Term Paper Classical[1]

8/3/2019 Term Paper Classical[1]

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TABLE OF CONTENTS

Table of Figures (ii)

Abstract (1)

Introduction (1)

Bicycle Geometry (2-5)

Roll angle (5-7)

Gyroscopic effect (8-9)

Trail (10-11)

Frictional forces (11-13)

General equation of motion (13)

The Model (13)

Model assumption (13-14)

Table of data (14)

Simulation (14-15)

Results (16-21)

Discussions (22)

Conclusion (22)

References (23)

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TABLE OF FIGURES

Figure

No

Title of figures Page

1 Bicycle geometry 1

2 Turning geometry of bicycle 23 Yaw motion of bike 4

4 Rear view of project showing roll angle 5

5 Rear view of bike showing sideways accelerations 5

6 Free Body Diagram and Mass Acceleration Diagram

of rolled bicycle

6

7 Bike in a steady circle 7

8 Change in angular momentum of front wheel with a

roll deflection

8

9 Effect of trail 10

10 Drop in CG with steering angle and roll 10

11 Frictional forces on tires 11

12 Moment of FB about steering axis 1213 The Simulink Model 15

14 Dependence of angle of lean (above) and steering

angle (below) on time at a speed of 1.5m/s for zero

gyroscopy

16


angle (below) on time at a speed of 6m/s for zerogyroscopy

17


angle (below) on time at a speed of 1.5m/s for with

gyroscopic effect

18


angle (below) on time at a speed of 6m/s with

gyroscopic effect

19


angle (below) on time at a speed of 6m/s for lowtrail

20


angle (below) on time at a speed of 6m/s for high

trail

21

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ABSTRACT

The purpose of this paper is to examine the dynamics of a bicycle as it affects stability,based on previous work especially as presented by Lowell and Mckell in their 1982 paper

on Stability of bicycles, and to assess their relative importance with the help ofmeasurement of real bicycle. The paper examine the dynamics of the bicycle based onkinetic and kinematic consideration, and treated the bicycle as a simple model.

This work focuses on the fact that the appreciable angular momentum of the front wheelincreases the steering angle when the rolling (lean) angle increases. If the wheels wererolling alone, this increase would ensure its stability if the speed were large enough.

This paper concluded that of several factors that govern the stability of bicycles, the Trailis the most important factor.

INTRODUCTION

Bicycle is a single track vehicle and is fundamentally different and more difficult to studythan other wheeled vehicles. Bicycles lack lateral stability when stationary, and onlyremain upright when in motion (1). Experimentation and mathematical analysis haveshown that a bicycle stays upright when it is steered to keep its centre of mass over itswheels (2). The steering is usually supplied by the rider or in certain circumstances by thebicycle itself (3).

The control of the bicycle is a rich problem offering a number of considerable challengesof current research interest in the area of mechanics and robot control (4). To better satisfygeneral curiosity about bicycle balance and perhaps contribute to the evolution of bicycledesign, the aim of this work is to settle some basic, and largely previously presentedbicycle stability science. The core of this paper is to set and easy-to-use and thoroughlychecked linearized dynamics equations of a well defined simplified model.

Lateral dynamics has proven to be very complicated, requiring three-dimensional, multi-body dynamic analysis with at least two generalized coordinate to analyze (5). At aminimum, two coupled second-order differential equations are required to capture the

principal motions. Exact solutions are not possible. Hence, numerical methods must beused instead (6).

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BICYCLE GEOMETRY

Figure 1 shows the geometry of the bicycle with important parameters indicated.

ABC

x

y

z

a

b

h

Figure 1 Bicycle geometry

The bicycle has a coordinate system with its origin at the contact point of the rear wheel(point A). The x axis passes through the contact point of the front wheel. The z axis is

vertical, leading up from point A. The y axis leads to the right side of the bicycle.

The main parameters that govern the bikes dynamics are the wheelbase, a, The height ofthe center of mass above the ground, h, the distance of the center of mass forward of theorigin, b, the head-tube angle, , and the trail, . Note that B is the contact point of thefront wheel on the ground and point C is the point where the steering axis intersects theground. As we shall see, trail is very important for stability. Point C must be ahead ofpoint B for a stable bicycle, a bicycle that wants to return upright after it has been rolledoff the vertical plane.

In Figure 2 the bicycle is shown from

above. The bike is proceedinggenerally to the left with a velocity v.The front fork is deflected through anangle . In this figure this angle isexaggerated for illustration purposes.If the tires track true, the velocity ofthe frame is in the x direction and thevelocity of the front fork is in

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Figure 3 shows the accelerations actingon the mass center of the bike due toroll acceleration, yaw acceleration, andyaw velocity. These accelerations are

all substantially perpendicular to thebicycle frame. They are:

a - Roll acceleration. If the angularroll velocity is increasing ordecreasing, a 0. The doublederivative signifies that this is atangential acceleration associated with .

a - Normal acceleration toward

center of turn. At any instant the bikeis yawing about the instant center.Even if the yaw rate, , is constantand the bike is traveling in a circularpath, this acceleration will exist.Normal acceleration is alwaysassociated with the direction change ofthe velocity.

a - Yaw acceleration. If the yawrate, , is not constant, then 0.

This means that the curve followed bythe bike is not circular but tightens orloosens. The double derivativesignifies that this is a tangentialacceleration associated with .

a and a are perpendicular to the

bike frame, so in the y direction. a isdirected toward the instant center. Butas can be seen in the drawing, it is a

good approximation to assume it is alsocollinear with the other twoaccelerations.

B A

y

x

z

R

R

/c

os

I n s t a n tC e n t e r

a

b

vv / c o

s

a

a

a

Figure 3 Yaw motion of bike

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A

xy

z , ,

Figure 4 Rear view of

project showing roll angle

Figure 4 shows the bicycle roll angle, velocity, andacceleration. Notice that the roll angle represents anegative rotation about the x axis.

Roll Angle

Lets look at the stability of the bicycle from anupright, straight-ahead path with velocity v.Figure 5 shows the bike from the back with thethree previously described accelerations of themass center.

From a consideration of Figures 3 and 4, itshould be clear that the accelerations have thefollowing values:

ha =

R

va

2

=

ba =

A

xy

za aa

h

Figure 5 Rear view of bike

showing sideways accelerations

Now give the bike a slight roll angle and impose Newtons Second Law, summingmoments about the x axis.

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xy

z

M a

M a

M a

hM g

F

F

f y

f z

F B D M A=

=

J x

Figure 6 Free Body Diagram and Mass Acceleration Diagram of rolled bicycle

The moment equation is thus

+

xxJhMahMahMaMghM +++= cossin:

Applying the small angle approximation for , and simplifying the equation,

Mh

Jaaag x+++=

Substituting the values for the accelerations, ignoring the Jx term, and rearranging,

02

2

=++

++=

gR

vbh

bR

vhg

From the yaw relationships (R

v= and /aR = ),

a

v

a

v

dt

d

R

v

dt

d=

=

=

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if we consider the case of constant speed (v is constant). Thus

02

=++ h

g

ha

v

ha

bv (1)

This is a second order ordinary differential equation in , the roll angle. But , thesteering angle, is also in the equation. In fact the equation expresses the fact that thesteering angle and the roll angle are coupled. Thus is the bicycle is proceeding straightahead with no roll angle and a steering angle of 0, the equation states that a suddensteering angle input will induce a roll. Also a sudden roll angle input, for example if thecyclist encounters a sudden sideways puff of wind that causes a roll, will also result in asteering angle deflection. At this point we can start to gauge the stability of the bicyclefrom this equation.

Imagine the steady state condition where the steering angle is constant ( 0= ) and theroll angle is constant ( 0= and 0= ). In this case from equation (1) we see that

R

vMMg

R

v

a

vg

2

22

sin =

==

This represents the state where the bicycle is traveling around in a circle (with centripetal

acceleration,R

v2

, or centrifugal force,R

vM

2

) at a constant roll angle. The weight

tending to roll the bicycle inward toward the center of the circle is counteracted by thecentrifugal force tending to make the bike roll over to the outside. That the modelcorrectly represents this known situation gives us some confidence in the model. Alsonote that the greater the bike velocity, the larger must be the angle of roll. Also thetighter the turn radius, the larger must be, again what we would expect.

This represents the state where the bicycle is traveling

around in a circle (with centripetal acceleration,R

v2

, or

centrifugal force,R

vM

2

) at a constant roll angle (See

Figure 7). The weight tending to roll the bicycle inwardtoward the center of the circle is counteracted by the

centrifugal force tending to make the bike roll over tothe outside. That the model correctly represents thisknown situation gives us some confidence in the model.Also note that the greater the bike velocity, the larger mustbe the angle of roll. Also the tighter the turn radius, thelarger must be, again what we would expect.

x

M g

F

F

f y

f z

M vR

2

Figure 7 Bike in a steady

circle

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Lets look closer at what happens to the bike if it suddenly rolls from the vertical due to asudden wind gust. Specifically, lets look at the gyroscopic effect on the front wheel andhow it responds to a sudden change in the axis of rotation. This will be done to assess thebikes self-stability, i.e. the hands-off stability. This is the ability of the bicycle to rightitself without intervention of the rider.

The steering axis is out in front of the wheel axle to give positive or trail. Trail is thedistance on the ground from the point of intersection of the steering axis (point C) andpoint B, the contact point of the front wheel (See Figure 1). Usually point C is forward ofpoint B.

Gyroscopic effect

The gyroscopic effect is encapsulated in the vector relation from 3-D kinetics

HMW =

where H is the angular momentum of the front wheel,is the rotation through which

H goes, and WM is the corresponding moment. In our analysis, H is the angularmomentum of the front wheel about its axle, so IH = , where is the rotation rate of

the front wheel. If the bicycle is traveling straight ahead,r

v= , where r is the wheel

radius.

H

H

Figure 8 Change in angular momentum of front

wheel with a roll deflection.

Thus

=

0

0

r

vIH

and

=

0

0

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So

==

r

vI

HMW

0

0

This represents a moment in the negative z direction. If we ignore the rake of the steeringaxis and take it to be vertical (which it almost is for most bikes), this would represent aclockwise moment about the steering axis, looking from above.

If we consider the moment to be a cause for the change in the angular momentum withthe roll, then this moment has been imposed on the wheel to cause the change. Themoment was delivered by the fork. The wheel thus delivers an equal and oppositemoment to the fork. So the moment on the fork is

==r

vI

MM WF

0

0

Considering the moment equilibrium of the fork about the steering axis,

zF IM =

where Iz is the mass moment of inertia of the wheel about a diameter. Thus

rI

vI

Ir

vI

z

z

=

=

This shows that the gyroscopic effect will accelerate the wheel counterclockwise aboutthe steering axis looking from above. Thus if the roll velocity is positive, that is the riderrotates to the left, the front wheel accelerates to the left, i.e. into the direction of the roll.This steers the bike to the left, thus up under the roll. Therefore it is stabilizing.

As it turns out, this effect is minor (negligible) for most bikes. This is because the massof the front wheel is small. But for motorcycles, where the wheel mass is higher, this

effect is more pronounced.

Trail

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Notice what happens to the bicycle when the front fork is turned to the left (Figure 9).

B

C

A

z

a

b

v

Figure 9 Effect of trail

This figure shows the steering angle highly exaggerated to show its effect. Recall thatpoint B is the contact point of the front wheel with the ground. C is the intersection ofthe steering axis with the ground. The x axis is defined by the two contact points. Sincethe steering axis is bound to the bicycle frame, the bicycle plane is actually along AC, notAB. Thus the center of gravity is shown on this line and the velocity is from A to C. Thedistance of point B off AC is

=a

ora

=

A

x

b

Figure 10 Drop in

CG with steering

angle and roll

Also note that the sideways movement of the center of gravity isb . If the bicycle is at a certain roll angle, then thissideways movement of the CG involves a lowering of the CGand a loss of potential energy. Figure 10 shows this. Note thatthis drop, , is

bb == sin

for small roll angles. Thus

a

b =

We equate this drop in potential energy with work done aboutthe steering axis.

SMa

bMg =

Where MS is the moment about the steering angle as it moves through . Thus

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a

bMgMS

=

If we sum moments about the steering axis,

a

bMgIz

=

Notice that this represents a coupling between and . A positive roll motioncorresponds with a positive acceleration of the steering angle. Thus the bike turns intothe roll, which is stabilizing.

Frictional Forces

Lets look also at the moment aboutthe steering axis caused by thefrictional forces on the tires. Theseforces are directed toward the centerof curvature of the bike path(perpendicular to the tire planes, asshown in Figure 11). If we summoments around a vertical axisthrough the center of mass,

b

baFF

baFbF

BA

BA

)(

0)(

=

=+

B A

xz

a

FF AB

b

Figure 11 Frictional forces on tires

Notice that we have used the distance (a-b) for the moment arm ofFB , which assumes asmall angle for .

These frictional forces keep the bicycle moving in a curved path. They cause theacceleration of the bicycle toward the instant center. With a small a these forces areroughly parallel. The normal acceleration toward the instant center is

R

va

2

=

Thus if we apply force equilibrium in a direction perpendicular to the bike frame,

R

vMaMFF BA

2

==+

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Thus

2

22

2

1)(

a

bvM

aR

bvMF

RvM

baF

bbaFFF

B

BBBA

==

== +=+

BC

c o

FB

Figure 12 Moment of FB about steeringaxis

The moment of this force, the sidewaysforce on the front wheel, shown coming outof the page at B in Figure 12, is

)cos(2

2

=

a

bvMMFB

The negative sign here indicates that ifis positive (to the left), this force tends toreduce .

These three moments about the steering axis are separate, caused by differentphenomena: 1) the gyroscopic effect, 2) the reduction in height of the CG, and 3) thesideways friction force of the tire. So the total moment about the steering axis is the sumof all these effects. Thus, applying Newtons Second Law, the moment about thesteering axis is

zIa

bvM

a

Mgb

r

vI=+ )cos(

2

2

or

02

=

v

rI

I

ga

v

aI

Mgb

zz

(2)

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Thus we have a second, second-order ODE, this one the a equation. This is the matchingequation for equation (1), the q equation. Together these two equations model thebicycle, especially the coupling between and .

General Equation of motion

Summarizing, the dynamic model consists of two second-order coupled equations in and . These equations are those of a free system (only 0s on the right-hand side ofthese equations). The equations are:

equation: 02

=++ h

g

ha

v

ha

bv

equation: 02

=

vga

v

whereaI

Mgb

z

= and

rI

I

z

= . Also in the Lowell paper 00 =W andg

h= .

THE MODEL

To keep the order of the system low and allow us to directly see the affects the frontsteering angle has on the system, we will model the bicycle and rider as a single rigidbody. 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 inherent stabilitycharacteristics associated with an offset angled front fork and the rotating wheels will beneglected. The following list details the assumptions made in the development of theequations of motion of the basic bicycle model.

Model assumptions

1. The bicycle consists of one rigid body. This includes the rear frame, rider, thefork/handlebar assembly, and the two wheels. Rider position is fixed.

2. The wheels are treated as rigid knife-edge disks, which roll without friction on a flatsurface with no slippage from torque or steer. They are assumed to have no moments ofinertia. This is acceptable for light wheels that dont rotate at very high speeds.

3. The linear equations are valid only for small angles of lean and steer.

4. Fixed external forces are limited to gravitational forces and constraint forces from theground acting on the wheels.

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5. The bike-rider system is symmetrical about a plane passing lengthwise through the rearframe.

.

The Table

Quantity Description Magnitudea Wheel base 1.0mb Distance from real-wheel

hub to CM0.33m

h Height 1.5m

ZI Couple on front

wheel/angle of lean8Nm rad-1

r Wheel radius 0.33mm Wheel mass 1.6KgM Mass of cycle + rider 80Kg

hg/ 6.5s-2

I Polar M. of I. of front wheel 0.08Kgm2

ZI M. Of I. of front wheel

about diameter.0.06Kgm2

133s-2

4.0s-1

The values of the table are given for rough guidance with no great accuracy. Because,most of the parameters vary from one bicycle to another and many of them depend on theweight of the rider and on the riding position he adopts (7).

The Simulation

The simulation of this model is done using MATLAB Simulink. Simulink is anenvironment for multi-domain simulation and model-based design for dynamic andembedded systems. It provides an interactive graphical environment and a customizableset of block libraries that let you design, simulate, implement, and test a variety of timevarying systems.

Simulink uses a solver to perform tasks. For this model, I employed the Ode-45 variablestep continous solver which is based on explicit Runge-Kutta(4,5) formula. To examinestability, I assume that the bicycle is initially in equilibrium and upright, that is setting

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my initial conditions as ( 0.....

===== ) and study the effect of an impulse which

impart an initial rate of falloW=

.

.

Scope

1

s

Integrator3

1

s

Integrator2

1

s

Integrator1

1

s

Integrator

-K-

Gain5

-K-

Gain4

u

Gain3

-K-

Gain2

-K-

Gain1

g/h

Gain

Figure 13 The Simulink Model

RESULTS

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Figure 14 Dependence of angle of lean (above) and steering angle (below) ontime at a speed of 1.5m/s for zero gyroscopy

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Figure 15 Dependence of angle of lean (above) and steering angle (below) ontime at a speed of 6m/s for zero gyroscopy

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Figure 16 Dependence of angle of lean (above) and steering angle (below) ontime at a speed of 1.5m/s for with gyroscopic effect

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Figure 17 Dependence of angle of lean (above) and steering angle (below) ontime at a speed of 6m/s with gyroscopic effect

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Figure 18 Dependence of angle of lean (above) and steering angle (below) on

time at a speed of 6m/s for low trail

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Figure 19 Dependence of angle of lean (above) and steering angle (below) ontime at a speed of 6m/s for high trail

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DISCUSSION

From figure 14 & 15, while neglecting the gyroscopic effects of the bicycle, thedependence of the Roll angle and steering angle on time shows that If someperturbation causes the bicycle to state falling at t=0, the fall does not continue, instead

and asymptotically approaches a steady value. Also, at higher speed of 6m/s, smallsteering angles quickly move the ground contact, while at lower speed of 1.5m/s, largesteering angles are required to achieve the same results in the same amount of time.Therefore, it is easy to maintain balance at high speed.

Figure 16 & 17 examine the gyroscopic effects. The results show that gyroscopic effectsresult in a smaller mean deflection of the bicycle for a given perturbation. However, theyproduce oscillatory instability which increases at higher speed. This suggests thatgyroscopic effects do not have much impact on stability of the bicycle (8).

Figure 18 & 19 shows the effect of Trail, which is incorporated in the value ofand

is usually large. For this model, 133= .The effect of Trail is examined at low trail (5.66= ) and at high trail ( 266= ). From the figures, it is evident that Oscillatory

instability decrease with increasing trail. That is, the more trail a bicycle has, the morestable it feels (9).

CONCLUSION

It can be concluded that several factors including geometry and gyroscopic effectscontribute in varying degree to the self stability of a bicycle. The steering angle and thelean angle (Roll) play a major role in ensuring self stability. A bicycle must lean in order

to maintain balance in a turn and the higher the speed or smaller the radius, the more leanis required.

At high speed, small steering angle quickly ensure stability while at low speed largesteering angle is required to achieve the same result in the same amount of time. Becauseof this, it is easier to maintain balance at high speed.

Also, gyroscopic effect makes oscillations grow more rapidly than they would otherwisedo. This produces oscillatory instability which increases at high speed.

Finally, the Trail of the front wheel is the most important factor in the self stability of

bicycles. A fairly large trail of real bicycles ensures a slow growth of the oscillatoryinstability, so that the rider can be quite leisurely in his intervention.

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References

Karnopp, D. Vehicle Stability. Marcel Dekker. USA. 2004

Olsen, J. J.M. Papadopoulos. Bicycle Dynamics: The MeaningBehind the Math. BikeTech, Dec 1988

Watkins, G. K. The Dynamic Stability of Fully Faired Single Track Human PoweredVehicle.Ph.D. Dissertation, The University of North Carolina at Charlotte. 2002

1. Fajans, J. Steering in bicycles and motorcycles. Am. J. Phys. 68 (7), July 2000.2. strm, K.J. Control System Design. Preprint. 2002.3. strm, K.J. et al. Bicycle dynamics and control. IEEE Control Systems

Magazine. 25 (4). 26-47. 2005

4. Tewari, K. S. Modern Control Design with Matlab and Simulink. John Wiley &sons. 20025. F. Whipple, The stability of the motion of a bicycle, Quarterly

Journal of Pure and Applied Mathematics, vol. 30, pp. 312348,1899.

9. J. P. Meijaard and A. L. Schwab, Linearized equations for anextended bicycle model, III European Conference on ComputationalMechanics, Structures and Coupled Probelms in Enginering, pp. 118, 2006.

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Term Paper Classical[1]