Essay: An investigation of what effect braking force has on the braking time and the energy caused in the process

Abstract
This is an attempt to try and find the relationship between the heat radiated by a disc brake and the time it takes to stop, at difference brake forces.
I did this by using a horizontally rotating disc with a model of a frictional brake, measuring the change in temperature with an infrared thermometer.
I found a relationship with the gradient of 0.03848 ±0.00335 for 1/Time vs brake force and an equal, and decreasing gradient for the change in temperature, meaning that the change in time is equal to the change in temperature (ΔTime≈ΔTemp.)
Introduction
Most vehicles of any sort have brakes. Anything from a simple bicycle to an everyday sedan, to advanced racing cars, use the same principle of disk-brakes. The first comparable model of a disc brake was used by the manufacturing company ‘Lanchester’ in the year 1902 (Lentinello). This was a rudimentary model, using wires to activate the brake pads on a nearly paper thin, copper disc. For the 116 years to follow, braking systems got continuously more advanced. Now disc brakes have evolved to becoming omnipresent throughout almost any vehicle, with varying degrees of quality and performance. Yet, there is still a large amount of aspects that can be investigated to improve the safety and reliability of these brakes.
The most important factor in making this concept more secure is to reduce braking time, the time it takes for a brake to come to stand-still. A low braking time will cause the vehicle to come to a halt faster. This can not only decrease the risk of a crash in an emergency but also increase the handling and efficiency of the vehicle.
The basic principle of a disk brake consists of a disk, with caliper-like brake pads, that are hydraulically pressed against the disk. In order to increase the performance of said disk break, it is evident that the force applied by the brake pads has to be increased. In addition to this, the function of the disc brake depends on an interaction that dissipates the kinetic energy of the spinning brake disk, into thermal-, sound- and, in extreme cases, light-energy. This energy is then either conducted or radiated out of the system, meaning that it is not used within the car.
With the use of a concept called regenerative braking, the otherwise lost energy is used to either charge up the car’s battery or used to speed up the turbo, both of which can help the vehicle use energy more effectively and accelerate out of corners faster, while even becoming more environmentally friendly. This technology is also widely spread throughout more everyday technology, like E-bikes and electrical skateboards (“Regenerative Braking”).
For my investigation, I wanted to find how the brake force affected the braking time, as well as the efficiency of the regenerative braking system. This leads to my research question:
Research Question: How does the braking force affect the braking time, and how much energy could be converted in the process?
Exploration
In an attempt to answer the question at hand, I made a model of a disc brake, off of which I could take measurements in a more controlled environment like a laboratory. I set up the model of my disc brake in a vertical orientation. This allowed me to conveniently use the weight of controlled masses as my brake force. In this set up, the brake disc is a rotating plate, spun by a pulley system and an electric motor.
I used this setup as a preliminary experiment, where I included an infrared thermometer to measure the change in temperature. The setup of this is shown in Figure 1.
Figure 1: Diagram to show the friction brake setup of the experiment
Mathematical model
To mathematical illustrate the theory of my investigation, I constructed this mathematical model. This starts off with the formula for kinetic rotational energy, shown below.
E=1/2 lω^2
This can also be stated as the work done:
F_f d=1/2 lω^2
F_f in this case is the friction force. To incorporate the normal reaction force (F_R), the equation has to be rewritten as shown below in equation 3.
F_f= μ_d F_R
Combining these two equations:
μ_d F_R d= 1/2 l〖ω_o〗^2
The next step is now to find the inertia of the rotating disk. The equation of inertia of a solid disk when spinning on its axis of symmetry is shown in equation 5. Here M is the mass of the disk and r is it’s radius
I= 1/2 Mr^2
Substituting this into the previous equation:
μ_d F_R d= 1/4 Mr^2 〖ω_o〗^2
As F_R can also be expressed as weight, and the controlled variable (independent variable) is the Mass, F_R can also be noted as mg, where m is the mass, and g is the acceleration due to gravity.
mg=(Mrω_o)/(2μ_d t)
Therefore, it can be said that mg∝1/t, where the slope is (Mrω_o)/(2μ_d ). mg being equal to the braking force.
Then, through the law of conservation of energy, I can deduce that the difference between my expected value and gathered value is ‘wasted’ energy. Through then measuring the change in temperature of the brake disk, I will investigate how much of this energy is thermal energy.
Finding the coefficient of friction
To calculate the frictional force between the simulated braking pads and the brake disc with equation number 3, the coefficient of friction (μ) has to be calculated. This is done by the deriving the formula from a free-body diagram shown in figure XX
Figure XX: Free-body diagram to derive the coefficient of friction
Assuming, that the frictional force and the acceleration due to gravity are equal, the following equation can be derived.
mg sin⁡〖θ=μmg cos⁡θ 〗
Then, using video analysis, and ruler swift, the exact angle at which the friction is overcome () is measured. for the smallest mass used throughout the experiment (m=0.1000±0.0001kg), =15°.
This means that the coefficient of static friction (μ) is equal to 0.2700 ±0.0001 (μ=tan ), which is greater than or equal to the coefficient of dynamic friction.
Method
In order to collect my data, I have set up a lab with which I can measure both the change in temperature of the braking disc, and simulate different severities breaking strengths. A Picture of this is depicted in figure 3.
For the investigation I connected the electrical motor to a VariVolt power supply. The power supply was set at a constant 8 volts throughout the entire data collection process. To the motor, the brake disk was attached through the use of a pulley system. The motor and the wheel of the model brake disc were connected with a large (and wide) rubber band, as conventional rubber bands would break very quickly, and therefore cause random error throughout the data collection (this was established through a preliminary test of the setup). The motor was then turned on for 60 seconds, to allow the disc to accelerate to a terminal velocity. After exactly one minute, the mass was dropped onto the spinning disc from a height of 0.01 meters. The acceleration due to the drop will affect the system to a negligible amount. Then the time taken until the disc comes to a stand-still was measured. At the same moment, when the mass was dropped onto the model brake disc, the power supply for the motor was stopped, allowing the model to spin as freely as possible, and be affected by as little resistance as possible. In addition to this, the change in the temperature of the spinning disc was recorded with an infrared thermometer a set distance away from the model. The initial temperature was measure off of the spinning disc immediately before the masses were dropped onto it, the final temperature was measured when the disc had just come to a halt. The time taken until the disc comes to a stop, as well as the increase in temperature was then recorded in a Microsoft excel spreadsheet. For each different mass I recorded 5 trails to take the average from, as the process of data collection may well result in random error.
Preliminary research
With the same setup as described in the method, I conducted a preliminary test, in which I found out that due to the increase fatigue on conventional rubber bands, the braking times would decrease with the amount of trials done with the same rubber band. This is most likely due to the fact that the rubber band started slipping on both the model brake disc, as well as the motor. This cause the acceleration of the model brake disc to become much slower, therefore resulting in a lower rotational velocity at the end of the 60 seconds. In addition to this, the preliminary test has helped me with the measurement of the kinetic energy that is dissipated throughout the braking process. Due to Newton’s second law, the law of conservation of energy, the kinetic energy of the spinning disc must be equal to the radiated forms of energy. In the current setup (metal masses directly on metal plate), the increase in thermal energy is very minute. This is because the interaction also creates a lot of sound and very little friction. Deriving from this, the kinetic energy of the brake disc is proportionate to the sound energy and the thermal energy emitted through the reaction, given that all other ways of emitting energy can be ignored. Consequently, a decrease in the sound when ‘applying’ the brakes, results in a more measurable increase in temperature. I tested this, by including a lightweight (as to not dramatically increase the brake force) sheet of card at the bottom of the masses. This resulted in less sound being created, while keeping a comparable amount of friction. Throughout the data collection process the card was kept on the model brake calipers / the masses.
Apparatus setup
Figure 2: Photograph of the entire apparatus setup showing the brake system from in figure 1
The Practical apparatus setup includes a VariVolt power supply, powering the vertical brake-model.
To establish the angular velocity of the spinning disc, the video-tracking tool, logger pro, came in use. A slow-motion video was taken of the spinning disc, with a distinct mark on it. The video slowed the events down to 12.5% of its original speed. Using the video-tracking feature, the video was then used to find the time period at maximum angular velocity at 8 Volts. This was 0.233 seconds. Multiplying this by 0.125 to get the original speed, the time period for 1 full rotation at maximum velocity is 0.03 seconds.
To calculate the angular velocity (ω), the following formula (formula 9) is used.
ω=2π/T
Inserting the time period found with logger pro, ω=209.4 rad s-1
According to primary research, a typical street car has a tire with a total diameter between 58cm, and 70cm, with 64cm, or 0.64m, being the most common. This means that, when traveling at 100kmh-1, the tire spins at 2,604 rpm. In this model, the brake disc rotates at around 2,000 rpm, simulating a similar stress of the brakes as a car without mass.
Predicted data
In order to graph the final formula (mg=(Mrω_o)/(2μ_d t) ), t is represented on the y-axis, Mg is represented on the x-axis, and (Mrω_o)/(2μ_d ) is the slope. A sample calculation of how the hypothesized results are derived is shown below:
M=0.77kg m=0.1kg g=9.81 r=0.05m μ_d≤0.27 ω=209.4rad s^(-1)
(Mrω_0)/(2μ_d mg)=t (0.77×0.05×209.4)/(2×0.27×0.10×9.81)=15.2
Figure XX: Graph of the hypothesized results
The prediction shows, that the brake force has an inversely proportional to the time. The predicted graph has the x-axis as a horizontal asymptote. Having an x-axis intercept would imply, that at a certain brake force, the brake disc would come to an instantaneous stop, which is impossible.
In order to analyze this relationship a linear graph is formed by comparing the brake force to 1/time.
Figure XX: linear graph of the hypothesized results
Raw data
Table XX: Data collection table of the time taken to stand still for every mass
Mass
(± 0.0001 kg) Time it takes to stand still (±0.01 s)
trial 1 trial 2 trial 3 trial 4 trial 5
0.1000 9.54 9.85 8.97 9.64 9.67
0.2000 8.45 7.63 8.70 7.70 7.68
0.3000 5.95 5.97 4.99 5.01 4.98
0.4000 3.97 4.84 4.93 4.75 4.66
0.5000 4.35 3.48 3.68 4.36 3.98
0.6000 3.23 3.38 3.14 3.70 4.01
0.7000 3.17 3.37 2.81 3.01 2.98
0.8000 2.93 2.87 2.95 2.89 2.96
0.9000 2.49 2.52 2.58 2.46 2.49
1.0000 1.99 2.17 2.26 2.31 2.26
Table XX: Data collection table of the temperature before and after the masses were place on the disk
Mass
(±0.0001 kg) Temperature increase (± 0.1°C)
trial 1 trial 2 trial 3 trial 4 trial 5
before after before after before after before after before after
0.1000 20.6 21.3 20.8 21.2 20.7 21.0 20.8 21.5 21.3 21.7
0.2000 20.8 21.0 20.9 21.2 20.8 21.3 21.0 21.4 21.3 21.7
0.3000 21.2 21.5 21.1 21.4 21.4 21.6 21.3 21.6 21.5 21.7
0.4000 21.5 21.7 21.7 21.8 21.8 22.0 21.5 21.6 21.5 21.6
0.5000 22.4 22.9 22.6 22.9 22.7 23.0 22.8 23.2 23.0 23.3
0.6000 21.0 21.4 21.4 21.6 21.5 21.8 21.5 21.7 21.5 21.9
0.7000 21.5 21.6 21.6 21.8 21.8 22.0 21.6 21.7 21.6 21.6
0.8000 22.1 22.2 22.3 22.4 22.3 22.3 22.3 22.4 22.4 22.5
0.9000 21.8 21.9 21.8 21.9 21.8 21.8 21.8 21.9 22.0 22.1
1.0000 22.1 22.1 22.1 22.2 22.1 22.1 22.1 22.2 22.2 22.2
Data processing
1/(mean values) will be used to construct the linear graphs.
In order to better represent the collected data, error bars have to be included into the graphs. These represent the margin of error, helping to form a more accurate conclusion. As there are not an excessive amount of data values, the following formula can be utilized to calculate the uncertainty for each data group. The error for the 1/Time values is found by setting each data point under one, then following the dame procedure.
((Max. Value+uncertainty)-(Min. Value-uncertainty))/2
Ex. M=0.1000 kg ((1/9.85 +0.1)-(1/8.97 -0.1))/2=0.26 → ± 0.3°
Table XX: Data processing table showing the mean time and mean temperature change
Break Force
(± 0.1 N) Mean time to stand still
(±0.01 s) 1/Mean time to stand still
(±0.01 s) Mean temperature change
(± 0.1°C)
Values Uncertainty Values Uncertainty Values Uncertainty
0.981 10 3 0.10 0.00 0.5 0.3
1.962 8 2 0.12 0.01 0.4 0.2
2.943 5.4 0.7 0.19 0.02 0.3 0.2
3.924 4.6 0.9 0.22 0.02 0.1 0.2
4.905 4 1 0.25 0.03 0.4 0.2
5.886 3 1 0.29 0.03 0.3 0.3
6.867 3 1 0.33 0.03 0.1 0.2
7.848 3 1 0.34 0.01 0.1 0.3
8.829 2.5 0.9 0.40 0.01 0.1 0.2
9.810 2.2 0.9 0.45 0.03 0.0 0.2
Due to the large amount of trials, the standard deviation can be used to calculate the uncertainty.
The error bars for the brake force are too small to be visible on the graph.
Data presentation:
Figure X1: Graph of 1/(mean time) to the brake force
Figure X2: Graph of the mean change in temperature to the brake force
Conclusion:
Figure X1 shows that the relationship between 1/Time and the Brake Force forms a clear linear relationship with a positive gradient of 0.03849. This means that the time is inversely proportional to brake force, as predicted by the hypothesized results in table XX. Figure X1 also shows, that the line of best fit has a Y-intercept of 0.06133, indicating a systematical error of the setup (a directly proportional relationship would have the Y-intercept at the origin of the graph.
The quality of this observed data is also very good, as the line of best fit intersects all but one set of error bars. The fact that the maximum and minimum gradients are within ±0.01 of the line of best fit, shows that the random errors within the individual trials do not cause large error bars. The line of best fit doesn’t intersect the error bar for a brake force of 7.848N, as this error bar specifically is very small, suggesting accurate measurement or a lack of collected data values, to increase the margin of error.
In order to test how accurately the observed values, follow the trend from the predicted values, both sets of data are graphed in figure X3.
Figure X3: Graph of the hypothesized values as well as 1/timevs the brake force
Figure X3 shows that the observed results have a much smaller gradient, meaning that braking times are longer than predicted. The gradient of the linear trendline of the predicted values is 0.06694, whereas the gradient for the linear trendline of the observed values is 0.03849, meaning that the predicted gradient is roughly 1.7 times greater than that of the observed values.
There are 15 predicted values in order to better show the trend of the experiment. After a brake force of 14.715N (10 masses), however the errors in the collected data are too high to be included into the data processing.
Compares the predicted results and the observed results to highlight the difference between them. This graph includes the 15 predicted values as well as the 10 measurable values.
There was an unexpected intercept between the predicted and observed values in figure X3. This was because the mean time for 0.981N of braking force was shorter than the hypothesized values. This was not expected before the conducting of the experiment. A reason for this deviation may be that the friction caused by the pulley system and motor had a larger impact of the braking time than the difference between the calculations. As the time decreased, the difference between the hypothesized and the actual values got larger, due to the decreasing impact of the friction of the motor / pulley system.
Figure X2 shows that the change in temperature and the time are also inversely proportional to one another. With a gradient of -0.05066, this is almost proportional to the relationship between time and brake force. The error bars on the graph in figure X2 are very large, due to the limitation of the equipment, as the digital infrared thermometer only measured with an uncertainty of ±0.1. Nevertheless, the line of best fit almost touches every collected data point. As these error bars even allow for a line of fit with an equal gradient as in figure X1, the conclusion can be made, that the time spent decelerating can be most efficiently used to gain back more of the KE otherwise lost to the environment as the graphs show, that ∆Time≈∆Temp.
Evaluation:
The results obtained from the experiment were expected. The braking time should be longer that the predicted values, as the coefficient of static friction (μ_s), which was used to calculate the hypothesized values, is larger than μ_d, the coefficient of dynamic friction, which actually applies within this experiment.
This is also shown by the difference in the gradient. As figure XX compares the two 1/Time graphs, the discrepancy in the gradient means that there is an increasing difference in the Time vs. Brake Force graphs. That is the effect that the difference between the coefficient of static and dynamic friction has on the experiment, linking back to the mathematical model.
Limitations:
Throughout the conducting of this experiment, there were multiple limitations, the first one being the coordination of the method. Although having one person to operate the timing, the applying of the simulated brake as well as the disconnecting of the motor, may allow for more coordination, the random errors caused throughout this procedure are immense. A solution to this problem could be motion capture. Using video analysis of the mass hitting the model brake disk, the exact time can be established until the disc comes to a halt, eliminating a large portion of the human error.
Related to this, another limitation is the modeling of the stress on the disk, more specifically the initial angular velocity which it should have as the mass is dropped onto it. In order to simulate the stress caused by 100kmh-1, the disc had to be spinning at 2,000 rpm. At this rate, the individual frames of the slow-motion video were very blurry as shown below in figure XX, with the tracking mark highlighted.
Figure XX.1 and XX.2: Screenshots of the video analysis from logger pro with the marked area of uncertainty in and out of motion
This causes for a large systematic margin of error throughout the entire experiment. In order to improve this, a photogate system could be used to measure the angular velocity before each trial, leading to more consistent data and further eliminating some random error.
Although the disc is spinning at roughly the same rotations per minute as a car tire at 100kmh-1, the inertia of the dis itself is nowhere near that of an actual car, due to the remaining mass of said car. In addition to this, the simulation applies the brake from only one side, whereas a disc brake used in a car typically uses a set of calipers pressing against both sides of the brake disc. Due to these limitations, the application from the conclusion of this experiment are limited.
Another limitation was the friction caused by the rubber band and the motor. Although the motor was turned off as soon as the ‘brakes’ were applied, it and the pulley system still provided a large amount of additional friction, possibly decreasing the time. A solution to this problem would be to cut the rubber band before applying the ‘brakes’ at every trial. This would not only result in a large amount of non-biodegradable waste, but also require multiple people, which in turn will lead to more human error.
Extensions:
In order to extend this investigation, I could research in which ways the result would vary if the rotational velocity (rpm) of the brake disc was changed.
In addition to this, the use of a more accurate thermometer would increase the quality of the results found. Rather than using an infrared thermometer, the use of a thermal camera would allow for the visualization of the heat radiation, rather than the increase in the temperature of the brake disc through conduction.