iii

Abstract

The objective of this project is to consider alternative methods for measuring tyre rolling resistance. This project focuses particularly on testing heavy vehicle tyres under heavy load conditions in the region of 4 tonne.

Chapter 1 presents a background to the rolling resistance phenomenon and explains the importance of measuring it, particularly for tyre design. A review of the standard methods for measuring rolling resistance is given, and a laboratory method for testing small tyres is presented which lends itself to being extended for use with larger tyres under higher load.

The design problem is defined in more detail in Chapter 2, and four conceptual solutions for the problem are introduced.

Chapter 3 analyses the case of a rolling axle pendulum, which is one of the considered solutions. A dynamic model is suggested, and several aspects such as angular velocity and contact forces are simulated for certain design choices.

Chapter 4 draws conclusions of the project and gives suggestions for future work, which include further investigation of the candidate solutions, and designing and building a prototype of a measuring rig.

iv

Table of contents

Nomenclature .......................................................................................................................................... v

Chapter 1 - Introduction .......................................................................................................................... 8

1.1. Background.............................................................................................................................. 8

1.2. Standard methods for measuring rolling resistance ................................................................. 9

1.3. Previous work ........................................................................................................................ 10

1.4. Project objective .................................................................................................................... 12

1.5. Conclusions ........................................................................................................................... 13

1.6. Figures ................................................................................................................................... 13

Chapter 2 - Conceptual design of a rolling resistance measuring rig .................................................... 17

2.1. Introduction ........................................................................................................................... 17

2.2. Specification .......................................................................................................................... 17

2.3. Embodiment design ............................................................................................................... 17

2.4. Suggested concepts ................................................................................................................ 18

2.5. Summary and conclusions ..................................................................................................... 20

2.6. Figures ................................................................................................................................... 20

Chapter 3 - Dynamics of a rolling axle pendulum ................................................................................ 23

3.1. Introduction ................................................................................................................................ 23

3.2. Mass distribution along the axle ................................................................................................ 23

3.3. 2D dynamic model of rigid eccentric pendulum ........................................................................ 24

3.4. 3D dynamic model of rigid eccentric pendulum ........................................................................ 29

3.5. Conclusions ................................................................................................................................ 35

3.6. Figures........................................................................................................................................ 35

Chapter 4 - Conclusions and future work ............................................................................................. 43

4.1. Conclusions ................................................................................................................................ 43

4.2. Future work ................................................................................................................................ 43

Appendix A ........................................................................................................................................... 44

Appendix B ........................................................................................................................................... 46

References ............................................................................................................................................. 47

v

Nomenclature

rF : Rolling resistance force

rC : Rolling resistance coefficient

: Rotation angle. Zero angle is defined when the centre of gravity is aligned under

the axle. And positive value is defined in Figure 3.4

min i , max i , max 1i : Rotation angle, see Figure 1.5

cp : A rolling angle which is account for travel of contact patch length.

1R : external radius of tyre

2R : radius of dead weight cylinder

e : distance between axle and centre of gravity, radius of eccentricity

.c mJ : moment of inertia of the whole system about the centre of mass yyI

g : earth gravity coefficient

DWe : distance between centre of gravity of dead weight and the axle, see Figure 3.3

1 2,m m : mass of concentric parts and mass of eccentric part of a pendulum. see Figure 3.3

1 2,U U : velocity of mass 1m and 2m

1 2,J J : moment of inertia of concentric and eccentric part of a pendulum. see Figure 3.3

1L : distance along y axis between centre of gravity and test tyre

2L : distance along y axis between centre of gravity and rigid wheel

1 2,X X : longitudinal contact forces acting on tyre and on rigid wheel, respectively

1 2,Y Y : lateral contact forces acting on tyre and on rigid wheel, respectively

1 2,Z Z : vertical contact forces acting on tyre and on rigid wheel, respectively

X : total contact forces acting on the system in x direction

Z : total contact forces acting on the system in y direction , ,x y z : inertial coordination frame

Ω : vector of rotational velocity about c.g, described in body frame

I : tensor of inertia of the whole system about c.g, described in body frame

,xy zyI I : tensor of inertia components

vi

x : vector of unknown forces

: steady-state rolling resistance coefficient

: transient rolling resistance coefficient

: Rolling resistance coefficient as affected from both transient and steady-state

factors

E : mechanical energy (gravitational and kinetic)

m : total mass of the system

d : distance travelled

cpL : contact patch length

1 2,d d : distance between the tyre and the outer weight and of the inner weight respectively

(see Figure 3.1)

totalL : total length of the axle

aL : distance between the test tyre and the outer weight (see Figure 3.1)

bL : distance between the inner weight and the outer weight (see Figure 3.1)

cL : distance between the solid wheel and outer weight (see Figure 3.1)

1 2,a a : width of the outer weight and of the inner weight respectively (see Figure 3.1)

1 2,W W : gravity force of the outer weight and the inner weight respectively (see Figure 3.1)

F : total force on the systes in Figure 3.4 and in Figure 3.10

,x z

F F : components x and z of the total force on the system in Figure 3.4,

m : total mass on the system in Figure 3.4

W : system total weight, equal to mg

.c gr : location of the centre of gravity

, ,,equ x equ zr r : 2D location vector of centre of gravity when the system is in equilibrium 0

, , , ,,equ x equ y equ zr r r : 3D location vector of centre of gravity when the system is in equilibrium 0

Z : normal force acting on the wheel at the contact area with the floor (see Figure 3.4)

X : longitudinal force acting on the wheel at the contact with the floor (see Figure 3.4)

y : total torque about y axis

.c gτ : total torque vector about the centre of gravity

1rC

2rC

1

,1 1 1

1 1r rrr fitted

F dx Z C dxFC

Z Z Zdx dx

vii

yH : momentum about y axis

.c gH : momentum vector about the centre of gravity

,A B : expressions of the differential equation in the 2D approach

,A B : matrixes expressing the linear equation system in the 3D force calculation

t : time from motion initiation

0 : initial rotation angle

,maxkinT : maximal kinetic energy throughout a cycle

,maxgravT : maximal gravitational potential energy throughout a cycle

,maxs : maximal static friction coefficient

frF : static friction force

1stasicZ : vertical load in the test wheel when the system is stationary

f : motion frequency

8

Chapter 1 -

Introduction

1.1. Background

Fuel cost is one of the major expenditures for heavy goods vehicle operators. Fuel consumption is also

the direct cause of vehicle carbon emission. Accordingly there is continuous pressure to improve

vehicle fuel economy and at the same time to reduce their environmental damage per freight task [1].

Fuel is the vehicle energy source, whereas several factors serve as the vehicle energy sinks. The major

energy consumers on a vehicle are engine thermodynamic loss, braking losses, rolling resistance of

the drive train and the wheel bearings, as well as tyre rolling resistance and aerodynamic drag. Each

of these factors contributes to the total fuel consumption. This project focuses particularly on the tyre

rolling resistance. Given any road freight task, minimizing the rolling resistance of truck tyres

contributes to both fuel saving and emissions reduction.

In order to reduce energy loss from tyres, it is necessary to understand the nature of rolling resistance.

Rolling resistance originates in internal forces in the tyre material. When a tyre is loaded and is rolled

against a hard surface, such as a road, its rubber deforms. Rubber is a viscoelastic material. As such,

its elastic deformation stores energy, while its viscous deformation dissipates energy as heat.

Irrecoverable loss of energy is mainly caused by hysteresis and friction [1].

Rolling resistance force is defined by considering a vehicle driving at a constant speed in a straight

line on a flat and horizontal road. The horizontal force that opposes the vehicle’s motion, acting at the

contact between the tyre and the road is known as the rolling resistance force [2]. Rolling resistance

force rF is defined as the energy consumed per unit distance of travel. According to the International

System of Units (SI), the unit conventionally used for rolling resistance force is the /N m N , which

is equivalent to a drag force in Newtons.

A number of factors influence the rolling resistance force of a tyre. The dominant ones are vertical

load, inflation pressure, tyre structure and tyre material. Steer angle and speed also affect the

instantaneous rolling resistance as it changes during a journey. Other factors such as road camber,

temperature and rim width have a lesser effect on the rolling resistance force [1].

For comparison purposes a non-dimensional measure of rolling resistance is used: the rolling

resistance coefficient rC . It is defined as the ratio of rolling resistance force to vertical load on a tyre. This metric is dimensionless [3]. Assuming the rolling resistance coefficient is uniform and can be scaled across different loads, it may serve as a single criterion to compare different tyres [4].

9

Recent regulations enforce tyre manufacturers to label their products with their rolling resistance

coefficient rating. The European Union, for example, has applied such regulations from November

2012 [5]. The law refers to a standard practice, which describes how the rolling resistance coefficient

should be measured. Accordingly, tyres are classified into a number of bands, from the least efficient

to the most, depending on their rolling resistance coefficient. This labelling scheme allows consumers

to consider tyre rolling resistance when purchasing tyres, taking fuel consumption into account.

Figure 1.1 shows an example of a European tyre label. The fuel efficiency section on the left is based

on the rolling resistance coefficient.

1.2. Standard methods for measuring rolling resistance

Procedures for measuring the rolling resistance of pneumatic tyres are specified by international

standards. The International Organization for Standardization (ISO), the Society of Automotive

Engineers (SAE) and the United Nations Economic Commission for Europe (UNECE), all publish

methodologies for measuring rolling resistance. All three organisations suggest similar test equipment

and methods [6][7][3].

According to standard practices, rolling resistance is measured by a laboratory test. A tyre is rolled

against the outer surface of a large drum. An example is shown in Figure 1.2 . A tyre, fitted on a

wheel, is loaded radially against the outer surface of a relatively large drum. The drum is then rotated

by a driving motor at a controlled velocity and the tyre and the drum roll against each other without

slip. While the tyre and the drum rotate, a rolling resistance force develops in the tyre contact patch

and it applies on both the tyre and the drum. There are four ways to measure the rolling resistance

force:

i. Force - The force at the tyre spindle is measured when the drum is rotated in a constant

velocity.

ii. Power - The electric power needed to maintain the drum rotation at a constant speed is

measured.

iii. Torque - The input torque needed in order to maintain the drum rotation at a constant speed.

iv. Deceleration - The drum is firstly rotated up to a certain speed. The driving motor is then

detached from the drum, and the decay in the angular velocity is measured [3]

In each of these approaches, the raw measurement is converted into rolling resistance and the parasitic

loss is subtracted. Parasitic loss is the energy consumed by the system per unit distance, excluding

internal losses in the tyre. It includes sources of energy loss such as aerodynamic drag and bearing

friction [3].

The European standard includes guidelines of how to measure parasitic loss with sufficient accuracy

to compare results from different laboratories [3]. For methods i-iii above, the parasitic loss is

10

measured using a similar experiment as the one used to measure the rolling resistance, but with a

smaller load. The normal load applied on the contact surface is the smallest load needed to perform

no-slip rolling. The relevant measurement (force, power or torque) is used to calculate the parasitic

loss, which is later subtracted from the rolling resistance force measurement. The parasitic loss is

measured differently for the deceleration method (iv). When the drum and wheel rotate separately,

their angular deceleration is measured. The deceleration magnitudes are taken into account when

calculating the rolling resistance force with the presence of load.

Using the methodology of a rolled tyre against a spinning drum for measuring rolling resistance has

several drawbacks. First and primarily, it requires expensive equipment. A large drum with diameter

of 1.7 m [3], a firm frame to assure aliment of radial load and a motor that rotates both a drum and a

heavy vehicle wheel are estimated as the most costly components. Another weakness of this method is

that it might be particularly sensitive to sensor inaccuracy, since the rolling resistance force is

measured in the presence of much larger forces [6].

1.3. Previous work

An alternative laboratory experiment for measuring rolling resistance was suggested by Santin [2]. In

this test two wheels are rigidly fixed to an eccentric shaft, as shown in Figure 1.3. Since the system is

eccentric, it behaves like a rolling pendulum when placed on a flat floor. The experiment is initiated

by rolling the pendulum to a certain angle and releasing it, as shown in Figure 1.3(a). As it is

perturbed from its equilibrium position, it oscillates with decaying amplitude and eventually comes to

rest. The decay time indicates the tyre rolling resistance. The faster the decay, the larger the rolling

resistance is. In comparison with the drum method, the pendulum method requires much cheaper

equipment and its results are hardly affected by parasitic loss.

Recent research at Cambridge University has used a similar principal in order to assess the rolling

resistance of small tyres [8][9][10]. The same procedure as described by Santin has been conducted,

using an eccentric pendulum as shown in Figure 1.4. The oscillatory motion has been measured by an

accelerometer placed on the axle. The change in the rotation angle throughout the experiment can be

calculated from the accelerometer data. Figure 1.5(a), shows measurements from such an experiment.

The experimental conditions of the case shown are detailed in Table B-1 in Appendix B.

To calculate the rolling resistance coefficient based on the change in rotation angle, it is assumed that

the gravitational energy dissipated between successive peaks in rotation angle is purely due to tyre

rolling resistance. Furthermore, it was suggested that the rolling resistance coefficient is affected by

two different forms of rolling motion: steady-state and transient. For the parts of every cycle of

oscillation during which the tyre is changing direction, the contact stress distribution changes from the

steady-state distribution for rolling in one direction to the steady-state distribution for rolling in the

11

other direction. This is modelled by assuming that the rolling resistance coefficient during the

transition is different to the steady-state value shows the model based on the assumptions that

two different rolling resistance phenomena occur alternately during each cycle. represents the

transient coefficient, and - the steady-state coefficient. [9].

The algorithm for calculating the rolling resistance coefficient from the angles recorded contains

several major steps:

i. The signal is filtered and the DC offset is subtracted, to obtain a smooth signal with zero

mean.

ii. The local extreme points are detected, as illustrated by the star markers in Figure 1.5(a).

iii. The fitted rolling resistance coefficient for each interval between two maximum points is

calculated using Equations (1-1)-(1-4) [10][9].

iv. An iterative trial-and-error process is conducted in order to find values for 1rC and 2rC

which fit a curve to the measured points, using Equation (1-6)[9].

Gravitational energy:

1 cosE mge (1-1)

Distance travelled:

1d R 1(1-2)

Average rolling resistance force during an interval from max i to max 1i :

max 1 max

max min max 12i i

r

i i i

E EF

d d d

(1-3)

Rolling resistance coefficient:

1

rr

FC

Z (1-4)

Rolling resistance measured coefficient is separated into two coefficients using the model described in Figure 1.6 by the following calculation:

1 It is suggested that using 1 sind R e , would be more accurate, while e is the eccentricity radius and d is the distance travelled by the centre of mass. See also Figure 3.5.

1rC

2rC

1rC

2rC

12

1,

1 1 1

1 1r rrr fitted

F dx Z C dxFC

Z Z Zdx dx

Under the assumption that 1Z is constant:

rC dx

dx

area calculationrC dx

dx

Assuming that the rolling resistance coefficient is during periods when the tyre is a

distance cpL from its maximum or minimum points, and 2rC elsewhere ( Figure 1.6), this gives

2 2 1,

4r r r cp

r fitted

C dx C C LC

dx

2 12

max, min, max, 1

42

cp r r

r

i i i

R C CC

R R R

(1-5)

1 2, 2

max, min, max, 1

42

cp r r

r fitted r

i i i

C CC C

(1-6)

Where /cp cpL R and 1 2,r rC C are fitted coefficients.

Figure 1.5(b) presents the measured rolling resistance coefficients calculated using equations (1-3)

and (1-4) based on the recorded data in Figure 1.5(a). The measured rolling resistance coefficient

increases with the angle amplitude. Also shown is a line represents equation (1-6), which was

obtained by fitting coefficients 1rC and 2rC . This fitting procedure enables Cr2 to be determined.

The fitted shows good agreement with the measured coefficients. Furthermore, the transient

coefficient ( 1rC ) is smaller than the steady state coefficient ( 2rC ).

Previous experimental results of this method showed a reliability and reputability in measuring

various combinations of tyre types and load levels [8].

This method uses low cost equipment and it is mechanically simple. However, adaptation of such

method for heavy vehicle tyres is not straightforward.

1.4. Project objective

Improving the tyre rolling resistance characteristics would enable operators to reduce their fuel

consumption and carbon footprint. The existing methods for measuring tyre rolling resistance all have

drawbacks such as requiring an expensive test facility and an arguable lack of accuracy. Therefore

1

,1 1 1

1 1r rrr fitted

F dx Z C dxFC

Z Z Zdx dx

1rC

13

this project aims to develop a means of cheaply and easily assessing the rolling resistance of heavy

vehicle tyres for comparative purposes.

1.5. Conclusions

i. Rolling resistance is one of the vehicle energy consumers. Reducing the rolling resistance of

tyres leads to a reduction in fuel consumption. Therefore there is a need for measuring tyre

rolling resistance.

ii. A literature review on measuring tyre rolling resistance was conducted. This suggests there is

potential to develop alternative methodologies for measuring tyre rolling resistance but there

is a lack of research in this area.

iii. Existing methods either require expensive laboratory equipment or are not suitable for high

load conditions. A cheaper method for measuring the rolling resistance of a heavy vehicle tyre

is required.

iv. The project objective was identified as developing a means for measuring rolling resistance of

heavy vehicle tyres.

1.6. Figures

Figure 1.1: European Union tyre label [5]

14

Figure 1.2: the standard method for measuring rolling resistance

(a)

(b) _

Figure 1.3: an eccentric pendulum suggested for rolling resistance measurement of two tyres [2]

15

Figure 1.4: an eccentric pendulum for a small tyre, Cambridge University [9]2

Figure 1.5: sample results of an eccentric pendulum for small tyres (case 7 in Appendix B)

2 with minor changes

16

Figure 1.6: model for rolling resistance coefficient in oscillatory rolling for angles higher than the

contact patch angle (based on [9])

17

Chapter 2 -

Conceptual design of a rolling

resistance measuring rig

2.1. Introduction

The need for an inexpensive and simple mean for assessing the rolling resistance of heavy vehicle

tyres has been justified. Here, the design problem is detailed by listing requirements and criteria

regarding the required solution. Additionally, some conceptual solutions are introduced along with a

preliminary comparison between them.

2.2. Specification

2.1.1. Requirements The following characteristics are essential (‘demands’):

1. It should be safe to use

2. It should provide accurate and repeatable measurement

3. It should provide adjustable normal load up to 4 tonnes

4. It should be suitable for heavy vehicle tyres of 1 m in diameter

2.1.2. Criteria The following are optional characteristics (‘wishes’), which are used to compare design concepts:

1. It shall be conducted in an indoor laboratory

2. It shall have low initial cost

3. It shall have low cost per test

4. It shall be easy to operate

5. It shall be mechanically simple

6. It shall emulate road driving conditions

7. It shall have minimal parasitic loss

8. It shall provide measurement with good accuracy

2.3. Embodiment design

The problem may be broken into four main aspects: the load source, the measurement method, the

shape of the contact surface and whether the motion is oscillatory or continuous. Table 2.1 details

different ideas to separately address each part of the problem.

18

Table 2.1: Embodiment design

For example, a standard apparatus for measuring rolling resistance as described above, might use

hydraulic pistons to apply the load on the tyre, and may use a reading from a force sensor.

Furthermore, the motion, in this case, is continuous and the road is emulated by a drum external

surface.

2.4. Suggested concepts

The numbered lines represent design concepts.

Figure 2.1 shows five paths throughout the suggested design grid above. Accordingly five conceptual

solutions are introduced, including the existing commercial method (concept 1, Figure 2.2) and four

other ideas. The concepts are detailed in the following sections.

2.1.3. Sprung axle pendulum (Concept 2, Figure 2.3)

The test tyre is fitted on a wheel rim which is rigidly fixed to its axle. The axle is also fixed to a solid

wheel of the same radius as the tyre. The axle is loaded by a spring against a road surface, on which

the wheels can roll. Like the rolling pendulum described in previous work, the system is initially

rolled to a certain angle and then released, and oscillates until it comes to rest. The equilibrium point

is where the spring is at its shortest. The rotation angle may be measured as well as the strain in the

spring cable and the rolling resistance force can be calculated based on the assumption that the rolling

resistance force is the most significant cause of energy dissipation. A desired load distribution

between the two wheels may be achieved by fixing the spring closer to the test tyre than to the solid

wheel.

Advantages: small parasitic loss since there are no bearings

Disadvantages: large, dangerous due to high spring tension, possible parasitic loss in the spring joints

19

2.1.4. Eccentric drum pendulum (Concept 3, Figure 2.4)

The test tyre is loaded against the external surface of an eccentric weighted drum, where both the

wheel and the drum are able to roll around their hub. Each hub is connected with a bearing to a frame,

and the frames are pressed toward one another by a pneumatic mechanism. This structure creates

another type of pendulum, which is initiated by rotating the eccentric drum to a certain angle and then

releasing it. Rolling resistance force can be calculated based on the decrement of rotation angle, which

may be measured during oscillations.

Advantages: easy to vary the load, simple

Disadvantages: large, the drum might be expensive, parasitic loss at bearings of both rotating parts.

2.1.5. Dropped road plate (Concept 4, Figure 2.5)

Two tyres of the same tested type are fitted on two wheels. The wheels are assembled to frames by

bearings, so they are able to spin. Once load is applied on the frames by a pneumatic mechanism, both

tyres are pressed against a double-side road plate, located between them. The experiment begins when

more and more weight is gradually added to the plate, until this total weight reaches just over the

magnitude of the total rolling resistance force from both tyres. Then, the plate is expected to descend,

and the wheels to roll. A record of either the rotation angle or the plate acceleration, together with a

record of mass properties of the moving parts, can be used to calculate the rolling resistance force.

Alternatively, one wheel may be replaced by a low friction surface, such as a flat air bearing, in order

to avoid using two tyres as well as to avoid using two road surfaces.

Disadvantages: complex, parasitic loss of bearings

Advantages: moderate size, continuous motion, flat road surface

2.1.6. Eccentric axle pendulum (Concept 5, Figure 2.6)

As described in section 1.3, one axle contains two wheels: one wheel is fitted with a tested tyre and

the other wheel is solid. The shaft is rigidly attached to an eccentric weight, which provides the

desired load in the test wheel. The experiment is initiated by rolling the axle to a certain angle, and the

oscillations are measured by an accelerometer.

Advantages: simple, a little parasitic loss since there are no bearings, low cost since there are no

bearing and no motor

Disadvantages: large, potentially dangerous due to large weight, difficult to change tyres between

experiments

20

2.5. Summary and conclusions

The design problem was defined, and several solutions were suggested. In comparison with the

existing methods, the suggested concepts have several advantages, with their lower cost being the

most significant one. Further evaluation is recommended for the four concepts, but the eccentric axle

pendulum might be the simplest and cheapest measuring equipment of all.

2.6. Figures

The numbered lines represent design concepts.

Figure 2.1: Components combinations as a method for conceptual design

Figure 2.2: Concept 1: driven external drum

21

Figure 2.3: Concept 2: sprung axle pendulum

Figure 2.4: Concept 3: eccentric drum pendulum

Figure 2.5: Concept 4: dropped road plate

22

Figure 2.6: Concept 5: eccentric axle pendulum

23

Chapter 3 -

Dynamics of a rolling axle pendulum

3.1. Introduction

In previous research, the use of a rolling pendulum was shown to be a good method for measuring the

rolling resistance of small tyres [8]. In order to modify this solution to accommodate a truck wheel

and apply normal load of up to 4 tonnes on the tyre, several issues have to be considered. The

following sections discuss these issues.

The conceptual rig discussed below contains two wheels; one is solid and the other is fitted with a

tyre. In this report, the latter is referred to as the ‘test tyre’.

Section 3.2 discusses how to distribute mass along the axle to obtain a large load on the test tyre and a

much smaller load on the other end. Section 3.3 presents a dynamic analysis of the system, which

attempts to predict the characteristics of its oscillatory motion as well as the contact forces acting on

the system during a cycle. In addition, an evaluation of whether or not such motion is possible with no

slippage is included (section 3.4.3) as well as a prediction of the change in the vertical load during the

motion (section 3.4.4).

3.2. Mass distribution along the axle

A schematic of a proposed rig for measuring rolling resistance of heavy vehicle tyres is shown in

Figure 3.1. The assumptions and constraints for parameters on the rig are as follows:

i. When the system is static, the vertical load on the test tyre ( 1Z ) should be 39,240 N (4

tonnes).

ii. When the system is static, the vertical load on the solid wheel ( 2Z ) should be 196 N (20 kg).

iii. The test tyre width is 0.3 m

iv. Load and eccentricity is achieved by two cylindrical weights fixed on each side of the test

tyre. These have a diameter of 0.45 m and are made of steel or a solid material of similar

density (7850 3/kg m )

v. To keep the overall size manageable, the distance between the tyre and each the weights

( 1 2,d d ) needs to be at least 0.1 m to allow space for connections.

vi. The total length of the axle (totalL ) should be shorter than 3 m, and the distance between the

solid wheel and outer weight (cL ) is fixed to 2.7 m.

Figure 3.1 shows a free-body-diagram of the proposed rig and defines the relevant geometric

measures. The equations for static equilibrium of forces and moments are:

24

1 2 1 2 0Z Z W W

(3-1)

1 2 2 0a c bL Z L Z L W

(3-2)

According to the above list of constraints, 1 2, ,cZ L Z are fixed parameters, while the others are

unknown. In order to solve these equations, an array of combinations of aL and bL has been searched

and suitable solutions were identified. Each solution was checked against the four conditions, as

shown in Table A-1. The respective solutions for 1W and 2W are summarised in Table A-2 in

Appendix A. The conditions express the constraints listed above and physical feasibility, such as non-

negative mass. From the whole range of combinations in question, six were found to fulfil all

conditions. The appropriate lines in Table A-2 are highlighted. Furthermore, the highlighted lines are

demonstrated in Figure 3.2, which shows the proportions of the assembly for all six cases.

In summary, several arrangements of weights have been calculated using a systematic search over

combinations. This shows that 4 tonnes can be applied to a test tyre using just over 4 tonnes of dead

weight. The principal of the existing small scale rolling pendulum rig may be scaled up using one of

the suggested geometries. Such systematic search for calculating the masses and the geometry could

be adjusted by adding further constraints in the future. Additional constraints may include bending

moments and shear stresses along the shaft. Adding such analysis may further refine the design.

3.3. 2D dynamic model of rigid eccentric pendulum

There were safety concerns about the potentially large momentum in the test rig. There was also a

requirement to ensure no slip occurred at the contact patch. To address these issues, a dynamic

analysis of the system was completed using a simplified 2D model of the system.

3.3.1. Equation of motion For the 2D rolling pendulum, shown in Figure 3.3, the following assumptions were made:

1. The entire structure is a rigid body, meaning the deformation in the system is negligible. 2. Both wheels roll with no slip 3. The test tyre and the solid wheel have the same radius ( 1R )

According to the free-body-diagram in Figure 3.4, the system is affected by the following forces:

X - total longitudinal contact force between the floor and both wheels

Z - total normal force between the floor and both wheels

W - total weight, equal to mg

We define e as the radial distance between the centre of gravity of the whole system and the wheels centres, as shown in Figure 3.4. e is sometimes called the ‘eccentricity radius’.

25

Although in reality there are two wheels in contact with the floor (the test tyre and a solid wheel), in this approach both wheels are modelled as one rigid wheel with the same radius as both wheels (R1).

The equation of motion for translation is:

.c gmF r (3-3)

Considering Figure 3.5, is defined as the rotation angle, which is zeroed when the centre of gravity is aligned under the centre of the wheel. Under the assumed no-slip condition, the centre of gravity trajectory is a function of :

, 1

,

sin

1 cosx equ x

z equ z

r r R e

r r e

1 cos

sinx

z

r R e

r e

21

2

cos sin

sin cosx

z

r R e e

r e e

(3-4)

Where , ,,equ x equ zr r is the location of centre of gravity when the system is in equilibrium ( =0).

Separating the forces into x and y components gives: x x

z y

F X mr

F Z mg mr

(3-5)

Substitution of Equations (3-4) and (3-5) into the Equation (3-3) gives:

21 cos sinX m R e e

2sin cosZ m e e g

(3-6)

(3-7)

The equation of motion for rotation about the system centre of mass is:

y yH (3-8)

The torque acting about the centre of mass is:

1 cos siny X R e Ze (3-9)

The derivative of the angular momentum is:

.y c gH J (3-10)

Substitution of Equations (3-6) and (3-7) into the equation of angular motion (3-8) results the following:

26

(3-11)

(3-12)

(3-13)

A and B are defined such that:

0A B (3-14) Where

1 1 .

2 1

, , , ,

, , , , ,c mA f e R m J

B f e R m g

This results in an Ordinary Differential Equation (ODE) for as a function of time as follows:

B

A

(3-15)

The initial conditions are:

00

0 0

t

t

(3-16)

Once the physical parameters are set, this equation can be solved numerically by an explicit Runge-

Kutta (4,5) formula called the Dormand-Prince pair, practically using MATLAB ODE solver.

In order to validate this model, two checks were performed. Firstly, energy conservation was checked

for two different points during a simulation. Secondly, the model was used to simulate the existing

small tyre rig (Section 1.3) and the results were compared to the motion of the actual rig.

3.3.2. Validation check by energy conservation Assuming the only external forces on the system are the ones identified in Section 3.3.1, its energy

should be conserved. The following calculation checks that the kinetic energy of the system when

0 equals the gravitational energy at the initial angle using the model parameters from Table 3.1.

27

Total energy when 0 :

2 2 2 2,max 1 2 1 1 2 2

1 1 1 12 2 2 2kinT J J mU m U (3-17)

The velocities of the two masses are:

1 0 1

2 0 1 DW

U R

U R e

(3-18)

From Figure 3.6(b) 0 max 117

2 22 2,max 1 2 1 1 2 1

22 21 2 1 1 2 1

222

1 1 1 12 2 2 2121 117 0.1176 0.0996 3 0.28 12.8 0.28 0.1052 1801.76

kin DW

DW

T J J m R m R e

J J m R m R e

J

(3-19)

Total energy when: 0 :

,max 2 01 cos

12.8 9.81 0.105 1 cos 30

1.76

grav DWT m ge

(3-20)

This shows there is an equal amount of energy in both points.

3.3.3. Validation check by the frequency of the existing pendulum The parameter values detailed in the Table 3.1 were used to model the small tyre pendulum rig.

Table 3.1 Input data set – small pendulum

Parameter

Small Tyre Pendulum

0 30

1R m 0.28

1m kg ** 3.0

2m kg ** 12.8

DWe m 0.105

m kg * 15.8

2.c mJ kg m ** 0.2440

28

*measured from rig ** obtained from previous research [9]

Figure 3.6 shows the pendulum angle (a) and the angular velocity (b) from the simulation of the small

tyre rig. As seen in Figure 3.6, the model suggests a periodic motion. The simulated motion frequency

was extracted and compared with the frequency of the oscillations in the accelerometer measurements

shown in Figure 1.5. The model resulted in a frequency of 0.60 Hz, which is similar to the measured

frequency at a comparable angle of about 30° (0.61 Hz).

3.3.4. The effect of eccentricity on the system kinematics Table 3.2 details several sets of parameters for the proposed rig, which differ by their eccentricity.

Each case represents an alternative design for a rolling pendulum.

Table 3.2 Input data sets – Truck Tyre Pendulum

Parameter Truck Tyre Pendulum Case 1

Truck Tyre Pendulum Case 2

Truck Tyre Pendulum Case 3

Truck Tyre Pendulum Case 4

0 ** 105 105 105 105

1R m 0.50 0.50 0.50 0.50

1m kg 100 100 100 100

2m kg 4000 4000 4000 4000

DWe m 0.05 0.10 0.25 0.50

m kg 4100 4100 4100 4100

2.c mJ kg m *** 419.02 419.75 424.88 443.17

e m 0.048 0.098 0.244 0.488

2L m **** 1.708 1.708 1.708 -

1 2/L L 1/142 1/142 1/142 -

** chosen to initiate rolling distance of just over three times a typical contact patch length of 0.3 m.

e m 0.0851

29

*** based on estimation of 1 2,J J

**** based on proportion suggested in Figure 3.2 (a).

Figure 3.7 shows the three solutions of input cases 1, 3 and 4. The graphs in (a) and (b) illustrate the

expected change in the rotation angle ( ) and in the angular velocity ( ) respectively. Figure 3.8

shows the same for the case of smaller initial angle of 30°.

The conclusions from Figure 3.7 and Figure 3.8 are as follows:

i. The motion is periodic but not sinusoidal. Hence, in certain cases, the system behaviour

cannot be approximated by simple harmonic motion.

ii. Across the three cases, the motion frequency increases with the eccentricity.

iii. From Figure 3.7(b), across the three cases, a smaller eccentricity radius results in smaller

angular velocity peak ( ) and smaller angular acceleration peak (maximum of derivative).

iv. Comparing Figure 3.7(b) with Figure 3.8(b), a smaller initial angle results in motion which is

more sinusoidal giving a lower angular velocity peak and lower angular acceleration peak.

Figure 3.9 shows additional cases of eccentricity with the same mass and radius as detailed for all

cases in Table 3.2. The model graphs in Figure 3.9 show that the frequency of a rolling pendulum is

expected to peak when the radial distance between centres of gravity is approximately 25% longer

than the wheel radius 1( / 1.25)DWe R .

3.3.5. Validation check by the frequency of a simple pendulum Figure 3.9 includes the expected frequency of an equivalent simple pendulum as a reference. A simple

pendulum is the classic case of a point mass hanged on a negligible mass rod about a fixed point.

Assuming a small angle perturbation, its frequency is approximated by the following expression:

12 DW

gf

e (3-21)

In the problem of a rolling pendulum, if the initial angle is as small as 30°, and 1DWe R , the system

can be approximated to a simple pendulum. The fact that the model graph of 30° approaches the

simple pendulum graph as the eccentricity increases, gives an additional validation to the model.

3.4. 3D dynamic model of rigid eccentric pendulum

A 3D dynamic model was used to calculate the required contact force magnitudes for ensuring pure

rolling motion.

This section relies on the results of the 2D model in Section 3.3.

30

3.4.1. Equation of motion A rolling pendulum and a coordinate reference frame, xyz , are illustrated in Figure 3.10. The

following is assumed:

i. A rigid body

ii. Rolling with no slip

iii. Both the test tyre and the solid wheel share the same radius ( 1R )

iv. The components ,xy zyI I of the rig moment of inertia are negligible

The system is affected by the following forces:

1 1 1, ,X Y Z Contact forces between the floor and the tested wheel

2 2 2, ,X Y Z Contact forces between the floor and auxiliary wheel

W - Total Weight

Geometric measurements:

1L - Distance along y axis between c.g and the contact point of wheel 1

2L - Distance along y axis between c.g and the contact point of wheel 2

The equation of motion for translation is:

.c gmF r (3-22)

Similarly to the 2D approach discussed in section 3.3.1, the centre of gravity trajectory can be

expressed as follows:

, 1

. ,

,

sin

1 cos

equ x

c g equ y

equ z

r R e

r

r e

r , 1

.

cos0sin

c g

R e

e

r ,

21

.2

cos sin0

sin cosc g

R e e

e e

r

(3-23)

The force acting in all three directions are:

1 2

1 2

1 2

X X

Y Y

Z Z mg

F (3-24)

Substitution of (3-23) and (3-24) into the equation of motion (3-22) gives:

31

21 2 1

1 22

1 2

cos sin0

sin cos

X X R e e

Y Y m

Z Z mg e e

(3-25)

(3-26)

(3-27)

The equation of motion for rotation about centre of gravity is:

. .c g c gτ H (3-28)

From Figure 3.10 and Figure 3.12 (a) and (b), the total torque acting about the centre of gravity,

expressed in body frame is as follows:

1 1 1

. 1 1

1 1 1

1 2 2

2 2

1 2 2

sin cos sin

cos sin cos

sin cos sin

cos sin cos

c g

R X Z

L Y

e R X Z

R X Z

L Y

e R X Z

τ

(3-29)

. , 1 1 1 2 2 2 1 1 2

. , 1 1 1 2 2

1 1 1 2 2

. , 1 1 2 1 1 1 2 2 2

sin cos sin cos cos

cos cos sin cos sin

sin sin cos sin cos

sin cos sin cos sin

c g xbody

c g ybody

c g zbody

L X Z L X Z e R Y Y

e R X Z X Z

R X Z X Z

R Y Y L X Z L X Z

(3-30)

A body coordination frame is defined with the origin at the centre of gravity, as illustrated in Figure

3.11.

The derivative of angular momentum can be calculated by:

.c g H I Ω Ω I Ω (3-31)

While

Ω - Angular velocity vector described in body frame

I - Tensor of inertia of the whole system about c.g, described in body frame

Assuming no-slip rolling the angular velocity vector fulfils:

0 0,

0 0

Ω Ω (3-32)

The tensor of inertia about c.g is:

32

.

0

0c gJ

I (3-33)

The sign ( ) indicates an ineffective element.

Substitution of (3-32) and (3-33) into (3-31) gives:

. . . .

.

0 0 0 0 0 0 00

0 0 0 0 0 0 0

0

0

c g c g c g c g

c g

J J J

J

H

(3-34)

Substitution of the moments (3-32) and the momentum change (3-34) into the equation of motion

(3-28), for the x and z components, gives:

1 1 1 2 2 2 1 1 2sin cos sin cos cos 0L X Z L X Z e R Y Y

(3-35)

1 1 2 1 1 1 2 2 2sin cos sin cos sin 0R Y Y L X Z L X Z (3-36)

It is known from Equation (3-26) that 1 2 0Y Y , hence the previous two equations can be simplified

to:

1 1 1 2 2 2sin cos sin cos 0L X Z L X Z (3-37)

1 1 1 2 2 2cos sin cos sin 0L X Z L X Z (3-38)

Given the values of 1 2, , , ,L L e m g and the instantaneous values of , , as calculated in the 2D

approach, Equations (3-37), (3-38), (3-25) and (3-27), can represent a linear equation system for the

instantaneous forces 1 2 1 2, , ,X X Z Z . This system can be put in the following form:

211

22

1 2 1 2 1

1 2 1 2 2

cos sin1 1 0 00 0 1 1 sin cossin sin cos cos 0cos cos sin sin 0

m R e eX

X m g e e

L L L L Z

L L L L Z

(3-39)

The following manipulation can be done to simplify line (III) and line (IV) of the above equation

system:

33

cos

sin cos

III sin IV III

III IV IV

and that results in:

211

22

1 2 1

1 2 2

cos sin1 1 0 00 0 1 1 sin cos0 0 0

0 0 0

m R e eX

X m g e e

L L Z

L L Z

(3-40)

The above equation can be defined as Ax B , and thus can be solved numerically by matrix

manipulation: 1x A B (3-41)

Figure 3.13 (a), (b), (d) and (e) show the forces calculated by the above equation against the

pendulum rotation angle, , for eccentricities shown in Table 3.2 (cases 1-3). The graphs show the

results from a 20 second simulation, which includes several cycles of the motion in all cases.

3.4.2. Condition for continuous contact

To check that the wheels of the pendulum will not lose contact with the floor during rotation, the

expected normal forces on both wheels can be checked to see if they would be positive during the

motion.

Figure 3.13 (b) and (e) show that 1 0Z and 2 0Z during the whole period, hence the contact is

expected to be continuous between both wheels and the floor over the whole cycle, in all cases tested.

3.4.3. Condition for no slip

To check that the wheels of the suggested pendulum designs would not slip along the floor during its

motion, the friction forces were evaluated. This information can then be used to choose appropriate

materials which can provide sufficient friction, or to adjust the normal load on the wheels. Assuming

the maximum static friction force varies linearly with normal load ( ,maxfr sF Z ), and the lateral

forces 1 2,Y Y are negligible, the ratios 1

1

X

Zand 2

2

X

Z are the required friction coefficients to enable no-

slip motion of wheel 1 and wheel 2 respectively. Figure 3.13(c) and (f) show the change in these

ratios over a cycle for different choices of eccentricity radius (cases 1-3 in Table 3.2).

The conclusions from the graphs are as follows:

i. Although there is a significant difference in the load distribution at the two wheels, the same

friction coefficient is required for both wheels. The rationale for this is that the distance

between a wheel and the centre of mass have two opposing affects. The longer the distance,

the smaller is the load proportion it carries, but the higher is the torque about the centre of

mass per friction force unit. In other words less friction force is needed in order to obtain a

34

certain magnitude of torque, due to longer lever arm. The two factors above mean that the

further wheel experiences 2 1/L L times less vertical load but also 2 1/L L times less friction

force needed, than these of the closer wheel, and therefore both wheel require the same

friction coefficient.

ii. The minimum required friction coefficient is 0.35 for case 3. Cases 1 and 2 are even lower.

(The maximum static friction coefficient of steel-on-steel surfaces is 0.49 [11], and rubber on

concrete is approximately 1.0)

3.4.4. Load change on the test tyre The arrangements of cases 1-3 in Table 3.2 may provide a static vertical load of 40,021 N on the test

tyre. This is a result of the static analysis, by which the static load on the test tyre is:

21 1 2

1 2stasic

LZ m m g

L L

(3-42)

However, Figure 3.13(b) shows that the dynamic load changes as the pendulum oscillates, and its

change is influenced by the eccentricity. Table 3.3 summarises the range of the dynamic normal load

applied on the tyre over a motion cycle in the three cases. The numbers in brackets express the

relative change in respect to the static load 1stasicZ .

Table 3.3 Range of instantaneous load on the test tyre

Parameter Truck Tyre Pendulum Case 1

Truck Tyre Pendulum Case 2

Truck Tyre Pendulum Case 3

DWe m 0.05 0.10 0.25

e m 0.048 0.098 0.244

1minZ kN 39.7 (99%) 38.8 (97%) 32.4 (81%)

1maxZ kN 40.8 (102%) 43.6 (109%) 75.5 (189%)

It can be seen that in case 3, which represents a radial distance of 0.25 m between the centres of

gravity, the dynamic load varies significantly. Such variation is undesirable when measuring a tyre

rolling resistance. In contrast, cases of lower eccentricity radius (cases 1-2) showed almost uniform

levels of vertical load, with all loads staying within 10% of the nominal values.

35

3.5. Conclusions

Several design aspects of a scaled up rolling resistance measurement rig were examined by simplified

models: a static model and both two dimensional and three dimensional dynamic models. The analysis

showed the following:

i. Several mass arrangements were suggested which apply a desired load on each wheel.

ii. The dependency of the pendulum motion frequency on the eccentricity was evaluated,

and this dependency was shown graphically. It appears that the frequency is largest when

the radial distance between centres of gravity is approximately 25% longer than the wheel

radius 1( / 1.25)DWe R .

iii. In the design cases examined, the normal forces on both wheels are expected to remain

positive during the motion. This means loss of contact with the floor is not expected.

iv. In the design cases examined, the critical threshold for static friction coefficient is at least

0.35 to ensure pure rolling with no slip.

v. A choice of smaller eccentricity radius may be beneficial for several reasons: less load

variation, slower motion and lower required friction coefficient.

3.6. Figures

Figure 3.1: rolling pendulum sketch with suggested load distribution (side view)

36

Figure 3.2: six alternative proportions for mass along a rolling pendulum

Figure 3.3: geometric measures of 2D eccentric rolling pendulum

37

Figure 3.4: free body diagram in 2D for eccentric rolling pendulum

Figure 3.5: translation of the centre of mass for eccentric rolling pendulum

38

Figure 3.6: calculated behaviour of rotation angle and angular velocity for the existing rig, using model parameters from Table 3.1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-40

-20

0

20

40X: 0

Y: 30P

endulu

m A

ngle

[

°]

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-200

-100

0

100

200

X: 1.245

Y: 117.1

Rota

tional V

elo

city [

°/sec]

Time [sec]

(b)

39

Figure 3.7: simulated behaviour of rotation angle and angular velocity, for different eccentricity radii (initial angle 105°)

Figure 3.8: simulated behaviour of rotation angle and angular velocity, for different eccentricity radii (initial angle 30°)

0 1 2 3 4 5 6 7 8 9 10-150

-100

-50

0

50

100

150

Pendulu

m A

ngle

[

°]

(a)

0 1 2 3 4 5 6 7 8 9 10-1000

-500

0

500

1000

Rota

tional V

elo

city [

°/sec]

Time [sec]

(b)

e = 0.05 [m]

e = 0.25 [m]

e = 0.49 [m]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-40

-20

0

20

40

Pendulu

m A

ngle

[

°]

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-200

-100

0

100

200

Rota

tional V

elo

city [

°/sec]

Time [sec]

(b)

40

Figure 3.9: expected frequency for different eccentricity radii in two initial angles

Figure 3.10: free-body-diagram of 3D rolling pendulum

0 2 4 6 8 10 12 14 16 18 200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

eDW

/ R1

Fre

quency [

Hz]

model, 0=30°

simple pendulum

model, 0=105°

41

Figure 3.11: body frame definition

(a) (b)

Figure 3.12: torque calculation sketch in XZ plane

42

Figure 3.13: expected forces for different eccentricity radius cases

-100 -50 0 50 100-2

0

2x 10

4

X1 [

N]

Rotation angle [deg] [°]

(a)

-100 -50 0 50 1002

4

6

8x 10

4

Z1 [

N]

Rotation angle [deg] [°]

(b)

-100 -50 0 50 1000

0.2

0.4

|X2|/Z

2

Rotation angle [deg] [°]

(c)

-100 -50 0 50 100-100

0

100

X2 [

N]

Rotation angle [deg] [°]

(d)

-100 -50 0 50 100100

200

300

400

Z2 [

N]

Rotation angle [deg] [°]

(e)

-100 -50 0 50 1000

0.2

0.4|X

2|/Z

2

Rotation angle [deg] [°]

(f)

43

Chapter 4 -

Conclusions and future work

4.1. Conclusions

Investigating tyre rolling resistance may help in the attempt to reduce vehicle fuel consumption per

trip. The standard methods for measuring rolling resistance require large and expensive equipment, as

well as accurate sensors. In previous research a rolling pendulum has shown repeatable and accuracy

in measuring small tyres [8], but such pendulum has not been adapted yet for testing a heavy vehicle

tyre under a load of up to 4 tonne.

Four solutions were suggested as alternative measuring rigs to assess truck tyres: sprung axle

pendulum, eccentric drum pendulum, dropped road plate, and eccentric axle pendulum. The last one

has been further developed in this study.

For the eccentric weighted rolling pendulum, several arrangements of dead weight distribution were

introduced; these involve hanging two weights in both sides of the tested wheel, applying only minor

load on an auxiliary wheel. The motion of such pendulum was simulated, and in certain case studies,

the model showed the feasibility of no-slip rolling. The relationship between the mass eccentricity and

the pendulum oscillation frequency has been calculated.

4.2. Future work

i. The dynamic model for a rolling pendulum needs to be verified by further experimental study.

ii. More detailed evaluation of candidate solutions is required prior to rating and comparing

them.

iii. A detailed design of a measuring rig

iv. Prototype Building and testing

44

Appendix A Table A-1: Conditions for mass distribution design

Condition 1 1 20.1 0.1d m and d m

Condition 2 a bL L

Condition 3 1 20 0W and W

Condition 4 3totalL m

Table A-2: Mass distribution combinations

Figure 3.2

La (m)

Lb (m

)

W1

(N)

W2

(N)

d1

(m)

d2

(m)

a1 (m

)

con

ditio

n 1

con

ditio

n 2

con

ditio

n 3

con

ditio

n 4

All C

on

ditio

ns

0.10 0.50 30,498 8,898 -0.51 0.01 0.62 0 1 1 0 0

0.10 0.92 34,542 4,854 -0.55 0.47 0.71 0 1 1 0 0

0.10 1.33 36,059 3,337 -0.57 0.90 0.74 0 1 1 0 0

0.10 1.75 36,854 2,542 -0.58 1.32 0.75 0 1 1 0 0

0.10 2.17 37,343 2,053 -0.58 1.75 0.76 0 1 1 0 0

0.10 2.58 37,674 1,722 -0.59 2.17 0.77 0 1 1 0 0

0.10 3.00 37,913 1,483 -0.59 2.58 0.78 0 1 1 0 0

0.54 0.50 -4,129 43,525 0.28 -0.79 -0.08 0 0 0 1 0

0.54 0.92 15,655 23,741 0.08 -0.17 0.32 0 1 1 1 0

0.54 1.33 23,074 16,322 0.01 0.32 0.47 0 1 1 1 0

0.54 1.75 26,960 12,436 -0.03 0.78 0.55 0 1 1 1 0

0.54 2.17 29,352 10,044 -0.06 1.22 0.60 0 1 1 0 0

0.54 2.58 30,972 8,424 -0.08 1.66 0.63 0 1 1 0 0

0.54 3.00 32,142 7,254 -0.09 2.08 0.66 0 1 1 0 0

0.98 0.50 -38,756 78,152 1.08 -1.58 -0.79 0 0 0 1 0

0.98 0.92 -3,232 42,628 0.72 -0.80 -0.07 0 0 0 1 0

0.98 1.33 10,089 29,307 0.58 -0.25 0.21 0 1 1 1 0

a 0.98 1.75 17,067 22,329 0.51 0.24 0.35 1 1 1 1 1

b 0.98 2.17 21,361 18,035 0.46 0.70 0.44 1 1 1 1 1

d 0.98 2.58 24,270 15,126 0.43 1.15 0.50 1 1 1 1 1

0.98 3.00 26,371 13,025 0.41 1.58 0.54 1 1 1 0 0

1.43 0.50 -73,382 112,778 1.88 -2.38 -1.50 0 0 0 1 0

1.43 0.92 -22,119 61,515 1.35 -1.44 -0.45 0 0 0 1 0

1.43 1.33 -2,896 42,292 1.15 -0.82 -0.06 0 0 0 1 0

1.43 1.75 7,174 32,222 1.05 -0.31 0.15 0 1 1 1 0

c 1.43 2.17 13,370 26,026 0.99 0.18 0.27 1 1 1 1 1

e 1.43 2.58 17,568 21,828 0.95 0.63 0.36 1 1 1 1 1

1.43 3.00 20,600 18,796 0.91 1.08 0.42 1 1 1 0 0

1.87 0.50 -108,009 147,405 2.67 -3.18 -2.21 0 0 0 1 0

1.87 0.92 -41,007 80,403 1.99 -2.07 -0.84 0 0 0 1 0

45

Figure 3.2

La (m)

Lb (m

)

W1

(N)

W2

(N)

d1

(m)

d2

(m)

a1 (m

)

con

ditio

n 1

con

ditio

n 2

con

ditio

n 3

con

ditio

n 4

All C

on

ditio

ns

1.87 1.33 -15,881 55,277 1.73 -1.40 -0.33 0 0 0 1 0

1.87 1.75 -2,720 42,116 1.59 -0.85 -0.06 0 0 0 1 0

1.87 2.17 5,379 34,017 1.51 -0.35 0.11 0 1 1 1 0

f 1.87 2.58 10,866 28,530 1.46 0.12 0.22 1 1 1 1 1

1.87 3.00 14,828 24,568 1.41 0.58 0.30 1 1 1 0 0

2.31 0.50 -142,636 182,032 3.47 -3.97 -2.92 0 0 0 1 0

2.31 0.92 -59,894 99,290 2.62 -2.71 -1.23 0 0 0 1 0

2.31 1.33 -28,866 68,262 2.30 -1.97 -0.59 0 0 0 1 0

2.31 1.75 -12,613 52,009 2.14 -1.39 -0.26 0 0 0 1 0

2.31 2.17 -2,611 42,007 2.04 -0.87 -0.05 0 0 0 1 0

2.31 2.58 4,164 35,232 1.97 -0.39 0.09 0 1 1 1 0

2.31 3.00 9,057 30,339 1.92 0.08 0.19 0 1 1 0 0

2.75 0.50 -177,262 216,658 4.27 -4.77 -3.63 0 0 0 1 0

2.75 0.92 -78,781 118,177 3.26 -3.34 -1.61 0 0 0 1 0

2.75 1.33 -41,851 81,247 2.88 -2.55 -0.86 0 0 0 1 0

2.75 1.75 -22,506 61,902 2.68 -1.93 -0.46 0 0 0 1 0

2.75 2.17 -10,602 49,998 2.56 -1.40 -0.22 0 0 0 1 0

2.75 2.58 -2,538 41,934 2.48 -0.90 -0.05 0 0 0 1 0

2.75 3.00 3,286 36,110 2.42 -0.42 0.07 0 1 1 0 0

46

Appendix B

Table B-1: conditions of repeating small pendulum methodology

Surface: Smooth lab floor Tyre: Radial X

case no. Contact Patch length (cm) Inflation (psi) mass addition (kg)

1 5.7 60 9.07

2 6 60 20.39

3 7.2 40 20.39

4 6.6 20 20.39

5 3.8 20 9.07

6 5.5 40 9.07

7 4.3 40 0

8 4.2 60 0

9 6.5 20 0

Table B-2: size measurements of small pendulum

Recorded measurements

Radius of tyre R1 0.28m

Radius of small metal wheel RS 0.07m

Axle length l 1.51m

Mass of rig (axle, wheels and arm) M 15.82

Height from axle to weight hanger h 0.11m

Table B-3: mass property measurements of small pendulum

Cgh of weight 200N cgh(1) 0.03m

Cgh of weight 20lb cgh(2) 0.025m

Cgh of no weight cgh(3) 0m

Mass added m(1) 20.39kg

Mass added m(2) 9.07kg

Mass added m(3) 0kg

Table B-4: frequency measurements of small pendulum

Case frequency

4 0.66

7 0.44

47

References [1] D. E. Hall and J. C. Moreland, “Fundamentals of Rolling Resistance,” Rubber Chem.

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