PSI - Issue 77

Alessandro Zanarini et al. / Procedia Structural Integrity 77 (2026) 64–70 A. Zanarini / Structural Integrity Procedia 00 (2025) 1–7

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Inverse Vibro-Acoustic FRF in WHITE-NOISEstd amp.mod. Air. Pressures RR dof [1092]

Step[345]=289.062 [Hz] InvAmpDIC_r=6.535e+01 [m^2] [dB]

InvPhaDIC_r=2.211e+00 [rad]

3.142

Pha [rad]

-3.142

8.652e+01

DIC_r

Amp [m^2] [dB]

3.179e+01

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 4. Example of an inverse vibro-acoustic FRF graph in the frequency domain, evaluated as force in shaker 1 over the airborne acoustic pressure from dof 1092.

4.1. Meshing the acoustic domain For the aims of this paper, a squared mesh was generated, of size 0.5 m × 0.5 m, with 51 × 51 dofs ( N a = 2601, 10 mm as acoustic grid spacing), centred on the vibrating plate and positioned 1 m above it. The air parameters were fixed in c 0 = 340.27m / s and ρ 0 = 1.225kg / m 3 . 4.2. Evaluation of the vibro-acoustic transfer matrix The evaluation of the vibro-acoustic transfer matrix V aq ( ω ), directly from the experiment-based receptances as proposed in Section 3, is given without the need of any FE structural model, but with great detail and field quality. It is important to underline how the vibro-acoustic transfer matrix obtained from the experiment-based receptances preserves, with its complex-valued nature , the real life conditions of the test, without any simplification in the damping, nor in the materials’ properties, nor in the boundary conditions, nor in the modal base truncation or identification. As the frequency rises, more shape complexity pertains the receptance maps, as can be clearly seen in the red-toned tiles a - e of Fig.2. 4.3. Evaluation of the pseudo-inverse airborne vibro-acoustic FRFs Following the formulation of Eq.8, the pseudo-inverse vibro-acoustic FRFs V + fa ( ω ) of force over airborne sound pressure can be achieved, as shown in the single inverse vibro-acoustic FRF of Fig.4, where the airborne pressure field is considered acting on the single acoustic dof 1092 and the force in the structural dof 2611 of the shaker 1. It can be clearly appreciated how the whole complex-valued information is retained in the pseudo-inversion. 4.4. Identification of the force induced by the airborne acoustic field For the identification of the force ˆ F 1 ( ω ) in the structural dof 2611 of the shaker 1, by means of Eq.7, the whole airborne pressure field modelled by Eq.9, acting on all the dofs of the acoustic mesh, must be used, together with all the pseudo-inverse vibro-acoustic FRFs in Section 4.3. The white noise amplitude modulation was adopted to simulate the pressure field by Eq.9 ∈ R , with a frequency domain example in dof 1092 – located in the magenta dot of Fig.2: due to the specific modelling of Eq.9, there results a frequency-dependent R airborne pressure pattern. The identified airborne induced force is clearly complex-valued , as in Fig.5.

Identified Force from WHITE-NOISEstd amp.mod. Airborne Pressures RR at dof [2611]

Step[134]=124.219 [Hz] IdAmpDIC_r=-1.409e+01 [N] [dB] IdPhaDIC_r=2.762e+00 [rad]

3.142

Pha [rad]

-3.142

-4.782e+00

DIC_r

Amp [N] [dB]

-7.352e+01

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 5. The identified force graph in the frequency domain, evaluated as force in shaker 1 from the whole airborne acoustic pressure field.

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