Natural frequency is the number of oscillations per unit of time made by a resonant system when left to itself after excitation. Frequency is measured generally in hertz
(1 Hz = 1s^-1). Natural frequency and natural angular frequency are both used to describe oscillation phenomena. The following relationship exists:
w =2p · f

w = natural angular frequency

f = natural frequency

For a simple spring-mass oscillator with damping the following equation applies:

w = w₀ with w₀ =

d =

w_o = natural angular frequency of the undamped oscillation

c = spring rate

m = oscillating mass

u = damping ratio

d = decay factor

b = damping coefficient

No oscillation occurs with aperiodic damping (u>1). Oscillators with a continously distributed energy store (e. g. a rod or plate) have a theoretically infinite number of natural frequencies. A metal bellows (bellows mechanical seal) must be regarded as an oscillator of this type. If it is treated as a spring clamped at both ends, the initial natural frequency of the axial oscillations is calculated as:

f = 0,5 ·

c = spring stiffness

m = bellows mass

In the case of forced oscillations, the position of the resonant frequency and its amplitude are conditional on the damping. The smaller the damping ratio, the bigger the amplitude and the closer the resonant frequency approaches the natural frequency of the undamped oscillation. Resonant oscillations must be prevented at all costs. They cause damage to mechanical seals, leading to their failure (oscillations).