Difference between revisions of "User:Raxu"

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raxu is currently a [[Division C]] competitor at [[Scarsdale High School]] and Captain of the team in [[2017]] and [[2018]]. He also competed in Division B for the [[Scarsdale Middle School]] team in [[2014]].
====Material Considerations====
====Reducing weight====
====Friction in the wheel====
====Physics Theory of the wheel====
Below is the theory of physics of the wheel, which may help in the design of the device.
<div align = center>[math]E = I\omega ^2=\frac{mr^2v^2}{r^2}=mv^2.[/math]</div>
<div align = center>[math]f_s = m g \mu_s,\qquad ma=mg\mu_s,\qquad a=m\mu_s,[/math]</div>
If the scrambler hits the wall at faster than about 1 m/s, the egg is very likely to break. Therefore, once the car can travel the required distance, a brake should be used to stop the car before it reaches the wall.
Here are three commonly used braking systems, listed in order of difficulty of construction.
====String Type====
*Easy to build
*Poor Accuracy
====Threaded Rod Type====
*High accuracy
*No Backlash
*Somewhat complicated to build
*Added friction to the axles
====Brake Pad Type====
This type of breaking is '''forbidden''' under the [[2015]] rule: Nothing that was not in contact at the start of the run can ever touch the ground.
{{SpoilerBoxBegin}}'''Brake Pad Type (Illegal for Current Rules)'''
*High accuracy
*Reduced skid
*Most complicated to make
*Adds friction to the axles
Most attempts use a wingnut to trigger.
====Physics Theory of the break====
{{SpoilerBoxBegin}}'''Physics Theory of the Breaking System'''
We expand the theory of friction in wheels described in the wheels section. There are four main questions that must be considered during the design of a braking system of a scrambler: how long to make the car, where to put the center of mass, which wheels (front versus back) work better for braking and how to avoid skidding. To answer these questions, we shall consider a simple model of a braking car.
In the picture, [math]N_1, N_2[/math] are the normal forces exerted by the floor on the back and front wheels, respectively. [math]F_1, F_2[/math] are the resulting forces of friction. [math]b, B[/math] are the horizontal distance from the back axle to the center of mass of the scrambler and the front axle, respectively. [math]h[/math] is the height of the center of mass, [math]m[/math] is the mas of the scrambler. Since the scrambler is at equilibrium,
<div align = center>[math]\begin{cases} N_1 + N_2 = mg \\  F_1 + h F_2 + b N_1 = \left(B - b\right)N_2\end{cases}[/math]</div>
Furthermore, [math]F_1 = \mu_1 N_1, F_2 = \mu_2 N_2.[/math] Two coefficients [math]\mu[/math] are used, because one pair of wheels is used for friction. Solving for [math]N_1, N_2[/math],
[math]\begin{cases}N_1 = M g \frac {B - b - h k_2} {B + h\left(k_1 - k_2\right)} \\ N_2 = M g \frac {h k_1 + b} {B + h\left(k_1 - k_2\right)}\end{cases}[/math]
If one of the normal forces is calculated to be negative in the design, one of the axle loses contact and the scrambler is rotating - the scrambler is unstable.
The scrambler starts skidding if the force applied by the brake exceeds the maximal static friction, [math]F_1+F_2[/math]. If the scrambler skids, the kinetic friction is much smaller than the maximum static friction and is dependent on the floor of the competition. Thus, a breaking system with skidding is imprecise and ineffective.
We now determine which axle is better for breaking. If the front axle is used, [math]F_1=0,\ N_2 = M g \frac {b} {B - h k_2}[/math]
The maximum normal force occurs when b = B - h k<sub>2</sub>. This puts the center of mass as close as it can be without reducing [math]N_1[/math] to less than 0. [math]N_{2max} = M g \frac {B - h k_2} {B - h k_2} = M g[/math]
If the back axle is used, [math]F_2=0,\ N_1 = M g \frac {B - b} {B + h k_1}[/math]
In this case the maximal normal force happens when b = 0. [math]N_{1max} = M g \frac {B} {B + h k_1}[/math]
Since the ratio [math]\frac{B}{B + hk_1}[/math] is always less than 1, [math]N_{1max}<N_{2max}[/math]. Therefore, given a choice of using exclusively the front or the back axle, the front axle should be used. However, the scrambler may flip over before the maximal normal force can be reached. A lower height and longer length will prevent the cart from flipping over. However, in some cases it is impossible to have the center of mass be located anywhere near the front of the car (e.g. on a ramp scrambler). Furthermore, back breaking systems do not flip over as easily.
If both the back and front axles are used for breaking, for example if we set the k's to be equal, then the total normal force is [math]mg[/math] and we are back to the original equation. Notice that [math]N=N_{1,max}[/math], and the center of mass does not have to be in the front. Therefore, a design using both axles can be very beneficial.
To summarize, the best braking system involves the use of both axles. The next best choice is the front axle only braking system, which requires a car to be long and low. The least favorable choice is a back axle only system, but it is stable, like the one using both axles, and easier to design.
It should be noted that having high coefficients of friction (while staying in the rule constraints) will make even sub-optimal brake geometry perform well. This makes the choice of location for the brake pad (wheel based brakes cannot easily utilize adhesives) less important. It also prevents the scrambler from skidding.
Finally, the actual length of the car is not important in the stability of the system: the distance from axle to axle is.
===Mass Release Mechanism===
===Device Systems===
====Integrated Mass Type====
'''These types of vehicles have come into question as to being legal under the new rules'''
''An Introduction from Physics''
<ol start="9">
<li>[math]PE_{mass} = KE_{car}[/math]</li>{{Anchor|f9}}
<ol start="10">
<li>[math]m g h = 1/2 M v^2[/math]</li>{{Anchor|f10}}
Since in the integrated mass scrambler the mass, by definition, is a part of the car, M is always bigger than m. Let us take the limiting case of M being exactly equal to m (this is a good approximation for a ramp integrated mass type described later). If we assume that the mass falls exactly 1 (.75 in 2009) meter. Then, the m's in equation [[#f10|(10)]] cancel, and we can solve for the v.
:[math]v = 4.42 m/s[/math]
=====Tower Scrambler=====
*Reasonably easy to build
*Good theoretical performance
*Can be almost free to build - see [[Zero Cost Scrambler]]
*Manufacture of the variable radius axles can be challenging
*Severe dependence on mass used
<ol start="11">
<li>[math]M_g r_1 = F_f r_2[/math]</li>{{Anchor|f11}}
''M - the mass of the mass; r<sub>1</sub> - radius of the axle; r<sub>2</sub> - radius of the wheel; F<sub>f</sub> - force imparted onto the ground. ''
The F<sub>f</sub> force is the force that pushes the device forward. It would appear that ideally, you would have a very small radius of the wheel and a very large radius of the axle to maximize the F<sub>f</sub>. However, the F<sub>f</sub> can never exceed the force of static friction that the wheels feel touching the ground. If it exceeds it, then the wheels will skid and the scrambler will not move. This places a practical limit on how small one can make the wheels. This can be helped a bit by increasing the traction between the wheels and the ground, which will allow for smaller wheels. It should also be noted that smoother force curves are preferred for this scrambler type. Often by making the F<sub>f</sub> small and thus allowing for more time for the weight to accelerate the car one can avoid the problems associated with the imperfection of the materials: if you increase the F<sub>f</sub> too much, the strings may start acting as springs. It is, however, difficult to predict what will happen using physics alone and thus it is advisable to test various ratios of axle to wheel radii.
To improve performance, teams sometimes use variable radius axles. Generally they are set up to provide a large torque at the start to get the scrambler moving (i.e. a large r<sub>1</sub> from equation [[#f11|(11)]]). Then as the scrambler advances, the radius of the axle decreases thus providing more acceleration, as at this point the forces of static friction are less pronounced. This fact makes theoretical predictions to be very hard for these types of devices, and thus none are provided.
=====Ramp Scrambler=====
{{SpoilerBoxBegin}}'''Ramp Scrambler (Illegal for Current Rules)'''
{{SpoilerBoxContent}}'''*Not Legal For 2017 since the rules state nothing that was not in contact with the ground at the beginning of the run can touch the ground.'''
*Easy to build
*Good theoretical performance
*Performance largely independent of the mass used
*Theoretical performance difficult to achieve
*Often difficult to fulfill the egg on the starting line requirement
''(Note that the entire car is counted as the mass during impoundment)''
Note that this scrambler is often classified as a launcher type, citing the fact that the ramp acts as a launcher, and that this type was not allowed by the rules from years that did not allow for multipart launchers. However, this scrambler obeys the equation [[#f10|(10)]] which takes precedence.
{|class="wikitable sortable"
!Mass !!Idle Time !!Acceleration Time !!Exit Speed !!Travel Time !!Total Time
|0.5 kg||0.00 sec||0.59 sec||4.43 m/s||2.57 sec||3.16 sec
''*A note about these and future numbers:''{{Anchor|note}}
====Launcher Type====
''An Introduction from Physics''
Launcher type scramblers without exception are characterized by the existence of a device separate from the car that is designed solely for the purpose of converting the potential energy of the mass into the kinetic energy of the car. This, unsurprisingly, means that means that this type of scrambler is easily simulated by the equation [[#f10|(10)]]. The advantage that these systems possess over the integrated mass scrambler type is that the little m in that equation can be smaller than the big M.
=====Spring Launcher=====
*Nearly maximum theoretical efficiency possible in a scrambler
*Reasonably easy to achieve the theoretical performance
*Low dependence on mass used
*The locking mechanisms are often difficult to build
*Complexity of the device requires careful construction
*Rapid speed of the car makes it difficult to stop
The following are some numbers derived from physics. Note that this launcher pushes the car forward 1 meter before it is set free.[[#note|<sup>*</sup>]]
Assuming Vehicle Mass of 100 grams
Total Time is from .5m to 8.5m.
{|class="wikitable sortable"
!Mass !!Exit Speed !!Total Time
|0.5 kg||9.90 m/s||0.81 sec
|1.0 kg||14.00 m/s||0.57 sec
|2.0 kg||19.80 m/s||0.41 sec
=====Hammer Launcher=====
*High theoretical efficiency
*Very easy to build
*High dependence on mass used
*Theoretical efficiency is difficult to achieve, due to the non-truly elastic collisions
*The rapid acceleration is stressful on the car, requiring it to be sturdy
<ol start="12">
<li>[math]v_1 = \frac {2M} {M + m} \times v_2[/math]</li>{{Anchor|f12}}
''v<sub>1</sub> = velocity of the car; v<sub>2</sub> = velocity of the mass; M = mass of the mass; m = mass of the car''
It is clear from this formula that the larger the difference between the two masses is, the faster the car will go. The limiting scenario is when the mass of the car is nonexistent, which makes it go at double the speed of the mass. We know the speed of the mass from the equation [[#f10|(10)]] when we set the big M and small m to be equal: 4.42m/s. Thus, the maximum possible speed achievable by a car launched by this launcher is an amazing 8.84m/s.
Here are some theoretical data.[[#note|<sup>*</sup>]]
{|class="wikitable sortable"
||0.5 kg||0.59 sec||0.00 sec||6.81 m/s||1.46 sec||2.06 sec
||1.0 kg||0.59 sec||0.00 sec||7.70 m/s||1.29 sec||1.89 sec
||2.0 kg||0.59 sec||0.00 sec||8.24 m/s||1.21 sec||1.81 sec
=====Pulley Launcher=====
*Good theoretical efficiency
*Easy to achieve theoretical efficiency
*Simple construction
*Some dependence on mass used
''M = mass of the mass; g = acceleration due to gravity (9.8m/s<sup>2</sup>); m = mass of the car; a = acceleration of the car''
Here are some theoretical data from physics. Note that this launcher pushes the car about .7 meter before releasing it.[[#note|<sup>*</sup>]]
{|class="wikitable sortable"
|0.5 kg||0.00 sec||0.51 sec||3.88 m/s||2.32 sec||2.83 sec
|1.0 kg||0.00 sec||0.48 sec||4.13 m/s||2.18 sec||2.66 sec
|2.0 kg||0.00 sec||0.47 sec||4.27 m/s||2.11 sec||2.57 sec
=====Push Rod Launcher=====
{{SpoilerBoxBegin}}'''Push Rod Launcher (Illegal for Current Rules)'''
{{SpoilerBoxContent}}'''NOTE: THIS TYPE OF LAUNCHER IS ILLEGAL FOR THE 2017 SEASON. Nothing that was not in contact at the start of the run can ever touch the ground.'''
*Simple construction
*Poor theoretical efficiency
*Difficult to achieve the theoretical efficiency
Here are some theoretical data for this launcher. Due to the rod length, the car is pushed a full 1.4 meters forward before being released.[[#note|<sup>*</sup>]]
{|class="wikitable sortable"
|0.5 kg||0.00 sec||0.66 sec||3.88 m/s||2.21 sec||2.87 sec
|1.0 kg||0.00 sec||0.63 sec||4.13 m/s||2.08 sec||2.71 sec
|2.0 kg||0.00 sec||0.61 sec||4.27 m/s||2.01 sec||2.62 sec
[[Category:User Pages]]
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Latest revision as of 08:30, 24 July 2018

raxu (Richard Xu) was a Division C competitor at Scarsdale High School and Captain of the team in 2017 and 2018. He also competed in Division B for the Scarsdale Middle School team in 2014.