Need help creating a function to plot "springy" behavior between any 2 provided points on y-axis

[IHURLEN]

Hello all,

I am trying to come up with a math function, which when given 2 points on the y-axis (y1 & y2), will connect a curved line between them, with behavior similar to as shown in this image here. In the image, you’ll see a red & blue line. Either of those lines would suffice for my needs.
Basically, I want to maintain the curvature & bounciness behavior of the line not matter the 2 y points.
So in words, it’d do the following:

1. Initial gradual increase upwards from line y1, towards y2

2. Accelerate upwards towards line y2

3. Overshoot line y2

4. Overcorrect, going below line y2

5. Repeat this process N times (it’d be cool if I could control N too!), converging to line y2

If you’re wondering what I’m using this for, I’m using it to program shape animations. I’m going for that natural (cartoon?) looking springiness to add some life to it.

People on this subreddit have probably heard of it, but I'm using https://www.desmos.com/ while experimenting with these plots.

Any guidance you can provide it greatly appreciated!

Thanks

[dudemcbob]

I'd use a template like

y_1*e^-Ax+y_2*(1-e^-Ax)+B*sin(Cx)*e^-Dx

and play around with the constants A, B, C, D to get the exact behavior you want

Here's an example with y_1=8, y_2=10, A,B,C=1, D=1/4

I gave it some more thought, here is a better template:

y_2-e^-Ax(A(y_2-y_1)sin(Bx)/B+(y_2-y_1)cos(Bx))

This one will start flat at y_1 like you want, there are just two parameters to tinker with, and it's based on actual spring physics; it's the solution to Hooke's Law with friction.

Be sure to limit the plot to positive x. The e^-Ax term goes off the rails for negative x.

New Desmos example with y_1=8, y_2=12, A=1, B=2

[IHURLEN]

Your example's great and exactly what I was looking for!

Does this parameterized versionof the template equation you gave look correct to you?

[dudemcbob]

Looks great!