Exploring Simple Harmonic Motion Equations: Damping vs. Driving Force

[imagemania]

(This isn't homework)
I've been crating some notes, and noticed that we haven't been told one equation.
I know for:
[tex]\frac{d^2 x}{dt^2}[/tex] [tex] = \frac{-kx}{m} -\frac{b}{m} \frac{dx}{dt}[/tex]
I know the displacement is:
[tex] x = A \omega e^{-bt/2m} cos(\omega t+ \phi) [/tex]

[Ie damping]

I know for:
[tex]\frac{d^2 x}{dt^2}[/tex] [tex] = \frac{-kx}{m} -\frac{b}{m} \frac{dx}{dt} + \frac{F}{m}[/tex]
I know the displacement for this is:
[tex] x = Asin(\omega t + \phi) [/tex] (omega is for driving force).

But what about:
[tex]\frac{d^2 x}{dt^2}[/tex] [tex] = \frac{-kx}{m} + \frac{F}{m}[/tex]
I.e. No damping, what would x be for this?

Thank you!

[This is out of pure interest]

 

[AlephZero]

When [itex]\omega^2 \ne k/m[/itex], it makes no difference if b = 0, except that [itex]\phi[/itex] will always be 0 or [itex]\pi[/itex]

When [itex]\omega^2 = k/m[/itex] the solution is of the form

[tex]x = A t \cos (\omega t)[/tex]

In words, the response grows without limit, and it is 90 degrees out of phase with the applied force.

 

[mathman]

The solution for your second equation should have a b in it, but it does not appear. For the third equation use the second equation solution with b=0.

 

[imagemania]

Gah first off i made a mistake on the first one, should have no omega next to A (demensions wouldn't equate)

Ok it would seem i have some equations written down wrong (well two).

[tex]\frac{d^2 x}{dt^2}[/tex] [tex] = \frac{-kx}{m} + \frac{F}{m}[/tex]
Would
[tex] x = Asin(\omega t + \phi) [/tex] (i.e. the cos shifts by pi/2 to make sine?)

And
[tex]\frac{d^2 x}{dt^2}[/tex] [tex] = \frac{-kx}{m} -\frac{b}{m} \frac{dx}{dt} + \frac{F}{m}[/tex]

[tex] x = A e^{-bt/2m} sin(\omega t + \phi) [/tex]

Let me know if these are right :)

 

[mathman]

Have you tried verifying them yourself. Simply plug your answers back into the equations.