At the very highest level, a vehicle’s performance around any circuit applies Newton’s Second Law, F = ma, coupled with some equations of motion. The description that follows below is basically how vehicle dynamics simulations work.
Starting with the equations of motion, the motion of a vehicle around a circuit is dynamic, where the vehicle travels through three-dimensional space over time. If we break this motion through space into smaller and smaller time intervals, we can start to think about the vehicle’s state for each time interval as having a set of initial and final conditions.
When the time intervals are reasonably small enough, it is possible to approximate the change in the vehicle state from the initial to final conditions as a constant acceleration problem. Using this approximation, we can apply the SUVAT equations of motion from physics to get from the initial vehicle state to the final vehicle state. SUVAT is an acronym where s = displacement, u = initial velocity, v = final velocity, a = acceleration, and t = time.
For now, we will assume we already know the vehicle’s acceleration, so if we also know the initial velocity and time step, we can apply the second SUVAT equation to calculate the vehicle’s displacement during the time step, i.e. s = ut + ½ at2. In addition, we can apply the first SUVAT equation, v = u + at, to calculate the final velocity of the vehicle at the end of the time step. For the following time step, the initial velocity is the final velocity from the previous time step, and we can proceed to evaluate each time step sequentially. However, we cannot do this until we know the vehicle’s acceleration for each of these time steps!
We now need to consider Newton’s Second Law. From the above application of the equations of motion, we can see that the velocity of a vehicle at any point around a circuit is governed by the vehicle’s ability to accelerate; therefore, it is best to think of Newton’s Second Law expressed in terms of acceleration, i.e. a = F/m. We must consider this equation as the acceleration equaling the sum of all forces (total force) divided by the mass. These total forces include those available for propulsion and those forces resisting propulsion.
For example, a block sitting on an incline will have two forces acting on it: a gravitational force and a frictional force. The component of the gravitational force that is parallel to the incline’s surface will pull the block down the incline. Still, the friction force between the block and the incline’s surface will resist this gravitational force component. If the gravitational force component is smaller than the friction force, the block will not move. Only once the gravitational force component is greater than the friction force will the block begin to accelerate down the ramp. So, acceleration cannot happen until the total force – the sum of propulsive forces minus the sum of resistive forces – is great enough. Hang on to this concept as we start applying it to a vehicle.
The idea of breaking forces into directional components applies to the motion of vehicles. We cannot work with the SUVAT equations or Newton’s Second Law until we break the forces acting on a vehicle into two components. We need to break the equation a = F/m into its respective longitudinal and lateral components, yielding two equations: longitudinal acceleration ax = Fx/m, and lateral acceleration ay = Fy/m. Considering the concept of total forces, the longitudinal acceleration equals the total vector sum of all longitudinal forces divided by the vehicle mass. The lateral acceleration equals the total vector sum of all lateral forces divided by the vehicle mass.
Longitudinal Acceleration
The longitudinal acceleration, ax, is the acceleration that we feed into the SUVAT equations above and is the only acceleration we need to consider when looking at a straight-line acceleration problem. We know that longitudinal acceleration is the sum of longitudinal forces divided by the vehicle mass. The longitudinal propulsive force for a vehicle comes from the vehicle’s power unit. A typical internal combustion engine’s output torque is fed through a drivetrain (clutch, drive shaft, gearbox, differential, axles, hubs, wheels) to the vehicle’s tyres.
This torque acting through the tyres results in a force parallel to the road that attempts to drive the vehicle forwards. This is not the only force we need to consider, though. Just as there was a frictional force resisting the motion of a block on an incline, several forces resisted the propulsive force from the engine and tyres. These resistive forces include frictional losses from the drivetrain, rolling resistance from the interaction of the tyres with the road, aerodynamic drag, and any applied braking forces.
Lateral Acceleration
The lateral acceleration component, ay, does not directly impact the SUVAT equations. Still, it does indirectly impact them in that the longitudinal acceleration of a vehicle is limited by the total possible combined acceleration, i.e. the vector sum of the lateral and longitudinal acceleration. Before considering combined acceleration or forces, think about a pure cornering situation around a constant radius corner. In this scenario, the vehicle corners at a constant velocity related to the lateral acceleration through the equation ay = v2 / R, where v is the constant velocity around the corner, and R is the corner’s radius.
We are still dealing with lateral acceleration resulting from the total lateral force divided by the vehicle mass, and we still have propulsive and resistive forces in the lateral direction. The propulsive force, or the force that is driving or pushing the vehicle towards the instantaneous centre of curvature, comes from the ability of the vehicle’s tyres to generate a lateral force between the tyre and the road. This frictional force increases as the vertical load on the tyres increases. The resistive force comes from the inertia of the vehicle. This inertial force wants to push the vehicle back to travelling straight, pushing it away from the instantaneous centre of curvature.
Like all bodies in motion, the vehicle does not want to turn because it wants to keep travelling along happily in a straight line. When the lateral force from the tyres equals the lateral force from inertia, the vehicle is balanced and can travel around the curved path. If the inertial force exceeds the available tyre force, the vehicle leaves the curved path, which often ends spectacularly poorly for the vehicle’s occupants. When the inertial force is less than the lateral force potential of the tyres, the vehicle can speed up and travel around the corner faster or take a smaller radius line around the corner.
Combined Acceleration
On the topic of combined forces, where you have both lateral and longitudinal vehicle accelerations or lateral and longitudinal tyre forces, we are talking about the ability of a tyre to generate combined force. A tyre is just a big elastic, and an elastic generates force when it is stretched. Unfortunately, an elastic can only stretch so far before it fails. Longitudinal forces stretch the tire parallel to the direction of travel, while lateral forces stretch the tire perpendicular to the direction of travel.
The total stretch, or total force the tyre can generate, is the vector sum of the lateral and longitudinal components. This concept is demonstrated through a tyre’s friction ellipse, where the outer limits of the ellipse define how much combined stretch/force the tyre can handle. When the combined force exceeds this boundary, the tyre either loses grip by snapping back to a less strenuous amount of stretch or fails where the rubber in the contact patch falls apart. The point here is that a tyre can only generate a fraction of the maximum possible longitudinal force for a given amount of lateral force.
Returning to the SUVAT equations, we can now see how lateral force and acceleration impact the available longitudinal force a tyre can generate. This limits the longitudinal acceleration available to calculate each time step’s distance travelled and final velocity.
Summary
So why have we spent this much space discussing the physics of vehicle performance? How is this related to the balance of performance? Well, the balance of performance is simply a physics problem. When attempting to balance vehicles, we are manipulating a vehicle’s ability to generate longitudinal and lateral forces, which determines how the vehicle accelerates longitudinally and laterally. I want to emphasize this point because if we think of BoP as a physics problem, we can begin to have a much better understanding of how changes to vehicle parameters will influence the overall performance of a vehicle. And the better we understand physics, the better we will be at making changes!
