How to Build the Perfect Yo-Yo for a 30m Jump

[dislect]

Hi guys!

My undergrad school is having a "yo yo" competition.
We build the yo yo which is thrown from the height of 30m, connected by a 10mm thick elastic rope. The factors of success depend on who can make the yo yo jump back as high as possible (meaning the less energy lost the better), and then we start counting how many times it jump above the line of 5m hight. The more important factor is the initial back jump.

Can I use your assistance in figuring out the best way to create the yo-yo?
First, how should I tie the rope to the yo-yo in order to preserve as much energy and less friction?
Should I use a ball bearing in the yo yo?
What geomatry is the best for the yo yo? Do i want as little momentum inertia as possible?

Any other suggestions and idea will be more the welcomed :)

 

[AlephZero]

One important factor is what happens when the yoyo reaches the bottom of the rope.

The linear momentum has to be changed from going down to going up. That will lose a lot of energy. If you just tied a rock on the end of a rope and dropped it, it wouldn't "bounce" very high.

But the angular momentum does not have to change because the yoyo is rotating in the same direction as it goes down and back up again.

So you want as much angular momentum as possible compared with the linear momentum.

Find out what the breaking load of your 10mm rope is. I expect the winner of this competition will be a seriously large and heavy yoyo. (If you can make it say 2m in diameter and weighing 50kg without breaking the rope, then go for it!)

 

[dislect]

the restrictions are a weight of 10kg max and the yoyo has to be blocked inside a 1.2m imaginary cube

what about the geometry of the yoyo? should i go for minimal inertia momentum and max the weight as close as i can to the center? what about the way it is tied to the yo yo, or its radius?

 

[rcgldr]

what about the geometry of the yoyo? should i go for minimal inertia momentum and max the weight as close as i can to the center? what about the way it is tied to the yo yo, or its radius?

The previous posts mention that you want to maximize angular momentum. so spoked hubs (approximates a hollow cylinder) and a small axis would be best. Depending on your budget, you could try connecting two bicycle tires or rims to a common solid axis as a prototype, with an inner hub with flanges to constrain the rope and keep the rope from rubbing on the tires / rims. The fiction with the sides of what ever constrains the rope will probably be the main source of energy loss.

 

[dislect]

can you explain why spoked hubs are best? it gives a bigger momentum of inertia and therefor bigger angular momentum?
and what about using ball bearing? would it help?

i thought of maybe creating some sort of conic shaped axis so the rope is folded on tope of each other but still remains centered when it is fully streched, and choosing a smooth meterial to constrain the rope and minimize friction but there's still the friction of the rope rolled on itself

 

[rcgldr]

can you explain why spoked hubs are best? it gives a bigger momentum of inertia and therefor bigger angular momentum?

A hollow cylinder has the highest amount of angular inertia because all of it's mass is at the outer radius. Any other shape will have less angular inertia. A hollow cylinder needs some means to be connected to it's axis, so low mass spokes would need to be used to connect the hollow cylinder "rims" to the axis.

ball bearing

I don't understand where this would be used.

conic shaped axis

sounds like a good idea.

 

[dislect]

thanks a lot!
now has to calculations, we need to present some sort of calculation to the aproximate height we think the yoyo will bounce at the first time.
given that my intial energy its mgH, the rope is L and so the yoyo goes down to height H-L how can i find an expression to its angular momentum and linear momentum and come up with its height in his bounce-back? (lets say that the energy loss is E_loss and ill get into it in further detail later)

 

[dislect]

yet another question, how does the length of the axis affect the yo yo?

 

[sophiecentaur]

The change in translational direction at the bottom need not lose you much energy as long as you have a resilient mounting (a spring) which can return the Kinetic Energy. The spring could be part of the top mounting or a torsion spring at the bottom.
I think the major loss of energy could end up being the friction between string and sides of the spinning yoyo. This can be reduced if you have a pair of non rotating 'cheeks' between the rotating sides and the string as a 'guide' to feed the string in between the rotating sides without brushing against them.
The optimum may not be to have the yoyo rotating too fast because there could be significant turbulence in the air it will stir up. Maximising the Moment of Inertia sounds like a good idea.

A flexible steel wire or tape may be better than a string in the conventional sense.

 

[dislect]

spohie thank you for your comment
i was thinking about using a spring at the end of the fall to return kinetic energy and reduce the loss of linear momentum but i couldn't picture myself how the build it. could you explain a bit more about that/ilustrate how?
im not sure how to make non rotating cheeks at the sides when the entire yo yo is spinnig.. but i was thinking of creating a smooth funnel shaped cone from the side circles that create the yoyo to the yoyo axis so the rope 'lands' on the funnel and is force to go down to the center but with low energy loss due to friction

 

[sophiecentaur]

These cheeks could be free to rotate on the spindle (good bearings) and would tend to stay in one orientation because the string friction would be the only significant force on it, I guess the cheeks would rotate slightly as the paid-out string would be putting a tangential force on it. (slow speed, though, so little energy loss)
Your idea of a conical slot might be good, too.
I was wondering how much of a problem the wobbling of the yoyo might increase friction with the string against the sides and perhaps a three wheel system with two slots and two parallel strings could help stabilise and reduce wobble. Would rules permit?

A spring at the top could have a chosen spring constant to resonate with the mass on the end so that the 'jerk' would be smoothed out. The tension in the string will be increased. briefly, as the string passes over the top of the spindle and starts to coil up on the other side. This pulse of tension will be sinusoidal (I reckon) and will last for half a rotation of the yoyo. (see attachment). I think that the resonant frequency of the suspension should be equal to 1/(jerktime) so the spring would take up the energy and return it in a half cycle. When you've found the rotation speed of the yoyo at the bottom, adjust the suspension spring so that it 'tunes' to this and you should minimise energy loss by making sure the energy goes into the spring and not into the string- which may be lossy.

 

[dislect]

hi, thanks a lot for your help but I am having some difficulties picturing the system. could you make a drawing of the rotating cheeks on a bearing, the 3 wheel system and the spring?

also i tried calculating the angular speed at the bottom (without friction at the moment) using energy conservation: mgh=0.5I1*w^2 + 2*0.5*I2*w^2 + 0.5*m*v^2
when I1 is the moment inertia of the axis : 0.5*mR_axis^2 and I2 is the moment inertia of the 2 side wheels which are made like a bike wheel so most of the mass is far from the center : 0.5(mR1^2+R2^2)
and i got that w=2*sqrt{gh/(3R_axis^2 + 2R1^2 + 2R2^2)}
and i expect to get a result that suggests the bigger R1 and R2 (and the closest they are to each other in size) are the higher angular speed but that's not the result ..

 

[sophiecentaur]

This diagram shows how I see the cheeks mounted. They would be v. light and they would prevent the string from brushing against the wheels (massive torus on light disc, fixed to spindle and locked to drum). The bearings would allow the cheeks to rotate freely on the spindle. The proportions are just how it came out with a naff drawing package.
Losses will be mostly through string friction (within material and against sides) and air turbulence so spokes are not an option, I think.

I don't see how you expected to get the fastest rotation with a large drum diameter (that is what you were saying?). For fastest rotation you want the slowest translation - for total KE to be = mgh. That would be with a minimal drum radius. IS there a mention of translational KE in your calculation? It has to be a factor, surely, and equal to revs times drum circumference.

I actually think that the most efficient yoyo would have NO rotation and just an enormous long suspension spring. High speed rotation is a source of energy loss and the more elastic energy storage the better. You would have to read the rules carefully to see if anything as simple as a suspension spring is even allowed. If not, you'd have to have the resilience in the form of a torsion spring between the drum and the spindle.

 

[dislect]

thanks for you comments sophie
im having some difficulties understanding how the cheeks on bearings help reducing energy loss. is there a way to see that using equations? I am not sure as to how to approach this physics wise.

as i understood and by reading the other comments and other threads on yoyo motion, the bigger moment of inertia I the more angular speed (like bycicle wheels) but when i tried writting down the energy euqation as written above - i could see it happening. what do you mean by translational energy? could you help me with the intial energy equation (writting down all the factors)?

i don't think that it isn't allowed to use a spring inside the drum but dosent this mean a system of many springs inside? otherwise, how do i make sure that the spring is exactly colinear with the string at the end of the yoyo fall?

thanks a lot for your help!

 

[sophiecentaur]

In a toy yoyo, the string is constantly rubbing agains the rotating sides. That is a major loss of energy. 'My' discs will not be rotating significantly and will have little relative motion to the string. Whatever the friction force (F), the power loss will be Fv, where v is the relative speed of the two surfaces. v for the discs, when in contact with the string, will be much than what v would be for the string chafing against the rotating wheels.

Your comments about bicycle wheels, MI and speed are wrong, I think.
There will be a given amount of energy available, yes? The rotational energy is given by
I ω²/2, if I remember right.
That means, for a given amount of energy, more moment of inertia means less angular speed - not more. But the available proportion of the original mgh that's available for will depend on the radius of the drum (?) because the yoyo must be actually moving downwards too, with translational KE. I guess this must mean that, all other things being equal, you need a small radius of drum.
I'm sure I could but can't really be naffed to do the sums. You do it- HAHA. It's your project.

 

[rcgldr]

I actually think that the most efficient yoyo would have NO rotation and just an enormous long suspension spring.

In my opinion, the goal is minimum linear and rotational speeds, which would be accomplished with a small radius hub combined with large radius bicycle wheel like setup (most of the mass in the rims). The energy loss due to the transition at the bottom would be minimal, since the the linear speed and it's transition would be minimal, and the transition from unwinding to winding on a small radius hub would not create a lot of jerk.

Note with this setup, the descent and ascent rate of the yo-yo is going to be slow, since almost of the energy is going into angular energy.

If you can manage to avoid the rope scraping the sides, then most of the losses will be due to aerodynamic drag as the tires spin and losses related to rope flex, but since linear speed is relatively slow, the angular speed related drag would mostly setup a boundary layer, without a lot of drag. The rope flex losses are another issue. If the rope can't efficiently wind around a very small hub, there may be some optimal slightly larger hub that reduces losses related to rope flex, but then that increases linear speed and jerk at the bottom.

The other loss would be related to energy lost in the rope due to the jerk at the bottom due to the transition in linear speed, which is minimized with large wheels, small hub. If the rope itself is somewhat elastic, that would help.

Reducing jerk with a top mounted spring would required that the spring parameters be choosen to not result in the yo-yo bouncing during the transition at the bottom. Even subtle bouncing could result in the rope winding loosely at the transition during the 'free fall" portion of the bounce, then resulting in friction forces at tension returned.

About the inner bearing mounted hub sides, without some type of guide above or weighted hubs, the hubs will rotate due to the small amount of friction in the bearings, and the string could still be rubbing. The drawback is the inner hubs add non-rotating mass to the yo-yo, increasing the linear speed of the yo-yo and the jerk at the bottom. Another option would be a very light, smaller diameter, lower friction hub as part of the axis. It if was possible to construct this using teflon coated hubs, that would help. Another option would be some type of wax or lubricant on the inner hub surfaces.

I have no idea how you're supposed to estimate the losses without actually building a yo-you and measuring the height lost in each cycle.

 

[sophiecentaur]

I agree with most of that but why should the energy during the 'jerk' be lost? It can just as easily be stored and returned to the system using a suitable spring. By suitably 'tuning' the spring / mass combination, the majority of the translational KE at the bottom could be returned.
I think the problems of having a spindle that is very thin and the associated losses in the tight curvature of the string at the centre would make it unattractive.

 

[rcgldr]

I agree with most of that but why should the energy during the 'jerk' be lost?

The rope isn't 100% elastic, so during a transition when tension and length are increased, some of the energy is converted into heat. If the linear speed of the yo-yo is slow, then the losses due to jerk would be low, and a spring wouldn't be needed. The main resevior of energy is in the angular energy of the yo-yo, and it's not affected significantly by a small amount jerk.

I think the problems of having a spindle that is very thin and the associated losses in the tight curvature of the string at the centre would make it unattractive.

I agree. If it's too thin, flexing in the rope is an issue. If it's too thick, the linear speed increases and angular speed decreases, in which case jerk would be an issue. Using the thinnest and most flexible rope possible would allow a relatively thin hub.

 

[dislect]

hi guys, got some news
the rope used in the competition is mammut gym rope 10.2mm
http://www.mammut.ch/en/productDetail/201001170_v_0130_150_102_3/10.2+Gym+Rope.html;jsessionid=AA501869D461023EBBCA86620B505416

im confused about what's happening at the 'jerk'. how can you explain the energy loss there using equations? its a group project and i tried convincing my friend about the loss there but couldn't explain it mathematicly

 

[rcgldr]

Im confused about what's happening at the 'jerk'. how can you explain the energy loss there using equations?

An example might make it easier to under stand. Connect a weight to rope that attached securely, then drop the weight, it won't bounce back up because the rope is not elastic like a spring.

 

[dislect]

is there a name for the kind of energy loss the depends on the elasticity that i can google? i want to start using formulas so given elasticity K for the string how can i calculate its energy loss once the rope reaches its end (an estimation)?

 

[rcgldr]

is there a name for the kind of energy loss the depends on the elasticity that i can google?

Hysteresis. The force during deformation is greater than the force during recovery, with the energy lost being converted into heat. Also, if the forces involved are strong enough, there will be permanent deformation (the rope remains stretched even with no tension on it), another type of energy loss.

 

[sophiecentaur]

If you want to avoid energy loss in the stretched string then a spring with a low k value will store the energy preferentially. Steel springs have low hysteresis.

 

[dislect]

i was talking with my friends about using a spring but we couldn't figure out how and where to place it in the hub-axis ?
rcglder thanks for the Hysteresis ill do some reading :)

 

[sophiecentaur]

The way to connect a spring within the yoyo would be to have the drum on a bearing around the spindle and to connect the drum to the spindle with a clock-type coiled spring. The spring would be normally rotated by a constant amount corresponding to the weight of the yoyo but would rotate at the bottom as the impulsive load increased. Once the string had started to wrap in the other direction, the spring would return to the original position, returning its potential energy. If the rules permit a spring on the top support then that would be a much easier solution to implement.
I am convinced that this energy return system would be far more efficient than relying on the resilience of the string. When you consider how many oscillations it takes for a simple mass hanging on a spring to come to rest, you are talking of energy loss on only a few percent for each bounce. Compare this with the passive toy yoyo behaviour where the yoyo returns to, say 10 or 20% of its release height and hardly completes more than two cycles.

 

[dislect]

so with the clock spring assembly, at first when the yoyo is dropped only the hub moves and side wheels remain still. after a short amount of time the spring is completely drawn and entire yoyo (including the side wheels) begin to gain angular velocity. when the yoyo reaches its bottom i can't see how the spring helps, it even resists the current directon of angular velocity

 

[sophiecentaur]

The spring stores energy that can be returned. A piece of string will never be as 'elastic' as a spring so energy will be lost when it is stretched impulsively. You are right that the spring will stretch initially but only in response to the weight of the yoyo on a full spool (small force at large distance from CM) . It will stretch a lot more due to the impulse at the bottom.

I really don't know what this means: "it even resists the current directon of angular velocity". It merely connects the drum to the rest of the yoyo with a torque that is the same as that provided by the string (at all times) - needless to say, the very small 'unsprung MI' of the drum would still exert an impulse on the string.

 

[rcgldr]

If the axis (hub) of the yo-yo is relatively small, there isn't going to be that much force from the impulse when the yo-yo reaches the bottom. Think of what happens during the 1/2 revolution where the string transitions from being unwound to being wound at the axis, which smooths out the transition from the yo-yo descending to ascending, as well as the relatively slow rate of descent of the yo-yo just before the transition.

Using a spring to effectively replace a portion of the string may not make that much difference, and it may not be allowed, since the goal here is to build a yo-yo, not a spring and weight mechanism.

 

[sophiecentaur]

Yes I agree that a small drum has an advantage but only if bending and coiling the string involves no loss. Some actual measurements are needed to find how much is lost this way and also to find out the energy lost during the jerk.
As I said at the start - if no spring is allowed then you can't use a spring. But if you are, then it is worth considering as a way of limiting the energy loss during the jerk.
I actually wonder whether a steel wire might be the best solution for the 'string'. This has low hysteresis and low internal friction.

 

[rcgldr]

I actually wonder whether a steel wire might be the best solution for the 'string'. This has low hysteresis and low internal friction.

If you are allowed to use something else instead of a spring, then a strip of mylar tape would be better if it could be guided to wind up on the hub without a lot of friction. Mylar tape is very flexible, and could work with a relatively small hub radius.

 

[sophiecentaur]

Great for low friction and it might allow a small enough spindle to eliminate a significant jerk problem.