THESIS

DYNAMICS OF ROLLING OF ELASTIC WHEELS

Author: Dénes Takács, MSc student of Faculty of Mechanical Engineering

at Budapest University of Technology and Economics

Tutors: Professor Gábor Stépán, Department of Applied Mechanics

at Budapest University of Technology and Economics

Professor John Hogan, Department of Engineering Mathematics

at University of Bristol

May 20, 2005

1. CONTENTS

1. Contents ....................................................................................................................... 1

2. Introduction.................................................................................................................. 2

3. Elastic tyre model......................................................................................................... 3

3.1. Analytical calculation ........................................................................................... 3

3.1.1. Kinematical constraint ................................................................................... 4 3.1.2. Equation of motion ........................................................................................ 5 3.1.3. Delayed mathematical system........................................................................ 5 3.1.4. Sorting of the equation of motion .................................................................. 7 3.1.5. Stability Analysis ......................................................................................... 10

3.2. Simulation ........................................................................................................... 13

3.2.1. Construction of the simulation..................................................................... 14 3.2.2. Analysis of stability by simulation .............................................................. 15 3.2.3. Shape of the deformation function............................................................... 16 3.2.4. Investigation of frequencies......................................................................... 17

3.3. Experiment .......................................................................................................... 19

3.3.1. Basic questions............................................................................................. 19 3.3.2. Precedent measuring .................................................................................... 19 3.3.3. Design .......................................................................................................... 20 3.3.4. First run........................................................................................................ 21 3.3.5. Cycle of the measuring ................................................................................ 23 3.3.6. Measuring results ......................................................................................... 24

3.4. Summary ............................................................................................................. 26

4. Rigid tyre model......................................................................................................... 27

4.1. Analytical calculation ......................................................................................... 27

4.1.1. Constraints ................................................................................................... 28 4.1.2. Equations of motion..................................................................................... 29 4.1.3. Linear stability analysis of the undamped system ....................................... 32 4.1.4. Hopf bifurcation........................................................................................... 36 4.1.5. Equations of motion of the damped system................................................. 42 4.1.6. Linear stability analysis of the damped system ........................................... 46

4.2. Continuation........................................................................................................ 48

4.2.1. Undamped system........................................................................................ 48 4.2.2. Damped system............................................................................................ 52

4.3. Summary ............................................................................................................. 62

5. Conclusion ................................................................................................................. 64

6. References.................................................................................................................. 65

7. Supplement CD .......................................................................................................... 66

1

2. INTRODUCTION

Shimmy is the name for the lateral vibration of a towed wheel, which is a well

known phenomenon in vehicle systems. The name comes from a dance which was

popular in the early thirties. The first scientific study of this problem was made in 1941,

see [1]. The developments of more and more sophisticated wheel suspensions and tyres

in vehicles require a full analysis of this phenomenon. In many cases the appearance of

shimmy is very dangerous – for example in the nose gear of airplanes or the front wheel

of motorcycles etc. – so engineers have to eliminate it.

There are lot of models which describe shimmy but all of these can be separated in

two groups. On the one hand some of the descriptions assume elasticity of the

suspension system, on the other hand in some studies the tyre is modelled as elastic.

Both cases can cause lateral vibration but of course the real system is a combination of

the both descriptions. The analytical handling of a general model is very complicated.

This thesis investigates systems which describe shimmy by a low number of degrees

of freedom. The first part of the thesis assumes an elastic tyre model, see [2], which was

investigated analytically, numerically (with an indirect simulation) and experimentally.

The second part considers a rigid tyre model with elastic suspension; see [3], which is

capable to modelling shopping trolley wheel shimmy. This model was investigated

analytically and by numerical continuation. The effect of damping was also analysed in

both models.

2

3. ELASTIC TYRE MODEL

As mentioned in the introduction, we can model shimmy by assuming elasticity of the

wheel. Because the majority of vehicles are equipped with pneumatic tyres, this model

is important.

The model we will investigate is taken from [2]. As shown by Figure 1 the elastic

wheel is towed in the absolute ),,( ζηξ coordinate system with constant velocity. The

tyre adheres to the ground on a contact area. This area is modelled as a contact line

whose length is 2a. In this way, the deformation of the tyre can be modelled by the

lateral displacement of this contact line. ),( txq

zx

yv

ξ

η

ψ(t)

l

A

P x

a a

q(x,t)

c k

Figure 1: Model of elastic towed wheel

3.1. ANALYTICAL CALCULATION

In this section the equation of motion of the linear system will be presented and the

linear stability boundaries will be calculated. In the model we can have damping in

different places, as well as distributed damping in the tyre. It is possible locate a torsion

damper to the pin (A) too. In this thesis the first case will be calculated and the results

of the second case will be outlined.

3

3.1.1. KINEMATICAL CONSTRAINT

The applied model has a kinematical constraint. In particular the points in contact with

the ground have zero speed in the absolute coordinate system, namely they can not

slide. This is an approximation which is different from the real behaviour.

yψ (t)

η

ξ vAx

P

q(x,t)x

rP

O

A

v · t

Figure 2: Scheme of the system

The kinematical constraint can be written as

P Ptrans Prel= + =v v v 0

O

. (1)

The transport velocity can be determined if we start from the point A:

O A A= + ×v v ω r , (2)

O OPtrans P= + ×v v ω r , (3)

If we expand (2) we get the velocity of the centre of the wheel:

O

cos i j k cossin 0 0 sin0 0 0 0

v vv v

l

ψ ψlψ ψ ψ

⎡ ⎤ ⎡⎢ ⎥ ⎢= − + = − −⎢ ⎥ ⎢⎢ ⎥ ⎢−⎣ ⎦ ⎣

v ψ⎤⎥⎥⎥⎦

, (4)

where v is the towing speed. Substitute this into (3) and we get the transport velocity in

the moving ( , , )x y z relative coordinate system:

Ptrans

cos i j k cos ( )sin 0 0 sin

0 ( ) 0 0

v vv l v l x

x q x,t

q x,tψ ψ ψψ ψ ψ ψ ψ

−ψ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢= − − + = − − + ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣

v

⎦

. (5)

The location of the point P is determined by in the moving relative coordinate system

which is fixed to the caster. The relative velocity is given by derivation of this vector

with respect to time:

Pr

Prel P ( , ) ( , )0

xq x t q x t x⎡ ⎤⎢ ⎥′= = + ⋅⎢ ⎥⎢ ⎥⎣ ⎦

v r , (6)

4

where dots denote derivation respect to time and prime denotes partial derivation

respect to x. After substitution of the results into (1) we find:

P

cos ( , )sin ( , ) ( , )

0 0

v q x t xv l x q x t q x t x

ψ ψψ ψ ψ−⎡ ⎤ ⎡

⎢ ⎥ ⎢ ′= − − + + + ⋅ =⎢ ⎥ ⎢⎢ ⎥ ⎢⎣ ⎦ ⎣

v 0⎤⎥⎥⎥⎦

. (7)

From the first row of this vector equation, can be found. If we substitute this into the

second row and we order the equation we obtain the kinematical constraint in a form of

a partial differential equation (PDE):

x

( , ) sin ( ) ( , ) ( cos ( , ) )q x t v l x q x t v q x tψ ψ ψ ψ′= + − + ⋅ − , (8)

where [ ],x a a∈ − and . [ )0 ,t t∈ ∞

3.1.2. EQUATION OF MOTION

From Newton’s second law and Figure 1 then the equation of motion can be written in

the form:

A ( ) ( ) ( , )d ( ) ( , )da a

a a

t c l x q x t x k l x q x tθ ψ− −

= − − − −∫ ∫ x , (9)

in which c [N/m2] is the stiffness of the wheel distributed over its length, k [Ns/m2] is

the distributed damping of the wheel and Aθ [kgm2] is the mass moment of inertia of

the overall system with respect to the z axis at the articulation.

3.1.3. DELAYED MATHEMATICAL SYSTEM

The lateral deformation , which is in (9), is determined by (8). Together these

equations determine the system but in this form they are not manageable.

( , )q x t

The deformation function can be written as a travelling-wave like solution which

comes from the kinematical constraint. Because the points on the ground are not

moving in the absolute system, from the moment of the contact until separation, we can

add a time function ( )xτ to all contacted points, see Figure 3. This function determines

the elapsed time which is needed for the leading edge (x=a) to travel backwards

(relative to the caster) to the actual point P.

5

y

x

q(x,t)

xOOa

-a

t

tt-τ (x)

0τ (x)

P

Figure 3: Time delay in the system

The position of P is determined in the absolute system:

( , ) ( ) cos ( , ) sinx t v t l x q x tξ ψ ψ= ⋅ − − − , (10)

( , ) ( ) sin ( , ) cosx t l x q x tη ψ ψ= − − + , (11)

which comes from the geometry of the system. So in the absolute system can be written:

))(,(),( xtatx τξξ −= , (12)

))(,(),( xtatx τηη −= . (13)

If we write the linearized form of (10) into (12) we obtain the time delay:

vxax −

=)(τ . (14)

If we repeat the method with (11) and (13) we can take the deformation function of the

linearized system:

( ) ( )( , ) ( ) ( ) ( ) ( ) , ( )q x t l x t l a t x q a t xψ ψ τ τ= − − − − + − , (15)

in which the last part is the lateral deformation of the leading edge. This deformation

can be calculated by from elasticity theory or it can be approximated by different

functions which are presented in some publications, for example, see [4]. Now we

suppose that the tyre is made from separated elements. These separated elements do not

affect one another i.e. one deformed and contacted segment can not pull another, not

contacted, segment, see Figure 4. In this case the deformation at the leading edge is zero

namely ( , ( )) ( , ) 0q a t x q a tτ− ≡ ≡ . With this approximation and with (14) the travelling-

wave like solution of the linearized system is

( , ) ( ) ( ) ( ) a xq x t l x t l a tv

ψ ψ −⎛ ⎞= − − − −⎜ ⎟⎝ ⎠

, (16)

which of course satisfies the linearized form of the (8) kinematical constraint equation.

6

a a a a

q(a,t)≠0

Figure 4: Type of tyres

3.1.4. SORTING OF THE EQUATION OF MOTION

In the equation of motion there is the derivative of the travelling-wave like solution too.

So we need to take its derivative with respect to time:

( , ) ( ) ( ) ( ) a xq x t l x t l a tv

ψ ψ −⎛ ⎞= − − − −⎜ ⎟⎝ ⎠

, (17)

so in the equation of motion will appear the history of the angular velocity. Thus if

substitute (16) and (17) into (9) we obtain the following expression:

A ( ) ( ) ( ) ( ) ( ) d

( ) ( ) ( ) ( ) d .

a

aa

a

a xt c l x l x t l a t xv

a xk l x l x t l a t xv

θ ψ ψ ψ

ψ ψ

−

−

⎛ ⎞−⎛ ⎞= − − − − − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞−⎛ ⎞− − − − − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫

∫ (18)

We now make the substitution )(xvax τ−= and accordingly replace with xd

τdd vx −= . So integration limits will be changed:

at ax = ⇒ 0a x a av v

τ − −= = = ,

at ax −= ⇒ 2a x a a a

v vτ

v− +

= = = .

So the equation of motion becomes

7

( )2 2

2A

0 02 2

2

0 0

( ) ( ) ( ) d ( )( ) d

( ) ( ) d ( )( ) ( ) d .

a av v

a av v

t c l a v t v c l a l a v t v

k l a v t v k l a l a v t v

θ ψ τ ψ τ τ ψ τ τ

τ ψ τ τ ψ τ τ

= − − + + − − + −

− − + + − − + −

∫ ∫

∫ ∫

(19)

In this expression the first and the third integrals can be calculated. The first becomes: 2 2

2 2 2 2 2

0 02

2 3 22 2 2

0

( ) ( ) d ( 2 2 2 ) ( )d

( ) ( ) 2 ( ) 2 ( ) ,2 3 3

a av v

av

c l a v t v cv l a v la lv av t

acv t l a v l a v ac t l

τ ψ τ τ τ τ ψ

τ τψ τ ψ

− − + = − + + − + −

⎡ ⎤ ⎛= − − + − + = − +⎜ ⎟⎢ ⎥

⎣ ⎦ ⎝

∫ ∫ τ

⎞

⎠

(20)

and the third: 2 2

2 2 2 2 2

0 02

2 3 22 2 2

0

( ) ( ) d ( 2 2 2 ) (

( ) ( ) 2 ( ) 2 ( ) .2 3 3

a av v

av

k l a v t v kv l a v la lv av t

akv t l a v l a v ak t l

)dτ ψ τ τ τ τ ψ

τ τψ τ ψ

− − + = − + + − + −

⎡ ⎤ ⎛= − − + − + = − +⎜ ⎟⎢ ⎥

⎣ ⎦ ⎝

∫ ∫ τ

⎞

⎠

(21)

If we substitute (20) and (21) into (19) we obtain 2 2

2 2A

2 2

0 0

( ) 2 ( ) 2 ( )3 3

= ( ) ( ) ( ) d ( ) ( ) ( ) d

a av v

a at ak l t ac l t

l a vc l a v t l a vk l a v t

θ ψ ψ ψ

.τ ψ τ τ τ ψ τ

⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

− − + − + − − + −∫ ∫ τ

(22)

Then if we set:

ϑτ −= ⇒ d dτ ϑ= − ,

we have: 2 2

2 2A

0 0

2 2

( ) 2 ( ) 2 ( )3 3

= ( ) 1 ( )d ( ) 1 ( )da a

v v

a at ak l t ac l t

l v l va l a vc t a l a vk t .a a a a

θ ψ ψ ψ

ϑ ψ ϑ ϑ ϑ ψ ϑ− −

⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− − − + + − − − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫ ∫ ϑ

(23)

We can introduce new variables:

22 2

1

aa at tvv v

ϑϑ ϑ

ϑ

⎫= ⎪ ⇒ = ⇒ =⎬⎪= ⎭

. (24)

8

With these new variables the equation of motion can be rewritten: 2

22 22 2

A A

2 2

2 02

A 12

A

122 2 2 2( ) ( ) ( )

3 3

2 12 ( ) 1 2 ( ) d

2 1( ) 1 2

: / V: D / Va ak a a ac at l t lv v

: pα :

a la l a c tv a

a lav l a kv a

tψ ψ ψθ θ

α

ϑ ψ ϑ ϑθ

θ

−

==

⎛ ⎞ ⎛ ⎞⎛ ⎞′′ ′+ + + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

= =

⎛ ⎞ ⎛ ⎞= − − − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞+ − − −⎜ ⎟⎝ ⎠

∫0

1

( ) dt ,ϑ ψ ϑ ϑ−

⎛ ⎞ ′ +⎜ ⎟⎝ ⎠∫

(25)

where the primes denote derivation with respect to t .

As shown in the parentheses in (25), some new dimensionless parameters can be

introduced. The first parameter is the dimensionless towing speed:

α1

2⋅=

avV , (26)

in which

22

A

23

ac alαθ

⎛ ⎞= +⎜ ⎟

⎝ ⎠ ⎥⎦

⎤⎢⎣⎡

s1 (27)

is the natural angular frequency of the tyre about the articulation point (A) if the speed

is zero (v=0). We can introduce another parameter L, which we will call the

dimensionless towing length:

alL = . (28)

Because there is distributed damping in the model, so we can identify the relative

damping D, (which is zero if the model is undamped):

αpD21

= , (29)

where p is the proportional damping of the tyre material:

[ ]sckp = . (30)

After the introducing of the new dimensionless parameters we can write (25) in the

following form:

9

02

21

0

21

1( ) 2 ( ) ( ) ( 1 2 ) ( ) d1 3

1 2 ( 1 2 ) ( ) d1 3

LV t DV t t L tL

L DV L t .L

ψ ψ ψ ϑ ψ ϑ ϑ

ϑ ψ ϑ

−

−

−′′ ′+ + = − − ++

− ′+ − −+

∫

∫ ϑ+

(31)

In the following the tildes above the variables will be dropped, but it should not be

forgotten that t t≠ and ϑ ϑ≠ .

3.1.5. STABILITY ANALYSIS

The equation of motion (31) is an ordinary second-order differential equation so the

solution can be found in the form of ( ) tt Aeλψ = . If it is substituted into (31) then 2 2

0 0( ) ( )

2 21 1

2

1 1( 1 2 ) d 2 ( 1 2 ) d1 3 1 3

t t t

t t

V Ae DV Ae Ae

L LL Ae DV L AeL L

λ λ λ

λ ϑ λ ϑ

λ λ

.ϑ ϑ λ ϑ+ +

− −

+ +

− −= − − + − −

+ +∫ ∫ ϑ (32)

Both sides of the equation can be divided by tAeλ , so 2 2

0 0

2 21 1( ) ( )

2 1

1 1( 1 2 ) d 2 ( 1 2 ) d1 3 1 3

w w

V DV

L LL e DV L eL L

λϑ λϑ

ϑ ϑ

λ λ

,ϑ ϑ λ ϑ− −

+ +

− −= − − + − −

+ +∫ ∫ ϑ (33)

where ( )w ϑ is the weight function, which is displayed by Figure 5.

w( )

-1 0

L-1

L+1

Figure 5: Weight function of the integrals

The integrals of the equation can be calculated by partial integration:

10

0 0 0

1 1 10 0 0

11 10

2 21

( 1 2 ) d ( 1) d 2 d

( 1) 2 2 d

1 1 2 1 1 12 2

L e L e e

e e eL

L L e e e ee e L

λϑ λϑ λϑ

λϑ λϑ λϑ

.λϑ λ λ

λ λ

ϑ ϑ ϑ ϑ ϑ

ϑ ϑλ λ λ

λ λ λ λ λ λ λ

− − −

−− −

− −− −

−

− − = − −

⎡ ⎤ ⎡ ⎤= − − + =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤− − − + −⎛ ⎞ ⎛ ⎞− − + = − +⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦

∫ ∫ ∫

∫λ−

(34)

If we substitute this result into (33) the characteristic equation is

2 22 2

2 2

1 1 1 1( ) 2 1 21 3

1 1 1 12 21 3

L e eK V DV LL

L e e eDV L .L

λ λ

λ λ λ

λ λ λλ λ λ

λλ λ λ

− −

− − −

⎛ ⎞− − + −= + + − − +⎜ ⎟+ ⎝ ⎠

⎛ ⎞− − + −+ − +⎜ ⎟+ ⎝ ⎠

e λ−

(35)

The stability boundaries can be found if we write iλ ω= in the characteristic

equation. Equating real and imaginary parts, we find:

( )

( ) ( )( )

2 22

1 1 1Re ( ) 1 2(cos 1) ( 1) sin1 3

2 2sin cos 1 1 cos ,

LK i V LL

DV L

ω ω ωω ω

ω ω ω ω ω

− ⎡= − + − − + +⎢+ ⎣⎤+ − + + − ⎦

ω ω (36)

( ) (( )

( )

2

1 1 1Im ( ) 2 2sin cos 1 1 cos1 3

2 2(cos 1) ( 1) sin .

LK i DV LL

DV L

)ω ω ω ω ω ωω ω

ω ω ω

− ⎡= − − + + − −⎢+ ⎣

⎤+ − + + ⎦

ω (37)

On the stability boundaries

( ) 0( ) 0

Re K iIm K i

ωω

=⎧⎨ =⎩

(38)

have to be satisfied. These equations can not be solved analytically, but the stability

boundaries can be plotted numerically. In Figure 6 the relative damping D is zero

namely the system is undamped. In the figure, ω runs from 0 to 8π. If the caster length

is infinite then the system is stable. If we suppose the reversing of stability at the

crossing the stability boundaries then the stable areas can be determined.

11

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3

10 2

10 4

10 6

0 0.05 0.1 0.15 0.20

0.5

1

V

V

L

ω/α

Figure 6: Stability map of the undamped system

12

By increasing the relative damping, the instable areas get smaller i.e. the instability

of the system is decreasing, see Figure 7.

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

L

V

D=0.00D=0.01D=0.02D=0.03

D=0.04D=0.05

D=0.06

Figure 7: Stability map of the system

As mentioned earlier, the damping can be located in different places. The distributed

damping in the tyre or the torsion damper at the articulation gives the same results, as

discussed in [5]. The major difference is in the relative damping expression. If we apply

only a torsion damping at the articulation (A) then the relative damping is given by

A2TkD

α θ=

⋅ ⋅, (39)

where kT [Nm·s/rad] is the torsion damping.

3.2. SIMULATION

After the calculation of the stability map, new questions can emerge, for example the

shape of the deformation function. The answers to these questions give new information

about the behaviour of the system. One of the ways to obtain these results is simulation.

13

3.2.1. CONSTRUCTION OF THE SIMULATION

The nonlinear equations of the system consist of a partial differential equation (8) which

is the kinematical constraint and of an integral equation (9) which is the equation of

motion. The equation system which is determined by these two equations can be solved

with difficulty. Another possibility is to solve the linearized continuous time delayed

integral equation (18), which is good for small motions. Now there is not accessible

continuation software for continuous time delayed systems, so only an indirect

simulation was possible. But by this simulation the linear stabilities and another

interesting phenomena could be analysed.

In the simulation the major problem was the continuous time delay. During the

running of the simulation the past of the angle of the caster ψ and the past of the

angular velocity of the caster ψ are needed. Because of the discretization, which is used

in the solution of the differential equation, the past of the necessary values is calculated

only at discrete time moments. The calculated values are stored in vectors until they are

needed in the simulation. The deformation function consists of the history of the

angle of the caster so the deformation function is discretized too. With the

determination of the simulation time step the number of the segments by which the

deformation is built is determined too. So we have to be careful that the deformation

function will be smooth enough. Otherwise the results will be not precise.

( , )q x t

The simulation was preformed in Labview 4.1, which is a graphical program

language. The source code for the simulation will not be shown in this thesis because it

is very complicated. The desktop of the simulation is shown in Figure 8.

14

Figure 8: The simulation during running

3.2.2. ANALYSIS OF STABILITY BY SIMULATION

The simulation is capable of localising the linear stability boundaries of the system. The

runs were carried out with real system parameters, which came from measurements (see

3.3.2 and 3.3.4). The procedure of the stability analysis is similar to the experimental

measures (see 3.3.5). Namely after the declaration of all system parameters only the

towing speed was changed and the points of the stability boundaries were searched

where the amplitude of shimmy is constant. The stability can be found easily on the

both side of the point. A stability map at D=0.02 is shown in Figure 9. In the figure

there are the theoretical curves (with green) too and the similarity of the theoretical and

the numerical borders is conclusive.

15

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

L

V

D=0.02

TheoreticalSimulation

Stable

Unstable

Unstable

Figure 9: Stability borders founded by simulation

3.2.3. SHAPE OF THE DEFORMATION FUNCTION

Maybe the most important result found by simulation is the shapes of the travelling-

wave like solution. These shapes were not calculated analytically and the observation of

the shapes in an experiment is a very complicate exercise.

Simulations in the unstable holes of the theoretical stability map were carried out and

the different form of deformation was noted. These forms are shown by Figure 10.

16

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

L

V

D=0.02

Figure 10: Shapes of the deformation function

3.2.4. INVESTIGATION OF FREQUENCIES

As in Figure 6, at the intersection of two segments of the stability boundary there are

two frequencies of the lateral motion of the wheel. These quasi-periodic vibrations were

detected with the simulation. The angle-time diagram was recorded during the run close

to an intersection and the frequencies were calculated using Fourier transforms, see

Figure 11.

As the parameters are known the theoretical frequencies can be calculated. The

system parameters are:

2

2A

0.023[m],2 0.119 [m],

230459 [N/m ],0.33780 [kgm ].

la

cθ

==

=

=

So from (27) the natural frequency is

22

A

2 111.93 1.899 [Hz]3 s

ac alα αθ

⎛ ⎞ ⎡ ⎤= + = ⇒ =⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠.

17

The values of the two red points in the upper graph of Figure 11 are

1 0.47426ωα

= and 2 1.06520ωα

= .

So the frequencies of the vibration can be calculated as:

11 0.9 [Hz]f ω α

α= ⋅ ≅ , 2

2 2.0 [Hz]f ω αα

= ⋅ ≅ .

The values are in good agreement with the FFT diagram of the simulation.

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

L

V

D=0.02

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

V

ω/α

0 1 2 3 4 f [Hz]0

5

10

15

20

25

A

0 1 2 3 4 5 6 7 8 t [s]-0.03

-0.02

-0.01

0

0.01

ψ[rad]

Angle-time diagram

Angle-time FFT diagram

Figure 11: Detection of a quasi-periodic vibration

18

3.3. EXPERIMENT

A simulation is useful to find new information about a model and to check theoretical

calculations but it is not enough to verify the validity of the model that can only be done

when comparison is made with experiment. Accordingly we decided to build an

experiment to test our elastic tyre model.

3.3.1. BASIC QUESTIONS

Because of the complexity of the model we had to be careful about the experimental

design. First we had to choose a wheel what was a really heavy. The best tyre should

have the same behaviour as in chapter 3.1.3. First we wanted to build this type of tyre

but it was a very difficult exercise. We needed a wheel which can run with different

stiffness, damping and contact length. Therefore in the end we have chose a simple

wheel of a little bicycle. This wheel has a pneumatic tyre so by changing the air

pressure in the tube we can tune the system very easily.

In parallel with the correct choice of wheel we had to be look for a pulling device

which can move our wheel. We considered three possibilities. One of them was the

simple towing of the wheel on the ground. This solution has lot of problems, for

example the roughness of the ground. Another possibility is running on a revolving

drum. With this choice the wheel is running on a curved surface instead of a flat surface

and this can influence the measured results. The last possibility, which we have chosen,

is a running belt machine. With this option, the endlessness of the belt and the lateral

stiffness of the belt can cause problems.

3.3.2. PRECEDENT MEASURING

For the design we had to find the stiffness and the damping of the tyre because these

parameters affect the construction of the experimental device. For these measurements,

I had to create a test board which is shown in Figure 12. The wheel was placed between

on surface, which could be moved to achieve the required contact length. First the axle

of the wheel was pulled in the lateral direction and the force and the lateral

displacement were measured in order to calculate the stiffness. Second, the damping

19

was measured with a vibration detector in the same set-up. Both results are shown by

Table 1.

Figure 12: Test board for measuring of stiffness and damping

Contact length Along the length distributed stiffness

Along the length distributed dampingAir pressure

in the tube 2a [m] c [N/m2] k [Ns/m2]

0.112 250513 97 2 bar 0.051 425266 181 0.133 138495 85 1 bar 0.052 320668 250 0.178 166662 161 0 bar 0.087 112481 143

Table 1: Measured data of stiffness and damping

3.3.3. DESIGN

Knowing the parameters of the wheel and the running belt machine I can design the

experimental plant. I built the device virtually in a CAD program (SolidWorks), which

is displayed by Figure 13, so I got the approximate value of the mass moment of inertia

which is needed to check the operating region. Because we wanted to measure the

interesting region of the stability map, I had to scale the system to the required zone as

20

far as possible. Even so the minimum speed of the running belt machine prevented us

from measuring the whole stability map.

Figure 13: CAD model of the experimental plant

3.3.4. FIRST RUN

After the design I made all of the parts of the device myself. These are shown in Figure

14.

Figure 14: Parts of the experiment device

Two problems occurred immediately during the first experiment. One of them was

the lateral stiffness of the belt and the other was a cut under the belt. So we had to

relocate the suspension of the wheel and we supported the belt in the lateral direction.

After some running we found out that perhaps the relative damping of the system is

bigger than we had believed (namely the damping of the articulation increases the

relative damping). So we measured the damping of the tyre on the fitted device again,

21

see Figure 15, furthermore we measured the real mass moments of inertia. The newly

obtained damping data are shown by Table 2.

Figure 15: Measuring of the damping

Contact length Along the length distributed damping Air pressure

in the tube 2a [m] k [Ns/m2]

0.112 1077 2 bar 0.051 3732 0.133 764 1 bar 0.052 3022 0.178 1017 0 bar 0.087 1104

Table 2: Data of newly measured damping

As a consequence of the bigger values of the damping, we have to solve for the

changing of mass moment of inertia, because the value of the relative damping D can be

reduced with an increase in the mass moment of inertia θA , see (27) and (29). But the

increase of the mass moment of inertia causes another problem, since from (26) the

dimensionless towing speed V is increases also. Because of the minimum speed of the

running belt machine is not zero we can not measure the whole stability map. The

rebuilt device is displayed in Figure 16.

22

Figure 16: Experiment device ready to measure

3.3.5. CYCLE OF THE MEASURING

Because the system is very complex measurements are complicated. We have lot of

parameters, for example c, k, l, a, α, θA and v tuning the system is difficult. In addition

these parameters are not independent. So at the first run it was clear that we needed a

program which can help us in the measurements. The written software can interpolate

between the measured data of the stiffness and damping and gives the required value of

parameters, for example the mass moment of inertia. Because the changing of a

parameter causes a change in the relative damping, so we have to correct it. Of course

the software can store and plot the results too. A screenshot of the program is shown in

Figure 17.

From the measurements we select a relative damping and with this we select a

stability map too. Then we tune the system to the required constant L curve by changing

the caster length or the contact length. Next we correct the relative damping with the

adjusting of the mass moment of inertia. And now we can move along the selected

constant L curve from the point of the minimum speed. Our running belt machine has a

minimum speed of 0.5 [km/h] and a 0.1 [km/h] speed step size. As we sweep along the

line in 0.1 [km/h] steps we can find stable and unstable regions. The stability boundary

lies between stable and unstable measuring points.

23

Figure 17: A screenshot from the measurement program

3.3.6. MEASURING RESULTS

The experimental design was done very carefully but the built experimental plant has its

limits. Because of the minimum speed and the speed step size we can not measure all of

the stability map. Also we can not tune the system to any relative damping so

sometimes the required increasing of mass moment of inertia is overlarge.

The best choice of relative damping is 0.054 in our system. By tuning, we can hold

this value very easily and correctly in the lower region of the stability map. The

measured results of this run are plotted in Figure 18. The green plus marks mean a

stable system i.e. after deflecting the wheel from the longitudinal direction the shimmy

motion was decreased and some time later stopped. The red points mean an unstable

system i.e. after deflecting the wheel the amplitude of shimmy motion increased or even

sometimes the wheel started shimmying without being deflected. The blue circles mean

the first points along the constant L lines where unstable system was detected. The

fitting of the experimental points is conclusive. In Figure 18 three measured points are

highlighted. At these points quasi-periodic vibrations were observed. The video of the

quasi-periodic vibration is on the supplemental CD of this thesis.

24

0 0.05 0.1 0.15 0.2 V0

0.2

0.4

0.6

0.8

LD=0.054 Minimum speed

Stable

Unstable

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

ω/α

V

Quasi-periodic vibrations

Theoretical

Speed step size

Figure 18: Measurements results in the lower region

In Figure 19 the results of the upper region are plotted. We note that there are very

big differences between the theoretical curve and the measured points, namely from the

theory there can be no shimmy. In the upper region an increasing mass moment of

inertia had to be applied and some vibration appears in the suspension because it is not

stiff enough.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 V0

0.5

1

1.5

2

2.5

L D=0.054

StableUnstable

Theoretical

Figure 19: Measurement results in the whole map

25

3.4. SUMMARY

As noted in the introduction shimmying of pneumatic tyres can be modelled by in

different ways. The analysis of our model is complex because the analytic calculations

are difficult. We have considered only the linearized equations of motion and the

nonlinear behaviour of the system, which can be very interesting, is hidden.

The linear stability map of the damped system has great importance. The effect of the

damping (particularly of a torsion damper) is well known from practical experience. It

customary to locate the torsion damper at the articulation next to nose gears of airplanes

and on motorcycles, see Figure 20. So the theory confirms the practice, namely the

bigger the damping is the more stable is the motion.

Figure 20: Dampers in practice

From the simulation new useful knowledge was obtained about the deformation of

the tyre and the analytic stability boundaries were confirmed. The building of the

experiment was very interesting and enjoyable. It was a big challenge and its

development is a big challenge in the future. Some of the results are compatible with the

theory. Perhaps the major differences are caused by the elasticity of the running belt and

by the approximation of the tyre model. In future the relaxation length at the leading

edge has to be adverted by the theory. The elimination of the elasticity of the running

belt is a complicated challenge. Above all the increasing of the stiffness of the

suspension has to be solved.

26

4. RIGID TYRE MODEL

The elastic tyre model is of major importance because of the prevalence of pneumatic

tyres. But the shimmy motion of a rigid tyre is well-known also. A shopping trolley has

rigid tyres. Sometimes one of these wheels shimmys because of the soft stiffness of the

suspension. We can model the behavior of this suspension by the model which is shown

in Figure 21.

This model is derived from [3]. In the figure the wheel has a single contact point (P)

with the rigid ground because of its hard stiffness. The radius of the wheel is R. The

caster length is l, the distance between the mass center of the caster and the king pin is

lc. The system is towed in the horizontal ( , )ξ η surface with constant v velocity. The

king pin is supported by springs of overall stiffness s.

z

x

y

x

vξ

η

Rφ(t)

lc

ψ(t)

l

q(t)

AC

O,P

P

O

s/2

s/2

mw , θwy , θwz

mc , θcz

Figure 21: Model of rigid tyre

4.1. ANALYTICAL CALCULATION

In this section the equations of motion of the undamped and the damped system are

analysed. The linear stability of both systems is studied. The nonlinear behavior of the

undamped model is also considered.

27

4.1.1. CONSTRAINTS

Our system has a geometrical constraint because the king pin is towed with constant

speed so

A 0v tξ − ⋅ = , (40)

where ξA is the position of the king pin in the absolute ( , , )ξ η ζ coordinate system.

There is a kinematical constraint too because we suppose that the wheel does not

slip. As a consequence, our system is non-holonomic. So the contact point has zero

velocity in the absolute coordinate system namely

P =v 0 . (41)

The velocity of the contact point can be written by

P Atrans OArel POrel= + +v v v v , (42)

where

Atrans

0

vq⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

v , (43)

OArel

sincos0

llψ ψψ ψ

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

v , (44)

POrel

cossin

0

RRϕ ψϕ ψ

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

v . (45)

So we get two scalar equations from (41) and (42):

( ) ( )sin cos 0l Rψ ψ ψ ϕ v− + = , (46)

( ) ( )cos sin 0q l Rψ ψ ψ ϕ− − = . (47)

The system is determined by four coordinates ξA, q, ψ and ϕ so the number of

degrees of freedom is four. The geometrical constraint decreases the number of degrees

of freedom by one. Furthermore the kinematical constraint decreases the number of

degrees of freedom by another one. So the number of degrees of freedom is two.

Because of the geometrical constraint (40), only three (n=3) general coordinates are

chosen: , 1q q= 2q ψ= and 3q ϕ= .

Equations (46) and (47) can be rewritten as:

28

10

n

k kk

A q Aβ β=

+ =∑ , (48)

in which 1,...., hβ = where h is the number of the scalar kinematical constraints i.e.

h=2. So

( 1)β = , 11 12 13 10, sin , cos , ,A A l A R A vψ ψ= = = − =

)

( 2β = , 21 22 23 21, cos , sin , 0.A A l A R Aψ ψ= = − = − =

4.1.2. EQUATIONS OF MOTION

Since our system is non-holonomic the Routh-Voss equations, [6], are used which have

the following structure:

1

dd

h

k kk k

T T Q At q q β β

β

µ=

∂ ∂− = +

∂ ∂ ∑ , (49)

where , T is the kinetic energy of the system, Q1,...,k = n k is the general force and µβ

are the Lagrange multipliers.

The kinetic energy has two components, one is the kinetic energy of the wheel and

the other is the kinetic energy of the caster:

w cT T T= + , (50)

where

2 T OO

1 12 2w w w wT m= +v ω Θ ωw (51)

and

2 T CC

1 12 2c c c cT m= +v ω Θ ωc . (52)

The velocity of the mass center of the wheel is

O

sincos

0

v lq l

ψ ψψ ψ

+⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

v , (53)

so 2 2 2 2 2

O 2 ( sin cosv q l l v q )ψ ψ ψ= + + + −v ψ . (54)

The angular velocity vector of the wheel is

29

( , , )

0

w

x y z

ϕψ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

ω . (55)

The matrix of the mass moment of inertia of the wheel is

O

( , , )

0 00 00 0

wx

w wy

wz x y z

θθ

θ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Θ , (56)

where wx wzθ θ≡ .

The velocity of the center of mass of the caster is

C

sincos

0

c

c

v lq l

ψ ψψ ψ

+⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

v , (57)

so 2 2 2 2 2

C 2 ( sin cosc cv q l l v q )ψ ψ ψ= + + + −v ψ . (58)

The vector of the angular velocity of the caster is

( , , )

00c

x y zψ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

ω . (59)

The matrix of the mass moment of inertia of the caster is

C

( , , )

... ...... ...... ...

cx

c cy

cz x y z

θθ

θ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Θ . (60)

Hence the kinetic energy can be calculated. After some manipulation:

2 2

2 2 2 2

1 ( )( ) 2( )( sin cos2( ) .

w c w c c

w c c wz cz wy

T m m v q m l m l v q

m l m l

)ψ ψ ψ

θ θ ψ θ ϕ

⎡= + + + + −⎣

⎤+ + + + + ⎦

(61)

We can define new variables so that in the following it will be more easily to handle the

expressions. Thus:

0

12 2

2

,,

.

w c

w c c

w c c wz

M m mM m l m l

M m l m l czθ θ

= += +

= + + +

(62)

With these new variables the kinetic energy is simple:

30

2 2 2 20 1 2

1 ( ) 2 ( sin cos )2 wyT M v q M v q Mψ ψ ψ ψ θ ϕ⎡ ⎤= + + − + +⎣ ⎦ .

sq q

(63)

For the Routh-Voss equation we need the general forces too which come from the

virtual power P of the active forces. We have only one active force caused by the spring

at the king pin. If δ denotes virtual quantities, this calculation gives

[ ]TA A

00 0

0P sq qδ δ δ

⎡ ⎤⎢ ⎥= ⋅ = − = −⎢ ⎥⎢ ⎥⎣ ⎦

F v δ

q

, (64)

that is, the general force depends only on the general coordinates q as follows

1

2

3

,0,0.

Q sQQ

= −==

(65)

Now we have everything to write the equations of motion and so the Routh-Voss

equations with the two scalar kinematical constraints (46) and (47) become: 2

0 1 1cos sinM q M M sq 2ψ ψ ψ ψ µ− + = − + , (66)

2 1 1

1 2

( cos cos sin ) ( cos sin )sin cos ,

M M v q q M v ql l

ψ ψ ψ ψ ψ ψ ψ ψµ ψ µ ψ

ψ+ − + − += −

(67)

1 2cos sinwy R Rθ ϕ µ ψ µ= − − ψ , (68)

sin cos 0l R vψ ψ ϕ ψ− + = , (69)

cos sin 0q l Rψ ψ ϕ ψ− − = . (70)

From the (48) kinematical constraint we can express the angular velocity of the rotation

of the wheel:

sincos

v lRψ ψϕ

ψ+

= , (71)

and if we differentiate this with respect to time then we obtain the angular acceleration: 2 2tan sin tan sin cos

cosv l l l

Rψ ψ ψ ψ ψ ψ ψ ψ ψϕ

ψ+ + +

= . (72)

Now we substitute (71) in (70) and we order the equation to get

tancos

lq v ψ ψψ

= + , (73)

which, after derivation with to respect to time becomes

31

22 tan

cos cos cosv l lq ψ ψ ψψ ψ ψ

= + + ψ . (74)

We can express µ2 from (66) as follows: 2

2 0 1 1cos sinM q M M s qµ ψ ψ ψ= − + +ψ , (75)

and with this and (72) we can express µ1 from (68) as:

21 2 2

20 1 1

tan sincos cos

tan sin sin tan tan .

wy lv lR

M q M M s q

θµ ψ ψ ψ ψ ψ

ψ ψ

ψ ψ ψ ψ ψ ψ ψ

⎛ ⎞= − + +⎜ ⎟

⎝ ⎠− + − −

(76)

Now we consider the Lagrange multipliers in (67) and we eliminate all the using (73)

and all q using (74). After some ordering we obtain a second-order differential

equation, which is one of the three scalar equations of motion. The other two equations

of motion are (71) and (73) which are first-order differential equations. Therefore the

equations of motion become:

q

2 220

2 1 2 2

2 01 2 2

22 2

0 2 2

2 tan coscos

tancos

sincos

0,

wy

wy

wy

M l lM M lR

M lvlvM vR

lM lR

slq

θ ψ ψ ψψ

θ ψ ψψ

ψθ ψψ

⎛ ⎞− + +⎜ ⎟

⎝ ⎠⎛ ⎞

+ − + +⎜ ⎟⎝ ⎠⎛ ⎞

+ +⎜ ⎟⎝ ⎠

+ =

tancos

lq v ψ ψψ

= + ,

sincos

v lRψ ψϕ

ψ+

= .

(77)

The last differential equation is not dependent on the others and in the following we do

not need it.

4.1.3. LINEAR STABILITY ANALYSIS OF THE UNDAMPED

SYSTEM

We can search for the trivial solution of the system, where:

0 0q q q≡ ⇒ = and 0 0, 0.ψ ψ ψ ψ≡ ⇒ = = (78)

In this case the trivial solution is

32

00, 0, .vq tR

ψ ϕ ϕ= = = + (79)

which is the linear motion of the wheel along a line.

For the linear stability analysis we have to write the first equation of (77) as two

first-order differential equations and the third equation of the equations of motion can

be eliminated. So the equations of motion written in the form of first-order differential

equations become:

,ψ ϑ=

2 01 2 2

2 220

2 1 2 2

22

0 2 22

2 220

2 1 2 2

2 220

2 1 2 2

tancos 1

cos2 tan

cos

sincos 1

cos2 tan

cos1 ,

cos2 tan

cos

wy

wy

wy

wy

wy

M lvlvM vR

M l lM M lR

lM lR

M l lM M lR

sl qM l lM M l

R

θ ψψ

ϑ ϑψ

θ ψψ

ψθψ

ϑψ

θ ψψ

ψθ ψ

ψ

⎛ ⎞− + +⎜ ⎟⎝ ⎠= −

⎛ ⎞− + +⎜ ⎟

⎝ ⎠⎛ ⎞

+⎜ ⎟⎝ ⎠−

⎛ ⎞− + +⎜ ⎟

⎝ ⎠

−⎛ ⎞

− + +⎜ ⎟⎝ ⎠

tancos

lq v ψ ϑψ

= + .

(80)

If these equations are linearized about their trivial solution with respect to small

perturbations x1, x2 and x3, a three dimensional linear ordinary differential equation is

obtained in the form:

=x A x ,

where

0 12 2

2 1 0 2 1 0

0 1 0

02 2

0

M lv M v s lM M l M l M M l M l

v l

⎡ ⎤⎢ ⎥−⎢ ⎥− −=

− + − +⎢ ⎥⎢ ⎥⎣ ⎦

A . (81)

The characteristic equation

det( ) 0λ =I A- (82)

can be transformed into the 3rd degree polynomial equation

33

2 3 2 22 1 0 0 1( 2 ) ( )M M l M l M lv M v sl slvλ λ λ 0− + + − + + = . (83)

The real parts of all the three characteristic roots λ1,2,3 will be negative, and so the

stationary running of the towed wheel will be asymptotically stable, if and only if the

conditions of the Routh-Hurwitz criterion are fulfilled [6]. That means all of the

coefficients of (83) and the Hurwitz determinants – in this case only H2 – have to be

positive:

0, 0,1,2,3ia i> = ,

2 1 0 3 0a a a a− > . (84)

From and we find that the stiffness at the king pin s has to be positive

and the towing length l and the towing speed v have to be the same sign. The condition

implies that

0 0a > 1 0a >

2 0a >

cl l> , ( )0 1M l M> (85)

which means that the center of the mass of the caster has to be nearer to the king pin

than the center of mass of the wheel.

The condition does not give a new criterion if we suppose that the mass

moments of inertia and the masses are positive which is of course true. In this case

3 0a >

2( )c c wz czm l l θ θ 0− + + > (86)

is always fulfilled.

The Hurwitz determinant gives

21 ( )cr c c wz czc c

l l m lm l

θ θ> = + + , ( )2 2 1crM M M< = l (87)

which is always a more stringent condition than (85) if the mass moments of inertia and

the masses are positive. So we obtain a stability map with a stable region which is

bordered by 0 0a = and . The stability map is shown by Figure 22. 2 0H =

34

0

l [m]

v [m/s]

Unstable

StableH2=0

a0=0

a2=0

a2=0

Figure 22: Linear stability map of the undamped system

The condition (85) is necessary but not sufficient and has a big importance because it

can be checked very easily. If we are towing a trailer with our car we should not put the

loads into the back of the trailer, because the system will be unstable.

It is very important to realize that, on the basis of the choice of model, the critical

caster length do not depend on the towing speed. That problem will be solved by the

location of a damper into the system.

On the stability boundary we can calculate the eigenvalues if we write 2 0H =

2 2cr 1M M M= = l

0

in (83) to find:

2 3 2 21 0 0 1( ) ( )M l M l M lv M v sl slvλ λ λ− + + − + + = . (88)

In this case the eigenvalues are given by:

1,2 31 0

, ,sl vM M l l

λ λ= ± = −−

Because (85) is satisfied on the 2 0H = stability boundary the eigenvalues can be

written:

1,2 3, ;( )c c

sl vim l l l

λ ω ω λ= ± = = −−

. (89)

As a consequence a Hopf bifurcation can be expected.

35

4.1.4. HOPF BIFURCATION

The behavior of the system can be analytically approximated close to the critical length

in (87). The approximation is based on the truncated power series of the nonlinear terms

in (80) at the critical parameters. So the equations of motion when 2 2cr 1M M M= = l are

given by:

,ψ ϑ=

2 01 2 2

2 220

1 2 2

22

0 2 22

2 220

1 2 2

2 220

1 2 2

tancos 1

costan

cos

sincos 1

costan

cos1 ,

costan

cos

wy

wy

wy

wy

wy

M lvlvM vRM l lM l

R

lM lR

M l lM lR

sl qM l lM l

R

θ ψψ

ϑ ϑψ

θ ψψ

ψθψ

ϑψ

θ ψψ

ψθ ψ

ψ

⎛ ⎞− + +⎜ ⎟⎝ ⎠= −⎛ ⎞− + +⎜ ⎟⎝ ⎠

⎛ ⎞+⎜ ⎟

⎝ ⎠−⎛ ⎞− + +⎜ ⎟⎝ ⎠

−⎛ ⎞− + +⎜ ⎟⎝ ⎠

tancos

lq v ψ ϑψ

= + .

(90)

After the series expansion of the nonlinear terms:

,ψ ϑ=

21 0 02

22

2 2 21 0 02

22

0 2 3 22

2 2 21 0 02

22

2 2 21 0 02

112

43

11 ,2

wy

wy

wy

wy

wy

lvM v M lv M lvRlM l M l M lR

lM lR

lM l M l M lR

sl qlM l M l M lR

θ ψϑ ψ ϑ

θ ψ

θψ ψ ϑ

θ ψ

ψθ ψ

⎛ ⎞− + + +⎜ ⎟ ⎛ ⎞⎝ ⎠= − +⎜ ⎟⎛ ⎞ ⎝ ⎠− + + +⎜ ⎟⎝ ⎠

+ ⎛ ⎞− +⎜ ⎟⎛ ⎞ ⎝ ⎠− + + +⎜ ⎟⎝ ⎠

⎛ ⎞− +⎜ ⎟⎛ ⎞ ⎝ ⎠− + + +⎜ ⎟⎝ ⎠

3 21 13 2

q v l v lψ ϑ ψ ψ= + + + ϑ .

(91)

36

We have to expand in series the second and third fractional coefficients of the second

equation what can be made using

22

1 1 O1

aa

ψψ

≈ − ++

(4) . (92)

After some manipulation we obtain the equations:

2

0 1

2 22 2

0 1 02 2

2 2 2

0 1 0 1

2 2

3 2

2

2 2(

3 2

0 1 0

0

0

0

.wy wy

v s l

l M l M l

l lM l M l M lv R R

l M l M l M l M l

v l

q qv l

slqθ θ

ψ ψ

ϑ ϑ

ψ ϑ ψϑ ψ

ψ ψ ϑ

− −−

+ + +

− −

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥+ − − +⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦

2

)

(93)

The nonlinear analysis requires the use of the Poincaré normal form of the system.

This can be obtained by a linear transformation based on the eigenvectors of the

corresponding characteristic roots of (89). The required eigenvectors can be calculated

by

( )k kλ =I A s 0- , (94)

where A is the matrix of the linear part of (93) namely:

2

0 1

0 1 0

0

0

v s l

l M l M l

v l

− −−

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

A . (95)

The calculated eigenvectors: 2

2 2 2

2 3

2 2 21 2

1

l i v

v l

v i l

v l

ω ω

ω

ω ω

ω

+

+

−−

+

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

s s , 3 1

0

l

v−

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

s . (96)

The transformation matrix can be written as

37

[ ]1 1Re Im=T s s 3s , (97)

so 2

2 2 2 2 2 2

2 3

2 2 2 2 2 21

1 0

l v

v l v l v

v l

v l v l

ω ω

ω ω

ω ω

ω ω

−+ +

−+ +

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

T

0

l

. (98)

Let us introduce the new variables [ ]T1 2 3x x x=x by

1

2

3

xx

q x

ψϑ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢= ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣

T

⎦

. (99)

With this (93) can be written in the following form:

( )(3)f= +Tx ATx Tx . (100)

If we multiply this equation on the right with

2 2 2

2 2 2 2 2 2 2 2

1

0 0 1

0v l

l v v v

v l v l v l

ω ω

ω ω

ω ω ω

−

−+ + +

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢⎢ ⎥⎢ ⎥⎣ ⎦

T ⎥ , (101)

then

( )1 1(3)f− −= +x T ATx T Tx . (102)

Now the Poincaré normal form of the equation is calculated:

1 1 3 1 2, 0

2 2 3 1 2, 0

3 3

0 0 ...0 0

...0 0 ...

j kj k jkj k

j kj k jkj kv

l

x x a x x

x x b x x

x x

ωω

+ =>

+ =>−

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤+⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥= + ⎢ ⎥+⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦

∑∑ . (103)

Because the third eigenvalue is negative the system is stable in the third direction. So

we can eliminate the third row of (103) and we have to calculate only the first and

second rows of the transformed equation where x3 appears too. The centre of manifold

(CM) can be approximated by a second-degree surface: 2 2

3 1 2 20 1 11 1 2 02 2( , ) ....x h x x h x h x x h x= = + + + . (104)

38

After the writing of (104) into (103) we obtain fourth-degree elements too which are not

needed because by the calculation only the third-degree parts are necessary. See Figure

23 to understand the structure of the phase space easily.

ψ

q

ψ

x1

x2

x3

Re s1

Im s1

s3

.

Figure 23: The structure of the phase space

The coefficients of the Poincaré normal form which we need later can be calculated.

The calculation is straightforward and so only the results are presented. The coefficients

which are needed for the δ parameter are: 6 3

30 2 2 26( )v la

v lωω

= −+ 3 ,

3 3 3 3

12 2 2 2 3

( 22( )v l v la

v lω ω ω

ω+

=+

) ,

( )

42

12 02 2 2 3 20 1

20 1 0 1 2 2

1 0 21 0

( )( )

2 3/ 24 ,

2( )

wy

wy

v l lb M l vM l M v l R

M l M M l MlM M l lR M M l

ω θω

θ ω

⎛⎛ ⎞= +⎜ ⎟⎜− + ⎝ ⎠⎝⎞⎛ ⎞− +

+ + + + ⎟⎜ ⎟⎜ ⎟ ⎟−⎝ ⎠ ⎠

2402 2

03 2 2 2 30 1

/1( ) 2

wyM l l Rv lb vv l M l M

θω ωω

⎛ ⎞+= − +⎜ ⎟⎜ ⎟+ −⎝ ⎠

2l .

(105)

The δ parameter determines the sense of the Hopf bifurcation. It is given by [7]:

( )( ) ( )(

}

20 02 11 20 02 20 02 20 02 11

30 12 21 03

1 183 3 ,

a a a b b b b a a b

a a b b

δω⎧ )⎡ ⎤= + − + − + + − −⎨ ⎣ ⎦⎩

+ + + + (106)

which simplifies to:

39

{ }30 12 21 031 3 38

a a b bδ = + + + . (107)

If δ is negative/positive, the trivial solution of (103) is weakly stable/unstable and the

Hopf bifurcation of the corresponding system is supercritical/subcritical. In our case

( )( )4

12 22 2 20 1

0.8

wyv l lM

RM l M v l

ωδ θω

⎛ ⎞= + >⎜ ⎟⎝ ⎠− +

(108)

So the limit cycle is subcritical.

The amplitude of the limit cycle can also be calculated. Before that we have to

choose the bifurcation parameter. The caster length is an important parameter of the

system and is often used by other authors, so we adopt it here.

We need the derivative of (83), the characteristic polynomial equation, with respect

to the bifurcation parameter l. It can be calculated by implicit derivation:

( ) ( )( )( )

23 2

2 2

d2 3d

d d2 2d d

c c c c wz cz

c c c

m l l m l ll

m v m v l l s l s l s vl l

0.

λλ θ θ λ

λ λλ λ λ

− + − + +

+ + − + + + = (109)

We manipulate this equation, using the expression for λ1 from (89). Then we calculate

the real part of the expression and after that we write the critical value of the bifurcation

parameter which comes from (87). In the end we obtain the required formula:

( )( ) ( ) ( )(

( )( )( ))

2 2 21

3 6 2 2 2 2

2

2

4 4dRed 2 3

.3

cr

c c c c cz wz

l l cz wz c c cz wz c c c c

c c cz wz

cz wz c c cz wz

m l sv m ll m l s m l sl m v

m l

s m l

θ θλ

θ θ θ θ

θ θ

θ θ θ θ

=

+ += −

⎡+ + + +⎣

+ +

⎤+ + + + ⎦

(110)

Now we can calculate the amplitude of the limit cycle from:

( )1Re.crl l

crr lλ

δ=

′= − − l (111)

The formula will be lengthy after the substitution so it is not presented here. We can

transform the amplitude back into the ( ), ,qψ ϑ original coordinate system using (98).

The amplitudes of the variables are

40

2 22

2 2 2 2 2 2

l vA rv l v lψω ωω ω

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟+ +⎝ ⎠⎝ ⎠,

2 22 3

2 2 2 2 2 2

v lA A rv l v lϑ ψω ωω ω

⎛ ⎞ ⎛= = − +⎜ ⎟ ⎜+ +⎝ ⎠ ⎝

⎞⎟⎠

,

qA r= .

(112)

We can plot these functions using the following parameters of the model:

[ ]

[ ][ ]

[ ]

[ ]

-5 2

-5 2

-6 2

0.3519 kg ,

4.63 10 kgm ,

2.38 10 kgm ,

0.04 m ,

0.0668 kg ,

0.012 m ,

3.48 10 kgm ,

100 N/m .

w

wy

wz

c

c

cz

m

R

m

l

s

θ

θ

θ

=

⎡ ⎤= ⋅ ⎣ ⎦⎡ ⎤= ⋅ ⎣ ⎦

=

=

=

⎡ ⎤= ⋅ ⎣ ⎦=

(113)

The bifurcation diagrams are shown in Figure 24, Figure 25 and Figure 26. The

amplitude of the limit cycle was plotted by different towing speeds. The green line

denotes stable solutions, the red dashed lines denote unstable solutions.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Aq [m]

l [m]

v=5 [m/s]

v=0.1 [m/s]

lcr=0.04603 [m]

Figure 24: The amplitude of the displacement of the king pin

41

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

0.2

0.4

0.6

0.8

1

Aψ [rad]

l [m]

for all v

lcr=0.04603 [m]

Figure 25: The amplitude of the angle of the caster

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-5

0

5

10

15

20

25

30

35

40

45

50Aψ’ [rad/s]

l [m]

for all v

lcr=0.04603 [m]

Figure 26: The amplitude of the angular velocity of the caster

As can be seen in the figures, the amplitude of the angle Aψ does not depend on the

towing speed, nor does Aψ the amplitude of the angular velocity.

4.1.5. EQUATIONS OF MOTION OF THE DAMPED SYSTEM

In the rigid tyre model damping can occur at the king pin or at the articulation of the

caster. In this project we study only the first case. To calculate the equations of motion

of the undamped system, the Routh-Voss equations were used. These needed a lot of

manipulation. The simplest method to calculate the equations of motion is the Appell-

42

Gibbs equations, see [6]. The equations of motion of the damped system were

calculated by this method and so we can compare both procedures.

First the pseudo velocities have to be chosen. The number of these velocities is equal

to the difference between the number of general coordinates and the number of

kinematical constraints, that is, 3 2 1− = . The simplest choice for this pseudo velocity is

the angular velocity of the caster:

σ ψ= . (114)

The kinematical constraints (46) and (47) and the pseudo velocity definition (114)

can be arranged into a system of linear algebraic equations:

0 sin cos1 cos sin 00 1 0

l R ql R

ψ ψψ ψ ψ

v

ϕ σ

− −⎡ ⎤ ⎡⎢ ⎥ ⎢− − =⎢ ⎥ ⎢⎢ ⎥ ⎢⎣ ⎦ ⎣

⎤ ⎡ ⎤⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎦ ⎣ ⎦

. (115)

The solution of these equations gives the description of the general velocities in terms

of the pseudo velocity and the general coordinates:

tancos

tancos

lvq

v lR R

ψ σψ

ψ σϕ σ ψ

ψ

⎡ ⎤+⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥+⎢ ⎥⎣ ⎦

. (116)

The general accelerations can also be expressed in terms of the general coordinates, the

pseudo velocity σ and the pseudo accelerationσ :

22

22

tancos cos cos

tan tancos cos

v l lq

v l lR R R

ψσ σ σψ ψ ψ

ψ σϕ ψ σ σ σ ψ

ψ ψ

⎡ ⎤+ +⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥+ +⎢ ⎥⎣ ⎦

. (117)

For the Appell-Gibbs equation the so-called acceleration energy S is needed which

consists of the acceleration energies of the caster and the wheel:

2 T O OO

2 T C CC

1 1 ( )2 2

1 1 ( ) ... .2 2

w w w w w w w w

c c c c c c c c

S m

m

= + + ⋅

+ + + ⋅ +

a ε Θ ε ε ω Θ ω

a ε Θ ε ε ω Θ ω (118)

The acceleration of the centre of gravity of the wheel is

43

2

2O O

sin coscos sin

0

l lq l lψ ψ ψ ψ

ψ ψ ψ ψ

⎡ ⎤+⎢ ⎥= = − +⎢ ⎥⎢ ⎥⎣ ⎦

a v , (119)

and the acceleration of the centre of gravity of the caster is 2

2C C

sin coscos sin

0

c c

c c

l lq l l

ψ ψ ψ ψ

ψ ψ ψ ψ

⎡ ⎤+⎢ ⎥= = − +⎢ ⎥⎢ ⎥⎣ ⎦

a v . (120)

The angular velocities are given by (55) and (59). The matrixes of the mass moment of

inertia of the caster and the wheel are given by (56) and (60). The angular acceleration

of the wheel can be calculated by

( , , )

ˆw w w c w

x y z

ϕψϕψ

−⎡ ⎤⎢ ⎥= = + × = ⎢ ⎥⎢ ⎥⎣ ⎦

ε ω ω ω ω , (121)

where the hat denotes derivation with respect to time in the ( , , )x y z coordinate system.

The angular acceleration of the caster has a simple form:

( , , )

0ˆ 0c c c c c

x y zψ

⎡ ⎤⎢ ⎥= = + × = ⎢ ⎥⎢ ⎥⎣ ⎦

ε ω ω ω ω . (122)

Using the formulae (116) and (117), the acceleration energy can be written as a function

of the general coordinates, the pseudo velocity and the pseudo acceleration. Among the

general coordinates, only the angle ψ of the caster appears. For the Appell-Gibbs

equation we need only that part of the acceleration energy which depends on the pseudo

acceleration σ . So after some manipulation the final formula of the acceleration energy

takes the form:

( )

2 22

2 2

22 2 2

3 2

2

1 tan( , , ) tan 2 2 tan2 cos cos

1 1 1tan 1 2 22 cos cos cos

1 ... .2

w wy

c cc

wz cz

l vS m lR l

l lvm ll l l

ψ σσ σ ψ θ σ ψ σ σ ψψ ψ

ψtanψ σ σψ ψ ψ

θ θ σ

⎛ ⎞⎛ ⎞= + + +⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟+ + − + − +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

+ + +

σ σ (123)

The right hand side of the Appell-Gibbs equation is the so-called pseudo force Π

which has to be calculated via the virtual power of the active forces caused by the

spring and the damper at the king pin. The virtual power is given by:

44

[ ] ( )TA A

2

2

00 0

0

tan tancos cos

tan ,cos cos cos

P sq kq q sq kq q

lsq k v v

l lsq klv k

l

δ δ δ δ

ψ σ δ ψ σψ ψ

ψ σ δσψ ψ ψ

⎡ ⎤⎢ ⎥= ⋅ = − − = − −⎢ ⎥⎢ ⎥⎣ ⎦

⎛ ⎞⎛ ⎞ ⎛= − − + +⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞

= − − −⎜ ⎟⎝ ⎠

F v

⎞ (124)

where δ denotes virtual quantities. So the pseudo force is 2

2

tan( , , )cos cos cos

l lq sq klv kψσ ψ σψ ψ ψ

Π = − − − . (125)

The Appell-Gibbs equation can be written in the following form:

Sσ∂

= Π∂

. (126)

For the easy comparison with the undamped system we rewrite the pseudo velocity

( )ϑ σ ψ≡ ≡ and make use of the simplifications in (62). So the equations of motion of

the two degree of freedom damped system become:

,ψ ϑ=

22 0

1 2 2

2 220

2 1 2 2

22

0 2 22

2 220

2 1 2 2

2 220

2 1 2 2

tancos cos 1

cos2 tan

cos

sincos 1

cos2 tan

cos1

co2 tan

cos

wy

wy

wy

wy

wy

M lvlv klM vR

M l lM M lR

lM lR

M l lM M lR

slM l lM M l

R

θ ψψ ψ

ϑ ϑψ

θ ψψ

ψθψ

ϑψ

θ ψψ

θ ψψ

⎛ ⎞− + + +⎜ ⎟⎝ ⎠= −⎛ ⎞

− + +⎜ ⎟⎝ ⎠

⎛ ⎞+⎜ ⎟

⎝ ⎠−⎛ ⎞

− + +⎜ ⎟⎝ ⎠

−⎛ ⎞

− + +⎜ ⎟⎝ ⎠

2 220

2 1 2 2

s

tan 1 ,cos

2 tancos wy

q

kl vM l lM M l

R

ψ

ψψ

θ ψψ

−⎛ ⎞

− + +⎜ ⎟⎝ ⎠

tancos

lq v ψ ϑψ

= + ,

sincos

v lRψ ψϕ

ψ+

= .

(127)

45

4.1.6. LINEAR STABILITY ANALYSIS OF THE DAMPED

SYSTEM

Of course the damped system has the same trivial solution given by (79). As before we

do not need the fourth independent equation of motion, so the first three equations of

(127) when linearized can be written in the form

=x A x ,

where

20 1

2 2 22 1 0 2 1 0 2 1 0

0 1 0

2 2 20

M lv M v klk l v s lM M l M l M M l M l M M l M l

v l

⎡ ⎤⎢ ⎥− +⎢ ⎥− − −=⎢ ⎥− + − + − +⎢ ⎥⎣ ⎦

A . (128)

The characteristic equation (82) then becomes the following 3rd degree polynomial

equation 2 3 2 2 2

2 1 0 0 1( 2 ) ( ) ( )M M l M l M lv M v kl sl kl v slvλ λ− + + − + + + + = 0λ . (129)

The stationary running of the towed wheel is asymptotically stable, if and only if the

conditions of the Routh-Hurwitz criterion are fulfilled which means that all of the

criterion of (84) have to be fulfilled.

Because the damping factor k and the stiffness s are positive, from the

condition we obtain that the towing length l and the towing speed v have to have

the same sign. From the condition we do not get any important condition. The

condition is

0 0a >

1 0a >

2 0a >

2 0c c ckl m lv m l v+ − > . (130)

The condition is the same as in the undamped system. It is given in (86) and

is always fulfilled because of the positive signs of the mass moments of inertia and the

mass. The strongest condition which determines the linear stability comes from the

Hurwitz determinant. Namely

3 0a >

23 2 2 0c c c c c c c wz cz

k k k kl l m l m v l m l m l vv s s s

θ θ⎛ ⎞+ + + − − − − >⎜ ⎟⎝ ⎠

. (131)

46

The linear stability boundary of the damped system given by (131) is shown in Figure

27. The location of the maximum point of the curve – which is shown in Figure 28 as a

function of the damping – was calculated by implicit derivation:

( )crc c

s lv lm l l

=−

. (132)

1 2 3 4

0.01

0.02

0.03

0.04

00

l [m]

v [m/s]

k=0.25 [Ns/m]

Unstable

Stable

Figure 27: Linear stability of the damped system

1 2 3 4

0.01

0.02

0.03

0.04

00

l [m]

v [m/s]

k=0.25 [Ns/m]

k=0.05 [Ns/m]

k=1.0 [Ns/m]

k=5.0 [Ns/m]

k=0.0 [Ns/m]

vcr

Figure 28: Linear stability boundaries for different damping factors

The damped model is more similar to reality than the undamped one, because the

stable caster length depends on the towing speed which is similar to practical

experience.

The eigenvalues can be calculated at the stability boundary 2 0H = :

1,2 32 2 2

( ), ;( )c c wz cz

l sl vk svim l l sl kv

λ ω ω λθ θ

+= ± = = −

− + + +. (133)

So a Hopf bifurcation can be expected there.

47

4.2. CONTINUATION

The analysis of the nonlinear system has its limitations. The other possibility which can

give new information about the system is simulation. We can obtain bifurcation

diagrams by numerical continuation. The software called AUTO [8] was developed to

continue bifurcations in differential equations and has long been used in the Department

of Engineering Mathematics in Faculty of Engineering at the University of Bristol. The

Department has a lot of experience with AUTO and the importance of the numerical

analysis can not be queried. The analytical calculation of the Hopf bifurcation in the

undamped system was checked using AUTO and some very interesting results were

obtained for the damped system.

4.2.1. UNDAMPED SYSTEM

AUTO needs only the equations of motion of the system which, for the undamped

system, are given by (80). The parameters of the system are given by (113). The

bifurcation parameter was the caster length l. First the linear stability of the system was

checked and a Hopf point was detected. The location of the Hopf point was identical to

the location determined by the analytic formula (87). So the unstable periodic solution

was followed using AUTO. The path of the periodic solution is shown in Figure 29,

Figure 30 and Figure 31 at a speed of 5 [m/s].

0 0.05 0.1 0.15 0.2 0.25

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Aq [m]

l [m]

lcr=0.04603 [m]

Theore

tical

Numerical

Figure 29: The amplitude of the displacement of the king pin

48

0 0.05 0.1 0.15 0.2 0.25

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Aψ [rad]

l [m]

lcr=0.04603 [m]Th

eore

tical

Numerical

Figure 30: The amplitude of the angle of the caster

0 0.05 0.1 0.15 0.2 0.25

0

2

4

6

8

10

12

14

16Aψ’ [rad/s]

l [m]

lcr=0.04603 [m]

Theo

retic

al

Numerical

Figure 31: The amplitude of the angular velocity of the caster

In each figure the theoretical bifurcation curves are displayed too. The theoretic and

the numeric curves are tangential near to the Hopf point what confirms the analytical

calculations. The convergence of the numeric curves can be observed in Figure 32,

Figure 33 and Figure 34. In Figure 33 and Figure 34 there is a logarithmic scale on the

horizontal axis so it is easy to see that the curves converge to zero. The logarithmic

slopes of the curves at l=600÷800 [m] are nearly -0.5.

The time histories and trajectories of the two unstable periodic solutions marked 1

and 2 in Figures 32, 33 and 34 are shown together in Figure 35 and Figure 36.

49

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Aq [m]

l [m]

1

2

Figure 32: The amplitude of the displacement of the king pin

10 -1 10 0 1010

0.05

0.1

0.15

0.2

0.25Aψ [rad]

l [m]

1

2

Figure 33: The amplitude of the angle of the caster

50

10 -1 10 0 10 10

1

2

3

4

5

6

7

8

9

10 Aψ’ [rad/s]

l [m]

1

2

Figure 34: The amplitude of the angular velocity of the caster

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

q (t)

[ m]

Tnorm [1]

2

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

ψ (t)

[ rad

]

Tnorm [1]

2

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

-6

-4

-2

0

2

4

6

8

ψ’( t)

[ra d

/s]

Tnorm [1]

2

1

Figure 35: The time histories of the unstable periodic solutions

In the case of a long caster length the trajectories have an interesting meaning. From

the figures we can determine the motion of the system, which has a good agreement

51

with the practical experience, namely that in case number 2, the wheel has only a small

motion in the lateral direction and the king pin is vibrating. The contact point of the

wheel behaves approximately as an articulation. Because of the long caster length the

angular deflection of the caster is small, so the displacement of the end of the caster,

which is the displacement of the king pin, can be linearized. Therefore the trajectory of

this motion is the simple line in the first diagram of Figure 36. Of course we have to

realize that the unstable limit cycle of the system has very large amplitude.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

q(t) [m]

ψ (t )

[rad

]

1

2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-8

-6

-4

-2

0

2

4

6

8

q(t) [m]

ψ’( t)

[ rad

/ s]

1

2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15-8

-6

-4

-2

0

2

4

6

8

ψ’( t)

[ rad

/ s]

1

2

ψ(t) [rad] Figure 36: The trajectories of the unstable periodic solutions

4.2.2. DAMPED SYSTEM

The linear behaviour of the damped system is similar to practical experience. So the

results of the continuation of the damped system may have a greater significance. We

obtained new information about the damped model. We have found a fold in the

unstable periodic solution and we have followed the path of the stable periodic solution.

52

We have investigated the effects of the damping and the towing speed and we

discovered an isola. We have followed the separated isola and we have plotted the

locations of folds.

Bifurcation diagrams are shown by Figure 37, Figure 38 and Figure 39. Our model

has two stable motions if the caster length is longer than the critical length. If the caster

length is shorter then the system is linearly unstable. In this case the motion of the

wheel is determined by the stable periodic solution. The similarity of the results to the

practical experiences is arguable because the large value of amplitude of the angle of the

caster; it is nearly π/2. Probably by including the lateral sliding of the wheel would the

behaviour of the model would be more similar to reality.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

1

2

3

4

5

6Aq [m]

l [m]

k=0.2 [Ns/m]v=5 [m/s]

Fold

Unstable periodic solution

Stable periodic solution

Hopf point

Figure 37: The amplitude of the displacement of the king pin

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Aψ [rad]

l [m]

k=0.2 [Ns/m]v=5 [m/s]

Fold

Unstable periodic solution

Stable periodic solution

Hopf point

Figure 38: The amplitude of the angle of the caster

53

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

50

100

150

200

250Aψ’ [rad/s]

l [m]

k=0.2 [Ns/m]v=5 [m/s]

Fold

Unstable periodic solution

Stable periodic solution

Hopf point

Figure 39: The amplitude of the angular velocity of the caster

From these the bifurcation diagrams of the damped system it is possible that an isola

birth can occur if we increase the damping factor. After experimentation this was found

for a speed of 5 [m/s] and shown in Figure 40, Figure 41 and Figure 42. There are five

marked points in the figures. The time histories of the periodic solutions of these points

are plotted in Figure 43. The corresponding trajectories are shown by Figure 44.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.5

1

1.5

2

2.5

3

3.5

4

l [m]

Aq [m]

k=0.2688349 [Ns/m]v=5 [m/s]

1

2

3

4

5

Figure 40: The amplitude of the displacement of the king pin at the isola birth

54

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

l [m]

Aψ [rad]

k=0.2688349 [Ns/m]v=5 [m/s]

1

2

345

Figure 41: The amplitude of the angle of the caster at the isola birth

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

50

100

150

200

250

300

l [m]

Aψ’ [rad/s]

k=0.2688349 [Ns/m]v=5 [m/s]1

2

3

4

5

Figure 42: The amplitude of the angular velocity of the caster at the isola birth

55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3

4

q(t)

[m]

Tnorm [1]

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

ψ(t )

[rad

]

Tnorm [1]

1

2

3

45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-300

-200

-100

0

100

200

300

ψ’(t)

[rad

/s]

Tnorm [1]

1

2

3

4

5

Figure 43: The periodic orbits

-4 -3 -2 -1 0 1 2 3 4-1.5

-1

-0.5

0

0.5

1

1.5

ψ(t)

[ra d

]

q(t) [m]

1

2

3

45

-4 -3 -2 -1 0 1 2 3 4-300

-200

-100

0

100

200

300

ψ ’(t)

[ rad

/s]

q(t) [m]

1 2

3

45

56

-1.5 -1 -0.5 0 0.5 1 1.5-300

-200

-100

0

100

200

300

ψ’( t)

[ra d

/s]

ψ(t) [rad]

12

34

5

-1.5-1

-0.50

0.51

1.5 -300

-200

-100

0

100

200

300-3

-2

-1

0

1

2

3

q(t) [m]

ψ(t) [rad]ψ’(t) [rad/s]

1 2

3

4

5

Figure 44: The trajectories of the periodic solutions

The periodic solutions numbered 2 and 4 are for the same caster length. So if we

deflect the system more than the amplitude of that at point 2 then the motion will be

determined by trajectory number 4.

If we increase the damping factor a little bit then the periodic solution divides into

two branches. The isola on the right side has two folds and the branch on the left side

has one fold. The bifurcation diagrams are shown in Figure 45, Figure 46 and Figure 47.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

l [m]

Aq [m]

k=0.35 [Ns/m]v=5 [m/s]

Fold 1

Fold 2

Fold 3

Figure 45: The amplitude of the displacement of the king pin

57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

l [m]

Aψ [rad]

k=0.35 [Ns/m]v=5 [m/s]

Fold 1

Fold 2

Fold 3

Figure 46: The amplitude of the angle of the caster

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

l [m]

Aψ’ [rad/s]

k=0.35 [Ns/m]v=5 [m/s]

Fold 1

Fold 2

Fold 3

Figure 47: The amplitude of the angular velocity of the caster

The effect of changing the damping factor is shown in Figure 48, Figure 49 and

Figure 50. Note the location of the folds for different values of the damping.

58

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

l [m]

Aq [m]

k=0.2688349 [Ns/m]

k=0.25 [Ns/m]

k=0.3 [Ns/m]

k=0.35 [Ns/m]

v=5 [m/s]

Figure 48: The amplitude of the displacement of the king pin

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

l [m]

Aψ [rad]

k=0.2688349 [Ns/m

]

k=0.25 [Ns/m]

k=0.3 [Ns/m

]

k=0.35 [Ns/m

]

v=5 [m/s]

Figure 49: The amplitude of the angle of the caster

59

0 0.5 1 1.50

20

40

60

80

100

120

140

160

180Aψ’ [rad/s]

k=0.2688349 [Ns/m]

k=0.25 [Ns/m]k=0.3 [Ns/m]k=0.35 [Ns/m]

l [m]

v=5 [m/s]

Figure 50: The amplitude of the angular velocity of the caster

The effect of changing the towing speed is displayed in Figure 51, Figure 52 and

Figure 53.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

4v=5 [m/s]

v=4 [m/s]

v=3 [m/s]

v=2.2 [m/s]

v=1.5507 [m/s]

v=1 [m/s]

v=0.5 [m/s]

l [m]

Aq [m]

k=0.2688349 [Ns/m]

Figure 51: The amplitude of the displacement of the king pin

60

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

l [m]

Aψ [rad]

v=5 [m/s]

v=4 [m/s]

v=3 [

m/s]

v=2.2

[m/s]

v=1.5

507 [

m/s]

v=1

[m/s]

v=0.

5 [m

/s]

k=0.2688349 [Ns/m]

Figure 52: The amplitude of the angle of the caster

0 0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

200

Aψ’ [rad/s]

l [m]

v=5 [m/s]

v=4 [m/s]

v=3 [

m/s]

v=2.2

[m/s]

v=1.5

507 [

m/s]

v=1

[m/s]

v=0.

5 [m

/s]

k=0.2688349 [Ns/m]

Figure 53: The amplitude of the angular velocity of the caster

The dependence of the system on the damping and the velocity is complex. It can not

be claimed that a decrease in the towing speed causes more stability. The effect of the

damping is more concrete. The bigger the damping factor the more stable is the

structure. The behaviour of the system and the effect of the towing speed are well

represented in Figure 54 and Figure 55. It is worth noting, that the splitting of the

branch, that is the isola birth, is at the same caster length.

61

0 1 2 3 4 5 6 7 80

0.02

0.04

0.06

0.08

0.1

v [m/s]

Unstable

Bistable

Stable

S

k=0.2688349 [Ns/m]

k=0 [Ns/m]Isola birth

l [m]

(linearly unstable, stable limit cycle)

(linearly stable)

(linearly stable, stable limit cyle)

Fold 1

Fold 2

Fold 3

Figure 54: Stability map of the system 1

0 1 2 3 4 5 6 7 80

0.02

0.04

0.06

0.08

0.1

v [m/s]

Unstable

Bistable

Stable

S

k=0 [Ns/m]

l [m]

k=0.5 [Ns/m]

Isola birth

(linearly unstable, stable limit cycle)

(linearly stable)

(linearlystable)

(linearly stable,stable limit cyle)

Fold 1

Fold 2

Fold 3

Figure 55: Stability map of the system 2

The increase of the towing speed in linearly stable system first causes a nonlinear

instability because it generates a stable limit cycle. If we increase the speed again and

the isola is born then the system will have only one stable solution. So the system is

bistable only in a region of speed what is similar to that of practical experience.

4.3. SUMMARY

Anybody who has a pushed a supermarket shopping trolley is aware of shimmying

wheels. If that person has some technical interest and played with the pushing speed

62

during shopping, they may have tried pushing the trolley faster and slower and

observed the behaviour of the wheel. They may have then noticed that usually shimmy

appears at some range of speeds and disappears outside this region as it is in our system.

Of course engineers do not have to eliminate wheel shimmy in shopping trolleys.

Dangerous accidents do not usually occur and nobody gets injured.

But by modelling a pneumatic tyre as rigid we have obtained some useful

information about shimmying. We have analysed our model both theoretically and

numerically. Both methods give us a lot of information and we have obtained some new

results. The nonlinear stability charts of the system and the effect of the damping are

similar to practical experience. If we included the sliding of the wheel, the behaviour of

the model would be more similar to reality. In future the continuation of a sliding model

can be made by AUTO and perhaps we would have the ideal system which can model

the motion of a rigid tyre.

63

5. CONCLUSION

In this thesis we have studied two different models of shimmying, both of which can be

found in the literature of the phenomenon. Damping has been added and both analytic

and numeric methods have been used to investigate these systems. The elastic model

was also investigated by experiment. The acquired knowledge can be very useful for the

investigation of the longitudinal deformation of elastic tyre which has great importance

in the design of electronic vehicle stabilizers, for example the ABS algorithm. But it is

worth noting that the models in this thesis can describe only the simplest behaviour of

wheels. The analysis and understanding of these simple models are necessary for

investigation of the complex real systems.

Acknowledgements

In conclusion I would like to say special thanks to Professor Gábor Stépán, who was my

tutor during my time at university and to Professor John Hogan, who was my tutor my

time in Bristol. I am grateful for useful discussions with Gábor Orosz on the rigid tyre

model. Finally I thank Professor László Kocsis, Tibor Gáspár, Róbert Paróczai and

Attila Zsidai for help with the experimental measurements.

64

6. REFERENCES

[1] B. von Schlippe and R. Dietrich. Shimmying of a pneumatic wheel. Lilienthal-

Gesellschaft für Luftfahrtforschung, 140:125-160, 1941. (translated for the AAF

in 1947 by Meyer & Company).

[2] G. Stépán: Delay, nonlinear oscillations and shimmying wheels. F. C. Moon (ed.),

Applications of Nonlinear and Chaotic Dynamics in Mechanics, pp. 373-386.,

Kluwer Academic Publisher, Dordrecht, 1998.

[3] Gábor Stépán: Chaotic motion of wheels. Vehicle System Dynamics, 20(6), pp.

341-351. 1991.

[4] H. Pacejka: Modelling of the pneumatic tyre and its impact on vehicle dynamic

behaviour. Technical report, Technical University of Delft, The Netherlands,

1988.

[5] Dénes Takács: Instability of rolling wheel caused by time delay, Scientific

Conference of Students, Budapest November 11, (2003)

[6] Gantmacher, F.: Lectures in Analytical Mechanics, MIR Publishers, Moscow,

1975.

[7] G. Stépán: Retarded Dynamical System, Longman, Harlow Essex, 1989.

[8] E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, X. Wang:

AUTO 97: Continuation and bifurcation software for ordinary differential

equations (with HomCont), January 9, (2003)

65

7. SUPPLEMENT CD

On the supplement CD of this thesis there are some interest animations which may help

to understand the thesis. There are videos about the measurements, the simulation and

the shimmy motion. In addition the thesis can be found in electronic form.

To browse of CD an internet browser is needed. The site was developed using

Microsoft Internet Explorer. To play the videos Microsoft Media Player is

recommended.

66