Motion of particle dropped onto an elastic band

[polomeister]

Hi All,

This is part solution, part request for peer review.

I wasn't able to find a model for the motion of a particle dropped onto an elastic band so I've created my own in Excel (attached).

Note that the model is primarily focussed on the motion of the particle, and not the elastic band, although in reality the two are dependent.

Key Results

• Once the particle has hit the band the vertical motion is like a sine wave.
• If the particle is dropped off-centre there is also horizontal motion.
  • Movement towards the middle of the band might be expected – as this should minimise the potential energy.
  • However, the particle seems to move briefly away from the centre before returning. (I found this interesting and counter-intuitive.)
  • In general, the particle traverses a large proportion of the elastic band.
• For certain combinations of displacement, mass and elasticity, the particle bounces off the elastic band.

Requests from the Forum

• It would be great if someone could check over my working. This is one component of a larger simulation I'm working on.
• Tips for reducing the approximation errors in the working (see "Assumptions/Limitations" below).
• Experimental verification that a particle does initially move away from the centre. Might be a nice project for a physics student!

The attached file

• The format is the new Excel binary format .xlsb (even 1 row of simulation exceeds the 484k limit if saving as an .xlsx file). Send me a PM if you require the file in a different format.
• There are four tabs
  • "Diagram" – Shows the setup when the particle has displaced the elastic band by a vertical distance $y$. Forces induced in the two components of the elastic band are assumed to act on the particle.
  • "Particle Dropped on Band" – This is the main working.
    • Boxes highlighted in green are inputs
    • I've had to curtail the number of rows to allow upload. I recommend at least 10,000.
    • The time-step is 0.001s. This gives a relatively high error-rate (about 10% as measured by Energy). Reducing the time-step to 0.0005s or less gives much improved accuracy. (I was suprised that the algorithm required such small increments to calculate accurately.)
    • See "Algorithm" below for basis of calculation
    • There is a block on the right hand side that calculates the total Energy in the system, which I've used as a check.
    • The last row will always have an error in it and should be excluded from summary calculations.
  • "Graph of x, y and energy" – Graph showing $x$, $y$ and Energy with respect to time. Note: It won't look right until you complete enough rows.
  • "Forces on Elastic Band" – This was a bit of pre-working that I've left in. It calculates the forces in the elastic band just by pulling it down a vertical distance $y$ at a horizontal displacement $x$. Note $y$ is positive whereas in the main working it is negative.

Algorithm
There is a complication to the equations of motion in that, once the particle has hit the elastic band, the force exerted on the particle is a function of position.

• Position, velocity and force are calculated component-wise. The formulae below apply for both components.
• Position: $x_{t+\delta t} = x_t + 0.5(v_t + v_{t+\delta t})\delta t$
  • Change in position is an average of the starting and ending velocities. This improves the accuracy of the modelling.
• Velocity: $v_{t+\delta t} = v_t + F_t \delta t / m$
  • Change in velocity is based on the force at the start of the time-step. Because force depends on position, it's not possible to take an average without generating circular calculations.
• Force: $F_{t} = F(x_{t})$
  • Force is calculated using the starting position of the particle for each time-step.
• The forces from the elastic band only operate in the region $\{0 \leq x \leq l; y<0\}$. This is handled through the indicator function in Column U.
• The elastic extension forces are $|F_L|=k(a-x)$ and $|F_R|=k(b-(l-x))$, where
  • $a$ and $b$ are the hypotenueses generated by extending the elastic band vertically down at horizontal position $x$
  • $l$ is the length of the elastic band
  • $k$ is the elasticity coefficient

Assumptions/Limitations

1. Ends of band are fixed
2. Left peg of the band is the Origin
3. Band is weightless
4. Only two dimensions
5. The particle is updated before the elastic band
6. The elastic band is assumed to comprise of a left hand and right hand component with a vertex at the position of the particle.
7. No friction
8. Band is assumed to reset to a horizontal line once particle is higher than $y=0$ – in reality it would continue to oscillate. This probably means the model is only valid for the period up to the particle achieving its maximum height for the first time. One might expect the band to oscillate in height as much as the maximum depth
9. Looks like a fixed amount of energy is being added per cycle
10. I've only checked motion when the particle is dropped vertically
11. Note: You can waste a lot of iterations in the 'freefall' phase of the model. A quick workaround is to work out the velocity the particle would have at $y=0$ and start from there.
    

Improvements

• Sort out the Energy issue (but reducing the size of the time-steps mitigates this)
• Write a macro to animate the movement of the particle
• Improve the modelling of the elastic band:
  • First step would be to treat the components as two elastic rods, with a mass per unit length
  • Next step would be to further segment the elastic band into $n$ such rods
• Please suggest more!

Thanks & best wishes – I hope this is a worthy edition to the forum and perhaps it may even help someone out in future.

P

 

[mfb]

As far as I understand your formulas, you assume that the fraction of the band left/right of the mass is always determined by the position of the particle. This is certainly not true. A realistic (but massless) band will be somewhere between "same tension at both sides" (no friction) and "same proportion as at the first point of contact" (maximal friction).

It is odd that some formulas (for θ, for example) use the position of the following time step.
Your energy is not conserved.Edit: I found a glitch with the following values:

	horizontal	vertical
x0	0,5		0
v0	0,01		0
		
mass	1	
		
"Const Ext Force
per unit mass"	0	-9,8
		
time step	0,002	
		
Length of Band	1	
Elastic const	9

At step 7086, it "tunnels" through the stretched band.

 

[polomeister]

Thanks mfb - apologies for the delay - back at work now so less time for physics!

A few points:

1.

As far as I understand your formulas, you assume that the fraction of the band left/right of the mass is always determined by the position of the particle. This is certainly not true. A realistic (but massless) band will be somewhere between "same tension at both sides" (no friction) and "same proportion as at the first point of contact" (maximal friction).

The model is no-friction. You're right, this is a limitation but, with a small enough time increment, will it be material?

2. Will look into the θ issue you raise

3. Time step of 0.002 is quite large for the model - I found I needed 0.0001 (ie 20 times smaller) to get it to work satisfactorily. I was surprised at how sensitive the model was to the size of the time increment - if anyone knows of more efficient computational techniques, it would be good to hear

4. I hadn't tried it with horizontal motion - I was interested in a vertical particle falling on the band but will look into

Thanks very much for taking the time to reply & try out the model - much appreciated! I've got limited time to work on this now, so my replies might be a bit sparse.

 

[mfb]

The model is no-friction. You're right, this is a limitation but, with a small enough time increment, will it be material?

That issue has nothing to do with your timesteps.

Consider the following (extreme) example (black=rubber band):

Something like ~1/5 of the band is to the right of the particle - it is stretched, but only by a small factor relative to the rest position. In your calculations, you stretch the tiny red element by a large factor to get the right part of the band. This leads to very large forces from that side, and would require a lot of friction to prevent slipping of the rubber band.

 

[PJC01]

.S. I've also posted this in the Physics Forums under the username "CleverM" if anyone wants to follow the discussion there.

Hello,

Thank you for sharing your model and findings with the community. Your approach and methodology seem sound and well thought out. I appreciate your efforts to address the limitations and potential sources of error in your model, and your openness to peer review and suggestions for improvement.

One potential improvement could be to incorporate a more accurate representation of the elastic band, as you mentioned in your "Improvements" section. This could potentially involve breaking the band into smaller segments and accounting for the mass and elasticity of each segment. Additionally, incorporating friction into the model could also provide a more realistic representation of the motion.

Overall, I think your model is a valuable contribution to understanding the motion of a particle dropped onto an elastic band and I look forward to seeing how it progresses with further experimentation and improvements. Keep up the great work!