Forced Driven oscillator

1. Dec 27, 2009

drszdrsz

1. The problem statement, all variables and given/known data

http://fatcat.ftj.agh.edu.pl/~i7zebrow/rysunek.jpg
tring constant is k,
object mass is $$M_{1}$$
Damping friction is b
and we wiggle the top end of spring in the above diagram with amount Asin(wt)
(Where A is a amplitude and w is a frequency).
2. Relevant equations
Spring Force:
$$Fs=-kx$$
Damping force is:
$$Fb=-b\frac{dx}{dt}$$
3. The attempt at a solution
I don't need full solution.I just looking for differential equation for this problem.My attempt is:
$$M_{1}\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx=Asin(\omega t)$$
I am sure about left side of the eguation but the right side is propably wrong I hope that someone can tell me where I made an mistake.

Last edited by a moderator: Apr 24, 2017
2. Dec 27, 2009

RoyalCat

Look at the dimensions, you are comparing force with distance. That is dimensionally inconsistent!

Consider the extension of the spring as a function of time, how does the oscillation of the end-point affect it? How does this translate into the force the spring exerts on the mass?

3. Dec 27, 2009

drszdrsz

Do you mean that the force of spring should looks like this:

$$Fs=-k(x-Asin(wt))$$
and now the differential equation will be:
$$M_{1}\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx=k*Asin( \omega t)$$
Now the differential is dimensionally consistent.But, is that correct??

Last edited: Dec 27, 2009
4. Dec 27, 2009

RoyalCat

That's how I would approach the problem, you might want to wait for confirmation from someone else that that's the way to go.

As for actually solving the differential equation, assume that the response, $$x(t)$$ is also a sinusoid. (Think about it, you're driving it at a certain frequency, why would it do anything but oscillate at that frequency?)

That is to say, assume a general solution of the form $$x(t)=B_0\sin{(\omega t)}$$ and then find $$B_0$$ and the conditions under which that is indeed a solution.

5. Dec 27, 2009

drszdrsz

Ok thanks a lot.

ps.
I'm not sure about solution. there mus be transitory solution too(otherwise the stationary solution that is sinusoid).

6. Dec 27, 2009

RoyalCat

To find the transient solution, all you need to do is add 0 to both sides! Remember that the sum of solutions is also a solution for such linear differential equations!
Do you remember doing this to find transients?

If not, this is how it's done (Very long and interesting lecture on the subject!):
http://ocw.mit.edu/OcwWeb/Physics/8-03Fall-2004/VideoLectures/detail/embed04.htm [Broken]

Last edited by a moderator: May 4, 2017
7. Jan 1, 2010