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oddjobmj
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Homework Statement
A force Fext(t)=F0[1-e(-a*t)] acts, for time t>0, on an oscillator which is at rest at x=0 at time 0. The mass is m; the spring constant is k; and the damping force id -b dx/dt. The parameters satisfy these relations:
b=mq and k=4mq2 where q is a constant with units of inverse time.
Find the motion. Determine x(t); and hand in a qualitatively correct graph of x(t).
(B) Determine the final position.
Homework Equations
Typical form of a damped oscillator:
[itex]\ddot{x}[/itex]+2β[itex]\dot{x}[/itex]+w2x=0
The Attempt at a Solution
This question is from an online, self-study course and we have no lecture for forced oscillators. I am left to figure this out with my book and the internet. I would greatly appreciate a few pokes in the right direction. All the examples I am finding on forced oscillators are of the more common form of: [itex]\ddot{x}[/itex]+2β[itex]\dot{x}[/itex]+w2x=Acos(wt)
Here is what I am working with:
[itex]\ddot{x}[/itex]+2β[itex]\dot{x}[/itex]+w2x=F[1-e(-at)]
I understand that I need to find a solution to the homogeneous equation and also a particular solution and their sum will be my x(t). My solution to the homogeneous equation is simple if I can determine the nature of the damping (which I have problems with below). The tricky part then becomes the particular solution.
I also need to determine the nature of the damping. Is this overdamped, underdamped, or critically damped? How do I relate b=mq to b2=4mk which is the form I am more accustomed to if it were critically damped, for example? I believe that if someone helps me figure out the nature of the damping I will be able to piece together a particular solution.
(B) I see that the driving force quickly approaches F0 and I could take the limit as t->∞ to find the final position. I need x(t) first, though.
Thank you for your time and help!