# Mass dropped with preloaded spring

1. Oct 16, 2015

### carb

Hello everybody,
I'm trying to figure out how to calculate xadd, the additional compression of a preloaded spring (initially compressed of xp) on which a mass is dropped from a height "h".
I also wonder :
- if the final compressed spring length is the same with or without preload ?
- if the spring receive an additional compression xadd >0 whatever the drop height ? (even at h=0.001 ?)

I have some remembering about my academic studies (Energy conservation laws applied to spring) but I feel quite uncomfortable since I did not practice for a long time. Thanks a lot for any kind of help.

2. Oct 16, 2015

### Hesch

It is not the same.
Yes it does.

Consider the energy of the cylinder, just before the frame hits the ground:
E1 = m*g*(L-xp) + ½*m*v2
Now consider the energy, E2 , when the cylinder has been brought to a halt ( last figure ).

The difference in energies: ( E1 - E2 ) must have been absorbed by the spring, so:

( E1 - E2 ) = xadd Fspring(x) dx

Last edited: Oct 16, 2015
3. Oct 19, 2015

### carb

Hello Hesch !
Thanks to your help I did it this way :
E1 (just before hit) = m*g*(L-xp) + ½*m*v2 + ½*k*xp2
with v = √(2*g*h)
E2 (just before rebund phase) = m*g*(L -xp -xadd) + 0 + ½*k*(xp + xadd)2

Then E1 = E2 led me to this factorisation form (xadd as variable) :

but I actually changed my mind about the study result and decided to get xadd as an input and to find the corresponding height ("how high can I drop the cylinder to get exactly 100% of potential energy absorbed by the spring").

k: 10 000N/m
m: 1kg
g: 9.81 m/s2

Spring
L: 0.010 m
xp: 0.003 m
xadd_max: 0.006 m (considering a minimal spring contiguous wire height of 0.001 m)

calculation led to h_max = 0.0307m (max height avoiding potential energy causing impact on the ground)

to confirm the results, here is what I got using a mechanism simulation software :
(velocity as initial condition of the dynamic study (just before hit) : v = √(2*g*h) = 776.067 mm/s)

Good to see that lowest reached position is 1mm according to minimal spring height used is previous calculation (contiguous wire)

I hope the results are not accidentally good .
Thanks again Hesch !

4. Oct 19, 2015

### Hesch

What ?? So you gave up finding an algebraic solution, and started playing with numbers instead ?

Well, I calculated that

E1 = m*g*(L-xp) + ½*m*2*g*h = m*g*(h+L-xp)