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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reminding that the same structural dynamics can be sensed in complementary domains, which means frequency for SLDV & ESPI, time for DIC. Topology transforms were added to have the datasets in the same physical references. 2.1. Brief recall of a direct characterisation The formulation of receptance matrix H d ( ω ), taken from Ewins (2000); Heylen et al. (1998) as spectral relation between displacements and forces, will be used for the full-field FRF estimation , describing the dynamic behaviour of a testing system, with potentially multi-input excitation, here 2 distinct shakers, and many -output responses, here also several thousands, covering the whole sensed surface, as can be formulated in the following complex-valued equation: H d qf ( ω ) = N m = 1 S m X q F f ( ω ) N m = 1 S m F f F f ( ω ) ∈ C (1) where X q is the output displacement at q -th dof induced by the input force F f at f -th dof, while S m X q F f ( ω ) is the m -th cross power spectral density between input and output, S m F f F f ( ω ) is the m -th auto power spectral density of the input and ω is the angular frequency, evaluated in N repetitions. 3. Sound pressure & inverse vibro-acoustic formulation In the case of propagating waves as in Mas and Sas (2004), according to Kirkup (1994); Desmet (2004); Wind et al. (2006); Kirkup and Thompson (2007); Kirkup (2019), in the a − th point of global coordinates a a of the acoustic domain A , or air, the sound pressure p ( a a ,ω ) can be defined from the Helmholtz equation as: p ( a a ,ω ) = 2 i ωρ 0 S v n ( q q ,ω ) G ( r aq ,ω ) dS , G ( r aq ,ω ) = e − ikr aq 4 π r aq = e − i ω r aq / c 0 4 π r aq , (2) where i is the imaginary unit, ω is the angular frequency, ρ 0 is the medium (air) density, v n ( q q ,ω ) is the out-of-plane velocity of the infinitesimal vibrating surface dS located in the global coordinate q q , q representing the whole vector of coordinates of the vibrating surface S , k = ω/ c 0 = 2 π/λ is the wavenumber in the Helmholtz equation ( c 0 is the speed of sound at rest in the medium, λ is the acoustic wavelength), r aq = ∥ r aq ∥ is the norm of the distance r aq = a a − q q between the points in the two domains, and G ( r aq ,ω ) is the free space Green’s function as described in Eq.2. The normal velocities in the frequency domain are linked to the dynamic out-of-plane displacements over the static configuration q , by means of the relation v n ( q ,ω ) = i ω d n ( q ,ω ), which are expressions, by d n ( q ,ω ) = H d n q f ( ω ) · F f ( ω ), of the receptance FRFs H d n q f ( ω ) of size N q × N f – being N q the number of the outputs and N f of the inputs – and of the excitation signatures F f ( ω ). Eq.2 can be therefore rewritten in terms of a sum of discrete contributions, by means of a discretisation of the vibrating surface domain S ≈ q ∆ S q that scatters the sound pressure: p ( a a ,ω ) ≈− 2 ω 2 ρ 0 N q q H d n qf ( ω ) F f ( ω ) G aq ( r aq ,ω ) ∆ S q ∈ C , (3) with H d n q f ( ω ), F f ( ω ) and G aq ( r aq ,ω ) as complex-valued discrete quantities, r aq = ∥ r aq ∥ = ∥ a a − q q ∥ . Being G aq ( r aq ,ω ) and ∆ S q function of the locations of the N a discrete points in the acoustic domain and of the N q points on the structure, respectively, they can be grouped in a complex-valued di ff usion matrix T aq ( ω ), sized N a × N q , of element T aq ( ω ) = − 2 ω 2 ρ 0 G aq ( r aq ,ω ) ∆ S q , to transform Eq.3 into: p ( a a ,ω ) ≈ T a q ( ω ) H d n q f ( ω ) F f ( ω ) ∈ C . (4) If, di ff erently from the acoustic transfer vectors in Ge´rard et al. (2002); Citarella et al. (2007) between acoustic pressures and structural surface velocities, a vibro-acoustic transfer matrix V af ( ω ), sized N a × N f , is defined as: V af ( ω ) = T aq ( ω ) · H d n qf ( ω ) ∈ C , (5) Eq.4 can be easily rewritten as: p ( a a ,ω ) ≈ V af ( ω ) F f ( ω ) ∈ C , (6) relating pressures to excitations by the filter of the structure. It can be useful also in the cases where the structural response and acoustic domains are kept unchanged, while varying only the excitation signature to map the responses on the acoustic pressure field. 3