# Question about elastics: any good mathematical analysis out there?

I don’t find the physics involved with elastics to be intuitively obvious to the most casual of observers. To fully understand what’s going on, I feel obligated to break down all the forces into vectors and see what’s happening at some deeper level, but I’m lazy enough to wonder if anyone has already published some analysis of this sort. I’ve goggled around, looking for such an analysis, but so far have found nothing technical tailored to Vex.

I don’t mind my kids using rules of thumb or trial-and-error to get their machines working, but I always feel better if I have some sense of the physics involved.

Anyone know of a mathematical breakdown (vectors, etc.) of elastics used on, for example, reverse double 4 bars?

don’t know of anything so I just went ahead and did it. (warping up a trig based physics class at my local community collage so this stuff is fresh on my mind)

the first thing I did was to draw out a simple four-bar and put the elastics in a somewhat similar position to where I would put them in practice and then moved them to the corners to make the math easy. (rubber-bands in red)

then I took out the stuff that didn’t matter using my intuition. (middle left)

I then solved for angle ‘X’ (angle C-A) using vector addition. (lower-middle right and lower left)

https://vexforum.com/attachment.php?attachmentid=8795&stc=1&d=1418413610

after that I got board and made a lot of mistakes so I will just give you the diagram of whats going on. Fr will equal Fc sin(4) from which you could find the toque. or you could find the y component of Fr which would give you the upward force.
https://vexforum.com/attachment.php?attachmentid=8796&stc=1&d=1418413649

Hope this helps.  It’s certainly a start. I guess what I’m looking for is some kind of technical paper outlining the kinematics, vectors, maybe even some treatment of the energy transfers going on for various angles using good ole rubber bands, etc.

If it doesn’t exist, then it certainly would make a nifty project for a bright high school kid or college level VexU team. I’m just saying. What I do know is that every rubber band is different and that they do not follow hooke’s law. Perhaps their force to shape change function is a weird curve. I also know that as you add more rubber bands, the joints get tight and friction increases.

But generally, to make a rough estimation, I would stick to calculating torque they exert on a certain joint with respect to angle change ( or any joint if their angles change at the same rate, like in the case of a four bar or six bar) and know that this torque is added to the activated joint with the huge gear. Then I would see how much torque requirement is relieved from the lift’s breakpoint with the elastics, in the case of a bar lift, the point at which the lift extends farthest forward.

You might want to empirically test out the function between shape change and force for a rubber band and then use a model to estimate that curve in order to do theoretical calculation.

Again, rubber bands are not reliable. They break and get loose, and I am pretty sure different brands perform differently. I can do theoretical calculation, and I will probably enjoy it as I love physics. But the easiest way is to have some understanding about elastics and torque and then to start putting them on and tuning them.

Things to keep in mind:

1. The force function is different for every elastic, although I think it is close to hooke’s law in the middle of its range. This means you generally means you want the change in the length of the elastic to be a small proportion of the length of the elastic to get near-constant force.
2. The amount of elastic you can add is generally limited by the force when lowered (you want the lift to stay down at match start), but you normally need the force most when the bars are parallel to the ground (greatest torque for the motors)
3. The force at a given position is:
(number of elastics)*(force of elastic in given position)*sin(angle elastics meet what they pull)
4. Try disconnecting the motors and just feel where there is too much force, and where you need more

Just by being aware of these factors you can optimize fairly well. To really perfect it you can plug all everything into a geometric modeling tool like geogebra and adjust things until you get what you want.

https://www.rose-hulman.edu/~moloney/Ph425/0143-0807_33_4_771RubberBands.pdf Has some Nice info about the forces that the rubber bands provide at different lengths. I also remember a caltech analysis several years ago that gave info specific to the #32 rubber bands, ill try to find that and post it here.

Edit: http://design.caltech.edu/Archive/2001/handouts/elasticband32.html Here is the analysis from earlier. Hope it helps!

The most recently quoted papers look good from the elastic side.

In theory, you can make a “weightless” arm by exchanging gravitational PE=mgh with spring PE=kxx. The tricky part is to get the right amount of spring stretch at each point in the arm movement. For a simple arm, I proposed using a cam shaped like a cartoid curve, which wraps up a string pulling on the rubber band.
Once the cam shape is correct for the mechanism, the amount of force can easily be adjusted by increasing number of rubber bands, and the pre-stretch.

Very nice vector analysis. . I did notice a couple of thongs that did not align with what I learned in AP physics C. First, the Pe of a spring is 1/2K*x^2. However, you would want to ballance forces and torques across the axle to create a weightless system. I can try to do an analysis soon, when I’m bored in a history class Rolling 1/2 into K simplifies the eqn clutter, similar to xx as x^2.
But since we might need both F=kx and PE=kxx/2 at the same time,
its better to use the same value of k for both, which means we need /2 for PE. Its also a good practical example of integration/differentiation.

The spreadsheet shows that using a variable radius cam to balance the force at every cam rotation also exactly balances the PE.

What is the second thing you noticed?

First practical step might be to calculate these:

• How many rubber bands worth of PE do you need?
– How much PE=mgh do you want to cancel?
– What is the available PE in a rubber band?

The Two things that I noticed in the post above:
[LIST=1]
*]The 1/2 was missing from the PE equation. I understand why you did this now, and accept that, However,
*]You Used Potential energy to try to calculate for the number of rubber bands/force required.
[/LIST]

This is incorrect. Potential energy is not relevant to the lever arm, as the sum of the PE will not be 0. You have to use forces, and therefore, because it is on a rotating point, Torques because teh sum of the torques must = 0.

To start, I drew all of the torques that could act on a simple lever arm:
https://vexforum.com/attachment.php?attachmentid=8811&stc=1&d=1418759460

After drawing this, I put together the following equation for setting the torques equal to each other without the motor.
https://vexforum.com/attachment.php?attachmentid=8816&stc=1&d=1418759617
https://vexforum.com/attachment.php?attachmentid=8813&stc=1&d=1418758010
I then solved this for the number of rubber bands required to lift X amount of weight at a distance.
https://vexforum.com/attachment.php?attachmentid=8814&stc=1&d=1418758010
I then put the theoretical weights of the bar with a cube and simple claw into the equation and calculated that 4.26 rubber bands at 12 in stretch perpendicular to the lever arm would create a weightless system.
https://vexforum.com/attachment.php?attachmentid=8815&stc=1&d=1418758010     It always warms my heart to see anyone actually using some math on this forum.

I thought the goal was to help the motor move the arm from low to high,
not just hold it steady at one position.
What happens when the arm is not perpendicular to the rubber band?
When the arm moves up, the tension in the rubberband goes down, so there is less force and the motor has to do more work.
When the arm moves down, the motor is working against the rubberband to stretch it more.

If the gear had a smaller radius when the arm were down, and a bigger radius when the arm were up, maybe the forces and torques would be more balanced at all arm positions, rather than just the horizontal position.