Simulation and validation of tire deformation under certain load cases

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M A S T E R’S T H E S I S

HENRIK ANDERSSON

Simulation and Validation

of Tire Deformation under

Certain Load Cases

MASTER OF SCIENCE PROGRAMME Mechanical Engineering

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering Division of Computer Aided Design

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This master’s thesis deals with computer aided simulations of mechanical systems in the automotive industry. The specific target of simulation is the pneumatic tire and its behaviour. The aim is to establish a method to use computer simulations for shortening the development cycle and reducing the need for testing and physical prototypes. The work has been separated into several steps, starting with a thorough information study, continuing with creative methods and concept creation. Later on, an evaluation of the concepts has been performed, to find the best approach to continue working on. The selected concepts from the evaluation were further developed to result in the final simulation method. The results of the simulations have then been validated against measurements.

A proposal for further work in the subject has been made, as well as ideas for other projects.

Keywords

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The work presented in this masters thesis is to obtain the Master of Science degree in Mechanical Engineering, with specialization in Computer Aided Engineering. This the-sis has been written at the BMW Group Research and Innovation Center (FIZ) in Munich Germany, during the second half of 2006. The thesis project was initiated and granted by Mr. Robert Hartl, head of the wheel design team at the Wheel and Tire department EF-33 at BMW.

I wish to express my appreciation to my supervisor at BMW, Thomas Kellner for all his help and support during my time at BMW in Munich and to my examiner Tobias Larsson at Luleå University of Technology. Many thanks also goes to Erich Rott and Jens Holtschulze, as well as all the other colleagues at the BMW Research and Innovation Center for their help during my thesis.

Finally, I would like to thank my family and friends for their support during the entire thesis work.

Munich, November 30th, 2006

—————————————-Henrik Andersson

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List of Figures

1.1. Volumetric decomposition of a BMW 3 SERIES Sedan (E90) . . . 3

1.2. Overview of the tire clearance process . . . 4

1.3. ETRTO Standards for generating the static envelope contour . . . 5

1.4. Measurement of deformation on test vehicle . . . 6

1.5. Post-processing in Catia of the measured tire deformation . . . 6

1.6. Sectors for measuring tire deformation . . . 7

1.7. Different types of RHK geometries . . . 7

1.8. Example of SRHK, combination of RHK for different tire dimensions . . . 8

1.9. RGB for front and rear wheels . . . 9

1.10. Foam mounted on inside of wheel well . . . 9

1.11. Definition of tire deformation . . . 10

2.1. Schematic description of a simulation process . . . 17

2.2. Illustration of the axle loads . . . 25

2.3. Theory of cornering . . . 27

2.4. Design of a radial tire . . . 30

2.5. Illustration of the naming conventions for rims and tires . . . 31

2.6. Comparison of SAE and ISO axis systems . . . 32

2.7. Friction circle . . . 33

2.8. Tire/Road friction interaction . . . 34

2.9. Friction dependence on relative velocity . . . 35

2.10. Friction dependence on contact pressure . . . 35

2.11. Description of slip angle phenomenon . . . 36

2.12. Cambering effects . . . 38

2.13. MTS Flat-Trac CT tire test rig . . . 40

2.14. Pacejka’s Magic Formula . . . 42

2.15. Discretization of FTire flexible ring model . . . 44

2.16. Cross-section representation in the flexible ring model . . . 45

2.17. Structure representation in the FETire model . . . 45

2.18. Structure representation in the RMOD-K 7 Flexible belt model . . . 47

3.1. Simple tire model used in Volkswagen study . . . 51

3.2. Statistics of tire deformation . . . 56

3.3. Adams tire model . . . 57

3.4. Mechanical tire model . . . 58

3.5. FTire based tool for tire simulation . . . 61

3.6. RMOD-K tool for tire simulation . . . 61

3.7. ABAQUS FE modelling . . . 62

3.8. Measurement of forces acting on the wheels . . . 63

3.9. Measurement from the handling course at Aschheim . . . 63

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3.11. Validation of lateral slip . . . 65

3.12. Validation of longitudinal slip . . . 66

3.13. Measurement cycle on the test rig . . . 67

3.14. Mirrored deformations . . . 68

3.15. Post-processing of results in Catia V5 . . . 68

4.1. Comparison of simulations and measurements . . . 70

4.2. Comparison between measurements and Gnadler’s approximation for-mulas . . . 71

5.1. Influence of the rim flange to the deformations at large lateral forces . . . 74

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List of Abbreviations

Notation Description

4WD Four wheel drive

ABS Antiblockier System, Anti-lock brakes ALK Auslenkkontur, Deformation contour

CAD Computer Aided Design

CAE Computer Aided Engineering

CATIA Computer Aided Three-dimensional Interactive Ap-plication

cg Centre of gravity

CTI Cosin Tire Interface

ESP Electronic stability program

ETRTO European Tire and Rim Technical Organization FEA Finite Element Analysis

FIZ Forschungs- und Innovationszentrum FORTRAN Formula Translation Language

FWD Front wheel drive

ISO International Organization for Standardization MATLAB Matrix Laboratory

MBS Multibody Dynamics Simulations

NVH Noise, vibrations and harshness. Related to comfort and unwanted vibrations in a vehicle

PLM Product Lifecycle Management

rc Roll centre

RGB Reifengebirge, Wheel envelope RHK Reifenhüllkurve, Tire envelope RMOD-K Reifenmodell K, Tire model K

RWD Rear wheel drive

SRHK Summenreifenhüllkurve, Summary tire envelope SWIFT Short Wavelength Intermediate Frequency Tire

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List of Symbols

α Slip angle [deg]

β Sector angle on tire [deg]

∆K Tire deformation (Gnadler Formula) [mm] γ Camber angle [deg]

µ Coefficient of friction [−]

µ Population mean (Normal distribution) [−] Ω Actual wheel rotation speed [rad/s]

Ω0 Reference rotation speed at actual velocity, vertical force, slip angle and

inclina-tion angle and for longitudinal force Fx = 0 [rad/s]

σ2 Variance (Normal distribution) [−] θ Hill gradient angle [deg]

A Rim width [mm]

Amax Maximum rim width [mm]

ax Longitudinal acceleration [m/s2]

ay Lateral acceleration [m/s2]

B Stiffness factor (Magic Formula) [−] C Reference tread width [mm]

C Shape factor (Magic Formula) [−] cα Cornering tire stiffness [N/deg]

c∗_Rp Pressure dependent stiffness in Radial direction [N/mm/bar] c∗_Rv Speed dependent stiffness in Radial direction [N/mm/m/s] cR Vertical tire stiffness (Radial stiffness) [N/mm]

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cy Lateral tire stiffness [N/mm]

D Peak value (Magic Formula) [−] DA Aerodynamic drag [N]

DG Maximum tire overall diameter in service [mm]

dr Nominal rim diameter [mm]

E Curvature factor (Magic Formula) [−] E Young’s modulus of elasticity [N/mm2] ET Einpresstiefe, wheel inset distance [mm]

fRL Tire load factor, relative to tire load index [−]

Fs Side force [N]

FX Longitudinal force [N]

FY Lateral force [N]

FZ Normal (vertical) force [N]

G Shear modulus [mm]

h Center of gravity height [mm]

ha Aerodynamic force arbitrary height [mm]

L Length [m]

MX Overturning moment [Nm]

MY Rolling resistance moment [Nm]

MZ Self aligning moment [Nm]

OD Tire outer diameter [mm] Pi Tire inflation pressure [bar]

R Tire undeflected radius [mm]

r Radial distance from wheel center to point on sidewall (Gnadler Formula) [mm] RI Tire inner radius (Gnadler Formula) [mm]

RX Tire rolling resistance [N]

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Sv Vertical shift (Magic Formula) [−]

SG Maximum tire overall width in service on intended rim width Amax [mm]

SR Slip ratio [%]

tp Pneumatic trail [mm]

W Tire width [mm]

W Vehicle load (total) [N]

Wf Vehicle load on front wheel pair [N]

Wr Vehicle load on rear wheel pair [N]

xm Peak location (Magic Formula) [−]

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1. Introduction

The recent centuries of product development in the industry have seen a lot of revolu-tionizing changes. Not long ago product development and manufacturing were made by the local blacksmith or other skilled craftsmen. The lifecycle for a product was usu-ally very long and new products could take many years to develop.

Since the beginning of the industrialization and onwards, the globalization has affected how business is done and has led to increased competition between companies on a global market.

Nowadays, the time between idea and final product can be just a few months; this ap-plies especially to the computer and electronics industry where new products appear within months and older products become obsolete in the same short time.

For the vehicle industry the tough competition and demands on high profitability have led to big changes in recent years. Manufacturers merge and cooperate on the develop-ment of new vehicles to reduce the costs and to save time.

These big changes have led to the demands of shorter, more economic development cycles. To be able to achieve this, the companies wish to move the expensive and time-consuming prototyping to a very late step in the development process. By using com-puter aided tools for virtual product development these demands can be fulfilled. The introduction of virtual product development started in the 60s with the introduction of computer based tools and has since then gained further momentum due to the ever increasing capacity of computers.

1.1. Tire clearance

In the early steps of a vehicle development process, the specifications for chassis, sus-pension, drive train and other important system components are set.

At the tire development department the complete range of approved rims and tires is determined in close cooperation with the tire manufacturers and other internal depart-ments involved in vehicle development.

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To ensure fitment and clearance of all these wheel and tire combinations in the wheel housings of the vehicle, tire clearance investigations are performed.

In a simple static analysis, where all components are assumed to be infinitely stiff and to not deform, the clearance between different components is easily verified in the CAD system.

The case, however, is not this simple. During driving, i.e. dynamic conditions the com-ponents do not behave in the same way as for a simple static case. An example of this is the deformation of the tire structure due to the external forces generated by the in-teraction between tire and road. To ensure that the clearance can be guaranteed for all conditions, the deformation of the tire also has to be taken into consideration.

The first step in the clearance process is to create a so called wheel envelope for all possible tire and rim combinations. It contains information on rim size and positioning, tire dimension, worst case of tire deformation and also the snow chain geometry. These different geometries are superpositioned so that they represent the space required for the wheels in the suspension movement.

During the course of the development process, the wheel envelope is used for clearance investigations in different departments, including packaging, suspension, styling, etc. If there is a problem with the wheel envelope and this is discovered in a later step, during prototype testing, it will be very costly to go back in the development process and correct it.

1.1.1. Purpose

The packaging of components in cars today is becoming more and more complex. With increasing safety requirements, demands on weight distribution, more system compo-nents in the engine bay and also to give room for passengers and luggage, optimizing the use of space has a high priority.

When looking deeper into how much space each different subsystem on a car takes, calculations show that the passenger compartment allocates about 4.1m3, a little more than half the total volume of the car. Both wheel housings in the front require more space, (0.8m3) than the two rear wheel housings (0.5m3) due to the steering of the wheels in the front.

The four wheel housings account for about 17% of the total vehicle volume and much of this space is only used during large suspension and steering movements. It is therefore desired to use this space as efficiently as possible.

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Vehicle: 1.6m3 4.1m3 2.1m3 Wheels: 0.8m3 0.5m3 50% 25%

Figure 1.1.:Volumetric decomposition of a BMW 3 SERIES Sedan (E90)

Also for styling purposes, there is a strong need for more accurate knowledge of the tire clearance, so that the designers can have as much freedom as possible to design the wheel arches and the body panels around the wheels.

The local deformation of the tire is one of the components in the clearance process that can be further analysed, to improve the tire clearance process.

1.1.2. Schematic overview

The beginning of the development process for a new vehicle with respect to tire clear-ance starts with the concept design of possible tire and rim combinations for series and aftermarket equipment. As soon as the evaluation and selection of the different tire and rim concepts have been done, the creation of wheel envelopes for the upcoming vehicle can begin.

The first step is the creation of the "base tire profile" (RFNP: 1.1.2) and the "new tire limit profile" (NRGK: 1.1.2) for the range of approved tire and rim combinations.

The next step is to take into consideration the use of snow chains and the worst case of tire deformation (ALK: 1.1.2) to create the tire envelope (RHK: 1.1.2).

Combining the surfaces for all the approved tire and rim combinations results in the so called summary tire envelope (SRHK: 1.1.2). A SRHK is composed of several RHK’s. With the suspension geometry as basis for the kinematics, the SHRK is then superposi-tioned at all the possible positions in the suspension movement, to finally create the end result in the tire clearance process, the wheel envelope (RGB: 1.1.2).

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TyreClear-Module

TyreClear

Process overview

Oracle-DB for Tire-Data-Management / Guideline-Management (RDM) P R I S M A (PDM) Tire cross section profile (RFNP) New tire limit profile (NRGK) Tire envelope (RHK) Summary Tire envelope (SRHK) Wheel envelope (RGB) Tire deformation (ALK) Wheel Use • History • Read Administration / Settings • User-Admin • RDM-Admin • PRISMA/RDM-Login • RDM-Reconnect Driving maneuvers (max. lat force)

Snow chain geometry CAD Axle kinematics ... ... ... Control data F(x) Calculations Driving maneuvers (Compl. tyre Clearance )

Elast. Displacements

Tyre Industry

Figure 1.2.:Overview of the tire clearance process

RFNP

1

- Base tire profile

The base tire profile (RFNP) is the foundation of the tire clearance process. A problem with tires is that a specific tire dimension is not always exactly the same size depending on manufacturer and when the tire was produced.

To take this into consideration, the tire profile is created based on ETRTO2 standards [1]. These state exactly how to create the tire contour for any tire dimension, based on a design and maximum in service measurement of the base diameter and width for the tire (figure 1.3).

NRGK

3

- New tire limit profile

From experience it is known that the tire manufacturers rarely use the maximum al-lowed diameter and width for their tires, for economical and other reasons they try to stay below this limit.

By starting out from the ETRTO standards used to create the RFNP and then reducing the overall contour by a few millimetres, a better corresponding tire geometry is created, the NRGK.

1_{Reifenprofil - Base tire profile}

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S_G A_max Rf Rt Re B_max Gmin C P S P dr DG S T Rbf

Figure 1.3.:ETRTO Standards for generating the static envelope contour

ALK

4

- Deformed tire contour

The deformed tire contour, ALK is one of the most important parts in the tire clearance process. It contains information on how the tire deforms during driving, as this has a big effect on the required clearance.

The first step to create the ALK, is to drive a test vehicle on a track under varying con-ditions and expose the tire to the maximum lateral forces that can occur. The driving style is more extreme than during normal conditions, but still not misuse. The resulting vertical and lateral forces lead to a deformation of the tire sidewall.

The deformation is measured with a foam plate mounted to the spring strut (figure 1.4). The foam is positioned to sit firmly against the inner sidewall of the tire.

During driving, abrasion of material from the foam occurs, leaving an imprint of the tire deformation behind. The foam plate is then removed from the vehicle and measured with a laser scanner to create a geometry for import in the CAD-system (figure 1.5a). In the final step of creating the ALK, the contours of the deformations in the foam are extracted from twelve sectors around the tire circumference, according to figure 1.6. These extracted curves constitute the ALK (figure 1.5b).

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(a) Foam plate mounted on front suspension upright

(b) Front suspension and tire with foam plate mounted

Figure 1.4.:Measurement of deformation on test vehicle

(a) Scanned foam plate imported into Catia V5 (b) Extracted deformation curves from foam plate

Figure 1.5.:Post-processing in Catia of the measured tire deformation

RHK

5

- Tire envelope

The tire envelope, RHK contains information about tire deformation from the ALK and also snow chain geometry. It is created by sweeping a surface around the tire circumfer-ence using the curves from the ALK.

As tire deformation with respect to clearance is only seen as critical on the inside of the tire, the resulting RHK only contains information about the deformation on the inside (figure 1.7a).

The deformations for a new tire dimension can also be approximated, by using the ALK from one tire dimension to create the corresponding RHK for another dimension. For example; the deformation for a 195/65 R15 tire can be approximated from a 185/65 R15 tire.

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v0 45° 90° 105° 120° 135° 180° 225° 240° 255° 270° 315° 0/360° ω Direction of travel Direction of rotation Circumference angle β

Figure 1.6.:Sectors for measuring tire deformation

(a) RHK with tire deformation (b) RHK with tire deformation and snow chain geometry

Figure 1.7.:Different types of RHK geometries

SRHK

6

- Summary tire envelope

The summary tire envelope, SRHK is created by combining all tire envelopes (RHK) for the range of allowed tire and rim combinations.

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Figure 1.8.:Example of SRHK, combination of RHK for different tire dimensions

RGB

7

- Wheel envelope

The creation of the wheel envelope or the "tire mountain"8, RGB is a two step process where several parameters such as kinematics, elastokinematics, steering of the wheels, etc. are taken into consideration.

In the first step certain vehicle parameters are determined. This takes the following vehicle data into account, such as: drive train (4WD, RWD, FWD), suspension type (Independent, Solid axle), wheel travel and driving conditions (Off-road, etc.).

Based on these parameters, control data is retrieved from a database, which contains all the possible positions of the wheel in the suspension. The second step is then to super-position the SHRK geometry at these super-positions. The end result is the wheel envelope (RGB), illustrated in figure 1.9.

TyreClear Application

As an aid in the creation of the different geometries used in the tire clearance process, a special software called TyreClear is used by BMW. The application takes care of the creation of all curves and surfaces that make up the different geometries (RFNP, NRGK, ALK, RHK, SHRK and RGB) and the positioning of these in the the suspension travel.

7_{Reifengebirge - Wheel envelope}

8_{The expression "tire mountain" - Reifengebirge originates from the appearance of the tire geometries} looking from above, resembling the altitude-curves of a map.

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(a) Front wheel RGB (b) Rear wheel RGB

Figure 1.9.:RGB for front and rear wheels

Validation of tire clearance

The verification of the simulated tire clearance is performed on a prototype of the pro-duction vehicle, this time with pieces of foam attached to the inside of the wheel housing liner (figure 1.10).

Figure 1.10.:Foam mounted on inside of wheel well

In the same way as with the foam plate mounted to the inside of the wheel, material will be worn off from the foam during driving. By scanning the surfaces of the foam and comparing this with the computer analysis, the simulated clearance can be validated.

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Definition of tire deformation

The tire deformation is defined as the difference between the undeformed inflated tire profile and the deformed tire contour (figure 1.11). The outermost point on the unde-formed tire profile is used as the reference and a vertical line is created at this point. The tire deformation (∆K) is then the perpendicular distance from the maximum outer point on the deformed tire profile to the reference line. A positive deformation equals a deflection outwards with respect to the tire centre line.

Undeformed tire profile (inflated)

Deformed tire profile

Undeformed sidewall reference line

Tire deformation ∆K Tire centre line

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1.2. Observations

The following important observations of the current method for measuring the tire de-formation have been made:

• The foam measurements performed on the test vehicle only gives detailed infor-mation about the deforinfor-mation of the tire sidewall; the tread area cannot be mea-sured using this method. The deformation of the this area has to be extrapolated from the deformation of the sidewall to form the complete tire contour. From ex-perience and theory it is known that the tire not only deforms in lateral direction, but also in radial direction.

• Deformation measurements on a test vehicle cannot be performed for each and every approved tire/rim combination; it would mean too much work. Also, when measuring a specific dimension, tires from different manufacturers have to be tested, to be certain that the most compliant tire is used as the reference.

• There are certain cases, when a completely new tire dimension, that does not yet exist on the market, is developed for a new vehicle. For these unknown dimen-sions, the deformation has to be approximated from another tire.

• Efficient use of resources during the development cycle is important and measure-ments with test vehicles take a lot of time. To be able to perform these tests, a prototype of the car model in question has to exist. Waiting for this prototype is not always possible, so either another car or the measurements from another tire are used to approximate the deformations.

1.3. Problem formulation

The main objective for this thesis is to research and develop a method to simulate the behaviour of the tire and determine the deformation of the tire under different condi-tions. The aim is to replace the current method, which requires full vehicle testing and measurements, with simulations.

Different tools and approaches are to be investigated, to result in a final method for simulation of the tire deformations.

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2. Theory

This chapter brings the reader deeper into the theory of product development, simula-tion in the automotive industry, vehicle dynamics and the pneumatic tire. It can be seen as a kind of reference manual for the work performed in this thesis.

2.1. Product design & Development processes

Much of the information in this section has been found in the works by Ulrich & Ep-pinger [2] and Pugh [3].

2.1.1. Introduction to product development

What is a product? According to ISO1and the ISO9000 standards [4] the definition is: A product is an output that results from a process. Products can be tangible or intangible, a thing or an idea, hardware or software, information or knowledge, a process or procedure, a service or function, or a concept or creation.

A process uses resources to transform inputs into outputs. In every case, inputs are turned into outputs because some kind of work, activity, or function is carried out. Processes can be administrative, industrial, agricultural, governmental, chemi-cal, mechanichemi-cal, electrichemi-cal, and so on.

Ulrich & Eppinger [2] sums up all this to describe what product development really is about.

The set of activities beginning with the perception of a market opportunity and ending in the production, sale and delivery of a product

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The different phases in the development of a product are compiled in a "Product devel-opment process", which can be somewhat different depending on which kind of com-pany and what kind of product it concerns. In general, however, most product develop-ment processes are quite similar.

As described by the Participatory Innovation Process [5] used at Luleå University of Technology, the different steps or phases in product development can be described as:

• Design space exploration - Need finding, benchmarking and related technology • Roadmap - Mission statement, product characteristics

• Concept Design & Prototyping - Concept generation, evaluation and selection. • Detailed Design & Manufacturing - Virtual and physical prototypes

• Product launch

Using a well defined process ensures that lead times are kept short, development costs are reduced and a high quality of the end product is achieved.

2.1.2. Design space exploration phase

In the beginning of a product development process, it is usual that not much information is known about the product to be created. In the case of a completely new product, everything starts from scratch.

To determine what kind of product to create, a need finding analysis can be applied to research for market opportunity and customer needs. When an opportunity and needs have been found, existing products that fulfil these are benchmarked to see if there is any interesting information that can be of use.

Another step is to research related technologies for other interesting ideas that could be applied to the new product.

This results in the information describing market needs and the potential for creating a new product.

2.1.3. Roadmap phase

When the needs are clearly defined it its time to document the product description and characteristics to have a clear understanding of exactly what kind of product to create during the upcoming design phases.

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2.1.4. Concept Design & Prototyping

Using the product description, a lot of different product concepts are created, evaluated and finally a selection of the best ideas is made.

Ideas and concepts are generated with different creative methods, to come up with so-lutions to problems or creating complete concepts. Sometimes these creative methods give you a direct solution to the problem, other times you will end up with a list of ideas eventually resulting in a solution.

Creative methods

Random word

The random word method is perhaps the simplest of creative techniques. A person confronted with a problem is given a random word. The idea is that the random word could connect to the problem and make way for new ideas.

A variation of this is to use a random image or some other form of random information to associate to.

Brainstorming

Brainstorming is a creative method to generate ideas for a new product. It is most effi-cient when performed in a small group. The most important thing is not to judge any ideas in advance and let everybody present their thoughts. No ideas should be turned down this early in the process. Brainstorming is really more about quantity than having a single good idea.

Brainwriting

Brainwriting is another creative method, where in contrast to brainstorming ideas are written down on a paper, rather than being discussed openly. The participants can sit down and write down their ideas in peace and quiet, without being distracted by oth-ers.

The benefit of brainwriting is that quiet people, that have difficulties to verbally express their opinions, have the same chances, as more open people.

The first stage of a brainwriting session is to gather as many ideas as possible, without any restrictions. An evaluation of the ideas is later done in the second phase to narrow

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Rohrbachs method

Another form of brainwriting was developed by Rohrbach [6].

A brainwriting session with Rohrbach’s method can be described as sitting down with 6 people in a group, handing out a paper to each person. During 5 minutes, each person sketches down 3 ideas on a predefined theme. Each idea is then passed on to the 5 other people to develop the ideas further.

Advantages of this method is that it is easy to use, generates a lot of new ideas and no ideas will be criticized. Some drawbacks are that there is no real feedback on ideas and it can also inhibit creativity, due to the many restrictions.

The Rohrbach method is best suited as a compliment to a brainstorming session, for example to further develop already discovered ideas.

Concept evaluation

To determine which concepts that are most suitable, it is common to perform a concept evaluation. The evaluation is usually based on different criterion, which the concepts are compared against.

Pughs method

Pughs method [3], scores and evaluates the concepts relative to each other.

One concept is selected as a datum, for example the favourite candidate or maybe the current method that is to be improved on. The other concepts are then compared to the datum for each criterion and graded depending on if they are better (+), equal (S) or worse (-). The evaluation is performed using a simple matrix to group the data.

Pugh’s method will give an insight to the best alternatives, but not a conclusive candi-date for selection.

Quantitative method

Another form of evaluation is the quantitative method, where the concepts are scored against each other with weighted criteria.

The quantitative method is however, very sensitive to whom does the scoring and the degree of confidence in the results cannot really be justified. The danger is that you can be convinced to trust the numbers and say that a concept that has 1.32 points really is much better than the concept with 1.26 points.

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2.1.5. Detail design & Manufacturing

This is the phase of the development process where the selected concept is developed into a real product. The concept is optimized for product launch and manufacturing with high quality, few problems and low cost.

Much of the work is performed with computer aided tools, to aid the design and sim-ulation processes. This ensures that many of the problems can be corrected long before the product goes out to manufacturing and market launch.

A very important step in the development process is the testing. Using virtual and real prototypes, the product can be rigorously tested to comply with the specifications. Even though this step is desired to be entirely performed with computer aided tools, real testing is still today an important part of any development process.

2.2. Simulation in Engineering

As defined by Smith [7],

Simulation is the process of designing a model of a real or imagined system and conducting experiments with that model. The purpose of simulation experiments is to understand the behaviour of the system or evaluate strategies for the operation of the system. Assumptions are made about this system and mathematical algo-rithms and relationships are derived to describe these assumptions - this constitutes a "model" that can reveal how the system works. If the system is simple, the model may be represented and solved analytically.

2.2.1. The simulation process

A simulation process [7] can be described as:

• Define the problem, i.e. the system to be simulated

• Create a simple model representing of the system and collect input data for the model

• Create a computer software for the model

• Verify and validate the model, iterate back to correct errors in software & model. • Design model experiments & perform simulations

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• Collect output from the simulations, analyse the data and document the results • Expand and adapt model for other types of simulations

Real system Simulation model

Documented results Formal results

Design Experiments

Model optimizations Expand & Adapt

Evaluate and analyse results

•Graphical, tabular, animation, written evaluations

Analyze system and define model

•Collect input data •Create computer model •Verify and validate

Figure 2.1.:Schematic description of a simulation process

2.2.2. Simulation in Automotive Engineering

The use of simulations in the automotive industry is very common today. Charlés [8] describes how the digital revolution started in the 1960s with CAD/CAM and since then been extended to include full simulation of the entire design, manufacturing and assembly processes of vehicles.

The benefits of using simulations will increase further in the future as more advanced models are developed, taking advantage of increased computer performance.

2.3. Product Lifecycle Management

PLM is the term used for the process to manage the complete lifecycle of a product, from concept, design, manufacturing to service and finally end disposal. The term PLM covers several other capabilities such as CAD, CAE and CAM2.

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2.3.1. Computer Aided Design

CAD is a common term for the use of computer based tools for engineers, architects and designers in their design work. CAD is the tool used to create the 3D geometries within the PLM process.

Thirty years ago, most of the design drawings in the industry were created on paper, resulting in a lot of effort to change a design. Since the introduction of computers as the primary tool to create drawings, the process of designing products has changed drastically.

The first CAD systems were used to create 2D drawings. Later came the transition to 3D, which enabled creation of simple solid geometries and to combine them to form other solids. Today, the CAD technology fulfils all the demands for complete virtual product development.

CAD Software

CATIA

CATIA is a PLM/CAD/CAM/CAE software developed by Dassault Systemés [9]. CA-TIA is widely used in the engineering industry, especially in the aerospace and automo-tive sector. Many German automoautomo-tive companies such as BMW, Porsche and Daimler-Chrysler use CATIA. Other big users are the aircraft manufacturers Boeing and Air-bus.

The history of CATIA [10] began in 1969 when French aircraft company Avions Mar-cel Dassault needed a software to create drawings. In 1975 they bought the software CADAM3 from Lockheed. CADAM was a 2D software which was combined with in house developed 3D software called CATI. Dassault Systèmes was then founded in 1981. Around this time a deal was made with IBM to market CATIA to other companies. From 1982 until 1988 version 1 to 3 of CATIA was released. In 1993 came the release of version 4 (V4). The latest version of the software is version 5 (V5), introduced in 1999. V5 was a complete rework from the old V4, with availability for the Microsoft Windows platform.

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2.3.2. Computer Aided Engineering

CAE is the area within PLM that encompasses the simulation, validation and optimiza-tion of products and manufacturing tools. The use of CAE tools has drastically reduced the effort of testing and validation by using computer software for simulations. Es-pecially in the automotive industry, CAE tools have revolutionized the development process, making products cheaper but in the same time more durable and safer.

A CAE simulation process can be summarized in three steps:

• Pre-processing - creating the simulation model with geometry, loads and bound-ary conditions.

• Analysis - solving the equations behind the model

• Post-processing - analysis of the results using visualization tools The main uses of CAE are:

• Stress analysis on components using Finite Element Analysis (FEA);

• Thermal and flow simulations with Computational Fluid Dynamics (CFD); • Simulation of kinematics and dynamics with Multibody Simulation (MBS);

• Manufacturing process simulation such as casting, moulding, and die press form-ing.

Finite element analysis

A finite element analysis (FEA) uses the finite element method, which is a technique for solving complex engineering problems. It can be applied to a wealth of problems, including component strength, heat transfer and vibration, etc.

It is difficult to pinpoint the exact origin, due to different interpretations of what really counts as a finite element method. Oden [11] describes several different works, which all have contributed to the theory of the finite element method.

The ABAQUS documentation [12] gives a good introduction to the theory of the FE method.

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Finite element theory

The finite element method can be explained as dividing a geometric body into smaller parts, in the form of elements. Each element in the model is a discrete part of the complete structure, which is composed of several elements, connected to each other by nodes. The collection of elements and nodes is referred to as the mesh.

The elements can be of different type, size and shape, and depending on the number of elements the mesh can be a better description of the real physical problem. Increasing the number of elements and nodes in the mesh also means that the computational cost increases, giving longer simulation times.

Depending on the type of problem to solve, different equations that describe the phys-ical properties of these elements are used. An example is Hooke’s law, which describe the relation between stress and strain for a material using Young’s modulus of elasticity, σ =e∗E.

The analysis can be performed with mainly two methods, implicit and explicit. In an implicit analysis the solver must iterate to determine the solution to a problem, where in the explicit case the solver determines the solution without iterating, by explicitly advancing the kinematical state from the previous increment.

An implicit method is more efficient for solving smooth nonlinear problems, where the explicit is the choice for a wave propagation analysis, i.e. more dynamic problems. Hanley [13] explains the difference between these two methods, where the explicit solv-ing does not use the large stiffness matrices, therefore reducsolv-ing both the number of calculations and required disk space. A drawback of the explicit solving is the need of shorter time steps, to obtain numerical stability and good accuracy.

FE Software

ABAQUS

ABAQUS is a software package for finite element analysis, which is widely used in the automotive, aerospace and industrial product industries. The package is also very popular with academic and research institutions due to the wide material modelling capabilities.

The software package consists of two analysis components: ABAQUS/Standard and ABAQUS/Explicit, for implicit and dynamic problems.

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Multibody Dynamics

Multibody Dynamics (MBS) is a field of work which involves the kinematics and dy-namics of bodies connected to each other with joints. It can be used to build and sim-ulate mechanical systems and determine the system behaviour without building real prototypes.

Examples of the application for the MBS method are: • Analysis of complex mechanical systems

• Virtual prototyping of different product concepts

• Determining static/dynamic loads and stresses on structures • Load cases for Fatigue and FE analysis

• Design experiments to optimize certain product parameters

During the last four decades, the use of MBS in the industry has seen a big increase due to the introduction of virtual product development. The MBS tools enable to build models of complex systems in a fraction of the time it would take to create prototypes.

MBS theory

MBS describes the kinematics and dynamics of rigid bodies and has its foundation in the science of mechanics, which is the oldest and maybe the most important of the physical sciences in engineering today.

Archimedes was the first person to write down his discoveries in mechanics, in the form of the mathematics behind the lever and the principle of buoyancy. Further achieve-ments that led to progress in mechanics was the work of Stevin [14], who formulated much of the theory behind statics and how to combine forces with vector algebra. The first documented dynamic experiments date back to Galileo (1564-1642) [15]. He performed and documented experiments with falling stones and derived the physics and mathematics behind them.

Based on Galileo’s discoveries, Newton (1643-1727) [16] continued with this work and in 1687 the first edition of the book Principia was published. In Principia, Newton formu-lated the universal laws which govern the motion for almost all bodies. These include the law of gravitation and also the so called three laws of Newton[17], which are as fol-lows :

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Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is com-pelled to change that state by forces impressed thereon.

Second law

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Third law

To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

These three laws can all be summarized into the well known equation, which states that the force is directly proportional to the acceleration,

F =ma

By using the laws of Newton together with the work of people like Lagrange and Hamil-ton [18] [19], systems of equations describing the motion for bodies in a MBS system can be formulated.

Degrees of freedom

An important conception in three dimensional modelling is degrees of freedom (DOF). As the position of a body in space is described with three coordinates, the total number of natural DOF is six, i.e. both translation and rotation in all three coordinates (x,y,z). In a multibody system the total DOF can be calculated as follows:

Total DOF=6× (Number of parts−1) − (Number of constraints)

The reason of subtracting one part from the total number of parts has to do with the non moving ground, which is counted as a part in the system.

Joint types in a MBS System

Parts in a multibody system are all connected with joints. Depending on the type of joint, it will constrain and restrict the motion, reducing the number of DOF.

Something that might happen is the subject of over-constraining or redundant con-straints, i.e. when two or more constraints describe the same motion. This occurs when the total number of DOF in the system is less than zero.

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Solving of the multibody system

The formulation of the equations of motion for the parts in a multibody system often lead to a matrix with the larger part of the elements being zero. Such a matrix is referred to as a sparse matrix. Solving such a matrix is relatively fast, making it possible to analyse complex mechanical systems in relatively short time. Compared to solving finite element or fluid dynamics problems the time for solving a MBS system is much faster. Depending on if the problem is linear or not, the solver uses different techniques to compute the results. The Newton-Raphson [20] technique is one method that can be used for solving.

MBS Software

The history of computer simulations with MBS started with text based or simple graph-ical methods, based on the kinematics of the components. One of the first applications was KAM [21] developed by Cooper et al., which could solve simple mechanical sys-tems and determine displacements, velocity and acceleration of the components. COM-MEND [22], was another software for solving planar systems.

The Adams package by MSC.Software [23] is possibly the most used MBS software to-day. The origin of Adams can be traced back as far as 1963, to the work initiated by Chace [24] [25] [26] [27] at the University of Michigan.

Adams/Car is a part of the Adams software package. This software enables the use of advanced simulations for predicting and verifying vehicle dynamics and performance. It is used by many vehicle manufacturers as well as their subcontractors.

Adams/Car is also available in a special version, known as Adams/Car-AT [28]. It is the result of a cooperation between MSC.Software and BMW.

2.4. Theory of ground vehicles

The works from Gillespie [29] and Milliken et al. [30] have been used as reference for the theory of ground vehicles.

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2.4.1. Vehicle dynamics

The mechanics behind ground vehicles can be described using Newton’s laws. A vehi-cle, independent of configuration can be treated as a body moving in space, affected by forces and accelerations. The big difficulties come when you add aerodynamics and tire mechanics to the model.

History of road vehicles

The first motorized vehicles appeared in the end of the 18:th century, invented by people like Cugnot [31], Watt and Trevithick [32]. These vehicles were powered by steam and rather slow. In 1886, the first gasoline powered automobiles were created by Daimler and Benz [33]. Further developments to vehicle technology, including the introduction of the Model T and the moving assembly line, were made by Ford [34] in the beginning of the 20th century.

Due to the advancements in automotive engineering, the speed capability of vehicles soon exceeded what the roads of the time could handle. As a result of these higher speeds, the dynamics of vehicles became more important. Much of the understanding was however hindered by the lack of knowledge in tire mechanics. Then, in 1931, the first equipment to analyse tire behaviour was created [35]. This became the starting point for more research in the area, which led to much of the understanding of vehicle dynamics as we know it today.

Basic theory

To understand tire behaviour it is also important to have a basic knowledge in vehicle dynamics. By studying the forces that the vehicle applies to the tires, the behaviour of the tire can be determined.

Force equilibrium

Using the reasoning of Gillespie [29], a free body diagram for the forces acting on the vehicle can be drawn (figure 2.2).

The loads on the front and rear wheels during acceleration on an inclined road are de-scribed by equation 2.1:

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D_A ma x W cosθ W f Wr Fxr R_xr Fxr R_xr c b L h θ W W sinθ ha Z X

Figure 2.2.:Illustration of the axle loads

Wf = (W·c·cos(θ) −max·h−DA·ha−W·h·sin(θ))/L Wr = (W·b·cos(θ) +max·h+DA·ha+W·h·sin(θ))/L

(2.1)

To calculate the static load distribution on level ground, set DA, θ and axto zero.

The acceleration capability of a vehicle, with a combined tractive force FXon the wheels

is described by:

max =

W

g ax = Fx−Rx−DA−W·sin(θ) (2.2)

Roll centre

The roll centre, rc (figure 2.3), is defined as a virtual point in the suspension geometry which the chassis rotates (rolls) around under influence from lateral forces. The lateral force acts through the centre of gravity (cg) and results in roll moment through the lever between rc and cg. The location of the roll centre depends on the type of suspension. Roll centre can be defined for both the front and the rear suspension, drawing a line between these points give a roll axis, which the complete car wants to roll around.

Steady state cornering

To enable a vehicle to drive through a corner, the driver has to turn the steering wheel and follow the curvature of the road. The forces that are generated have to be reacted by the tires.

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Known from basic mechanics [36], the centripetal force is the force required to make a body move in a circular path. This can be any physical force (gravity, electromagnetic, friction, etc.). For ground vehicles this force is generated by the friction between the road and the tires.

The centripetal force, which depends on vehicle mass, speed and cornering radius is defined as:

Fc = −mv 2

r (2.3)

A fictitious force4 related to the centripetal force is the centrifugal force, which can be observed as the force pushing a passenger in a vehicle outwards in a corner. The cen-trifugal force is equal in magnitude and opposite in direction to the centripetal force. Following this reasoning, the centrifugal acceleration (lateral acceleration) in the vehicle y-direction can be defined as:

ay = v 2

r (2.4)

Due to the lateral acceleration in a corner, the vehicle will be affected by a force acting on the centre of gravity. As the centre of gravity usually lies higher than the ground plane, this will lead to a moment around the point defined as the roll centre.

Consider the following simple example with a BMW 325i 3 Series (E90) from 2006. The vehicle is assumed to have the following specifications, vehicle weight mtotal1500 kg , cg

location hcg500 mm from the ground plane, track width t 1500 mm, weight distribution

50/50 front/rear. The roll centre location (rc) is assumed to be in the road plane.

To simplify the problem, only the loads on the rear axle are studied. With the given weight distribution (50:50), this gives us half the total vehicle weight on the rear wheels, m=750kg. Each wheel will then have a load of 3679 N. Studying the lateral load transfer for the forces in y and z directions, as well as for the moment around the rc, gives us the equations for the wheel loads:

FZ1 = mg 2 + may·hcg t FZ2 = mg 2 − may·hcg t (2.5)

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cg Fz1 _Fz2 mg Fy1 Fy2 may rc hcg t Z Y

Figure 2.3.:Theory of cornering

With the given vehicle data, equation 2.5 gives FZ1 =4905 N and FZ2 =2453 N at a lateral acceleration of 1g. Compared to the static weight distribution this gives a 33% increase of the load on the outer wheel.

Vehicle simulation with MBS

Simulation of vehicle dynamics often requires solving multiple equations to get a re-sult. Simple static cases can be calculated by hand, but for more complex situations, a computer simplifies the simulation process drastically. The main types of tools for performing these simulations, according to Crolla [37] are:

1. Purpose designed simulation tools (MSC.Adams/Car) 2. Multibody simulation tools that are numerically based 3. Multibody simulations tools that are symbolic

4. General mathematic software tools (MATLAB)

Many of these tools, such as MSC.Adams/Car [23] [28] use predefined templates of the vehicle subsystems, such as suspension, steering, engine and transmission to create a full vehicle simulation model.

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2.4.2. Tires

Tire history

The wheel, maybe one of the greatest inventions in the history of man kind, was in-vented around 3500 BC [38]. The first models were simple and often made of a solid piece of wood or stone. Later developments were the spoked wheel and reinforced wheels with a steel ring around it for improved abrasion.

The invention of the modern wheel, consisting of a metal rim and a rubber tire, is rela-tively new. Early types of rubber were difficult to find a useful application for, as they did not retain shape and behaved very different depending on the environment temper-ature. It was not until Goodyear invented the vulcanization process in 1839 [39] that the widespread use and development of the modern tire really took of.

The vulcanization process transforms the sticky raw rubber together with addition of sulphur and other materials to a firm bendable material, perfect to make tires from. For Goodyear, things did not work out very well, despite this great invention. He died bankrupt at the age of 60. It was not until 1898 when the Seiberling brothers from Ger-many decided to name their company Goodyear [39] that he got some credit for his invention.

This new rubber technology was directly put to use in form of solid rubber tires [40]. These were strong, absorbed shocks and could resist all forms of cuts and abrasions. The disadvantage was that the tires were very heavy and did not give a smooth ride. Thomson [41] was the original inventor of the air filled tire in 1845, which was a big improvement of the solid rubber tire. Unfortunately his invention came too early and never really took off.

In 1888 Dunlop [42] also claimed to have invented the pneumatic air filled tire. At that time bicycles had started to become popular and the new tire gave a substantially im-proved ride, which gave the new invention good success.

Tire theory

From the beginning of the formulation of the mechanics and mathematics behind vehicle dynamics, the understanding of the tire has been very important.

Except for aerodynamic and gravitational forces, all the primary control and disturbance forces which affect the vehicle are generated in the contact patch of the tire. These forces

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act on an area not much larger than the palm of a hand. A good understanding of the behaviour of the tire is essential to determine the dynamics of the total vehicle.

The main functions of the tire as described by Pacejka[43] are:

• Support the vertical load of the vehicle, while also providing shock absorption. • Develop longitudinal forces for acceleration and braking.

• Develop lateral forces to enable cornering.

Tire design and construction

Basically, the tire is a simple visco-elastic toroid5. The structure of the tire, consists of a composite of several materials including different rubber compounds, steel and textile fibres, inflated with air. This gives the tire a very complex non-linear behaviour.

Most modern tires are composed of a flexible carcass of high tensile strength cords fas-tened to steel cable beads that attach the tire to the rim. The carcass is made from a rubberized fabric reinforced by cords of nylon, rayon, polyester or fibreglass. The inter-nal inflation pressure loads the structure and causes every exterinter-nal force deforming the tire to be reacted with a tire reaction force.

There are two main types of tire designs available, radial and bias-ply tires. The first tires used the bias-ply design and they were popular in the early years of the modern tire. The radial tire was invented by Michelin [44] in 1946 and soon became popular in Europe. Around 1960 the benefits of the radial tire was discovered in the USA and has since then gradually become the standard tire for use in passenger cars in all parts of the world.

Radial tire

The radial tire is characterized by parallel plies 6 (the carcass) running at a 90 degree angle to the circumference. This type of design makes the tire sidewall very flexible and gives a soft ride. To provide directional stability, a belt made from fabric or steel wire is placed between the tread and the carcass. Normal orientation of the belt is around±20 degrees from the tread.

5_{doughnut-shaped circular object} 6_{layers of reinforcement in tire structure}

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Sidewall Rib Sipes Tread blocks Cap plies 2nd Steel belt 1st Steel belt

2nd Carcass layer (plies) 1st Carcass layer (plies)

Bead reinforcement Tire shoulder

Tread side profile

Rim protector Bead filler Bead wire Tread groove Tread-base Bead Rubber liner Rim flange profile

Figure 2.4.:Design of a radial tire

Bias-ply tire

In a bias-ply tire design the carcass consists of plies running in alternating angles from the rim bead. The angle of the plies is around 35-40 degrees. Higher angles result in a soft ride, as for a radial tire. To give directional stability, smaller ply angles are used.

Benefits of the radial tire

The differences in tire design give the models both advantages and disadvantages [45] [46]. For passenger cars, the benefits of the radial tire clearly outperforms its drawbacks, hence the radial tire has become the standard. For other applications where it is impor-tant to have good sidewall cut resistance, the bias ply tire can be a better choice.

Tire and rim designation

Tires can be denoted in many different ways, passenger car tires usually follow the met-ric designation, for example 225/45 R17 denotes a tire with 225 mm overall section width, 45% aspect ratio, R for radial tire and 17 for the rim diameter in inches. The aspect ratio describes the side profile height in percent of the overall section width. This information is printed on the tire sidewall.

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Radial Bias Vehicle Steadiness - + Cut Resistance - Tread + -Cut Resistance - Sidewall - +

Reparability - + Self Cleaning - + Traction + -Heat Resistance + -Wear Resistance + -Floatation + -Fuel Economy +

-Table 2.1.:Benefits of radial and bias tires

G D B A Hmi n SG C D G

Rim flange Bead seat

Figure 2.5.:Illustration of the naming conventions for rims and tires

Rims are named after width and diameter. An example is a 8Jx18 ET32 rim, which stands for 8 inch wide, J profile, 18 inch nominal diameter and inset (ET) 32 mm. Recommen-dations for suitable rim and tire combinations can be found in the ETRTO Manual [1].

Tire terminology and tire axis systems

There are several very important variables (forces, moments and angles ) that are used to describe tire behaviour. To have a common description of these variables, SAE has created the tire axis system [47], which describes the forces, moments and angles in a predefined coordinate system.

There is also the ISO-tire system, which describes the same physical parameters, but with a different orientation.

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MX MY MZ ω FZ FY FX α γ

(a) SAE tire axis system

MX _M Y MZ ω FZ FY FX α γ

(b) ISO tire axis system

Figure 2.6.:Comparison of SAE and ISO axis systems

The systems describe forces in all three directions, Fx the longitudinal force (braking,

traction), Fy lateral force and Fz the vertical load (Normal force). Important moments

are: Mz aligning moment, Mx overturning moment and My traction/brake/rolling

re-sistance. The two angles in the tire axis system are camber γ and slip angle α.

The tire footprint and generation of grip

The generation of grip in the footprint, the area of the tread in contact with the road, is one of the most important aspects in tire mechanics. Without grip, there cannot be any transfer of force from the car to the road.

Friction

From the basic mechanics [48], it is known that the friction between solids is dependent on the properties of both contacting surfaces. In general, the frictional force is said to be proportional to the vertical load, but independent of the contact area between the two surfaces. It is also known that the static friction (stiction) is higher than the kinetic (sliding) coefficient of friction.

For tires, the laws of friction can be somewhat limited under certain conditions, as the coefficient of friction for tires depends both on the relative velocity and the ground pres-sure.

When the tire is rolling straight ahead with zero lateral force, the maximum longitudinal force (traction) is limited only by the coefficient of friction times the vertical tire load Fmax =Fx =µN.

Under combined driving and steering, both longitudinal and lateral forces are generated in the contact patch of the tire. In this case, the total amount of grip (total force) will be

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the resultant of these forces, according to: Fmax =

q

(Fx)2+ (Fy)2=µN. The maximum grip Fmax is still the same as for the straight ahead case, therefore reducing the grip

during combined braking and steering.

This phenomenon can be illustrated with a so called friction circle, where the outer ring can be used to represent the maximum acceleration (g) the tires can sustain. In some cases the circle can also be illustrated with an ellipse, when the tire can take higher forces in lateral direction. The horizontal axis corresponds to steering and the vertical axis is acceleration/braking.

In the simple example illustrated in figure 2.7, combined driving and steering in a corner limits the longitudinal force (braking) to 0.87g at a lateral force of 0.5g. If there is only steering, the lateral force can be higher, up to 1g.

Right High μ Acceleration Braking Left 0.5g Lowμ 0.87g

Figure 2.7.:Friction circle

Mechanisms

Gillespie[29] describes two main mechanisms responsible for generating friction be-tween the tire and the road surface. First there is the molecular adhesion mechanism, which creates molecular bonds between the rubber and the texture of the road surface. This component is predominant on dry roads and is heavily reduced if the road surface is covered with water, resulting in less grip on wet roads.

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The other mechanism is mechanical stiction7in form of hysteresis, where energy is lost when the rubber deforms. Hysteresis can be explained as a sort of memory effect in the rubber, where the material does not return to its original state directly after relaxation.

Hysteresis Adhesion

RUBBER

Binder Aggregate

V

Figure 2.8.:Tire/Road friction interaction

Sliding friction

Studies by Kummer [49] shows that the coefficient of friction is dependent on a relative sliding velocity between tire and road. Figure 2.9 shows the characteristic curve for this phenomenon.

The conclusion of this is that the generation of grip requires a relative velocity (slip) between the road and the tire. When the sliding velocity becomes to high, the friction decreases.

Tire load sensitivity

According to experiments by Mäckle and Schirle [50], the coefficient of friction is also dependent on the ground pressure between the road and the tire. This is not related to the inflation pressure of the tire. This phenomenon, where the coefficient of friction varies depending with the tire load is called tire load sensitivity.

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0 0.5 1 1.5 2 0.001 0.01 0.1 1 10 Relative velocity [m/s] Friction µ [-]

Figure 2.9.:Friction dependence on relative velocity

0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 ground pressure pB [bar]

maximum coefficient of friction

µ [−]

Figure 2.10.:Friction dependence on contact pressure

The curve depicted in figure 2.10 shows how the coefficient of friction decreases with higher loads (higher ground pressure). Having a tire with larger surface area gives lower ground pressure, resulting in a higher coefficient of friction and more grip.

Slip angle, aligning moment and pneumatic trail

During cornering, the driver turns the steering wheel and the wheels to keep the vehicle along the curvature of the road. At slow speeds and low lateral accelerations, only a small amount of steering is required.

At higher speeds, the centrifugal force increases (equation 2.4) and exposes the tires to a side force. To follow the curvature of the road, the driver has to increase the steering

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an-gle additionally. This means that there will be a difference between the vehicle heading and the tire rolling direction. This is called the slip angle.

α Mz Fy y x tp F_y F_s z y Direction of heading Direction of travel

(a) Illustration of slip angle

Cα La teral for ce Fy

Slip angle α [deg]

(b) Plot of slip angle as function of lateral force

Figure 2.11.:Description of slip angle phenomenon

It is difficult to translate the angle of the steering wheel to a slip angle, as it depends on both speed and corner radius. The Michelin tire handbook [51] states that 20 degrees on the steering wheel easily generates a slip angle of 1 degree.

As slip angles are defined as a difference between the rolling direction and the vehicle heading, it also means that slip angles can be generated on the rear wheels, even though these cannot be steered.

By measuring the lateral force on a tire test rig as a function of the slip angle (the slip angle is generated by steering the tire), a characteristic curve for the tire behaviour can be created (figure 2.11b).

The curve is linear in the beginning, but when the tire reaches the maximum coefficient of friction, the lateral force will fall of. The linear part of the curve is called cornering

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stiffness cα and it is an indication of how fast the tire can generate a lateral force when the slip angle increases.

When subjected to lateral forces, the tread elements of the tire are locally deflected side-ways. Along the contact patch this deformation continues until the lateral force is larger than the friction and slip occurs. At this point, the tread elements will quickly return to their undeformed position.

By integrating the lateral force along the contact patch, the force distribution results in a moment around the tire z-axis. This is known as the aligning moment Mz.

The moment arm that the lateral force acts on the centre point of contact is called pneu-matic trail tpand can be derived as:

tp = Mz

Fy

Depending on the suspension design, this aligning moment can have different magni-tude. It is an important design parameter, as it can be used to automatically realign the wheels to a straight ahead direction if the driver lets go of the steering wheel.

Longitudinal slip

Contrary to slip angles, which is related to the build-up of forces in the lateral direction, there is also something called longitudinal slip. This relates to the slip occurring from forces in the longitudinal direction during acceleration and braking.

A useful measure of the longitudinal slip is the so called slip ratio (SR), which is defined as the quotient between the wheel rotation and vehicle speed:

SR= Ω−Ω0

Ω0

= Ω∗Re

V∗cos(α) −1 (2.6)

The slip ratio zero corresponds to a free rolling tire, negative values braking.

According to Gillespie [29], the explanation of the longitudinal slip comes from the compliance of the tread elements. To create forces between tire and road, these tread elements have to bend, which can only occur if the tire is rotating faster than the tread circumference.

Simulation and validation of tire deformation under certain load cases

M A S T E R’S T H E S I S

HENRIK ANDERSSON