Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 May 2018 doi:10.20944/preprints201805.0074.v1 1 Technical Note 2 Modelling the Dynamics of a Three-wheeled Racecar: 3 A Pilot Study to Establish the Feasibility of 4 Developing a ‘Delta’ Configuration Performance Car 5 Rebecca Margetts 1,* 6 1 University of Lincoln; [email protected] 7 * Correspondence: [email protected]; Tel.: +44-1522-837951 8 9 Abstract: This paper presents a preliminary study of the dynamics of a ‘delta’ configuration three- 10 wheeled sports car, based on a Reliant Robin chassis. Stiffness and damping values to give this 11 vehicle ‘sporty’ ride dynamics have been proposed. Other dynamic qualities such as rollover and 12 steady-state cornering have also been investigated, and then verified using Carmaker® simulation 13 software. The car performs far better than the Robin in steady-state analyses, but simulation 14 indicates that transient maneuvers such as braked corners can still cause the vehicle to roll. Remedial 15 actions such as lowering the center of gravity are suggested. However, much of the data used in 16 these analyses is assumed, and should be updated as the design progresses. Since the standard 17 calculations used in this report are intended for four-wheeled vehicles, there is scope to develop a 18 more detailed transient model to fully describe the unique dynamics of three-wheeled vehicles and 19 develop design guidelines more suited to them. 20 Keywords: vehicle; dynamics; handling; ride; simulation; bond graphs; bond-graphs 21 22 1. Introduction 23 Electric cars are an active area of research and development, offering personal mobility with 24 higher efficiency and lower emissions than existing internal combustion engine vehicles [1]. 25 MIBRID Ltd. – an Electric Vehicle company situated in Lincolnshire - are developing a three-wheeled 26 electric sports car called the Mayfly, shown in Figure 1. It is designed to be ‘low-cost, stylish, strong, 27 and light … suitable for the unique roads of Britain’ [2] and it has the following specifications: 28 • Plug-in electric vehicle 29 • Hybrid capacitor/li-ion battery system 30 • Three wheels with three independent motors (3 x 10kw approx. 40bhp) 31 • Lightweight composite body (under 500kg) 32 • Two seats 33 • Hatchback with room for the everyday. 34 Model-based design offers the opportunity to investigate the dynamics of the car – and the 35 effects on comfort and handling – while the design is still at an early, fluid stage. Using computer 36 modelling techniques allows the design to be optimized before investing in physical prototype 37 vehicles, and minimizes the risk of the final product having unforeseen undesirable qualities. In the 38 case of the Mayfly, there are two immediate areas for concern: 39 • The use of in-hub motors yields a higher ‘unsprung mass’ than seen in conventional cars, which 40 can adversely affect handling. 41 • Three-wheeled vehicles are associated with rollover, tipping, and ‘spinning out’ in turns. 42 This report therefore details the development of handling models and proposes adjustments to 43 the suspension, tyres and weight distribution to yield performance appropriate to a sports car. 44 © 2018 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 May 2018 doi:10.20944/preprints201805.0074.v1 45 46 47 Figure 1. The Mayfly [2]. 48 49 2. Background 50 2.1. Three-Wheeled Vehicles 51 Three-wheeled cars can take one of two configurations: the ‘delta’ (one wheel at the front and 52 two at the back) or the ‘tadpole’ (two wheels at the front and one at the back), shown in Figure 2. 53 They can be highly economical: using three wheels instead of four can significantly reduce the weight 54 of a vehicle, improving its power consumption. For ‘delta’ cars, using one wheel at the front for 55 steering also simplifies the steering system, further reducing weight and cost. ‘Tadpole’ cars can be 56 extremely aerodynamic. 57 Three-wheelers are generally associated with instability, in particular rollover (although they 58 can be very stable) [3]. This is particularly true of the ‘delta’ configuration, which is associated with 59 rolling in braked turns or ‘spinning out’ when handled roughly, although the tendency to rollover 60 has been exaggerated in popular culture [4]. The ‘tadpole’ configuration is much more stable [5]. 61 Fwd ‘Delta:’ 1 wheel front, 2 wheels rear ‘Tadpole:’ 2 wheels front, 1 wheel rear 62 63 64 Figure 2. Configurations of Three-wheeled Vehicles. 65 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 May 2018 doi:10.20944/preprints201805.0074.v1 66 Dynamically, three-wheeled cars behave like a cross between a motorbike and a car [6]. A 67 tadpole car behaves like a motorbike at the rear (where a single wheel is connected to the drivetrain) 68 and a car at the front (where two wheels usually steer), allowing it to perform as a more stable 69 motorbike. Delta cars, however, behave like a car at the rear (typically connected to the drivetrain) 70 but a motorbike at the front, which is where the steering occurs. A motorcyclist would lean into 71 corners counteracting the lateral forces on the bike by wheel camber action as well as tire slip. This 72 does not occur in a three-wheeled car, where the chassis generally does not allow the front wheel to 73 tilt significantly. Narrow-track three-wheel vehicles have notoriously limited rollover stability, and 74 there is a body of work on ‘tilting three-wheelers’ where the vehicle leans into corners in the same 75 way that a motorcyclist leans [6]. ‘Tilting’ three wheeled cars such as the GM Green Machine [7] or 76 CLEVER [3,6,8] therefore improve stability by allowing the wheel(s) to lean as a motorbike would. 77 2.2. The Reliant Robin 78 The initial prototype is based on a Reliant Robin chassis. The Robin is a small 3-wheeled car in 79 the ‘delta’ configuration, which was produced in the UK between 1973 and 2002 and still has a cult 80 following. 81 The rear suspension utilizes semi-elliptic leaf springs and a DeDion axle [9]. The DeDion ‘tube’ 82 has a sliding joint to permit wheel track variation during suspension movement, but it keeps both 83 wheels parallel to each other under all conditions (so they are always perpendicular to the road 84 surface regardless of body roll, and hence more stable). This system is cheaper than most independent 85 suspensions, and offers superior handling [10]. The use of the DeDion tube means that the vehicle 86 has much less unsprung weight than a fixed axle would because the final drive / differential and 87 driving shafts are not rigidly attached to the wheels. The system is a popular choice for budget sports 88 cars and coupes. 89 The front suspension of the Reliant Robin is a leading arm with coil spring damper unit [11]. 90 With this design, the wheel remains parallel to the body and cambers with roll [10]. 91 In producing the models, data for the 1981 Mk 1 Reliant Robin was assumed as a starting point. 92 The Robin evolved somewhat over the years, including plans for an electric Robin prior to production 93 ceasing [12]. 94 Note that the Robin is rear wheel drive and the front wheel steers. The Mayfly has a hub motor 95 on each wheel, making it all-wheel-drive and offering the possibility for differential steering on the 96 rear wheel. 97 2.3. Electric Vehicles 98 The prototype is a fully electric BEV (Battery Electric Vehicle) with three hub motors i.e. one in 99 each wheel. This configuration has already been set: electric vehicles can have a variety of 100 configurations such as a single motor and transmission, or hub motors in only some wheels [13]. 101 The weight distribution will therefore be significantly different to that of the Robin, since the 102 conventional engine and drivetrain are replaced by the battery and hub motors as indicated in figure 103 3. The battery is assumed to be in a low central position, but could again be positioned in place of the 104 engine or in the boot. 105 3. Method 106 3.1. Modelling Approach 107 In constructing a model, a ‘crawl-walk-run’ approach is followed whereby a simplistic, single 108 degree-of-freedom initial model is created, analyzed, and then complexity is added to this model in 109 stages. In the case of a car, the ride dynamics (symmetric motions) are first investigated using the 110 ‘Quarter Car’ model. Side forces, slip and responsiveness are then investigated via the ‘Bicycle Model’ 111 to establish over/under-steer. Quasi-static rigid rollover is also assessed. The results are verified by 112 simulation, and should eventually be validated against physical tests. 113 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 May 2018 doi:10.20944/preprints201805.0074.v1 Transmission Gear- I.C.E box Hub motor Hub motor Battery Hub motor 114 115 Figure 3. The Robin’s drivetrain [top] and the prototype electric drivetrain [lwr]. 116 117 Additional analysis such as those for ‘wobble’ (seen on three-wheeled rickshaws) [14] are 118 neglected here. 119 3.2. Ride Dynamics: The Quarter Car Model 120 The standard initial model used for modelling vehicle dynamics is the quarter car, shown in 121 figure 4. This models the vertical motion of each suspension unit under the effective load of the body. 122 The suspension is abstracted to a linear spring and damper acting in the vertical sense only, and the 123 tire is also abstracted to a linear vertical spring (and sometimes a damper).