Driven un-damped harmonic motion

AI Thread Summary
The discussion focuses on solving a driven un-damped harmonic motion problem involving a mass 'm' subject to a spring force and a driving force. The approach involves guessing a particular solution of the form x(t) = A*cos(wd*t) + B*sin(wd*t) and substituting it into the governing equation. The participants derive the particular solution as x(t) = F/(w^2 - wd^2)*cos(wd*t) under the condition that w > wd, yielding C = F/|w^2 - wd^2| and φ = 0. In the case where w < wd, the solution takes the form x(t) = -F/|w^2 - wd^2|*cos(wd*t) + π, resulting in φ = π. The discussion emphasizes the importance of considering the phase constant based on the relationship between w and wd.
KaiserBrandon
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Homework Statement



A particle of mass 'm' is subject to a spring force, '-kx', and also a driving force, Fd*cos(wdt). But there is no damping force. Find the particular solution for x(t) by guessing x(t) = A*cos(wdt) + B*sin(wdt). If you write this in the form C*cos(wdt - \phi), where C > 0, what are C and \phi? Be careful about the phase (there are two cases to consider).

Homework Equations



basic inhomogeneous second-order equation solving.

The Attempt at a Solution



So I got the equation F = Fd*cos(wdt) - k*x = m*x'', and put it in the form x'' + w2*x = F*cos(wdt), where w2\equiv k/m, and F \equiv Fd/m. I took the first and second derivative of the equation they told us to use for our guess, and subbed them into x'' + w2*x = F*cos(wdt). I ended up with A(w2 - wd2)*cos(wdt) + B(w2 - wd2)*sin(wdt) = F*cos(wdt). I therefore deduced that F = A(w2 - wd2). Therefore the particular solution would be xp(t) = A(w2 - wd2)*cos(wdt). Is there anyway that this can somehow simplify for the form they want us to put it in, C*cos(wdt - \phi), or did I do something wrong?
 
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You know F and have to express A and B in terms of F, wd and w.

ehild
 
ok, so then I get x(t) = \frac{F}{w^2-wd^2}*cos(wd2*t) for the particular solution. I still can't figure out how this can take the form that is asked for.
 
KaiserBrandon said:
ok, so then I get x(t) = \frac{F}{w^2-wd^2}*cos(wd2*t) for the particular solution. I still can't figure out how this can take the form that is asked for.

It is x(t) = \frac{F}{w^2-wd^2}*\cos{(w_d*t)}

instead and has the desired form if w>wd.
C = {\frac{F}{|w^2-wd^2|}, \Phi =0

In the opposite case

x(t) =- \frac{F}{|w^2-wd^2|}*\cos{(w_d*t)}

you have to include a phase constant of pi:

C = {\frac{F}{|w^2-wd^2|}, \Phi =\pi

x(t) =\frac{F}{|w^2-wd^2|}*\cos{(w_d*t+\pi)}

ehild
 
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