# Advances in the Modelling of Motorcycle Dynamics

**ABSTRACT** Starting from an existing advanced motorcycle dynamics model, which allows simulation of reasonably general motions and stability, modal and response computations for small perturbations from any trim condition, improvements are described. These concern (a) tyre/road contact geometry, (b) tyre shear force and moment descriptions, as functions of load, slip and camber, (c) tyre relaxation properties, (d) a new analytic treatment of the monoshock rear suspension mechanism with sample results, (e) parameter values describing a contemporary high performance machine and rider, (f) steady-state equilibrium and power checking and (g) steering control. In particular, the Magic Formula motorcycle tyre model is utilised and complete sets of parameter values for contemporary tyres are derived by identification methods. The new model is used for steady turning, stability, design parameter sensitivity and response to road forcing calculations. The results show the predictions of the model to be in general agreement with observations of motorcycle behaviour from the field and they suggest that frame flexibility remains an important design and analysis area, despite improvements in frame designs over recent years. Motorcycle rider parameters have significant influences on the behaviour, with results consistent with a commonly held view, that lightweight riders are more likely to suffer oscillation problems than heavyweight ones.

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**ABSTRACT:**In this brief, we provide optimal control-based strategies to explore the dynamic capabilities of a single-track car model that includes tire models and longitudinal load transfer. First, we propose numerical tools to analyze the equilibrium manifold of the vehicle. That is, we design a continuation and predictor-corrector numerical strategy to compute the cornering equilibria on the entire range of operation of the tires. Second, as a main contribution of this brief, we explore the system dynamics by the use of nonlinear optimal control techniques. Specifically, we propose a combined optimal control and continuation strategy to compute aggressive car trajectories. To show the effectiveness of the proposed strategy, we compute aggressive maneuvers of the vehicle inspired to testing maneuvers from virtual and real prototyping.IEEE Transactions on Control Systems Technology 08/2013; · 2.00 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, to analyse the rider's effects on the motion of a motorcycle, we model a rider–motorcycle system by taking into account the leaning motion of the rider's upper torso and his/her arm connection with the handlebars. The nonlinearity of the tyre force is introduced by utilising hyperbolic tangent functions to approximate a Magic Formula tyre model. On the basis of a derived nonlinear state-space model, we analyse the effects of not only the rider's arms but also his/her postures during steady turning by simulations. The rider's postures including lean-with, lean-in and lean-out are realised by adding the lean torque to the rider's upper torso. The motorcycle motion and the rider's effects are analysed in the case where the friction coefficient of the road surface changes severely during steady turning. In addition, a linearised state-space model is derived during steady turning, and a stability analysis of the rider–motorcycle system is performed.Vehicle System Dynamics 02/2012; 50(8):1225-1245. · 0.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Improving braking skills of a rider supported by a real-time training device embedded in the motorcycle represents a possible strategy to deal with safety issues associated with the use of powered two wheelers. A challenging aspect of the braking trainer system is the evaluation of the adherence between tyre and road surface on each wheel. This paper presents a possible method to evaluate the current and maximum adherence during a braking manoeuvre. The proposed approach was positively validated through multi-body simulations and experimental data acquired in naturalistic riding conditions.Vehicle System Dynamics 09/2013; · 0.77 Impact Factor

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Multibody System Dynamics 12: 251–283, 2004.

C ?2004 Kluwer Academic Publishers. Printed in the Netherlands.

251

Advances in the Modelling of Motorcycle

Dynamics

R.S. SHARP, S. EVANGELOU and D.J.N. LIMEBEER

Electrical and Electronic Engineering, Imperial College London, South Kensington Campus,

London SW7 2AZ, UK; E-mail: robin.sharp@imperial.ac.uk

(Received: 7 July 2003; accepted in revised form: 7 June 2004)

Abstract. Starting from an existing advanced motorcycle dynamics model, which allows simulation

ofreasonablygeneralmotionsandstability,modalandresponsecomputationsforsmallperturbations

from any trim condition, improvements are described. These concern (a) tyre/road contact geometry,

(b) tyre shear force and moment descriptions, as functions of load, slip and camber, (c) tyre relax-

ation properties, (d) a new analytic treatment of the monoshock rear suspension mechanism with

sample results, (e) parameter values describing a contemporary high performance machine and rider,

(f) steady-state equilibrium and power checking and (g) steering control. In particular, the “Magic

Formula” motorcycle tyre model is utilised and complete sets of parameter values for contemporary

tyresarederivedbyidentificationmethods.Thenewmodelisusedforsteadyturning,stability,design

parametersensitivityandresponsetoroadforcingcalculations.Theresultsshowthepredictionsofthe

model to be in general agreement with observations of motorcycle behaviour from the field and they

suggest that frame flexibility remains an important design and analysis area, despite improvements

in frame designs over recent years. Motorcycle rider parameters have significant influences on the

behaviour, with results consistent with a commonly held view, that lightweight riders are more likely

to suffer oscillation problems than heavyweight ones.

Keywords: motorcycle, tyre, contact, monoshock, stability, response, sensitivity.

1. Introduction

The handling qualities of motorcycles are often of great importance. They affect

the pleasure to be gained from the rider–machine interactions and the safety of

the rider. Self-steering action is crucial with single track vehicles and rider control

is primarily by steering torque, so-called free-control [1]. A consequence of the

free steering system is that motorcycles are oscillatory. Several modes of motion

potentially have small damping factors. Therefore much attention must be directed

towardscontrollingtheoscillatorytendencies,throughouttheoperatingrange.Also,

it is desirable that motorcycles are responsive to the rider’s commands and stability

should not be pursued without reference to other qualities.

In straight running, motorcycles are substantially symmetric and in-plane and

out-of-plane motions are decoupled at first order level [1, 2]. In cornering, in-

planeandout-of-planecross-couplingmakesanyeffectiveanalysisofthedynamics

complicated. Automated multibody dynamics analysis software [3–7] has opened

upthetopicsignificantlyinrecentyears.Thesteadyturningproblemcanbesolved,

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possibly with the aid of a stabilising steering controller, and modal analysis can be

completed for small perturbations from any equilibrium “trim” state.

Accuracyofpredictedbehaviourdepends,notonlyoneffectiveconceptualmod-

elling and multibody analysis, but also on good parameter values. Central issues

in modelling include the representations of frame flexibilities, tyre–road contact

geometry and tyre shear forces. Many previous findings relate to motorcycle and

tyre descriptions which are now somewhat dated and to tyre models which have

a limited domain of applicability. It is therefore of interest (i) to obtain a para-

metric description of a modern machine, (ii) to utilise a more comprehensive tyre

force model, with parameter values to correspond to a modern set of tyres, (iii)

to determine steady turning, stability, response and parameter sensitivity data for

comparison with older information, to determine to what extent it remains valid,

and (iv) to better understand the design of modern machines. The paper is subse-

quentlyanaccountofsuchwork.Novelanalysisofa“monoshock”rearsuspension

system is also included.

2. Parametric Description of a Modern Motorcycle

The authors are currently engaged in a measurement campaign to obtain the rele-

vant parameters of a Suzuki GSX-R1000K1 machine. Such a motorcycle has been

disassembled and many of its parts have been measured, starting with the lighter

ones. At this stage, the campaign is incomplete. In particular, the frame stiffness

and damping parameters used and the location of the elastic centre are currently

only estimates.

2.1. GEOMETRY AND MASSES

The workshop manual for the motorcycle includes pictures to scale and key dimen-

sions, like the wheelbase and the steering head angle. Joints between components

at the steering head and the swing arm pivot can be identified there and many key

points, including those related to the monoshock rear suspension, can be located

with reasonable precision from these pictures. A scaled diagrammatic representa-

tion of the motorcycle is shown in Figure 1, the corresponding parameter values

being included in an Appendix. The front frame has been measured separately to

give the points p3 and p5. The point p4 is along the line of the lower front fork

translationrelativetotheupperforks.Theestimatedlocationp2istheelasticcentre

of the rear frame with respect to a moment perpendicular to the steer axis.

The rider’s total mass is taken as 72 kg, 62% of which is associated with the

upper body. The masses of the hands and half of the lower arms may be considered

to be part of the steering system. The rider parameters derive from bio-mechanical

data [8], accounting for his posture on the machine.

Circles representing the body mass centres are in proportion to the masses

concerned, which are known through straightforward weighing.

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ADVANCES IN THE MODELLING OF MOTORCYCLE DYNAMICS

253

Figure 1. Scaled diagrammatic motorcycle in side view.

2.2. INERTIAS AND MASS CENTRES

Wheel and tyre inertias have been obtained by timing oscillations of bi-filar and

tri-filar suspension arrangements, utilising axial symmetry in each set-up. Similar

bi-filar suspension systems have been used separately for the front and rear frames

(Figure 2). Each of these is assumed to have a plane of symmetry and it is clear that

thefrontframeprincipalaxes,intheplaneofsymmetry,arealongandperpendicular

to the line of the forks. Oscillation periods, geometric dimensions and the mass of

the suspended body lead simply to the moment of inertia about the rotation axis

and standard transformations allow the determination of principal inertias and axes

for the more complex rear frame [9].

Recent measurements on a driving simulator [10] provide estimates of the

contributions to the front frame inertia, steering stiffness and steering damping

that arise from the rider’s arms and hands, corresponding to relaxed and tense

riding. These can be added to the measured values if it is considered appropri-

ate [11]. The swing arm inertias are small enough to be obtained by estimation

based on the mass centre location and the dimensions. The wheels have their mass

centres at their geometric centres. Other mass centre locations were found using

plumb lines and taking photographs (Figure 2). Relevant values are given in the

appendix.

2.3. STIFFNESS AND DAMPING PROPERTIES

Springs and dampers were tested in a standard dynamic materials testing machine

[12]. The maximum actuator velocity available was about 0.25 m/s, which con-

strained the damper characteristic measurements. Uni-directional forcing of the

steering damper up to the maximum rate of the actuator yielded a substantially

linear force/velocity relationship with slope 4340 N/(m/s). Using the effective

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R.S. SHARP ET AL.

Figure 2. Bifilar suspended motorcycle rear frame for inertia measurement.

momentarmofthedamper(0.04m)toconvertthisvaluetoanequivalentrotational

coefficient gives a value, 6.944 Nm/(rad/s).

The dimensions of the single rear steel spring, from the monoshock suspension

were measured and the standard helical spring formula, k = Gd4/(64R3n), was

applied to calculating the rate, k, as 55 kN/m. The gas filled damper contributes

some suspension preload and a small rate, determined from the test machine via

static measurements as 3.57 kN/m. The damper unit was stroked at full actuator

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ADVANCES IN THE MODELLING OF MOTORCYCLE DYNAMICS

255

performance first in compression and then in extension, achieving velocities up to

about0.13m/s.Allowingforthegaspressureforcesintheprocessing,thedamping

coefficient in compression was 9.6 kN/(m/s) and in rebound 13.7 kN/(m/s). Front

springanddampercoefficientsareestimates,atthisstage.Suspensionlimitstopsare

included at each end, modelled as fifth powers of displacement from stop contact.

The relevant displacements are known from examination of the parts and from

information given in the workshop manual.

The torsional stiffness of the main frame, between the steering head and the

power unit, remains to be measured. It is clear from the structural design and

materials used that the frame is considerably stiffer than was the norm for tubular

framed motorcycles of some years ago. In those cases, it was established that the

frame flexibility was an essential contributor to the stability of the wobble mode,

in particular [13, 14]. It remains to be seen how significant this area is for modern

machines.Thetorsionalstiffnessassumed,at105Nm/rad,is3.5timesthatmeasured

statically for a Yamaha 650S [15] and 2.9 times that measured at about the same

time by Koenen [2]. Tyre radial stiffnesses come directly from [7].

The rider’s upper body has roll freedom relative to the main frame, while the

lower body is part of the main frame. The upper body is restrained by a parallel

spring damper system. Stiffness and damping parameters are chosen in alignment

with the experimental results of Nishimi et al [16], obtained by identifying “rider”

parameters in forced vibration on a mock motorcycle frame. The decoupled natural

frequency of the rider upper body in roll is 1.27 Hz and the corresponding damping

factoris0.489.Accordingtothismodel,riderresonancewillnotbeapparentdueto

the high damping factor and it will not be tuned to the machine oscillations, where

these are at all vigorous.

2.4. AERODYNAMICS

Aerodynamicdrag,liftandpitchingmomentdatacomefromaTriumphmotorcycle

of similar style and dimensions to the GSX-R1000 [1]. This is steady-state drag

force, lift force and pitching moment data from full scale wind tunnel testing, with

a prone rider.

3. Tyre–Road Contact Modelling

The geometry of the contact between the front tyre and the ground is a relatively

complex part of the motorcycle modelling. It is also important to the behaviour of

themachine.Ithasbeencommontorepresentthetyreasathindisc,withthecontact

point migrating circumferentially for larger camber and steer angles, but Cossalter

et al have pioneered the inclusion of tyre width in their descriptions [7, 17–19].

If a disc model is used, it needs to be augmented with an overturning moment

description [2, 5]. This is not necessary with a thick tyre model, since the lateral

migration of the contact point then occurs automatically and the overturning mo-

ment is a consequence of that movement. A wide tyre with a circular cross-section

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Figure 3. Diagrammatic three-dimensional front wheel contact geometry.

crown is now modelled. In addition to making the overturning moment automatic,

longitudinal forces applied to the cambered tyre will lead to realistic aligning

moments appearing automatically. A necessary test for the wide tyre model is

that it gives the same results as the thin tyre model, when physically equivalent

systems are being represented. This test has been applied, with some significant

consequences.

To define each tyre/ground contact point (Figures 3 and 4) the vertical and the

wheelspindledirectionsareusedinavector(cross)producttodescribethelongitu-

dinal direction, with respect to the wheel. Similarly, the wheel radial direction, OC

in Figure 3, comes from combining the longitudinal and wheel spindle directions.

ThevectorOCisoffixedlengthandsoiscompletelyspecified.Gisverticallybelow

C and the difference between the tyre crown radius and the distance CG defines

the change in the tyre carcass compression from the nominal state and hence the

change of the wheel load from the nominal, via the tyre radial stiffness. If the road

is profiled, the road height is accounted for in working out the wheel load. The

vector OG = OC + CG defines the contact point, which belongs to the wheel but

moves within it. G remains at road surface height but the tyre load cannot become

negative. If the tyre leaves the ground, the shear forces are zero, whatever the other

conditions are. Tyre forces are applied to the point G, in each case.

The longitudinal slip is the rearward component of the material contact point

velocity divided by the absolute value of the rolling velocity, the latter being the

forward velocity of the contact point (or the crown centre point, since these are

the same). The contact point is defined by its coordinates in the parent body of the

wheel and it is de-spun relative to the material contact point. Thus the longitudinal

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ADVANCES IN THE MODELLING OF MOTORCYCLE DYNAMICS

257

Figure 4. Diagrammatic two-dimensional front wheel contact geometry.

slip is given by an expression of the form:

κ = −(rollingvelocity + spincomponentoflongitudinalvelocity)

/abs(rollingvelocity)

The slip angle is the arctangent of the ratio of the (negative) lateral velocity of the

tyre contact centre point to the absolute value of the rolling velocity.

In developing this new model from the former one [5], in which the wheels

were represented as thin discs, subtle differences between the root locus predic-

tionsoftheoldandnewversionswereobservedincircumstanceswhichwereatthat

stage thought physically equivalent. Such differences were found to be associated

with the former description of the slip angles as deriving from the lateral veloc-

ity components of the disc tyre contact points. When the wheel camber angle is

changing, these points have a small lateral velocity component not connected with

sideslipping, since with the real tyre, the contact point moves around the circular

sectionsidewallofthetyre.Theformermodelwouldhaveprovidedamoreaccurate

description if it had used the crown centre point velocities to derive the slip angles.

4. Tyre Forces and Moments

The basis for the new tyre modelling is the “Magic Formula” [20–22]. The original

development was for car tyres [23], in which context, it has become dominant. The

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R.S. SHARP ET AL.

extension for motorcycle tyres is relatively recent, with substantial changes being

necessary to accommodate the completely different roles of sideslip and camber

forces in the two cases. In each case, the “Magic Formula” is a set of equations

relatingload,slipratio(longitudinalslip),slipangleandcamberangletolongitudi-

nal force, sideforce and aligning moment (and possibly overturning moment), with

constraintsontheparameterstopreventthebehaviourfrombecomingunreasonable

in any operating conditions. Only very limited parameter values can be found in

the literature, but a certain amount of relevant experimental data is available. Such

data can be used for parameter identification.

A complete set of parameter values for a given tyre will allow the calculation of

the steady-state force and moment system for any realistic operating condition. It

is required here to determine such a full set of parameters for modern front and rear

high performance motorcycle tyres, imposing the condition that the modelled tyres

have left/right symmetry. Test data used shows bias and it is necessary to ignore

such bias and to omit certain offset terms from the “Magic Formula” relations, in

order to model the generic, rather than the particular. Significant published data

can be found in [2, 20–25]. Naturally, the older data refers to older tyres, while the

newer data relates to contemporary ones. The main sources relied upon here are

[20, 23]. The other sources are used for checking purposes, as appropriate.

4.1. LONGITUDINAL FORCES IN PURE LONGITUDINAL SLIP

From Pacejka [23], with the simplifications explained above, the “Magic Formula”

expressions for the pure longitudinal slip case are:

d fz= (Fz− Fz0)/Fz0

Fx0= Dxsin[Cxarctan{Bxκ − Ex(Bxκ − arctan(Bxκ))}]

Dx= (pDx1+ pDx2d fz)Fz

Ex=?pEx1+ pEx2d fz+ pEx3d f2

Bx= Kxκ/(CxDx)

(1)

(2)

(3)

(4)

(5)

(6)

z

?· (1 − pEx4sgn(κ))

Kxκ= Fz(pKx1+ pKx2d fz) · exp(pKx3d fz)

which must satisfy the constraints Dx> 0 and Ex< 1.

Corresponding test results for a 160/70 ZR17 tyre are shown in [23]. The se-

quential quadratic programming constrained optimisation routine “fmincon” was

employed1to iteratively improve the elements of a starting vector of parameters

1Alternatively, for unconstrained optimization, the Nelder Mead Simplex routine “fminsearch”

was employed. Also occasionally, it was necessary to “invent” data, outside the range of experimental

results available, to force the identified parameters to give sensible predictions over a wide range of

operatingcircumstances,aproblemalsoreferredtoin[26].Often,reasonablyaccuratestartingvalues

for the parameters were needed to ensure convergence to the optimal solution. The methods need to

be judged by the results obtained.

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ADVANCES IN THE MODELLING OF MOTORCYCLE DYNAMICS

259

Table I. Best-fit parameter values for longitudinal force from 160/70 tyre.

Cx

pDx1

pDx2

pEx1

pEx2

pEx3

pEx4

pKx1

pKx2

pKx3

1.60641.2017

−0.0922 0.02630.27056

−0.0769 1.126825.94

−4.233 0.3369

Figure 5. Tyre longitudinal force results for a 160/70 tyre from [23] (thick lines) with best-fit

reconstructions (thin lines).

appearing in Equations (1)–(5). The nominal normal load Fz0was chosen to be

1600 N based on typical usage of such a tyre. That choice is far from critical,

in fact, a change leading to compensatory changes in other parameters. Optimal

parameters are given in Table I and the fits are illustrated in Figure 5. The two

constraints are satisfied for loads less than 20890 N, which includes all practical

circumstances.

Longitudinal force results are not available for any other tyres, so lateral forces

are considered next.

4.2. LATERAL FORCES IN PURE SIDESLIP AND CAMBER

In exactly the same way, the relevant equations for the lateral force are:

Fy0= Dysin[Cyarctan{Byβ − Ey(Byβ − arctan(Byβ))}

+Cγarctan{Bγγ − Eγ(Bγγ − arctan(Bγγ))}] (7)

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Figure 6. Tyre lateral force results for a 160/70 tyre from [23] (thick lines) with best-fit

reconstructions (thin lines). Camber angles 5, 0, −5, −10, −20, −30◦.

Dy= FzpDy1exp(pDy2d fz)/(1 + pDy3γ2)

Ey= pEy1+ pEy2γ2+ pEy4γsgn(β)

Kyα= pKy1Fz0sin[pKy2arctan{Fz/((pKy3+ pKy4γ2)Fz0)}]

/(1 + pKy5γ2)

By= Kyα/(CyDy)

Kyγ= (pKy6+ pKy7d fz)Fz

Bγ= Kyγ/(CγDy)

(8)

(9)

(10)

(11)

(12)

(13)

with the constraints Cy+ Cγ < 2, Cy> 0, Dy> 0, Ey< 1, Cg> 0, Eg< 1.

For the same tyre as before, the parameter optimisation process, with the effective

friction coefficient limited to values no greater than 1.3, gives the results illustrated

in Figure 6 with parameter values given below in Table II. For this particular tyre,

pKy7inEquation(12)wassettozero,becauseexperimentalresultsareonlyavailable

Table II. Best-fit parameter values for lateral force from 160/70 (top), 120/70 (middle) and

180/55 (bottom) tyres

Cy

pDy1

pDy2

pDy3

pEy1

pEy2

pEy4

pKy1

0.93921

0.8327

0.9

pKy2

1.0167

2.1578

1.6935

1.1524

1.3

1.3

pKy3

1.4989

2.5058

1.4604

−0.01794

0

0

pKy4

0.52567

−0.08088

0.669

−0.06531

0

0

pKy5

−0.24064

−0.22882

0.18708

−0.94635

−1.2556

−2.2227

Cγ

0.50732

0.86765

0.61397

−0.09845

−3.2068

−1.669

pKy6

0.7667

0.69677

0.45512

−1.6416

−3.998

−4.288

pKy7

0

−0.03077

0.013293

26.601

22.841

15.791

Eγ

−4.7481

−15.815

−19.99

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261

Figure 7. Tyre lateral force results for 120/70 tyre from [20] (thick lines) with best-fit recon-

structions (thin lines). Camber angles 0, 10, 20, 30, 40, 45◦.

at non-zero camber angle for one load. This is consistent with results obtained for

120/70 and 180/55 tyres (see below), for which pKy7is relatively small, being

positive in one case and negative in the other. All the constraints are satisfied for

camber angles less than 70◦in magnitude.

Next, the lateral force fitting is repeated for the experimental results included

in [20] for a 120/70 front tyre and a 180/55 rear tyre, first recognising that the

former results suffer from an unreasonable positive force offset, especially for the

smaller loads, which would imply a friction coefficient greater than 2, if they were

true. To avoid responding too strongly to these apparently spurious features, Dyis

allowed to be no greater than 1.3 times Fz. Also, the measurements for slip angles

greater than +5◦are ignored. The previous rear tyre value of Fz0as 1600 N is

retained while the non-critical value for the front tyre was chosen as 1100 N. Best-

fitparametersareshowninTableII,withFigures7and8showingthequalityofthe

fits for the front and rear tyres respectively. All the constraints are satisfied by these

parameters.

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R.S. SHARP ET AL.

Figure 8. Tyre lateral force results for 180/55 tyre from [20] (thick lines) with best-fit recon-

structions (thin lines). Camber angles 0, 10, 20, 30, 40, 45◦.

4.3. ALIGNING MOMENTS IN LATERAL SLIP AND CAMBER

Aligning moment results are included in [23] for the 160/70 tyre and in [20] for

120/70 and 180/55 tyres. Three loads are covered in the former but only two in

the latter, which makes the model very heavy in parameters for the amount of

experimental data available. In setting the parameters for the 160/70 tyre of [23]

assuming the full quadratic dependency of Bton load, the fitting is good within the

load range used for the measurements but the extrapolation is poor, with constraint

violationsatlowandhighloads.Withlineardependency,thefittingisalmostasgood

and the extrapolation problem can be eliminated. Consequently, Btis considered

linear with load. Even so, there are many parameter combinations which give

almost equally good fits to the limited data. It is advantageous to use some physical

reasoning to guide the choice between the alternatives. The product of Bt, Ctand

Dtis the aligning moment stiffness of the tyre. According to the “Brush Model”

[23], the aligning moment stiffness is proportional to load to the power 1.5, so that

featureisusedtoaidthechoiceofthesecondaryparametersqBz1andqBz2,see(18).

It turns out to be quite feasible to match that characteristic closely. Also, as before,

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263

TableIII. Best-fitparametervaluesforaligningmomentfrom160/70(top),120/70(middle)

and 180/55 tyre (bottom)

Ct

qBz1

qBz2

qBz5

qBz6

qBz9

qBz10

1.3115

1.0917

1.3153

qDz1

0.20059

0.19796

0.26331

qDz11

0

−0.17873

−0.40709

10.354

10.486

10.041

qDz2

0.05282

0.06563

0.030987

qEz1

−3.9247

−0.91586

−0.19924

4.3004

−0.001154

−1.61e-8

qDz3

−0.21116

0.2199

−0.62013

qEz2

10.809

0.11625

−0.017638

−0.34033

−0.68973

−0.76784

qDz4

−0.15941

0.21866

0.98524

qEz5

0.9836

1.4387

3.6511

−0.13202

1.0411

0.73422

qDz8

0.30941

0.3682

0.50453

qHz3

−0.04908

−0.003789

−0.028448

10.118

27.445

16.39

qDz9

0

0.1218

0.36312

qHz4

0

−0.01557

−0.009862

−1.0508

−1.0792

−0.35549

qDz10

0.10037

0.25439

−0.19168

right/leftsymmetryandzerooffsetsareassumed,makingqEz4,qHz1andqHz2zero.

The relevant “Magic Formula” Equations [23] are:

Mz0= Mzt0+ Mzr0

Mzt0= −Dtcos[Ctarctan{Btβ − Et(Btβ − arctan(Btβ))}]

/

1 + β2· Fy0,γ=0

Mzr0= Drcos[arctan(Br(β + SHr)]

SHr= (qHz3+ qHz4d fz)γ

Bt= (qBz1+ qBz2d fz)(1 + qBz5|γ| + qBz6γ2)

Dt= Fz(R0/Fz0)(qDz1+ qDz2d fz)(1 + qDz3|γ| + qDz4γ2)

Et= (qEz1+ qEz2d fz){1 + qEz5γ(2/π)arctan(BtCtβ)}

Br= qBz9+ qBz10ByCy

Dr= FzR0{(qDz8+ qDz9d fz)γ + (qDz10+ qDz11d fz)γ|γ|}

/

1 + β2

(14)

?

(15)

(16)

(17)

(18)

(19)

(20)

(21)

?

(22)

with the constraints: Bt > 0, Ct > 0 and Et < 1. For the 160/70 tyre, qHz4

in Equation (17) and qDz9and qDz11in Equation (22) are set to zero, because

experimental results are only provided at non-zero camber angle for one load.

The tyre crown radius, R0, for each tyre derives from the cross-sectional geome-

tryas0.08mfor160/70,0.06mfor120/70and0.09mfor180/55[7].Identification

of the remaining parameters using “fmincon” as before gives the values in Table

III. Constraint violations occur only for loads greater than 11 kN, sideslip angle

greater than 45◦or camber angle greater than 60◦. These violations are outside the

practical running range. The fit qualities are shown in Figures 9–11.

Page 14

264

R.S. SHARP ET AL.

Figure 9. Tyre aligning moment results for 160/70 tyre from [23] (thick lines) with best-fit

reconstructions (thin lines). Camber angles 5, 0, −5, −10, −20, −30◦.

Figure 10. Tyre aligning moment results for 120/70 tyre from [20] (thick lines) with best-fit

reconstructions (thin lines). Camber angles 0, 10, 20, 30, 40, 45◦.

Figure 11. Tyre aligning moment results for 180/55 tyre from [20] (thick lines) with best-fit

reconstructions (thin lines). Camber angles 0, 10, 20, 30, 40 and 45◦.

Page 15

ADVANCES IN THE MODELLING OF MOTORCYCLE DYNAMICS

265

4.4. COMBINED SLIP RESULTS

4.4.1. Longitudinal Forces

Inthe“MagicFormula”scheme,thelossoflongitudinalforceduetosideslippingis

described by a “loss function” to be applied to the pure slip force described above.

Presuming as before that the generic tyres of interest will be symmetric (SHxα= 0)

and, in the absence of any indication to the contrary, assuming that wheel camber

will not affect the loss of longitudinal force due to sideslipping (rBx3= 0), the

equations describing the loss are:

Fx= cos[Cxαarctan(Bxαβ)]Fx0

Bxα= rBx1cos[arctan(rBx2κ)]

with the constraints that Fx> 0 and Bxα> 0.

The only relevant combined slip data available is from [23] for the 160/70 tyre

for 3 kN load and zero camber angle. The same parameter identification process

as before yielded the best values as rBx1= 13.476; rBx2= 11.354; Cxα= 1.1231,

with the fit quality shown in Figure 12. The constraint on Bxαis always satisfied

while that on Fxis satisfied for slip angles less than 23◦, which is considered to

provide an adequate operating range.

(23)

(24)

4.4.2. Lateral Forces

In the same way (with SVyκ= SHyκ= rBy4= 0), the equations describing the loss

of lateral force due to longitudinal slip are:

Fy= cos[Cyκarctan(Byκκ)]Fy0

Byκ= rBy1cos[arctan{rBy2(β − rBy3)}]

with constraints Fy> 0 and Byk> 0.

Data again comes from Pacejka [23] and is for the 160/70 tyre at 3 kN and zero

camber. It yields the best-fit parameters as rBy1= 7.7856, rBy2= 8.1697, rBy3=

−0.05914 and Cyκ= 1.0533. The fit quality is shown in Figures 13 and 14.

4.4.3. Aligning Moments

(25)

(26)

The relevant equations (with s = SVyκ= SHyκ= 0) are:

Mz= −Dtcos[Ctarctan{Btλt− Et(Btλt− arctan(Btλt))}]

/

1 + β2· Fy,γ=0+ Mzr

Fy,γ=0= cos[Cyκarctan(Byκκ)] · Fy0,γ=0

Mzr= Drcos[arctan(Brλr)]

λt=

λr=

?

(27)

(28)

(29)

?

β2+ (Kxκκ/Kyα,γ=0)2sgn(β)

(β + SHr)2+ (Kxκκ/Kyα,γ=0)2sgn(β + SHr)

(30)

?

(31)