Trigonometric Regression of bouncing ball vertical displacement over time

[RyanPuccetti]

Not sure if this is the right spot to post this question, but here it is:

I am trying to create a trigonometric model for the vertical displacement of the bouncing of a ball with respect to time, however the period of the ball's bounce shortens over time, so I am unsure how to produce a function that properly models the harmonic decay.

Currently my function looks something like: s(t)= | a(e^-kt)cos(bt) |, where a and k are constants and b is some function that models the period over time.
Does anyone know how to model the decay over time using b? Please let me know if there is a way to do this.

.<- what i currently have

 

[Dr.Peterson]

Not sure if this is the right spot to post this question, but here it is:

I am trying to create a trigonometric model for the vertical displacement of the bouncing of a ball with respect to time, however the period of the ball's bounce shortens over time, so I am unsure how to produce a function that properly models the harmonic decay.

Currently my function looks something like: s(t)= | a(e^-kt)cos(bt) |, where a and k are constants and b is some function that models the period over time.
Does anyone know how to model the decay over time using b? Please let me know if there is a way to do this.

.View attachment 38184View attachment 38185<- what i currently have

The actual motion is not a decaying cosine, but a series of parabolas. Ignoring air resistance, the decay occurs entirely in each bounce, not throughout the motion.

Is there a reason you want to pretend it has the form you chose?

 

[blamocur]

Not sure if this is helpful, but the terms that come to mind are Chirps and Chirplet Transform.

 

[RyanPuccetti]

The actual motion is not a decaying cosine, but a series of parabolas. Ignoring air resistance, the decay occurs entirely in each bounce, not throughout the motion.

Is there a reason you want to pretend it has the form you chose?

Currently completing a maths investigation on damped harmonic motion, and the extension portion of the investigation is based on concepts that are harder to model (most are not harmonic, but some form of decaying motion), just trying to explain to model it using a harmonic decaying model to then explain a better way to do it later on (and compare), since this is what is required of me. Are there any ways for how I could deduce an approximation of the periodic decay/ an entire function that more closely resembles the data?

 

[Dr.Peterson]

Currently completing a maths investigation on damped harmonic motion, and the extension portion of the investigation is based on concepts that are harder to model (most are not harmonic, but some form of decaying motion), just trying to explain to model it using a harmonic decaying model to then explain a better way to do it later on (and compare), since this is what is required of me. Are there any ways for how I could deduce an approximation of the periodic decay/ an entire function that more closely resembles the data?

Are you trying to model the actual motion of a bouncing ball (which is not damped harmonic motion), or to find something else to model that is damped harmonic motion?

Can you show us (this part of) the actual assignment? And did the data come from that assignment, so that your goal is to match the data as closely as possible?

 

[RyanPuccetti]

Hi Dr. Peterson,
Very sorry for the late reply,

I have attached an image of the part 3 task sheet below, this investigation is quite flawed since half of the 'options' listed are not even damped harmonic motion, however I do not have enough time to go back and change to one that is harmonic motion. Parts 1 and 2 required me to model a pendulum in undamped and damped systems using a cosine function and decaying amplitude cosine function respectively. For part 3:
1. What would be the best way to get a regression that almost fits the changing period and rest of the bouncing ball motion?
2. How would I best find a series of parabolas to model it (so that I can provide a better approach than the cosine model)?

I used MATLAB demo bouncing ball to collect data, while noise has been added to the data, it is free from air resistance, parallax error etc.
For parabolas, since acceleration is constant at -9.8, then the a(t)=9.8, thus s(t) must atleast be s(t)=-(9.8/2)t^2 + ___ + ____ (by integration), but i can't figure out the others, nor a general case for the nth bounce.

Hope this helps,

Kind Regards,
Ryan

 

[Dr.Peterson]

this investigation is quite flawed since half of the 'options' listed are not even damped harmonic motion

I think the bouncing ball is the only one that is not reasonably modeled as damped harmonic motion!

1. What would be the best way to get a regression that almost fits the changing period and rest of the bouncing ball motion?

How are you doing regression?

2. How would I best find a series of parabolas to model it (so that I can provide a better approach than the cosine model)?

You might want to first gather data only on the height or time of each bounce, to determine for yourself what happens on each bounce -- is the height multiplied by a constant fraction, or something else? That would let you decide how to model the transition from one bounce to the next.

But since what you will be learning by doing this will not be about the given subject, you might want to ask your teacher what to do. At the least you should point out to them that this option is not a good one. But they might tell you just to pretend it is harmonic, so you can practice the intended skills.

 

[RyanPuccetti]

I think the bouncing ball is the only one that is not reasonably modeled as damped harmonic motion!


How are you doing regression?

You might want to first gather data only on the height or time of each bounce, to determine for yourself what happens on each bounce -- is the height multiplied by a constant fraction, or something else? That would let you decide how to model the transition from one bounce to the next.

But since what you will be learning by doing this will not be about the given subject, you might want to ask your teacher what to do. At the least you should point out to them that this option is not a good one. But they might tell you just to pretend it is harmonic, so you can practice the intended skills.

I am doing regression by plotting the period at different times, I defined the period at time t, to be the time at the start of the period. each peak to peak is half a period (since every second maxima is technically a cosine troph), therefore the red distance indicates half the initial period, blue indicates 1/2 the period at the time of the second peak (t=1.9), green distance is 1/2 the period at t=3.5s, so on so on. I plotted all of these points, (period at time t) into desmos and then did different regressions, such as linear regression and exponential regression. I plug these functions into b = 2pi/period, to get b in |ae^-kt(cos(bt))| but none seem to produce a period function of time that actually matches the decay.

Is there anything I'm doing wrong with my process, also are there any more advanced methods to better model the period decay/ the entire bouncing ball?



Kind Regards,
Ryan

 

[Dr.Peterson]

I defined the period at time t, to be the time at the start of the period. each peak to peak is half a period (since every second maxima is technically a cosine troph), therefore the red distance indicates half the initial period, blue indicates 1/2 the period at the time of the second peak (t=1.9), green distance is 1/2 the period at t=3.5s, so on so on.

Properly speaking, there is no period! The time between bounces is not constant, so it is not a period. (And one reason this is not damped harmonic motion is that harmonic motion must have a fixed period.)

Each parabola has its own interval between bounces (horizontal intercepts); the interval between two peaks belongs to two different bounces, making it sort of the average of two successive "periods". I would not use that.

And there is no cosine "trough" here. It is not a cosine.

I plotted all of these points, (period at time t) into desmos and then did different regressions, such as linear regression and exponential regression. I plug these functions into b = 2pi/period, to get b in |ae^-kt(cos(bt))| but none seem to produce a period function of time that actually matches the decay.

That's because it can't be accurately modeled that way. So don't expect it. Whatever you get will have a constant period, and therefore will look wrong.

Tell your teacher what you are doing. If you can't start over with a more suitable problem, your teacher should understand that your model will not work well, but give credit for doing the best you can.

If a physicist were experimenting with a wrong assumption about the physics of his problem, then the fact that the results don't match would be a success, demonstrating, correctly, that the model is wrong and needs to be changed. If he faked the data or something to make it look right, he would be doing bad science and producing a false result. Show the truth!

If you want a function you can use to model bouncing, I think it would have to be expressed as a piecewise function, or just an algorithm to generate each parabola in turn. It will not look anything like $\left|ae^{-kt}\cos(bt)\right|$.

So, where did the data in that plot come from?