PSI - Issue 77

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2026) 000 – 000

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Procedia Structural Integrity 77 (2026) 1–2

International Conference on Structural Integrity Editorial Pedro Moreira, Paulo Tavares

INEGI – Institute of Science and Innovation in Mechanical and Industrial Engineering, Porto, Portugal

During this two-year span from ICSI2023 our planet witnessed unexpected turmoil in several respects, none of which beneficial to scientific conferences. From war spreading in the EU and the Middle-East, to EU economy restart push efforts with the NextGen initiative driving essentially industrial-focused projects or even, more recently, Economies focus redirection to Defence Industry, where previously Social Welfare was a primary concern and the U.S. Administration politics change in regard to long standing EU economic partners, all forced us rethink our priorities and somehow redirect our work. During this period, we also witnessed the exponential growth of generative pre- trained transformers use, with the amazing capability to make us “talk to computers”, a sking for knowledge support, irrespective of the subject we search, albeit the known limitations of the supporting LLMs. At this day and age, our model for knowledge update and scientific networking in conferences may well be at risk, in the twilight of a golden knowledge era. Our gathering in ICSI2025 will be a clear statement that this isn’t so! These last two years also witnessed an exponential increase in research activity in Structural Integrity, spilling over to several exciting areas in materials, methods and applications. The green energy transition, in face of the pressing evidence of climate change, and the required technological developments seem to have been the main drivers for these changes. Research into metal behaviour in the presence of Hydrogen, to cite just one example, which has long been an important topic in Structural Integrity, gradually came under the focus of a growing number of scientists due to its importance in H2 storage and distribution. Novel applications for validated simulation models, which are currently driving a large amount of work into Digital Twins and sensor virtualisation, together with disruptive sensing technologies, have influenced the R&D activities in the entire world, with large implications in the Engineering fields. Current research topics in the realm of Structural Integrity targeted by this year’s ICSI include, but are not limited to Fracture and Fatigue, Stress Analysis, Damage Tolerance, Durability, Crack Closure, Joining Technologies, Nano mechanics and Nanomaterials, Ageing, Coatings Technology, Environmental Effects, Structural Health Monitoring, New materials, Surface Engineering, Integrity of biomechanics structures in and many other exciting research topics. This year, we have also received a faint expression of work directed at Defence needs, expected to grow substantially in future editions.

2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers

2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers 10.1016/j.prostr.2026.01.085

Pedro Moreira et al. / Procedia Structural Integrity 77 (2026) 1–2 Barile and Kannan/ Structural Integrity Procedia 00 (2026) 000 – 000

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In 2025, the ICSI organization focused on inviting lecturers related to topics that dominate the contemporary status in Structural Integrity, such as Prof. Virgínia Infante from Instituto Superior Técnico, working in the field of Failure Analysis, Prof. Alicia Salazar from Rey Juan Carlos University in Madrid, devoted to the non-conventional mechanical characterization and Structural Integrity of composite materials or Prof. Humberto Varum from Porto University, fully engaged in infrastructures Structural Integrity. Similar to previous editions, ICSI2025 has been organized into a general track and a number of thematic symposia. In addition to Procedia Structural Integrity, special issues of several important well-known journals will cover ICSI2025. In 2025, ICSI is also hosting the REOTech International Conference on Renewable Energies and Ocean Technologies, which is expected to provide a forum for interdisciplinary, cross-fertilization of innovative thinking. The response to the organisation’s efforts has been outstanding: ten symposia were organized, and the number of abstract submissions was maintained at a similar level to previous editions, with around 180 approved for oral communication. These are very challenging times for organizing meaningful conferences, but the organisers strived to make this 6th Edition a memorable one, which will stimulate both young and well-known researchers in the field to contribute further to ICSI.

Structural Integrity Procedia Structural Integrity Procedia Structural Integrity Procedia Structural Integrity Structural Integrity Procedia Structural Integrity Procedia

Structural Integrity Procedia 00 (2025) 1–7

Available online at www.sciencedirect.com

ScienceDirect Structural Integrity Procedia 00 (2025) 1–7 International Conference on Structural Integrity Structural Integrity Procedia 00 (2025) 1–7 Structural Integrity Procedia 00 (2025) 1–7 Structural Integrity Procedia 00 (2025) 1–7

Procedia Structural Integrity 77 (2026) 64–70 International Conference on Structural Integrity International Conference on Structural Integrity About airborne fatigue life predictions by means of full-field receptances. Part A: retrieving structural forces from pressure fields. International Conference on Structural Integrity About airborne fatigue life predictions by means of full-field receptances. Part A: retrieving structural forces from pressure fields. International Conference on Structural Integrity About airborne fatigue life predictions by means of full-field receptances. Part A: retrieving structural forces from pressure fields. Structural Integrity Procedia 00 (2025) 1–7 International Conference on Structural Integrity About airborne fatigue life predictions by means of full-field receptances. Part A: retrieving structural forces from pressure fields. About airborne fatigue life predictions by means of full-field receptances. Part A: retrieving structural forces from pressure fields. About airborne fatigue life predictions by means of full-field receptances. Part A: retrieving structural forces from pressure fields. Alessandro Zanarini ∗ DIN, Industrial Engineering Dept., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Alessandro Zanarini ∗ DIN, Industrial Engineering Dept., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Alessandro Zanarini ∗ DIN, Industrial Engineering Dept., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Alessandro Zanarini ∗ DIN, Industrial Engineering Dept., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Alessandro Zanarini ∗ DIN, Industrial Engineering Dept., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers Abstract By means of a simplified inverse vibro-acoustic relation – based on the Rayleigh integral approximation of the sound propagated by a vibrating surface, modelled by the testing-based full-field receptances – in Part A the spectrum of the induced force by the airborne pressure fields can be retrieved, taking into account all the hardly modellable conditions of the set-up realisation. This part starts therefore from the modelling of the acoustic pressure field, in its spatial and broad frequency band contents. By means of the simplified receptances-aided inverse vibro-acoustics, the complete spectrum of the induced force is retrieved in the location of the impedance head of the direct characterisation testing of the surface. Details and considerations on the airborne force retrieval method, together with examples coming from a real thin plate tested, are provided in this Part A of the work. Keywords: airborne pressure fields; Rayleigh integral approximation; inverse vibro-acoustics; airborne force identification; full-field receptance FRFs; full-field dynamic testing. 1. Introduction Airborne pressure fields, with their variable spectral content in the acoustic and frequency domains, can become a threatening dynamic distributed loading for many surfaces in industrial applications, thus leading to airborne fatigue. The latter can be dangerous especially for lightweight parts with tight structural dynamics in the frequency range of interest, typical of aerospace and automotive engineering. The working life of these components can be severely short ened – also with catastrophic failures – due to unexpectedly high dynamic responses to the airborne pressure fields. The latter may indeed excite excessively the modal base or may shorten the life of the actual realisation. Advanced de sign and manufacturing methodologies need therefore to consider carefully any generalised damping distribution and specific boundary condition in the structural dynamics of the actual realisation of the mounted component, especially for mission critical ones. When simplified numerical models can not cope with real-life complex structural dynamics, experiment-based full-field optical techniques can substitute – by means of full-field receptances - the former in hybrid modelling of the vibro-acoustic coupling, at the base of airborne fatigue. Furthermore, experiment-based full-field op tical techniques can provide advanced benchmarks for the improvement of the more traditional modelling tools, such as FEM / BEM (see Wyckaert et al. (1996); Kirkup and Thompson (2007)), for the accurate design and manufacturing Abstract By means of a simplified inverse vibro-acoustic relation – based on the Rayleigh integral approximation of the sound propagated by a vibrating surface, modelled by the testing-based full-field receptances – in Part A the spectrum of the induced force by the airborne pressure fields can be retrieved, taking into account all the hardly modellable conditions of the set-up realisation. This part starts therefore from the modelling of the acoustic pressure field, in its spatial and broad frequency band contents. By means of the simplified receptances-aided inverse vibro-acoustics, the complete spectrum of the induced force is retrieved in the location of the impedance head of the direct characterisation testing of the surface. Details and considerations on the airborne force retrieval method, together with examples coming from a real thin plate tested, are provided in this Part A of the work. Keywords: airborne pressure fields; Rayleigh integral approximation; inverse vibro-acoustics; airborne force identification; full-field receptance FRFs; full-field dynamic testing. 1. Introduction Airborne pressure fields, with their variable spectral content in the acoustic and frequency domains, can become a threatening dynamic distributed loading for many surfaces in industrial applications, thus leading to airborne fatigue. The latter can be dangerous especially for lightweight parts with tight structural dynamics in the frequency range of interest, typical of aerospace and automotive engineering. The working life of these components can be severely short ened – also with catastrophic failures – due to unexpectedly high dynamic responses to the airborne pressure fields. The latter may indeed excite excessively the modal base or may shorten the life of the actual realisation. Advanced de sign and manufacturing methodologies need therefore to consider carefully any generalised damping distribution and specific boundary condition in the structural dynamics of the actual realisation of the mounted component, especially for mission critical ones. When simplified numerical models can not cope with real-life complex structural dynamics, experiment-based full-field optical techniques can substitute – by means of full-field receptances - the former in hybrid modelling of the vibro-acoustic coupling, at the base of airborne fatigue. Furthermore, experiment-based full-field op tical techniques can provide advanced benchmarks for the improvement of the more traditional modelling tools, such as FEM / BEM (see Wyckaert et al. (1996); Kirkup and Thompson (2007)), for the accurate design and manufacturing Abstract By means of a simplified inverse vibro-acoustic relation – based on the Rayleigh integral approximation of the sound propagated by a vibrating surface, modelled by the testing-based full-field receptances – in Part A the spectrum of the induced force by the airborne pressure fields can be retrieved, taking into account all the hardly modellable conditions of the set-up realisation. This part starts therefore from the modelling of the acoustic pressure field, in its spatial and broad frequency band contents. By means of the simplified receptances-aided inverse vibro-acoustics, the complete spectrum of the induced force is retrieved in the location of the impedance head of the direct characterisation testing of the surface. Details and considerations on the airborne force retrieval method, together with examples coming from a real thin plate tested, are provided in this Part A of the work. Keywords: airborne pressure fields; Rayleigh integral approximation; inverse vibro-acoustics; airborne force identification; full-field receptance FRFs; full-field dynamic testing. 1. Introduction Airborne pressure fields, with their variable spectral content in the acoustic and frequency domains, can become a threatening dynamic distributed loading for many surfaces in industrial applications, thus leading to airborne fatigue. The latter can be dangerous especially for lightweight parts with tight structural dynamics in the frequency range of interest, typical of aerospace and automotive engineering. The working life of these components can be severely short ened – also with catastrophic failures – due to unexpectedly high dynamic responses to the airborne pressure fields. The latter may indeed excite excessively the modal base or may shorten the life of the actual realisation. Advanced de sign and manufacturing methodologies need therefore to consider carefully any generalised damping distribution and specific boundary condition in the structural dynamics of the actual realisation of the mounted component, especially for mission critical ones. When simplified numerical models can not cope with real-life complex structural dynamics, experiment-based full-field optical techniques can substitute – by means of full-field receptances - the former in hybrid modelling of the vibro-acoustic coupling, at the base of airborne fatigue. Furthermore, experiment-based full-field op tical techniques can provide advanced benchmarks for the improvement of the more traditional modelling tools, such as FEM / BEM (see Wyckaert et al. (1996); Kirkup and Thompson (2007)), for the accurate design and manufacturing ∗ Corresponding author. Tel + 39 051 209 3442. Email address: a.zanarini@unibo.it (Alessandro Zanarini) 1 Abstract By means of a simplified inverse vibro-acoustic relation – based on the Rayleigh integral approximation of the sound propagated by a vibrating surface, modelled by the testing-based full-field receptances – in Part A the spectrum of the induced force by the airborne pressure fields can be retrieved, taking into account all the hardly modellable conditions of the set-up realisation. This part starts therefore from the modelling of the acoustic pressure field, in its spatial and broad frequency band contents. By means of the simplified receptances-aided inverse vibro-acoustics, the complete spectrum of the induced force is retrieved in the location of the impedance head of the direct characterisation testing of the surface. Details and considerations on the airborne force retrieval method, together with examples coming from a real thin plate tested, are provided in this Part A of the work. Keywords: airborne pressure fields; Rayleigh integral approximation; inverse vibro-acoustics; airborne force identification; full-field receptance FRFs; full-field dynamic testing. 1. Introduction Airborne pressure fields, with their variable spectral content in the acoustic and frequency domains, can become a threatening dynamic distributed loading for many surfaces in industrial applications, thus leading to airborne fatigue. The latter can be dangerous especially for lightweight parts with tight structural dynamics in the frequency range of interest, typical of aerospace and automotive engineering. The working life of these components can be severely short ened – also with catastrophic failures – due to unexpectedly high dynamic responses to the airborne pressure fields. The latter may indeed excite excessively the modal base or may shorten the life of the actual realisation. Advanced de sign and manufacturing methodologies need therefore to consider carefully any generalised damping distribution and specific boundary condition in the structural dynamics of the actual realisation of the mounted component, especially for mission critical ones. When simplified numerical models can not cope with real-life complex structural dynamics, experiment-based full-field optical techniques can substitute – by means of full-field receptances - the former in hybrid modelling of the vibro-acoustic coupling, at the base of airborne fatigue. Furthermore, experiment-based full-field op tical techniques can provide advanced benchmarks for the improvement of the more traditional modelling tools, such as FEM / BEM (see Wyckaert et al. (1996); Kirkup and Thompson (2007)), for the accurate design and manufacturing Abstract By means of a simplified inverse vibro-acoustic relation – based on the Rayleigh integral approximation of the sound propagated by a vibrating surface, modelled by the testing-based full-field receptances – in Part A the spectrum of the induced force by the airborne pressure fields can be retrieved, taking into account all the hardly modellable conditions of the set-up realisation. This part starts therefore from the modelling of the acoustic pressure field, in its spatial and broad frequency band contents. By means of the simplified receptances-aided inverse vibro-acoustics, the complete spectrum of the induced force is retrieved in the location of the impedance head of the direct characterisation testing of the surface. Details and considerations on the airborne force retrieval method, together with examples coming from a real thin plate tested, are provided in this Part A of the work. Keywords: airborne pressure fields; Rayleigh integral approximation; inverse vibro-acoustics; airborne force identification; full-field receptance FRFs; full-field dynamic testing. 1. Introduction Airborne pressure fields, with their variable spectral content in the acoustic and frequency domains, can become a threatening dynamic distributed loading for many surfaces in industrial applications, thus leading to airborne fatigue. The latter can be dangerous especially for lightweight parts with tight structural dynamics in the frequency range of interest, typical of aerospace and automotive engineering. The working life of these components can be severely short ened – also with catastrophic failures – due to unexpectedly high dynamic responses to the airborne pressure fields. The latter may indeed excite excessively the modal base or may shorten the life of the actual realisation. Advanced de sign and manufacturing methodologies need therefore to consider carefully any generalised damping distribution and specific boundary condition in the structural dynamics of the actual realisation of the mounted component, especially for mission critical ones. When simplified numerical models can not cope with real-life complex structural dynamics, experiment-based full-field optical techniques can substitute – by means of full-field receptances - the former in hybrid modelling of the vibro-acoustic coupling, at the base of airborne fatigue. Furthermore, experiment-based full-field op tical techniques can provide advanced benchmarks for the improvement of the more traditional modelling tools, such as FEM / BEM (see Wyckaert et al. (1996); Kirkup and Thompson (2007)), for the accurate design and manufacturing Alessandro Zanarini ∗ DIN, Industrial Engineering Dept., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Abstract By means of a simplified inverse vibro-acoustic relation – based on the Rayleigh integral approximation of the sound propagated by a vibrating surface, modelled by the testing-based full-field receptances – in Part A the spectrum of the induced force by the airborne pressure fields can be retrieved, taking into account all the hardly modellable conditions of the set-up realisation. This part starts therefore from the modelling of the acoustic pressure field, in its spatial and broad frequency band contents. By means of the simplified receptances-aided inverse vibro-acoustics, the complete spectrum of the induced force is retrieved in the location of the impedance head of the direct characterisation testing of the surface. Details and considerations on the airborne force retrieval method, together with examples coming from a real thin plate tested, are provided in this Part A of the work. Keywords: airborne pressure fields; Rayleigh integral approximation; inverse vibro-acoustics; airborne force identification; full-field receptance FRFs; full-field dynamic testing. 1. Introduction Airborne pressure fields, with their variable spectral content in the acoustic and frequency domains, can become a threatening dynamic distributed loading for many surfaces in industrial applications, thus leading to airborne fatigue. The latter can be dangerous especially for lightweight parts with tight structural dynamics in the frequency range of interest, typical of aerospace and automotive engineering. The working life of these components can be severely short ened – also with catastrophic failures – due to unexpectedly high dynamic responses to the airborne pressure fields. The latter may indeed excite excessively the modal base or may shorten the life of the actual realisation. Advanced de sign and manufacturing methodologies need therefore to consider carefully any generalised damping distribution and specific boundary condition in the structural dynamics of the actual realisation of the mounted component, especially for mission critical ones. When simplified numerical models can not cope with real-life complex structural dynamics, experiment-based full-field optical techniques can substitute – by means of full-field receptances - the former in hybrid modelling of the vibro-acoustic coupling, at the base of airborne fatigue. Furthermore, experiment-based full-field op tical techniques can provide advanced benchmarks for the improvement of the more traditional modelling tools, such as FEM / BEM (see Wyckaert et al. (1996); Kirkup and Thompson (2007)), for the accurate design and manufacturing 2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers 10.1016/j.prostr.2026.01.010 ∗ Corresponding author. Tel + 39 051 209 3442. Email address: a.zanarini@unibo.it (Alessandro Zanarini) 1 ∗ Corresponding author. Tel + 39 051 209 3442. Email address: a.zanarini@unibo.it (Alessandro Zanarini) 1 ∗ Corresponding author. Tel + 39 051 209 3442. Email address: a.zanarini@unibo.it (Alessandro Zanarini) 1 ∗ Corresponding author. Tel + 39 051 209 3442. Email address: a.zanarini@unibo.it (Alessandro Zanarini) 1 ∗ Corresponding author. Tel + 39 051 209 3442. Email address: a.zanarini@unibo.it (Alessandro Zanarini) 1

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a

b

c

Figure 1. The lab in the TEFFMA project (see Zanarini (2014a,b, 2015a,b,c,d, 2018, 2019a,b, 2022b)): aerial view in a , restrained plate sample in b , 2 shakers on the back of the plate in c .

of relevant components. Many times the sound radiation simulations from structural vibrations in NVH studies are run with linear structural FE models, potentially oversimplified on the treatment of boundary conditions, frictions, damp ing, mistuning from actually produced parts and non-linearities. Instead, working with full-field optical receptances , coming from broad frequency band real testing (see Van der Auweraer et al. (2001); Zanarini (2014a,b, 2015a,b,d, 2018, 2019a,b, 2020, 2022b) for enhanced structural dynamics assessments and model updating; see instead Zanarini (2008a,b, 2015c, 2022f,e,c,a, 2023c) for enhancements of fatigue spectral methods and failure risk grading), may represent a viable path in order to have the best achievable representation of the real behaviour of manufactured and mounted components around their working load levels, also with modally dense structural dynamics and complex patterns in the dynamic signature of the excitations. A recall of the experiment-based FRF modelling is sketched in Section 2, with a brief description of the testing set-up of Fig.1. The specimen under test was the simple thin rectangular plate of the TEFFMA project, designed as a lightweight structure to retain a complex structural dynamics within the operative ranges of the used measurement technologies, with its real constraints, realisation, materials and damping characteristics. The experiment-based optical full-field receptances proved to work (see Zanarini (2022d, 2023a,b)) also in the Rayleigh integral approximation of the sound propagated in the free-field acoustic domain by the characterised surface, for the numerical approximation of the spectral relation among the sound radiation field, the structural dynamics and excitation forces. The same background (see also Wind et al. (2006)), reformulated in Section 3 with notes for the inverse vibro-acoustics and acoustic pressures’ spatial and spectral modelling, is here followed by means of full-field experiment-based receptances , with the aim to identify the broad frequency band force that is transmitted to the excitation points used in the direct FRF problem. This identification may permit the airborne structural dynamics’ characterisation of the components under test for further dynamic displacement and strain / stress distribution studies. The induced airborne force spectrum will be used in Part B as the excitation in airborne fatigue life assessment. In Section 4 the needed examples, in the space and frequency domains, are given to achieve the airborne induced structural force identification, once the airborne acoustic pressures are synthesised: notes on the meshing of the acoustic domain, on the contribution of the experiment-based full-field receptance maps to vibro-acoustic direct FRFs, and on the inverse acoustic pressure FRFs. Section 5 contains the final conclusions. 2. Full-Field FRFs: direct experimental modelling To the interested reader, the most detailed notes on the test campaign appeared in Zanarini (2019a), with further suggestions in Zanarini (2019b, 2020, 2022b, 2024a, 2025a), but here is a brief summary of what was available at TU Wien as in Fig.1: a dedicated seismic floor room; a mechanical & electronic workshop with technicians; traditional tools for vibration & modal analysis; but, in particular, there were SLDV, Hi-Speed DIC and ESPI measurement instruments. Accurate studies were needed to understand each technological limit and if a common test for concurrent usage might have been really possible. All this brought to a unique set-up for the comparison of the 3 optical technologies in full-field FRF estimations ; great attention was paid on the design of experiments for further research in modal analysis. After an accurate tuning, a feasible performance overlapping was sought directly out of each instrument, 2

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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

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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

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

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

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

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

(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

(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

(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

b

c

d

e

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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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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5. Conclusions This paper has highlighted the chance to retrieve airborne structural forces from known acoustic fields, opening inquiry’s possibilities in NVH and coupled fluid-structural dynamics, thanks to experiment-based optical full-field tools . The unprecedented mapping ability, in both spatial and frequency domains, opens new cross vibro-acoustic prediction scenarios, as the real-life structural dynamics of the radiating surface is entirely retained in the receptances with great accuracy, but without assumptions nor errors in any type of structural virtual modelling. Acknowledgements The European Commission Research Executive Agency is acknowledged for having funded in 2004-05 the HPMI CT-1999-00029 Speckle Interferometry for Industrial Needs Post-doctoral Marie Curie Industry Host Fellowship project at Dantec Ettemeyer GmbH and in 2013-15 the project TEFFMA - Towards Experimental Full Field Modal Analysis at the TU-Wien, Austria, by the Marie Curie FP7-PEOPLE-IEF-2011 PIEF-GA-2011-298543 grant. References Citarella, R., Federico, L., Cicatiello, A., 2007. Modal acoustic transfer vector approach in a FEM-BEM vibro-acoustic analysis. Engineering Analysis with Boundary Elements 31, 248–258. doi: 10.1016/j.enganabound.2006.09.004 . Desmet, W., 2004. Boundary element method in acoustics. Technical Report. Katholieke Universiteit Leuven, Belgium, Mechanical Engineering Department, Noise & Vibration research group, URL: https: // www.mech.kuleuven.be / en / research / . 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Vibro-acoustical modal analysis: Reciprocity, model symmetry and model validity. J. Acoust. Soc. Am. 100, 3172–3181. doi: 10.1121/1.417127 . Zanarini, A., 2008a. Fatigue life assessment by means of full field ESPI vibration measurements, in: Sas, P. (Ed.), Proceedings of the ISMA2008 Conference, September 15-17, Leuven (Belgium), KUL. pp. 817–832. doi: 10.13140/RG.2.1.3452.9365 . Condition monitoring, Paper 326. Zanarini, A., 2008b. Full field ESPI vibration measurements to predict fatigue behaviour, in: Proceedings of the IMECE2008 ASME International Mechanical Engineering Congress and Exposition, October 31- November 6, Boston (MA) USA, ASME. pp. 165–174. URL: https: // www.researchgate.net / publication / 267591013 Full Field ESPI Vibration Measurements to Predict Fatigue Behavior, doi: 10.1115/IMECE2008-68727 . paper IMECE2008-68727. Zanarini, A., 2014a. On the estimation of frequency response functions, dynamic rotational degrees of freedom and strain maps from di ff erent full field optical techniques, in: Proceedings of the ISMA2014 including USD2014 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 15-17, KU Leuven. pp. 1177–1192. URL: http: // past.isma-isaac.be / downloads / isma2014 / papers / isma2014 0676. pdf. Dynamic testing: methods and instrumentation, paper ID676. Zanarini, A., 2014b. On the role of spatial resolution in advanced vibration measurements for operational modal analysis and model updating, in: Proceedings of the ISMA2014 including USD2014 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 15-17, KU Leuven. pp. 3397–3410. URL: http: // past.isma-isaac.be / downloads / isma2014 / papers / isma2014 0678.pdf. Operational modal analysis, paper ID678. 6

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Zanarini, A., 2015a. Accurate FRFs estimation of derivative quantities from di ff erent full field measuring technologies, in: Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, September 7-10, Univ. Ljubljana & IFToMM. pp. 1569– 1578. URL: https: // www.researchgate.net / publication / 280013778 Accurate FRF estimation of derivative quantities from di ff erent full field measuring technologies. ID192. Zanarini, A., 2015b. Comparative studies on full field FRFs estimation from competing optical instruments, in: Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, September 7-10, Univ. Ljubljana & IFToMM. pp. 1559–1568. URL: https: // www.researchgate.net / publication / 280013709 Comparative studies on Full Field FRFs estimation from competing optical instruments. ID191. Zanarini, A., 2015c. Full field experimental modelling in spectral approaches to fatigue predictions, in: Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, September 7-10, Univ. Ljubljana & IFToMM. pp. 1579–1588. URL: https: // www. researchgate.net / publication / 280013788 Full field experimental modelling in spectral approaches to fatigue predictions. ID193. Zanarini, A., 2015d. Model updating from full field optical experimental datasets, in: Proceedings of the ICoEV2015 International Conference on Engineering Vibration, Ljubljana, Slovenia, September 7-10, Univ. Ljubljana & IFToMM. pp. 773–782. URL: https: // www.researchgate.net / publication / 280013876 Model updating from full field optical experimental datasets. ID196. Zanarini, A., 2018. Broad frequency band full field measurements for advanced applications: Point-wise comparisons between optical technologies. Mechanical Systems and Signal Processing 98, 968 – 999. doi: 10.1016/j.ymssp.2017.05.035 . Zanarini, A., 2019a. Competing optical instruments for the estimation of Full Field FRFs. Measurement 140, 100 – 119. doi: 10.1016/j. measurement.2018.12.017 . Zanarini, A., 2019b. Full field optical measurements in experimental modal analysis and model updating. Journal of Sound and Vibration 442, 817 – 842. doi: 10.1016/j.jsv.2018.09.048 . Zanarini, A., 2020. On the making of precise comparisons with optical full field technologies in NVH, in: ISMA2020 including USD2020 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 7-9, KU Leuven. pp. 2293–2308. URL: https: // past.isma-isaac.be / downloads / isma2020 / proceedings / Contribution 695 proceeding 3.pdf. Optical methods and computer vision for vibration engineering, paper ID 695. Zanarini, A., 2022a. About the excitation dependency of risk tolerance mapping in dynamically loaded structures, in: ISMA2022 including USD2022 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 12-14, KU Leuven. pp. 3804–3818. URL: https: // past.isma-isaac.be / downloads / isma2022 / proceedings / Contribution 208 proceeding 3.pdf. paper ID 208 in Vol. Structural Health Monitoring. Zanarini, A., 2022b. Chasing the high-resolution mapping of rotational and strain FRFs as receptance processing from di ff erent full-field optical measuring technologies. Mechanical Systems and Signal Processing 166, 108428. doi: 10.1016/j.ymssp.2021.108428 . Zanarini, A., 2022c. Introducing the concept of defect tolerance by fatigue spectral methods based on full-field frequency response function testing and dynamic excitation signature. International Journal of Fatigue 165, 107184. doi: 10.1016/j.ijfatigue.2022.107184 . Zanarini, A., 2022d. On the approximation of sound radiation by means of experiment-based optical full-field receptances, in: ISMA2022 including USD2022 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 12-14, KU Leuven. pp. 2735–2749. URL: https: // past.isma-isaac.be / downloads / isma2022 / proceedings / Contribution 207 proceeding 3.pdf. paper ID 207 in Vol. Optical Methods. Zanarini, A., 2022e. On the defect tolerance by fatigue spectral methods based on full-field dynamic testing. Procedia Structural Integrity 37, 525–532. doi: 10.1016/j.prostr.2022.01.118 . paper ID 105, ICSI 2021 The 4th International Conference on Structural Integrity. Zanarini, A., 2022f. On the exploitation of multiple 3D full-field pulsed ESPI measurements in damage location assessment. Procedia Structural Integrity 37, 517–524. doi: 10.1016/j.prostr.2022.01.117 . paper ID 104, ICSI 2021 The 4th International Conference on Structural Integrity. Zanarini, A., 2023a. Experiment-based optical full-field receptances in the approximation of sound radiation from a vibrating plate, in: IMAC XLI - International Modal Analysis Conference - Keeping IMAC Weird: Traditional and Non-traditional Applications of Structural Dynamics, Austin (Texas), USA, Springer Nature Switzerland AG & SEM Society for Experimental Mechanics. pp. 1–13. doi: 10.1007/978-3-031-34910-2_ 4 . paper ID 14650 - chapter 4, in J. Baqersad, D. Di Maio (eds.), Computer Vision & Laser Vibrometry, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series. Zanarini, A., 2023b. On the influence of scattered errors over full-field receptances in the Rayleigh integral approximation of sound radiation from a vibrating plate. Acoustics 5, 948–986. URL: https: // www.mdpi.com / 2624-599X / 5 / 4 / 55, doi: 10.3390/acoustics5040055 . Zanarini, A., 2023c. Risk tolerance mapping in dynamically loaded structures as excitation dependency by means of full-field receptances, in: IMAC XLI - International Modal Analysis Conference - Keeping IMAC Weird: Traditional and Non-traditional Applications of Structural Dynamics, Austin (Texas), USA, Springer Nature Switzerland AG & SEM Society for Experimental Mechanics. pp. 43–56. doi: 10.1007/ 978-3-031-34910-2_9 . paper ID 14648 - chapter 9, in J. Baqersad, D. Di Maio (eds.), Computer Vision & Laser Vibrometry, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series. Zanarini, A., 2024a. Assessing the retrieval procedure of complex-valued forces from airborne pressure fields by means of DIC-based full field receptances in simplified pseudo-inverse vibro-acoustics. Aerospace Science and Technology 157, 109757. doi: 10.1016/j.ast.2024. 109757 . Zanarini, A., 2024b. On the use of full-field receptances in inverse vibro-acoustics for airborne structural dynamics. 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