13-09-2014, 12:54 PM
KING FAHD UNIVERSITY OF PETROLUM AND MINERALS
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Abstract
In this article a simplified equation of motion (1 "def") describing the bicycle
dynamics has been developed using Lagrangian approach. After defining the
required parameters and determining the velocity components, the potential
and kinetic energy expressions have been constructed and then inserted into
Lagrange equation
Introduction
A bicycle is the most efficient vehicle ever designed. Its cost, in
terms of energy spent in carrying a comparable load over a
comparable distance, is one-tenth that of the most efficient jet
aircraft and about one-twentieth of the best automobile.
The humble bicycle has a glorious past. First conceived as a
plaything of the rich, it soon evolved into an efficient and
convenient means of transport. The coming of the automobile,
however, relegated it to a role of an exerciser or a
sportsmachine, though in large parts of the world, notably
China and Southeast Asia, it is still used as the primary means
of daily transportation. In the industrialized world it seems to
have come a full circle. The bicycle is re-emerging as a vehicle
of choice for short runs in urban areas. It does not pollute the
atmosphere. It is almost noiseless
Literature review
The bicycle was invented in the middle of the 19th century [1].
Ever since its development, Engineers, Mathematician, and
physicists have been attempting to predict or model the
handling behavior using mathematics. In many areas of
engineering, mathematical modeling is regarded as a valuable
and practical design tool. Unfortunately, the mathematical
description of bicycle handling is a challenging task, even if
simplifying assumptions are made. [3]
With some success, this task has been assumed by number of
researchers throughout cycling history. For instance, accurate
equations describing the bicycle motion were derived as early
as 1899 by Wipple. However, no one has translated
mathematical descriptions into practical design rules
Bicycle Geometry
The parameters that describe the geometry of a bicycle are
defined in Figure 1. The key parameters are wheelbase b, head
angle λ, and trail c. The front fork is angled and shaped so that the
contact point of the front wheel with the road is behind the
extension of the steer axis. Trail is defined as the horizontal
distance c between the contact point and the steer axis when the
bicycle is upright with zero steer angle. The riding properties of the
bicycle are strongly affected by the trail. In particular, a large trail
improves stability but makes steering less agile. Typical values for
c range 3-8 cm
System description
The analyzed system consists of a bicycle-rider system like the
one shown in Figure 2. It consists of a front frame and the front
wheel. The rear section consists of the rear frame assembly, the
rear wheel, and the rider.
There are several parameters that affect a bicycle-rider system
behavior. In order to determine those parameters that were
needed to restraint the lean angle of the system, a reviewing of
some models available in the literature [1, 3, 4, 6] was made. The
models show different complexity, according to the amount of
parameters included in each system. However, it was found that
two variables have the most effect on the bicycle-rider lean angle
namely the steering angle/steering torque, and the rider’s lean
angle (riding without hands [9]).
Fajans (1999) agrees that these two parameters can be
considered as independent inputs (parameters) when it comes to
restraining or controlling the lean angle because either one of them
can be used to control the bicycle without including the other [9].
Thus, either one of them could be included in the model in order to
control the lean angle of the system.
Model Assumptions
In order to keep the order of the system low, and enable us to see
the affects that the front frame steering angle has on the system
directly, we will model the bicycle and rider as a single rigid body.
The rear velocity will be assumed to be constant and the wheels
are assumed to be in pure rolling contact with the ground with no
sideslip. The innate stability characteristics associated with an
offset angled front fork and the rotating wheels will be neglected.
Successful control and maneuvering of a bicycle depend critically
on the forces between the wheels and the ground. Acceleration
and braking require longitudinal forces, whereas balancing and
turning depend on lateral forces. A good understanding of these
forces is necessary to make appropriate assumptions about valid
models of the rolling conditions. The following list details the
assumptions made in the development of the equations of motion
of the basic bicycle model.