Measuring the Rolling Resistance of Heavy Vehicle Tyres

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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.

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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

Nomenclature

Fr : Rolling resistance force

Cr : 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

mini ,maxi ,maxi 1 : Rotation angle, see Figure 1.5

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

R1 : external radius of tyre

R2 : radius of dead weight cylinder e : distance between axle and centre of gravity, radius of eccentricity J cm. : moment of inertia of the whole system about the centre of mass I yy  g : earth gravity coefficient eDW : distance between centre of gravity of dead weight and the axle, see Figure 3.3 mm12, : mass of concentric parts and mass of eccentric part of a pendulum. see Figure 3.3

UU12, : velocity of mass m1 and m2

JJ12, : moment of inertia of concentric and eccentric part of a pendulum. see Figure 3.3

L1 : distance along y axis between centre of gravity and test tyre

L2 : distance along y axis between centre of gravity and rigid wheel

XX12, : longitudinal contact forces acting on tyre and on rigid wheel, respectively

YY12, : lateral contact forces acting on tyre and on rigid wheel, respectively

ZZ12, : 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

IIxy, zy : tensor of inertia components

x : vector of unknown forces C r1 : steady-state rolling resistance coefficient C r 2 : transient rolling resistance coefficient

F 11Frr dx Z1 C dx C r   r, fitted ZZZ : Rollingdx resistance dx coefficient as affected from both transient and steady-state 1 1 1 factors E : mechanical energy (gravitational and kinetic) m : total mass of the system d : distance travelled

Lcp : contact patch length dd, 12 : distance between the tyre and the outer weight and of the inner weight respectively (see Figure 3.1)

Ltotal : total length of the axle

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

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

Lc : distance between the solid wheel and outer weight (see Figure 3.1) aa12, : width of the outer weight and of the inner weight respectively (see Figure 3.1)

WW12, : 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 FF xz, : components x and z of the total force on the system in Figure 3.4, : total mass on the system in Figure 3.4 W : system total weight, equal to mg rcg. : location of the centre of gravity

rrequ,, x, equ z  : 2D location vector of centre of gravity when the system is in equilibrium   0

requ,,,, x, r equ y r equ z  : 3D location vector of centre of gravity when the system is in equilibrium

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

τcg. : total torque vector about the centre of gravity

H y : momentum about y axis

Hcg. : momentum vector about the centre of gravity AB, : expressions of the differential equation in the 2D approach

AB, : matrixes expressing the linear equation system in the 3D force calculation t : time from motion initiation

0 : initial rotation angle T kin,max : maximal kinetic energy throughout a cycle T grav,max : maximal gravitational potential energy throughout a cycle

s,max : maximal static friction coefficient

Ffr : static friction force

Z1stasic : vertical load in the test wheel when the system is stationary f : motion frequency

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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 Fr 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 coefficientCr . 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].

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

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

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 Cr1 and Cr 2 which fit a curve to the measured points, using Equation (1-6)[9].

Gravitational energy:

E  mge 1 cos  (1-1)

Distance travelled:

1(1-2) dR   1

Average rolling resistance force during an interval from maxi to maxi 1 :

EEmaxii 1   max  Fr  (1-3) dmaxi 2 d  mini  d  maxi 1 

Rolling resistance coefficient:

Fr (1-4) Cr  Z1

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 d   R1  esin  , 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.

F dx Z C dx (1-5) Fr 11rr1 C    r, fitted ZZZdx dx 1 1 1

Under the assumption that Z1 is constant:

FC dx11FCrr dx dx Z1 C dx  r r  r Cr, fitted      ZZZdxarea dx calculation 1dx 1  dx 1

Assuming that the rolling resistance coefficient is during periods when the tyre is a distance Lcp from its maximum or minimum points, and Cr 2 elsewhere ( Figure 1.6), this gives

C dx4 C C  L r 2  r2 r 1 cp 4cpRCC r21 r  Cr, fitted C dx r2 RRR    max,i 2 min,i max,i 1

4cpCC r12 r  CCr,2 fitted r (1-6) max,i 2  min,i  max, i 1

Wherecp LR cp / and CCrr12, 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 Cr1 and . This fitting procedure enables Cr2 to be determined. The fitted shows good agreement with the measured coefficients. Furthermore, the transient coefficient ( ) is smaller than the steady state coefficient ( ).

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

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]

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]

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

Figure 1.6: model for rolling resistance coefficient in oscillatory rolling for angles higher than the contact patch angle (based on [9])

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.

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

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

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

Figure 2.3: Concept 2: sprung axle pendulum

Figure 2.4: Concept 3: eccentric drum pendulum

Figure 2.5: Concept 4: dropped road plate

Figure 2.6: Concept 5: eccentric axle pendulum

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 ( Z1 ) should be 39,240 N (4 tonnes).

ii. When the system is static, the vertical load on the solid wheel ( Z2 ) 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 kg/ m3 ) v. To keep the overall size manageable, the distance between the tyre and each the weights

( dd12, ) needs to be at least 0.1 m to allow space for connections.

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

solid wheel and outer weight ( Lc ) 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:

ZZWW    0 (3-1) 1 2 1 2 LZLZLW   0 (3-2) a1 c 2 b 2

According to the above list of constraints, ZLZ12,,c are fixed parameters, while the others are

unknown. In order to solve these equations, an array of combinations of La and Lb 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 W1 and W2 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 ( R1 )

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. is sometimes called the ‘eccentricity radius’.

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:

Fr m (3-3)  cg.

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 :

r r  R  e 2 (3-4) x equ,1 x sin rx   R1  e cos  rx   R1  e cos   e  sin  2 rz r equ z  e1  cos  re sin r esin  e  cos  , z z

=0 Where rrequ,, x, equ z  is the location of centre of gravity when the system is in equilibrium ( ).

Separating the forces into x and y components gives: (3-5)  Fxx X mr

 Fzy Z  mg  mr

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

X m  R  e cos   e 2 sin  (3-6)  1 

Z m esin   e 2 cos   g (3-7)

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

(3-8)  yy H

The torque acting about the centre of mass is:

(3-9)  y  X R1  ecos   Ze sin  The derivative of the angular momentum is:

(3-10) HJy c. g  

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

(3-11)

(3-12)

(3-13)

A and B are defined such that:

AB 0 (3-14) Where

A f1 e,,,, R 1 m Jcm .  

B f21 e,,,,, R m g 

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

B (3-15)   A The initial conditions are:

t 0  0 (3-16)

 t 00  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.

Total energy when   0 :

1 1 1 1 T2 J 2 J  mU2  m U 2 (3-17) kin,max 21 22 2 1 1 2 2 2 The velocities of the two masses are:

UR1  0 1 (3-18) U2  0 R 1 eDW

From Figure 3.6(b)  0 max 117 

22 122 1 1 1 Tkin,max  J1   J2  m 1 R 1   m2 R 1   eDW   2 2 2 2 1 2  22J  J  m R  m R  e  1 2 1 1 2 1 DW   2 (3-19) 2 1  2 2 117  0.1176 0.0996  3  0.28  12.8 0.28  0.105  2 180   1.76J 

Total energy when:  0 :

(3-20) Tgrav,max m 2 geDW 1 cos0 

12.8  9.81  0.105 1  cos 30    1.76 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

Rm1   0.28

m1  kg  ** 3.0

m2  kg  ** 12.8

emDW   0.105 m kg  * 15.8

2 Jcm.  kg m  ** 0.2440

0.0851 *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 Truck Tyre Truck Tyre Truck Tyre Pendulum Pendulum Pendulum Pendulum Case 1 Case 2 Case 3 Case 4

0   ** 105 105 105 105

Rm1   0.50 0.50 0.50 0.50 m1  kg  100 100 100 100 m2  kg  4000 4000 4000 4000 emDW   0.05 0.10 0.25 0.50 m kg  4100 4100 4100 4100

2 Jcm.  kg m  *** 419.02 419.75 424.88 443.17 em  0.048 0.098 0.244 0.488

Lm2   **** 1.708 1.708 1.708 -

LL12/ 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.

*** based on estimation of JJ12, **** 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 (eRDW /1  1.25) .

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:

1 g f  (3-21) 2 eDW

In the problem of a rolling pendulum, if the initial angle is as small as 30°, and eRDW  1 , 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.

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 ( R1 )

iv. The components IIxy, zy of the rig moment of inertia are negligible The system is affected by the following forces:

XYZ1,, 1 1 Contact forces between the floor and the tested wheel

XYZ2,, 2 2 Contact forces between the floor and auxiliary wheel

W - Total Weight

Geometric measurements:

L1 - Distance along y axis between c.g and the contact point of wheel 1

L2 - Distance along axis between c.g and the contact point of wheel 2

The equation of motion for translation is:

Fr m cg. (3-22) Similarly to the 2D approach discussed in section 3.3.1, the centre of gravity trajectory can be expressed as follows: r R esin Re cos  equ,1 x 1 rc., g  requ y , rcg.  0 , re e equ, z 1 cos  sin

R  e cos   e 2 sin  (3-23) 1 rcg.  0 2 eesin   cos 

The force acting in all three directions are:

XX12  YY F 12 (3-24)  Z12 Z mg Substitution of (3-23) and (3-24) into the equation of motion (3-22) gives:

2 X1 X 2  R1  e cos   e  sin  (3-25)  Y Y m 12 0 (3-26) Z Z  mg e  e 2  12 sin cos (3-27)

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

τHc.. g c g (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:

RXZ1 sin   1 cos1 sin        LY   τcg. 1   1      e R1 cos    X1 sin   Z1 cos   (3-29) RXZ1 sin   2 cos2 sin       LY 22        e R1 cos    X2 sin   Z2 cos  

 c., g xbody  L1  X 1 sin  Z1 cos    L2   X 2 sin  Z2 cos     e  R1 cos  Y1  Y 2 

 c., g ybody e  R1 cos   X1 cos  Z1 sin    X2 cos   Z2 sin   (3-30) RXZXZ1 sin 1 sin  1 cos    2 sin   2 cos  

 c., g zbody  RYYLXZLXZ1 sin 1  2   1  1 cos 1 sin    2  2 cos 2 sin   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:

(3-31) HIcg.  ΩΩIΩ     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: 00        ΩΩ  ,     (3-32)     00    The tensor of inertia about c.g is:

0  J  I cg. (3-33) 0

The sign ( ) indicates an ineffective element. Substitution of (3-32) and (3-33) into (3-31) gives: 0   0 0 0   0  0  0           J     J    J   Hc. g c. g     c. g   c. g  0            0   0 0 0   0  0  0 (3-34) 0  J  cg.  0

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:

L  Xsin  Z cos    L   Xsin  Z cos     e  R cos  Y  Y   0 (3-35) 1 1 1 2 2 2 1 1 2

RYYLXZLXZ1sin  1  2   1  1 cos 1 sin   2  2 cos 2 sin    0 (3-36)

It is known from Equation (3-26) thatYY120 , hence the previous two equations can be simplified to:

LXZLXZ1  1 sin 1 cos   2   2 sin 2 cos    0 (3-37)

LXZLXZ1 1 cos1 sin   2  2 cos 2 sin    0 (3-38)

Given the values of L12,,,, 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 XXZZ1,,, 2 1 2 . This system can be put in the following form:

2 X m R1  e cos   e  sin    1 1 0 0 1    2 0 0 1 1 X 2 m g esin  e  cos     (3-39) LLLLZ     1 sin2 sin1 cos2 cos 1   0 LLLLZcos cos  sin  sin    1 2 1 2 2  0 The following manipulation can be done to simplify line (III) and line (IV) of the above equation system:

cosIII   sin   IV    III 

sinIII   cos   IV    IV  and that results in:

2 X m R1  e cos   e  sin    1 1 0 0 1    2 0 0 1 1 X 2 m g esin  e  cos     (3-40) LLZ 00 1 2 1   0 LL 00Z   1 2 2  0

The above equation can be defined as Ax B , and thus can be solved numerically by matrix manipulation: x A1 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 Z1  0 and Z2  0 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 ( FZfr  s,max ), and the lateral

X1 X 2 forces YY12, are negligible, the ratios and are the required friction coefficients to enable no- Z1 Z2 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

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

further wheel experiences LL21/ times less vertical load but also 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:

L2 Z1stasic  m1 m 2  g (3-42) LL12

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 Z1stasic .

Table 3.3 Range of instantaneous load on the test tyre

Parameter Truck Tyre Truck Tyre Truck Tyre Pendulum Pendulum Pendulum Case 1 Case 2 Case 3

emDW   0.05 0.10 0.25 em  0.048 0.098 0.244

Z1min  kN  39.7 (99%) 38.8 (97%) 32.4 (81%)

Z1max  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.

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 (eRDW /1  1.25) . 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)

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

Figure 3.3: geometric measures of 2D eccentric rolling pendulum

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

X: 0 (a) 40 Y: 30 [°]  20

-20 Pendulum Angle Pendulum -40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b) 200

100 X: 1.245 Y: 117.1 0

-100

Rotational VelocityRotational [°/sec] -200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [sec]

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

(a) 150

100 [°]  50

-50

Pendulum Angle Pendulum -100

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

(b) 1000

e = 0.05 [m] 500 e = 0.25 [m] e = 0.49 [m]

-500 Rotational VelocityRotational [°/sec]

-1000 0 1 2 3 4 5 6 7 8 9 10 Time [sec]

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

(a) 40 [°]  20

-20

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

(b) 200

100

-100

-200 Rotational VelocityRotational [°/sec] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [sec]

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

1 model, =30° 0 0.9 simple pendulum model,  =105° 0.8 0

0.7

0.6

0.5 Frequency [Hz]Frequency 0.4

0.3

0.2

0.1 0 2 4 6 8 10 12 14 16 18 20 e / R DW 1

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

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

Figure 3.11: body frame definition

(a) (b)

Figure 3.12: torque calculation sketch in XZ plane

4 x 10 (a) (d) 2 100

[N] 0 [N] 0 1 2 X X

-2 -100 -100 -50 0 50 100 -100 -50 0 50 100 Rotation angle  [deg] [°] Rotation angle  [deg] [°]

4 x 10 (b) (e) 8 400

6 300 [N] [N] 1 2

Z 4 Z 200

2 100 -100 -50 0 50 100 -100 -50 0 50 100 Rotation angle  [deg] [°] Rotation angle  [deg] [°]

|/Z 0.2 |/Z 0.2 2 2 |X |X

0 0 -100 -50 0 50 100 -100 -50 0 50 100 Rotation angle  [deg] [°] Rotation angle  [deg] [°]

Figure 3.13: expected forces for different eccentricity radius cases

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

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

Condition 1 d120.1 m and d 0.1 m Condition 2 LLab Condition 3 W1200 and W Condition 4 Lmtotal  3

Table A-2: Mass distribution combinations Figure Figure La Lb W1 W2 d1 d2 a1 All Conditions

condition 1 condition 2 condition 3 condition 4 ( ( ( ( (

m m m m m ( ( N N ) ) ) ) )

) )

3 . 2

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

Figure Figure La Lb W1 W2 d1 d2 a1 All Conditions

condition 1 condition 2 condition 3 condition 4 ( ( ( ( (

m m m m m ( ( N N ) ) ) ) )

) )

3 . 2

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

Appendix B

Table B-1: conditions of repeating small pendulum methodology Surface: Smooth lab floor Tyre: Radial X Inflation case no. Contact Patch length (cm) (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

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