Straight-Line dynamics

MAY 17th, 2019

Introduction       top

Straight-line behavior is very important for a sports vehicle. On every track, the curves are connected by straight sections. The faster these sections are driven, the shorter the lap time. It is necessary to optimize the geometry in order to maximize acceleration and deceleration: it is useless to have a powerful engine and brakes and the best tires on the market, if the geometry of the vehicle does not allow to exploit this potential.

In the study of acceleration/deceleration of a vehicle, many aspects come into play, such as tire slippage (I am referring to the slippage necessary to obtain maximum grip, not to the loss of grip), suspension behavior, torque between engine and chassis and/or chain tension, chassis stiffness (which can trigger vibrations), etc.

It is much more useful to begin the study by simplifying the model as much as possible, to understand the basic concepts. Details can be added later.

Simplified model       top

We remove everything that can distract: the tires do not slip and their coefficient of friction is constant, there is no suspension, there are no rotating masses, there are no aerodynamic forces, the chassis and every other component is a rigid body. Furthermore, we study the behavior in the 2D plane. All the forces acting in this vehicle are shown in Figure 1.

Figure 1

The forces acting on this vehicle are:

• m*g, representing the weight force;
• Ra and Rp are the front and rear vertical reactions;
• Fa and Fp represent the longitudinal forces applied to the wheels (thus, they will be the traction and braking forces).
Geometry, on the other hand, is represented by:
• h, the height of the center of gravity;
• p, wheelbase;
• b, the longitudinal distance of the center of gravity from the rear wheel.
The ratio of b to p is the percentage of weight acting on the front axle.

Being in a plane, the vehicle has three degrees of freedom: a longitudinal displacement (the forward motion), a vertical displacement (may be the suspension motion, or the slope of the road), and a rotation around its center of gravity.

We can write a system of three equations, representing the balance of forces along each of the three degrees of freedom:

This system can be solved in some special cases:

Static       top

The vehicle is stationary, there are no longitudinal forces. Ground reactions can be calculated:

These values can be measured with two scales, one on each axis, and using one of the two formulas above, the value of b can be determined.

For example, suppose we have a motorcycle that weighs 200 kg, we put a scale under the front wheel and see that it weighs 90 kg. The wheelbase of this motorcycle is 1450 mm. Then:

Acceleration: general       top

If the values of Fa and Fp are known, we can solve the equation:

The forces Ra and Rp have the same equation as in the static case, with the addition of a term: (Fa + Fp)*h/p. This term is called Load Transfer, and represents the mass that, under the effect of a driving force, is transferred from one axis to another. In fact, we notice that it is negative for Ra and positive for Rp: in case Fa + Fp is positive (i.e. under acceleration), the front axle gets lighter and the rear axle gets heavier. Vice versa in the case of braking.

Analyzing the influence of the geometry in the load transfer, we notice that this will be greater the higher the center of gravity and/or the shorter the wheelbase.

Using these equations, and knowing the geometry of the vehicle, I can calculate the loads acting on the wheels and the traction forces. It is much more interesting, however, to calculate the maximum transmissible forces given the geometry, or again, to optimize the geometry so as to obtain the greatest possible acceleration.

Acceleration: rear wheel drive       top

Neglecting friction, there are no forces acting on the front wheel (Fa = 0). The two limiting cases are:

1. Wheelie

If the front wheel lifts, Ra = 0. I find:

If the vehicle has an excessive tendency to wheelie, in order to increase the force transmitted (and therefore acceleration) I must advance and/or lower the center of gravity.

2. Loss of traction

The maximum force that can be transmitted by the rear wheel will be equal to the load on the wheel multiplied by the coefficient of friction μ. Solving the system I find the following equations:

If the vehicle has an excessive tendency to lose grip, I must shift the weight to the rear axle or increase the height of the center of gravity.

Geometry optimization

The changes to be made to the geometry to reduce the tendency to wheelie are the exact opposite of those to reduce the tendency to lose grip. There is then a meeting point, where wheelie and loss of grip occur simultaneously. In this case, the acceleration is the maximum acceleration. Equalizing the Fp forces in the two cases I find the optimal geometry:

Acceleration: front wheel drive       top

Similar to the previous case, if we neglect friction there are no longitudinal forces on the rear wheel (Fp = 0). The limiting acceleration is given by the loss of grip. The maximum force that can be transmitted will be equal to the load on the front wheel multiplied by the coefficient of friction μ. I find the equations:

If the slip is excessive, I can move more weight to the front or lower the center of gravity. In this case there is no optimal geometry, the more the center of gravity is advanced and lowered, the less the slip will be.

Acceleration: all wheel drive       top

There is both force Fa and force Fp, and they will be equal to the load on the respective wheels, multiplied by the coefficient of friction μ. Solving the system I find:

The interesting thing is that the acceleration depends only on the coefficient of friction, so it will always be the maximum allowed by the conditions of the tires and the ground.

Maximum acceleration allowed by the engine       top

The forces calculated so far are the ideal ones, assuming that the engine is able to make the chassis reach those limits. Consider now the maximum force that the engine can transmit. The power is considered constant, as if you had a gearbox with infinite ratios able to turn the engine always at its maximum power. The force transmitted to the ground is:

The force decreases with the speed with a hyperbolic trend. There is then a velocity, which depends on the geometry of the vehicle, above which the limit is given by the power of the engine. To find this velocity, I must equal the equation for the maximum force that the engine can transmit with one of the limit equations seen above.

Let's take for example a front-wheel drive car, weighing 1000 kg, having a power of 100 hp, with 65% of the weight on the front axle, a wheelbase of 2500 mm and with the center of gravity at 600 mm from the ground. Suppose that the coefficient of friction is μ = 1:

This car, above 51.5 km/h, will no longer have the tendency to lose grip, because the limit will be given by the power of the engine.

Gear ratio considerations

An ideal vehicle is always able to express maximum power at any speed. As we have just seen, this translates into the equation of a hyperbola.

In a real vehicle, there is always a gearbox having a finite number of ratios (except for CVT gearboxes, which are very close to the ideal gearbox).

The graph above represents force as a function of speed for a six-speed transmission. The dashed line is the constant power hyperbola, which is the force that would be produced if the vehicle were equipped with an infinite ratio transmission.

Even assuming that the gear change occurs instantaneously, the average power between 0 and the maximum speed (corresponding to the area under the curve divided by the speed range) is still less than the maximum power, and the "lost" power corresponds to the dashed area in the graph. This will be discussed later.

Maximum braking       top

To find the limit equations, we proceed in exactly the same way as seen so far for acceleration, but now the forces Fa and Fp will be negative.

We can divide braking into 3 cases. Since the considerations are the same as those already made for acceleration, only the final equations will be given.

Front brake only

Rear brake only

Both brakes

Geometry optimization: braking

As in acceleration geometry optimization, the limit is reached when rear wheel lift and front wheel slip occur simultaneously:

Depending on the coefficient of friction, the optimal positions of the center of gravity can be found.

Geometry optimization: acceleration + braking

Unless it is an acceleration race, it is good that the vehicle is designed to minimize the space for both acceleration and braking. Equating the equation of optimal geometry in acceleration with that in braking, I find:

The formula tells me that the weight distribution must be 50/50. Then, to find the optimal height of the center of gravity, I have to use the value of b thus found and reenter it into one of the two formulas for optimal geometry (under acceleration or braking), which at this point will give me the same result.

The higher the coefficient of friction, the lower the center of gravity must be placed.

Complete model       top

Introduction       top

In order to calculate the actual acceleration and braking of a vehicle, many more variables must be considered:

• Aerodynamic effects, which include:
• Resistance, the force that opposes motion;
• Portance, the force that tends to lift or push the vehicle to the ground;
• Aerodynamic momentum, the force that tends to cause the vehicle to yaw or pitch.
All three of these effects have an equation like:

Where ρ is the air density, A the frontal area, v the vehicle speed and C a coefficient, which can be CD (drag, "drag"), CL (lift, "lift") or CM (momentum).
• The real behavior of tires: usually, when we talk about grip, we talk about coefficient of friction, but tires have a strongly nonlinear behavior. In order to generate grip they must slip, and the coefficient of friction varies as the slip varies. This is a vast topic, to which I will devote an article in the future.
• The actual torque characteristic of the engine.
• Gearbox ratios.
• The equivalent mass: in the previous simplified analysis, only the total mass of the vehicle, the one that can be measured with a scale, was considered for the calculation of acceleration. In reality, the engine must not only translate the mass of the vehicle plus that of the driver, it must also rotate all the rotating parts: wheels, gear shafts and crankshafts in primis. The inertia of these parts can be transformed into an equivalent mass to be added to the total mass, as if the whole vehicle were heavier. Let's consider, for example, a cylinder that is rotating and translating (it could be a wheel), and let's calculate its total kinetic energy:

The ratio of the rotation speed to the forward speed of this cylinder will be equal to the gear ratio:

If we replace this equation with the starting equation, we find:

Thus we have found the value of the equivalent mass, which must be added to the total mass of the vehicle. Since it is a function of the gear ratio squared, its value will be greater the shorter the ratio, so the greatest influence will be in the lower gears.

The resulting equation is nonlinear, and therefore difficult (or impossible) to solve analytically. Thus, it was placed within a computational code that can be run with GNU Octave or Matlab software.

In this code, the length of the straight line, the initial speed and the final speed are chosen. The vehicle in question, starting from the desired initial speed, will be made to accelerate and then brake in order to reach the end of the straight at the final speed set. The travel time will be calculated as well as other values that may be of interest, such as the percentage of opening of the throttle, the load on the wheels and so on. In the examples presented here, we have considered in all cases a 400 m long straight run starting from a speed of 60 km/h, and then braking until reaching again the speed of 60 km/h at the end of the straight. You can find the program to download in the download section.

Optimal center of gravity position       top

The effect of variation in center of gravity for four vehicle types was analyzed:

• Bike 125cc;
• Bike 600cc;
• Front-wheel drive car;
• Rear-wheel drive car

The two cars considered have exactly the same characteristics, with the only exception in the drive. In the case of the motorcycles, the test was done either using only the front brake or using both brakes. All tests were done considering a coefficient of friction μ = 1.

First of all, let's analyze the graphs of the two bikes considering the use of the front brake only. In the 600cc motorcycle, it is clear that the shortest time is obtained when the center of gravity is in a very restricted area of the position calculated considering the simplified model. This confirms what we have already seen.

On the 125cc motorcycle, however, there is a "band" of excellent center of gravity positions. What these positions have in common is the relationship between the height of the center of gravity and its distance from the front wheel. That is, the angle of load transfer under braking remains constant. This is due to the low power of the engine: not having problems of wheelie or loss of grip, the position of the center of gravity is almost irrelevant during acceleration, but it becomes important during braking.

Using both brakes expands the area of optimal center of gravity positions. This is because, in the simulation carried out, the distribution of braking is always the optimal one, which allows to exploit all the grip available. Of course if the center of gravity is too high or too advanced, the limit is always represented by the lifting of the rear wheel, and this is why the area at the top right in the two graphs does not change.

In the case of cars, however, the graphs don't allow for much consideration.

Optimal gear ratio       top

Let's consider again the graph seen above:

Maximum acceleration occurs when the engine is running at constant power. In a real gearbox, with a finite number of ratios, during each gear change the engine drops in rpm and takes some time before returning to maximum power. This implies that the average power, during acceleration, is less than the actual maximum power of the engine.

To increase the acceleration, we must try to reduce the dashed area in the graph as much as possible. To do this, we can:

• Increase the overdrive limit of the engine: the more revs an engine has, the closer we will be to maximum power at the time of the gear change. This is why, to maximize acceleration, the gear change should not be done at maximum power, but in overdrive.
• Increase the number of ratios: the more ratios we have, the smaller the drop in speed between one ratio and the next. However, this also means more effort for the driver, as well as the fact that gear changes take time (unless you use a seamless gearbox), so don't overdo it.
• Find a mathematical progression between the ratios that minimizes the area of loss.

The last point is the most interesting, and also the only one you can usually work on in the design (or tuning) phase.

Two paths were followed to find the optimal ratio, both involving the use of a genetic optimization algorithm:

• Finding the shortest time under acceleration, using the same program used to find the optimal position of the center of gravity;
• Search for the largest area in the rpm-velocity graph.

Since both approaches led to the same result, only the second one will be described here. However, in the download section there are also all the results obtained with the first one.

A genetic optimization algorithm was chosen because, in addition to being simple to implement in a computational code, it can optimize a large number of variables without increasing computational complexity. The rpm-velocity graph is as follows:

A normalized graph was considered, i.e., having a maximum rpm equal to 1, and a maximum speed also equal to 1. We want to find the speeds v1, v2, ......, vn (with n = number of gear ratios) that maximize the dashed area. The larger the area, the higher the average RPM, and consequently the average power during acceleration.

Having found these velocities, we looked to see if they belonged to any known mathematical series. An almost perfect match was found with the generalized harmonic series, having exponent 1.6:

Where Xn is the gear ratio for each gear. Knowing the first and last ratios (which may be imposed by the track, for example) all others can be calculated using this formula.

Bibliography       top

1. Motorcycle Dynamics, Vittore Cossalter, Grafiche Gemma Borgoricco, 2008
2. Effetto Moto - Dinamica e Tecnica della Motocicletta, Gaetano Cocco, Giorgio Nada Editore, 2008
3. Appendici - Motori ad Alta Potenza Specifica, Giacomo Augusto Pignone, Ugo Romolo Vercelli, Giorgio Nada Editore, 1995
4. On the Braking Behaviour of Motorcycles, Vittore Cossalter, Roberto Lot, Fabiano Maggio, 2004