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 Donadio 1 Roberto Bortolussi 1 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 (X 1 , Y 1 , Z 1 ) = rotating coordinate system (1) (x d , y d , z d ) = front reference coordinate system (x t , y t , z t ) = rear reference coordinate system a = mechanical trail A = origin of coordinate system (t) a g = centre of mass acceleration a n = front wheel normal trail a t = tire trail b t = longitudinal position of rear centre of mass C = turning centre point d = coordinate system (d) d p = forward displacement of the tire contact point e d = eccentricity of front centre of mass E d = front tire longitudinal force E t = rear tire longitudinal force F = lateral force F A = aerodynamic force on the rear frame F D = aerodynamic drag force F d = lateral front tire force F Gd = gravity forces on the front frame F Gt = gravity forces on the rear frame F L = aerodynamic lift force F Pd = road reaction, front F Pt = road reaction, rear F S = aerodynamic side force F t = lateral rear tire force g = acceleration due to gravity G d = front center of mass G t = rear center of mass h t = height of rear centre of mass I CXZd , I CYZd = components of inertia tensor of front frame with respect to (X 1 , Y 1 , Z 1 ) I CXZt , I CYZt = components of inertia tensor of rear frame with respect to (X 1 , Y 1 , Z 1 ) I wd = front wheel inertia I wt = rear wheel inertia I xd , I yd , I zd = components of inertia tensor of front frame with respect to (x f , y f , z f ) I xt , I yt , I zt = components of inertia tensor of rear frame with respect to (x t , y t , z t ) K d = angular momentum of the front frame K t = angular momentum of the rear frame K Wd = angular moment of the front wheels K Wt = angular moment of the rear wheels l z = z d position of front centre of mass m = total motorcycle mass M A = torque of aerodynamic forces M Ax , M Ay , M Az = components of aerodynamic torque m d = front mass M Gd = torques of gravity forces, front frame M Gt = torques of gravity forces, rear frame M Rd = torques of reaction forces, front M Rt = torques of reaction forces, rear m t = rear mass M Tz = twisting torque Mx = overturning torque M xd , M yd , M zd = torques on front wheel M xt , M yt , M zt = torques on rear wheel M y = rolling resistance torque M z = yaw torque N = vertical force N d = front wheel load N t = rear wheel load p = wheelbase P d = front tire contact point P t = rear tire contact point Q = point on steering axis R = circle radius R d = front wheel radii R Gd = path radius of front centre of mass with respect to (X 1 , Y 1 , Z 1 ) R Gt = path radius of rear centre of mass with respect to (X 1 , Y 1 , Z 1 ) R t = rear wheel radii S = longitudinal force S d = longitudinal front tire force s p = lateral deformation S t = longitudinal rear tire force t = coordinate system (t) t d = front tire head radii t t = rear tire head radii V = forward speed X Gd , Y Gd , Z Gd = coordinates of centre of mass of front frame in (X 1 , Y 1 , Z 1 ) X Gt , Y Gt , Z Gt = coordinates of centre of mass of front frame in (X 1 , Y 1 , Z 1 )
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.
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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.
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
Motorcycle Cornering Behavior Modeling
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.