DYNAMIC SIMULATION OF A FORMULA SAE RACING CAR FOR DESIGN AND DEVELOPMENT PURPOSES

Transcription

1 MULTIBODY DYNAMICS 29, ECCOMAS Thematic Conference K. Arczewski, J. Frączek, M. Wojtyra (eds.) Warsaw, Poland, 29 June 2 July 29 DYNAMIC SIMULATION OF A FORMULA SAE RACING CAR FOR DESIGN AND DEVELOPMENT PURPOSES Alfonso Callejo, Eduardo Elipe, Michele Macchi and Román Bravo INSIA - University Institute for Automobile Research Universidad Politécnica de Madrid, Ctra. Valencia km 7, 2831 Madrid, España s: a.callejo@alumnos.upm.es, eduardo.elipe.jorquera@alumnos.upm.es, michele.macchi@alumnos.upm.es, romanbg@alum.etsii.upm.es web pages: Keywords: Multibody Systems, Dynamic simulation, Vehicle Dynamics, Formula SAE, Racing Car, Virtual Tests. Abstract. This article presents a multibody model of a Formula SAE racing car, which has been implemented in Matlab and has had a key role in the early design and development stages of the vehicle construction. It uses a fast semi-recursive method which takes advantage of the system s topology and it has some of its core functions written in C/C++ so that its efficiency is even higher. The computational and theoretical effort needed to develop it, as the article shows with some examples, has been relatively small. Its versatility, which comes from the fact that it has been entirely programmed, and its realism, proved by the comparison with the real vehicle and by a certain identification of the parameters, allow performing virtual tests which would be difficult (or at least costly) to carry out with the real vehicle. With those tests, the overall dynamic behaviour of the vehicle, some of its internal magnitudes and its global stability can be easily evaluated, as a detailed example shows at the end of the article. 1

2 1 INTRODUCTION The Formula SAE competition challenges teams of university undergraduate and graduate students to conceive, design, fabricate and compete with small, formula style, autocross racing cars [1]. Among the four mentioned steps of the process, the design is often the biggest challenge, mainly because of the lack of experience and the limited material resources of the teams. Moreover, trying to directly apply a finite element method analysis over the complete assembly of bodies which form the suspension might be impossible because most of the parts have not been designed yet and the element constraints and forces have to be defined in a quite arbitrary way. In order to improve the development of the racing car suspension and to have a better understanding of its dynamic behaviour and stability, the INSIA s Computational Mechanics research group has created a multibody model of the racing car, which can accurately simulate it. The use of this model has several advantages in the early design and development stages. It can predict the ground contact forces on the wheels and the overall dynamic behaviour of the racing car, by means of a relatively low computational and conceptual work. The model does not need to contain the exact geometry of all the components, but just a simplified definition of the rigid bodies, the joints and the global geometry. (a) (b) Figure 1: (a) CAD and (b) MBS geometries of a suspension rocker. Among the many multibody formulations that have appeared since they begun to be exhaustively studied forty years ago [4], topological formulations seem to be more appropriate than global formulations in this particular case. The reason is that, as explained in [2], in complex mechanisms as the racing car suspension, topological formulations make use of the system s topology and result in a more efficient calculation. Specifically, the semi-recursive approach made by García de Jalón et al. [2] has been followed. The article describes the way in which the model has been defined and implemented in Matlab, including the enhancement of some of the core functions made by compiling them as MEX-functions in C/C++. Then, some ideas about parameter identification, which is necessary to tune the model up, are explained. Eventually, the efficiency and utility of the model are assessed by performing a virtual test with two differently set up vehicles running in parallel and by making a comparison with a model created with commercial software. 2 MODEL DESCRIPTION The aforesaid semi-recursive method works with an open-loop spanning tree, which grows from the ground and is made up by the linked rigid bodies. As the racing car suspension, like many other systems, is a closed-loop mechanism, the closed loops have to be opened and then enforced by means of constraints in the numerical integration. A similar strategy is followed with the rods. So the steps for a correct model definition would be: definition of the bodies and the joints between them, opening of the closed loops and removal of the rods, definition of the external forces and moments, setting of the coordinates which will operate the vehicle and execution of the numerical integration. 2

3 2.1 Geometry and joints definition The geometry has been defined parametrically to allow rapid modifications of the dimensions and, once the basic points are defined, it is easy to add new bars, modify their sizes and redefine the joints. Thus, the design process is simplified and the designer can evaluate the influence of changes in suspension over dynamic properties like the roll center of the vehicle, the damping behaviour or the suspension installation ratio. (a) (b) Figure 2: (a) CAD model; (b) Multibody model. The racing car suspension is a double wishbone with push rod actuator, as can be glimpsed in Figure 2. The model contains the necessary bodies and rods (wishbones, push rods, torsion bars, etc.) and joints (spherical, revolute, etc.) to exactly resemble the real suspension morphology. For instance, the wishbones are connected to the chassis by revolute joints and to the carriers by spherical joints. All of the joints must be defined as a combination of revolute and prismatic joints, using auxiliary elements with no mass when necessary. Following this approach, a spherical joint would be the sum of 3 revolute joints (one for each direction). In the following lines an example of the definition of a body (the front right triangle) can be seen. The user needs to define the geometrical properties (points, unit vectors), the inertial properties (mass, inertia tensor, external forces) and other graphic properties of the solid. 3

4 % Lower front right triangle nelem=nelem+1; LFR_TRIANG=nelem; Tbody(LFR_TRIANG).name = 'LFR_TRIANG'; Tbody(LFR_TRIANG).localPoints = [... Tbody(CHASSIS).localPoints(:,7)'; % control arm - chassis front Tbody(CHASSIS).localPoints(:,6)'; % control arm - chassis rear Suspdd(3,:); % lower ball joint ]'; Tbody(LFR_TRIANG).localPoints(:,4)=generateCOG(Tbody(LFR_TRIANG).localPoints); Tbody(LFR_TRIANG).localUnitVectors = [... 1.,.,.; ]'; Tbody(LFR_TRIANG).cogPoint = 4; Tbody(LFR_TRIANG).Aini = eye(3); Tbody(LFR_TRIANG).mass =.3; Tbody(LFR_TRIANG).Jlocal = Tbody(LFR_TRIANG).mass*diag([.13^2,.7^2,.15^2]); Tbody(LFR_TRIANG).fExt = Tbody(LFR_TRIANG).mass*gravity; Tbody(LFR_TRIANG).nExt = [,,]'; Tbody(LFR_TRIANG).linesToDraw=[1 3; 3 2]; Tbody(LFR_TRIANG).colorToDraw='g'; Regarding the definition of the joints, it is necessary to set the type ( R revolute or P prismatic), the linked bodies, their connecting points and vectors, and the initial velocity and position of the joint. The joint between the lower front right triangle and the chassis would be: % Joint between front right triangle and chassis njoint=njoint+1; J_LFRTR_CHASSIS=njoint; Tjoint(J_LFRTR_CHASSIS).name = 'J_LFRTR_CHASSIS'; Tjoint(J_LFRTR_CHASSIS).type = 'R'; Tjoint(J_LFRTR_CHASSIS).bodies = [CHASSIS LFR_TRIANG]; Tjoint(J_LFRTR_CHASSIS).points = [ 7 1 ]; Tjoint(J_LFRTR_CHASSIS).vectors = [ 1 1 ]; Tjoint(J_LFRTR_CHASSIS).zPosIni = ; Tjoint(J_LFRTR_CHASSIS).zVelIni = ; 2.2 Closed loops and rods As it has been said, the four bar linkages of the suspension are closed-loop mechanisms and need to be opened so that the mechanism is a tree structure. This means some of the joints will be temporarily disabled and enforced later on in the numerical integration. For instance: % Open loop between front left carrier and lower front left triangle openclosedloops(2).joint = [J_FL_CARR_LTRIANG]; openclosedloops(2).bodies = [FL_CARR LFL_TRIANG]; openclosedloops(2).points = [ 2 3 ]; openclosedloops(2).unitvectors = [ ]; Rods like the actuators and the toe control arms also have to be temporarily removed. They are not considered as normal bodies and they do not belong to the system tree: they act just as constraints and forces in their ends. They are defined in the following way: % Front left rod actuator nrod=nrod+1; FL_ACT=nrod; Trod(FL_ACT).name='FL_ACT'; Trod(FL_ACT).connectedBodies = [FL_BAL LFL_TRIANG]; Trod(FL_ACT).connectedPoints = [ 2 4 ]; Trod(FL_ACT).mass =.25; Trod(FL_ACT).length = distance; Trod(FL_ACT).externalForce = gravity*trod(fl_act).mass; 4

5 With this technique the racing car system tree is now perfectly open, as showed in the Figure 3, and the semi-recursive algorithm can make the most of the system s topology. Auxiliary elements have been omitted to clarify the figure. FL means front left, RR means rear right, and so on. Figure 3: Model open-loop tree. The reason why the chassis is divided into two pieces is that doing so the model can take into account the racing car s torsional behaviour, which is important in this kind of vehicles. 2.3 Applied forces and moments Some of the forces like the weights of the bodies and rods, as well as the springs and dampers acting directly on the joints, are straightforwardly added and computed by the method, but other forces like contact forces, propulsion, standard springs and standard dampers have to be introduced in a more detailed way. The contact forces between the wheels and the ground are a critical issue of all vehicle models because of their complex nature and the high force gradients they cause. They have been modelled using Pacejka s formulas [6] with SAE s coefficients, and they have been tested in varied situations: running through a straight path, a curved path and a pothole. The calculation needs some geometric parameters to compute the contact forces between the wheels and the ground: angular velocity of the wheel, nominal and loaded ratios, normal forces on the pneumatic, longitudinal and transversal velocities of the centre of the wheel, camber angle and Pacejka s tire coefficients. Thanks to these parameters, it is possible to estimate the values of the contact forces, the slip angle and the longitudinal slip ratio. One of the calls to the tyre forces function is: 5

6 [Fx(2),Fy(2),My(2),Mz(2),x(2),alpha(2)] = magicformula (w(2),wheel.r,va2x,va2y,fn(2),r(2),gamma(2),wheel.wheel_coefficient(2)); As far as the propulsion is concerned, it is modelled by applying a moment over the relative coordinates between the rear wheels and the carriers, that is, in the joints which connect the rear wheels with the carriers. A more realistic option, which would be interesting in order to study the propulsion chain, would be to include the engine, the gearbox, the differential and the drive shafts. The last group of applied forces is the one formed by spring and damper forces, which obviously are proportional to the elongation and to the relative velocity between the ends respectively, and are applied in the connecting points of the two linked bodies. Spring forces might include an initial load. Note that there is a great freedom to define these loads and any others the user wants to put in, on the basis of the positions and velocities of the bodies, the current time, etc. 2.4 Driven coordinates The independent coordinate (degree of freedom) of the steering wheel, which acts over the rack and therefore over two of the front wheel rods, is a kinematically driven coordinate so that it operates the racing car in the desired way. There is also a great freedom to set the value of this roll (or any other coordinate) along time, for example in a way in which the vehicle performs the moose test (see Figure 4) or a slalom..6.4 Driven coordinate value Numerical integration Time (s) Figure 4: Driven coordinate values for a moose test. The method yields a system of DAEs, sum of the open-loop equations and the closed-loop constraints, which characterize the mechanical system. It is converted into an ODE system by keeping a subset of independent variables. To follow the evolution of the system on time, it is integrated using a variable order Adams-Bashforth-Moulton or Runge-Kutta algorithm [3]. One of the most interesting features of the model is that it is very close to achieve real time integration, thanks to a partial implementation in C/C++ using Matlab s MEX-functions. As an example to assess this improvement, the time spent to compute 2, calls of the function which evaluates the state vector derivative can be seen in Table 1 with and without C/C++ code. The enhanced code spends 143 times less time on the calculation. 6

7 Evaluation of the state vector derivative (2, calls) Only Matlab Matlab and C/C s 8.19 s Table 1: Matlab and C/C++ computation times. In addition to this, the integration generates completely configurable visual and written information of the model along time in order to fulfil the needs of the designer, as all the variables can be recorded and shown in plots. 3 RESULTS Now that the model works properly, it is time to check its realism with a certain identification of parameters and to carry out virtual experiments to see what the model can give. 3.1 Validation The validity of the model mainly depends on the validity of the method and the validity of the physical model. During the development, the global energy balance has been constantly checked, which guarantees that the method is consistent and that the defined mechanical system is being solved correctly. 3.5 x Energy [J] Time [s] Figure 5: Kinetic (red), gravitational potential (blue), dissipative (cyan), driven coordinate (magenta) and total (black) energies against time. Energy is calculated as a post-process so that the numerical integration is not affected. It is a sum of the kinetic and potential energies, the energy of the non-conservative forces and the one related with the driven coordinate. This sum shall remain constant during all the integration, and is a good way to assess the consistency of the process. Figure 5 shows the values of all this energies during a simulation of 6 seconds. However, a good identification of the physical parameters is essential to rely on the specific results returned by the method. During the development stage, many of the model parameters (inertias, masses, stiffness values, etc.) have been set up with estimated values measured on similar parts and vehicles. Once the elements have been designed and mechanized, the parameters have been adjusted with the real values. 7

8 3.2 Anti-roll virtual test As a specific use of the racing car model, which can help to understand its potentiality, this section shows a virtual test carried out with two racing cars driving in parallel. The objective was to evaluate the benefits of a specific suspension setup, that is to say, of certain geometry and parameter values, when performing a manoeuvre. Provided that the virtual test is much quicker and cheaper than the real test, besides its higher safety and versatility, this tool is clearly beneficial in a Formula SAE development team, in order to optimize time and material resources. The model may be big, but it is conceptually simple and bodies and joints are added as the user wants. Many other features or aspects can be studied, like for example the roll center, the anti-roll bars stiffness, the damping coefficients, etc. In this case, the only different parameter between the two vehicles was the diameter of the rear anti-roll or stabilizer bar, which, in the red one (see Figure 6), was 18 mm, while in the black one was 1 mm, which meant the anti-roll bar of the red vehicle was much more stiff. Consequently, its rear load transfer was also bigger than the one of the black vehicle. The tyre normal forces have been drawn vertically in blue. (a) (b) X (m) Figure 6: Perspective (a) and side (b) view of the manoeuvre with two vehicles in parallel (end of the simulation). The result was that the red vehicle had a stronger oversteer behaviour, as it could be expected, because for the same steering roll it covered a bigger distance in the slalom manoeuvre both of the vehicles undertook, arriving behind the black vehicle to the finishing line (see Figure 6). Therefore its yaw angles during the route where bigger. In order to assess the differences in the dynamic behaviour of both vehicles, some variables have been measured during the manoeuvre, and they are presented in the following figures. The front and rear roll angles are measured separately because, as it has been said, the model considers the torsional behaviour of the chassis and consequently the front and rear parts of the chassis have different roll angles. 8

9 6 4 soft racing car 8 6 stiff racing car 4 Load Transfer (N) Rear Front Roll Angle (º) Roll Angle (º) -6 Figure 7: Load transfer against roll angles. Increasing the stabilizer bar stiffness, as it can be seen in Figure 7, makes the race car have a higher rear roll stiffness, which is one of the desired effects. Rear Front 6 4 soft racing car 8 6 stiff racing car 4 Load Transfer (N) Front Rear Time (s) Time (s) Figure 8: Load transfers against time. Figure 8 shows how the stiff car load transfer is much more centred on the rear part of the chassis, while in the case of the soft car the load transfer is almost equally distributed. Angles (º) soft racing car Torsion Front Roll Rear Roll Time (s) Figure 9: Torsion angle and roll angles against time. stiff racing car Front Rear Torsion Front Roll Rear Roll Time (s) 9

10 With respect to the angles, Figure 9 shows that the roll angles are much smaller in the stiff racing car, which reaffirms the aim of the anti-roll bar, and that the torsion angle between the front and the rear parts of the chassis increases, which means that the chassis undertakes an extra torsion load. 3.3 Comparison with commercial software During the development of the model, a parallel model has been created with Adams. It has been useful to prove the numerical efficiency of the presented model and make several comparisons between them. The geometry of the Adams model can be seen in Figure 1. Figure 1: Adams multibody model of the FSAE racing car. With respect to the computation times of both models, the presented one is achieving real time computation velocities, while the Adams model is generally far from those times, specially when the manoeuvres are complex. In the following table the approximate computation times of several manoeuvres can be seen. Real time Adams time Presented model time Braking 1 s 13 s 8 s Acceleration 1 s 3 s 9 s Circular manoeuvre with constant radius 2 s 5 s 23 s Table 2: Adams and presented model approximate computation times. 1

11 4 CONCLUSIONS This multibody model has proven to be very useful to analyse the dynamic behaviour of the vehicle in the design and development stages, and can be a key complement to define loads in the finite element method analysis. With this model, the developers can run virtual tests to evaluate changes on the suspension and to have a global idea of the racing car dynamic behaviour without investing additional material or economical resources. The specific semi-recursive Matlab implementation has additional advantages against other commercial software based models. It is extremely configurable and adaptable to the user s necessities because there is access to all model variables in every moment of the simulation. Moreover, this simple model, as a programmed model, can be integrated in other simulation frameworks like the ones making finite element method analysis. The model, thanks to its partial implementation in C/C++, achieves real time in its simulations, and is as efficient as other commercial software models. REFERENCES [1] 29 Formula SAE rules, SAE International, 29. [2] J. García de Jalón, E. Álvarez, F. A. de Ribera et al. A Fast and simple semi-recursive formulation for multi-rigid-body systems. Advances in Computational Multibody Systems, Springer, 25. [3] L. F. Shampine and M. K. Gordon. Computer solution of ordinary differential equations: the initial value problem. W. H. Freeman, San Francisco, [4] J. García de Jalón and E. Bayo. Kinematic and dynamic simulation of multi-body systems the real-time challenge. Springer, New York, [5] W. F. Milliken and Douglas L. Milliken. Race car vehicle dynamics. Society of Automotive Engineers, [6] H. B. Pacejka. The tyre as a vehicle component. XXVI Fisita Congress, Prague,