3

4

5

V2/R

R=30 mR=50 mR=100 m

0 5 10

0

1

2

3

4

5

V2/R

Case 6

4 4 4

Case 2

0 200

0.5

1

1.5

2

2.5

3

3.5

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 100

0.5

1

1.5

2

2.5

3

3.5

V2/R

R=30 mR=50 mR=100 m

0 5 100

0.5

1

1.5

2

2.5

3

3.5

V2/R0 20

0

1

2

3

4

5

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 100

1

2

3

4

5

V2/R

R=30 mR=50 mR=100 m

0 5 100

1

2

3

4

5

V2/R

Case 3

Case 4

0 10 20 300.5

0

0.5

1

1.5

2

2.5

3

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5-0.5

0

0.5

1

1.5

2

2.5

3

V2/R

R=30 mR=50 mR=100 m

0 5-0.5

0

0.5

1

1.5

2

2.5

3

V2/R

- 0 10 20 30

0

1

2

3

4

5

6

Speed (m/sec)

Ste

erin

g to

rque

(N

m)

0 5 10

0

1

2

3

4

5

6

V2/R

R=30 mR=50 mR=100 m

0 5 10

0

1

2

3

4

5

6

V2/R

Case 5

Figure A1. Numerical verification of approximate theory for steering torques: eight cases.

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0 10 20 30-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Speed (m/sec)

Ste

erin

g an

gle

(Deg

.)

0 5 100.5

0

0.5

1

1.5

2

2.5

3

3.5

V2/R0 5 10

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

V2/R

R=30 mR=50 mR=100 m

Case 5

Ste

erin

g an

gle

(Deg

.)

0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

3

Speed (m/sec)0 5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

V2/R0 5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

V2/R

R=30 mR=50 mR=100 m

Case 3

0 20-3

-2

-1

0

1

2

3

Speed (m/sec)

Ste

erin

g an

gle

(Deg

.)

0 5 10-3

-2

-1

0

1

2

3

V2/R0 5 10

-3

-2

-1

0

1

2

3

V2/R

R=30 mR=50 mR=100 m

Case 8

0 20-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Speed (m/sec)

Ste

erin

g an

gle

(Deg

.)

0 5 10-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

V2/R0 5 10

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

V2/R

R=30 mR=50 mR=100 m

Case 9

Figure A2. Numerical verification of approximate theory for steering angles: four cases.

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Common underlying steering curves for motorcycles in steady turns

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Transcript of Common underlying steering curves for motorcycles in steady turns