Race Car Design - Guia Completo para Projetos de Carros de Corrida

RACE CAR DESIGN DEREK SEWARD Race Car Design Derek Seward Emeritus Professor of Engineering Design Department of Engineering Lancaster University Derek Seward 2014 All rights reserved No reproduction copy or transmission of this publication may be made without written permission No portion of this publication may be reproduced copied or transmitted save with written permission or in accordance with the provisions of the Copyright Designs and Patents Act 1988 or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency Saffron House 610 Kirby Street London EC1N 8TS Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages The author has asserted his right to be identified as the author of this work in accordance with the Copyright Designs and Patents Act 1988 First published 2014 by PALGRAVE Palgrave in the UK is an imprint of Macmillan Publishers Limited registered in England company number 785998 of 4 Crinan Street London N19XW Palgrave Macmillan in the US is a division of St Martins Press LLC 175 Fifth Avenue New York NY 10010 Palgrave is the global imprint of the above companies and is represented throughout the world Palgrave and Macmillan are registered trademarks in the United States the United Kingdom Europe and other countries ISBN 9781137030146 paperback This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources Logging pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin A catalogue record for this book is available from the British Library A catalog record for this book is available from the Library of Congress Printed and bound in Great Britain by Lavenham Press Ltd Lavenham Suffolk Contents Preface iv Symbols vi Acknowledgements viii 1 Racing car basics 1 2 Chassis structure 33 3 Suspension links 61 4 Springs dampers and antiroll 90 5 Tyres and balance 119 6 Front wheel assembly and steering 154 7 Rear wheel assembly and power transmission 176 8 Brakes 193 9 Aerodynamics 201 10 Engine systems 227 11 Setup and testing 241 Appendix 1 Deriving Pacejka tyre coefficients 250 Appendix 2 Tube properties 262 Glossary of automotive terms 265 References 269 Index 271 Preface The aim of this book is to explain the fundamentals of racing car design using the basic principles of engineering science and elementary mathematics There is already an extensive list of books that purport to explain this topic However with a few honourable exceptions they tend to fall into one of two camps either they deal with a highly theoretical and narrow aspect of the subject in a very mathematical way and contain little practical design guidance or they are written by enthusiastic drivers or constructors who resort to formulas or rules of thumb gained through experience without proper explanation of the underlying theory Hopefully this book avoids both of these pitfalls by aiming at a deeper understanding of the principles and avoidance of the black art approach to design That is not to say that we fully understand every aspect of topics such as tyreroad interaction or aerodynamics Theory can only take us so far Even the best designs will require optimisation on the track or in the wind tunnel where the car is tuned to meet the detailed requirements of the specific circuit driver tyre compound and weather conditions The objective of the initial design is therefore to produce a robust solution which is close enough to optimum so that it can be readily tuned to a wide range of specific conditions This book is intended for students on motorsport degree courses those involved in Formula StudentFSAE and practising car designers and constructors It will also be of interest to racing drivers and the general reader who is interested in understanding why racing cars are the way they are and why they perform so much better than normal cars on the track The book is based on the principles of engineering science physics and mathematics and hence some previous knowledge of these subjects is required but only at a relatively elementary level The book covers the design of most elements of a car including the chassis frame suspension steering brakes transmission lubrication and fuel systems however the internal components of such elements as the engine gearbox and differential are beyond the scope of this short text Where relevant emphasis is placed on the important role that computer tools play in the modern design process In many ways the design process for a racing car is much simpler than that for a conventional passenger car because the racing car has a highly focused mission to propel a driver around a circuit in the shortest time possible A passenger car on the other hand has a wider remit It must cope with a varying range of loads from people and luggage be easy and safe to drive and be comfortable and economical The narrow focus of the racing car enables iv Preface the designer to concentrate almost exclusively on performance issues The racing car design process can be described as a highly multivariable problem and inevitably the solution of such problems involves compromise and tradeoffs between competing objectives Resolving these design conflicts presents the skilled designer with the greatest challenges and pleasure The companion website link to this book can be found at wwwpalgravecomcompanionSewardRaceCarDesign Derek Seward Note on glossary terms The first main use of each glossary term is shown in bold italic typeface in the text The glossary is located on pages 265268 Note on plates The plates referred to throughout the text are located between pages 54 and 55 v Symbols This list does not include Pacejka tyre model symbols which are defined in the text A crosssectional area mm² amplitude mm Aₘ brake mastercylinder piston area mm² Aₛ brake slavecylinder piston area total on one side mm² a acceleration ms² C damping coefficient roll couple Nm Ccrit critical damping coefficient CD drag coefficient CL lift coefficient C₀ bearing basic load rating kN Cr bearing dynamic load rating kN D diameter mm downforce N E modulus of elasticity Nmm² F force N Fφ lateral load transfer at wheel from roll couple N f frequency Hz fₛ sprung mass natural frequency Hz fᵤ unsprung mass natural frequency Hz G maximum number of g forces shear modulus Nmm² g acceleration due to gravity 981 ms² H horizontal component of force N h height mm hₐ distance from sprung mass to roll axis mm I second moment of area mm⁴ KR suspension ride rate Nmm KT tyre stiffness Nmm KW wheel centre stiffness rate Nmm L wheelbase mm l length mm M moment or couple Nmm MR roll couple Nmm m mass kg P power W Pi absolute pressure Nmm² Pe Euler buckling load N Symbols Pm bearing mean equivalent dynamic load kN P0 maximum radial load on bearing kN R radius of curve m Rm motion ratio RR rolling radius of tyre mm Re Reynolds number rb brake pad radius mm s distance travelled m s0 bearing static safety factor T wheel track width mm torque Nm Ti absolute temperature C t pneumatic trail mm time s u initial velocity mms Vi volume m³ v velocity ms W weight or wheel loads N Z elastic section modulus mm³ α tyre slip angle deg or rad δ displacement mm δφ wheel displacement from roll mm ξ damping ratio θ angle deg θφ roll angle rad μ coefficient of friction viscosity Pa sec ρ density kgm³ φ wheel camber angle deg Acknowledgements The publisher and author would like to thank the organisations and people listed below for permission to reproduce material from their publications Avon Tyres Motorsport for permission to reproduce the graphs in Figures 58a 58b A11 and A14 adapted by the author Avon Tyres Motorsport for permission to reproduce the data in Tables 51 and 52 and the Avon column in Table A13 Caterham F1 Team for permission to reproduce the photographs in Figures 320 517 and 91 Mike Pilbeam for permission to reproduce the photograph in Figure 211 which was taken by Rick Wilson of Redline Design The publisher and author would like to acknowledge the companies listed below for the use of their software Figure 106 uses software under licence from DTAfast Figures 12 and 121 use software under licence from ETB Instruments Ltd DigiTools Software Figures 23ac 26ab 29 214 215ac 319 and Plates 1 2 3 and 5 use software under licence from LISA Figure 103 uses software under licence from Lotus Engineering Norfolk England Figures 13 515 516 Table 61 Figures 72 74 75 A12 and A15 use Excel under licence from Microsoft The following figures use Visio under licence from Microsoft Figures 11 12 14 15 16 18 19 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 21ac 22 23ac 24 25 26ab 27 28 29 210 211 212ab 213ab 217 218 32 35 38ac 39 310 311 312 313 43 44 47 48 49ac 410 413 51ad 52ab 53 54 55 56 57 510 511 512 513 514 518 62 64 65 66 67 68 73 76 83 92 93 94 95 96ab 97 98 99 910 102 104 107 108 109 and 111 Acknowledgements The following figures use software under licence from SketchUp Figures 14 15 16 18 19 110 112 113 115 116 117 122 123 28 38ac 39 310 311 312 313 51ad 52ab 65 66 83 92 97 98 99 910 and 111 Plates 6 and 7 use software under licence from SolidWorks The following figures and plate use software under licence from SusProg Figures 31ad 33ab 34ac 36ab 37ab 314ad 315 316 317 318 320 321 322 323 69 and Plate 4 The following figures use ViaCAD software under licence from PunchCAD Figures 21ac 22 23ac 24 25 27 29 210 211 212ab 213ab 45 46 411ad 412 59 63 610 611 612 613 614ab 615ab 78 79 710ab 81 and 101 This page intentionally left blank 1 Racing car basics LEARNING OUTCOMES At the end of this chapter You will understand the basic elements of car racing You will be able to calculate the varying loads on the wheels of a racing car as it accelerates brakes and corners and appreciate how these loads are influenced by aerodynamic downforce You will be able to identify some important design objectives for a successful racing car 11 Introduction This chapter introduces many of the key concepts that must be grasped to obtain a good understanding of racing car design It also contains signposts to later chapters where topics are covered in more depth By its nature racing is a highly competitive activity and the job of the designer is to provide the driver with the best possible car that hopefully has a competitive advantage To do this we need answers to the following questions What does a racing car have to do What is the best basic layout of a car for achieving what it has to do How can the car be optimised to perform better than the competition What loads and stresses is the car subjected to and how can it be made adequately safe and robust This chapter will start to provide some of the answers to these questions 12 The elements of racing Motor racing can take many forms ranging from short hill climbs and sprints where the driver competes against the clock to conventional headtohead circuit racing such as Formula 1 and IndyCar however there are common elements to all forms In general the aim of all racing is to cover a particular piece of road or circuit in the shortest possible time To do this the driver must do three things Race car design Accelerate the car to the fastest possible speed Brake the car as late as possible over the minimum possible distance Go round corners in the minimum time and more importantly emerge from corners carrying the maximum possible velocity so that a speed advantage is carried over the ensuing straight From the above it can be seen that the competitive driver will spend virtually no time cruising at constant velocity The only time this will occur is either queuing in traffic or flatout on a long straight Also of course the skilled driver may combine these basic elements by accelerating out of a corner or braking into a corner This is illustrated in Figures 11 and 12 which show the layout of a circuit together with a plot of speed data for one lap The labels indicate matching points on both figures Note that the slope of the curve in Figure 12 is steeper during braking than during acceleration This is for three reasons firstly at faster speeds the rate of acceleration is limited by the power of the engine secondly braking uses the grip from all four wheels whereas in this case acceleration uses only rear wheel grip thirdly at fast speeds the car develops significant aerodynamic drag forces which assist braking but impede acceleration Figure 11 Brands Hatch circuit UK The three basic elements of racing all involve a form of acceleration or change in velocity In the case of cornering this is lateral acceleration and braking can be considered to be negative acceleration We know from Newtons first law of motion that An object in motion stays in motion with the same speed and in the same direction unless acted upon by an external force 2 Chapter 1 Racing car basics Figure 12 Brands Hatch speed data produced with ETB Instruments Ltd DigiTools Software Consequently in order to accelerate or change direction the car must be subject to an external force and the principal source of such a force is at the interface between the tyres and the road known as the tyre contact patch Clearly external aerodynamic forces also exist and these will be considered later Thus it can be concluded that the ability of a car to accelerate brake and change direction depends upon the frictional force developed between the rubber tyre and the road surface This force is normally referred to as traction or grip and its maximisation is an important design criterion for a competitive car Classical or Coulomb friction has a simple linear relationship between the applied normal load and a constant coefficient of friction μ mu Friction force normal load x μ As we shall see when we look at tyre mechanics in more detail later the contact patch between a tyre and the road does not follow this simple law Figure 13 shows the relationship between vertical wheel load and maximum lateral grip for a typical racing tyre and compares it to simple Coulomb friction with μ 1 dashed line We will see later that the lack of linearity ie the coefficient of friction not being constant provides a powerful means by which a cars handling is tuned for peak performance It can be concluded from Figure 13 that As the vertical load is increased on the wheel the grip increases but at a progressively slower rate This is known as tyre sensitivity 3 Race car design Figure 13 Typical racing tyre grip Eventually the level of grip peaks and then starts to fall with increasing wheel load The tyre has become overloaded The value of grip divided by vertical wheel load at a specific point in Figure 13 can be considered to be an instantaneous coefficient of friction It is clear that knowledge of the normal force at each tyre contact patch ie the individual vertical wheel loads is vital for many aspects of racing car design They are used to determine the loads in the chassis brake components suspension members transmission etc as well as for tuning the fundamental handling and balance of the car We will look at static wheel loads and then see how they change when the car is subjected to the three elements of racing braking acceleration and cornering First it is necessary to determine the position of the cars centre of mass which is often referred to as the centre of gravity The centre of mass is the point where all of the mass can be considered to be concentrated Knowledge of its location is important to car designers as this determines the weight distribution between the front and rear wheels Also the height of the centre of mass above the ground influences the degree to which the car rolls on corners as well as the amount of weight that transfers between the wheels during braking acceleration and cornering 4 13 Position of centre of mass of a vehicle At the preliminary design stage it is necessary to estimate the centre of mass of each major component as it is added to the scheme The final positional relationship between the components and the wheels can then be adjusted to achieve the desired frontrear weight distribution To illustrate the process Figure 14 shows just a couple of components together with distances from their individual centres of mass to a common point In this case the common point is the front contact patch x Chapter 1 Racing car basics Figure 14 Calculating the position of the centre of mass The magnitude m and location lh of the centre of mass of each individual component is either measured or estimated The objective is to find the value of the combined mass mm and its location relative to the common point lm and hm The combined mass is simply the sum of the individual components For a total of n number components this is shown mathematically as mm Σm1 m2 mn 11 The location of the combined centre of mass is given by lm Σm1 l1 m2 l2 mn lnmm 12 hm Σm1 h1 m2 h2 mn hnmm 13 The above process simply ensures that the combined mass of the components exerts the same moment about the front contact patch as the sum of all the individual components EXAMPLE 11 The following data is relevant to the two components shown in Figure 14 Determine the magnitude and location of the combined centre of mass Item Mass kg Horiz dist from x mm Vert dist from ground mm Engine 120 2100 245 Driver 75 1080 355 Solution From equation 11 Combined mass mm 120 75 195 kg From equation 12 Horiz distance to combined mass lm 120 2100 75 1080195 1708 mm Race car design From equation 13 Vertical distance to combined mass hm 120 245 75 355195 287 mm Answer Combined mass 195 kg acting 1708 mm horizontally from x and at a height of 287 mm above ground Clearly in the case of a real car there are many more components to consider and the use of a spreadsheet is desirable Table 11 shows such a spreadsheet which can be downloaded from wwwpalgravecomcompanionSewardRaceCarDesign for your own use Your own data can be input into the shaded cells Once a car has been constructed the position of the centre of mass should be confirmed by physical measurements and this is discussed in Chapter 11 as part of the setup procedure 14 Static wheel loads and frontrear weight balance The static case refers to the loads on the car when it is not being subjected to accelerations from accelerating braking or cornering The car should be considered when fully laden with driver and all fluids These are the loads that would be measured if the car was placed on level ground in the pits Up to now we have referred to the mass of components in kilograms However the terms load and weight actually imply force which is of course measured in Newtons Consequently from now on we will consider the forces W on the car where force N mass kg acceleration ms2 where for vertical loads the acceleration g 981 ms2 Figure 15 Calculating static wheel loads Figure 15 shows a car where the magnitude and position of the centre of mass has been determined We wish to find the static wheel loads Knowing Chapter 1 Racing car basics Element Mass kg Horiz dist front axle mm H moment kgm Vert dist ground mm V moment kgm Car Front wheel assemblies 324 0 0 280 9072 Pedal box 5 0 0 260 1300 Steering gear 5 300 1500 150 750 Controls 3 200 600 400 1200 Frame floor 50 1250 62500 330 16500 Body 15 1500 22500 350 5250 Front wing 5 450 2250 90 450 Rear wing 5 2700 13500 450 2250 Fire extinguisher 5 300 1500 260 1300 Engine assembly oil 85 1830 155550 300 25500 Fuel tank full 25 1275 31875 200 5000 Battery 4 1200 4800 120 480 Electrics 4 1500 6000 200 800 Exhaust 5 1750 8750 350 1750 Radiator water 10 1360 13600 150 1500 Rear wheel assemblies drive shafts diff 58 2300 133400 280 16240 Reversing motor 6 2500 15000 280 1680 Ballast 0 1200 0 0 0 Other 1 0 0 0 0 Other 2 0 0 0 0 Other 3 0 0 0 0 Other 4 0 0 0 0 Other 5 0 0 0 0 Total car 3224 1454 468825 282 91022 Driver Weight of driver 80 Distance front axle to pedal face 50 Mass kg Horiz dist sole foot mm Horiz dist front axle mm H moment kgm Vert dist ground mm V moment kgm Feet 28 40 90 250 310 859733333 Calves 77 350 400 3072 360 27648 Thighs 173 760 810 13997 295 50976 Torso 369 1050 1100 40597 300 11072 Forearms 32 800 850 2720 400 1280 Upper arms 53 1100 1150 6133 420 2240 Hands 13 650 700 896 510 6528 Head 55 1200 1250 6933 670 3716266667 Total driver 80 5950 6350 745984 346 276832 Grand total 4024 1350 543423 295 1187052 Rear axle load 236 Front axle load 166 Ratio FR 413 587 Table 11 Spreadsheet for calculating centre of mass Race car design the wheelbase and the horizontal position of the centre of mass we can simply take moments about the front axle to find the rear axle load WR Rear axle load WR Wx lm L From vertical equilibrium Front axle load WF W WR It should be noted that Figure 15 is a freebody diagram If the car is considered to be floating weightlessly in space the three acting forces W WF and WR must keep it in static equilibrium ie the downward force from gravity W must be equal and opposite to the sum of the wheel reaction forces WF and WR This is why the wheel forces are shown upwards They represent the forces from the road acting on the car We will make extensive use of freebody diagrams throughout this book EXAMPLE 12 For the car shown in Figure 16 a Determine the static axle loads b Calculate the percentage frontrear distribution c Estimate individual static wheel loads Figure 16 Position of centre of mass a Weight of car W 7543 x 981 7400 N Rear static axle load WR 7400 x 1920 3235 4392 N Front static axle load WF 7400 4392 3008 N b to front 3008 7400 x 100 406 to rear 100 406 594 c A circuit racing car is usually expected to have good leftright balance and therefore the individual wheel loads can be assumed to be half of the axle loads Hence Rear static wheel loads WRL and WRR 4392 2 2196 N Front static wheel loads WFL and WFR 3008 2 1504 N Answer Static axle loads 4392 N rear and 3008 N front Distribution 406 front and 594 rear Static wheel loads 2196 N rear and 1504 N front Figures 17a d Weight distribution as a function of driven wheels engine position and tyre selection a VW Scirocco front engine frontwheel drive equal tyres b BMX 3 series front mid engine rearwheel drive equal tyres c Formula car mid engine rearwheel drive wider rear tyres d Porsche 911 rear engine rearwheel drive much wider rear tyres Race car design Clearly the designer can influence the frontrear weight balance by moving certain components such as the battery or hydraulic pumps A significant change results from modifying the location of the front andor rear axles relative to the significant mass of the engine and gearbox In addition competitive cars are invariably built significantly lighter than the minimum weight specified in their formula technical regulations The difference is then made up by the addition of heavy ballast which is strategically placed to give the best frontrear balance What then is the optimum frontrear weight ratio From a handling point of view it can be argued that a 5050 ratio is optimum However as we shall see shortly for accelerating off the line there is a clear advantage in having more weight over the driven wheels Racing cars typically aim for about a 4455 frontrear ratio and address the handling issue by means of wider rear tyres Figures 17ad show how different weight distributions have resulted from particular combinations of driven wheels engine position and tyre selection The position of the fuel tanks presents a challenge as the weight of fuel clearly varies throughout the race In Formula 1 where refuelling is no longer allowed the cars start with up to 170 kg of fuel The solution is to put the fuel tanks as close to the centre of mass as possible so that as it is used the balance of the car does not change The three elements of racing will now be considered in more detail 15 Linear acceleration and longitudinal load transfer The starting point for understanding linear acceleration is Newtons second law of motion The acceleration a of an object is directly proportional to the magnitude of the applied force F and inversely proportional to the mass of the object m This can be written as a F m 14 As the mass of a car can be considered constant the rate of acceleration is dependent upon the force available to propel the car forward Figure 18 shows this traction force acting at the contact patch of the driven rear wheels We can convert a dynamic analysis into a simple static analysis by invoking dAlemberts principle which states that the car effectively resists forward acceleration with an imaginary inertial reaction force that acts through the centre of mass This is equal and opposite to the traction force and is shown as the resistive force in Figure 18 The fact that the traction force occurs at road level and the resistive force at the level of the centre of mass means that an outofbalance couple or moment is set up This causes changes to the static axle loads WF and WR The magnitude of the change ΔWx is known as longitudinal load transfer and it is added to the static rear axle load and subtracted from the static front axle load This explains why when accelerating hard the front of a car rises and the rear drops known as squat By taking moments about the front contact patch F hm ΔWx L Longitudinal weight transfer ΔWx FhmL 15 Incidentally if the force accelerating the car was applied at the level of the centre of mass instead of at road level say by a jet engine there would be no longitudinal weight transfer as all the forces act through one point and there is no outofbalance couple When accelerating a car from the startline up to its maximum speed we can consider two distinct stages Stage 1 Traction limited During initial acceleration off the startline the value of the traction force F is limited by the frictional grip that can be generated by the driven tyres The problem at this stage for the driver is to avoid wheelspin Stage 2 Power limited As the speed of the car increases the point will be reached where the engine cannot provide enough power to spin the wheels and from this point onwards maximum acceleration is limited by engine power As speed increases still further aerodynamic drag force and other losses build until all of the engine power is needed to overcome them At this point further acceleration is not possible and the car has reached its maximum speed or terminal velocity 151 Tractionlimited acceleration The initial tractionlimited stage produces the highest levels of traction force and hence longitudinal load transfer It is the case that for design purposes this produces the highest loads on the rear suspension and transmission Figure 19 shows the same car with static loads added and the imaginary resistive force removed From equation 15 Longitudinal load transfer ΔWx FhmL Traction force F WR ΔWx μ F WR FhmL μ F FhmμL WRμ F 1 hmμL WRμ F WRhL 1 hmμL At this stage it is necessary to assume a value for the coefficient of friction μ It has already been stated that the value for the tyre contact patch is not in fact a constant see Figure 13 however an appropriate average value for a warm racing slick tyre is generally assumed to be in the range 14 to 16 This compares to about 09 for an ordinary car tyre Once equation 16 is solved for F it is an easy matter to substitute back into equation 15 to obtain the longitudinal weight transfer ΔWx This is demonstrated in the following Example 13 EXAMPLE 13 For the car shown in Figure 110 a Estimate the individual wheel loads during maximum acceleration assuming an average coefficient of friction μ between the tyre and the road of 15 b If the rear tyres have a rolling radius of 275 mm estimate the peak torque through the transmission when accelerating offtheline c Calculate the maximum acceleration in both ms2 and equivalent g force a Weight of car W 7350 N Static rear axle load WR 7350 1950 3215 4458 N Static front axle load WF 7350 4458 2892 N From equation 16 Traction force F WRμ 1 hmμL 4458 15 1 325 15 3215 7882 N From equation 15 Longitudinal load transfer ΔWx FhmL 7882 325 3215 797 N Rear wheel loads WRL and WRR 4458 797 2 2628 N Front wheel loads WFL and WFR 2892 797 2 1048 N b Peak torque at rear wheels Twheels WRL WRR rad μ 2628 2628 275 15 2 168 000 Nmm 2168 Nm Race car design c Mass of car 7350 981 7492 kg From equation 14 Acceleration a F m 7882 7492 1052 ms2 1052 981 1072g Answer Wheel loads 2628 N rear and 1048 N front Torque through transmission 2168 Nm Acceleration 1052 ms2 1072g Comment The above rear wheel loads and torque represent an important load case for the design of the transmission rear wheel assemblies and suspension components It should be recognised that if a car is to achieve the above peak values of traction force and acceleration then it is necessary for it to have an adequate powertoweight ratio rarely a problem for a racing car suitable transmission gearing and a driver capable of appropriate clutch and throttle control or an automated traction control system These issues will be dealt with later in the book 152 Powerlimited acceleration When a force such as the traction force causing acceleration moves through a distance it does work Work force distance Nm or Joules 17 Power is the rate of doing work Power force distance time force speed Nms or Watts or Force power speed N 18 Chapter 1 Racing car basics It can be seen from equation 18 that if power is limited the traction force must reduce as speed increases In this case it is not appropriate to consider absolute peak engine power as this is generally only available at specific engine revs The average power available at the wheels as the driver moves through the gears will be a bit less In addition some power is lost in spinningup the transmission components and wheels as well as overcoming transmission friction Furthermore not all of the traction force is available for accelerating the car Some of it must be used to overcome further losses The two principal additional losses are rolling resistance from the tyres aerodynamic drag Rolling resistance largely results from the energy used to heat the tyre as the rubber tread deforms during rolling The degree of resistance is related to the vertical load carried by each tyre as well as the rolling velocity It depends upon the tyre construction wheel diameter and the road surface but for racing tyres it can be approximated to 2 of the car weight Aerodynamic drag is dependent on the frontal area of the car and the degree of streamlining It increases with the square of the velocity and hence becomes the dominant loss at high speeds Aerodynamic forces are dealt with in more detail in Chapter 9 Figure 111 shows how the net force available to accelerate the car reduces as speed increases When this force is zero the car has reached its maximum or terminal speed Figure 111 The force available for acceleration Acceleration will be considered again in Chapter 7 where we will consider the implications of choosing the best gear ratios for peak performance Race car design 16 Braking and longitudinal load transfer As all four wheels are braked the braking force can be considered to be simply the weight of the car N multiplied by an assumed average tyreground coefficient of friction µ Braking force F W µ 19 Figure 112 Braking and longitudinal load transfer It can be seen from Figure 112 that the forces are reversed compared to the acceleration case shown in Figure 18 In this case load is transferred from the rear wheels to the front causing the nose to dip known as dive From equation 15 Longitudinal weight transfer ΔWx F hmL W µ hmL 110 EXAMPLE 14 For the same car as Example 13 and shown in Figure 113 a Estimate the individual wheel loads during maximum braking assuming an average coefficient of friction µ between the tyre and the road of 15 b Calculate the maximum deceleration in both ms2 and g force Figure 113 Calculating wheel loads during braking Chapter 1 Racing car basics δvv dR But Distance travelled d v δt Therefore δvv v δtR Divide both sides by δt δvv δt vR Multiply both sides by v δvδt v²R Hence a v²R For an object with mass m Centripetal force F ma mv²R 111 Centripetal force is the force that the string exerts on the mass The equal and opposite force that the mass exerts on the string is the socalled centrifugal force and this will act through the centre of mass In the case of a car the centripetal force is provided by lateral grip from the tyres as shown in Figure 115 This is often referred to as the cornering force The equal and opposite centrifugal force passes through the centre of mass The fact that the centre of mass is not in this case in the centre of the Figure 115 Racing car cornering 27 Chapter 1 Racing car basics a Weight of car W 7350 N As before Static rear axle load WR 7350 19503215 4458 N Static front axle load WF 7350 4458 2892 N From equation 19 Braking force F W μ 7350 15 11025 N From equation 15 Longitudinal weight transfer ΔWx FhmL 11025 3253215 1115 N Front wheel loads WFL and WFR 2893 11152 2004 N 55 Rear wheel loads WRL and WRR 4458 11152 1672 N 45 Mass of car 7350981 7492 kg b From equation 14 Deceleration a FM 110257492 1472 ms² 1472981 15g Answer Wheel loads 2004 N front and 1672 N rear Deceleration 1472 ms² 15g Comments It can be seen that 1 The above wheel loads represent an important load case for the design of the brake system and the front wheel assemblies and suspension components 2 Because braking involves grip from all four wheels and aerodynamic drag a car decelerates at a higher rate than that achieved during acceleration 3 During maximum braking the front wheel loads and hence brake forces are usually greater than the rear which explains why road cars often have bigger brake discs on the front This is despite the fact that in this case the static rear wheel loads are larger than the front 28 Race car design 4 The deceleration in terms of g force magnitude is equal to the average friction coefficient μ However this is only true if aerodynamic drag and downforce are ignored Details of brake system design will be considered in more detail in Chapter 8 17 Cornering and total lateral load transfer Cornering can be considered to be the most conceptually challenging element of racing It is not immediately obvious why a car travelling at constant speed around a corner should be subject to acceleration The key lies in the fact that velocity is a vector quantity A vector has both magnitude and direction unlike speed which is a scalar quantity and has only magnitude Although the magnitude may remain constant a cornering vehicle is subject to changing direction and hence changing velocity Changing velocity is acceleration and because a car has mass this requires a force socalled centripetal force Figure 114 Deriving the centripetal force formula Consider the familiar problem of a mass m on the end of a string and being swung in a circle Figure 114 In a small increment of time δt the mass moves from point A to B distance exaggerated for clarity The arrows emanating from A represent the vectors of velocity for points A and B ie they are the same length magnitude but the directions are the tangents to the circle at A and B The dashed arrow indicates the change in velocity δv As the increment becomes small the direction of this velocity change vector points to the centre of rotation O Also as the increment becomes small the lines from A and those from O form similar triangles Hence 18 Race car design wheelbase means that the tyre lateral grip forces are unequal The designer must therefore provide additional grip to the rear wheels in this case say by using wider tyres For peak cornering performance the front wheels must give way at roughly the same time as those at the rear This is what is meant by a balanced car If the front wheels give way before the rears the car is said to understeer and the car will refuse to turn and carry straight on at a bend If the rear wheels give way before the fronts the car is said to oversteer and the car is likely to spin These issues will be considered in much more detail in Chapter 5 where a more rigorous definition of understeer and oversteer is provided and we will see how calculations can be applied to achieve the necessary balance However finetuning for balance is invariably required by driving on the circuit where adjustments are made to suit the particular driver tyres road surface and weather conditions More is said about this in Chapter 11 It is clearly an easy matter to use the centripetal force equation 111 to find the required cornering force for a car going round a specific corner at a specific speed However the designer is more interested in maximising the cornering force and expressing the cornering performance in terms of the number of lateral g forces that the car can attain As with braking we can approximate this by estimating the average coefficient of friction µ at the tyre contact patch For a car with no aerodynamic downforce Maximum cornering force F W µ N 112 where W is the weight of the car It can be seen from Figure 116 that because the centrifugal force passes through the centre of mass which is above the road surface an overturning moment or couple is created which causes lateral load transfer Wy When cornering the load on the outer wheels increases and the load on the inner wheels decreases by the same amount Total lateral load transfer Wy Fhm T 113 where T is the distance between the centre of the wheels or track Figure 116 Lateral load transfer during cornering Chapter 1 Racing car basics EXAMPLE 15 For the same car as in Examples 13 and 14 and shown in Figure 117 a Calculate the cornering force F assuming an average coefficient of friction µ between the tyre and the road of 15 b Determine the maximum total lateral load transfer c Estimate the velocity that the car can travel around a 100 m radius corner Weight of car W 7350 N a From equation 112 Maximum cornering force F W µ 7350 15 11 025 N b From equation 113 Total lateral weight transfer Wy Fhm T 11 025 325 1500 2389 N c From equation 111 F mv2 R Hence v2 FR m 11 025 100 7350 981 14715 v 384 ms 138 kmh Answer Cornering force 11 025 N Total lateral load transfer 2389 N Corner speed 384 ms 138 kmh Figure 117 Calculating total lateral load transfer during cornering Race car design 171 Cornering and tyre sensitivity So far we have referred to total lateral load transfer ie the total load that is transferred from the inside wheels to the outside wheels when a car corners The distribution of this load between the front and rear axles is complex and depends upon the relative stiffness of the front and rear suspensions suspension geometry relative track widths antiroll bars etc Altering the proportion of lateral load transfer between the front and rear wheels is an important means by which the suspension is tuned to achieve a balanced car Consider again the tyre grip curve previously considered in Figure 13 and shown again in Figure 118 Lines A and B represent the equal vertical loads on say the front wheels during straightline travel When the car enters a corner lateral load transfer Wy takes place and this is added to the outer wheel and subtracted from the inner wheel A and B represent the wheel loads during cornering The significant point to note is that because of the convex nature of the curve the sum of the grip at A and B after lateral load transfer is significantly less than that at A and B In this particular case Combined front wheel grip without lateral load transfer 2 4900 N 9800 N Combined front wheel grip after lateral load transfer 2600 5800 8400 N Furthermore it can be seen that if the lateral load transfer is increased still further the grip at A would continue to reduce significantly and the grip at B also starts to reduce as the tyre becomes overloaded We can conclude that as lateral load transfer increases at either end of a car the combined grip at that end diminishes Although total lateral load transfer remains as calculated in Example 15 the proportion of lateral load transfer at each end of the car can as already indicated be engineered to achieve an optimum balance for the car This topic is dealt with in more detail in Chapter 5 Figure 118 Cornering and tyre sensitivity 18 The gg diagram A very useful conceptual tool for visualising the interaction between cornering braking and acceleration is the gg diagram It comes in various forms and is alternatively known as the friction circle or the traction circle Figure 119 shows a simple form of the gg diagram for an individual tyre The diagram indicates the upper boundary to traction in any direction In this case it indicates that the tyre can support pure acceleration or braking at 15g and cornering at 14g but where cornering is combined with braking or acceleration such as at point A this figure is reduced In this case if the car is accelerating at say 075g it is only able to corner at 13g Figure 119 The gg diagram for an individual tyre Figure 120 Traction circle for whole car As in this example the diagram is not a perfect circle as most tyres will support a little more traction in braking and acceleration than cornering Also the diameter of the gg diagram will swell and contract as the individual tyre is subjected to lateral and longitudinal load transfer A more meaningful form of the gg diagram is that shown in Figure 120 for the whole car which is obtained by summing the diagrams for the four wheels The individual wheel loads shown in Figure 120 are for a rearwheeldrive car which is accelerating out of a lefthand turn ie load transfer to the rear and the right The wholecar diagram represents the maximum g value that can be achieved in any direction The flattop curves in the acceleration zone are the result of rearwheeldrive and power limitations It is the aim of the designer to maximise the size of the traction circle It is the aim of the driver to keep as close to the perimeter of the circle as possible Figure 121 shows driver data logged during a race It can be seen from the above that the driver was braking at up to 17g cornering at 22g and accelerating at up to 10g 23 Race car design Figure 121 Real traction circle data produced with ETB Instruments Ltd DigiTools Software 19 The effect of aerodynamic downforce Over the last forty years the performance of racing cars has advanced significantly as evidenced by continually reducing lap times The biggest single cause is the development of effective aerodynamic packages to produce downforce The objective is to increase traction by increasing the downward force on the contact patch but without the addition of extra mass The three major elements that produce downforce are the front wing the rear wing and the underbody The design of these elements will be considered in detail in Chapter 9 However it is important to appreciate two points Aerodynamic forces are proportional to the square of the velocity of the airflow relative to the car This means that the designer may need to consider load cases at different speeds as the forces change The penalty paid for downforce generation is increased aerodynamic loss or drag The designer must decide upon the amount of engine power that can be sacrificed to overcome this drag This means less power available for accelerating the car and consequently a reduced top speed It follows therefore that relatively lowpower cars can only run lowdownforce aerodynamic setups As the power of the engine increases a more aggressive downforce package can be adopted 24 Table 12 shows some examples with approximate figures for guidance Table 12 Typical downforce classification Level of downforce Engine power bhp Max speed kmh mph Downforce in g values at 180 kmh 110 mph Downforce in g values at max speed Example car Low 200 225 130 05 07 Motorcycleengined single seater Medium 200350 250 150 075 14 F3 High 350700 275 170 085 20 F2 Very high 700 320 200 10 33 F1 The above two points also lead to different setups for different circuits A typical highdownforce circuit contains lots of fast sweeping corners A lowdownforce circuit consists of tight hairpin corners joined by fast straights In such circumstances even with aggressive wings little downforce is generated at the corners because the velocity is low and the presence of drag reduces the topspeed on the straights FSAEFormula Student cars present an interesting case in relation to downforce Average speeds in the range 48 to 57 kmh and maximum speeds of only 105 kmh are at the lower limit of where aerodynamic devices start to become effective There have been successful teams with wings and successful teams without wings It is the authors view that welldesigned and engineered devices of lightweight construction are despite a small weight penalty almost certainly beneficial in the hands of the right driver Throughout this book we will occasionally pause and reflect upon the implications of aerodynamic downforce on the design process We will now consider the effect of aerodynamic downforce on the three elements of racing acceleration braking and cornering 191 Acceleration and downforce Downforce has only a relatively small effect on acceleration For lowpower cars the tractionlimited stage is relatively short and the car is likely to move into the powerlimited stage by the time it has reached about 90 kmh 55 mph At this speed there is relatively little downforce Thereafter increased traction provides no benefit for acceleration and the small increase in drag will actually reduce performance Consequently for lowpower lowdownforce cars the critical transmission loads will be close to those that occur when accelerating offtheline For highpower highdownforce cars the situation is a little different Such a car will not reach its powerlimit until about 150 kmh 90 mph by which time it will have developed significant downforce Transmission loads will increase alongside the growing traction forces If at 150 kmh the downforce 25 produces say an additional vertical force of 07g the transmission loads will be 70 higher than those offtheline 192 Braking and downforce Aerodynamic downforce has a significant effect on braking from high speeds Consider the F1 car in Table 12 when braking from 320 kmh the car is effectively subjected to 10g from gravity plus 33g from downforce Its effective weight therefore becomes its mass 43g Because of tyre sensitivity the average coefficient of friction μ may be reduced from say 15 to 12 In addition to braking from friction at the tyre contact patch the car will also be slowed by airbraking from aerodynamic drag As soon as the driver lifts off the throttle at 320 kmh the car will slow at about 15g even without touching the brakes This braking force is applied at the centre of pressure on the front of the car and not at road level As this is likely to be close to the centre of mass position it would have little effect on longitudinal weight transfer EXAMPLE 16 Repeat Example 14 for a F1 car shown in Figure 122 braking from 320 kmh assuming aerodynamic downforce of 33g which is divided between the wheels in the same proportion as the static loads drag braking of 15g applied at the centre of mass a Estimate the individual wheel loads during maximum braking assuming an average coefficient of friction μ between the tyre and the road of 12 b Calculate the maximum deceleration in both ms2 and g forces Figure 122 Braking with downforce a Effective total weight of car WT static weight downforce 7350 24 255 31 605 N As before Rear axle load WR 31 605 1953215 19 169 N Front axle load WF 31 605 19 169 12 436 N From equation 19 Braking force F WT μ 31 605 12 37 926 N From equation 15 Longitudinal weight transfer ΔWx FhmL 37 926 3253215 3834 N b Front wheel loads WFL and WFR 12 436 38342 8135 N Rear wheel loads WRL and WRR 19 169 38342 7668 N Air braking force 7350 15 11 025 N Total braking force FT 11 025 37 926 48 951 N Mass of car 7350981 7492 kg From equation 14 Deceleration a FTm 48 9517492 653 ms2 653981 67g Answer Wheel loads 8135 N front and 7668 N rear Deceleration 653 ms2 67g Comments 1 Braking at 67g is clearly impressive and stressful for the driver However it falls rapidly as the velocity and hence the downforce and drag reduce 2 The wheel loads calculated above are an important load case for the braking system and the front wheel assemblies bearings suspension members etc a Effective total weight of car WT static weight downforce 7350 24 255 31 605 N As before Rear axle load WR 31 605 1953215 19 169 N Front axle load WF 31 605 19 169 12 436 N From equation 19 Braking force F WT μ 31 605 12 37 926 N From equation 15 Longitudinal weight transfer ΔWx FhmL 37 926 3253215 3834 N b Front wheel loads WFL and WFR 12 436 38342 8135 N Rear wheel loads WRL and WRR 19 169 38342 7668 N Air braking force 7350 15 11 025 N Total braking force FT 11 025 37 926 48 951 N Mass of car 7350981 7492 kg From equation 14 Deceleration a FTm 48 9517492 653 ms2 653981 67g Answer Wheel loads 8135 N front and 7668 N rear Deceleration 653 ms2 67g Comments 1 Braking at 67g is clearly impressive and stressful for the driver However it falls rapidly as the velocity and hence the downforce and drag reduce 2 The wheel loads calculated above are an important load case for the braking system and the front wheel assemblies bearings suspension members etc Chapter 1 Racing car basics In terms of lateral g forces 185407350 252g b From equation 113 Total lateral weight transfer ΔWy FhmT 15450 3251500 3348 N c From equation 111 F mv²R v² FRm 18540 1007350981 2475 v 497 ms 179 kmh Answer Cornering force 18540 N 252g Total lateral load transfer 3348 N Corner speed 497 ms 179 kmh Comment It can be seen that compared to the zerodownforce car the corner speed has increased from 138 kmh to 179 kmh ie a 30 increase 194 Effect of downforce on gg diagram We have seen that downforce has relatively little effect on acceleration but produces significant increases in braking and cornering ability Figure 124 shows how this may be represented on the gg diagram It can be seen that as speed increases attainable g forces in braking and cornering increase This should be compared to Figure 120 which represents the zero downforce case Figure 124 A gg diagram for a whole car with downforce Race car design 110 Racing car design issues The purpose of this section is to review the preceding topics and to extract those features necessary for a competitive car 1101 Mass We have seen that the three elements of racing acceleration braking and cornering all involve either longitudinal or lateral acceleration and we know from the Newton equation 14 that in order to maximise these accelerations we need to maximise the force and minimise the mass The origin of this force in all cases is the contact patch between the tyre and the road We have also seen the phenomenon of tyre sensitivity Figure 13 which indicates that the effective coefficient of friction between the tyre and the road decreases as the load on the tyre increases This means that minimising the mass improves all three elements of racing A light car will accelerate brake and corner better than a heavy one The Lotus designer Colin Chapman is credited with saying Adding power makes you faster on the straights Subtracting weight makes you faster everywhere Provided that a car is adequately stiff robust and safe it is imperative that the mass is minimised Where a formula specifies a minimum weight it is preferable to build the car below that weight and then add strategically placed ballast Weight reduction requires discipline in the design and build process and great attention to detail Stress calculations should be carried out for all key components to optimise the shape and thickness of material The number and size of bolts should be questioned Where possible components should perform more than one function for example a stressed engine can replace part of the chassis A bracket or tag can support more than one component Every gram needs to be fought for and unfortunately this can become expensive as it inevitably leads down the road of expensive materials such as carbon fibre composites 1102 Position of the centre of mass The positioning of the wheels and other components needs to ensure that the centre of mass of the car is in the optimum location Clearly the centre of mass needs to lie as close as possible to the longitudinal centreline of the car so that the wheels on each side are evenly loaded This can usually be achieved by offsetting small components such as the battery The fronttorear location of the centre of mass should ensure more weight over the driven wheels to aid traction during acceleration Thus for a rear Chapter 1 Racing car basics wheeldrive car the centre of mass should be towards the back of the car A 4555 or 4060 front to rear split is generally thought to be about optimum however this requires wider rear tyres to balance the car during cornering With regard to the height of the centre of mass above the ground we have seen that because of tyre sensitivity overall grip reduces as a result of weight transfer during cornering and from equation 113 Total lateral weight transfer ΔWy FhmT This indicates that the height of the centre of mass hm should be as small as possible to minimise weight transfer A low centre of mass also produces less roll when cornering which as we shall see later means there is less risk of adversely affecting the inclination camber of the wheels The lowest possible centre of mass is therefore the aim Lastly items which change their mass during a race such as the fuel should be positioned as close to the centre of mass as possible so that the overall balance of the car is not affected as the mass changes 1103 Enginedrive configuration Virtually all modern singleseat openwheel racing cars adopt the rearmid engine position with rear wheel drive This facilitates good location of the centre of mass short and hence light transmission of power to the rear wheels and a small frontal area for aerodynamic efficiency Such cars generally benefit from wider tyres at the rear 1104 Wheelbase and track The optimum wheelbase ie length of a car between the centrelines of the axles is somewhat difficult to define In general we know that shortwheelbase cars are nimble and hence good at cornering on twisty circuits whereas longwheelbase cars are more stable on fast straights Hillclimbsprint cars generally need to negotiate narrower roads with tighter hairpins and hence have evolved relatively short wheelbases 2025 m Circuit racing cars spend more time at higher speeds on wider roads and hence have evolved longer wheelbases 2528 m Modern F1 cars are particularly long 3132 m and the main motivation for this is that the extra length provides a longer floor with which to generate vital downforce However all things being equal a short car is obviously lighter than a long one The optimum track ie width of a car between the centres of the wheels is easier to define From equation 113 above we can see that weight transfer reduces as track T increases consequently it usually pays to adopt the widest track that formula regulations allow A wide track also reduces cornering roll The regulations are often couched in terms of the maximum overall width of the car which means that the track of the wider rear wheels is a little less than the front FSAEFormula Student presents a special case as far as wheelbase and track are concerned The narrow and twisting circuits demand a light and highly nimble car Experience has shown that compact cars perform best with the wheelbase in the range 1517 m and the track around 12 m SUMMARY OF KEY POINTS FROM CHAPTER 1 1 Racing involves optimum acceleration braking and cornering which all require maximum traction at the contact patch between the tyre and the road 2 Many aspects of racing car design require knowledge of individual vertical wheel loads and the static values are governed by the position of the centre of mass of the car It is important to be able to calculate this position 3 The wheel loads change as the car accelerates brakes or corners as a result of load transfer 4 The coefficient of friction between a tyre and the road is not a constant but reduces as the load on the tyre increases This is known as tyre sensitivity 5 The traction circle is a useful tool to show the interaction between acceleration braking and cornering 6 The use of aerodynamic downforce is vital for improving track performance and its presence increases wheel loads and hence traction Its effect is proportional to the square of the velocity of the car 7 We conclude that for optimum performance racing cars should be as light as regulations permit with a low centre of mass and a wide track A rearmid engine configuration with rearwheel drive is favoured 2 Chassis structure LEARNING OUTCOMES At the end of this chapter You will be able to define the key requirements of a racing car chassis structure You will know the basic types of structure spaceframe monocoque and stressed skin and the features of each type You will be able to specify the loads on the chassis structure and understand the need for safety factors You will be aware of analysis techniques for the chassis frame You will know about crash safety structures 21 Introduction The term chassis can be used to refer to the entire rolling chassis ie including the suspension and wheel assemblies however this chapter is concerned only with the structural frame of the car The basic requirements of the structural chassis are to comply with relevant formula regulations to provide secure location for all the components of the car such as the engine fuel tank battery etc to provide sufficient strength and stiffness to resist the forces from the suspension and steering components when the car accelerates brakes and corners at high g forces to accommodate and protect the driver in the case of a collision and to provide a secure anchorage for the safety harness to support wings and other bodywork when subject to high aerodynamic forces The chassis structure is analogous to the human skeleton which keeps all the vital organs in the correct location and provides anchorage for tendons and muscles so that useful movement and work can be carried out Over the last fifty years or so there have really been only two major forms of chassis structure 1 The spaceframe which is a 3D structure formed from tubes This is then clad with nonstructural bodywork 2 The monocoque which consists of plates and shells constructed to form a closed box or cylinder The monocoque can thus replace some of the bodywork Modern monocoques are invariably made from carbon fibre composite A third form worth mentioning is stressedskin construction which is a hybrid of the first two This can be an alternative name for a monocoque but here it is used for a spaceframe where some of the members are replaced or supplemented by a skin which is structurally fixed to the tubes 22 The importance of torsional stiffness The torsional deformation of a chassis refers to twisting throughout the length of the car Figure 21a shows an undeformed spaceframe chassis while Figure 21b shows this chassis subjected to twisting in a torsion test in the laboratory In practice a chassis will be put in torsion whenever one wheel encounters a high or low spot on the track It can also occur during cornering as lateral forces cause both horizontal bending and twisting of the chassis as shown in Figure 21c There are two important reasons why a racing car chassis needs to be torsionally stiff The first reason concerns the ability to tune the balance of the car effectively ie to achieve a car that has fairly neutral handling without excessive understeer or oversteer As an example consider a car that suffers from excessive oversteer This means that when cornering at the limit the backend of the car loses grip first and a spin is the likely outcome We saw from section 171 that because of tyre sensitivity as increasing load is transferred across the car the combined grip at that end of the car is progressively reduced and vice versa Consequently if we could engineer more load transfer at the front of the car and less at the rear this would reduce front grip and increase rear grip thus reducing or eliminating the oversteer problem We can achieve this by stiffening the front suspension in roll and softening the rear suspension in roll This does however mean that the chassis needs to transport lateral cornering loads from say the weight of the driver and engine to the front suspension connection points This causes the chassis to twist If the chassis has low torsional stiffness it behaves like a spring in series with the front suspension thus seriously reducing its effective roll stiffness The ability to tune the balance of the car is thus compromised It has been shown that for tuning to be at least 80 effective the torsional stiffness of the chassis needs to be at least approaching the total roll stiffness of the car ie the combined roll stiffness of the front and rear suspensions ref 6 Milliken and Milliken ref 15 add Chapter 2 Chassis structure Figure 21a Undeformed chassis Figure 21b Chassis subjected to torsional twisting Equal and opposite horizontal loads applied here at nodes This end rigidly fixed to support Figure 21c Chassis subjected to torsional twisting and lateral bending from cornering Front tyre load Centrifugal loads from weight of engine and driver Rear tyre load 35 Race car design Predictable handling can best be achieved if the chassis is stiff enough to be safely ignored In practice the total roll stiffness of a racing car increases in proportion to the amount of downforce and hence lateral g forces encountered For a low downforce softly sprung car the total roll stiffness can be as low as 300 Nmdegree whereas for a F1 car the figure is as high as 25 000 Nmdegree The required torsional stiffness of the chassis measured over the length between the front and rear suspension connection points is thus also in the range 30025 000 Nmdegree The higher figures are probably only achievable with carbon fibre monocoque construction Stiffer is always better provided not too much weight penalty has been paid and a 1000 Nmdegree minimum is recommended The second reason that torsional stiffness is important is because a flexible chassis stores considerable strain energy Thus during hard cornering a flexible chassis can wind up like the spring in a mechanical watch This energy then returns to unsettle the car at the critical point when the driver wants to straightenup and accelerate out of the corner Energy is also stored in the suspension springs however this is not a problem as it is controlled by dampers The absence of chassis damping could result in repeated torsional oscillations 23 The spaceframe chassis structure 231 Principles The essence of good spaceframe design is the use of tubes joined together at nodes to form triangles Each node should have at least three tubes meeting at it Three such triangles form a tetrahedron as shown in Figure 22 The centroidal axis of each tube that meets at a node should pass through a single point All major loads applied to the chassis should be applied only at nodes and ideally also pass through the same single intersection point The beauty of such a structure is that the tubes are loaded almost entirely in either Figure 22 Tetrahedron with load at node 36 Force N Chapter 2 Chassis structure pure tension or compression ie there is virtually no bending The use of almost and virtually in the preceding sentence is due to the fact that in practice the nodal joints are generally welded rather than pure pin joints The members in a truly pinjointed structure would not contain any bending Additional nodes can be created by progressively connecting three more members to existing nodes The power of triangulation is clearly illustrated by the 2D frames shown in Figure 23 1 kN Supports Deflection 7 mm Figure 23a Unbraced 2D frame Max stress 190 Nmm2 1 kN Supports Deflection 0024 mm Figure 23b 2D frame with diagonal bracing Max stress 14 Nmm2 1 kN Supports Deflection 08 mm Figure 23c 2D frame with nonnodal loading Max stress 118 Nmm2 37 Race car design Figure 23a shows an unbraced square 2D tubular frame supported at the bottom corners and loaded with a 1 kN load at the top corner The frame is 500 mm square and made from a typical mild steel tube The lefthand picture shows the undeformed shape and load The righthand picture shows the exaggerated deflected shape from a finite element analysis It indicates the magnitude of the maximum deflection and tensile stress ie 7 mm and 190 Nmm² respectively Figure 23b shows the effect of forming triangles by adding a diagonal bracing member The deflection is reduced to a staggering 03 of the unbraced case Also the maximum tensile stress is reduced to about 75 As the strain energy stored in the structure is proportional to the deflected distance of the load this is also reduced to 03 by introducing the diagonal brace The members are subjected to almost pure tensile or compressive forces Figure 23c shows how even with a triangulated structure much of the good work is undone if the load is not applied at a node Both the maximum deflection and the maximum stress are intermediate between the two former cases Although the formation of strong triangulated tetrahedrons is desirable they are often not ideally suited to the flatsided shape of most modern racing cars Another good principle is to aim for prisms of rectangular or trapezoidal crosssection where each face of the prism is itself a triangulated rectangle Such a buildingblock is shown in Figure 24 Figure 25 shows several such building blocks with the addition of a rollbar and it can be seen that a racing Figure 24 Triangulated prism as a chassis building block Figure 25 Emerging spaceframe chassis but some diagonal members had to be removed 38 Chapter 2 Chassis structure car shape is starting to emerge However for practical reasons several key diagonal bracing members have been removed Most significantly the cockpit opening cannot be diagonally braced Two diagonal members have been removed to permit the drivers legs to extend forwards A diagonal has been removed to allow the engine to be installed and removed All of these factors will reduce the torsional stiffness of the chassis and remedial action should be considered Figure 26a shows a plan view of a chassis subjected to torsional deformation where it can clearly be seen that most of the deformation occurs in the cockpit area Actions that can be considered are increasing the bending stiffness of the upper cockpit side members particularly in the horizontal plane adding additional higher members between the front and main rollhoops reducing the effective length of the cockpit side members by adding short diagonals across the corners providing some form of picture frame around the cockpit possibly in the form of structural sidepods as shown undeformed in Figure 26b In this case the torsional stiffness was increased by about 50 Figure 26a Deformed chassis under torsion indicating most movement at cockpit opening Figure 26b Addition of structural sidepod members increases torsional stiffness by 50 39 Race car design With regard to the front of the chassis around the drivers legs it is possible to introduce one or more bulkheads as shown in Figure 27 This is particularly desirable if they need to resist nonnodal loads from suspension members Also many formulae require a substantial front rollhoop as part of the driver protection structure Openings to permit the fitting and removal of the engine can be strengthened either by the addition of removable usually bolted members andor by adding bracing to rigid points on the engine itself Consideration can also be given to making the engine a stressed part of the chassis provided suitable strong connection points exist In this case the chassis is likely to be split into a front section and a rear section Figure 27 Typical front rollhoop bulkhead 232 Spaceframe design process and materials Because it is important to apply all major chassis loads at or close to nodal points it is necessary to know the location of suspension and enginemounting points before the chassis frame layout can be finalised It has been said that the process is one of joining the dots In practice the design of the chassis frame and the suspension tend to proceed in parallel however the frame designer must be prepared to adjust node positions to suit the suspension The initial layout of the chassis frame members must therefore take account of 1 The particular formula regulations appropriate to the car These are likely to require minimum ground clearance size and height of rollhoops and possibly other driver protection structures Some formulae such as F1 and FSAEFormula Student require minimum cockpit sizes by requiring cockpit profiles to be inserted into the finished car 2 The driver dimensions Figure 28 shows rough dimensions as a starting point to design the cockpit area The final dimensions used will depend upon the degree to which the driver is reclined the amount of bend in the knees and the height of the front floor where the pedals are mounted To maintain an acceptable front lineofsite the front rollhoop should not be above the height of the drivers mouth 3 The provision of nodes at or close to the points where all high loads attach to the chassis This includes suspension components engine mounts and seat belt fixings 40 Chapter 2 Chassis structure Figure 28 Standard driver dimensions for cockpit design Materials We have seen that for a welltriangulated spaceframe with loads applied at the nodes the members are in almost pure tension or compression Also as we shall see many members will be subjected to load reversal under different load cases Consequently all members need to be designed to carry compressive loads and the only real option for this is the hollow tube Table 21 lists the main options for spaceframe members Most options are available in several material strength grades and are also delivered in various conditions such as hardened tempered or annealed Tubes in the annealed state are easier to bend but usually significantly weaker The sizes and properties of standard tubes are given in Appendix 2 At the time of writing it was easier to source imperial size tubes than metric However note that in Appendix 2 all the imperial dimensions are given in equivalent metric units 24 Stressedskin chassis structure Plate 1 shows the same square frame as Figure 23 but with the diagonal member replaced by a 1 mm thick aluminium sheet It can be seen that the maximum deflection and stress are of a similar order to the frame with the diagonal member Clearly the attachment of the plate to the tube requires a highquality structural bond and this is plainly easier to achieve with square 41 Race car design Table 21 Options for spaceframe tubes Name Standard or type Yield strength Nmm2 Comment Steel Density 7850 kgm3 Elastic modulus 205 000 Nmm2 Seamless circular tube BS EN 102971 2003 235470 The UK Motor Sports Association MSA requires a minimum yield strength of 350 Nmm2 for roll cages and bracing Seamless is also recommended for high stress components such as suspension members E355 Welded circular tube BS EN 102961 2003 175400 Cheaper than the above and now a consistent and reliable product Welded square tube BS EN 103055 2010 190420 Square tubes are generally easier to join Also good for attaching sheets such as floors More compact than circular tubes and slightly stronger in bending but about 20 less efficient in compressive buckling Alloy steel tube BS4 T45 A1S1 4130 Osborne GT1000 6201000 Generally referred to as aerospace tubes these are made from steel alloys containing elements such as chrome molybdenum and nickel They are much more expensive and can require special welding These will produce the lightest solution where regulations require the chassis frame to resist specific loads Aluminium alloy Density 2710 kgm3 Elastic modulus 70000 Nmm2 6082T6 BS EN 7542 2008 260 Although this aluminium alloy is about the same strength as mild steel and is only about a third of the weight the elastic modulus stiffness is also only a third This reduces its effectiveness in compressive buckling and hence erodes most of the weight advantage Also welding is more difficult and results in significant strength reduction 50 in the heat affected zone HAZ Ideally significant heat treatment is required after welding section tubing The key is good surface preparation appropriate curing conditions and the use of a highquality structural adhesive often supplemented with rivets or selftapping screws The use of structural flat plates in this way can produce weight saving for such elements as the floorpan however wider use as an alternative to bodywork can compromise access aesthetics and aerodynamics 25 The monocoque chassis structure 251 Monocoque principles The monocoque chassis is now universally adopted at professional levels of racing as it has the potential to be lighter stiffer stronger and hence safer It is however more expensive and difficult to build Whereas the spaceframe was based on triangulated tubes in tension and compression the monocoque is based on plates and shells largely in shear The triangulated prism building block of Figure 24 is replaced by the plated prism of Figure 29 Provided it is well restrained at one end such a structure is very stiff in torsion As with the 42 Chapter 2 Chassis structure Figure 29 Monocoque plated box building block spaceframe for practical reasons some faces must be removed or perforated to allow access for the driver and other components Plates 2a and b show the effect on torsional stiffness of removing the end plate It can be seen that the deflection is increased fourfold implying that the torsional stiffness is reduced to a quarter The outer skin of a monocoque is therefore supplemented with internal stiffening ribs and bulkheads Also thin plate structures are not good for resisting concentrated point loads so metallic inserts are introduced as reinforcements at key connection points such as where the suspension wishbones attach 252 Monocoque materials Aluminium alloys In the past solid aluminium alloy sheet around 1 mm thick has been used to form monocoques The sheet can be bent to form longitudinal boxes or tubes that run the length of the car each side of the driver These are in turn joined by floor plates bulkheads and the firewall to form a strong overall structure Sheets can be joined by rivets welds or adhesive bonding A more modern approach is to fabricate the chassis from aluminium honeycomb composite sheet This consists of two sheets of aluminium bonded to a core of aluminium honeycomb as shown in Figure 210 Overall thickness is generally about 1015 mm This produces a strong and lightweight panel that can be cut bent and bonded Clearly it is not possible to produce curves in more than one plane so aesthetics and aerodynamics may be compromised without the use of separate external bodywork Figure 210 Aluminium honeycomb sheet 43 Race car design Carbon fibre composites Carbon fibre composite is now universally adopted at the professional level as the material of choice for chassis tubs Figure 211 Per unit weight carbon fibre composites are about three times as stiff and strong as structural steel or aluminium As the word composite suggests it is made up from two distinct materials The reinforcement provides most of the strength of the composite and consists of fibres laid in various directions to suit the applied loads For our purposes the fibres are either carbon or Aramid also known as Kevlar or Nomex They generally come in the form of a mat of woven or unidirectional fibres Fibres are available in many grades of varying stiffness and brittleness The resin matrix provides the body of the material It bonds and protects the reinforcement and distributes loads to the fibres The type of resin determines the capacity of the composite to resist heat Phenolic resins have better fire resistance but epoxy resins are stronger and tougher and hence are the obvious choice for a chassis Polyester resins are cheap but lack strength and toughness Vinyl ester resins are intermediate between epoxy and polyester in terms of mechanical strength and cost Carbon fibre composite construction is invariably anisotropic ie the fibres are deliberately run in specific directions to produce the most advantageous strength properties Much of the construction consists of a thin skin of carbon fibre composite each side of a lightweight core of Aramid or aluminium honeycomb The first step in manufacture is the production of a pattern which is a fullsized copy of the chassis in wood or resin Female moulds or tools made of carbon fibre are then cast around the pattern Chassis tubs are usually made in two halves an upper and a lower part which are subsequently structurally bonded together All composite work should be carried out under carefully controlled clean room conditions Several processes are available for manufacture of the finished parts two of them being Figure 211 Carbon fibre tub Example shown is a chassis from a Pilbeam MP97 hillclimb car reproduced with kind permission from Mike Pilbeam 44 Chapter 2 Chassis structure Wet layup which is the simplest approach Alternate layers of resin and reinforcement are added to the mould and compressed with a handroller The whole is then enclosed in a vacuum bag to compress the composite as the resin cures The resin is designed to cure at room temperature Prepreg is the professional approach The reinforcement mats are already impregnated with resin but remain flexible until hotcured Different layers are carefully built up before vacuum bagging The whole is then placed in an autoclave which is a pressurised oven for curing at a specified temperature pressure and duration depending upon the resin used This process produces the lightest and strongest components but requires expensive equipment The production of a highquality carboncomposite chassis requires considerable skill and experience to the extent that even Formula 1 teams often use external specialist companies On the other hand several FSAEFormula Student teams have demonstrated considerable success 26 Chassis load cases and safety factors 261 Load factors If a chassis is adequately stiff in torsion it is likely to be adequately strong Nevertheless it is good practice to check the strength of the structure when subjected to peak loads from several load cases A further complication arises from the fact that a racing car is a highly dynamic object It is not sufficient to consider the stresses from only the static loads on the car When cars become airborne or go over bumps and kerbs shock loads are transmitted through the suspension springs and dampers The actual magnitude of these dynamic loads is very difficult to determine however the normal design procedure is very simple It is common practice to apply a dynamic multiplication factor to the static loads For vertical loads a typical multiplication factor is 3 This is essentially saying that the mass of the car is subjected to a vertical acceleration of 3g For loads that arise from frictional grip such as those during cornering or from aerodynamics a suitable factor is 13 Table 22 lists suggested load cases together with appropriate dynamic multiplication factors The factors given in Table 22 are based on refs 2 and 19 together with the authors experience An interesting question arises is it necessary to check for load case combinations such as max vertical load max cornering In general structural engineers consider that the likelihood of the same maximum values occurring during such combinations is less likely and hence tend to use reduced multiplication factors say 20 and 11 This means that combined cases are not usually critical 45 Race car design Table 22 Load cases and dynamic factors Load case Dynamic multiplication factor Max vertical load 30 Max torsion diagonally opposite wheels on highspots 13 on vertical loads Max cornering 13 on vertical and lateral loads Max braking 13 on vertical and longitudinal loads Max acceleration 13 on vertical and longitudinal loads 262 Material factors In addition to the factors on loads given in Table 22 it is necessary to apply a safety factor to the material strength A value of 15 is suggested This takes account of factors such as quality of the material small errors in component dimensions defects in the component such as lack of straightness loads not being applied on the centroid of the component Some designers may use a slightly higher safety factor for safety critical and mission critical components and a lower factor for other components say 16 and 14 respectively An example of a component in the former category is a suspension wishbone member and in the latter category is a diagonal bracing member whose primary function is to increase torsional stiffness It may also be appropriate to vary the safety factor for different materials Thus steel and aluminium obtained from a reliable source may attract lower safety factors than say homemade fibre composites 27 Design of structural elements 271 Components in tension We have seen that triangulated structures with nodal loads contain members that are largely loaded in pure tension or compression For components in tension the above material safety factor say 15 is simply applied to the material yield stress σy or in the case of aluminium which does not have a clearly defined yield point to the 02 proof stress Hence Tensile stress σt Force Ft Area A Yield stress σy 15 Min area A 15 Ft σy 21 46 272 Components in compression Slender compression members or struts fail in buckling often long before the yield stress is reached A reasonable indication of the strength of such members is given by the Euler buckling load For pinended components this is Euler buckling load PB π2EI L2 where E Modulus of elasticity I Second moment of area L Effective length In this case the material safety factor say 15 is applied to the Euler formula Allowable Euler buckling load PB 15 π2EI 15L2 22 For pinended struts the effective length is the distance between nodes In a frame with fully welded joints the effective length of compression members can be taken as 085 distance between nodes An application of this formula is demonstrated in Example 21 on page 49 273 Components in bending As in Figure 23c if a force is applied to a member some distance away from a triangulated node it will cause a bending moment in that member which in turn generates a bending stress Again for a material safety factor of 15 we get Bending stress σb bending moment M elastic modulus Z yield stress σy 15 Min elastic modulus 15 M σy 23 The elastic modulus is a geometric property based on the crosssection of the member Values are given in tables for standard tubes If a structural member is subjected to both a bending moment and a tension force at the same time we can combine equations 21 and 23 to get Max stress σb Ft A M Z Yield stress σy 15 24 For more information on the calculation of bending moments and the design of structural elements see ref 22 47 28 Chassis stress analysis 281 Hand analysis Simple hand analysis techniques can be useful for checking the strength of principal members in a spaceframe chassis Consider the chassis shown in Figure 212a The aim is to determine the forces in the main members a b and c when the chassis is subjected to the maximum vertical load case Figure 212a Hand analysis of spaceframe chassis Figure 212b Application of the method of sections The first step is to determine the weight of the car which is applied at the centre of mass taking account of the dynamic multiplication factor It is appropriate to consider the sprung mass ms in this case ie the mass of the wheel assemblies with half the wishbones excluded as these are carried directly through the tyres to the ground without passing through the chassis frame The front and rear wheel loads are then determined For symmetrical 48 vertical loading half the car design weight is then apportioned to the nearest adjacent frame nodes at one side of the car The load at each node W1 and W2 is inversely proportional to its distance from the centre of mass This is a conservative approach as in reality the sprung mass of the car is distributed more widely throughout the frame For a dynamic multiplication factor of 3 Design weight of car W ms 3 981 N Moments about WF Rear wheel load WR W 2 lm L From vertical equilibrium Front wheel load WF W 2 WR Load at x W1 W 2 l2 l1 l2 Sum vertical forces Load at y W2 W 2 W1 The next step is to use a technique known as the method of sections to determine the forces in a b and c The members in question are assumed to be cut by an imaginary line as shown in Figure 212b The structure to the right of this line is ignored All forces to the left of the line must be in equilibrium with the forces shown as arrows on the cut members a b and c All three of these forces are unknown at this stage but if we take moments about a point where two of them intersect ie z these two are eliminated leaving the force in a as the only unknown Moments about z WF ly W1 lx Fa h Hence Fa Moments about x WF ly lx Fc h Hence Fc If members a and c are horizontal then the vertical component of the force in b must provide vertical equilibrium Sum vertical forces WF W1 Fb cos θ Hence Fb EXAMPLE 21 Figure 213a shows a spaceframe racing car chassis It has a fully laden sprung mass of 530 kg 1 Determine the forces in members a b and c 2 Check the suitability of using 25 mm outside diameter circular tubes with a wall thickness of 15 mm given Crosssectional area 1107 mm2 49