Mass Spring System: Modeling & Trajectory Analysis

[carb]

Hello,
I'm now working on a charge limiting system. I tried modeling it an easy way to be studied.

https://imageshack.us/a/img842/4572/3zvv.png

A steel ball (mass "m") moves horizontaly at a continuous speed to the right.
The ball has to pass over the obstacle (king of cam) which implies the compression of the spring (stiffness "k"). The spring is prestressed and applies a force Fp to the ball before passing the cam.

I'd like to figure out the ball trajectory (ball inertia has to be considered), to determine the maximum height reached by the ball, and the distance "d" of the ball jump before coming back to the ground.
I made some research on the web but I feel lost... Which differential equation to start with ? Is it possible to use a spreadsheet to plot the trajectory ?

Many thanks for any kind of help.

 

[Simon Bridge]

You start by drawing a free-body diagram for the mass for when it leaves the end of the ramp.

You will need the initial velocity as the ball leaves the ramp, and yo want to consider what speed is being maintained constant and when? i.e. is the horizontal speed maintained constant at all times? Maybe the total speed is a constant up until the ball leaves the ramp, then it is allowed to change freely?

 

[carb]

Hello Simon and thanks for your answer.
To make a first approximation I indeed consider that the horizontal speed is constant at all times. The translation motion is done by an electric motor, which one is switch-off when the ball pass over the cam (sensor), but due to the mechanical clearance between parts (reduction gearbox) I think I can consider that the speed actually start decreasing after the ball come back to the ground.

 

[Simon Bridge]

So draw a free-body diagram for the ball once it is released - the vertical force will depend on the height (because of the spring).

If you release the ball at the top of the ramp, then I'd set:
- negligible air resistance
$t=0$ when the ball is at the top of the ramp.
$S=$uncompressed length of spring (which you don't have - but you do have $F_p$?)
$C=$ceiling height
- the ceiling is furthest the ball could go up without hitting anything.
- the ceiling height is the distance from the floor to the ceiling.
$y=$vertical displacement, measured downwards from the ceiling.
... so - before mounting the ramp, the ball has $y=C$, $\dot{y}=0$, and $\dot{x}=u$
... at the top of the ramp, $y(t=0)=C-L\sin\alpha$ etc. You do it.

1. Use trig to find the vertical component of the velocity at the top of the ramp.

2. Use Hook's law to find expressions for $F_p$ and $F(y)$ due to the spring - eliminate S to get an expression for $F(y)$ in terms of things you know.

3. Use $\sum F=ma$ to find a 2nd order DE in $y$
... solve it as an initial value problem for speed $\dot{y}(t>0)$ and displacement $y(t>0)$.

4. From there you can solve for the things you want:
i.e. max height is when vertical speed is zero.If the spring is very strong the ball will just slam into the floor at the base of the ramp, very weak and the ball will hit the ceiling, either situation breaks this model. I'm guessing you have some means to adjust Fp?

 

[carb]

Thanks a lot for this resolution method. I feel closer to the results now.
Indeed, I know the value of Fp and can adjust it. I can also change the mass of the ball (however in a quite small range, something like +/-10%). I have to go sleeping now but I'll work on it tomorrow and let you know how it goes !