Acoustic modal analysis: eigenfrequencies, damping, participation

The eigenfrequencies and mode shapes of the air (and the packings) inside the model, from the same finite-element system as the frequency sweep: undamped modes, optionally the damped (complex) eigenfrequencies with their damping ratio, and optionally how much each mode carries of the swept response. Switched on in Run Settings, panel "Additional acoustic results", before the run.

Last updated 2026-11-11

Theory

Undamped modes solve equation (1) by shift-invert Lanczos. The matrices are those of the gas plus the real parts of the perforate and packing terms: a packing adds $1/\rho_{eq}$ to the stiffness and $1/K_{eq}$ to the mass of its elements, a perforate adds $i\omega/Z_p$ to the coupling of its two sides. These coefficients depend on frequency, so each mode is refined at its own frequency (at most three passes). The modes are scaled so that $\phi^T M \phi = 1$ .

The ports can be rigid (closed ends; mode 0 is then the uniform pressure at 0 Hz) or open (zero pressure on the port faces). Anechoic ports absorb, so they need the damped modes.

Damped modes solve the complex, frequency-dependent problem (2) mode by mode, starting from the undamped mode and repeating until the eigenfrequency changes by less than one part in a million. With anechoic ports the port term of the sweep, $U\,\mathrm{diag}(i k_z/(\rho S))\,U^T$ , is added. The time convention is $e^{+i\omega t}$ , so a decaying mode has $\mathrm{Im}\,\omega > 0$ . Equation (3) gives the damping ratio and quality factor.

Participation projects the swept pressure of the run's own excitation on the mass-normalised undamped modes at every frequency, equation (4). This is computed inside the sweep. The projection uses the mass matrix at mid-band, so with strongly frequency-dependent packings the shares are approximate.

A run with mean-flow convection gives the modes of the medium at rest: the convection terms are not in the modal problem (noted in the results).

Equations

K_t\,\phi = \omega^2 M_t\,\phi

(1)

\left[K + \sum_i c_i(\omega) T_i - \omega^2\left(M + \sum_j m_j(\omega) T_j\right)\right]\phi = 0

(2)

\zeta = \frac{\mathrm{Im}\,\omega}{|\omega|}, \qquad Q = \frac{1}{2\zeta}

(3)

\alpha_n(f) = \phi_n^T M_t\, p(f), \qquad \text{share}_n = \frac{|\alpha_n|^2}{\sum_m |\alpha_m|^2}

(4)

where	meaning	unit
$K_t, M_t$	Stiffness and mass with the real parts of the absorber terms	various
$\phi$	Mode shape	-
$\omega$	Angular eigenfrequency (complex for damped modes)	rad/s
$c_i, m_j$	Coefficients of the perforate / packing stiffness and mass terms	various
$\zeta, Q$	Damping ratio, quality factor	-
$p(f)$	Swept nodal pressure	Pa

Source

The undamped modes of a straight duct closed at both ends match $f_n = n c/(2L)$ within 0.2 %, and with open ends within 0.3 %. A packed resonator gives positive damping ratios.

Valid range

Modes found

The lowest modes up to the sweep's highest frequency, at most the number asked for.

Mean flow

The modes are those of the medium at rest.

Theory

Equations

Source

Valid range

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