Vex Note: How a single flywheel ball shooter minimizes the effect of ball mass variations

May 28, 2015

Nothing but Net 2015/2016 competition game involves shooting 4 inch balls that can have a 10% variation in mass.    We know that trajectory range ,R = V^2/g*sin(2*theta) so it  is dependent upon the square of the ball release speed , V, and shooter elevation, theta.   Mass does not enter into the equation unless it affects V.

Ball release energy :

Suppose we use a Vex 5″ diameter wheel as a flywheel and rotate it a w_wheel angular speed.      As the ball leaves the shooter, it will have a V = r_wheel*w_wheel/2.  e.g. half of the flywheel tangential speed.    The ball will have a spin rate , w_ball = V/r_ball.    The energy of the ball, E_b , is the sum of the ball translational energy and rotational energy.

E_b = 1/2*m_ball*V^2 + 1/2*I_ball*w_ball^2

where I_ball = 2/5*m_ball*r_ball^2 (solid sphere of uniform density).

so Eb =  1/2*m_ball*V^2( 1+2/5)  .   (corrected 5/29 Was 1/2*m_ball*V^2( 1+4/5)  So the rotational energy adds  40% more to the translational energy.  Rewriting in terms of w_ball gives

E_b = .7*m_ball*w_ball^2*r_ball^2  

Wheel Energy:

E_wheel = .5*I_wheel*w_wheel^2.  where

I_wheel = m_wheel*(r_wheel*.84)^2  (ref blog post https://vamfun.wordpress.com/2015/05/17/finding-the-moment-of-inertia-of-a-vex-wheel-using-parallel-axis-theorem/)

Energy Conservation:

E_wheel_initial = E_wheel_final + E_ball     This assumes that the wheel is not being powered by the motor during launch and that the extra energy needed for the ball comes from the flywheel.   Also, friction and ball compression energy losses are assumed zero to simplify this analysis but can be significant in actual percentages derived.   I am focusing  on how increasing flywheel mass lowers the percentage range errors caused by ball mass variations.

E_wheel_initial/E_wheel_final = (1 + E_ball/E_wheel_final)

Lets expand E_ball/E_wheel_final

E_ball/E_wheel_final = (.7*m_ball*w_ball^2*r_ball^2)/(.5*I_wheel*w_wheel_final^2)

= 1.4*m_ball*w_ball^2*r_ball^2/(m_wheel*r_wheel^2*.84^2*(2*w_ball*r_ball/r_wheel)^2)

= .4954*m_ball/m_wheel

SInce  m_ball = 60 g and m_wheel = 180 g   m

_ball/m_wheel = 1/3

So  E_ball/E_wheel_final = .165    for a single 5″  wheel flywheel     .165/n for n flywheels.    So the ball energy is almost equal to the 1/6 final energy of the wheel

Range Tolerance analysis:

So how does R vary with m_ball from all this.   Well , we know the range is proportional to V^2 which is proportional to w_wheel_final^2 which is proportional to E_wheel_final.

From above E_wheel_final = E_wheel_initial/(1+ .4954*m_ball/m_wheel)

So due to proportionality of R and E_wheel_final we can say

R/R_0 = ((1+ .4954*m_ball_0/m_wheel)/(1+ .4954*m_ball/m_wheel))

where R_0 and m_ball_0 are the nominal values without errors.

We can use R range= R_0(1+ %e_r)   and m_ball = m_ball_0*(1 + %e_m_ball) to work with % changes.

Then with some manipulation we can get %e_r as a function of %e_m_ball

%e_r  = -%e_m_ball/(2.02*m_wheel/m_ball_0 +1 + %e_m_ball)

Now m_wheel = n*.180 kg   and m_ball= .06 kg  so we can write an approx.

%e_r = -%e_m_ball /( n*6.06 +1)     where n is the number of 5″ vex wheels.

Lets put in a few numbers:

Assume %e_m_ball = 10%  then the range error is

n = 1, %e_r =  -1.42%

n = 2, %e_r =  -.76%

n = 3, %e_r =  -.52%

n = 4, %e_r =  -.40%

n = 5, %e_r =  -.32%

So you see the benefits of having a higher  flywheel mass to ball mass ratio.   The use of  two 5″ wheels in a single wheel design can reduce a potential 10% range error from ball mass variations  to  1% ( less than a ball radius).   To keep the spin up time to a reasonable number of seconds requires about 2 393 motors per wheel so 2 wheels costs 4 motors.   So there is a motor tradeoff to get that  higher accuracy with heavier flywheels.

Advertisements

Proposed Towed Ocean Debris Location and Evaluation Robot (TODLER)

May 22, 2013

Algalita has an informal sensor working group to help them define requirements for a 2014 voyage to sample plastic debris in the Eastern Pacific ocean.    I had proposed using robotics to assist them in some way such as a ROV or possibly R/C boat or helicopter with cameras.     These are local aids but the general problem of mapping the ocean debris remains largely unsolved due to inadequate sensors.   I began thinking there would be a need for a coarse debris ocean plastic sampler that could be towed by any ship or research vessel in the ocean including the Liquid Robotics Wave Glider which would be cheap, reliable and easily deployed.     So I wanted to start a requirements study for a proposed Towed Ocean Debris Location and Evaluation Robot (TODLER)

Why TODLERTotal debris weight data can be useful in estimating plastic content:  The plastic in the ocean is now reaching weights that are 6 to 40 + times more than the  dry biomass floating in the ocean.   E.g.  Algalita reported in 2001 that the plastic to plankton dry weight of the  was 6.1:1.   Subsequent voyages found much larger ratios… nearer to 40:1.   The ratio is increasing every year due to the influx of plastic from the rivers, ship dumping and natural disasters such as the Japanese tsunami.    Although we are interested in the amount of plastic in the ocean… measuring the total debris weight would give a reasonably accurate assessment due to the large plastic to biomass weight ratios.    It would avoid the tedious job of carefully separating biomass and plastic in the lab and give many more opportunities to collect samples world-wide.  The samples would measure the weight of wet biomass plus the debis so the ratios would be slightly lower than those mentioned above.

A total debris sample might have one additional data point… the difference between the dry and wet weight of the sample.   This could give an indication of the amount of biomass present.    The usefulness of this would vary depending upon the ratio of plastic to biomass.   On tows that do not sample a lot of plastic one could reduce the error in the plastic weight estimate by about 16% (in 6:1 ratio sample) but this would be of little use in a 40:1 .    The wet/dry ration would require some type of air or centrifugal water extraction device.   A tradeoff study would determine the cost effectiveness of the wet-dry weighing.

Concept:  This is a small towed robotic vehicle that contains  a mini  Manta plankton trawl net capability which can collect a debris surface sample, weigh the contents , record and transmit data to the towing vessel and then clean the net for another sample.   The sample time would be programmable and be based upon a flow sensor to ensure that the ocean area covered is consistent for each sample.  The total debris weight would be used to estimate the plastic debris weight.

There would be several versions of the system each with different capabilities. The baseline version would sample only the surface at <5 knots and be towable by small craft say less than 50 ft.  Follow-on versions would be capable of  sampling at greater depths and at higher speeds.   The higher speeds would allow TODLER to operate during normal ocean cruising speeds for small yachts or research vessels.  This could allow a large  amount of data to be taken by volunteers willing to tow the robot.  Automatic data logging would be a useful feature to simplify the tasks of the volunteer.     If proven successful, it might be adapted later for large cargo ships  to take data during normal voyages.     These added capabilities would change the design significantly due to the weight and stress on the towing tether.  However, possibly adding an intermediate small craft like pontoon boat/raft which had the main tether load attached to it could mean the TODLER would have a uniform interface for all its tow boats.

Prototype design driving requirements:

I)Sizing:

I.1)Towable by a  Wave Glider which can patrol the oceans at speeds from .4 to 1.5 knots using the power of the waves.   The Wave Glider weighs about 200 lbs and displaces a maximum 300 lbs.  If we assume that the drag is proportional to the displacement and we don’t want the Wave Glider to slow down too much.. then perhaps we should keep the TODDS at 30 lb limit and require it to have an aerodynamic shape.

I.2)Portable enough to fit on the Algalita  25ft x50ft ORV .  Perhaps a volume of a large duffel bag including its tow ropes and electronics.

I.3)Max off-board sensor power  13.3v at 3 amps or 40 watts. (Wave Glider driven)

II)Performance

II.1) Initial tow speed capability: 5 knots

II.2)Final tow speed capability TBD knots:   near the maximum speed of the Algalita ORV. (although Manta nets are typically towed at a maximum speed of 2.5 kts we would want the capability to collect plastic on outward and inward journeys without slowing down.  This could drive biomass into the mesh possibly making the scrubbing process more complex.)

II.3)  Net area:  TBD   I would like this to be small to make cleaning easier and to keep the robot volume small.   If it was 10% of the area of a Manta Trawl (209 sqin) this would make it around 20 sqin or the area of a 5 in diameter circle.  To match the ocean area of a manta trawl the tow distance would have to be increased from  about .7 km to 7 km.  If towed by a Wave Glider there could be a series of circular tows made during a voyage that would allow the sample taken to be constrained to a 1 sqkm area.

II.4 Net samples before replacement:

Prototype 40 samples

Wave Glider improvement:  Last 6 months (180 days x 6 samples per day)    ~1000 samples

This might involve having spare nets that can be changed periodically.

II.5  Measure only the wet weight of the sample.

More later:

Is this a viable thing to do??  Your comments are welcome.