Sound radiation estimate from vibration measurements with multiple cameras Paolo Gardonio 1 , Gianluca Guernieri, Roberto Rinaldo, Andrea Fusiello, Emanuele Turco Università degli Studi di Udine DPIA – Via delle Scienze 208 33100 – Udine (I) ABSTRACT This paper presents experimental results on the estimation of the sound radiation by a thin flat plate from flexural vibration measurements taken with multiple cameras. The study considers the sound radiation into free-field by a baffled rectangular plate fixed on a rigid frame. The plate is excited by a tonal point force exerted by a shaker at the first two resonance frequencies of the plate flexural response. The resulting vibration field is measured with a set of six cameras suitably synchronised. More specifically, 3D point tracking is implemented on a regular 11 ൈ7 grid of small circular points marked on the plate. The transverse displacements at these points are thus reconstructed from triangulation with multiple, i.e. 6 , view-points. The transverse displacements are then used to derive the sound radiation from a direct boundary integral formulation. More specifically, for the flat rectangular plate at hand, the sound radiation is derived from the Rayleigh integral, which is approximated into a finite sum over the mesh of rectangular elements centred at the 11 ൈ7 grid of points depicted on the plate. The flexural vibration field and the sound radiation field derived from the camera measurements are contrasted with those derived from vibration measurements taken with a laser vibrometer. The initial results presented in this paper demonstrate the feasibility of the proposed measurement approach at low audio frequencies. 1. INTRODUCTION

Measuring sound radiation from machinery or sound transmission through wall structures is a rather challenging task, which may require quite expensive facilities and complex sound detection probes. Indeed, sound radiation and sound transmission are normally measured either in situ or in special acoustic rooms. In the former case the radiated sound field is measured at few specific points with microphones, or phonometers [1-3]. Alternatively, given portions of the radiated sound field can be measured with sound intensity probes, which, normally, rely on microphone and hot-wire transducers to measure the sound pressure and acoustic particle velocity at a point [1-5]. Nowadays, these systems are equipped with video-cameras that automatically detect the position of the probe such that the radiation field can be reconstructed automatically [6]. Acoustic cameras have also been developed, which rely on microphone arrays to detect the source of noise coming from a vibrating machinery or wall structure through beamforming methods [7-9]. In situ measurements are often affected by uncertainty of the radiating source operation conditions, by flanking noise sources and by low repeatability of the measurement configuration [1-3]. Accurate quantitative measurements of sound

1 paolo.gardonio@uniud.it

radiation by machineries or sound transmission by wall structures require quite complex and expensive test facilities, which normally involve rather large reverberant and/or anechoic rooms equipped with microphones located in well-defined positions according to international guidelines [1-6]. For instance, the radiated sound power of machines is normally measured in large reverberant rooms [10,11]. Alternatively, the sound transmission through wall structures is measured with reverberant-reverberant or reverberant-anechoic facilities, where the structure is mounted on an opening of a common wall between the two cameras [12]. The aim of this paper is to investigate experimentally the possibility of using photogrammetry to assess the sound radiation into free-field produced by flexural vibrations of distributed structures. More specifically, the free-field sound radiation is derived from the Rayleigh integral equation [13] with respect to the transverse vibration velocity field of the structure measured using multiple, i.e. more than 2, cameras operating synchronously. As summarized in Refs. [14,15], during the past two decades, vibration measurement with optical cameras has been the subject of quite a vast bulk of studies, which have covered several photogrammetry approaches, principally based on 2D point tracking (2DPT) [16,17] and 3D point tracking (3DPT) [18], on 3D digital image correlation (3D-DIC) [19,20] and on target-less approaches [21]. The early works considered single-camera arrangements, where the camera should be oriented at a grazing angle with the surface of the flexural vibration field. Later, vibration measurements with stereo-camera setups using triangulation techniques [22-24] were also exploited. More recently, stereo acquisitions based on setups with moving single-cameras or with multiple mirrors focusing multi-images on single cameras have also been investigated [25,26]. In general sound radiation in air covers quite a wide range of frequencies comprised between 20 Hz and 20 kHz and even for typical measurements of interest in engineering applications it reaches 2 to 5 kHz. Therefore, rather high frame-rate cameras should be employed to secure accurate measurements. Moreover, to guarantee a correct estimate of the sound radiation, the spatial measurement of the vibration field should be quite accurate [13] and thus the cameras should have a high spatial-resolution too. In this respect, it should be noted that the past few years have seen the rise of new processing and acquisition technologies [27,28], which can be used to increase spatial and temporal resolutions. In this paper, a simple sound radiation case-study is examined, which is composed by a thin flat rectangular plate fixed on a rigid baffle and excited in bending by a transverse point force. The flexural deflection shape [29] produced by a tonal excitation at the natural frequencies of the first two flexural modes is measured at a 𝑁 ௫ ൈ𝑁 ௬ grid of target points with a set of 6 synchronous cameras using triangulation. The sound radiation into free-field is then derived using the Rayleigh integral formulation [13], which is approximated into a finite sum assuming the plate is discretised into a 𝑁 ௫ ൈ𝑁 ௬ mesh of elements whose centres correspond to the grid of target points where the flexural vibrations are measured with the cameras [30]. 2. PLATE AND CAMERAS EXPERIMENTAL SETUP

Figure 1 shows the plate and cameras setup built up for this study, whose principal properties are summarised in Table 1. The thin rectangular plate is made of steel and is fixed onto a rigid frame. The plate is excited in bending by a shaker, which is connected to the plate via a stinger equipped with a force cell. As can be noticed in Figure 1a, the plate is divided into a mesh o 𝑁 ௫ ൈ𝑁 ௬ ൌ11 ൈ7 rectangular elements, whose centres are highlighted by small circular markers, hereafter called dots. The 3D positions of the markers are defined with reference to the global (or world) coordinate system ሺ𝑜, 𝑥, 𝑦, 𝑧ሻ , which, as shown in Figure 1a, is located on the centre of the plate and oriented with the x and y axes to the longer (horizontal) and shorter (vertical) edges of the plate. More precisely, the coordinates of the centre position of the markers at time 𝑡 ௞ will be specified as 𝑋 ௝ ሺ𝑡 ௞ ሻ, 𝑌 ௝ ሺ𝑡 ௞ ሻ, 𝑍 ௝ ሺ𝑡 ௞ ሻ , where, assuming the cameras have a 1 𝑇 ⁄ frame rate, 𝑡 ௞ ൌ𝑘𝑇 .

Figure 1: Measurement test rig. (a) panel with the 11 ൈ7 array of markers placed at the centers of the mesh of 77 radiating elements. (b) rigid frame structure used to hold the cameras and scanner laser vibrometer. (c) camera fixed to the frame structure via an articulated arm. Table 1: Plate dimensions and physical properties and cameras optical characteristics.

Parameter Plate Cameras dimensions 𝑙 ௫௣ ൈ𝑙 ௬௣ ൌ668 ൈ443 mm thickness ℎ ௣ ൌ1.8 mm density 𝜌 ௣ ൌ7200 kg/m ଷ Young’s modulus 𝐸 ௣ ൌ14 ൈ10 ଵ଴ N/m ଶ Poisson ratio 𝜈 ௣ ൌ0.3 modal damping ratio 𝜉 ௣ ൌ0.002 Positions of the shaker force excitation 𝑥 ௣ , 𝑦 ௣ ൌെ86 , െ111 mm Model COOAU Action Cam 4K Nativo 60fps 20MP Spatial resolution 1920 ൈ1080 px Temporal resolution 120 fps Positions of cameras with respect to the centre of the plate Radial distance 𝑑 ௝ Azimuthal angle 𝜃 ௝ Elevation angle 𝜙 ௝

𝑑 ଵ ൌ1.20 m ; 𝜃 ଵ ൌെ60 deg ; 𝜙 ଵ ൌ16 deg 𝑑 ଶ ൌ1.245 m ; 𝜃 ଶ ൌെ58 deg ; 𝜙 ଶ ൌ40 deg 𝑑 ଷ ൌ1.135 m ; 𝜃 ଷ ൌെ45 deg ; 𝜙 ଷ ൌ65 deg 𝑑 ସ ൌ1.155 m ; 𝜃 ସ ൌെ10 deg ; 𝜙 ସ ൌ60 deg 𝑑 ହ ൌ1.10 m ; 𝜃 ହ ൌ20 deg ; 𝜙 ହ ൌ60 deg 𝑑 ଺ ൌ1.085 m ; 𝜃 ଺ ൌ60 deg ; 𝜙 ଺ ൌ35 deg As shown in Figures 1b,1c, a set of 6 cameras are arranged via small articulated arms on a stiff truss structure with the optical axes pointing to the centre of the plate. The structure is oriented parallel, and placed about 0.7 m away from, the surface of the plate. The cameras are arranged on the plane of the truss structure along an imaginary half-circumference, which is centred along the z -axis of plate and has a radius of about 1 m . To brake unwanted symmetries, the cameras are scattered unevenly along the circumference and are also slightly displaced with random offsets from the plane of the truss structure. 3. MEASUREMENT METHOD

Since the aim of the study is to investigate the intrinsic properties of the proposed flexural vibration measurement and sound radiation estimate with image based systems, a 3D-Point Tracking (3DPT) [15,18] approach is employed. It should be emphasised though, that in practice 3D-Point Digital Image Correlation (3D-DIC) [15,19,20] can be used too, provided a regular grid of target positions can be identified from the pattern of the radiating surface and thus used to construct the mesh of sound radiating elements. The sound radiation into free-field is then estimated using the Rayleigh integral [13], approximated into a Riemann sum by considering the sound radiation of the mesh of rectangular elements centred at the 𝑁 ௫ ൈ𝑁 ௬ grid of points [30,31]. To properly assess the accuracy of the proposed method, the time-harmonic plate flexural vibration field and sound radiation field have been respectively measured and estimated for the cases where the plate is excited by a tonal force at the first two resonance frequencies of the plate flexural vibration, that is at 𝑓 ଵ ൌ32.6 Hz and 𝑓 ଶ ൌ 50.5 Hz . In this way, both the plate flexural vibration field and the plate sound radiation field

resemble the typical shapes for the first two flexural natural modes of the plate, that is the (1,1) and (1,2) modes, which are characterised by a single lobe and a two-lobes shapes. The following subsections briefly summarise the principal steps and features of the proposed measurement approach. In particular, Section 2.1 introduces the approaches used to detect the markers and to track the points in the sequence of images acquired by the cameras. Then, Section 2.2 describes the reconstruction of the marker coordinates 𝑋 ௝ ሺ𝑡 ௞ ሻ, 𝑌 ௝ ሺ𝑡 ௞ ሻ, 𝑍 ௝ ሺ𝑡 ௞ ሻ from triangulation of the image points. Finally, Sections 2.3 and 2.4 describe the reconstruction of the plate flexural vibration field, 𝑤൫𝑥 ௝ , 𝑦 ௝ , 𝑡 ௞ ൯ , at the marker reference positions ൫𝑥 ௝ , 𝑦 ௝ ൯ from the 3DPT and the reconstruction of the sound radiation field, 𝑝൫𝑥 ௙ , 𝑦 ௙ , 𝑧 ௙ , 𝑡 ௞ ൯ , at the field positions ൫𝑥 ௙ , 𝑦 ௙ , 𝑧 ௙ ൯ , from the plate transverse displacements at the marker positions using the Rayleigh integral. A detailed description of the 3DPT camera measurement and Rayleigh integral derivation can be found in Refs. [31,32].

3.1. Marker detection and image point tracking The first task of the measurement procedure concerned the detection and image point tracking (e.g. see Refs. [14,15]) of the 11 ൈ7 grid of dots bonded on the surface of plate (see Figure 1a). To this end, the following sequence of steps was implemented.  First, a 2D transformation 𝐻 ௜ , called homography , was applied to each image acquired by the cameras, which ensures the image of the plate frame has a rectangular shape. As a result of this transformation, the image of each marker was mapped from that of an ellipse to a circle.  Second, tentative dots were detected by template matching (a.k.a. normalized cross-correlation) and the coordinates of the centers were extracted for each frame [32].  Third, these tentative grid-points were then matched to the actual grid points at their expected positions by solving an assignment problem based on their reciprocal distance (in pixels) with the Hungarian method [33].  Fourth, a sub-pixel refinement of the center of the dots was carried out by fitting a parabola to the point of maximum correlation and to its two-neighbor points along the horizontal and vertical axes.  Fifth, since the coordinates of the detected points are referred to a transformed image space, the inverse of 𝐻 ௜ was then applied to bring them back to the original image space.  Sixth, assuming time-harmonic small vibrations, the coordinates of the points follow with good approximation a sinusoidal trajectory in time, therefore, their amplitude and phase were finally derived using the procedure described in Ref. [32] to reduce jitter and noise. The implementation of the marker detection and the subsequent image point tracking was implemented in MatLab using routines from Ref. [34].

3.2. Marker 3D position reconstruction The cameras are characterized by two sets of parameters, which are customarily called interior parameters and exterior parameters (see Ref. [24] for a general reference on cameras and 3D reconstruction). The former include the focal length and other parameters that do not change when the camera moves. The latter describes the position and angular attitude of the camera in object space, which are collectively referred to as the exterior orientation. In this study, the interior parameters, including radial distortion, of each camera were calibrated separately using the Sturm-Maybank- Zhang method [35,36]. The exterior orientation of the six cameras was recovered during the measurements, as part of an optimization process called Bundle Adjustment (BA), which optimizes a cost function with respect to the orientation of the cameras and the 3D coordinates 𝑋 ௝ ሺ𝑡 ௞ ሻ, 𝑌 ௝ ሺ𝑡 ௞ ሻ, 𝑍 ௝ ሺ𝑡 ௞ ሻ of the points. This is equivalent to a simultaneous points triangulation and camera parameters estimation [24]. An initial rest frame taken with no vibrations of the plate was used to derive the true static deflection of the plate, which was found not exactly planar due to the buckling in-plane stress generated by temperature variations in the laboratory after the plate was clamped on the rigid frame. In this respect, since the aim of the study is to measure and display the flexural vibrations of the plate only, the 𝑍 ௝ ሺ𝑡 ௞ ሻ

coordinates of the dots derived from the 3DPT were adjusted in such a way as to eliminate the fraction due to the static deflection identified from the rest frame. To initialize the exterior orientation of the cameras, six separate resections (with the Direct Linear Transform algorithm) [37] were carried out using the dots detected in the rest frame and the (approximate) coordinates of the grid points, followed by a BA, which optimizes only the exterior orientation and takes the approximate coordinates of the grid of points as given. In a subsequent BA comprising all the frames, the dots coordinates were considered among the unknowns, and their approximate positions were used only for initialization. The output of the BA are the exterior orientation of the cameras and the 𝑋 ௝ ሺ𝑡 ௞ ሻ, 𝑌 ௝ ሺ𝑡 ௞ ሻ, 𝑍 ௝ ሺ𝑡 ௞ ሻ coordinates of the dots derived from each frame of the cameras.

3.3. Plate transverse displacement reconstruction For the thin plate structure at hand and the small-amplitude flexural vibrations, the plate transverse displacement at the j-th grid point can be reconstructed from its 𝑍 ௝ ሺ𝑡 ௞ ሻ coordinate obtained from the BA described above. For instance, assuming time-harmonic vibrations with circular frequency 𝜔 , as shown in Ref. [29], the time-harmonic transverse displacement of the j -th marker point can be expressed as

𝑤൫𝑥 ௝ , 𝑦 ௝ , 𝑡 ௞ ൯≅Re ቄ𝑤 ଴ ሺ𝑥 ௝ , 𝑦 ௝ , 𝜔ሻ𝑒 ௝ቀఠ௧ ೖ ାఝ బ ൫௫ ೕ ,௬ ೕ ,ఠ൯ቁ ቅ , (1)

where 𝑤 ଴ ሺ𝑥 ௝ , 𝑦 ௝ , 𝜔ሻ and 𝜑 ଴ ൫𝑥 ௝ , 𝑦 ௝ , 𝜔൯ are the amplitude and phase of the displacement at each marker point and 𝑡 ௞ is the k-th time sample. As shown in Ref. [29], since the cameras are operated at a frame rate 1/𝑇 , the amplitude and phase terms, 𝑤 ଴ ሺ𝑥 ௝ , 𝑦 ௝ , 𝜔ሻ and 𝜑 ଴ ൫𝑥 ௝ , 𝑦 ௝ , 𝜔൯ , can be derived from the sequence 𝑍 ௝ ሺ𝑡 ௞ ሻ of the j -th marker 𝑧 -coordinate, assuming 𝑘ൌ0, … , 𝑁െ1 and 𝑁൒2 .

3.4. Plate sound radiation reconstruction with Rayleigh integral Once the transverse vibration field is derived from triangulation of the measurements taken with the six cameras, the sound pressure field generated by the plate flexural vibration can be derived straightforwardly from the Rayleigh integral [13]. Indeed, for time-harmonic vibrations, as shown in Ref. [29], the sound pressure at a given position ൫𝑥 ௙ , 𝑦 ௙ , 𝑧 ௙ ൯ can be approximated in terms of the following Riemann sum:

𝑝൫𝑥 ௙ , 𝑦 ௙ , 𝑧 ௙ , 𝑡 ௞ ൯≅Re ൜ ିఠ మ ఘ బ ଶగ ∑ ௪ బ ൫௫ ೕ ,௬ ೕ ,ఠ൯௘ షೕೖೃ೑ೕ

ோ ೑ೕ ே ௝ୀଵ Δ𝑆𝑒 ௝൫ఠ௧ାఝ బ ሺ௫ ೕ ,௬ ೕ ,ఠሻ൯ ൠ , (2)

where 𝑤 ଴ ൫𝑥 ௝ , 𝑦 ௝ , 𝜔൯ and 𝜑 ଴ ሺ𝑥 ௝ , 𝑦 ௝ , 𝜔ሻ are derived from the plate transverse displacement

reconstruction described above and 𝑅 ௙௝ ൌ ට ൫𝑥 ௙ െ𝑥 ௝ ൯ ଶ ൅൫𝑦 ௙ െ𝑦 ௝ ൯ ଶ ൅𝑧 ௙ ଶ is the distance between

the free-field point ൫𝑥 ௙ , 𝑦 ௙ , 𝑧 ௙ ൯ and the j -th marker point ൫𝑥 ௝ , 𝑦 ௝ ൯. 4. VIBRATION AND SOUND RADIATION MEASUREMENTS

This section shows the plate flexural vibration and sound radiation obtained from the measurements taken with the six cameras. As highlighted in the figures, the sound radiation is assessed on two cross planes defined by the axes xz and yz respectively. For instance, Figure 2 shows the flexural vibration and sound radiation fields when the plate is excited by a time-harmonic force at the first resonance frequency. The vibration graph shows the typical volumetric transverse vibration field, which is characterised by a sort of bell shape. Also, the sound radiation graphs show the typical near-field effect, which resembles the volumetric sound radiation by a monopole source. Alternatively, Figure 3 shows the flexural vibration and sound radiation fields when the plate is excited by a time-harmonic force at the second resonance frequency. In this case the vibration graph shows the typical two bell- shaped lobes vibration fields that oscillate in counter phase. The sound field in the xz plane is also characterised by a near field with two lobes that oscillates in counter phase. Instead the sound field in the yz plane shows a very low sound pressure since it defines a nodal plane for the sound radiation.

It is important to highlight here that the two vibration fields do not exactly match the first two flexural natural modes of the plate. Indeed, these measurements show deflections shapes at the first two resonance frequencies, which are therefore dominated by the shapes of the first and second mode respectively, but anyhow include the effects of neighbour modes too. In fact, since the plate is excited by a point force, the two deflection shapes are bound to be not exactly symmetric and also encompass the typical nearfield response in the vicinity of the excitation.

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Figure 2: Camera vibration and sound radiation measurements for harmonic excitation at the first natural frequency of the plate flexural vibration.

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Figure 3: Camera vibration and sound radiation measurements for harmonic excitation at the first natural frequency of the plate flexural vibration.

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Figure 4: Laser vibrometer vibration and sound radiation measurements for harmonic excitation at the first natural frequency of the plate flexural vibration.

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Figure 5: Laser vibrometer vibration and sound radiation measurements for harmonic excitation at the first natural frequency of the plate flexural vibration. The experimental results obtained from the camera measurements were validated from measurements taken with a scanner laser vibrometer, which are shown in Figures 4 and 5. Contrasting Figures 2 and 4 and Figures 3 and 5, it is clear that the shapes and amplitudes of the flexural vibration fields and the sound radiation fields obtained with the camera measurements closely resembles those obtained from the measurements with the laser vibrometer.

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5. CONCLUSIONS

This paper has presented a new approach for the estimation of the flexural vibration field and the sound radiation field of a thin plate structure using photogrammetry. The paper has described in details the camera setup, the measurement approach and the reconstruction methods employed to obtain the flexural vibration field, using bundle adjustment and 3D point tracking, and to generate the sound radiation field, from a boundary integral formulation, which for the baffled plate at hand reduces to the so called Rayleigh integral. The experimental work has shown that the initial tests carried out with six cameras on a thin plate structure, have provided with good accuracy both the vibration field and sound radiation field when the plate is excited harmonically at the first two resonance frequencies. Moreover, the results obtained from the camera measurements have been validated from measurements taken with a laser vibrometer. This paper presents the first experimental results of a new approach for the measurement of vibro-acoustic response of thin structures, which should be refined to extend the frequency range of the measurement to the whole audio range. This aspect is however strongly linked to the spatial and time resolution of the cameras. Therefore it is expected that this feature can be met in the near future, particularly with the help of enhanced processing and acquisition technologies. 6. REFERENCES

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