Motorcycle cornering behavior modeling - Paper

Proceedings of the XIV International Symposium on Dynamic Problems of Mechanics (DINAME 2011). Donadio, R. N., ABCM, São Sebastião, SP, Brazil, March 13th - March 18th, 2011. [email protected]

Motorcycle cornering behavior modeling.

Rafael Donadio1

Roberto Bortolussi1

1 Centro Universitário da FEI

Abstract: The market for motorcycles has been showing a continuous increase in sales in last years. This result is

driven by the change of perception by the consumers not to despise the two-wheeled vehicle as a transport. Fuel

economy, parking easiness and speed of locomotion confirm the absorption of this product on the market. But the

growth of scientific research in motorcycles dynamics do not grow in the same market rate, making it an issue to be

exploited to improve the safety of the rider or assist in new projects development. This work uses a multi body

motorcycle model containing 4 rigid bodies connected by revolution joints parameterized by 7 degrees of freedom.

The model includes the major geometric and inertial characteristics of the motorcycle. It was used in the

mathematical model nonlinear algebraic equations. The model is subjected to curvilinear trajectory with constant

radius and speed, allowing to know the behavior of the motorcycle on a steady state maneuver, using two input

parameters imposed by the pilot: angle of steering and roll angle. The simulation results are discussed and presented

in graphical form. Aiming to validate the mathematical model, using an instrumented motorcycle with data acquisition

equipment and comparing the actual values with those obtained in the mathematical model.

Keywords: Motorcycle lateral dynamics. Multi body system. Steady state cornering. Data acquisition

NOMENCLATURE

reference coordinates systems:

(X, Y, Z) = ground coordinate system

(X1, Y1, Z1) = rotating coordinate

system (1)

(xd, yd, zd) = front reference

coordinate system

(xt, yt, zt) = rear reference coordinate

system

a = mechanical trail

A = origin of coordinate system (t)

ag = centre of mass acceleration

an = front wheel normal trail

at = tire trail

bt = longitudinal position of rear

centre of mass

C = turning centre point

d = coordinate system (d)

dp = forward displacement of the tire

contact point

ed = eccentricity of front centre of

mass

Ed = front tire longitudinal force

Et = rear tire longitudinal force

F = lateral force

FA = aerodynamic force on the rear

frame

FD = aerodynamic drag force

Fd = lateral front tire force

FGd = gravity forces on the front

frame

FGt = gravity forces on the rear frame

FL = aerodynamic lift force

FPd = road reaction, front

FPt = road reaction, rear

FS = aerodynamic side force

Ft = lateral rear tire force

g = acceleration due to gravity

Gd = front center of mass

Gt = rear center of mass

ht = height of rear centre of mass

ICXZd, ICYZd = components of inertia

tensor of front frame with respect

to (X1, Y1, Z1)

ICXZt, ICYZt = components of inertia

tensor of rear frame with respect

to (X1, Y1, Z1)

Iwd = front wheel inertia

Iwt = rear wheel inertia

Ixd, Iyd, Izd = components of inertia

tensor of front frame with respect

to (xf, yf, zf)

Ixt, Iyt, Izt = components of inertia

tensor of rear frame with respect

to (xt, yt, zt)

Kd = angular momentum of the front

frame

Kt = angular momentum of the rear

frame

KWd = angular moment of the front

wheels

KWt = angular moment of the rear

wheels

lz = zd position of front centre of mass

m = total motorcycle mass

MA = torque of aerodynamic forces

MAx, MAy, MAz = components of

aerodynamic torque

md = front mass

MGd = torques of gravity forces, front

frame

MGt = torques of gravity forces, rear

frame

MRd = torques of reaction forces,

front

MRt = torques of reaction forces, rear

mt = rear mass

MTz = twisting torque

Mx = overturning torque

Mxd, Myd, Mzd = torques on front

wheel

Mxt, Myt, Mzt = torques on rear wheel

My = rolling resistance torque

Mz = yaw torque

N = vertical force

Nd = front wheel load

Nt = rear wheel load

p = wheelbase

Pd = front tire contact point

Pt = rear tire contact point

Q = point on steering axis

R = circle radius

Rd = front wheel radii

RGd = path radius of front centre of

mass with respect to (X1, Y1, Z1)

RGt = path radius of rear centre of

mass with respect to (X1, Y1, Z1)

Rt = rear wheel radii

S = longitudinal force

Sd = longitudinal front tire force

sp = lateral deformation

St = longitudinal rear tire force

t = coordinate system (t)

td = front tire head radii

tt = rear tire head radii

V = forward speed

XGd, YGd, ZGd = coordinates of centre

of mass of front frame in (X1, Y1,

Z1)

XGt, YGt, ZGt = coordinates of centre

of mass of front frame in (X1, Y1,

Z1)

Motorcycle cornering behavior modeling [email protected] XPd = coordinates of Pd in (X1, Y1, Z1)

XPt = coordinates of Pt in (X1, Y1, Z1)

YPd = coordinates of Pd in (X1, Y1, Z1)

YPt = coordinates of Pt in (X1, Y1, Z1)

∆ = effective steering angle

δ = steering angle

ε = caster angle

λd = front tire side slip angle

λt = rear tire side slip angle

ρd = front tire centre-line radius

ρt = rear tire centre-line radius

φ = roll angle

Ψ = yaw angle

µ = pitch angle

µ f = rolling friction coefficient

ωd = front wheel spin rate

ωt = rear wheel spin rate

INTRODUCTION

The technical description for one vehicle "single track", as the motorcycle is called in the literature, it is tied to

single impression it leaves behind as it passes over the sand, for example. This peculiarity is the source of everything

that makes the study of the vehicle undeniably complex, and yet at the same time so fascinating.

Another factor is that the means of transport commonly used in day-to-day is so familiar that they are driven with

ease which can essentially be reduced to two vehicle categories, two and four wheels. The first category is the bicycles

and the motorcycles, which are equivalent in cinematic terms and the second the cars, which certainly is the most

studied vehicle today, with extensive bibliography.

A crucial consideration on these vehicles is that when a car is at rest, with or without passengers aboard, it remains

in stable equilibrium. However, a motorcycle upright tends to fall, unless a suitable support or supported by the rider.

A little observation brings to light some fundamental differences in the comparison of the two vehicles in motion:

An inexperienced person driving a motor vehicle, intuitively and quickly realized that when the steering wheel is

turned one direction, the vehicle is oriented in the same direction, so they can drive the car precisely in the direction

they want to go.

However, even an adult inevitably involves potential embarrassment and difficulty associated with attempting to

ride a bike for the first time - beginners are forced to put their feet on the ground, trying to maintain balance while

trying to keep the bike in the right direction. Initially, the bike is ridden supporting themselves with their feet, avoiding

a fall, but after some training, it appears that the faster the bike is conducted, the easier it is to keep it balanced.

Controlling a two-wheeled vehicle is, in fact, nothing simple and intuitive, but there is no doubt that the motorcycle

is a functional means of transport and it is also an exciting source of entertainment.

In the past, some studies were developed using single-track vehicles. Whipple (1899) studied the stability of motion

assuming bicycle with rigid tires. Sharp (1971) was among the first to investigate the stability of the motorcycle using

the tire properties. In 1980, Koenen published a stability study that caters to large lateral accelerations involving large

rolling angles. As the vehicle models became more complex with the interaction between the tire and the ground it was

necessary to develop more detailed tire models. Iffelsberger (1991), Wisselman et al. (1993), Breur (1998), Sharp et al.

(2001) and Berrita et al. (2000) produced works in this direction. In 1999 Cossalter published a work developing

nonlinear dynamic equations in steady state cornering.

Meijaard (2006) presented a single track model with a linear model of four rigid bodies, very close to the model

studied in this work, but the author decided that the tires have ideal contact with the ground (sharp edge). This model

was discarded since it does not slip angles.

The model developed in this paper was presented by Cossalter (1999). In it the motorcycle is modeled with the

nonlinear algebraic equations, considering the lateral and longitudinal slip of the driven wheel. The model presented is

valid for large values of motorcycle roll angle.

Motorcycle inertial and geometric properties, slip curves of the front and rear tire, the kinematic equations and

nonlinear algebraic equations were programmed using the Matlab. Like a motorcycle, the system input is the roll angle

and steering angle. The capacity of acceleration and braking of the motorcycle were discarded because the maneuver is

performed under steady state. The simulation results are represented by graphs where there are the values of angle of

cinematic steering, vertical and lateral force and lateral tire slip angle.

Model description

The motorcycle comprises a system of four rigid bodies: rear structure (including chassis, engine, the fuel tank and

rider), front structure (handlebars and fork) and front and rear wheel, as previously mentioned. The front and rear

structures are connected by a revolution joint. The front and rear wheels are connected respectively to the rear frame

and fork for revolution joints. The effect of front and rear suspension is not taken into account, since in a steady curve

the suspension deflection does not change. The rider is considered a rigid body securely attached to the rear structure.

The aerodynamic force distribution on the motorcycle is: drag, lift, lateral forces (acting at the center of mass of the rear

structure) and three torques.

D.Rafael, B. Roberto

The contact between the tire and the track is described by means of linking. If the wheel slips both in longitudinal

and in lateral, the restrains allow five degrees of freedom (two translational and three rotational). The lateral forces

exerted on the tires around the track are very important in the dynamic and steady state and they are related to slip angle

and roll angle. The front and rear tire side slip are described by λd and λt respectively. In relation to the longitudinal slip,

the front wheel does not slip, not producing longitudinal tire force as the rolling resistance effect was neglected, in

contrast with the rear wheel produces longitudinal tire force causing longitudinal slip (COSSALTER, 2006).

Three coordinate systems are introduced to describe the dynamic properties and kinematics of the vehicle.

Coordinate system t (xt, yt, zt,) as in Figure 1 is fixed to the structure and the rear plane xt, zt is the symmetry plane of

rear structure. When the vehicle is upright and the steering angle is zero, axis xt and yt are on plan horizontally and xt

points straight ahead, zt axis is vertical and points downward, the origin and the point Pt contact the rear wheel overlap.

Figure 1: Motorcycle t coordinate system, in upright position (a) and any position (b).

The coordinate system d (xd, yd, zd), as shown in Figure 2 is fixed on to front structure and it is described as follows:

the source is located at point Q, which is the point of intersection between the axis of rotation of the steering system and

the plane perpendicular to the axis of rotation direction, which passes through the center of the rear wheel axle zd and it

is aligned with the axis of rotation direction pointing downward; yd axis is parallel to the axis of rotation of the front

wheel; axis xd is in the plane of symmetry of the front structure.

Figure 2: Motorcycle geometry and d coordinate system.

Another coordinate system, according Figure 3, which is useful in the development of dynamic equations in steady

state is a rotating coordinate system 1 (X1, Y1, Z1). The source is located in the center of rotation of the motorcycle (C)

The Z1 axis is vertical and points downward (Z axis is parallel to the ground). The axis X1 is in the XY plane and parallel

to symmetry plane of rear structure. The Y1-axis completes the coordinate system.

Pt = A

zt = Z

yt = Y

xt = X

zt

yt

xt

A

µ < 0

Roll angle

φ

Yaw angle

ψPitch angle

µ

bt

Gt

ht

xd

ed

ε

td

Rd

ρd

zd

xtPt

zt

Rt

ρt

tt

Pd

Gd

a fork offset

a b

Motorcycle Cornering Behavior Modeling

Figure 3: Coordinate system 1 and main forces and momentum’s.

Steady state equation

In a steady state cornering, the speed of yaw, the roll, the steering angle and the slip (longitudinal in the rear wheel

and lateral in the rear and front wheels) are constants. Thus, the dynamic equations are composed of algebraic equations.

By Newton's 2nd

Law for the system of four rigid bodies:

�� + �� + �� + �� + �� = � (1)

−�� + � + � cos∆ − �� sin∆ = −�� +��Ψ�� (2)

�� sin� + �� cos � + �� cos∆ + �� + �� sin∆ = −�� +��Ψ� � (3)

−�� cos � + �� sin � + �� + �� + �� +�� = 0 (4)

The angular momentum equation around C for the system is:

� + �� + �� + !� + !� = "�� + "�� + "�#� + "�#� (5)

The angular momentum equation is expressed in the form of components in a coordinate system 1, as Cossalter

(1999, p.11).

�$ − �� cos � + %�� sin�� + ��−%�� cos � + �� sin�� + �� +�� + ��+ �� − &� sin∆ = �'()*� + '()*��Ψ� �

− +'#�,� cos � + '#�,�-cos . cos � − sin�/ + 0� sin . sin �12Ψ� (6)

�& cos � − �3 sin� + �� cos � − ��%�� − �� sin � − �� +�� − ��− �� + &� cos∆ + &� = −�'(4*� + '(4*��Ψ� � − I67Ψ� ω7 cos�µ+ ε� sin δ (7)

�3 cos � + �& sin� + �� cos � + �� sin � + �� + +�� cos∆ + �� sin∆�+ �� sin∆ − Y97 cos∆� + �� − �� + *� + *� = 0 (8)

It is necessary to calculate the inertia tensor for front and rear structure with respect to coordinate system 1 (X1, Y1,

Z1):

'(4*� = ��%�� + cos �-�'$� − '3�� cos / sin / + '4*��cos� / − sin� /�1 (9)

'(4*� = ��%�� + :'$� cos� . + '&� sin� . − '3�; cos�/ + 0� sin�/ + 0� cos �+ :'&� − '$�; cos�/ + 0� cos . sin . sin � (10)

X1

Y1

Z1

turning center point C

∆+λt-λd

ψ&turning radius R

λtλd

∆

Nd

Pd

Sd

Fd

ωdGd

mdg

Qsymmetry plane of front

frame

Symmetry plane of rear

framesteering

torquevertical plane

φ

ωt

Ft

St

Nt

Pt

V

mtg

FS

FL

FD

MAz

MAy

MAx

mtRGtψ².

mdRGdψ².

λt λd


'()*� = ��%�� + cos � sin ��'*� cos� / + 2'$3� cos / sin /� + cos � sin �:'$� sin� / − '&�; (11)

'()*� = ��%�� + cos � sin � �/ + 0�:'$� − '&�;�cos� � − sin� ��+ cos� sin � +'$�-cos� . sin��/ + 0� − sin� .1=+ '&�-sin� . sin��/ + 0� − cos� .1 + '3� cos��/ + 0�2 (12)

Tire modeling

The forces and moments produced by the tire as Cossalter (2006) are illustrated in the following figures: Figure 4a

forces acting on the intersection point between the plane of symmetry and the track in Figure 4b forces acting on the

point of tire contact Figure 4c and production of the moment Mx, My, Mz.

Figure 4: (a) Forces at the contact point, and the main moments. (b) Forces acting on intersection point between

the plane of symmetry and the track. (c) Tire contact point details.

Longitudinal front wheel slip is zero because the wheel is not driving. The longitudinal force is related to rolling

friction only, the longitudinal force on front tire is determined by:

� = −/>� (13)

The moment of rolling resistance is calculated by:

& = ?@� (14)

The torque Mx (overturning torque) is caused by lateral deformation of the tire sp.

$ = −A@� (15)

As Cossalter (1999) sp displacement is usually small due to high lateral stiffness of the tire, so the moment Mx is

zero. Torque Mz is produced by the lateral force F, longitudinal force (S > 0 propulsion, S < 0 braking) and MTz (twisting

torque):

3 = −��B�� − A@� + C3�� (16)

The first term due to lateral force tends to align the wheel in the direction of movement of the motorcycle. The offset

t (λ), whose distribution depends on the distribution of lateral force is called the tire trail (Figure 1). It is calculated as

the ratio between the torque Mz and lateral force and longitudinal force when the roll angle is zero, a good

approximation according to the experimental results (COSSALTER, 1999):

� = −DE F1 − H BBIJ$HK (17)

The second term of equation 16, because the longitudinal force, just tends to align the wheel if the longitudinal force

is tractive. As the displacement sp is usually very small, this term can be considered zero (COSSALTER, 1999). The

third term is the twisting torque, which arises due to the roll angle and tends to align the wheel. As Cossalter (1999)

assume a linear function based on experimental results, where M1 is 0.024 to front tire and 0.028 to rear tire.

yaw torque

overturning

torque

rolling resistence

torque

NS

F

contact

point

Z

Y

X

φ

P

MzMx

My

NS

F

contact pointY

X

Z

P

MTz

sp

P

NS

Y

X

Z

F

at

dp

a b c


C3 = L� (18)

Equation solving

The equations are nonlinear (due to the formulas of the tires and kinematic equations) and are solved numerically for

specific values assigned to roll angle and steering angle. First, the tire slips were set equal to zero and the equations

become a linear system of six equations with six unknowns: Nt, Nd, Ft, Fd, St, M� 2. After the first calculation, ignoring the side slip, values of normal and lateral forces are obtained and used for the

side slip angles of front and rear tire. With the slip obtained, the calculation is done again obtaining a new set of lateral

forces, vertical, propulsive force and angular velocity.

The equations were organized to solve the system of the form A.X = B each calculation step is defined by the range

of values attributed to steering and rolling and the value of the six unknowns was obtained.

NOOOOPLL L� LQ LR LS LTULL UL� ULQ ULR ULS ULTVLL VL� VLQ VLR VLS VLT?LL ?L� ?LQ ?LR ?LS ?LTWLL WL� WLQ WLR WLS WLTXLL XL� WLQ XLR XLS XLT Y

ZZZZ[

NOOOOP \�� YZZZZ[=

NOOOOP��]]^^__`` YZZZZ[ (19)

Due to lack of data related to the tires (the parameters are normally confidential and not published by manufacturers),

the values for forces and side slips were obtained directly from the curve of the tire thus decreasing the error in the

calculation of the forces produced by the tire. Using the Pacejka Magic Formula (2002) these curves were obtained in

tire test equipment and the result is found in Cossalter (2008). The test result allows composing curves as a function of

lateral slip, normalized lateral force (lateral force / vertical force on the tire) and camber angle. These curves have been

programmed along with the other equations. The curves related to the front and rear tire are shown in Figure 5.

Figure 5: (a) front tire side slip curve, (b) rear tire side slip curve

MATERIALS AND METHODS

Aiming to validate the mathematical model used in this study, tests were done using a Suzuki Bandit N650

motorcycle in stock configuration. These validations enabled also to check whether the simplifications of the

mathematical model used are satisfactory.

The center of mass of the motorcycle was obtained using a load cell on each wheel and a signal conditioner (Figure

6a). The methodology used was proposed by Milliken and Milliken (1995, p. 669) to get the height and longitudinal

position of center of mass motorcycle. The motorcycle was leaned over load cells with an angle of 27.3°. This angle

serves the recommendation of Reimpell, Stoll and Betzler (2001, p.390) reducing the calculation error. The inertia

moments used were not measured due to the difficulty in disassembling the motorcycle. The values were obtained from

Cossalter (1999), because, the motorcycle has similar mass and inertia characteristics.

For the motorcycle speed, an inductive sensor from AIM in each motorcycle wheel was used. This sensor captures

the transition metal in the face of the sensor without contact (Figure 6a and b).

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 4 8 12 16 20 24 28 32 36 40 44 48

no

rmal

ize

d l

ate

ral

forc

e

camber angle [°]

front tire - 120/70R17

-1,0°

-0,5°

0°

0,5°

1,0°

side slip angle [°]

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0 4 8 12 16 20 24 28 32 36 40 44 48

no

rmal

ize

d t

late

ral

forc

e

camber angle[°]

rear tire - 180/50-R17

-1,0°

-0,5°

0°

0,5°

1,0°

side slip angle [°]a b

Figure 6: (a) center of mass, (

The steering angle was measured using a potentiometer installed between the front structure and the steering system

and interconnected with pulleys and belt (Figure 7

AIM feature for calculating the radius of curvature (feature already implemented in the

the module and GPS antenna on the motorcycle fuel tank

and closing the lap was defined. An infrared sensor

transmitter installed on the test track (figure 7

Figure 7: (left) steering angle installed, (

The signal acquisition system chosen was the

motorsport. This system was chosen because it has good performance, expandability; possess all the functions of a

signal conditioner and its own software for analysis and storage of data with graphical interface.

installed in the center of mass of the motorcycle (Figure

equipment to obtain longitudinal and lateral acceleration for further studies. The equipment normally operates at a

temperature zone which was installed as in

Figure 8: AIM installed in motorcycle center of mass.

To measure the motorcycle roll angle was used an articulated arm

ball joints. At the other end of the articulated

caster), ensuring contact between the tire and the ground during the circular path. There is a LVDT (linear variable

displacement transducer) in the arm that provides the location

roll angle of the motorcycle.

Detailed roll angle meter device (Figure 9

1) ø150 mm caster, with self directional system;

2) 2x M8 screw;

3) 20x20 mm steel bar with 2 mm thickness;

4) Linking bracket

5) LVDT, Penny and Giles, MLS 130/150/R/N.

ba

a


) center of mass, (b) rear wheel speed sensor, (c) front wheel speed sensor


(Figure 7a). With an objective to verify the motorcycle trajectory and use the

feature for calculating the radius of curvature (feature already implemented in the AIM) it was necessary to install

antenna on the motorcycle fuel tank (figure 7b). To control all the experiment a point of opening

n infrared sensor was installed on the motorcycle which receives the signal from the

(figure 7c).

) steering angle installed, (center) GPS antenna, (right) lap sensor.

The signal acquisition system chosen was the AIM Evo3 Lane of Italian manufacturing and used in professional

his system was chosen because it has good performance, expandability; possess all the functions of a

signal conditioner and its own software for analysis and storage of data with graphical interface. The AIM module was

e motorcycle (Figure 8) with the goal of using inertial accelerometers built into


in the installation manual.

: AIM installed in motorcycle center of mass.

To measure the motorcycle roll angle was used an articulated arm (figure 9a) fixed to the motorcycle structure with

he other end of the articulated arm there is a tire with directional system of its own (commercially called


displacement transducer) in the arm that provides the location of the bar against the motorcycle, thus determining the

Detailed roll angle meter device (Figure 9b):

ø150 mm caster, with self directional system;

20x20 mm steel bar with 2 mm thickness;

nny and Giles, MLS 130/150/R/N.

c

b c


) front wheel speed sensor


to verify the motorcycle trajectory and use the

) it was necessary to install

. To control all the experiment a point of opening

receives the signal from the

) lap sensor.

of Italian manufacturing and used in professional

his system was chosen because it has good performance, expandability; possess all the functions of a

The AIM module was

) with the goal of using inertial accelerometers built into


fixed to the motorcycle structure with

(commercially called


of the bar against the motorcycle, thus determining the


Figure 9: (a) motorcycle fully instrumented with roll meter bar, (b) roll meter device detailed.

The system was calibrated with the aid of a digital angle meter fixed to the front frame of the motorcycle,

determining each sign for each roll angle of the motorcycle. The calibration curve is shown in Figure 10:

Figure 10: LVDT Calibration curve.

A table was created with the roll angle values by the input signal in signal conditioner (AIM). This curve was

approximated by a polynomial of third degree, and used the function obtained in the visualization software of acquired

data.

The entire test was performed at FEI University campus. Cones were used for marking the track trajectory. The rider

was instructed not to move the body during the motorcycle rolling, thereby ensuring the initial condition of the model

proposed (rider fixed to motorcycle rear structure). For space limitation reasons, the curvature radius of 20 m were used

and varied the speed until the limit which was necessary to ensure the motorcycle balance.

Figure 11: Test performing at FEI campus.

y = -0,0001x3 - 0,001x2 - 0,3617x + 1,8372

-5

0

5

10

15

20

25

30

35

40

45

-60 -50 -40 -30 -20 -10 0 10

roll

an

gle

[]̊

LVDT signal

Roll device calibration curve

Polynomial

Polinômio (Polynomial)

ba


The test was performed on two different days, the first with 14 laps and second with 10 laps. The first day was vital

for the decision to create the acquisition arm for roll angle measurement. It was discovered that the accelerometers and

gyroscopic inside the signal acquisition module were not efficient to manage the values tested (40°). The second day of

testing was characterized by the arm implementation and partial tests to check whether the system was safe for the rider

and whether the arm had sufficient bending stiffness to avoid damage to the LVDT and ensure results accuracy.

RESULTS

Simulation results

The simulation results are presented in graphs where the x-axis represents the motorcycle roll angle and y-axis

represents the steering angle. The curves in full lines represent speed curves of the motorcycle and the dotted line

represents the radius of curvature. The graph colors represent a simulated variable (effective steering angle, pitch angle,

vertical force on the front and rear tire, lateral force on the front and rear tire, slip angle of the front and rear tire) and

the results are provided by the variation of colors and with the descriptions given by the top bar or side bar of the chart.

As shown in Figure 12, it appears that even for small values of steering angle δ there is the front tire slip represented

by the color gradient and scale at the top of the chart, thus producing lateral force needed to keep the motorcycle in

circular path. With constant motorcycle speed by 40 km/h, there is between point K and L a variation of the radius of

curvature of the motorcycle 20-70 m. In point L, it has a 30° roll angle, 3.4° of steering angle and a side slip angle of

0.7°. Point K, it has a 9° roll angle, 1.2° of steering angle and a slip angle of tire sidewall of 0.25°. It appears that at the

point K, due to the large radius of curvature the motorcycle tire rolling is roughly as cinematic, causing a smaller slip

angle side compared with that determined in point L.

Figure 12: Front tire side slip angle variation.

Experimental results

Initially the accuracy of data collected in the test was confirmed. With the aid of GPS, the circular path of the

motorcycle was found. Two tools were used to do the verification: the first was overlapping the coordinates exporting

the GPS data and importing it into Google Earth (Figure 13a) and checking even the circular path than maneuvers

diameter (40 m).;s

φ ângulo de rolagem[º]

δ â

ngulo

de e

ste

rço[º

]

ângulo de escorregamento lateral do pneu dianteiro [°]

10

20 30

40

50

60

100

80

60

50

40

30

20

5 10 15 20 25 30 35 401

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0.2 0.4 0.6 0.8 1 1.2

velocidade [km/h]

raio de giro [m]

front tire side slip angle [0]

δsteeringangle[0]

Φ roll angle [0]

speed [km/h]

turning radius [m]

L

K


Figure 13: (a) Trajectory verification using the GPS output, (b) satellite orientation.

Another tool used was the motorcycle satellite orientation (figure 13b). There are variations between 0 and 180 ̊

linear, confirming the quality of the circular path.

The test results on the 14th

lap are shown at figure 14. The horizontal axis of the graph represents the extension of

the lap (about 130 m). The vertical axes represents from left to right: speed, steering angle and roll angle. The vertical

bar identifies the time when the data was chosen. The red line represents the steering angle in degrees (3.02°), the blue

line represents the motorcycle roll angle (34.36°) and green to the front wheel speed (42.5 km/h).

Figure 14: Experimental test results.

The figures represented above were overlaid with the graphs presented in the result simulation graphs and shown in

Figure 15. First crossed 3° steering angle with 35° of roll angle on the graph in Figure 15. This overlap allows two

important parameters, the speed of the motorcycle and the radius of curvature. Observe that the intersection radius of

curvature is equal to 20 m theoretical and speed 45 km/h. There was a small deviation in the value of speed obtained in

the practical test (42.5 km/h). This deviation was considered acceptable for low speed involved in the test.

Figure 15: Experimental results overlay.

Model modification

A constant need for vehicle designers is to understand the behavior of the vehicle during the design phase, avoiding

undesirable reactions or functions. This prior knowledge is only possible using computer simulations, or prototype

construction, the latter usually very expensive and time consuming.

roll angle [o] 34.25

steering angle [o] 3.6

front wheel speed [km/h] 41.8


δ â

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]

10

20

30

40

50

60

7060

50

40

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20

∆ â

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efe

tivo d

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]

5 10 15 20 25 30 35 401

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1

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velocidade [km/h]

raio de giro [m]

45 km/h

42.5 km/h

speed [km/h]

turning radius [m]

Φ roll angle [0]

δsteering angle [0]

∆effective steering angle [0]

ba


Another major function of the mathematical models is to obtain improvements of the system to certain reactions. In

the specific case of this work was chosen to alter the tires characteristics and check the consequences of this in the

motorcycle directional behavior. This parameter was chosen because, among the possible parameters to change in a

motorcycle the tire is the simplest parameter to be changed (from the manufacturer, for example). The other parameters

(geometric, for example) require changes in the mechanical characteristics of the motorcycle.

The first change was to increase by 10% the camber stiffness coefficient of the front tire, impacting on adherence of

the tire to the ground; the second change was to reduce by 10% the camber stiffness coefficient of the front tire. The

camber stiffness coefficient directly influences the lateral force produced by the tire.

In Figure 16, at point A on a curve with a speed of 40 km/h and curvature radius of 30 m the motorcycle must be

with a steering angle of 2.5° and 22.0° roll angle. Also to do a curve with a speed of 60 km/h and radius of curvature of

40 m (B) requires a steering angle of 1.5° and roll angle of 34.0°. The speed and roll angle will be presented as a

reference for future analysis. The motorcycle is equipped with tires in the "standard" configuration. The difference

between the side slip of the tires provides the directional behavior of the motorcycle. It is also observed that the

motorcycle has under steer behavior in under roll angles below 7° as indicated in the chart (red triangular area on the

bottom left graph) and over steer above this value, characterizing the motorcycle as over steer most use.

Figure 16: Variation of the difference between the front and rear tire side slip (front and rear tire "standard").

In Figure 17, the motorcycle front tire with camber stiffness coefficient increased by 10% and keeping the rear tire

in the configuration standard and using the speed and radius of curvature as previously proposed (40 km/h ; 30 m 60

km/h ; 40 m) the motorcycle meets the condition of equilibrium with a steering angle of 2.3° and roll angle 22° to the

first condition (A) and 1.2° of steering angle with 34° roll angle. Note a reduction in the steering angle required to meet

the speed and maneuvering within the radius of curvature proposed, because the increase of camber stiffness coefficient,

the tire provides greater lateral force compared with the tire in "standard" configuration.


δ â

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]

comportamento direcional λtraseiro - λdianteiro

10 20

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100 9080 70

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velocidade [km/h]

raio de giro [m]

A

B

speed [km/h]

turning radius [m]

directional behavior λt-λd

δsteeringangle[0]

Φ roll angle [0]

understeer


Figure 17: Variation of the difference between the front and rear tire side slip (front tire with the camber stiffness

coefficient increased by 10% and rear standard).

In Figure 18, used the velocity and radius of curvature (40 km/h ; 30 m and 60 km/h ; 40 m), but the front tire had a

camber stiffness coefficient reduced by 10% in the front tire and remained the rear tire to the standard configuration, the

motorcycle meets the condition of equilibrium with a steering angle 2.7° and 22° of roll angle for the first hypothesis (A)

and 1.7° angle of steering with 34° roll angle for the second hypothesis (B). In this configuration, the steering angle are

greater in both situations, because the configuration of the tire provides less lateral force than the configuration used in

"standard", requiring a steering angle increase.

Figure 18: Variation of the difference between the front and rear tire side slip (front tire with the camber stiffness

coefficient decreased by 10% and rear standard).

As result of the simulation there is a large zone that the motorcycle have neutral directional behavior (λt-λd = 0),

featuring a motorcycle with directional behavior safer in steady state cornering. This zone can be observed in the area

outlined in red dashed line in Figure 18, this area includes the regime of common use of motorcycles.

CONCLUSION

This work allowed interacting directly with a multibody model of a motorcycle in a steady state curve. This model is

useful in the development phase of a motorcycle anticipating directional characteristics and behaviors reducing

development time and prototype test vehicles, which are usually expensive and require long construction time.

The graphical results allowed a unified view of the simulated variation parameters versus the input parameters

(steering angle and roll angle) and speed of the motorcycle and radius of curvature.


δ â

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]


10

15

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35

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45

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8070

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5 10 15 20 25 30 35 401

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

velocidade [km/h]

raio de giro [m]

A

B

speed [km/h]

turning radius [m]

δsteeringangle[0]

Φ roll angle [0]



δ â

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ulo

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]


10

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70100 90

8070

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5 10 15 20 25 30 35 401

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-0.1 0 0.1 0.2 0.3 0.4

velocidade [km/h]

raio de giro [m]

A

B

δsteeringangle[0]

Φ roll angle [0]


speed [km/h]

turning radius [m]


The mathematical model has consistent appropriate results for the proposed changes, as theory suggested by the

references used in this work.

There was also a need to adapt the automotive dedicated data acquisition equipment to use on motorcycles, because the

high angles of roll (about 40°) present a problem for conventional signal acquisition "hardware" used in automotive

applications, where angles are no bigger than 7°. The construction of a specific device for reading the roll angle of the

motorcycle was of vital importance to the experiment and made the data acquisition system more flexible as to the

applicability. Even the system added to a point of contact between the motorcycle and to the ground due to contact of

rotation on the ground which did not influence the lateral dynamics of the motorcycle and so little on results.

The results of the mathematical model are compatible with the experimental results, for the range of speeds and radii

of curvature.

The study confirmed the different directional motorcycle behavior to vary the camber stiffness coefficient to the

front tire. The directional behavior is an important feature in the development of a motorcycle making it safer and easier

the riding.

ACKNOWLEDGMENTS

To my beloved parents, my wife and Dr. Agenor de Toledo Fleury.


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Motorcycle cornering behavior modeling - Paper

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