Natural Frequency vs. Resonant Frequency vs. Damped Frequency

The main difference is that resonant frequency is where a system prefers to vibrate or where it absorbs the most energy, while damped frequency is the actual frequency at which a system oscillates when left to its own devices after being disturbed.

1. Natural Frequency ( $ω_{n}$ )

This is the inherent frequency at which a system vibrates when it is disturbed and then left to move on its own without any damping or external force.

Context: Free, undamped vibration (an ideal, "perfect" world scenario).
Behavior: It is determined solely by the system's internal properties, such as its stiffness (how "springy" it is) and its mass (how heavy it is).
In Undamped Systems: It is the only frequency that exists; the system will oscillate at this rate forever.
In Damped Systems: It serves as the "baseline" or reference point. The actual oscillation frequency (damped frequency) and the frequency of max energy absorption (resonant frequency) are both calculated relative to this value.

2. Damped Natural Frequency ( $ω_{d}$ )

This is the inherent frequency at which a system vibrates when it is disturbed and then left to move on its own with damping. This is the frequency at which a system actually vibrates during its free decay (after you hit it once and let go).

Context: Free-vibration systems (like a guitar string fading out).
Behavior: Because the damping (friction/resistance) acts as a "drag" on the motion, it physically slows down the oscillations.
Relation to Natural Frequency: It is always lower than the undamped natural frequency ( $ω_{n}$ ).

3. Resonant Frequency ( $ω_{r}$ )

This is the frequency at which an external driving force causes the maximum amplitude of oscillation.

Context: Driven/Forced systems (like pushing a swing or tuning a radio).
Behavior: At this exact frequency, the system's impedance is at its lowest, and it "soaks up" the most energy from the source.
In Undamped Systems: It is identical to the natural frequency ( $ω_{n}$ ).
In Damped Systems: As damping increases, the peak of resonance actually shifts to a lower frequency than the natural frequency.

Comparison Table

Feature	Natural Frequency $ω_{n}$	Damped Frequency $ω_{d}$	Resonant Frequency $ω_{r}$
System State	Free & Undamped	Free & Damped	Driven / Forced
Physical Meaning	The "ideal" speed of vibration based on mass/stiffness.	The actual speed it vibrates as it dies out.	The speed that causes maximum amplitude.
Equation	$ω_{n} = \sqrt{k / m}$	$ω_{d} = ω_{n} \sqrt{1 - ζ^{2}}$	$ω_{r} = ω_{n} \sqrt{1 - 2 ζ^{2}}$
Relation	The baseline value.	Lower than $ω_{n}$ .	Lowest of the three.

As you can see from the formulas, as the damping ratio ( $ζ$ ) increases, both the actual vibration ( $w_{d}$ ) and the peak resonance ( $w_{r}$ ) shift further and further away from the original natural frequency ( $w_{n}$ ).

Simple Analogy

Imagine a pendulum:

Natural Frequency: If you pull it back and let go, it swings back and forth. That speed is the natural frequency.
Damped Frequency: If you put that pendulum in water, pull it back and let go, it swings back and forth slower than it would in air. That slower speed is the damped frequency.
Resonant Frequency: If you try to push the pendulum with your hand to keep it moving, the specific timing (tempo) of your pushes that creates the biggest swing is the resonant frequency. This tempo will be higher in air than it is in water.

When $ζ$ Reaches $1 / \sqrt{2}$

When the damping ratio $ζ$ reaches $\frac{1}{\sqrt{2}} \approx 0.707$ , the system becomes maximally flat. At this point, the "peak" of the resonance curve disappears because the damping is strong enough to suppress any amplification beyond the initial displacement.

Here is why that happens:

Critical Resonance Damping and Vanishing Resonance

1. Natural Frequency (ωn)

2. Damped Natural Frequency (ωd)

3. Resonant Frequency (ωr)