What is vibroacoustic coupled analysis?

Analysis solving structural vibration and the surrounding acoustic field (air or water) together, covering panels radiating sound and sound shaking panels and passing through them. It's used for vehicle interior noise, wall sound insulation, loudspeaker enclosures and appliance noise. Structure is solved with finite elements and the acoustic field with finite or boundary elements, coupled at the interface.

How is panel insulation estimated?

At low frequency the panel's mass mainly reflects sound, and transmission loss is about TL = 20 log₁₀(f·m″) − 47 dB (m″ surface density) — the mass law. 2 mm steel (15.7 kg/m²) gives 31 dB at 500 Hz and 43 dB at 2 kHz, rising 6 dB per doubling of mass or frequency.

Bending-wave speed in a panel rises with frequency and at some frequency equals the speed of sound in air. Near this coincidence frequency, sound and panel vibration couple strongly and sound passes through easily, so insulation falls sharply. 6 mm glass coincides near 2 kHz, 2 mm steel near 6 kHz and 1 mm steel near 12 kHz; thicker, stiffer panels coincide lower.

Vibroacoustic Coupled Analysis

Category: Coupled Analysis – Acoustics | Revised 2026-10-01

Theory: mass reflects, matched wave speed lets sound through

Overview

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Do panels of the same weight insulate equally?

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Nearly, at low frequency. Sound striking a panel tries to shake it, but heavier panels move less and reflect the sound, so low-frequency insulation depends mostly on surface density (mass per m²). At higher frequency stiffness matters. Bending waves in the panel speed up with frequency and at some frequency match the speed of sound in air. At this coincidence frequency sound and panel waves move in step and sound passes through easily. Stiffer, thicker panels coincide lower, so among equal-weight panels, one whose coincidence lies in an audible band insulates worse in practice.

Mass law

$$ TL \approx 20\log_{10}(f\,m'') - 47\ \mathrm{dB},\qquad m'' = \rho h $$

f is frequency (Hz) and m″ surface density (kg/m²). A field-incidence approximation valid well below the coincidence frequency.

Coincidence frequency

$$ f_c = \frac{c^2}{2\pi}\sqrt{\frac{m''}{D}},\qquad D = \frac{E h^3}{12(1 - \nu^2)},\qquad c_B = \left(\frac{D}{m''}\right)^{1/4}\sqrt{\omega} $$

c is the speed of sound in air (343 m/s), D bending stiffness, h thickness and c_B bending-wave speed; c_B = c at f_c.

Coffee Break Trivia

Laminated glass and double glazing

Window glass tends to coincide in the speech and traffic band (1–3 kHz), a weak point for insulation. Laminated glass bonds two panes with a soft resin interlayer that damps bending vibration and shallows the coincidence dip. In double glazing the air between panes acts like a spring, creating a low-frequency mass–air–mass resonance, so air gap and pane thicknesses are varied to shift resonance and coincidence to less audible frequencies.

Worked examples

Example: surface density, coincidence and mass-law transmission loss

Panel	Surface density	Coincidence frequency	TL at 500 Hz	TL at 2 kHz
Steel 1 mm	7.9 kg/m²	12.0 kHz	24.9 dB	36.9 dB
Steel 2 mm	15.7 kg/m²	6.0 kHz	30.9 dB	42.9 dB
Aluminium 2 mm	5.4 kg/m²	6.0 kHz	21.6 dB	33.7 dB
Glass 6 mm	15.0 kg/m²	2.0 kHz	30.5 dB	(near coincidence)
Gypsum board 12.5 mm	10.0 kg/m²	2.9 kHz	27.0 dB	39.0 dB

Mass-law TL rises 6.0 dB per doubling of surface density or frequency. For 6 mm glass the mass law would give 42.5 dB at 2 kHz, but coincidence pulls the real value well below.

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6 mm glass and 2 mm steel weigh about the same, yet differ at 2 kHz.

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Glass is thick and stiff in bending, so it coincides low, at 2 kHz, where its bending waves move in step with sound in air and sound passes through easily, dropping well below the mass-law 42 dB. Thin steel is compliant in bending and coincides high at 6 kHz, so at 2 kHz it performs close to the mass law. 2 kHz matters for speech and traffic noise, so this dip governs window insulation. The dip depth depends on panel damping — the less damping, the deeper. Vibroacoustic analysis therefore includes structural modes, damping and coupling to the acoustic field to obtain transmission loss by frequency.

Analysis workflow

Use the mass law and coincidence frequency to judge problem bands and material suitability.
Get structural modes and damping from a finite-element model.
Model the acoustic field with FEM or BEM, coupled to the structure at the boundary.
Compute transmission loss or radiated sound by frequency, checking coincidence and resonances.
Calibrate damping and boundary conditions against measurements such as reverberation-room tests.

Coffee Break Trivia

“Thicker glass didn't make it quieter”

In one building, window glass was thickened from 5 to 10 mm to cut road noise, but it didn't get as quiet as expected. Doubled mass should give 6 dB by the mass law, but 10 mm glass coincides near 1.2 kHz, actually worsening insulation in a key traffic-noise band. Switching to laminated glass of the same weight to damp vibration shallowed that dip and reached the target insulation.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Mass law at all frequencies	High-frequency TL overestimated	Check coincidence
Assuming thicker is always better	Coincidence moves lower	Damping or laminated build-ups
No damping	Exaggerated peaks and dips	Measure damping
Ignoring leaks through gaps	Real insulation lower	Assess gaps
Solving structure and air separately	Coupling effects missed	Solve coupled

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