On Thursday 19th November, I will be delivering part of a presentation by Southampton University Formula Student team. We are holding an event for IMechE members, university students, and staff, describing the processes undertaken to design, build, and race a Formula Student car. More details about the event can be found here.
My section of the talk will cover the design and modelling work required to produce a working suspension system. Putting together the talk has been a great opportunity to focus some of my thoughts from the last few weeks. This article covers the design process in a little more detail than I will have time to go through in the talk.
The Challenge
The first thing to consider when designing any part, component, system, procedure, or assembly, is the target outcome. This is the driver for all other design decisions. When we build a Formula Student car, the target outcome is something very simple: winning.
This target outcome is then broken down into smaller targets, until we reach a series of aims which form the design brief of the suspension system. That process normally happens automatically, through a series of highly developed engineering thought processes, but it is important to take note of nonetheless. For us, the first stage of breaking down the ‘winning’ criterion is to split out three separate design philosophies – safety, reliability, and time.
First of all, the car must be safe, to protect any driver and any marshal or spectator if things go wrong. The car must pass scrutineering in order to race, and in order to pass scrutineering it needs to be safe. Motorsport is dangerous, but all reasonable precautions are taken to minimise the risk to spectators and competitors.
The car also needs to finish the race. Not to be overlooked, this criterion requires that the car can last, unassisted by any mechanic or marshal, for about 30 minutes of flat-out driving. The magnitude of that challenge will become clear later, but suffice to say it puts significant constraints on the design process.
Any part that is put on the car is there for at least one of these three philosophies, often combining all three of them in order to come up with an optimum design. The third challenge, time, is by far the greatest avenue of development and consumer of design hours.
Time
To win any dynamic event means finishing it in the shortest possible time, and, as was realised long ago by a foolish hare, that means maintaining the highest possible average speed. We can investigate the course that we will race on to put this in context, and to determine what maintaining a high average speed requires from the car.
The Formula Student endurance and sprint track is an autocross circuit, with slaloms, chicanes, and hairpins interspersed between short straights. The net result is that we will spend a lot of time going around corners, and a lot of time accelerating away from corners and braking into corners. To win, we need to maintain the maximum acceleration in the appropriate direction – something Peter Wright of Lotus F1 famously stated in the early days of F1.
We design a car which is very light, very short, very wide, and as powerful as the regulations and budgets allow, and the net result is a car which is optimised for going around an autocross track quickly. A low weight means that the available forces are put to maximum effect generating accelerations. A low centre of gravity, wide track, and short wheelbase allows the car to generate the maximum possible yaw rates and lateral accelerations, getting around corners quickly, and the power-weight ratio is better than anything which isn’t a supercar.
Performance
The time constraint on our vehicle has now become a performance target – our cornering and acceleration rates characterise the car’s performance, and the car’s performance needs to be increased as much as possible.
The current world record for an electric car’s 0-100kph time is 1.779s, and it is held by
Stuttgart’s Formula Student team. The acceleration during that run is an average of over 1.5 times the force of gravity, which means the car will out-accelerate anything on two wheels or four wheels that isn’t designed for a drag strip. If lined up side-by-side, it will beat a Saturn V away from the lights, although if you’re planning on going to space or racing your Formula Student car again, this experiment is generally ill-advised.
The cornering numbers are even more impressive. Formula Student cars, thanks to the short wheelbase and low weight, achieve yaw rates of over 70 degrees per second. This is faster than anything on four wheels, including go-karts, anything on two wheels, and even faster than the roll rate of a Eurofighter Typhoon.
Design
When it comes to the specification and design of the suspension system, all of the comparisons and superlatives go away. We instead turn to the tyres, and a graph known as a traction circle, or a g-g diagram. This graph shows the maximum possible force available from the tyres in any direction, normalised by the vertical load. Our aim for the suspension is two-fold – to stay on the very edge of this circle for as much of the race as possible, and to maximise the size of the circle by keeping tyre parameters in their optimum conditions.
The first of these points is the reason we have suspension on the vehicle at all. To help the drivers stay on the limits of friction at all times requires a consistent and predictable car, and requires the car to not be unsettled by bumps, jolts, and accelerations from the vehicle itself. The suspension will compress and rebound in order to keep the chassis as level as possible, absorbing bumps and jolts and dissipating the energy in dampers so that the tyre loads are smooth and continuous, maintaining traction with the road surface for the maximum possible time.
The second of these points is massively complicated by the fact we have suspension; the fact that the wheels move relative to the chassis means that the tyre parameters – camber, toe, contact patch area, slip angle, slip ratio, and vertical load – are all continuously changing and highly variable.
Most Formula Student cars use a double-wishbone suspension setup and coil-over-dampers. This setup is a proven, effective way of transferring loads and providing adjustability in the suspension, while maintaining independent wheel travel on all four corners to absorb bumps. The layout push/pull rod system and spring/damper arrangement is then chosen to make the handling consistent and predictable, so that the driver can drive to the maximum of the circle at all times.
To try and optimise the second point requires delving deeper into the tyre parameters. Tyres will provide different amounts of longitudinal and lateral force depending on their alignment with the road (slip angle), angle to the vertical (camber), pressure, and vertical force. To a greater or lesser extent, these are the parameters which it is our task as suspension designers to keep in their optimal performance window.
To do this, a tyre model is developed. For us, this means taking raw data from rig testing and analysing it to create a model of how the tyre behaves under different conditions. In our case, we used tyre data provided and developed MATLAB scripts to extract various parameters and trends. This involves selecting appropriate data points and plotting the relationship between one parameter and another, by making educated guesses about what could and could not affect each parameter.
The trends we are interested in are the ones which show how normalised lateral and longitudinal force vary with the above parameters, since these normalised forces are what make up the traction circle. To maximise the circle size, we need to maximise these parameters. Once the trends have been characterised, we can find the parameter values at which the maximum points occur.
Thus, the challenge has been further reduced into maintaining as close to optimum parameter values as possible. To do this, we use a combination of further models. We model the vehicle in Adams, to see how the vehicle accelerations are likely to change during particular manoeuvers on track, and we use a bespoke model, developed in-house and detailed in other posts, to see what these accelerations translate to in terms of suspension forces and displacements.
There is some crossover between these models, so one can be used to validate the other. This validation is very important because the whole process is built on assumptions which may or may not be valid. Understanding how the model could be wrong is just as important as understanding the results it provides.
Finally, with our ideal parameter values and the variations in them identified, we build a wireframe model of all of the suspension nodes. All of the suspension positions can be defined using solely the parameters we have optimised for the car, and as such once the parameters are defined, the suspension design comes together quite quickly.
To actually keep these parameters as consistent as possible requires modifying swing arm lengths and instant centre locations, and setting geometric roll centre heights and anti-pitch features. These are all numbers which relate loads on the wheels to the behaviour of the chassis. We also set ideal ‘static’ values of the suspension setup, so that when the car is finally built, everything is in place and adjustments can be used for fine-tuning.
Using our model of the suspension, we can build components to specified sizes, design the shape of rockers and uprights in line with the dimensions required for the optimisation process, and specify the stiffness of springs, dampers, and anti-roll bars to deal with the expected loading conditions adequately.
Validation
There is an extensive validation process to go through once this has been completed. In particular, the suspension design will be passed back through our bespoke model to evaluate the loads that will be carried by each component, and these components will be tested using a combination of physical testing and finite element analysis to ensure that they are strong enough.
The model will also go back into Adams to predict vehicle behaviours and ensure that nothing has changed since the design was specified, and it will be tested in lap time simulators so that the parameters which remain adjustable – camber, toe, spring rate and tyre pressure – can be chosen appropriately for each of the different events that we will compete in.
Modelling
The word ‘model’ has been mentioned a lot here, and for good reason. Nothing that we have done to design the suspension has gone anywhere near the physical car – since it has not been built yet. We cannot build the car before it has been designed, but more than this, the design has been a process entirely driven by analysis of data and equations.
The design process has been, in this respect, entirely mathematical, based on principles which are universal. The work we have done could be applied to any Formula Student car, or further, any component in any location in the universe.
Models are the present and the future in engineering, and they are developed out of mathematics which is universal and fundamental. I find the idea that our perception of a Formula Student car can be reduced to pure mathematics very exciting, and it is what drives my curiosity to understand more about the world.
More Information
To find out more about Formula Student, the Southampton Formula Student Team, and the work that is going on behind the scenes in the team on a regular basis, see the links below:
- Formula Student: http://events.imeche.org/formula-student/
- Southampton University Formula Student Team website: http://www.sufst.com/
- Updates on Formula Student from this blog: https://amccormick21.wordpress.com/category/formula-student/
Alex McCormick 