Show that there are two values of $\omega$ at which resonance occurs

2,787

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.

Share:
2,787

Related videos on Youtube

user1038665
Author by

user1038665

Updated on October 31, 2020

Comments

  • user1038665
    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?

  • user1038665
    user1038665 about 11 years
    So resonance will occur at 1/4 and 3/4?
  • Robert Israel
    Robert Israel about 11 years
    No. The important thing is the frequency, not the amplitude.
  • user1038665
    user1038665 about 11 years
    So is this when $3\omega = 2$, or $\omega = \frac{2}{3}$, and when $\omega = 2$?
  • Robert Israel
    Robert Israel about 11 years
    Yes, that's right.