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I'm trying to find the amplitude of steady state response of the following differential equation:

$$\ddot{x}+2p\dot x + {\omega_0}^2x=\cos(\omega t)$$


A particular solution is $$x_p=\Re{\dfrac{e^{i\omega t}}{\omega_0^2 - \omega^2 + i2p\omega}} $$

The amplitude at steady state is then $$A=\dfrac{1}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2p\omega)^2}}$$

The denominator has minimum value when $\omega^2 =\omega_0^2 - 2p^2 $: $$A=\dfrac{1}{2p\sqrt{\omega_0^2-p^2}}$$

This expression seems to suggest that the amplitude goes to infinity as $p$ approaches $\omega_0$.

But amplitude has to be finite(from other examples of LRC tank circuit etc).
Pretty sure I'm wrong but not able to see where. Any help?

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  • $\begingroup$ @Qmechanic Ah right.. $p=\omega_0$ gives a single real root for homogeneous part: $\ddot{x}+2p \dot{x}+\omega_0^2x=0$. I also notice that $p=\omega_0$ is $0$ input frequency AND $p^2 > w_0^2/2$ corresponds to negative input frequency $\omega$ .. but that's it... not so sure what it means :( $\endgroup$
    – across
    Feb 8 at 14:12
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    $\begingroup$ The denominator is a sum of two positive quantities: $(\omega^2-\omega_0^2)^2$ and $(2p\omega)^2$: thus it cannot go to $0$ unless $p$ has an imaginary part. $\endgroup$ Feb 8 at 15:40
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    $\begingroup$ Linked. Show your work. The linked question has answers writing the denominator in a form minimized by inspection and thus preventing your algebra error. $\endgroup$ Feb 8 at 15:41

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You algebra is wrong. If $\omega^2-\omega_0^2=-2p^2$ you get $(-2p^2)^2+(2 p\omega )^2= (2p^2)^2+(2 p\omega )^2$= under the square root and, being the sum of two postive numbers, this can never be zero.

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Without math (or almost ;-)- a system driven by an external forcing function at resonance is accepting energy input that the system cannot easily get rid of. This makes the energy pile up in the system which makes the amplitude of the oscillations grow over time and get big enough to blow it up.

In electrical systems like underdamped RLC circuits, the energy piles up to the point where it makes the system driver emit smoke, or shut itself down through the use of an automatic circuit called a fault detector or foldback which is intended to prevent smoke emission.

This is a big deal because the process of catching the smoke and pumping it back into the system is difficult and expensive.

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