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]].
''This section contains some physics concepts in areas. The physics is intended to help in the design of the device, especially for Division C competitors, and is not a required part of the event.''
====Material Considerations====
Given the limited budgets of most teams, physics has to be eclipsed by the availability of the materials. Most teams use some pre-made disks for their wheels, for example, CD/LP based wheels. Some use roller blade wheels. Better funded teams may wield custom made acrylic wheels, or even those cut from sheets of balsa.
====Reducing weight====
Lighter wheels are generally better (see physics explanation below). Therefore, given good drilling equipment, it is advisable to drill a series of holes in the wheel to make it lighter. Take care to do it in a symmetrical fashion, and not to weaken the wheel too much. By the physics, the greatest effect shall be seen from reducing the weight of the rims of the wheel. Look at the professional wheels for guidance. For example, high-speed bicycles often have very efficient designs for their wheels.
Most designs drill up to four holes on each wheel, all of the same size. Drilling holes work well with materials like Plexiglas, milled aluminum, steel, etc. However, it does not work with materials like CD and fibrous alloys because it ruins the structural integrity of the wheel.
====Friction in the wheel====
To reduce the axle friction, teams use ball bearings. These gadgets, while sometimes expensive and tough to find, will often nullify any problems associated with small wheels. Keep these free from dust and well lubricated with grease or other lubricants.
To increase traction, many teams use rubber bands around their wheel rims. Latex gloves as a faster and easier alternative to rubber. The rules prohibit any lasting glues from being used for traction purposes.
In general, it is good to make sure that the wheels are straight, the axles are straight, and the bearings are mounted square to the car's chassis. This helps the scrambler move in a straight line.
====Physics Theory of the wheel====
The theory of physics of the wheel, which may help in the design of the device, can be found here:
{{SpoilerBoxBegin}}'''Physics Theory of the wheel'''
The wheel's mass, specifically its rotational inertia, is one of the most important for a scrambler: the rotational energy of the wheel does not accelerate the scrambler forward. Let [math]\omega, v, r, m, I[/math] denote the angular velocity, translational velocity, radius, mass and moment of inertia of the wheel, respectively. Since [math]\omega = \frac{v}{r}[/math] and [math]I=mr^2[/math] if we approximate wheels as hollow cylinders,
<div align = center>[math]E = I\omega ^2=\frac{mr^2v^2}{r^2}=mv^2.[/math]</div>
Therefore, given a constant starting energy, an increase in mass will cause the scrambler to have less translational kinetic energy. Since [math]m\propto r[/math], smaller wheels make the car go faster. Wheels that are solid disks have moment of inertia [math]I=\frac12mr^2[/math], but have much more mass and is therefore ineffective.
A larger wheel have different benefits. Smaller wheels allow for your scrambler to move faster, but they are more susceptible to axle friction because the wheel-to-axle ratio is lower. Scramblers with small wheels may fail to reach the target distance just because of friction. Ways to reduce this friction will be discussed shortly. Teams with large wheeled scramblers move slower, but at roughly constant speeds. The wheel radius is also important to one of the integrated mass scramblers discussed later.
The second important aspect of a wheel in a scrambler is its traction. Traction, also known as static friction, is important for the car's stability, and when the wheels are used as brakes, for its braking. The friction between the wheel and the ground is static friction, unless the wheel is skidding. Since the weight of the car is supported by the wheels, the traction is at most
<div align = center>[math]f_s = m g \mu_s,\qquad ma=mg\mu_s, a=m\mu_s,[/math]</div>
where [math]f[/math] is the force of friction, [math]m[/math] is the mass that the wheel supports, [math]\mu_s[/math] is the coefficient of static friction and [math]a[/math] is the acceleration of the scrambler. Beyond the maximum acceleration, the cart will start skidding.
Therefore, the wheel should have higher coefficient of friction to increase its maximum possible acceleration before skidding.
While some teams cannot even get their scrambler to travel the announced distance, those that do face the problem of the egg traveling more than the 1 m/s (the average upper limit for the speed that the egg can survive decelerating from) towards a very solid wall. Generally, a brake is used to stop the car before it reaches the wall.
====Braking Systems====
=====String Type=====
*Easy to build
*Poor Accuracy
This system relies on the braking power of the rims of the wheels, which may lead to skid, as this design does not allow for gradual locking of the wheels. That being said, this is the easiest system to implement. So, if a quick solution is needed, this is certainly an option.
=====Threaded Rod Type=====
*High accuracy
*No Backlash
*Somewhat complicated to build
*Added friction to the axles
The basic concept of this design is using a threaded rod for the axle, placing a nut (usually a wing nut is used) on the axle. As the wheels rotate the rotating motion is transferred into horizontal motion of a wing nut moving it along the axle. When the wing nut reaches a barrier, it will no longer be able to move and thus stops the axle from turning. The distance is set by setting how far the wing nut starts from the barrier and is usually measured in rotations of the wheels.
=====Brake Pad Type=====
{{SpoilerBoxBegin}}'''Brake Pad Type (Illegal for Current Rules)'''
{{SpoilerBoxContent}}'''NOTE: THIS TYPE OF BRAKING IS ILLEGAL FOR THE 2015 SEASON. Nothing that was not in contact at the start of the run can ever touch the ground.'''
*High accuracy
*Reduced skid
*Most complicated to make
*Adds friction to the axles
Another advantage of this method is that attempting this design carries a fairly low risk because it can be reduced to the basic threaded rod design fairly easily if you cannot get the system to work.
Most attempts use a wingnut to trigger.
The theory of physics of the breaking system, which may help in the design of the device, can be found here:
{{SpoilerBoxBegin}}'''Physics Theory of the Breaking System'''
While the equation [[#f5|(5)]] is a good approximation for a single wheel scrambler, such things are a rarity. A more careful analysis is required.
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.
As can be seen from the picture, N<sub>1</sub> and N<sub>2</sub> are the normal forces exerted on the car by the floor for the back and front wheel axles respectively. F<sub>1</sub> and F<sub>2</sub> are the resulting forces of friction. The center of mass is positioned between the wheels a distance b from the back axle (if it wasn't then the car would tip over even when stationary). The total length of the card from axle to axle is assumed to be B. The effective height of the car is h and is measured from the bottom of the floor to the height of the center of mass. The car mass is M. And, obviously, F<sub>g</sub> equals Mg. The system is assumed to be in an equilibrium (cases where it is not shall be discussed later on), so a balance of torques and a balance of forces are established:
:[math]N_1 + N_2 = M g[/math]
<ol start="6">
<li>[math]h F_1 + h F_2 + b N_1 = \left(B - b\right)N_2[/math]</li>{{Anchor|f6}}
Using the equation [[#f5|(5)]] F<sub>1</sub> and F'_2_' can be expressed in terms of N'_1_' and N'_2_' respectively.
:[math]F_1 = k_1 N_1[/math]
:[math]F_2 = k_2 N_2[/math]
Note that two different k's have been used: this shall be used to simulate the effects of separately using only one wheel pair for braking. We are interested in the expressions for N<sub>1</sub> and N<sub>2</sub> mostly, since they will allow us to determine the answers to the questions posed in the beginning of this section. We have a total of two unknowns and two equations (the equations for F<sub>1</sub> and F<sub>2</sub> are readily eliminated by substituting them into [[#f6|(6)]], which means that this system can be solved exactly. The equations for N<sub>1</sub> and N<sub>2</sub> are as follows:
:[math]N_1 = M g \frac {B - b - h k_2} {B + h\left(k_1 - k_2\right)}[/math]
:[math]N_2 = M g \frac {h k_1 + b} {B + h\left(k_1 - k_2\right)}[/math]
Another comment is the reason why we want the expressions for these two values. When the wheels are rolling, the coefficients of friction that get plugged into the above formulas are the static kind since the contact patch between the wheel and the ground does not move. As most people know from physics once the force on an object exceeds the static friction that is acting upon it, it begins to move. In this case, this would mean that the car begins to skid. This happens when the force that the brakes apply to the wheel exceeds the force of friction between the wheel and the ground. Also, from the diagram, it can be clearly seen that the sum of F<sub>1</sub> and F<sub>2</sub> equals the total force that decelerates the car. Thus, the expressions above give us the maximum acceleration a car can have during braking. Thus it is important to maximize the normal force to be able to utilize stronger brakes and stop the car more rapidly.
A question arises: why don't we want the car to skid? The answer is simple, the braking of the car will still be simulated by the equations above during skidding, but since the coefficients of kinetic friction are generally much smaller than the coefficients of static friction the car will not decelerate as quickly. Also, when the car does not skid the static friction supplies exactly as much force as it can to accommodate the friction of the braking system. Since the braking surfaces in the braking system can be well machined and can be controlled this results in an accurate and reproducible braking action. When the car skids, however, the kinetic friction is no longer determined by the friction supplied by the braking system. Instead, it is dependent on the coefficient of friction between the floor and the wheel, which varies greatly depending on the floor surface. This makes for an imprecise braking action. Lastly, due to the reduced friction between the wheels and the ground the car will be more likely to swerve sideways due to the decreased stabilizing friction force.
With that covered, let us determine which axle is the best for housing the braking system. Let us simplify the analysis by first assuming that only the front wheels have a friction coefficient. This makes N<sub>1</sub> irrelevant (since F<sub>1</sub> becomes 0) and N<sub>2</sub> becomes:
:[math]N_2 = M g \frac {b} {B - h k_2}[/math]
It is obvious that the maximum normal force then happens when b = B - h k<sub>2</sub>. This puts the center of mass as close at it can be without reducing the N<sub>1</sub> to less than 0. Thus:
<ol start="7">
<li>[math]N_{2max} = M g \frac {B - h k_2} {B - h k_2} = M g [/math]</li>{{Anchor|f7}}
Keeping that result in mind let us now simulate the back axle only braking system by setting k<sub>2</sub> to 0. As before, only the braking axle normal force matters:
:[math]N_1 = M g \frac {B - b} {B + h k_1}[/math]
<ol start="8">
<li>[math]N_{1max} = M g \frac {B} {B + h k_1}[/math]</li>{{Anchor|f8}}
If we compare the equations [[#f7|(7)]] and [[#f8|(8)]] it can be seen that the bigger force occurs when the front brakes are used alone since the ratio of B / (B + h k<sub>1</sub>) is always smaller than 1, thus making the N<sub>1max</sub> always be smaller than N<sub>2max</sub>. Therefore, given a choice of using exclusively the front or the back axle, the front axle should be used. In this case, it is better to put the center of mass as far in the front of the car as possible without making it flip over during the braking. Also, we can examine the effects of the length of the car. If one examines the formula for N<sub>1</sub> then it can be naively thought that it does not matter. However, since the numerator is will probably be less than this optimal value it is clearly seen that a large length and low height would be optimal. Experimentation is required as always since to achieve optimal performance the knowledge of the coefficients of friction must be known.
One case was ignored in the above discussion when both the front and back axles are used for braking. If we set the k's to be equal, then the total normal force simply equals to Mg and we are back to the original equation. Note that this is the same answer that we got for the optimal front axle braking system. There, however, we had to reduce the N<sub>1</sub> to 0, which would make a car unstable. This is not necessary in this case: any reasonable location for the center of mass will yield the maximum possible normal force.
One final point. The length of the car in these calculations was measured from axle to axle, which implies that the actual car length does not matter all that much. That is if your car has a short wheel base but a long rod behind it (which was say used during the launching process) the car is still short in terms of the analysis above.
===Mass Release Mechanism===
Many of these designs involve rollers that redirect strings. They are commonly referred to as pulleys as well, although that is somewhat dubious. The important consideration for these devices is their friction, or rather, its absence. This cannot be the case if the roller does not contain a bearing in it. The advantage of a bearing is that the friction remains roughly constant irrespective of the load that the roller is forced to bear. If a bearing is absent, then the equation [[#f3|(3)]] applies, which can prove catastrophic is large tensions are used (this assumes an unlubricated bearing, with a lubricated bearing the friction, will decrease once the speed of the bearing surfaces is high enough to cause hydrodynamic forces to float the bearing surfaces apart). Whether a roller has a bearing in it or not can easily be tested. Take a string and try using the roller to redirect the string while pulling on it. If when varying the tension the resistance also varies, then there is no bearing in that roller. Avoid it and buy a roller that has one.
===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}}
''m = mass of the mass; g = gravity constant (9.8 m/s2); h = the height that the mass falls during the launch; M = the mass of the car; v = the velocity (in this case simply the speed) of the car''
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]\left(9.8m / s^2\right)\left(1m\right) = 1/2 v^2[/math]
:[math]v = 4.42 m/s[/math]
Thus, friction aside, the best possible speed this scrambler can achieve is 4.42 meters per second. This number is actually a fair bit higher than possible this year due to the reduced height requirement of the launcher.  I can't find what mass is being used in these equations but they may also be incorrect for this year. You might think that this is insanely good, but as we will later see for other scrambler types, it is far from the best you can do.
=====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
A relic of the times when only single part devices were allowed, this integrated mass launcher consists of a car that is propelled by a falling mass that uses a pulley system or otherwise to directly convert the motion of the weight into the motion of the car. This scrambler obeys the equation [[#f10|(10)]] reasonably well. However, the approximation of equating the mass of the car to the mass of the mass is impossible to make in this case. The support structure for the mass and the pulley system (not to mention the wheels) often drive the masses of these scramblers above 5 kilograms. This makes these scramblers very slow. As such, it is difficult to build the scrambler to cover the whole distance. Also, they are normally slower than launchers.
<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.
Here are some numbers that can be calculated from simple physics. These calculations assume a perfect design and no friction. Distance is set at 10 meters. The mass of the car is set at the mass of the weight. They also assume that all of the mass of all three vehicles is centered at the same single point in space, actually, the more massive vehicles are likely to have a higher center of mass so they would actually exit the launcher at a higher speed. [[#note|<sup>*</sup>]]
{|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
Spring launcher is one of the rarest launcher scrambler types, often sighted alone in competitions. Being a very complicated design, it is often misbuilt by teams, and thus gets a bad reputation. However, it easily achieves the highest speeds for scrambler cars, at all distances and most masses.
Another problem unique to this launcher is that the car is launched at such a great speed, that most brakes used for other, slower, launchers are inadequate for this design. Due to the rarity of this design, no general solution to this problem has been posed, but the theory does not forbid a solution from existing.
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
Look at Jdogg's post on the Scrambler forum page for more info on this chart above.
=====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
*There is a chance that (due to the force of impact), the yolk of the egg is ejected from the shell at the moment of impact, resulting in a tiered run
<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.
The mass of the entire arm must be weighed during the impounding since most designs have the arm lose its potential energy as a whole. From physics it is known that to achieve optimum efficiency, most of the mass of the arm must be concentrated at its distal end, thus making most teams using a hammer/mallet like arrangement.
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
:[math]M g = \left(M + m\right) a[/math]
''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''
The fact that this launcher involves no true energy transfers, any deviation from the theoretical performance is explained away by friction in the rollers and wheels. Since those can be easily fixed, these scramblers are often the faster half of launchers in any given competition.
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
*Actual speeds a somewhat slower this year due to .75 meter height restriction.  Maximum possible speed for this launcher with a zero mass scambler is about 3.75 m/s.  with a 1.5KG mass and a 100-gram scambler, the max speed of the scrambler would be less than 3.5 m/s, although as the mass of the vehicle gets small compared to the falling mass there become less and less change in velocity as the mass of the scramble decreases.
=====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
The 2015 rules prohibit any part of the Scrambler (energy providing device and car) cannot touch the floor at any point after the launch. Violating this rule will result in a construction violation, which is a penalty of 3000 points. As shown by the above animations, there is a chance that the rod will contact the ground after the vehicle is moving of its own accord. Teams should take this rule change into consideration if this is the final design chosen (i.e. have the push rod attached to the vehicle so when the car is launched, the rod comes off the mass and with the car, not touching the floor).
[[Category:User Pages]]
[[Category:User Pages]]

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.