PDA

View Full Version : A Differential Equations Question



Mars
2004-Nov-17, 05:14 PM
This is for extra credit, so I don't need the answer I just want to make sure I am doing this right.


A mass of 1 slug, when attached to a spring, stretches 2 feet and then comes to rest in the equilibrium position. Starting at t=0, an external force of f(t)=e^-t*sin(4*t) is applied to the system. Find the equation of motion if the surronding medium offers a damping force numericall equal to 8 times the instantaneous velocity.

Analyze the displacements for t->infinity

So I have the equation:

x" + 8*x' + 16*x = e^-t*sin(4*t)


Am I correct this far?

I have W=mg which would be W= 1*32ft/sec

W=32

since ks=F and F=W and W=mg I get ks=32

k should = 16 because 32ft/2ft = 16

I used m^2+8*m+16=0 to find m=-4 with multiplicity 2 making

Yc=C1e^-4t+C2t*e^-4t

The rest I can get using the Superposition approach....

The other part the stumped me is the t->infinity part, I have forgotten how this goes.

Anyone help please!!

frogesque
2004-Nov-17, 05:46 PM
Arrrrgghhh! Slugs? Who is still teaching fps out there?

Mars
2004-Nov-17, 06:26 PM
LOL

Normandy6644
2004-Nov-17, 06:49 PM
The t-> infinity bit is the difference between steady state and transient solutions. For a damped system with a driving force, you should get a term that goes to 0 as t goes to infinity. That's the transient. The steady state is always present, and (IIRC) generally comes from the solution to the homogeneous part of the equation. Hope that helps. :D

Mars
2004-Nov-17, 07:13 PM
Yes it does, because I got 0. Thanks!

Normandy6644
2004-Nov-17, 07:51 PM
Welcome. :D