Half Power Bandwidth⚠ unverified

Understanding the Parameters

• Natural frequency $\omega_n$ — the centre of the resonance peak. Bandwidth scales with it, so at a given damping ratio, higher-frequency modes have proportionally wider peaks.

• Damping ratio $\zeta$ — bandwidth is directly proportional to damping: $\Delta\omega = 2\zeta\omega_n$. This is the inverse of the quality factor's message — high damping means a broad, low peak.

• "Half power" / 3 dB — the band edges are where the response amplitude falls to $1/\sqrt2$ (0.707) of the peak, i.e. where the power (amplitude squared) is halved — a 3 dB drop. The frequencies are $\omega_1, \omega_2 \approx \omega_n(1\mp\zeta)$.

• Measuring damping — from a measured FRF, read the peak frequency $\omega_n$ and the two half-power frequencies; then $\zeta = \Delta\omega/2\omega_n = (\omega_2-\omega_1)/2\omega_n$. This is the standard half-power method, the frequency-domain counterpart of log decrement.

• Link to $Q$ — the quality factor is $Q = \omega_n/\Delta\omega = 1/2\zeta$ (Quality Factor Q); bandwidth and $Q$ are reciprocal descriptions of the same peak.

Derivation (Approaching a Proof)

The magnification factor of a forced SDOF system peaks near $r = \omega/\omega_n = 1$. The half-power points are the frequencies where the response amplitude is $1/\sqrt2$ of the peak, so the squared response (power) is halved. Setting the magnification $M(r) = M_{\max}/\sqrt2$ and solving the resulting quadratic in $r^2$ (for light damping) gives the two roots

$$r_{1,2}^2 \approx 1 \mp 2\zeta \;\Longrightarrow\; \omega_{1,2} \approx \omega_n(1 \mp \zeta).$$

The bandwidth is their difference:

$$\Delta\omega = \omega_2 - \omega_1 \approx \omega_n(1+\zeta) - \omega_n(1-\zeta) = 2\zeta\omega_n.$$

The approximation is excellent for the light damping ($\zeta < 0.1$) typical of structures; it is where the half-power method gets its accuracy.

Dimensional check. $\Delta\omega = 2\zeta\omega_n = (\text{–})\cdot(\text{rad/s}) = \text{rad/s}$ — a frequency, as required ($\zeta$ dimensionless).

History and Development

The half-power (3 dB) bandwidth is borrowed from electrical resonant-circuit theory, where the same $Q = f_0/\Delta f$ relation defines filter selectivity, and applied to mechanical resonance. It became the standard frequency-domain damping measurement in experimental modal analysis: sweep or impact-test a structure, plot the FRF, and read damping from each peak's width. It complements the time-domain Logarithmic Decrement method, with each preferred in different test setups.

Related Concepts: Quality Factor Q, Damping Ratio, Magnification Factor, Natural Frequency Mass Spring, Logarithmic Decrement, Damping From Log Decrement

Notes: 3 dB (half-power) points at amplitude $1/\sqrt2$ of peak, $\omega_{1,2}\approx\omega_n(1\mp\zeta)$. $\zeta = \Delta\omega/2\omega_n$ (half-power method). $Q = \omega_n/\Delta\omega = 1/2\zeta$ (Quality Factor Q). Light-damping approximation.