PSI - Issue 77

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Alessandro Zanarini et al. / Procedia Structural Integrity 77 (2026) 64–70

A. Zanarini / Structural Integrity Procedia 00 (2025) 1–7

Shakers:active #1[2611] mute #2[931] Frequency step [303] = 256.250 Hz Airborne Acoustic PressuresR_WHITE-NOISEstd amp. mod. = 50 [µPa]

Shakers:active #1[2611] mute #2[931] Frequency step [630] = 511.719 Hz Airborne Acoustic PressuresR_WHITE-NOISEstd amp. mod. = 50 [µPa]

Shakers:active #1[2611] mute #2[931] Frequency step [959] = 768.750 Hz Airborne Acoustic PressuresR_WHITE-NOISEstd amp. mod. = 50 [µPa]

Shakers:active #1[2611] mute #2[931] Frequency step [1285] = 1023.438 Hz Airborne Acoustic PressuresR_WHITE-NOISEstd amp. mod. = 50 [µPa]

Shakers:active #1[2611] mute #2[931] Frequency step [1] = 20.312 Hz Airborne Acoustic PressuresR_WHITE-NOISEstd amp. mod. = 50 [µPa]

Complex amplitude [projection angle 0 deg] Dof [1092] DIC_r

(c) ALESSANDRO ZANARINI Spin-off activities from the researches in Marie Curie FP7-PEOPLE-IEF-2011 PIEF-GA-2011-298543 Project TEFFMA - Towards Experimental Full Field Modal Analysis a

(c) ALESSANDRO ZANARINI Spin-off activities from the researches in Marie Curie FP7-PEOPLE-IEF-2011 PIEF-GA-2011-298543 Project TEFFMA - Towards Experimental Full Field Modal Analysis

Figure 2. Examples of simulated white-noise amplitude-modulated R pressure field patterns acting on a flexible plate and its full-field C recep tances , inducing the force in shaker 1, at 20 Hz in a , at 256 Hz in b , at 512 Hz in c , at 768 Hz in d and at 1024 Hz in e .

Acoustic Pressure in WHITE-NOISEstd amp.mod. Air. Pressures RR at dof [1092]

Step[303]=256.250 [Hz] AmpDIC_r=-8.840e+01 [N/m^2] [dB] PhaDIC_r=-0.000e+00 [rad]

3.142

Pha [rad]

-3.142

-8.604e+01

DIC_r

Amp [N/m^2] [dB]

-1.238e+02

20.000

Frequency [Hz]

1023.000

Shakers: active #1[2611] mute #2[931]

(c) ALESSANDRO ZANARINI Spin-off activities from the researches in Marie Curie FP7-PEOPLE-IEF-2011 PIEF-GA-2011-298543 Project TEFFMA - Towards Experimental Full Field Modal Analysis

Figure 3. Example of an airborne acoustic pressure graph in the frequency domain, simulated from Eq.9, in the acoustic dof 1092.

3.1. Indirect excitation force retrieval from sound pressure fields As commented in Zanarini (2024a,b, 2025a,b,c), by reversing Eq.6, with the use of the pseudo-inverse of the vibro-acoustic transfer matrix V af ( ω ) of Eq.5, the forces induced on the structure at the excitation / shaker head by a known complex-valued pressure field ˆ p ( a ,ω ) can be retrieved: ˆ F f ( ω ) ≈ V + fa ( ω ) ˆ p ( a ,ω ) ∈ C . (7) with the pseudo-inverse of the vibro-acoustic transfer matrix V af ( ω ), sized N f × N a and callable V + fa ( ω ), precisely as: V + fa ( ω ) = [ V H fa ( ω ) V af ( ω )] − 1 V H fa ( ω ) ∈ C . (8) The matrix V H fa ( ω ) V af ( ω ), to be inverted at each angular frequency ω , is a complex-valued square matrix of size N f × N f , but this time N f = 1, with a strong simplification of the inversion, as already proofed in Zanarini (2024a,b, 2025b,c). 3.1.1. Modelling of the pressure field As in Eq.7, it is straightforward to obtain the induced force once the pressure field ˆ p ( a ,ω ) is known in its spatial pattern and in the frequency domain. In this work, ˆ p ( a ,ω ) is built as real-valued : the first four spherical Bessel functions of the first kind J b are positioned by the b -th functional S b – with frequency dependent wavelengths – in the spatial pattern, whose overall amplitude is modulated by coloured noises in the frequency domain. There follows: with A 0 as the reference amplitude for the modulation and with α ∈ [ − 2 , 2] indicating the specific noise colour : α = − 2 for violet , α = − 1 for blue , α = 0 for white , α = 1 for pink and α = 2 for red noise , as in Figs.2-3. 4. Full-field receptances in the numerical mapping of airborne acoustic pressures and inverse vibro-acoustics The relevance of the defined acoustic transfer matrix V af ( ω ) should be clear in sight of its pseudo-inverse V + fa ( ω ) evaluation in Eq.8, before the adoption of a specific airborne acoustic pressure model in Eq.9, to obtain the airborne identified force in Eq.7. Examples with the full-field receptances are here given. 4 ˆ p ( a ,ω ) = A 0 ω α 4 b = 1 J b ( S b { a ,ω } ) ∈ R , (9)

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