Motorbike Dynamics and Simulation

This assignment delves into the dynamic modeling of a single-degree-of-freedom motorbike system. It begins by outlining the equation of motion and reaction force for the motorbike, followed by a Laplace transformation to represent the system in the frequency domain. The focus then shifts to Simulink simulations, where various riding techniques and speeds are tested to validate the model's accuracy. The assignment highlights the advantages of frequency domain analysis over time domain analysis and emphasizes the importance of feedback control in achieving stability for the motorbike system. Finally, it suggests incorporating functional design principles inspired by mass-spring-damper systems for improved performance.

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ENEM14015 Dynamic System Modelling and Control
Project 2
Single Degree of Freedom Systems
Student Name
Professor (Tutor)

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Introduction
The motorbike is a two-wheeled automobile with rolling motion and gyroscopic moments.
Motorbikes are a convenient mode of transport and they are environmentally friendly. The
concept of motorbikes modelling and stabilization has been widely researched (Cossallter, 2006).
(i) System data
Parameter Data
Bike mass (without rider) 0.2 kg
Bike mass (with rider) 0.38kg
Seat height 50cm
Fuel capacity 1000cc
Front wheel diameter 45cm
Rear wheel diameter 45cm
Wheel masses 0.15kg
Front suspension travel, spring and damper rates 0.003456
Rear suspension travel, spring and damper rates 0.004523
(ii) Assumptions
It is assumed that the mass of the driver not only acts at the steering but also
displacing his body and it is a substantial fraction of the total mass. It is similarly
Σ
1−
? Frame
Frontfork
T
?
δ ϕ

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assumed that the two-wheeled vehicle is intrinsically unstable. The driver has to
perform as a stabilizer for the capsize mode. The body of the driver acts as an
aerodynamic brake or it can be a control surface that contributes in a substantial way
to the aerodynamic forces.
Analysis of a Motorbike as a Dynamic System with a single degree of Freedom
(i) Mathematical model schematic of the motorbike
The key concept behind the construction and system modelling of motorbikes is the
idea of stabilization. Some linearized momenta are developed around the tilt
dynamics about the ζ axis as shown in the equation below,
J d2 φ
d t2 =mgιφ+ιF
The stabilization concept is borrowed from the inverted pendulum model.
2P1P
O
ϕ
δ
b
a
ζ
η
η
ξ

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The lateral force, F, is obtained as shown in the kinematics equation,
F=m ( V 0
2
r + d V y
dt )=m¿
(ii) 1 DOF free body diagram and modelling equation
The modelling equation is obtained using the Lagrangian approach. The first step of
the Lagrangian approach requires that the free Lagrangian computes the total kinetic
energy of the motorbike. This accounts for all movements caused by any moving
component such as the wheels, frames, and the engine. The translational and
rotational kinetic energies are given as shown in the equations below,
T t= 1
2 m V G
2 … .translational KE
T r= 1
2 {Ωx
Ωy
Ωz
}T
[ J x 0 J xz
0 J y 0
J zz 0 J z ] {Ωx
Ωy
Ωz
}
As a result, the sum total kinetic energy of the steering system owing to the
movement of the motorbike is given as,
T r 1= 1
2 Ω1
T J1 Ω1
λ
3P2P1P
2C
1C
cb
a
h

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To determine the kinetic energy of a given wheel, i, the locked controls motion is
given by,
T ri=1
2 Ωwi
T J wi Ωwi
(iii) Free vibration response
The roll dynamics of a motorbike correspond to those of an inverted pendulum with
an acceleration influence applied at the vehicle’s base and are given by the following
equation,
Furthermore, there are four more equations that describe the lateral free vibration
response of the motorbike in motion when tested under different environments.

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(iv) Forced vibration
The tests on this section impact on the elasticity of carcass and the static behavior of
the tire (Genta & Morello, 2009). The vertical force N and horizontal force F is given
as shown in the following set of equations.
As a result, a relationship is developed between the longitudinal force, S and other
forces as describe by the equations above. The deformation of the resulting effect is,
(v) Road surface induced vibration (road speed of up to 100kph)

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When riding the motorbike, one is bound to encounter frictional forces on the road.
The rider may pull of different stunts were the motorbike is no longer in a
perpendicular position to the road surface but at an angle less than 900. The
longitudinal slip is the ratio between the slip velocity and the forward velocity (Lot,
2004). It is given as,
The slip angle is given as,
And the sideslip angle is calculated using the wheel kinematics as,
The illustration below describes the idea of the wheel when a rider is riding at high
speed and still about to negotiate a bend (Corno & Savaresi, 2010).
(vi) Magnification factor and transmissibility factor plots

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(vii) Laplace transform of the modelling equation
The modelling of a motorbike’s single degree of freedom follows the same
computation as that of simulating an inverted pendulum. The inverted pendulum
operates on the following system equations,
(a) The equation of motion
M ¨x +b ˙x + N=F … … ..(i)
(b) The expression for the reaction force
N=M ¨x+ M l ¨θ cosθ−ml ˙θ2 sinθ … .(ii)
(c) The resulting equation
( M +m ) ¨x +b ˙x +ml ¨θ cosθ−ml ˙θ2 sinθ=F

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(d) The Laplace transformation
( I +ml2 ) ¨ϕ−mglϕ=ml ¨x
( M +m ) ¨x +b ˙x−ml ¨ϕ=u
The Laplace transform of the equations above is,
( I +ml2 ) Φ ( s ) s2−mgl Φ (s)=ml X ( s ) s2
( M +m ) X ( s ) s2 +b X ( s ) s2−ml Φ ( s ) s2 =U (s)
(viii) System model in Simulink
The models developed for the motorbike as a dynamic system were simulated in
Matlab. The complete model included the suspension and pneumatic shape effect.
The Simulink performed tests where the motorbike was tested for different speeds
and on different riding techniques. The main characteristics of the simulation results
seemed to remain the same as the conventional motorbikes. The dynamics model was
validated using the control analysis. A number of tests were performed such as the
spectrum of the simulation that understood that different torques were applied by the
driver for different starting conditions and riding environment (Lot, 2004).
Discussion
(i) Frequency domain analysis vs. time domain
The relationship between frequency and time is as shown in the equation below.
f = 1
T
Signals and wave are said to run on periods. The horizontal axis of the waveform
depicts the time range as the wave traverses. A period refers to the time taken for a
wave to repeat itself. It is closely related to frequency as shown in the equation above.
There are two ways of describing a signal, that is, time domain or frequency domain.
In practice, the frequency domain is much more preferred than the time domain.
Frequency domain is derived from time domain analysis using the Fourier series and

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Fourier transform techniques. The frequency domain gains precedence to the time
domain in practice as a result of the following merits,
(i) It is used to determine the behavior of the linear time invariant systems and
formulate transfer functions which are difficult to achieve in time domain
(ii) It is easy to determine the corrective measurement for noises created within a
system due to parameter variation.
(iii) It is possible to determine the absolute and relative stability of the closed loop
system.
(iv) The frequency domain can establish the transfer functions and techniques for
control in nonlinear systems.
(ii) Design analysis and critic
The system is unstable and it is not capable of maintaining the equilibrium position
for the ride environment. The system is improved by setting up a feedback control on
the one side to eradicate the difference between the torque and also due to the
centrifugal effect and the effect of the gyroscope when the torque is applied.
Designers are yet to realize an important yet ubiquitous concept of using functional
design in modelling dynamic systems. They can benchmark on such kind of design
from the mass-spring-damper system design. The mass-spring-damper system
working principle is as illustrated by the mathematical equations below,