Show that there are two values of $\omega$ at which resonance occurs
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Hint: $\cos^3(\omega t) = \cos(3 \omega t)/4 + 3\cos(\omega t)/4$. You get resonance in $m y'' + k y = \cos(r t)$ when $\cos(r t)$ is a solution of the homogeneous equation.
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user1038665
Updated on October 31, 2020Comments
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user1038665 about 3 years
Consider a mass-spring system with $m = 1$, no damping, and $k = 4$. Find the general form of the mass’ motion if there is no forcing. Now suppose that we apply an external force $F(t) = 3\cos^3(\omega t)$ for a constant $\omega$. Show that there are two values of $\omega$ at which resonance occurs, and find both.
$m = 1$
$k = 4$
$my'' + ky = 0$
$y'' +4y = 0$
$y(t) = c_1\cos (2t) + c_2\sin (2t)$
Now I'm unsure how to show/find the two values for resonance. I know the equation is
$y'' + 4y = 3\cos^3(\omega t)$
but what does that tell me?
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user1038665 about 11 yearsSo resonance will occur at 1/4 and 3/4?
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Robert Israel about 11 yearsNo. The important thing is the frequency, not the amplitude.
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user1038665 about 11 yearsSo is this when $3\omega = 2$, or $\omega = \frac{2}{3}$, and when $\omega = 2$?
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Robert Israel about 11 yearsYes, that's right.