Attempts at Pseudo-Inverse Vibro-Acoustics by Means of SLDV-Based Full-Field Mobilities

Abstract

Lightweight components can have structural integrity and reliability concerns, coming from dynamic airborne pressure fields. Hardly tuned numerical structural models may enter into vibro-acoustic simulations of the pressure fields radiated by vibrating plates, potentially masking the forecast of severe outputs. Instead, this paper proposes—for the direct and inverse vibro-acoustic approaches—to characterise the broad frequency band structural dynamics of radiating surfaces by means of experiment-based full-field contactless techniques, with increased spatial resolution, but without the inertia-related distortions of traditional measurement transducers. The SLDV-based mobilities bring the real-life behaviour of the component into the vibro-acoustic simulations, with the actual realisation-related complete structural dynamics and broad frequency band excitation. The paper aims at assessing the procedure for the estimation, in the whole spectrum, of the airborne force, which can be transmitted by an airborne pressure field to known structural locations. The simulation tools revisit the simple Rayleigh integral approximation of sound radiation from a vibrating surface, a real thin flat plate, describable by SLDV-based complex-valued full-field mobilities. Airborne pressure fields and excitation forces concern the early attempts of direct and pseudo-inverse vibro-acoustics. Details, examples and considerations about the whole procedures are thoroughly provided: on the simulation of the vibro-acoustic transfer matrix and of the radiated sound pressures with given excitation forces; on the retrieval of the airborne forces in restraining locations, together with the assessment of the numerical precision of the retrieving procedure.

1. Introduction

Prohibitive operational dynamic states can be reached from distributed dynamic loading, in the shape of airborne pressure fields. Lightweight components—especially in wind turbines for energy harvesting and in vehicle engineering for aircraft, space launchers, high speed trains and automotive shields—can decrease their forecast exercise life, when reduced by unexpected strain distributions from not fully understood airborne pressure fields. A possible reason for this partial lack of knowledge might be searched in the simplifications of the numerical models at the base of many NVH studies (see e.g., [1,2,3,4,5,6,7,8,9,10,11]). The latter simulate the sound radiation from vibrating parts by means of linear structural FEM, which might be improperly tuned for the specific manufactured realisation, or much simplified about non-linearities, specific damping-friction distributions and real boundary constraints. The real-life structural dynamics is strongly simplified also in fluid-structure interaction studies (see, in the frequency domain, [12,13,14,15,16]). In the literature mostly focused on fluid motion (outside the target of this paper, but examples can be found in [17,18,19,20,21,22,23,24,25]), again the numerical structural model seems simplified, disregarding the real-life physical properties distribution and, in particular, the damping modelling, albeit relevant for the fluid-structure interactions and fluttering phenomena. Unfortunately, the most general modal modelling approach—based on complex-valued eigensolutions (see [26,27])—and model updating/tuning do not seem to receive the deserved attention. The above suggested examples simplify the structure by a limited modal base from a linear FEM with proportional damping only, which means a truncated real-valued eigensystem. Broad frequency band experiment-based full-field optical FRFs (either receptances or mobilities) can be complex-valued alternatives to extract the most realistic linearisation of the real-life and complete structural dynamics directly from contactless testing, with great accuracy and enhanced spatial resolution, with no modal model identification nor truncation. This statement is based on the experience of the author, as later briefly sketched, in full-field measurement techniques—boosted by the TEFFMA project1—and on inspiring researches [28,29,30,31]. As proposed in this work, the complex-valued experiment-based optical full-field mobilities substitute any numerical modelling (e.g., by FEM) of the structural part, with no need of any model updating, material properties’ and boundary conditions’ guess.

Among the optical—therefore contactless, not distorting with sensors & cabling inertia—technologies, SLDV has permitted to add dofs and discrete spatial resolution in established NVH approaches in broad frequency ranges, but it can be better defined as non-native full-field measurement technique, due to its scanning nature. SLDV has therefore become a reference in contactless measurements for standing vibrations, but not for transients, not acquiring synchronously the whole mapping. SLDV is the full-field technology here adopted for the real-life mobility (velocities over forces) testing of the vibrating structural part. Instead, the techniques based on the sensing of photons across the whole sensor matrix can be called as optical native full-field measurements, or also image-based, with an even higher dofs’ density and field-wise accuracy/continuity than that of SLDV, as proved extensively in [28,29,30,31,32,33,34], especially with stroboscopic ESPI. Although limited to standing vibrations in the frequency domain, ESPI can measure extremely detailed fields with a noise floor of the order of 50 nm and a ceiling of some μm, exploiting the interference phase maps and the precision of the constant wavelength of the coherent laser light, modulated up to several kHz in stroboscopic acquisitions. While quality-wise unsurpassed, for broad frequency band measurements ESPI has heavy drawbacks in the time consuming test, coherent lighting, recording size and environmental vibration insulation needs, hardly met outside a specific research laboratory. Another native full-field approach is that of DIC with good spatial detail (generally better than SLDV, but lower than ESPI) in the time histories of displacement maps. However, even with the usage of proper high-speed cameras, related to the frequency band of interest, DIC is generally more limited in the upper frequencies than SLDV, due to the different frequency domain amplification of displacements against velocities. But, with DIC, transients are easily acquired in the time domain if the on-board memory of the cameras allow long recordings; furthermore, the evaluation of multiple series of correlated displacements might be computationally intensive even nowadays. A relevant advantage of DIC systems is certainly their usage flexibility—therefore of increasing interest also in industrial environments—as some in-situ advanced applications have proved, as in [35,36,37,38,39]. Note how SLDV, Stroboscopic ESPI and Hi-speed DIC give complex-valued datasets in the frequency domain as in the TEFFMA project.

Concise notes about the author’s experience in optical full-field measurements follow. The first hints of the TEFFMA project appeared in 20142, consolidated by more enhanced reports in 20153. The first works were briefly gathered in [32]; in [33], instead, extended notes on the full-field receptances’ evaluation followed; in [34] model updating exploited the full-field eigenshapes coming from the EFFMA. The quality of native (ESPI and DIC) full-field datasets—in broad frequency band dynamic testing—was underlined in [40]. Relevant achievements for rotational4 and strain FRF high resolution maps were compared in [45]. The high quality strain FRF mapping further led to structural integrity5 inquiries: that attention—to the dynamic responses and structural integrity from specific excitations—is here recalled in the assessment of the airborne distributed loading.

The author proved how—nowadays—full-field techniques already provide, even if not fully explored, clear improvements in EFFMA, but even more in advanced model updating, derivative calculations (rotational dofs, strains, stresses, risk index maps) and vibro-acoustics, thanks to the higher continuity and consistency of the fields in the datasets when compared to more traditional sensors. Therefore, a highly reliable experiment-based behavioural modelling of complex components—in their manufacturing and mounting state—may come from full-field optical mobilities, the latter retaining any modally dense structural dynamics and being able to deal with any dynamic signature of the excitation, under the linearity assumption. Section 2.1 recalls, therefore, the basis of experimental FRFs in the TEFFMA project, with a brief summary of the rig and testing procedures.

In Section 2.2—inspired by the works in [1,2,3,4,5,6,7,8,9,11]—a full-field complex-valued re-elaboration of the Rayleigh integral approximation for the direct vibro-acoustic modelling is proposed. Indeed, the vibro-acoustic transfer matrices in this formulation are obtained by means of the experiment-based full-field mobilities from SLDV technique as frequency domain relations between the excitation forces on the structure and the acoustic pressures in the acoustic domain, similarly to the experiment-based full-field receptances already employed by the author in former works6 about vibro-acoustics; no numerical structural model, of any type (e.g., FEM or boundary element modelling, wave-based or any functional expansion), is used. By using a specified excitation signature, the direct vibro-acoustic FRFs can map the pressure field radiated by the vibrating plate. Another aim of this work is to assess the accuracy of an inverse vibro-acoustic approach in the interaction of airborne pressures with the structural dynamics. Therefore, expanding what already advanced in [55], a simplified pseudo-inverse airborne vibro-acoustics is developed in Section 2.2 by means of SLDV full-field experiment-based mobilities from the TEFFMA project. Starting from a known spectrum of the pressure field, the exerted force is reconstructed on the same excitation locations of the direct vibro-acoustic modelling. If the former pressure field was originated by a known force spectrum, the differences with the reconstructed force highlight the accuracy of the pseudo-inverse airborne vibro-acoustic procedure. With Section 2.3 a possible representation of broad frequency band forces is given by randomly contaminated coloured-noises, within the linearity assumption, as a realistic and enhanced complex-valued excitation broad frequency band spectrum that can fully promote the qualities of this experiment-based full-field approach.

Figure 1. Main steps for the direct and pseudo-inverse airborne vibro-acoustics approaches.

Figure 1. Main steps for the direct and pseudo-inverse airborne vibro-acoustics approaches.

The direct and pseudo-inverse vibro-acoustic approaches are simulated in Section 3, after notes on the computations and acoustic domain meshing. Specifically, the shapes of the vibro-acoustic transfer matrices are commented in relation to the SLDV experiment-based full-field mobility maps. Furthermore, by adopting specific excitation signatures, the radiated pressure fields are computed. These latter are feed back to the pseudo-inverse vibro-acoustic approach, after the comments on its FRFs, to assess the exerted airborne force on shaker head and the quality of the simplified identification procedure.

Section 4 gives a broader perspective on the results. Section 5 concludes by summarising the shown achievements. The reader can check in “Abbreviations” any nomenclature item that is used in the manuscript without explicit explanation.

2. Materials and Methods

The backbone of the experiment-based full-field vibro-acoustic approaches is sketched in Figure 1, as detailed in this Section. Details about the experiment-based modelling of lightweight structures by means of SLDV mobilities are given in Section 2.1, where the testing comes from the TEFFMA project. The direct and pseudo-inverse simplified vibro-acoustic approaches are presented in details in Section 2.2, where the Rayleigh integral approximation—of the relation between structural excitations and acoustic pressure fields—is reformulated to take advantage of the complex-valued SLDV mobilities. In this perspective, a general modelling of the excitation is given in Section 2.3.

Figure 2. Full-field optical measurement instruments set-up in front of the specimen on the anti-vibration table with shakers on the backside: the instruments in (a) with SLDV head close to the border of the table, the restrained thin plate in (b) and the 2 shakers in (c).

Figure 2. Full-field optical measurement instruments set-up in front of the specimen on the anti-vibration table with shakers on the backside: the instruments in (a) with SLDV head close to the border of the table, the restrained thin plate in (b) and the 2 shakers in (c).

2.1. Full-Field SLDV Mobilities from the TEFFMA Project

At the Technische Universitaet Wien (Austria), the TEFFMA fundamental research project was led by the author. In it, SLDV was taken as the reference technology (albeit as non-native) in the comparisons among state-of-the-art optical native, or image-based, full-field approaches (Stroboscopic ESPI and Hi-speed DIC), in advanced structural dynamics and NVH studies. The TEFFMA project was based on the Post-doctoral Marie Curie Industry Host Fellowship project HPMI-CT-1999-00029 ‘Speckle Interferometry for Industrial Needs’7 on ESPI-based testing, specifically for enhanced structural dynamics and fatigue life assessments, by means of spectral methods. During the TEFFMA project, as can be seen in Figure 2, the full-field measurement instruments (of SLDV, DIC and ESPI) were pivotal for all the activities; the test rig was assembled in a dedicated underground seismic-floor room, on an anti-vibration air-cushioning table, for the best insulation from environmental influences. These activities were favoured by the technicians of the mechanical & electronic workshop for customised tools. More detailed notes on the test campaign and on its achievements can be found in [33,34,40,45,54].

To make the unique comparisons in the TEFFMA project about the measurements for NVH and structural dynamics by the 3 full-field technologies, a common set-up was compulsory, but it only came after careful considerations on the limits of each approach. This unique set-up required a very fine tuning of even the smallest detail (revealed mostly by ESPI), but brought to a feasible performance overlapping of the different approaches. The same structural dynamics emerged in the frequency domain for ESPI and SLDV, but in the time domain for DIC. Although certainly not precisely super-imposable, the comparisons seemed certainly promising—from a qualitative point of view—already from each instrument software. Instead, to reach quantitative comparisons in the TEFFMA project—not here of specific interest—accurate topology transforms were implemented in the custom processing code, in order to have all the 3 different techniques’ datasets cast on the same physical (geometry and frequency, as noted in the 2D graphs) references.

2.1.1. Characterisation of the Structural Dynamics by Means of Full-Field Mobilities

The mobility matrix ${\mathit{H}}_{\mathit{v}}\left(\omega \right)$ follows the formulation for the H1 estimator—noise only in the outputs—found in [26,27]: it is estimated as the spectral relation between multi-output velocities and multi-input forces. In this full-field extension, the inputs are just 2 shakers, while thousands (2907) of responses are acquired on the whole surface of interest. The structural domain is represented by the vibrating surface S of the plate, whose q-th point has global 3D coordinates ${\mathit{q}}_q$, while $\mathit{q}$ represents all the field coordinates of the active structural domain. Each element of the mobility matrix ${\mathit{H}}_{\mathit{v}}\left(\omega \right)$ can be expressed, in the complex-valued representation of the frequency domain, as:

$$H_{v_{qf}}\left(\omega \right)={\displaystyle \frac{{\sum }_{m=1}^N{S}_{{\dot{X}}_q{F}_f}^m\left(\omega \right)}{{\sum }_{m=1}^N{S}_{F_f{F}_f}^m\left(\omega \right)}}\in \mathbb{C}$$

(1)

where ${\dot{X}}_q$ is the output velocity at q-th dof induced by the input force $F_f$ at f-th dof, while $S_{{\dot{X}}_q{F}_f}^m\left(\omega \right)$ is the m-th cross-power spectral density between input and output, $S_{F_f{F}_f}^m\left(\omega \right)$ is the m-th auto-power spectral density of the input, evaluated in N repetitions.

2.1.2. Brief Notes About the Tested Plate in the TEFFMA Project

To avoid any confusion of static deformations with dynamic ODSs, a flat pristine thin aluminum rectangular plate—with external dimensions of 250 × 236 × 1.5 mm—was chosen as the specimen. The latter, to restrain any dangerous rigid-body movement for the ESPI technique, was fixed by wires to the rigid frame—bolted on the air-spring optical table—in Figure 2a,b. According to DIC requirements, a noise patter of random black & white droplets was sprayed on the plate face towards the instruments. The plate was excited orthogonally, from its back side, by the two properly positioned shakers of Figure 2c. The force signal—from the impedance heads at each shaker-structure interface—was recorded, for the estimation of the mobility FRFs ${\mathit{H}}_{\mathit{v}}\left(\omega \right)$ in Equation (1). Stroboscopic ESPI-based measurements—in their phase-shifting procedures [56]—required shakers as controllable excitations, but the latter were fully compatible also with DIC and SLDV. The shakers were driven, one at a time, by an external sine-waveform generator in ESPI, on the contrary by LMS Test.Lab system in DIC and SLDV acquisitions.

2.1.3. Notes on the Processing of the Full-Field Mobilities

The formulation of Equation (1) permitted—in the TEFFMA project—the accurate estimation of mobility FRF maps from SLDV, and, similarly in [33], of full-field receptances for ESPI and DIC. Especially in ESPI datasets, the enhanced consistency and continuity of the spatial data in the sensed structural dynamics was appreciated, with remarkably clean ODSs, sharp nodal lines, and high-quality Coherences (see [26,27]). Each of the SLDV-based datasets—from the two shakers, differently positioned on the structure—retains $N_q$ = 2907 dofs ($N_q$ = 57 × 51, about 4.39–4.64 mm grid spacing in the structural domain). Each dataset contains 1285 frequency lines, detached by a frequency space of 0.78125 Hz; the frequency range is limited to [20–1024] Hz, the lower end as imposed by ESPI technique. The number of averages is N = 50.

SLDV-based datasets are here employed in their raw and undistorted estimations, with no down-sampling nor interpolations, thanks to unneeded topology transforms. Instead, if cast to the SLDV references, the higher spatial resolution image-based techniques of the TEFFMA project (DIC & ESPI) are forced to contain additional computational noise—albeit very limited in the customised routines—from the topology transforms.

2.2. Simplified Formulation for Direct and Pseudo-Inverse Vibro-Acoustics

At the core of a simplified vibro-acoustics is the relation between the structural motion—induced by excitation forces on the vibrating surface—and the radiated pressures in the facing medium. The latter definition pertains the here called direct vibro-acoustics, whereas by inverse vibro-acoustics the relation, between medium-borne pressure fields and induced forces on specific structural locations, is addressed.

The structural domain was already described and modelled in Section 2.1. The acoustic domain is here called A and its a-th point has global 3D coordinates ${\mathit{a}}_a$, while $\mathit{a}$ contains the whole field coordinates. Joining the results in literature [4,7,8,11,57] for propagating pressure waves, the sound pressure $p({\mathit{a}}_a,\omega )$—output in ${\mathit{a}}_a$, by a vibrating surface—can be defined in the frequency domain from the Helmholtz equation as:

$$p({\mathit{a}}_a,\omega )=2i\omega {\rho }_0{\int }_S{v}_n({\mathit{q}}_q,\omega )G(r_{aq},\omega )d{S}_q\in \mathbb{C},G(r_{aq},\omega )={\displaystyle \frac{e^{-ik{r}_{aq}}}{4\pi r_{aq}}}={\displaystyle \frac{e^{-i\omega r_{aq}/c_0}}{4\pi r_{aq}}}\in \mathbb{C},$$

(2)

where i is the imaginary unit, ${\rho }_0$ is the medium (air) density, $v_n({\mathit{q}}_q,\omega )$ is the normal (out-of-plane) velocity of the infinitesimal vibrating surface $d{S}_q$ located in the ${\mathit{q}}_q$ global coordinate, $k=\omega /c_0=2\pi /\lambda$ is the wavenumber in the Helmholtz equation ($c_0$ is the speed of sound at rest in the medium, $\lambda$ is the acoustic wavelength), $r_{aq}=\parallel {\mathit{r}}_{aq}\parallel$ is the norm of the distance ${\mathit{r}}_{aq}={\mathit{a}}_a-{\mathit{q}}_q$ between the points in the two domains, and $G(r_{aq},\omega )$ is the free-space Green’s function as described on the right of Equation (2).

2.2.1. Sound Pressures in a Direct Formulation of Vibro-Acoustics by Full-Field Mobilities

The mobilities of Equation (1) can be combined with the dynamic signature ${\mathit{F}}_f\left(\omega \right)$ of the excitation (later modelled in Section 2.3) to obtain the out-of-plane velocities of the vibrating surface S. By using only the projection towards the surface normal, it yields:

$${\mathit{v}}_n(\mathit{q},\omega )={\mathit{H}}_{{\mathit{v}}_n\mathit{q}f}\left(\omega \right)·{\mathit{F}}_f\left(\omega \right)\in \mathbb{C},$$

(3)

where the mobility FRFs ${\mathit{H}}_{{{\mathit{v}}_n}_{\mathit{q}f}}\left(\omega \right)$ is of size $N_q\times N_f$, $N_q$ being the number of the outputs and $N_f$ of the force inputs ${\mathit{F}}_f\left(\omega \right)$.

The Rayleigh integral approximation of the Helmholtz equation works as substituting the integral of Equation (2) by the sum of the $N_q$ discrete contributions $\Delta S_q$ of the pumping surface S around the structural mesh points, discretised as $S\approx {\sum }_q\Delta S_q$. By revisiting the Rayleigh integral approximation with the achievements of the FRF formulation of Equation (3), Equation (2) yields:

$$p({\mathit{a}}_a,\omega )\approx 2i\omega {\rho }_0\sum _q^{N_q}{\mathit{H}}_{{{\mathit{v}}_n}_{qf}}\left(\omega \right){\mathit{F}}_f\left(\omega \right)G_{aq}(r_{aq},\omega )\Delta S_q\in \mathbb{C},$$

(4)

with ${\mathit{H}}_{{{\mathit{v}}_n}_{\mathit{q}f}}\left(\omega \right)$, ${\mathit{F}}_f\left(\omega \right)$ and $G_{aq}(r_{aq},\omega )$ as complex-valued discrete quantities.

$G_{aq}(r_{aq},\omega )$ and $\Delta S_q$ are direct function of the discretisation of both domains, specifically subdivided in $N_a$ discrete points in the acoustic domain and of the $N_q$ points on the vibrating surface. $G_{aq}(r_{aq},\omega )$ and $\Delta S_q$ can therefore contribute to define a complex-valued diffusion matrix ${\mathit{T}}_{\mathit{aq}}\left(\omega \right)$, sized $N_a\times N_q$, with each element composed by:

$$T_{aq}\left(\omega \right)=2i\omega {\rho }_0{G}_{aq}(r_{aq},\omega )\Delta S_q.$$

(5)

Equation (4) can be algebraically rearranged as:

$$p({\mathit{a}}_a,\omega )\approx {\mathit{T}}_{a\mathit{q}}\left(\omega \right){\mathit{H}}_{{{\mathit{v}}_n}_{\mathit{q}f}}\left(\omega \right){\mathit{F}}_f\left(\omega \right)\in \mathbb{C}.$$

(6)

Contrary to the acoustic transfer vectors (ATVs) in [5,58,59] between sound pressures and surface velocities, a vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$, sized $N_a\times N_f$, can be defined between acoustic pressures and excitation forces as:

$${\mathit{V}}_{\mathit{af}}\left(\omega \right)={\mathit{T}}_{\mathit{aq}}\left(\omega \right)·{\mathit{H}}_{{{\mathit{v}}_n}_{\mathit{qf}}}\left(\omega \right)\in \mathbb{C}.$$

(7)

Therefore, Equation (6) can be easily compacted in:

$$p({\mathit{a}}_a,\omega )\approx {\mathit{V}}_{af}\left(\omega \right){\mathit{F}}_f\left(\omega \right)\in \mathbb{C}.$$

(8)

Equation (8) is the pivot of the direct vibro-acoustic formulation here proposed, since it is able to map each sound pressure field, as the excitation varies, while the structural dynamics in the mobilities and the acoustic mesh do not change.

2.2.2. Induced Forces in a Pseudo-Inverse Formulation of Vibro-Acoustics by Full-Field Mobilities

The direct reformulation in Equation (8) of the Rayleigh integral approximation by the use of mobilities can be seen in the reverse way of an inverse formulation. The latter starts from known complex-valued pressure fields $\widehat{\mathit{p}}({\mathit{a}}_{\mathit{a}},\omega )$ and aims at reconstructing the forces ${\widehat{\mathit{F}}}_f\left(\omega \right)$ induced in the structural locations of the direct problem’s excitations, similarly to what found in [57,60,61,62,63,64,65,66], but knowing exactly the points. At the core of the inverse formulation stays the pseudo-inversion of the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$ in Equation (7). Therefore, once the pressure field $\widehat{\mathit{p}}({\mathit{a}}_{\mathit{a}},\omega )\in \mathbb{C}$ is used as the acoustic excitation, the forces ${\widehat{\mathit{F}}}_f\left(\omega \right)$ at the shakers’ heads can be retrieved by:

$${\widehat{\mathit{F}}}_f\left(\omega \right)\approx {\mathit{V}}_{f\mathit{a}}^{+}\left(\omega \right)\widehat{\mathit{p}}({\mathit{a}}_{\mathit{a}},\omega )\in \mathbb{C},$$

(9)

where ${\mathit{V}}_{\mathit{fa}}^{+}\left(\omega \right)$ is the pseudo-inverse of the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$, sized $N_f\times N_a$, definable as acoustic-vibrational FRF matrix between the acoustic pressures in the medium and the forces on the structure—retaining its dynamics—and defined by:

$${\mathit{V}}_{\mathit{fa}}^{+}\left(\omega \right)={\left[{\mathit{V}}_{\mathit{fa}}^H\left(\omega \right){\mathit{V}}_{\mathit{af}}\left(\omega \right)\right]}^{-1}{\mathit{V}}_{\mathit{fa}}^H\left(\omega \right)\in \mathbb{C},$$

(10)

${\left(\right)}^H$ being the Hermitian operator. Note that the internal inversion in Equation (10) is operated on the complex-valued kernel, sized only $N_f\times N_f$, which dramatically simplifies the pseudo-inversion when $N_f$ is very small, like here with $N_f=1$ with only one shaker at a time.

In order to assess the quality of the force retrieval procedure of Equation (9), an error function ${\mathit{Err}}_{{\mathit{F}}_f}\left(\omega \right)\in \mathbb{C}$ can be defined. It can be based on the difference between the original forces ${\mathit{F}}_f\left(\omega \right)$ that generated the pressure fields $\mathit{p}({\mathit{a}}_{\mathit{a}},\omega )$ in Equation (8), later fed back—$\widehat{\mathit{p}}({\mathit{a}}_{\mathit{a}},\omega )\equiv \mathit{p}({\mathit{a}}_{\mathit{a}},\omega )$—to the acoustic excitation fields $\widehat{\mathit{p}}({\mathit{a}}_{\mathit{a}},\omega )$ in the retrieval of medium-borne forces on the structure ${\widehat{\mathit{F}}}_f\left(\omega \right)$ of Equation (9), simply by:

$${\mathit{Err}}_{{\mathit{F}}_f}\left(\omega \right)={\mathit{F}}_f\left(\omega \right)-{\widehat{\mathit{F}}}_f\left(\omega \right)\in \mathbb{C}.$$

(11)

2.3. Simple Broad Frequency Band Modelling of the Excitation Forces

The strength of this experimental mobilities-based approach comes from its avoiding simplifications of the linear responses of the real structure in a broad frequency range. Therefore, the modelled excitations must be defined in the same manner, same frequency domain and complex-valued nature. Coloured noises can generally represent the frequency domain deployment of broad band signals, with a definition in the whole range. The general shape of the coloured noises depends on the amplitude modulator parameter $\alpha$, with $\alpha \in [-2,2]$ defining the noise colour ($\alpha =-2$ for violet noise as in Figure 10, Figure 16 and Figure 21; $\alpha =-1$ for blue noise as in Figure 8, Figure 14 and Figure 19; $\alpha =0$ for white noise as in Figure 6, Figure 12 and Figure 17; $\alpha =1$ for pink noise as in Figure 7, Figure 13 and Figure 18; and $\alpha =2$ for red noise as in Figure 9, Figure 15 and Figure 20). Among all of them, the white noise keeps an even treatment of the amplitudes in all the frequency lines of the spectrum, to which some randomness might be added to better represent reality. Furthermore, proper phasing can be also added, with the same considerations about randomness. The other variances give more emphasis to the lower part of the spectral lines when $\alpha >0$; instead, with $\alpha <0$, more predominance is given to the higher frequency components of the spectra.

Complex-Valued Coloured Noise with Random Amplitude and Phase Variations

In order to appreciate the contribution of experiment-based full-field mobilities, thanks to their complex-valued representation of the real-life structural dynamics, different excitation signatures for $F\left(\omega \right)\in \mathbb{C}$ components can be discussed in the general formulation here proposed. The type of signal that better describes what can be generally acquired for forces in any real testing—with generically varied complex-valued spectrum—is that of a complex-valued coloured noise excitation, therefore with variable complex amplitude and phase in a fully populated broad frequency band, potentially with random variations in the amplitude & phase. Only the linearity—between forces and structural responses—needs to be respected. Here labelled as “rap”, which stands for random variations in the modulated amplitude and sinusoidal phase, $F\left(\omega \right)$ can be defined as:

$$F\left(\omega \right)={\displaystyle \frac{F_{0_r}}{{\omega }^{\alpha }}}{e}^{i{\theta }_r\left(\omega \right)}\in \mathbb{C},$$

(12)

with $F_{0_r}=F_0(1+{\beta }_{F_0}(Ran{d}_{F_0}-0.5))$, ${\theta }_r\left(\omega \right)=\theta \left(\omega \right)(1+{\beta }_{\theta }(Ran{d}_{\theta }-0.5))$ with $\theta \left(\omega \right)$ as the chosen phase function, contaminated by amplitude & phase random variations. In Equation (12) the other quantities are: $F_0\in \mathbb{R}$, the reference amplitude; ${\beta }_{F_0}\in \mathbb{R}$, $0\le {\beta }_{F_0}\le 1$, the level of randomness in the amplitude; $Ran{d}_{F_0}$, or $Ran{d}_{\theta }$, a function returning a pseudo-random number in the range 0 to 1, respectively for the amplitudes or for the phases; $\theta \left(\omega \right)=S_{0}sin(\pi N_p(\omega -{\omega }_0)/\Delta \omega +{\theta }_0)$, a selected phase function; $S_0$, the sinusoidal phase range multiplier; $N_p$, the number of half cycles of the phase in the range; ${\omega }_0$, the starting frequency; $\Delta \omega$, the frequency range; ${\theta }_0$, a reference phase; ${\beta }_{\theta }$, the level of randomness in the phase. For all the examples here provided, the following inputs were taken: $F_0=0.075$ N, $S_0=0.75$ rad, $N_p=6.5$, ${\theta }_0=\pi /4$ rad, ${\beta }_{F_0}=0.125$, ${\beta }_{\theta }=0.25$, with the respective quantities explained above.

Equation (12) clearly appears much more adherent to real-life excitations, as found in any random-noise shaker testing, with a generically varied complex-valued spectrum. In the procedures here proposed, based on full-field mobilities from testing, therefore, no limitations nor assumptions are imposed on the excitations, except the respect of the linearity of the structural responses with the shaking forces.

3. Results

This section is devoted to putting together the real-life testinThis Section is devoted to put together the real-life testing of Section 2.1—by examples of full-field SLDV-based mobilities in Equation (1)—with the acoustic radiation simulations formulated in Section 2.2. In particular, as summarised respectively in

Table 1. Summary of the proposed examples of vibro-acoustic transfer matrix.

Section	Equation	Quantity	Structural	Acoustic	Frequency	Figure
			Excitation	Domain	Domain
3.2	(7)	$V_{a{S}_1}\left(\omega \right)$	S1, –	single dof [374]	[20–1024] Hz	3a
3.2	(7)	$V_{a{S}_2}\left(\omega \right)$	S2, –	single dof [374]	[20–1024] Hz	3b
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	121.1 Hz	4a
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	127.5 Hz	4b
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	250.0 Hz	4c
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	284.4 Hz	4d
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	335.9 Hz	4e
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	496.1 Hz	4f
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	754.7 Hz	4g
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_1}\left(\omega \right)$	S1, –	whole mesh	990.6 Hz	4h
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	121.1 Hz	5a
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	127.5 Hz	5b
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	250.0 Hz	5c
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	284.4 Hz	5d
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	335.9 Hz	5e
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	496.1 Hz	5f
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	754.7 Hz	5g
3.2	(7)	${\mathit{V}}_{\mathit{a}{S}_2}\left(\omega \right)$	S2, –	whole mesh	990.6 Hz	5h

and

Table 2. Summary of the proposed sound pressure results.

Section	Equation	Quantity	Structural	Acoustic	Frequency	Figure
			Excitation	Domain	Domain
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S1, white noise-rap	single dof [374]	[20–1024] Hz	6a
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S2, white noise-rap	single dof [374]	[20–1024] Hz	6b
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S1, pink noise-rap	single dof [374]	[20–1024] Hz	7a
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S2, pink noise-rap	single dof [374]	[20–1024] Hz	7b
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S1, blue noise-rap	single dof [374]	[20–1024] Hz	8a
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S2, blue noise-rap	single dof [374]	[20–1024] Hz	8b
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S1, red noise-rap	single dof [374]	[20–1024] Hz	9a
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S2, red noise-rap	single dof [374]	[20–1024] Hz	9b
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S1, violet noise-rap	single dof [374]	[20–1024] Hz	10a
3.3	(8)	$p({\mathit{a}}_a,\omega )$	S2, violet noise-rap	single dof [374]	[20–1024] Hz	10b

, attention is given to the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$ in Equation (7) to simulate the acoustic pressure $p({\mathit{a}}_a,\omega )$ in Equation (8), thanks to the excitation signatures proposed in Section 2.3. Furthermore, as reported in Table 3, the acoustic-vibrational FRF matrix ${\mathit{V}}_{\mathit{fa}}^{+}\left(\omega \right)$ of Equation (10) is calculated, in order to estimate the airborne force in Equation (9) and related errors of Equation (11). The latter are useful for assessing the quality of the force retrieval procedure, with also its computational precision. All these results are accompanied by the related details and comments.

Some general notes about all the 2D frequency domain graphs are here given. At the top of each figure, generally three lines of coloured text are annotated. The first contains, on the left side, the frequency step and the corresponding value in Hertz, in dark yellow, linked to the vertical line in the graph, or cursor; at the centre, in magenta, the dof of inquiry is proposed before the name of the main function of the graph. The second line, on the right side, shows in blue the value of the main function at the cursor’s abscissa, with related bracketed quantities; eventually, in the case of complex amplitude and phase, this information is split also to the third line. The third line can be instead taken, when only the amplitude of the functions is shown, by the description of the eventually added function and value at the cursor position, in black with left margin. The sketched functions, or complex-valued parts, are framed by grey lines, with the main functions’ extremes and bracketed dimensions on the right side in grey text, while the frequency range is annotated in grey text at the very bottom of the frame. Another text line, at the very bottom, indicates the geometrical and frequency references (here both reporting SLDV), together with the used shaker in the specific structural dof, marking it with dark yellow, whereas the mute shaker is in dark grey. Inside the frame, the main function is drawn in pure blue, and the measurement technology (here SLDV) is recalled in blue text at the right of the frame. The dark yellow vertical cursor intercepts the main function in a point through which an horizontal light blue line is added to show the reached main function value. If another function is superimposed—with its own ordinate quantities, scales and ranges annotated on the right of the frame—the chosen colour is pure black, but the value at the cursor position is only reported in the text above.

Some notes on the proposed maps of the figures follow to help the reader better understand the information they provide. First, the top-line text describes briefly the sketched function, followed by the indication of the active (in dark yellow text) and mute (in dark grey text) shakers, with corresponding structural dofs, also reported on the map by a big dot. For SLDV datasets, shaker 1 was in structural dof 2611, shaker 2 in structural dof 931. The third grey text line contains the frequency step and the corresponding value in Hertz. All the maps are 3D complex-valued fields, but projected with the same geometrical view and phasing angle to keep the comparisons viable. Therefore, the fourth text line says which complex-valued part is figured, while the fifth line gives the projecting phase angle. For the mobility ODS of the structural domain (positioned in front of the acoustic mesh to give a hint of the radiation source), blue tones are associated to the out-of-plane motion, giving brightest blue (rgb = [0,0,1]) to the maximal value of the velocity in the range, while giving darkest blue (rgb = [0,0,0], or pure black) to the lowest value in the range. Also the triangles of the maps, rendered by OpenGL primitives, are coloured accordingly with the colours of the defining points. Instead, for the acoustic mesh, grey tones are associated to the shown function, with pure white (rgb = [1,1,1]) in the maximum value, while pure black(rgb = [0,0,0]) is associated to the minimal value in the range. The ranges of the sketched vibro-acoustic function are drawn on the left side of each map, with the linearly varying grey tones and the extremes of the whole range. On each acoustic mesh, an inquiry acoustic dof is highlighted by a magenta big dot; its number (here 374) is written in the magenta text (sixth line) above the maps, not to be confused with the structural dofs of the shakers on the mobility ODS; the value of the function reached in the inquiry dof is reported in blue in the seventh text line, but also by a magenta dash in the side range bar.

Table 3. Summary of the proposed pseudo-inverse vibro-acoustic quantities: acoustic-vibrational FRFs, estimated airborne forces and identification errors.

Section	Equation	Quantity	Acoustic Domain,	Structural	Frequency	Figure
			Excitation	Domain	Domain
3.4	(10)	$V_{S_{1}a}^{+}\left(\omega \right)$	single dof [374], –	S1	[20–1024] Hz	11a
3.4	(10)	$V_{S_{2}a}^{+}\left(\omega \right)$	single dof [374], –	S2	[20–1024] Hz	11b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, white noise-rap	S1	[20–1024] Hz	12a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, white noise-rap	S1	[20–1024] Hz	12b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, pink noise-rap	S1	[20–1024] Hz	13a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, pink noise-rap	S1	[20–1024] Hz	13b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, blue noise-rap	S1	[20–1024] Hz	14a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, blue noise-rap	S1	[20–1024] Hz	14b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, red noise-rap	S1	[20–1024] Hz	15a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, red noise-rap	S1	[20-1024] Hz	15b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, violet noise-rap	S1	[20–1024] Hz	16a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, violet noise-rap	S1	[20–1024] Hz	16b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, white noise-rap	S2	[20–1024] Hz	17a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, white noise-rap	S2	[20–1024] Hz	17b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, pink noise-rap	S2	[20–1024] Hz	18a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, pink noise-rap	S2	[20–1024] Hz	18b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, blue noise-rap	S2	[20–1024] Hz	19a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, blue noise-rap	S2	[20–1024] Hz	19b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, red noise-rap	S2	[20–1024] Hz	20a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, red noise-rap	S2	[20–1024] Hz	20b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, violet noise-rap	S2	[20–1024] Hz	21a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, violet noise-rap	S2	[20–1024] Hz	21b

Table 3. Summary of the proposed pseudo-inverse vibro-acoustic quantities: acoustic-vibrational FRFs, estimated airborne forces and identification errors.

Section	Equation	Quantity	Acoustic Domain,	Structural	Frequency	Figure
			Excitation	Domain	Domain
3.4	(10)	$V_{S_{1}a}^{+}\left(\omega \right)$	single dof [374], –	S1	[20–1024] Hz	11a
3.4	(10)	$V_{S_{2}a}^{+}\left(\omega \right)$	single dof [374], –	S2	[20–1024] Hz	11b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, white noise-rap	S1	[20–1024] Hz	12a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, white noise-rap	S1	[20–1024] Hz	12b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, pink noise-rap	S1	[20–1024] Hz	13a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, pink noise-rap	S1	[20–1024] Hz	13b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, blue noise-rap	S1	[20–1024] Hz	14a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, blue noise-rap	S1	[20–1024] Hz	14b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, red noise-rap	S1	[20–1024] Hz	15a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, red noise-rap	S1	[20-1024] Hz	15b
3.5	(9)	${\widehat{F}}_{S1}\left(\omega \right)$	whole mesh, violet noise-rap	S1	[20–1024] Hz	16a
3.5	(11)	${Err}_{F_{S1}}\left(\omega \right)$	whole mesh, violet noise-rap	S1	[20–1024] Hz	16b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, white noise-rap	S2	[20–1024] Hz	17a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, white noise-rap	S2	[20–1024] Hz	17b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, pink noise-rap	S2	[20–1024] Hz	18a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, pink noise-rap	S2	[20–1024] Hz	18b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, blue noise-rap	S2	[20–1024] Hz	19a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, blue noise-rap	S2	[20–1024] Hz	19b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, red noise-rap	S2	[20–1024] Hz	20a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, red noise-rap	S2	[20–1024] Hz	20b
3.5	(9)	${\widehat{F}}_{S2}\left(\omega \right)$	whole mesh, violet noise-rap	S2	[20–1024] Hz	21a
3.5	(11)	${Err}_{F_{S2}}\left(\omega \right)$	whole mesh, violet noise-rap	S2	[20–1024] Hz	21b

Figure 3. SLDV-based examples of vibro-acoustic FRF graphs in the frequency domain evaluated in acoustic dof 374, with excitation from shaker 1 in (a), from shaker 2 in (b).

Figure 3. SLDV-based examples of vibro-acoustic FRF graphs in the frequency domain evaluated in acoustic dof 374, with excitation from shaker 1 in (a), from shaker 2 in (b).

Figure 4. SLDV-based examples of vibro-acoustic transfer matrix mesh, with the highlighted (magenta) acoustic dof 374, excitation from shaker 1, in its complex-valued amplitude evaluated at different frequencies: 121 Hz in (a), 127 Hz in (b), 250 Hz in (c), 284 in (d), 336 Hz in (e), 496 Hz in (f), 755 Hz in (g), 991 Hz in (h).

Figure 4. SLDV-based examples of vibro-acoustic transfer matrix mesh, with the highlighted (magenta) acoustic dof 374, excitation from shaker 1, in its complex-valued amplitude evaluated at different frequencies: 121 Hz in (a), 127 Hz in (b), 250 Hz in (c), 284 in (d), 336 Hz in (e), 496 Hz in (f), 755 Hz in (g), 991 Hz in (h).

3.1. Brief Notes on the Acoustic Domain Modelling

In all the simulations of the interaction between the structural domain—extensively defined in Section 2.1—and the acoustic domain, the latter, among the many choices, is modelled as follows: a squared mesh, of size 0.5 m × 0.5 m, spatially sampled every 10 mm by 51 × 51 dofs ($N_a$ = 2601), distant 0.25 m from the vibrating plate, on which it is centred.

Figure 5. SLDV-based examples of vibro-acoustic transfer matrix mesh, with the highlighted (magenta) acoustic dof 374, excitation from shaker 2, in its complex-valued amplitude evaluated at different frequencies: 121 Hz in (a), 127 Hz in (b), 250 Hz in (c), 284 in (d), 336 Hz in (e), 496 Hz in (f), 755 Hz in (g), 991 Hz in (h).

Figure 5. SLDV-based examples of vibro-acoustic transfer matrix mesh, with the highlighted (magenta) acoustic dof 374, excitation from shaker 2, in its complex-valued amplitude evaluated at different frequencies: 121 Hz in (a), 127 Hz in (b), 250 Hz in (c), 284 in (d), 336 Hz in (e), 496 Hz in (f), 755 Hz in (g), 991 Hz in (h).

During the TEFFMA testing, unfortunately, a precise recording of the ambient temperature was not made, therefore here the assumption of a likely room temperature of 15 °C is made. This serves to obtain from [67] the medium data, matching the sensing of the structural dynamics in Section 2.1: the speed of sound $c_0$ = 340.27 m/s and the air density ${\rho }_0$ = 1.225 kg/m³. For the sake of showing the potential of these hybrid vibro-acoustic simulations, the assumed values can be considered sufficiently realistic. Indeed, if testing the components for different air conditions8, the structural testing should be run in a climate-chamber, to fully replicate the material behaviour in those conditions.

3.2. Vibro-Acoustic Transfer Matrices from Experiment-Based SLDV Mobilities

This section, by means of the formulation of Section 2.2, wants to highlight the high-quality results, in terms of field continuity and detail, of the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$, as obtained from accurate experiment-based SLDV mobilities in optical testing, but without any numerical structural model, which instead might be hard to tune and give a limited representation of the real-life behaviour, especially regarding the damping distribution, materials’ characteristics, simplifications on the mounting/boundary conditions and dynamics truncation at higher frequencies.

In Figure 3 examples of the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$ are reported as a frequency domain relation from shaker 1 (Figure 3a), or shaker 2 (Figure 3b), and acoustic dof 374, here selected in the squared acoustic mesh. It can be clearly seen how the data retain completely the complex-valued nature of the experiment-based full-field mobilities, especially in the damping around resonances and anti-resonances, or in the blending at any other frequency line. This type of results is hardly obtainable by synthetic structural models without a complete tuning or model updating, as in [34,43].

In Figure 4 and Figure 5 the whole acoustic mesh of the vibro-acoustic transfer matrix is displayed in grey tones behind the estimated mobility matrix ${\mathit{H}}_{{{\mathit{v}}_n}_{\mathit{q}f}}\left(\omega \right)$ of the vibrating & radiating plate in blue tones at 121.1, 126.6, 250.0, 284.4, 335.9, 496.1, 754.7, 990.6 Hz, respectively, with the complex-valued amplitudes of both the structural and acoustic domains. Beside each tile, the range scale in grey tones allows to locate, by means of a magenta line, the level reached by the vibro-acoustic transfer matrix in the acoustic dof 374, which is the magenta dot, highlighted on the acoustic mesh. As anticipated, darker tones correspond linearly to the minimal values in the range, while full brightness pertains to the maximal values. Instead, the dark yellow dot on the vibrating plate ODS locates the active shaker, while the dark grey dot references the mute one.

In Figure 4 and Figure 5 the evaluations of the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$ are shown over the entire acoustic mesh, retaining the complex-valued relations and phase delays, coming from the complex-valued mobility matrix ${\mathit{H}}_{{{\mathit{v}}_n}_{\mathit{q}f}}\left(\omega \right)$, but blended in the complex-valued summation by Green’s function. The distance of the acoustic mesh plays a relevant role in this blending, or averaging with fading effects, of the contributions9 of specific areas on the vibrating plate. Indeed, due to the greater distance—between the acoustic mesh and the vibrating plate—here simulated, in the complex amplitude of Figure 4 and Figure 5, the many discontinuities in the corresponding mobility maps are evened in a nearly constant field for the vibro-acoustic transfer matrix. The complex-valued mixing of the diffusion matrix ${\mathit{T}}_{\mathit{aq}}\left(\omega \right)$ in Equation (7) blends the contribution of the mobility maps in an evened field. Therefore, as the frequency rises, indeed more shape complexity pertains the mobility maps, as can be clearly seen in Figure 4a–h and in Figure 5a–h as well; but the resulting complex-valued blending in each point of the vibro-acoustic transfer matrix field, taking into account all the $N_q$ = 2907 contributions across the radiating surface, properly phased, has quite a different shape now, coming from the complex-valued summation, driven by Green’s functions in Equations (4) and (7).

Figure 6. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a white noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 6. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a white noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

3.3. Acoustic Pressure Spectra from Experiment-Based SLDV Mobilities and Complex-Valued Forces

The full retention of the complex-valued nature of the vibro-acoustic transfer matrix is displayed together with the specific signatures of the excitations, e.g., as modelled in Section 2.3, for the deployment in the whole frequency domain here analysed ([20–1023] Hz). The display of the fields of acoustic pressure in the acoustic mesh is disregarded, because the modulation of any excitation’s variant leads only to a scaling of the relative distribution in the complex amplitudes, shown for the vibro-acoustic transfer matrix, but with a specific phasing. Instead, the complex-valued spectra are here of major interest.

Compared to the shape of the vibro-acoustic transfer matrix shown in Figure 3, it is clear in Figure 6 how the white noise-rap excitation (in black) introduces a nearly constant—due to the limited randomness—complex amplitude scaling in the whole spectrum of the acoustic pressure (in blue) at dof 374, whereas the phase modifications appear stronger. Indeed, the slight randomness—of $\pm 6.25\%$ in the amplitudes—of the white noise-rap excitation, introduces a moderate variability of the scaling, as can be appreciated in Figure 6, appearing as a limited increase in thickness of the amplitude uncertainties in both the black and blue functions. Instead, the chosen $\theta \left(\omega \right)$ function—with its parameters and a scattering up to $\pm 12.5\%$—determines the marked complex phase variabilities.

Figure 7. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a pink noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 7. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a pink noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 8. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a blue noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 8. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a blue noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 9. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a red noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 9. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a red noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 10. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a violet noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

Figure 10. SLDV-based examples of acoustic pressure complex-valued spectrum graphs in the frequency domain evaluated in acoustic dof 374, from a violet noise-rap excitation, by shaker 1 in (a), shaker 2 in (b).

With the other noise colours, instead, the complex amplitude modulation given by each excitation becomes self-evident, with clear repercussions on the acoustic pressures from the original shapes of the vibro-acoustic transfer matrix, recalled from Figure 3. Besides, the complex phase variabilities replicate what commented above for the white noise-rap excitation, the phase parameters remaining exactly the same. The pink noise-rap excitation gives in both tiles of Figure 7a moderate (in logarithmic scale) decrease of the complex amplitudes in the acoustic pressure spectra. With an opposite effect, the blue noise-rap excitation raises moderately the higher frequency contributions of the complex amplitudes in the acoustic pressure in Figure 8. The stronger descent of the complex amplitudes is due to the red noise-rap excitation in Figure 9. Specular effect is contributed to the complex amplitudes of $p({\mathit{a}}_a,\omega )$ by the violet noise-rap excitation in Figure 10.

3.4. Airborne Acoustic-Vibrational FRFs from Experiment-Based SLDV Mobilities

Following the formulation of Equation (10), the pseudo-inverse vibro-acoustic FRFs ${\mathit{V}}_{f\mathit{a}}^{+}\left(\omega \right)$ (or acoustic-vibrational FRF matrix) of force over airborne sound pressures can be achieved, as shown in the single pseudo-inverse vibro-acoustic FRF of Figure 11, where the airborne pressure field is considered acting on the single acoustic dof 374 and the force in the structural dof 2611 of shaker 1 in tile a, in the structural dof 931 of shaker 2 in tile b. The same can be repeated for all the $N_a$ = 2601 acoustic dofs in the acoustic mesh. The comparison between the vibro-acoustic transfer matrix of Figure 3 and the acoustic-vibrational transfer matrix of Figure 11 is straightforward: there is a clear reflection of all the complex-valued data parts, without distortions. How the whole complex-valued information is retained in the pseudo-inversion, up to the numerical precision of the routines, can be clearly appreciated. The successful evaluation of the pseudo-inverse vibro-acoustic FRFs ${\mathit{V}}_{f\mathit{a}}^{+}\left(\omega \right)$ plays, therefore, a crucial role in the following force retrieval from airborne acoustic fields.

3.5. Airborne Structural Force Evaluation as Induced by Known Pressure Fields

For the identification of the force ${\widehat{F}}_{S1}\left(\omega \right)$ in the structural dof 2611 of shaker 1 (f = 2611 = $S1$), or ${\widehat{F}}_{S2}\left(\omega \right)$ in the structural dof 931 of shaker 2 (f = 931 = $S2$), by means of Equation (9), the whole airborne pressure field acting on all the $N_a$ dofs of the acoustic mesh must be used, together with all the corresponding pseudo-inverse vibro-acoustic FRFs in Section 3.4. The coloured noise excitation, in the “rap” variations, on shakers 1–2 was adopted to simulate the pressure fields with the different complex amplitude modulations and selected phasing modifications in the whole frequency range of interest.

In Figure 12a the nearly-even white noise-rap excitation $F\left(\omega \right)$ (in black)—with $\pm 6.25\%$ of amplitude scattering around the value of 7.500 $\times {10}^{-2}$ N and sinusoidal phase lag with $\pm 12.5\%$ scattering around the sine magnitude of 0.75 rad in the whole frequency domain—and the identified force ${\widehat{F}}_{S1}\left(\omega \right)$ (in blue) are shown together. Apparently, the functions are completely superimposed in Figure 12a, where the ranges are adapted to the actual functions; however, the truncation of the values at the 3rd decimal can not highlight tiny discrepancies. Instead, as summarised in the second and third columns of Table 4 and Table 5 for the errors in the complex amplitudes and complex phases, the punctual values tell the very small difference between the original excitation and the identified one, close to computational10 precision, which only Figure 12b can reveal by displaying the differences evaluated by Equation (11) against the functions’ ranges (fourth and fifth columns of Table 4 and Table 5).

The same considerations can be made about the white noise-rap excitation $F\left(\omega \right)$ when acting on shaker 2, with the retrieved force ${\widehat{F}}_{S2}\left(\omega \right)$ in Figure 17a, and with the errors ${Err}_{F_{S2}}\left(\omega \right)$ in Figure 17b. Table 4 and Table 5 are useful to summarise the ranges, in the complex amplitudes and complex phases, of both the retrieval errors and the forces.

Figure 11. SLDV-based examples of inverse vibro-acoustic FRF graphs in the frequency domain evaluated as force in shaker 1 (structural dof 2611) in (a), in shaker 2 (structural dof 931) in (b), over the airborne acoustic pressure from dof 374.

Figure 11. SLDV-based examples of inverse vibro-acoustic FRF graphs in the frequency domain evaluated as force in shaker 1 (structural dof 2611) in (a), in shaker 2 (structural dof 931) in (b), over the airborne acoustic pressure from dof 374.

Figure 12. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as white noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 12. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as white noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 13. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as pink noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 13. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as pink noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 14. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as blue noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 14. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as blue noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

In Figure 13a the original pink noise-rap excitation $F\left(\omega \right)$ (in black)—with the complex amplitude and phase contaminated by the random variations in the whole frequency domain—and the identified force ${\widehat{F}}_{S1}\left(\omega \right)$ (in blue) are shown together, again totally superimposed, because in Figure 13a the precision is truncated only at the 3rd decimal in the punctual values, while the ranges are now orders of magnitude greater than the real difference between the curves. Also in this case, only from Equation (11)—depicted in Figure 13b—can the right information be drawn. However, it is clear how the magnitude of the functions expands the error ranges of both the complex amplitude and complex phase. The example of ${\widehat{F}}_{S2}\left(\omega \right)$ force retrieval, in the chain due to the original pink noise-rap excitation $F\left(\omega \right)$ in shaker 2, is sketched in Figure 18a,b, with the extreme values as reported in Table 4 and Table 5.

Figure 15. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as red noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 15. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as red noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 16. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as violet noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

Figure 16. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as violet noise-rap force in shaker 1 from the whole airborne acoustic pressure field.

The retrieval of the original blue noise-rap excitation $F\left(\omega \right)$ in shaker 1, with related errors, is sketched in the tiles of Figure 14, while the same is made in Figure 19 for shaker 2 excitation, with minimal variations in the values of Table 4 and Table 5.

Similar deployment of the original red noise-rap excitation $F\left(\omega \right)$ in shaker 1 is made in the tiles of Figure 15, instead in Figure 20 for shaker 2 excitation.

Finally, the original violet noise-rap excitation $F\left(\omega \right)$ in shaker 1, and its retrieved $\widehat{F}\left(\omega \right)$, are shown in the tiles of Figure 16, while the same excitation in shaker 2 is shown in Figure 21, with the same approach and similar conclusions on the values of Table 4 and Table 5, which gather the data about all the excitations of these simulations.

4. Discussion

Section 3 has clearly shown how this simplified experiment-based approach—fully formulated in Section 2—can simulate the direct and pseudo-inverse vibro-acoustic problems of flat surfaces. In particular, as clearly deployed in Section 3.2, the augmented detail—coming from full-field datasets—gives an unprecedented mapping continuity in the complex-valued blending by the Green’s functions in the diffusion matrix of Equation (5) to obtain the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$, with no sacrifice for the experiment-based structural dynamics underneath. The latter, instead, enriches the simulations by extracting the real-life behaviour of the set-up realisation. Note that the measurement errors enter in the estimation of the full-field mobilities at the core of the approach, therefore strong care must be paid during the testing. This is even more important for SLDV, here used, which proved (see [40,53]) to suffer from hardly distinguishable scanning-related faults, which can ruin the whole mapping, while the native/image-based full field technologies have a better continuity in the spatial domain. Although simplified, this achievement deserves attention, also because it can be used to raise the benchmarks in purely numerical simulations, and to let them assess better when their models need better and refined updating or tuning.

Section 3.3 has shown clearly how the complex-valued formulation of the whole approach is fundamental to retain the whole structural dynamics in the broad frequency band spectra of the acoustic pressure $p({\mathit{a}}_a,\omega )$ simulations, while changing the complex-valued excitation spectrum. This accurate formulation can clearly work with any measured force from real testing, again with no assumptions, except for the respect of the linearity, under which the full-field mobilities were estimated. But in the case a new excitation level is needed, in a recursive manner, new mobilities must be acquired to make the approach valid.

Section 3.4 has shown, in this simplified—but mathematically sound and rigorous for the complex-valued formulation widely adopted—approach, how effective can the pseudo-inverse be. For the shake of compactness, the acoustic-vibrational FRF ${\mathit{V}}_{f\mathit{a}}^{+}\left(\omega \right)$ in Equation (10) was evaluated only between the one acoustic dof and the two structural dofs of the shakers here used, to make the appropriate comparison with the function already shown for the direct problem of the vibro-acoustic transfer matrix ${\mathit{V}}_{\mathit{af}}\left(\omega \right)$. Nonetheless, the pseudo-inversion worked well with no computational distortions, therefore the quality of the SLDV-based mobilities is again of uttermost importance, as the biggest errors may come from a sub-optimal testing campaign, rather than from rounds-off in the approach.

Figure 17. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as white noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 17. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as white noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 18. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as pink noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 18. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as pink noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Section 3.5 has revealed the high precision of the approach, by means of the ${\mathit{V}}_{\mathit{fa}}^{+}\left(\omega \right)$ and the acoustic pressure fields evaluated in the direct vibro-acoustic problem, in reconstructing the original excitation force, in all its complex-valued spectrum, again with no simplifications, except for linearity, as quoted above. While looking at Table 4, the distribution of the errors in the 10 examples of Section 3.5 seems broader and linked to the colour of the noise, especially regarding the second column, the one with the lower bound of the amplitude errors. Indeed, the range for it is a bit broader than expected [2.794e−20, 4.178e−17] N, but always of many orders lower than the reference amplitude $F_0$ of 7.500e−02 N, or than the values in the fourth and fifth column of Table 4. The upper bound of the identification errors is more compressed for all the simulated initial excitations, specifically in the range of [3.963e−15, 4.899e−15] N in the third column of Table 4. Thus, in the ten simulated examples, the errors of the complex amplitude of the identified force stay in the wide range of [2.794e−20, 4.899e−15]. But, if one normalises them against the extremes of the complex amplitudes of each function, as in the sixth and seventh column of Table 4, the relative errors result more constrained, independently of the excitation case, in a tighter range of [5.582e−16, 6.148e−14].

Figure 19. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as blue noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 19. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as blue noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 20. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as red noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 20. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as red noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 21. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as violet noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Figure 21. SLDV-based example of identified force complex-valued spectrum graph in (a), and related errors in (b), in the frequency domain evaluated as violet noise-rap force in shaker 2 from the whole airborne acoustic pressure field.

Instead, Table 5 just reports variations in the complex phase due to the random function evaluations in the adopted sinusoidal phase descriptor—common to all the signals, with the widest range in [−6.184e−14, 6.406e−14]. When the normalisation is made against the extremes of the complex phase wrapping ($\pm \pi$ rad), the relative phase errors stay therefore in a very similarly bounded range of [−1.901e−14, 2.039e−14].

Fusing together the extremes of these relative errors, the computational relative error of the identification approach in the pseudo-inversion of Equation (10) has a maximum of 6.148e−14, which can be considered remarkably low and close to the 64-bit machine- and numerical data-storage- precisions, especially if the thousands of parallel operations involved in the airborne force identification approach11 are taken into account. In [55], the relative errors of the amplitude were in the range [1.402e−14, 2.207e−14] with pure white noise excitation, without random variations nor phasing, at a lower amplitude of 1.0e−02 N and 0.1 m of distance between the acoustic mesh and the plate, but from the reduced DIC dataset, interpolated to SLDV references, again with $N_q=2907$ structural dofs and of $N_a=2601$ acoustic dofs. In [54], the relative error range had the widest extremes in [3.391e−20, 1.018e−14], but from raw DIC datasets and excitation at a lower amplitude of 5.0e−02 N and 0.2 m of distance between the acoustic mesh and the plate. Such accurate results were obtained from raw DIC-based datasets with the contributions of $N_q=\mathrm{11,998}$ structural dofs and of $N_a=\mathrm{10,201}$ acoustic dofs, with all the complex-valued computation involved.

Which role is played by the distance—between the structural domain and the acoustic mesh—on the accuracy of the whole approach should be more investigated, as well as which role is played by the number of dofs in both the acoustic and the structural domains. But, of much greater relevance, remains the quality of the measurements for estimating the receptances or the mobilities, which are responsible of sensing the real-life structural dynamics of the actual realisation of the set-up. Therefore, the clear lesson learned is to invest an expert’s time in the set-up preparation/tuning and resources in a high-quality measurement campaign (here, around 3.36 days just for each SLDV-mobilities’ scanning), because the proposed approach proved to be very accurate and very computationally fast (here around only 1 min, in the parallel computing custom coding), but it is not designed to correct the potential issues of the initial test measurements.

Table 4. Summary of the computational errors in the complex amplitude of force identification from airborne acoustic pressure fields.

Noise Colour	Min Amp Err	Max Amp Err	Min Amp	Max Amp	Min Amp Err	Max Amp Err
+ Shaker	[N]	[N]	[N]	[N]	/ Min Amp	/ Max Amp
white−rap S1	3.925e−17	4.816e−15	7.031e−02	7.969e−02	5.582e−16	6.043e−14
white−rap S2	4.178e−17	4.899e−15	7.031e−02	7.969e−02	5.942e−16	6.148e−14
pink−rap S1	1.999e−18	4.336e−15	1.412e−03	7.819e−02	1.416e−15	5.545e−14
pink−rap S2	4.671e−18	4.336e−15	1.412e−03	7.819e−02	3.308e−15	5.545e−14
blue−rap S1	1.122e−17	4.580e−15	1.552e−03	7.909e−02	7.229e−15	5.791e−14
blue−rap S2	1.122e−17	4.316e−15	1.552e−03	7.909e−02	7.229e−15	5.457e−14
red−rap S1	2.794e−20	4.336e−15	2.814e−05	7.819e−02	9.929e−16	5.545e−14
red−rap S2	5.421e−20	4.336e−15	2.814e−05	7.819e−02	1.926e−15	5.545e−14
violet−rap S1	4.282e−19	3.991e−15	3.080e−05	7.873e−02	1.390e−14	5.069e−14
violet−rap S2	4.282e−19	3.963e−15	3.080e−05	7.873e−02	1.390e−14	5.034e−14

Table 4. Summary of the computational errors in the complex amplitude of force identification from airborne acoustic pressure fields.

Noise Colour	Min Amp Err	Max Amp Err	Min Amp	Max Amp	Min Amp Err	Max Amp Err
+ Shaker	[N]	[N]	[N]	[N]	/ Min Amp	/ Max Amp
white−rap S1	3.925e−17	4.816e−15	7.031e−02	7.969e−02	5.582e−16	6.043e−14
white−rap S2	4.178e−17	4.899e−15	7.031e−02	7.969e−02	5.942e−16	6.148e−14
pink−rap S1	1.999e−18	4.336e−15	1.412e−03	7.819e−02	1.416e−15	5.545e−14
pink−rap S2	4.671e−18	4.336e−15	1.412e−03	7.819e−02	3.308e−15	5.545e−14
blue−rap S1	1.122e−17	4.580e−15	1.552e−03	7.909e−02	7.229e−15	5.791e−14
blue−rap S2	1.122e−17	4.316e−15	1.552e−03	7.909e−02	7.229e−15	5.457e−14
red−rap S1	2.794e−20	4.336e−15	2.814e−05	7.819e−02	9.929e−16	5.545e−14
red−rap S2	5.421e−20	4.336e−15	2.814e−05	7.819e−02	1.926e−15	5.545e−14
violet−rap S1	4.282e−19	3.991e−15	3.080e−05	7.873e−02	1.390e−14	5.069e−14
violet−rap S2	4.282e−19	3.963e−15	3.080e−05	7.873e−02	1.390e−14	5.034e−14

Table 5. Summary of the computational errors in the complex phase of force identification from airborne acoustic pressure fields.

Noise Colour	Min Pha Err	Max Pha Err	Min Pha	Max Pha	Min Pha Err	Max Pha Err
+ Shaker	[rad]	[rad]	[rad]	[rad]	/ Min Pha	/ Max Pha
white−rap S1	−5.973e−14	5.596e−14	$-\pi$	$+\pi$	−1.901e−14	1.781e−14
white−rap S2	−6.073e−14	6.017e−14	$-\pi$	$+\pi$	−1.933e−14	1.915e−14
pink−rap S1	−5.596e−14	5.496e−14	$-\pi$	$+\pi$	−1.781e−14	1.749e−14
pink−rap S2	−5.640e−14	6.406e−14	$-\pi$	$+\pi$	−1.795e−14	2.039e−14
blue−rap S1	−5.707e−14	5.562e−14	$-\pi$	$+\pi$	−1.817e−14	1.770e−14
blue−rap S2	−5.917e−14	5.329e−14	$-\pi$	$+\pi$	−1.883e−14	1.696e−14
red−rap S1	−5.174e−14	5.806e−14	$-\pi$	$+\pi$	−1.647e−14	1.848e−14
red−rap S2	−5.507e−14	5.329e−14	$-\pi$	$+\pi$	−1.753e−14	1.696e−14
violet−rap S1	−5.873e−14	5.607e−14	$-\pi$	$+\pi$	−1.869e−14	1.785e−14
violet−rap S2	−6.184e−14	5.607e−14	$-\pi$	$+\pi$	−1.968e−14	1.785e−14

Table 5. Summary of the computational errors in the complex phase of force identification from airborne acoustic pressure fields.

Noise Colour	Min Pha Err	Max Pha Err	Min Pha	Max Pha	Min Pha Err	Max Pha Err
+ Shaker	[rad]	[rad]	[rad]	[rad]	/ Min Pha	/ Max Pha
white−rap S1	−5.973e−14	5.596e−14	$-\pi$	$+\pi$	−1.901e−14	1.781e−14
white−rap S2	−6.073e−14	6.017e−14	$-\pi$	$+\pi$	−1.933e−14	1.915e−14
pink−rap S1	−5.596e−14	5.496e−14	$-\pi$	$+\pi$	−1.781e−14	1.749e−14
pink−rap S2	−5.640e−14	6.406e−14	$-\pi$	$+\pi$	−1.795e−14	2.039e−14
blue−rap S1	−5.707e−14	5.562e−14	$-\pi$	$+\pi$	−1.817e−14	1.770e−14
blue−rap S2	−5.917e−14	5.329e−14	$-\pi$	$+\pi$	−1.883e−14	1.696e−14
red−rap S1	−5.174e−14	5.806e−14	$-\pi$	$+\pi$	−1.647e−14	1.848e−14
red−rap S2	−5.507e−14	5.329e−14	$-\pi$	$+\pi$	−1.753e−14	1.696e−14
violet−rap S1	−5.873e−14	5.607e−14	$-\pi$	$+\pi$	−1.869e−14	1.785e−14
violet−rap S2	−6.184e−14	5.607e−14	$-\pi$	$+\pi$	−1.968e−14	1.785e−14

5. Conclusions

The complex-valued formulation of the direct and the pseudo-inverse vibro-acoustic problem, by means of the simplification of the Rayleigh integral approximation, has here proved to be accurate and computationally fast also for the use of SLDV-based mobilities, which—in some cases—have shown some unexpected noise from scanning-related issues.

The experiment-based full-field mobilities permitted to fully retain, with high quality mapping in the spatial and frequency domains, the structural dynamics of the actual realisation of the testing set-up, when evaluating vibro-acoustic transfer matrices and pressure fields, with no further assumptions nor simplifications, except for linearity. They become a valid alternative for those cases in which a purely numerical model of a complex structural dynamics cannot be easily tuned; at the same time, new benchmarks for synthetic digital-twins can be set. The procedure to retrieve an airborne force—impressed by airborne pressure fields on the flexible flat surface—proved to be reliable also with SLDV-based mobilities, as extensively documented by the different simulated examples.

The quality of the obtained methodological achievements fosters future investigations on more complex vibrating structures, also coupled with different fluids, for NVH and reliability studies, together with further experimental evidences from real-life challenging environments.