# First-order spring/damper system in parallel

1. Oct 13, 2015

### FissionChips

1. The problem statement, all variables and given/known data
A viscous damper, with damping constant b, and a spring, with spring stiffness k, are connected to a massless bar. The bar is displaced by a distance of x = 0.1m when a constant force F = 500N is applied. The applied force F is abruptly released from its displaced position if the displacement of the bar is reduced from its initial value of 0.1m at t = 0 to 0.01m at t = 10m find the values of b and k.

2. Relevant equations
The general differential equation for a spring and damper in parallel with a constant force is given as

$F=b\frac{dx}{dt}+kx$

3. The attempt at a solution
Since the displacement and time conditions are given when the force has been released, I rewrote the differential equation:

$0=b\frac{dx}{dt}+kx$

$-b\frac{dx}{dt}=kx$

Upon integration and simplification,

$x=e^{(-k/b)t+C}$

Applying the constraint $x=0.1m$ when $t=0$, $C=ln(0.1)$

Rewriting the displacement equation,

$x=e^{(-k/b)t+ln(0.1)}$

$x=0.1e^{(-k/b)t}$

Applying the second constraint $x=0.01m$ when $t=10$,

$\frac{k}{b}=-\frac{ln(0.1)}{10}$

Alas, this is where I've run out of steam. I have the ratio of the constants, but I can't solve for either one of them. I'm brand new to differential equations and these spring and damper contraptions (at least analytically), so I'm not sure how to proceed (or if my work up to here is even correct). I have a suspicion that k can be found by dividing the force by the initial displacement, but I'm not positive.

Can anyone point out what I'm failing to see?

Any help is greatly appreciated.

Last edited: Oct 13, 2015
2. Oct 13, 2015

### FissionChips

I've thought about this more since posting, and I'm confident that k can be solved by k = F/x since the damper provides no resistance when the velocity is zero. This whole thread can be deleted if a moderator wishes. Sorry for the waste of bandwidth!