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Energy ratio method for quantifying effect of ball mass variations on range error of a flywheel ball shooterMay 30, 2015
The new Vex game “Nothing but Net” requires a ball to be shot across the field into a goal. The ball has a mass specification that allows up to 10% variation. I derived equations that related the range error to mass error for a single flywheel shooter here and wanted to generalize the analysis to any flywheel system that uses the energy of the flywheel to impart energy to the ball. It turns out that the error in range due to mass is a simple function of the ratio of flywheel energy to the energy of the ball.
i.e. %range error = %mass of ball error /(1+ ratio)
where ratio = flywheel energy/ ball energy.
Following is a derivation of this formula. To simplify the work I will define a term called equivalent mass, Meq, for the flywheel which is a mass that would give the same kinetic energy of the flywheel if it were traveling linearly at the ball speed, V.
I. e. we can write the flywheel energy , Ew = 1/2*Meq*V^2
We know that Ew =1/2* I_wheel*w_wheel^2 so
So Meq = I_wheel*w_wheel^2/V^2
For a single wheel shooter, w_wheel = 2*V/r_wheel so
Meq_1 = 4*n*I_wheel/r_wheel^2
where n = number of wheels.
For a two wheel shooter, w_wheel = V/r_wheel so
Meq_2 = n*I_wheel/r_wheel^2
If we write I_wheel in terms of its radius and mass then
I_wheel = c*m_wheel*r_wheel^2. Substituting in the equivalent mass equations gives
Meq_1 = 4*n*c*m_wheel
Meq_2 = n*c*m_wheel
.Ball Energy, Eb
Eb_1 = 1/2 *m_b*V^2*(1.4) (1.4 factor is for added spin energy)
Eb_2 = 1/2*m_b*V^2
Conservation of Energy
The initial energy of the flywheel is equal to the final energy of the wheel plus the ball energy.
Ew_initial = Ew_final + Eb
Ew_initial = Ew_final*(1+ Eb/Ew_final) = Ew_final*(1 + 1/ratio)
ratio = Ew_final/Eb
Lets look at Eb/Ew_final
single wheel shooter
Eb_1/Ew_final_1 = 1/2*m_b*V^2*1.4/(1/2*Meq_1*V^2) = 1.4*m_b/Meq_1
ratio_1 = 5/7*Meq_1/m_b
two wheel shooter
Eb_2/Ew_final_2 = 1/2*m_b*V^2/(1/2*Meq_2*V^2)= m_b/Meq_2 so
ratio_2 = Meq_2/m_b
So it looks like ratio is only a function of m_b and not a function of V.
Rewriting the energy equation to isolate Ew_final
Ew_final= Ew_initial/(1 + 1/ratio)
Since Ew_final is proportional to V^2 which is proportional to range R
we can say that
R/R_0 = (1+1/ratio_0)/(1+ 1/ratio)
and introducing % changes R=R0 *( 1 + %e _r), m_b = m_b_0*(1 + %e_m_b)
we can after some manipulation show that
%e_r = %e_m_b *( 1/(1+ ratio) = %e_m_b*factor
where the mass % error reduction factor=1/(1+ ratio) and ratio . is defined above for each type of shooter.
factor vs ratio is simple to compute
.5 , 2/3
Single wheel shooter
ratio_1= Meq_1/1.4/m_b_0= 4*n*c*m_wheel/m_b_0/1.4
Two wheel shooter
ratio_2 = Meq_2/m_b_0= n*c*m_wheel/m_b_0
Lets put some numbers for the 5″ vex flywheel.
m_wheel = .18 kg, m_b_0 = .05 kg, c = .84^2 = .71
Single wheel shooter factor:
ratio_1 = 4*n*1.79/1.4
ratio_1 = 4*1*1.79/1.4 = 5.1 => factor = .163 (assumes 1 wheel for flywheel)
ratio_1 = 4*2*1.79/1.4 = 10.2 => factor = .089 (assumes 2 wheels for flywheel)
Two wheel shooter factor:
ratio_2 = n*1.79
ratio_2 = 2*1.79 = 3.6 => factor = .217 (assumes 1 wheel on each side)
ratio_2 = 4*1.79 = 7.2 => factor = .12 (assumes 2 wheels on each side)
ratio_2 = 6*1.79 = 10.74 => factor = .085 (assumes 3 wheels on each side)
Observation…. although the two wheel shooter flywheel has 1/4 the energy due to its slower speed, the ball energy is also less because it requires no spin energy. The energy ratio is only different by a factor of 2.85 rather than the full 4. Still, the single wheel shooter has about 1.3% less range error than the two wheel shooter with the n=2 configuration. This is not that big a deal since we are talking 1.4 inches error (compared to a 4 inch ball diameter).
The new vex game , Nothing but Net, could utilize a design similar to a two wheel tennis ball launcher.
Question: how many motors are required on the launcher?
The after a launch, the energy lost from the spinning wheels is transformed into ball kinetic energy and heat due to friction and ball compression.
After each shot the wheels are brought back to initial spin speed by the power of the motors. The maximum time allowed for respinning is the cycle time of the firing sequence. Lets take a look at the Vex game derived requirements:
Ball mass, m = 60 grams
Ball launch Speed, v = 6 m/s
Ball kinetic energy:K = 1/2*mass*v^2 = .5*.06*6^2 = 1.08 joules
Energy loss due to compression : E_c
Energy loss due to friction : E_f
Time between shots: 1 sec
Average power required p_avg = ( K + E_c + E_f)/ t
Force on ball during acceleration:
F = d(m*v)/dt or the change in momentum of the ball over the time of acceleration., dt.
dt can be approximated as the contact distance / tangential speed of the wheel , v.
The contact distance is about 3 cm so
dt= .03/6= .005 s
d(m*v) = .06*6 = .36 kg*m/s
hence F = .36/.005 = 72 newtons
Normal force on ball F_n = F/u_friction . The normal (compression) force on the ball is then
F_n = 72 newtons assuming a u_friction = 1 which is possible with a sticky wheel.
The assumed compression distance is about 1 in or 2.54 cm. (To be verified later)
Hence Ec = F_n*d/2 = 72*.0254/2 = .91 joules.
With good design, the friction loss in the drive train can be small (maybe .1 joules) so lets assume that E_c + E_f are about equal to the K= 1.06 j so
p_avg = 2*K/t = 1.06*2= 2.12 watts or 1.06 watts per motor.
We know the vex 393 motors have a max power = max_speed*max_torque/4 or about 4.5 watts but they will overheat if run continuously at this power. The PTC fuses will stop the motors if they run continuously with currents equivalent to more than 25% maximum torque (speeds less than 75% max speed). At this operating point, the motors only deliver 3/4 max power or 3.4 watts.
There are also friction losses from the teeth of the spur gears. I usually assume about 5% per 5:1 ratio. A shooter wheel with a 25:1 gearing would lose 10% torque or energy at a given speed.
So the net power to the shooter wheel will be .9*3.4 = 3.0 watts which is more than the 1 watt that we require.
If the gear train has pressure on the axles from bearing blocks and possibly the collars are too tight so the wheels slow down quickly when coasting with motors disconnected, then over heating can easily occur. e.g. if friction uses up just 15% of the available torque, the motors will have to provide about 1 watt extra. which cuts our margin considerably.
Faster shot rate?
We assumed 1 shot per second…what happens with 2 shots per second…. Well, the power requirements almost double since we are using twice as much energy per unit time. We would likely have to add extra motors.
Our Robodox Algalita OpenROV build team attended the subject conference and gave Captain Charles Moore and crew their first look at the OpenROV that will be used to survey fish habitat located in plastic pollution during their July expedition to the Pacific Gyre. See this video for Capt Moore interview.
Here is a link to the team summit report posted in our blog “Robodox Engineering ROV for ORV”
We had a short tank demo with a missing thruster… but at least we were able to go in circles:) Thanks to Dave L. for trying to get us new motors on the spur of the moment. We now have things back to normal and will proceed with salt water testing next week.
Here are all the images from the conference…. sure was a beautiful place to meet.
GHCHS Team 599 Robodox attended the 2014 Inland Empire Regional FRC competition. We collected the Engineering Design award and also the competition finalist award and medals. It was a well run competition and we had a great time. Once again we had a top performing robot that got beat by a stronger no 1 alliance. We made an operational mistake in the finals that cost us the first match by breaking communication during setup and causing our Crio to need resetting after the match started and we sat out the Autonomous plus some seconds. Our alliance partners 294 and 4139 were having functional problems in the second so we were soundly trounced by the winning alliance led by 1678 citrus circuits who teamed up 399 and 4161. The 2nd final match (see video) was a thing of beauty with 1678 and 399 performing two truss catches and racking up a 229 to 72 score. Hopefully we can redeem ourselves at our next regional in Sacramento where we will once again tangle with 1678 citrus circuits from Davis. Also, thanks again to 294 for selecting us for the direct eliminations.
Winning the engineering design award means a lot to us since this year we focused on doing a 3D Solidworks design supported by solid prototyping. All fabrication was done based upon automated drawings made from the 3D model. This the best looking robot we have done in years and clearly the most durable. See this post for picture.
Lots more pictures on my facebook page.
The choo-choo catapult reset mechanism performed well so long as we kept the linkages in good order. The high forces caused holes in the linkages to elongate after a day worth of shooting. This was anticipated so we brought three spares and used them all. We will use steel linkages rather than aluminum at our next competition so they should last longer.
Robodox also ran the robot First Aid Station and the spare parts booth. We also had on display our underwater ROV which will be used by Algalita Research Foundation to do plastic pollution exploration in the Pacific Gyre this summer.