How to Determine u(t) for a Damped System with Step Function Force?

[Ploppen]

Hi everybody!
I have a big question for you, I have been staring me blind on this problem I got.
And wonder if someof you could help mee with it?

the guestion is this:

"Find the time dependent oscillation u(t) for time t>0 for a damped system with one degree of freedome where c=50 kg/s, k=20 000 N/m, m=2 kg and which suddenly affected by a step function force p(t)=po H(t). (that is, the force in zero when t<0 and constant po=1000 N when t>0.) At time t=0 the mass is in the equilibrium position with zero velocity. (easiest to not use Laplace, even though it is one possible way)."

I whould be soo nice if somebody could help me with this task!

thanks!

best regards
//Tobias

 

[Mindscrape]

Well, if you know how to do it with Fourier Transform then go for it, but I can tell you how to do it otherwise. Let's work through the algebra first, and put in numbers later because numbers are boring. Fill in between my dots...

I like working with the ODE in the form

$$\ddot{x} + 2\Beta \dot{x} + \omega_0^2 x = H(t_0)$$

So for time less than zero it is trivial...

and for greater than zero, you get...

Which when you solve for the ODE will be...

Applying the initial conditions shows...

And plugging them back in gives...

(Possibly expressed fancily by...)

 

[Ploppen]

I don't have the paper here but I will take a look at it and get back to you in about a week beacuse I have and exam I need to make. And if i don't know how to do it I will ask you for help again if that's okej?!
Thanks for now!