# VEHICLE SPEED ESTIMATION USING KINEMATICS

Project Brief:

Select any passenger or F1 race car, collect data from the web information, Identify all the parameters to work out the speed of 0-100 KM/Hr in 7 seconds and 4 seconds for Passenger and F1 race car respectively.

You are designing a car to attain speed of 100 KM/Hr. Calculate the possible variable, write a report as outlined below.

Apply principles of Kinematics and Kinetic of Particle and rigid body motion, list and calculate possible variable.

Develop a report as mentioned below.

Outline of Final Report

The final written report should include the following:

1. A title page containing project and names of all group members.
3. An Abstract

A single paragraph that describes the overall project and the results.

1. An Introduction

Describe the overall problem and solving method.

Include an outline of the rest of the report.

1. A section that describes the mechanical aspects of the project.

Include figures that illustrate the project.

Write appropriate mechanics equations used in your project. Calculate the Velocity, Acceleration, Work, Power, Energy, Impulse and Momentum.

1. A Conclusion
2. List of References
3. Appendices that contain all C code and any appropriate data sheets.

A)    Submit hard copy to the reception at SoE (TAFE) Level 5 with cover sheet

On or before 30th May 2012

b)  Submit soft copy on black board using the link.

Note:

I have uploaded reference material on black board; keep in mind tis project carries only 20% of your assessments do not spend too much time. You will be assessed based on the year2 Mechanics of Machines requirement.

SOLUTION

# An Abstract

The formula one (F1) race is the most competitive and highly prestigious of all the races on the planet. These are high performance cars with a single seated and open cockpit. The tires are large and open, which make very high grip over the tracks. The front and rear of the F1 cars are significantly different from the passenger cars, as they are designed to produce down thrust for the car which is required to prevent them from skidding. The engines are placed at the back of the cars. The drivers of such car races are highly skilled which makes the possible use of aerodynamics, engine power, the grip of the tires and above all speed to keep him into the race. The Federation Internationale de I’ Automobile or FIA (France), the regulating body for these races makes it sure that the electronic help in these races are avoided and the driver makes the best possible use of his skills.

# Introduction

## Problem Statement

The formula one, cars move at the highest possible speeds to remain in the race. These speeds cannot be achieved by the normal cars. They have powerful engines to accomplish this task. They move in the predesigned tracks to test the skills and capabilities of the driver.  The tires rotate at pretty high number of rotations which are not possible to produce in any other practical conditions. (Jerry,2010)The brakes working over the tires are designed to withstand the high temperatures. The body of these cars is designed to minimize the air drag and keep the car stuck to the ground. The cars use every possible mechanical optimizations to win the race, like the brakes are not only used to produce a sharp turn or reduce the speed of the cars but also they store the energy from the change of momentum to provide a future boost to the speeding car.

## Outline

The study here focuses on the kinematics and the kinetics governing the motion of these cars. We would consider these bodies as point masses and rigid bodies to evaluate the various parameters of motions these cars. The kinematics replaces the real objects, with an equivalent point mass at the center of gravity. The behavior of the whole object can be then predicted as the motion of the center of the gravity. Such an analysis yields results which are suitable for doing a high level study of a system. Then, there are the rotational motion of the tires (other than tires, the can be the rotational motion of the parts of the engine, which are not in the scope of the study) which have a different impact from the translational motion of the same. We consider take a look at the various physical parameters like the velocity, acceleration, momentum, work, energy and power. There plays a lot of dynamics of the tires and the power of the engine and can be best understood only with the computational technologies.

# Mechanical Aspects

As discussed above the car would be represented by an equivalent point mass located at the center of gravity (COG) of the body, in order to let us do a Kinematic study. For the same purpose, we have presented a diagram in figure 1 to clearly imagine the scenario.

Figure 1: Shows the simplified view of different parts of a F1 race car.

We would go through the various mechanical aspects of the race car to let us make our divide the whole task of analysis.( http://www.formula1.com/inside_f1/understanding_the_sport/5294.html,2012)

## Kinematics

The kinematics study helps us make the study of the acceleration model of a moving body. This model let us represent the mathematical relationship between acceleration, velocity and the distance traversed by the object. The simplest among the model would be the constant acceleration model. It is true that race cars go through high acceleration and deceleration as and when required, but we can progress to get some results with constant acceleration model, as the cars can be assumed to move with constant acceleration in a period of time. Because mathematically, acceleration is represented as time rate of change of velocity, so we have the relations:

The four mathematical relations are used to get the position, velocity and acceleration of the body in case the other variables are known. Let us describe the names of the variables used in the above relations:

= the velocity of the body

= the acceleration of the body

= initial velocity of the body

= instant of time or the time elapsed

=position of the body

Suppose, we take the instance of the Ferrari 599 GTB Fiarano, which takes a mere 3.7 sec to achieve 100 km/h from an initial speed of 0 km/h. Thus, we have:

= 7.51 m/

(7.51) = 51.41m (This is the distance travelled by the car in such an amount of time)(Jerry,2010)

Given the mass of the car is 1,688 kg; we would calculate the momentum, energy, work and power for this case.

Momentum is given as the product of the mass and the velocity of the object.

, where  represents the momentum, m is the mass of the body, is the velocity as usual.

Initial momentum of the system would be zero on account of the zero initial speed.

Hence, final momentum =1.6 kg-km/h=46889 kg-m/s

Change in the momentum is the impulse applied on the system thus impulse

==12673 N

Energy in this case is the kinetic energy of the system, given as:

== 647592.76 J

Where, the variables have their meanings in usual sense.

We know that the work done is the change the kinetic energy. Here, the kinetic energy change from zero to 647592.76 J so, the work done =647592.76 J.

Now, power is the work done per unit time, so we have power as:

Power=== 175025 watt.

## Rigid Body analysis

The tires of the race cars, similar to the other cars and the vehicle are in the state of rotation. But, the only difference being, their high speed of rotation. Similar to the kinematics, we have analogous equations for the rotational motion:

Now, we explain the variables as under:

= the angular velocity of the body mostly expressed in rpms.

= the initial angular momentum of the rigid body.

= the change in the angular position of the body

the angular acceleration of the rigid body.

Another quantity of interest which is analogous to the force equivalent of Kinematics is Torque.

Torque= , where I is the moment of inertia and  is the angular acceleration.(Snare,2011)

Now, for the race car in the previous instance we have =5600 rpm in 3.7 sec. Hence the angular acceleration is given as:

= 33600

Now if the mass of the tire id give then the moment of inertia (I) of each tire can be calculated. Hence, the torque on the wheel can be easily calculated using the above equations.

By now, we have gone through the Kinematic and the rigid body analysis of the race cars has been done. We would now take the other mechanical aspects of the race cars.

## Cornering

The cornering is the vital to the concept of racing. The race on the straight tracks is determined by the power of the engine and the brakes but the small area of the corners gives advantage to the skilled driver. Around the corner the two concepts of Over-steer and under-steer come into picture. The two simply denote the fact that which one of the ends of the car runs out of grip first. The under-steer is the situation in which the front end of the car runs out of grip first but in case of over-steer the rear part of the car runs out of the grip first

The under-steer id the safe and stable one, the grip of the car gets restored once the driver reduces the speed. This is not preferred in formula one race as reducing the speed may result in a loss. The over-steer is unstable and may result in a skid if the driver does not make use of a skillful steering. The chassis of the race car supports the over-steer round the corner. We would analyze the cornering using the Kinematics.

In the figure below, we show a point mass (imagine the center of gravity of a race car), moving from the points A to B round a corner of circular path with approximate radius  .Even if the speed of the car is constant, there is change in the velocity of the body, due to change in the velocity. This can be easily understood from the figure as the tangential velocities shown are in two different directions. Thus, the acceleration is given as:

Here,   is the final velocity while  is the initial velocity. Similarly,  and  are the final and the initial time respectively. The direction of the acceleration is towards the center of the circle. This acceleration is known as the centripetal acceleration which has the tendency to pull the car towards its center.

Figure 2: The centripetal acceleration

## Aerodynamics

The chassis design of the formula one race car is much similar to the jet fighter. It is designed such with two major concerns: to improve the down force in order to help the car tires push onto the tracks and to reduce the air drag which reduces the speed of the cars.

The wings of the race cars play the same role as the wings in an aircraft. The properly designed wings of the cars let the difference in the air contours passing under the wings, thus creating a difference in the air pressure under the wings. This can be easily followed by the Bernoulli’s theorem. The Bernoulli’s theorem is given as:

Where, the variables are:

= pressure

=the density of fluid

=velocity of the fluid

g= the acceleration due to gravity

h= height of the fluid above the point of reference.

Now, since the average track in the formula one race is mostly level so we remove the third term of pressure due to height. (http://theory.uwinnipeg.ca/mod_tech/node30.html ,2012)Hence for the formula one race car has the two terms only, the first term representing the static pressure and the second term representing the dynamic pressure. Thus the static pressure decreases with speed while the dynamic pressure increases.

Figure 3: The air lift by Bernoulli’s principle.

Thus a low pressure is created by the faster moving air on the wings. Thus, the wings of the aircraft tend to move upwards and hence an upward lift is felt by them. The wings of the formula one race cars are inverted, so there is a downward pressure on the wings. This pressure is useful to prevent the race cars from skidding and keep them stuck to the ground. The streamline body of the race cars is also intended to make them feel less slowed down by the air.

The drag force is given by:

The Drag force is dependent on the density of the fluid (air in this case),  is the density of the fluid,  id the drag coefficient and A is the reference area. It should be noted that the drag force increases with the increase in the velocity of the body.

# Conclusion

We discussed the few Kinematics and the kinetics features of the race cars in formula one races. The equations used here are the basic to the course of mechanics for the high level study of the race cars. We can use these equations without any major changes. The Kinematics equations shown fit well in case even when we have a finite initial velocity, that is when we calculate the parameters while the car is at some point of time in the race. The rigid body calculations can be done for any other given race car.

# References

1. Aerodynamics , viewed on 29th may 2012,<http://www.formula1.com/inside_f1/understanding_the_sport/5294.html>

2. Cornering ,viewed on 29th may 2012, <http://www.formula1.com/inside_f1/understanding_the_sport/5294.html>

3. Matthew C. Snare 2011, Dynamics model for predicting maximum and typical acceleration rates of passenger vehicles

4. Raymond and Jerry 2010, College Physics

5. Kinetic Energy ,viewed on 29th may 2012 <http://theory.uwinnipeg.ca/mod_tech/node30.html>

6. The basics of flight, viewed on 29th may 2012, < http://library.thinkquest.org/3142/lift.htm >

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