# Resonant frequency vs Natural frequency

1. Jun 2, 2010

### sgsawant

I happened to sit in a presentation where the presenter displayed 2 separate values for resonant frequency and natural frequency. It was my understanding that the both are same and now I am in doubt.

Please explain - if there is and you know - the difference between resonant frequency and natural frequency.

I am specifically referring to beam resonance (and by extension - through its lumped element model - resonance in electrical circuits).

Regards,

-sgsawant

2. Jun 2, 2010

### SimonRoberts

Here's a +1 for them being the same, as far as I know.

3. Jun 2, 2010

### Stonebridge

Resonance is a condition in which a vibrating system responds with maximum amplitude to a periodic driving force.
Mechanical systems (beams, pendula, springs, wine glasses, guitar strings etc) will have a number of possible frequencies at which this occurs. These are the system's natural frequencies of vibration. For example, a guitar string will have a series of possible frequencies where this happens, the lowest is called the fundamental frequency. The other frequencies are at values which are whole number multiples of the fundamental.
When resonance occurs, the frequency is often called a resonant frequency. This is just saying that resonance occurs when the driving force has the same value as one of the natural frequencies.
A beam can have more than one natural frequency, and therefore can be made to resonate at more than one frequency.
An (LC) series electrical circuit will resonate at a frequency given by f= (1/2π)√LC
This could be called its natural frequency or its resonant frequency. It doesn't really matter. (It's usually called its resonant frequency.)

4. Jun 2, 2010

### sgsawant

Thanks a lot!

5. Jun 2, 2010

### uart

The presenter was most likely referring to the damped natural frequency versus the undamped natural frequency. When the damping factor is small the two are very similar, but as the damping is increased the oscillation frequency decreases.

For example in a parallel LRC circuit the undamped natural frequency is :

$$\frac{1}{\sqrt{LC}}$$

whereas the actual oscillation frequency of the natural response's damped sinusoid is :

$$\sqrt{\frac1{LC} - \frac{1}{(2RC)^2}}$$

Last edited: Jun 2, 2010
6. Jun 2, 2010

### sgsawant

@uart
Well that's exactly what I was looking for. Thanks! Can you point me to your source or give me a link that explains the concept?

Regards,

-sgsawant

7. Jun 2, 2010

### uart

The characteristic equation for a second order system is of the form :

$$s^2 + 2 \alpha s + w_0^2$$

For example in a parallel LRC circuit this would correspond to a function of the form :

$$s^2 + \frac{1}{RC} s + \frac{1}{LC}$$

If the damping factor (alpha) is zero then the roots are at

$$\pm j \sqrt{1/(LC)}$$

and it follows that the natural response is an undamped sinusoid of frequency 1/sqrt(LC).

When alpha is non zero the roots of the (quadratic) characteristic equation are

$$-\alpha \pm j \sqrt{(w_0^2 - \alpha^2)}$$

from which it follows that the natural response is a damped sinusoid of frequency $\sqrt{(w_0^2 - \alpha^2)}$.

Last edited: Jun 2, 2010
8. Jun 2, 2010

Great!