Damped and forced oscillations: does amplitude really go infinite at resonance, and which frequency does the system oscillate at?

6 answers & replies

• [+][–]Kabir Joshi11thSample· 1mo ago

  Starting from the amplitude formula in NCERT for the driven oscillator:

  $A = \frac{F_0}{\sqrt{m^2(\omega^2-\omega_d^2)^2 + \omega_d^2 b^2}}$

  At $\omega_d = \omega$ the first term inside the root vanishes and $A = \frac{F_0}{\omega_d b}$. It's infinite only if $b = 0$, i.e. no damping at all. In any real system $b > 0$ so the amplitude is large but finite, and smaller damping gives a taller, sharper peak. Your energy line is right too: energy goes as amplitude squared, so $A^2 \propto e^{-bt/m}$.

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• [+][–]Rohan Sharma11thSample· 1mo ago

  resonance = amplitude infinite, bridge collapses (the famous soldiers-marching one, we saw the video in class). and with damping the frequency becomes more than the natural one because damping "pushes" it, so $\omega' > \omega_0$. I think 😅 pls correct me if I'm wrong, I mostly remember the video part

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  • [+][–]Farhan ShaikhDropperSample· 1mo ago

    Rohan, half of this is wrong. Look at the formula Ananya typed: you subtract $b^2/4m^2$ under the root, so $\omega' < \omega_0$. Damping slows the oscillation a bit, the period gets slightly longer. For light damping the change is tiny so people say 'about the same'.

    And amplitude is not infinite, it is just maximum. Bridges and glasses break because the amplitude gets large enough to exceed what the material can take, not because it's infinite.

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    • [+][–]Ananya Reddy11thSample· 1mo agoAuthor

      ahh okay so my energy line was right but my "ω′ greater" guess would have been wrong too. fixing the orange box now. thank you everyone 💛

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• [+][–]Kavya NaikDropperSample· 1mo ago

  Swing: think of the swing as having its own favourite rhythm. When you start pushing, it is confused for a while (transient) but after a short time it settles and swings at the rhythm you are pushing, not its own. That is the steady state of forced oscillation: frequency = driving frequency. Only when your pushing rhythm matches the swing's natural rhythm do all your pushes add up in phase, and the amplitude builds up. Amma used to push me at random on the swing and I went nowhere, the one time she timed it properly I almost flew off. 😄

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• [+][–]Ishita GoyalDropperSample· 1mo ago

  Quick summary I keep on my planner:

  • Free damped: amplitude $\propto e^{-bt/2m}$, energy $\propto e^{-bt/m}$, $\omega' < \omega_0$.
  • Forced steady state: oscillates at $\omega_d$, not $\omega_0$.
  • Resonance when $\omega_d \approx \omega_0$: amplitude is max, but finite if $b>0$.
  • More damping: peak lower and broader.

  NEET likes 'a system oscillating under periodic force: its frequency is that of the ___' - the answer is the driving force.

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