# Why is the resonance frequency of an undamped oscillator equal to the undamped resonance?

I have read this post: 'How do you define the resonance frequency of a forced damped oscillator?'

And I see that the resonant frequency occurs at the undamped oscillation frequency $\omega_0$ as opposed to the damped oscillation frequency $\omega_d$. I don't understand why this is the case though? In the post, it stated that at resonance the ' energy flow from the driving source is unidirectional', and I'm sure this is the reason why it is the natural frequency of the system not the driving frequency at resonance, but I didn't really understand the rest of the post to see if it answered this question.

• Two uses of the word "undamped" in the title? Jan 1 '17 at 17:14
• Hi, I'm the author of that other question. Please define $\omega_d$. There is no single way to define "damped oscillation frequency": there's the frequency of free oscillations, the frequency at which the amplitude maximizes, and other things and they're not the same frequency. Jan 1 '17 at 17:24

Confusion arises because there are a number of ways in which to define resonance.

In mechanical systems, e.g. a spring-mss system, it is much easier to measure amplitudes rather than velocities and so graph to illustrate forced oscillations and resonance are of amplitude of driveN system against frequency of constant amplitude driveR.
When the amplitude of the driveN is a maximum there is amplitude resonance.
The frequency at which amplitude resonance occurs decreases as the damping of the driveN system increases.
For small amount of damping that change in amplitude resonance frequency is small and the amplitude resonance frequency is approximately equal to the free oscillation frequency of the undamped driveN system.

In electrical systems, e.g. an LCR series circuit, current is easier to measure than charge and so it is current which is usually measured to investigated forced oscillation and current resonance.
In that case the maximum current in the driveN system occurs at the same frequency irrespective of the amount of damping of the driveN system.
The current resonant frequency is equal to the free oscillation frequency of the undamped driveN system.
This is also energy resonance where the maximum power is transferred from the driveR to the driveN system.

Velocity resonance for a mechanical system is equivalent to current and energy resonance for an electrical system and charge resonance for an electrical system is equivalent to amplitude resonance for a mechanical system.

Now amplitude $A$ and maximum velocity $v$ are connected $v=\omega A$ where $\omega$ is the frequency of the oscillations.

So the amplitude resonance graph is of amplitude of the driveN $A_{\rm N}$ against frequency of the driveR $\omega_{\rm R}$ whereas velocity resonance graph is of maximum velocity of the driveN $\omega_{\rm R}A_{\rm N}$ against frequency of the driveR $\omega_{\rm R}$.
It perhaps should be of little surprise that the two types of resonance occur at different driveR frequencies.

• There is a simple way to define the resonant frequency which cuts through all of this potential confusion: For steady state response, the resonant frequency is when the displacement is in quadrature with the applied force. For non-mechanical systems, choose an appropriate analogy to "force" and "displacement." IMHO, if people choose to define it differently, they can lie in the bed of their own making! Jan 2 '17 at 3:59
• @alephzero I think that your idea is a good one but you will still have different resonant frequencies depending on which parameters you choose to compare the phase of. Jan 2 '17 at 4:33
• For a sinusoidal motion, displacement, velocity and acceleration are necessarily in a fixed phase relationship with each other. Of course if the external forces on the body are not all in the same phase, you have to make some arbitrary assumption. One advantage of this definition is in looking at energy flows: when the external force is in quadrature with the displacement (or equivalently in phase with the velocity), the work done by the external force is $>=0$ at all times (that fact is one justification for using the definition), and the average power is easy to calculate. Jan 2 '17 at 5:42