Motorcycle Dynamics

A Motorcycle Multi-Body Model for Real Time

Simulations Based on the Natural

Coordinates Approach

VITTORE COSSALTER1_{and ROBERTO LOT}2

SUMMARY

This paper presents an eleven degrees of freedom, non-linear, multi-body dynamics model of a motorcycle. Front and rear chassis, steering system, suspensions and tires are the main features of the model.

An original tire model was developed, which takes into account the geometric shape of tires and the elastic deformation of tire carcasses. This model also describes the dynamic behavior of tires in a way similar to relaxation models.

Equations of motion stem from the natural coordinates approach. First, each rigid body is described with a set of fully cartesian coordinates. Then, links between the bodies are obtained by means of algebraic equations. This makes it possible to obtain simple equations of motion, even though the coordinates are redundant.

The model was implemented in a Fortran code, named FastBike. In order to test the code, both simulated and real slalom and lane change maneuvers were carried out. A very good agreement between the numerical simulations and experimental test was found. The comparison of FastBike's performance with those of some commercial software shows that ®rst is much faster than others. In particular, real time simulations can be carried out using FastBike and it can be employed on a motorcycle simulator.

1. INTRODUCTION

The use of computer simulations in motorcycle engineering makes it possible both to reduce designing time and costs and to avoid the risks and dangers associated with experiments and tests. The multi-body model for computer simulations can be built either by developing a mathematical model of the vehicle or by using commercial software for vehicle system dynamics. Even though the ®rst method is more dif®cult and time consuming than the second, maximum ¯exibility in the description of the features of the model can be obtained only by using a mathematical model. In particular, it makes it possible to properly describe the tire behavior at large camber 1_{Department of Mechanical Engineering, University of Padova, Italy.}

2_{Corresponding author: Roberto Lot, Department of Mechanical Engineering, University of Padova, Via}

Venezia 1, 35131 Padova, Italy. Tel.: ‡39 049 8276806; Fax: ‡39 049 8276785; E-mail: [email protected]; website: www.dinamoto.mecc.unipd.it

angles, whereas multi-body codes such as ADAMS, DADS or Visual Nastran lack such a feature. Moreover, mathematical modeling has a high computation ef®ciency, while multi-body software require a lot of time to carry out simulations.

For the reasons above, the focus of this study was to develop mathematical models of a tire and motorcycle. The tire model properly describes the shape of the carcass and the position of the contact point. Moreover, it takes into account the sliding of the contact patch and the deformation of the tire carcass. The motorcycle model was developed based on the natural coordinates approach [1], which makes it possible to obtain simple equations of motion and hence high computation ef®ciency.

2. MOTORCYCLE AND RIDER DESCRIPTION

The motorcycle is modeled as a system of six bodies: the front and rear wheels, the rear assembly (including frame, engine and fuel tank), the front assembly (including steering column, handle-bar and front fork), the rear swinging arm and the unsprung front mass (including fork and brake pliers). The driver is considered to be rigidly attached to the rear assembly; front and rear assembly are linked by means of the steering mechanism. The front suspension is a telescopic type and the rear suspension is a swinging arm type.

This vehicle model has eleven degrees of freedom, which can be associated to the coordinates of the rear assembly center of mass, the yaw angle, the roll angle, the pitch angle, the steering angle, the travel of front and rear suspension and the spin rotation of both wheels (see Fig. 1).

The following forces act on the motorcycle elements: suspensions forces due to springs and shock-absorbers, tire forces and torques, aerodynamic forces, rider steering torque, steer damper torque, rear and front brake torques and ®nally propulsive torque, which is transmitted from the sprocket to the rear wheel by means of the chain.

The rider's actions on the motorcycle determine both the direction of the vehicle and the forward speed. In this model, the rider is considered to be a rigid body attached to the rear assembly, so that the rider's movement away from the saddle and the corresponding control action are neglected. In this way the motorcycle's direction is controlled only by the torque exerted on the handlebars (steering torque). The forward speed is controlled by applying the brakes (rear and front brake torques) and by acting on the accelerator lever (propulsive force).

3. TIRE MODEL

In motorcycles the roll angle can reach 50±55_{, hence it has a signi®cant in¯uence}

both on tire forces and torques and on the contact patch. In this model, the actual shape of the tire is described in detail and the deformation of the tire carcass is taken into account. The road±tire contact is assumed to be dot-shaped and the position of the contact point depends on the roll angle. Tire forces and torques are applied in the contact point. The tire forces include the vertical load N, the lateral force F and the

longitudinal force S; the tire torques include the rolling friction torque Myand the yaw

torque Mz.

The tire reference frame Tw is de®ned by using 4  4 transformation matrix

notation [2], as shown in Figure 2: its origin is located in wheel center G, plane XwZw

is the symmetry plane of the wheel, the Xwaxis is horizontal and points forwards, the

Yw axis is parallel to the wheel spin axis and points rightwards and the Zw axis

completes the reference frame. The frame T0_{has its origin located in contact point C,}

the road plane X0_Y0_{is horizontal, the X}0_{axis is parallel to Xw, points forwards and has}

unit vector s, the Y0_{axis points rightward and has unit vector n, the Z}0_{axis is vertical}

and points downwards.

As it is well known, horizontal tire forces depend on tread deformation and slide, i.e., they depend on sideslip angle l, longitudinal slip k, camber angle j and vertical load N as follows

S ˆ Sslip…k; l; j; N†

F ˆ Fslip…k; l; j; N† …1†

In several tire models [3±5] the sideslip angle and longitudinal slip are de®ned according to wheel kinematics, without taking into account the deformation of the tire carcass. On the contrary, in this model slip quantities are de®ned considering the actual contact point, which moves with respect to the rim because of the deformation

of tire carcass. Deformability of the tire carcass is taken into account as shown in Figure 3. The contact point lies on the vertical plane which passes through the wheel

spin axis. The tire de¯ection with respect to the rim consists of radial displacement r,

lateral displacement l and rotation x around the wheel spin axis. Moreover, it is

assumed that tire deformations do not alter the mass properties of the wheel.

Fig. 2. Tire kinematics and tire forces.

The position of the contact point is expressed by means of its coordinates yc; zc

with respect to frame Twas follows

C ˆ Twf0; yc; zc; 1gT …2†

Thus, the instantaneous sideslip angle is de®ned as:

l ˆ ÿarctanV_VY

X ˆ ÿarctan

_C
n

_C
s …3†

where VXis the forward speed, VYthe lateral speed, s and n the unit vectors of axis X0

and Y0_{respectively.}

The instantaneous longitudinal slip is de®ned as:

k ˆ ÿ1 ÿV_VR

X ˆ ÿ1 ÿ

zc…_y ‡ _x†

_C
s …4†

where VRis the rolling speed which depends both on spin velocity _y and rotational

deformation rate _x.

On the other hand, tire forces depend on carcass deformation and camber angle, as shown in experimental tests [6, 7]

S ˆ Selastic…x; j†

F ˆ Felastic…r; l; j†

N ˆ Nelastic…r; l; j†

…5† In absence of tire forces, no tire de¯ection is present and the contact point

coincides with the point of tangency between the tire surface and road plane C0. Thus,

the position of the contact point only depends on the tire shape and the coordinates of

C0with respect to frame Twcan be de®ned as a function of the roll angle, as follows

C0ˆ Twf0; yt…j†; zt…j†; 1gT …6†

where functions yt…j† and zt…j† make a parametric representation of the lateral pro®le

of the carcass. In order to guarantee the condition of tangency between tire and road plane, functions must satisfy the following relation

tan…j† ˆ ÿdz_djt

dyt

dj

Lateral and radial deformation can be calculated by subtracting expression (6) from expression (2), obtaining

lˆ ycÿ yt… †j

This model is able to properly describe tire behavior both in steady state and transient conditions. Indeed, by coupling Equation (1), which describe the behavior of the contact patch during sliding, with Equation (5), which describe elasticity properties of the tire carcass

Sslip…k; l; j; N† ÿ Selastic…x; j† ˆ 0

Fslip…k; l; j; N† ÿ Felastic…r; l; j† ˆ 0 …8†

one obtains a description of tire behavior which is equivalent to relaxation tire models [8±11]. To proof this, let us de®ne a linear relation between longitudinal force and longitudinal slip

S ˆ Ksk …9†

and a linear relation between longitudinal force and rotational deformation

S ˆ Kxx …10†

where Ksand Kxare respectively the longitudinal slip stiffness and rotational stiffness

of tire. By substituting Equation (4) in Equation (9) and by rearranging terms, one obtains: S ˆ KS ÿ1 ÿz_Vc_y X ! ÿ KSz_Vc_x X ˆ KSk0ÿ KS zc_x VX …11†

where k0is the steady state value of longitudinal slip, which corresponds to the steady

state value of longitudinal force S0: The time derivation of expression (10) yields:

_x ˆ_K_S

x …12†

By replacing Equations (12) in Equation (11) and by rearranging the terms, one obtains:

KSzc=Kx

VX _S ‡ S ˆ S0 …13†

which is a ®rst order relaxation equation, where relaxation length is s ˆ KSzc=Kx. The

equivalence between this tire model and the relaxation model can be found for lateral force as well.

This approach presents several advantages with respect to relaxation models. First, it explains the physical behavior of the tire in more detail, by highlighting both the deformability of the carcass and the sliding of the tread. Furthermore, with this tire model only static and steady state experimental tests are required in order to characterize tire behavior in both static and dynamic conditions.

In order to complete the model it is necessary to de®ne tire torques with respect to the contact point. The rolling resistance torque is assumed to be proportional to the wheel load

Myˆ N d …14†

where d is the rolling friction parameter.

Yaw torque Mzis generated by lateral force F, tire trail t and twisting torque MTzas

follows [12±14]:

Mzˆ ÿt l… †F ‡ MTz… †j …15†

The ®rst term depends on the sideslip angle and tends to align, the second term depends on the roll angle and tends to self-steer.

Finally, it is not necessary to take into account overturning moment Mx, because

tire forces are applied in the actual contact point [3, 13, 14]. 4. MULTI-BODY MODEL

The mathematical model of the motorcycle was developed based on the natural coordinates approach [1]. Natural coordinates consist of cartesian coordinates of points or direction cosines of vectors belonging to the bodies of the system. With this approach, kinematic relationships and equations of motion are very simple. However, the number of variables required for describing a system is larger than the number of degrees of freedom and so additional constraint equations must be introduced.

The equations were derived using Maple1_{, a software which makes it possible to}

perform symbolic manipulation ef®ciently and to avoid calculation errors. Moreover, it generates automatically the Fortran code.

4.1. Kinematic Description

Equations of motion were derived in the inertial reference frame XYZ: axes X and Y are horizontal and lie on the road level, the Z axis is vertical and points downwards;

the unit vectors of inertial frame are, respectively, cx, cyand cz.

A body-®xed frame Ti is attached to each rigid body. The elements of the

transformation matrix are used as generalized coordinates, i.e., the con®guration of each body is described by means of the coordinates of origin and direction cosines of the body-®xed frame (see Fig. 4).

The rear tire reference frame Tw1 has its origin in the center of the wheel

G1ˆ fx1; y1; z1; 1gT and is de®ned as shown in Section 3, as well as the reference

frame T0

1. Moreover, the rear wheel ®xed-frame T1 is obtained from frame Tw1

by a rotation of spin angle y1 around Yw1 axis. It is useful to de®ne the

w1ˆ fwx1; wy1; wz1; 0gT parallel to axis Yw1, v1ˆ fvx1; yy1; vz1; 0gT parallel to axis

Zw1 and n1ˆ fÿsy1; sx1; 0; 0gT parallel to axis Y01.

The rear assembly ®xed-frame T2 has its origin in the swinging arm pin joint

P2ˆ xf 2; y2; z2; 1gT; plane X2Z2is parallel to plane X1Z1, the X2axis is perpendicular

to the steering axis, points forwards and has unit vector u2ˆ fux2; uy2; uz2; 0gT, the Y2

axis has unit vector w2ˆ w1and ®nally the Z2axis is parallel to the steering axis and

has unit vector v2ˆ fvx2; yy2; vz2; 0gT.

The front assembly ®xed-frame T3has the origin in the point P3ˆ fx3; y3; z3; 1gT,

which is the intersection between the steering axis and its perpendicular plane passing

through P2. The X3Z3plane is parallel to the symmetry plane of the front wheel, the

X3 axis is perpendicular to the steering axis, points forwards and has unit vector

u3ˆ fux3; uy3; uz3; 0gT, the Y3axis is parallel to the front wheel spin axis and has unit

vector w3ˆ fwx4; wy4; wz4; 0gT, ®nally the Z3axis has a unit vector v3ˆ v2.

The front tire reference frame Tw4 has its origin in the center of the wheel

G4ˆ fx4; y4; z4; 1gT and is de®ned as shown in Section 3, as well as the reference

frame T0

4. Besides, the front wheel ®xed-frame T4 is obtained from frame Tw4by a

rotation of spin angle y4 around Yw4 axis. The following unit vectors are de®ned:

s4ˆ fsx4; sy4; 0; 0gT parallel to both Xw4 and X04axis, w4ˆ w3 parallel to Yw4axis,

v4ˆfvx4; yy4; vz4; 0gTparallel to Zw4axis and n4ˆfÿsy4; sx4; 0; 0gTparallel to Y04axis.

The swinging arm ®xed-frame T5has its origin in the rear wheel center G1, the X5

axis is parallel to vector G1P2and has unit vector u5ˆ fux5; uy5; uz5; 0gT, the Y5axis

has unit vector w5ˆ w1and the Z5 axis has unit vector v5ˆ fvx5; yy5; vz5; 0gT.

The front unsprung mass ®xed-frame T6 has the origin on the center of mass

G6ˆ T4fGx6; Gy6; Gz6; 1gT; X6; Y6and Z6axes are parallel respectively to X3; Y3and

Z3 and their unit vectors are u6ˆ u3, w6ˆ w4, v6ˆ v2.

The con®guration of the motorcycle is described by means of a set of n ˆ 45

coordinates, including the coordinates of points G1, P2, P3, G4, direction cosines of

unit vectors s1, v1, w1, u2, v2, u3, s4, v4, w4, u5, v5and spin rotations of both wheels:

q ˆ fx1; y1; z1; sx1; sy1; wx1; wy1; wz1; vx1; vy1; vz1; y1; x2; y2; z2; ux2; uy2; uz2; vx2; vy2;

vz2; x3; y3; z3; ux3; uy3; uz3; x4; y4; z4; sx4; sy4; wx4; wy4; wz4; vx4; vy4; vz4; y4; ux5;

uy5; uz5; vx5; vy5; vz5gT …16†

The motorcycle has only f ˆ 11 degrees of freedom, thus it is necessary to formulate a set of m ˆ n ÿ f ˆ 34 independent constraint equations:

f_jˆ 0; j ˆ 1 . . . m …17†

By imposing the unit length condition to all unit vectors, the following 11 independent constraint equations are obtained:

f1ˆ s1
s1ÿ 1 f₄ˆ u2
u2ÿ 1 f₇ˆ s4
s4ÿ 1 f₁₀ˆ u5
u5ÿ 1 f₂ ˆ w1
w1ÿ 1 f₅ ˆ v2
v2ÿ 1 f₈ ˆ w4
w4ÿ 1 f₁₁ ˆ v5
v5ÿ 1 f₃ ˆ v1
v1ÿ 1 f₆ ˆ u3
u3ÿ 1 f₉ ˆ v4
v4ÿ 1 …17:1ÿ11†

By imposing the orthogonal conditions to every couple of unit vectors which belong to the same reference frame, 15 more independent constraint equations are obtained:

f₁₂ ˆ s1
w1 f₁₅ ˆ u2
w1 f₁₈ ˆ u3
v2 f₁₂ ˆ s4
w4 f₂₄ ˆ u5
w1 f₁₃ ˆ s1
v1 f₁₆ ˆ v2
u2 f₁₉ ˆ u3
w4 f₂₂ ˆ s4
v4 f₂₅ ˆ v5
w1 f₁₄ ˆ v1
w1 f₁₇ ˆ v2
w1 f₂₀ ˆ v2
w4 f₂₃ ˆ v4
w4 f₂₆ ˆ w5
v5 …17:12ÿ26†

The remaining 8 constraint equations are the following:

 vector G1P2must be perpendicular to the f27 ˆ G1P2
w1 (17.27)

rear wheel spin axis Y1

 the magnitude of vector G1P2must be f28ˆ G1P2
G1P2ÿ l2f (17.28)

equal to the swinging arm length lf

 vector v5must be perpendicular to the f29ˆ G1P2
v5 (17.29)

 the magnitude of vector P2P3must be f30ˆ P2P3
P2P3ÿ l223 (17.30)

equal to l23

 vector P2P3must lie on the X2Z2plane f31 ˆ P2P3
w1 (17.31)

(thus it must be perpendicular to the f₃₂ˆ P2P3
v2 (17.32)

vectors w1and v2)

 the point R3ˆ G4ÿ l1u3 must lie on the f33ˆ P3R3
w4 (17.33)

steering axis Z3

(thus it must be perpendicular to vectors f₃₄ˆ P3R3
u3 (17.34)

w4and u3)

It is worth pointing out that the natural coordinates approach made it possible to obtain simple constraint equations, which are quadratic with respect to the coordinates. 4.2. Lagrange's Equations

Due to the presence of constraints, the Lagrange's equations become d dt @K @ _qiÿ @K @qi‡ Xm jˆ1 lj@f_@qj iÿ Qiˆ 0; i ˆ 1: : n …18†

where K is the kinetic energy, liare the Lagrange multipliers and Qithe generalized

forces.

By coupling the de®nition of kinetic energy to the transformation matrix notation, the kinetic energy of ith rigid body is

Kiˆ1₂ Z m_P 2_{dm ˆ}1 2 Z mfx; y; z; 1g _T T i _Tifx; y; z; 1gTdm …19†

where fx; y; z; 1gTare the coordinates of point P with respect to frame Ti. Assuming

that the origin of the reference frame is the center of mass of the body and expanding the previous equation, one obtains:

Tiˆ1₂ Z mfx; y; x; 1g _u2 i _ui
_wi _ui
_vi _ui
_Gi _wi
_ui _w2i _wi
_vi _wi
_Gi _vi
_ui _vi
_wi _v2i _vi
_Gi _Gi
_ui _Gi
_wi _Gi
_vi _G2i 2 6 6 6 6 4 3 7 7 7 7 5fx; y; x; 1g T_dm ˆ1₂ _G2 i Z mdm ‡ 1 2_u2i Z mx 2_{dm ‡}1 2 _w2i Z my 2_{dm ‡}1 2_v2i Z mz 2_dm ‡ _ui
_wi Z mxy dm ‡ _ui
_vi Z mxz dm ‡ _wi
_vi Z myz dm ‡ _ui
_Gi Z mx dm ‡ _vi
_Gi Z my dm ‡ _wi
_Gi Z mz dm

By substituting the integral terms in the previous equation with moments and products of inertia with respect to the center of mass, the kinetic energy of each rigid body can

be calculated as a function of the elements of transformation matrix Ti, as follows

Kiˆ1₂mi_G2i ‡₄1Ix;i ÿ _u2i ‡ _w2i ‡ _v2i ÿ
‡1₄Iy;i _u2i ÿ _w2i ‡ _v2i ÿ
‡1₄Iz;i _u2i ‡ _w2i ÿ _v2i ÿ

‡ Cxz;i_ui
_vi‡ Cxy;i_ui
_wi‡ Cyz;i_wi
_vi …20†

If the body center of mass does not coincide with the origin of the reference frame, it

is necessary to replace _Giˆ f_xi; _yi; _zi; 1gT with _Giˆ _TifGxi; Gyi; Gzi; 1gT in the

previous equation. Thus, the kinetic energy of the whole system is:

K ˆ1₂m1_G21‡1₂Iy1 _s21ÿ _w21‡ _v21‡ _y1…s1
_v1ÿ _s1
v1† ‡ _y21 h i ‡1₂Id1_w21 ‡1₂m2_G22‡1₄Ix2 ÿ _u22‡ _w21‡ _v22 ÿ
‡1₄Iy2 _u22ÿ _w21‡ _v22 ÿ
‡1₄Iz2 _u22‡ _w22ÿ _v22 ÿ
‡ Cxz2_u2
_v2‡ Cxy2_u2
_w2‡ Cyz2_w2
_v2 ‡1 2m3_G23‡1₄Ix3 ÿ _u23‡ _w24‡ _v23 ÿ
‡1 4Iy3 _u23ÿ _w24‡ _v23 ÿ
‡1₄Iz3 _u23‡ _w24ÿ _v23 ÿ
‡ Cxz3_u3
_v3‡ Cxy3_u3
_w4‡ Cyz3_w4
_v3 ‡1₂m4_G24‡1₂Iy4 _s24ÿ _w42‡ _v24‡ _y4…s4
_v4ÿ _s4
v4† ‡ _y24 h i ‡1₂Id4_w24 ‡1 2m5_G25‡1₄Ix5 ÿ _u25‡ _w21‡ _v25 ÿ
‡1 4Iy5 _u25ÿ _w21‡ _v25 ÿ
‡1₄Iz5 _u25‡ _w21ÿ _v25 ÿ
‡ Cxz5_u5
_v5‡ Cxy5_u5
_w1‡ Cyz5_w1
_v5 ‡1₂m6_G26‡1₄Ix6 ÿ _u23‡ _w24‡ _v23 ÿ
‡1₄Iy6 _u23ÿ _w24‡ _v23 ÿ
‡1 4Iz6 _u23‡ _w24ÿ _v23 ÿ
‡ Cxz6_u3
_v3‡ Cxy6_u3
_w4‡ Cyz6_w4
_v3 …21†

where the terms relative to wheels (i ˆ 1 and i ˆ 4) are slightly different from the terms relative to other bodies because of the axial symmetric structure of the wheels

(Ix;iˆ Iz;iˆ Id;i and Cxz;iˆ Cyz;i ˆ Cxy;iˆ 0) and because spin velocity _y1; _y4 has

been used.

The generalized forces expression can be obtained from the virtual work dW of the forces acting on the vehicle

dW ˆXm

iˆ1

In order to determine virtual works, it is necessary to calculate the virtual rotation dYi

of each rigid body with respect to its reference frame Ti. By extending the concept of

angular velocity matrix [2] to virtual rotation matrix dY ˆ TT_{dT and by extracting}

the components of virtual rotation from dY, the following virtual rotation operator can be de®ned:

dY T… † ˆ vi f i
dwi; ui
dvi; wi
dui; 0gT …23†

Virtual work contains the following terms:

dW ˆ dWg‡ dWS‡ dWA‡ dWt‡ dWB‡ dWt;F‡ dWt;T‡ dWP …24†

 The virtual work due to the gravity force:

dWg ˆ

X6

iˆ1

mig
dGi …24:1†

where g ˆ f0; 0; g; 1gT _{is the gravity acceleration.}

 The virtual work due to front suspension force FSf, which acts between the front

assembly and front wheel, and virtual work due to rear suspension force FSr, which

acts between the rear assembly and swinging arm:

dWsˆ FSfv2
…dP3ÿ dR3† ‡ tsFSrcy
dY T‰ … † ÿ dY T2 … †5 Š …24:2†

where tsˆ @yr=@zris the velocity coef®cient between spring de¯ection zrand arm

rotation yr.

 The virtual work due to drag, side and lift aerodynamics forces FA ˆ

Tw1fFD; FS; FL; 0gT, which are applied on point CAˆ T2fXCA; 0; ZCA; 1gT:

dWA ˆ FA
dCA …24:3†

 The virtual work due to rider steering torque t and steer damper torque tD, which

are applied between the rear and front assembly:

dWtˆ t ‡ t… D†cz
dY T‰ … † ÿ dY T3 … †2 Š …24:4†

 The virtual work due to rear brake torque MBr, which acts between the rear wheel

and swinging arm, and the virtual work due to front brake torque MBf, which acts

between the front wheel and front unsprung mass:

dWB ˆ MBrcy
dY T‰ … † ÿ dY T1 … †5 Š ‡ MBfcy
dY T‰ … † ÿ dY T4 … †6 Š …24:5†

 The virtual work due to rear tire force FT1ˆ T01fS1; F1; ÿN1; 0gT, which is applied

on rear contact point C1 ˆ Tw1f0; yc1; zc1; 1gT, and the virtual work due to front tire

force FT4ˆ T04fS4; F4; ÿN4; 0gT, which is applied on front contact point C4ˆ

Tw4f0; yc4; zc4; 1gT:

dWt;F ˆ FT1
dG1‡G1C1FT1
T1dY T… †‡F1 T4
dG4‡G4C4 FT4
T4dY T… †4

 The virtual work due to rear tire torque MT1ˆ T01f0; My1; Mz1; 0gT and front tire

torque MT4ˆ T04f0; My4; Mz4; 0gT:

dWt;Mˆ MT1
T1dY T… † ‡ M1 T4
T4dY T… †4 …24:7†

 The virtual work due to the propulsive torque, which is transmitted from the drive sprocket to the wheel by means of the chain. As shown in Figure 5, the drive

sprocket center is R ˆ T2fRX; 0; RZ; 1gT, whereas the chain angles are:

yc1ˆ arctan _GG1R
s1 1R
v1   ÿ arcsin rcÿ rp G1R j j   yc2ˆ arctan G_G1R
u2 1R
v2   ‡ arcsin rc_Gÿ rp 1R j j  

The chain tension FCˆ T1fTcsin…yc1†; 0; Tccos…yc1†; 0gT acts between point

P7ˆ T1frccos…yc1†; 0; rcsin…yc1†; 1gT and point P8ˆ T2RX‡ rpcos…yc2†; 0; RZ‡

rpsin…yc2†; 1gT, thus the virtual work is

dWpˆ Fc
dP… 7ÿ dP8† ÿ Tcrcdy1 …24:8†

Explicit Lagrange's equations are not shown because of their large number, while their compact form is the following:

F q; _q; q; k; t… † ˆ Mq ‡ _M _q ‡ FT_{k ÿ Q ˆ 0 …25†}

where M is the mass matrix, F is the Jacobian matrix of constraint equations (17), k is the vector of Lagrange multipliers and Q is the vector of generalized forces. Due to the natural coordinates approach, the mass matrix is very sparse and has only 9% non-zero elements; moreover the evaluation of Equation (25) require less than 2,000 multiplications and less than 1,000 additions.

4.3. Tire Equations

As seen in Section 3, tire deformation is described by means of three coordinates, hence for both the rear and front tires the following six coordinates should be de®ned:

q0_{ˆ y}

c1; zc1; x1; yc4; zc4; x4

f gT …26†

The tire behavior must be described by means of as many equations as coordinates. Equation (8) can be re-written as follows

p1 ˆ Sslip;1…k1; l1; j1; N1† ÿ Selastic;1…x1; j1† ˆ 0 p2 ˆ Sslip;4…k4; l4; j4; N4† ÿ Selastic;4…x4; j4† ˆ 0 p3 ˆ Fslip;1…k1; l1; j1; N1† ÿ Felastic;1 r;1; l;1; j1 ÿ
ˆ 0 p4 ˆ Fslip;4…k4; l4; j4; N4† ÿ Felastic;4 r;4; l;4; j4 ÿ
ˆ 0 …27:1ÿ4† Equations (3), (4) and (7) make it possible to express slip quantities and tire deformations as a function of generalized coordinates, whereas camber angles can be calculated as follows:

j₁ ˆ arcsin w… z1†

j₄ ˆ arcsin w… z4† …28†

The remaining equations are obtained by imposing the contact between the tire and road plane Z ˆ 0, as follows:

p5ˆ C… †1 zˆ z1‡ wz1yc1‡ vz1zc1ˆ 0

p6ˆ C… †4 zˆ z4‡ wz4yc4‡ vz4zc4ˆ 0 …27:5ÿ6†

It is worth pointing out that Equations (27.1±4) are differential equations because

slip quantities (3) and (4) contain time derivation of coordinates x and x0_{. On the}

contrary, Equations (27.5±6) are algebraic. 4.4. State Space Formulation

Lagrange's Equation (25), constraint Equation (17) and tire Equation (27) form a set of 85 second order differential-algebraic simultaneous equations (DAEs) of index 3 [15], with the following unknowns: 51 generalized coordinates and 34 Lagrange multipliers.

In order to obtain a 1 index DAEs problem, algebraic constraint Equation (3) should be replaced by differential equations using the Baumgarte stabilization method [16], as follows:

/0_{ˆ / ‡ 2&o _/ ‡ o}2_{/ …29†}

The DAEs problems of index 1 can be numerically solved using the DASSL solver [17], however the transformation of DAEs into a set of ordinary differential equations (ODEs) makes it possible to increase integration speed. For this purpose, the Lagrange multipliers are replaced with the following differential expression:

k ˆ c ‡ t0_{_c …30†}

where constant t0_{is properly chosen. Moreover, tire Equation (27) should be replaced}

by the following set of ODEs

p0_{ˆ p}

1; p2; p3; p4; p5‡ t0_p5; p6‡ t0_p6

f gT …31†

In addition, the 2nd order Lagrange's Equation (25) should be reduced to a 1st order ODEs. The system is then described by means of the following 2n ‡ m ‡ 6 ˆ 130 state variables

y ˆ x; v; c; x_f 0_gT _32†

and the following state space equations

G y; _y; t… † ˆ F v ÿ _q /0 p0 8 > > < > > : 9 > > = > > ;ˆ 0 …33†

Although the number of equations is rather high with respect to the number of degrees of freedom, each equation is simple and the evaluation of expression (33) require less than 3,000 multiplications and less than 2,000 additions. These equations have been implemented in a Fortran code, using the implicit solver DASSL for numerical integration.

5. COMPARISON BETWEEN COMPUTER SIMULATIONS AND EXPERIMENTAL MEASUREMENTS

In order to validate the multi-body model, some experimental tests were carried out on an Aprilia RSV 1000 motorcycle; they were then compared to the simulation results. The geometrical and inertial characteristics of the motorcycle and the non-linear elastic and damping characteristics of the suspensions were measured at the Department of Mechanical Engineering (DIM) at the University of Padua [18, 19]. Tire parameters were also measured with department's equipment [20], whereas the driver inertia properties were estimated as shown in reference [21]. The charac-teristics of the motorcycle are given in Appendix and in Figures 10, 11 and 12.

The motorcycle was equipped with a measurement system: roll and yaw rate, steering angle, spin velocity of both wheels and steering torque were measured and

stored on a data recorder [19]. Data post-processing made it possible to calculate vehicle forward speed and roll angle as well.

In order to reproduce the experimental maneuvers by means of numerical simulations, steering torque t was calculated according to measured steering torque

tmand measured roll angle jm, as follows:

t ˆ tm‡ kj…jmÿ j† …34†

where j is the simulated roll angle and kjthe control gain. The chain propulsive force

and the front brake torque were calculated based on measured speed um, as follows:

S ˆ mr‰_um‡ ku…umÿ u†Š Tcˆr_r1 cS MFf ˆ 0; S  0 …acceleration† Tcˆ 0 MFf ˆ ÿr4S; S < 0 …braking† 8 < : …35†

where S is the longitudinal thrust, mrthe generalized mass, u the simulated speed and ku

the control gain. Rear brake was not used in either the real or simulated maneuvers. Figure 6 shows the comparison of the experimental measurements with the numerical simulation for a lane change maneuver. The lane change width was 3.6 m and the lane change length was 40 m. It was not possible to measure the trajectory of the motorcycle, so the experiments were compared with simulations by analyzing steering torque (Fig. 6a), vehicle speed (Fig. 6b), roll angle (Fig. 6c) and steering angle (Fig. 6d). The ®gure shows that at the beginning of the maneuver the rider is driving straight and increasing speed. When he starts to apply positive steering torque (point A), the vehicle begins to capsize on the left-hand side. Afterwards, when the steering torque is zero (point B) the magnitude of roll angle is still increasing; when the steering torque reaches its minimum (point C), the roll angle is increasing and the vehicle begins to capsize on the right-hand side. Then, the rider straightens the vehicle (from point D) and ®nally decreases the speed.

The agreement between experimental and simulated data is very good: the overall error (RMS) of steering torque is less than 3% of its peak value, the overall error of vehicle speed is less than 0.5% of its peak value, the overall error of roll angle is about the 9% of its peak value and the overall error of steering angle is about 26% of its peak value. The steering angle has the maximum error, because of some steering oscillations that are present in the simulation but that were not found in the experimental test.

Figure 7 shows the comparison of a real slalom maneuver with a simulated one, by representing steering torque (Fig. 7a), vehicle speed (Fig. 7b), roll angle (Fig. 7c) and steering angle (Fig. 7d). The pylon distance is 14 m and the vehicle speed is about 13.5 m/s. During the slalom maneuver, both roll and steering angles are delayed in

phase from steering torque of about 90_{. Once again, the agreement between}

Fig. 6. Lane cha nge man euv er: co mparison between ex per imenta lmeasu rements and nume rical simul ations.

Fig. 7. Slalom man euv er: com parison betwee n experime ntal measur ements and numer ical simula tions.

Fig. 8. Lane change man euv er: com parison of nume rical simul ations carrie d out using dif ferent multi-bo dy sof tware.

less than 10% of its peak value, the overall error of vehicle speed is less than 3% of its peak value, the overall error of roll angle is about the 15% of its peak value and the overall error of steering angle is about the 13% of its peak value.

6. COMPARISON OF THE PERFORMANCES OF THE MULTI-BODY MODEL WITH PERFORMANCE OF MULTI-BODY COMMERCIAL SOFTWARE In this section simulations carried out using FastBike are compared with simulations carried out using Dads1 and Visual Nastran1.

The features of Visual Nastran and Dads motorcycle models are about the same as FastBike. It is worth pointing out that these multi-body software do not have a suitable tire model, so it was necessary to implement the tire model presented in [14] and [13]. In this model the tire is rigid and has a toroidal shape.

The Figure 8 shows simulations of a lane change maneuver carried out using different codes. The agreement between the data is excellent, both for the steering torque (Fig. 8a) and roll angle (Fig. 8b).

Even if commercial software for multi-body analysis greatly reduces the time needed for modeling systems, the time required for simulation is greater. Figure 9 compares the CPU time needed to carry out 1 s of simulation on a AMD K7 800 MHz processor. The only code that allows real time simulation is Fast bike, which is about 10 times faster than Dads and about 100 times faster than Visual Nastran.

7. CONCLUSIONS

An original mathematical model of a tire and motorcycle was presented.

The tire model was developed in order to describe tire behavior at a large camber angle. The shape of the tire and position of the contact point were described in detail. The model is based on the physical description of tire forces genesis: the sliding of the contact patch generates tire forces, which produce a deformation of the carcass of the tire. By taking into account simultaneously both phenomena, an accurate description of tire properties is obtained. It was demonstrated that this model is equivalent to relaxation tire models.

The motorcycle multi-body model has eleven degrees of freedom and includes the main features of a motorcycle, taking into account the non-linear properties of tires and suspensions. The very good agreement between the numerical simulations and experimental tests demonstrates the feasibility and correctness of the model.

The equations of motion were developed based on the natural coordinates approach. This method made it possible to obtain simple equations of motion and hence high computation ef®ciency was obtained. The comparison of the per-formances of the FastBike code with the performance of DADS and Visual Nastran showed that the ®rst is much faster than the others. In particular, real time simulations can be carried out using FastBike and it can also be used on a motorcycle simulator. For the same reason, it can be useful for solving optimization problems.

ACKNOWLEDGEMENTS

The authors would like to thank A. Doria for his suggestion regarding the organization of the paper and D. Bortoluzzi and N. Ruffo for their contribution during the experimental tests.

This research was partially supported by funds from the Italian Ministry for Universities and for Scienti®c and Technological Research (MURST 40% funds).

REFERENCES

1. Jalon, J.C. de and Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. Springer, 1994. 2. Sush, H. and Radcliffe, C.W.: Kinematics and Mechanism Design. Wiley, New York, 1978, Chapter 3. 3. Sakay, H.: Study on Cornering Properties of Tire and Vehicle. Tire Science and Technology 18 (1990),

pp. 136±139.

4. Pacejka, H.B. and Bakker, E.: The Magic Formula Tyre Model. Vehicle System Dynamics 21 (1991), pp. 1±18.

5. Pacejka, H.B. and Sharp, R.S.: Shear Force Development by Pneumatic Tyres in Steady State Conditions: A Review of Modelling Aspects. Vehicle System Dynamics 20 (1991), pp. 121±176. 6. Wang, Y.Q., Gnadler, R. and Schieschke, R.: Vertical Load-De¯ection Behaviour of a Pneumatic Tyre

Subjected To Slip And Camber Angles. Vehicle System Dynamics 25 (1996), pp. 137±146. 7. Berritta, R., Cossalter, V. and Doria, A.: Identi®cation of The Lateral and Cornering Stiffness af

Scooter Tyres Using Impedance Measurements. Proc. 2nd International Conference on Identi®cation in Engineering Systems, Swansea, UK, 1999, pp. 669±678.

8. De Vries, E.J.H. and Pacejka, H.B.: Motorcycle Tyre Measurements and Models. Proc. 15th IAVSD Symposium: The Dynamics of Vehicles on Road and Tracks. Budapest, Hungary, 1997, pp. 280±298. 9. Maurice, J.P. and Pacejka, H.B.: Relaxation Length Behavior of Tyres. Vehicle System Dynamics 27

(1997), pp. 339±342.

10. Zegelaar, P.W.A. and Pacejka, H.B.: Dynamic Tyre Responses to Brake Torque Variations. Vehicle System Dynamics 27 (1997), pp. 65±79.

11. Guo, K., Liu, Q. and Yangpin, H.: A Non-Steady Tire Model for Vehicle Dynamic Simulation and Control. Proc. AVEC International Symposium on Advanced Vehicle Control: AVEC'98, Nagoya, Japan, 1998.

12. Fujioka, T. and Goda, K.: Tire Cornering Properties at Large Camber Angles: Mechanism of the Moment around the Vertical Axis. JSAE Review 16 (1995), pp. 257±261.

13. Berritta, R., Cossalter, V., Doria, A. and Lot, R.: Implementation of a Motorcycle Tyre Model in a Multi-Body Code. Tire Technology International, March 1999.

14. Cossalter, V., Doria, A. and Lot, R.: Steady Turning of Two Wheel Vehicles. Vehicle System Dynamics 31 (1999), pp. 157±181.

15. Gear, C.W.: Differential-Algebraic Equation Index Transformations. SIAM Journal on Scienti®c and Statistical Computing, 9 (1988), pp. 39±47.

16. Baumgarte, J.: Stabilization of Constraints and Integrals of Motion in Dynamical Systems. Computer Methods in Applied Mechanics and Engineering 1 (1972), pp. 1±16.

17. Petzold, L.R.: A Description of DASSL: A Differential/Algebraic System Solver. In: R.S. Stepleman (ed.): IMACS Transactions on Scienti®c Computation 1, 1982, pp. 430±432.

18. Da Lio, M., Doria, A. and Lot, R.: A Spatial Mechanism for the Measurement of the Inertia Tensor: Theory and Experimental Results, ASME Journal of Dynamic Systems, Measurement and Control, 121 (March 1999), pp. 111±116.

19. Bortoluzzi, D., Doria, A., Lot, R. and Fabbri, L.: Experimental Investigation And Simulation Of Motorcycle Turning Performance. 3rd International Motorradkonferenzen, Monaco, 2000.

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APPENDIX

MOTORCYCLE CHARACTERISTICS Motorcycle Geometric and Mechanical Properties

m1 16.2 kg Rear wheel mass

Ia1 0.66 kgm2 Rear wheel axial inertia

Id1 0.33 kgm2 Rear wheel diametrical inertia

m2 223 kg Rear assembly mass (including rider)

(Gx2; Gy2; Gz2) (0.255, 0.000, ÿ0.0202) m Coordinates of rear assembly CoM with

respect to frame T2

Ix2; Iy2; Iz2 (24.4, 26.2, 30.3) kgm2 Rear assembly moments of inertia

l23 0.730 m Distance between rear arm pin and steer

axis

m3 8.75 kg Front assembly mass

(Gx3; Gy3; Gz3) (0.023, 0.000, ÿ0.098) m Coordinates of front assembly CoM with

respect to frame T3

Ix3; Iy3; Iz3 (0.29, 0.14, 0.21) kgm2 Front assembly moments of inertia

Cxz3; Cyz3; Cxy3 (0.0, 0.0, 0.0) kgm2 Front assembly products of inertia

10 Nms Damping coef®cient of steering damper

m4 12.0 kg Front wheel mass

Ia4 0.47 kgm2 Front wheel axial inertia

Id4 0.22 kgm2 Front wheel diametric inertia

l1 0.034 m Front wheel offset

ZF;0 0.517 m Center of wheel position (with respect to

frame T3) when the suspension is

completely extended

lf 0.535 m Rear arm length

m5 10.0 m Rear arm mass

(Gx5; Gy5; Gz5) (0.275, 0.000, ÿ0.052) m Coordinates of rear arm CoM with

respect to frame T5

Ix5; Iy5; Iz5 (0.20, 0.80, 0.80) kgm2 Rear arm moments of inertia

y5,0 ÿ.165 rad Rear arm rotation (respect frame T2)

when the suspension is completely extended

zrˆ 0:13526
y5ÿ 0:138
y25ÿ 0:036
y35 Relation between spring travel zr and

arm rotation y5

m6 7.00 kg Unsprung front mass

Gx6; Gy6; Gz6 (ÿ0.029, 0.000, ÿ0.189) m Coordinates of unsprung mass CoM

with respect to frame T3

Ix6; Iy6; Iz6 (0.22, 0.18, 0 .07) kgm2 Unsprung mass moments of inertia

rp 0.041 m Sprocket radius

rc 0.104 m Wheel sprocket radius

(ap; bp) (0.080, 0.030) m X±Z coordinates of sprocket center with

respect to frame T2

CDA 0.28 Ns2/m2 Drag force coef®cient (FDˆ CDA
u2)

Global Properties

m 276.8 kg Total mass

p 1.421 m Wheel base

e 0.43 rad Castor angle

h 0.636 m Height of the center of mass

b 0.675 m Horizontal position of the center

(with respect to the rear wheel)