Improving sound transmission loss at ring frequency of a curved panel using tunable 3D-printed small-scale resonators

The design of lightweight and efficient soundproofing packages is one of the top priorities in the general transportation industry for noise, vibration, and harshness objectives. A very common solution to this weight-efficiency compromise is the use of light porous material attachments on structures usually simplified or modelled as flat whereas they can have a variable curvature radius in reality (a ship hull, an aircraft cabin). In usual sound transmission loss (STL) results for cylindrical shells under an oblique incident plane wave¹ or under a diffuse acoustic field (DAF) excitation,² a first noticeable STL reduction occurs at the cylinder ring frequency which is a function of the shell radius and longitudinal wave velocity. A second dip in STL results happens at the coincidence frequency where structural and acoustic wavenumbers coincide (common to both curved and plane structures). Classical porous treatments that mainly provide broadband sound attenuation are not optimal for solving these reductions of sound insulation performance in narrow frequency bandwidths. Moreover, for typical aircraft sidewalls, the ring frequency-related dip occurs at low frequency (typically 400 to 800 Hz).

The design of ribbed or stiffened structures can be optimized to modify the frequency bandwidth affected by the two previously cited specific frequencies.³ A large variety of passive or active solutions have also been proposed. A typical example of an active approach for controlling acoustic transmissibility of aircraft panels is decentralized active control with distributed units.⁴ Concerning passive solutions and apart from the classical solution of increasing the panel's structural damping, innovative approaches to tonal or narrow-band problems have been largely inspired by recent research on locally resonant metamaterials and often involve the use of periodic add-on units. The locally resonant bandgaps obtained using either Helmholtz resonators,^5,6 periodically distributed tuned mass dampers,⁷ or surface-mounted resonators^8,9 can be used to improve the STL. For composites plates in which the coincidence frequency zone is usually larger than for a single layer homogeneous plate, a possible solution is to use multi-modal resonators to broaden stop-bands.¹⁰ Aside from Ref. 5, the previously cited references^6–10 all deal with plane structures and hence, only coincidence frequency is considered. The use of distributed vibration absorbers (DVAs) with no specific periodicity was also studied on cylindrical structures. The reduction of sound transmission into a cylindrical shell excited by an incident acoustic plane wave using DVAs and Helmholtz resonators was numerically investigated in Ref. 11. A review concerning applications of DVAs made up of lumped mass inclusions in poroelastic media (a heterogeneous blanket) can be found in Ref. 12. The two previously cited works mostly targeted sound insulation applications at low frequency (0–200 Hz).

To the best of the authors' knowledge, the reduction of the negative impact of the ring mode on the STL of a curved panel by means of a passive, tunable, and multi-resonant approach has not yet been experimentally studied. In this work, a tunable resonator concept involving three-dimensional- (3D-) printed cantilever beams with interchangeable tip end magnets is studied on an aircraft sidewall panel including stringer and ring frame attachments under a DAF excitation. Several spatial distributions of the resonators including a multi-resonant configuration as well as possible addition of this narrow-band solution to a broadband sound absorbing treatment are investigated.

The proposed small-scale resonators are based on a 3D-printed core structure using polycarbonate polymer consisting of a beam of section 10 × 4 mm² and length 23 mm supported by a stiffener of section 4 × 3.5 mm² [see various illustrations in Figs. 1(a)–1(e)]. The base part is a rectangular prism of dimensions 10 × 10 × 8 mm³ on which the beam and the stiffener are both connected. The connection to the structure is made at the base using an adhesive [see Fig. 1(d)]. The overall unit volume occupied by a resonator without a tuning mass is finally 33 × 10 × 8 mm³ (see Fig. 1), with a unit weight of 2.72 g.

A 1 mm-thick base magnet is glued at the tip end and the resonators are then tuned using different neodymium magnets of known masses. The tuning masses and corresponding resonance frequencies for the resonators shown in Figs. 1(a)–1(c) are 2.24 g and 670 Hz (F1), 1.11 g and 820 Hz (F2), and 0.74 g and 980 Hz (F3), respectively. These average resonance frequencies were estimated from a few resonators once installed on the tested panel using a laser Doppler vibrometer and a shaker excitation.

According to standards,^13,14 STL is determined in coupled reverberant-anechoic rooms using measurements of the spatially averaged sound pressure level in the source room L_p and of the spatially averaged average sound intensity level L_i over a scanning surface S_m on the receiving side (both in dB), $STL=Lp−Li−6−10 log10(Sm/S)$ (with S the effective panel area, considered equal to the scanning area S_m so that the last term was neglected). All measurements were conducted in the coupled reverberant-anechoic rooms at groupe d'acoustique de l'Université de Sherbrooke. A DAF excitation was generated in the reverberant room (7.2 × 6.5 × 3 m³) using a loudspeaker with a white noise input in the 50–5000 Hz frequency range. The average sound pressure level in the reverberant room L_p was obtained by rotating a half-inch PCB microphone for more than a complete turn of its supporting arm during the signal acquisition time of 120 s for each test. The average radiated sound intensity level L_i was measured in the anechoic room (6.8 × 6.5 × 3 m³) using a Bruel & Kjaer sound intensity probe composed of two half-inch microphones and a 12 mm spacer. Manual scanning was performed at a distance of 10 cm from the panel surface following recommended scan patterns.^13,14 Standards^13,14 also recommend not exceeding a pressure-intensity indicator value of 10 dB for a sound-reflecting test specimen which was verified in each measurement. Finally, all the results presented hereinafter are provided in 1/12th octave bands between 150 and 5000 Hz.

The structure tested is a curved rectangular fuselage panel equipped with axial and circumferential stiffeners, all riveted to the panel skin [see Fig. 2(a)]. The panel is made of 1.27 mm-thick aerospace grade aluminum with a total weight of 21.4 kg. Its length is 1.7 m with outer and inner circumferences of 1.45 and 1.3 m, respectively. Its curvature radius is approximately 1.34 m. The equivalent mass per unit area of the fuselage equipped with stiffeners is approximately 8.68 kg/m² while the bare aluminum shell has a theoretical 3.43 kg/m² mass per unit area. Additional details and results can be found in Ref. 15, in which this panel was tested in coupled rooms and also numerically modeled. In the present work, the panel was mounted in the test window using a frame made of plywood with acoustic sealant made of neoprene adhesive and silicone (only the panel skin is actually clamped in the mounting frame, not the stiffeners). The frame and surrounding surfaces were covered with a flexible decoupled barrier material composed of an open-cell foam and a heavy PVC layer.

The tested configurations are visually displayed and summarized in Figs. 2(a)–2(f). Figure 2(a) corresponds to the reference configuration, i.e., the bare panel, with measurements conducted both at the beginning and end of the series of experiments in order to verify repeatability and data consistency. The maximum level difference between those two tests was below 0.5 dB for every 1/12th octave band. For all tests, the resonators were bonded on the panel skin with wax typically used to mount accelerometers. Figure 2(b) illustrates the D1 distribution of resonators. In this case, 246 resonators are placed on the central portion of the panel (21 or 18 resonators per bay on 12 bays). The D2 distribution of resonators is depicted in Fig. 2(c). The available 246 resonators are now distributed on the panel with the exception of the upper and lower parts of the panel, with 7 to 10 resonators per bay on 28 bays. Figure 2(d) illustrates the D3 distribution of resonators that now fully cover the panel area. The number of resonators per bay varies between 5 and 9 for a total number of 246 resonators on 32 bays (including one additional resonator on each of the eight uppermost and lowermost bays). The choice of these configurations is mainly empirical for this first proof-of-concept.

The positioning of a sound absorbing treatment over the panel area is finally shown in Fig. 2(e) (including possible combination with the D3 distribution of resonators). A melamine foam with a 50.8 mm thickness was chosen to study its effect on measured STL when used alone or combined with resonators. This material has a mass density of 6.1 kg/m³, a static air flow resistivity of 7920 Ns/m⁴, a tortuosity of 1, a porosity of 0.98, and viscous and thermal lengths of 132 and 149 μm, respectively. These properties were measured using dedicated methods in the acoustic materials characterization labs at Université de Sherbrooke. Four foam layers of dimensions 1.34 × 0.37 m are inserted between the circumferential stiffeners, supported by stringers, and kept in place with tape [see Fig. 2(e)]. Since the resonators protrude less than the stringers, the resonators are not in contact with the foam layer. The total added mass from the foam is 0.6 kg. Figure 2(f) presents a close-up view of the panel equipped with resonators prior to complete installation of the melamine foam.

All the tested configurations are summarized in Table 1 with the corresponding mass percentages and figure result numbers. Using only F1 resonators, the D1, D2, and D3 distributions are first tested [see Figs. 2(b), 2(c), and 2(d), respectively]. Based on the D2 distribution [Fig. 2(c)], combinations of F1, F2, and F3 resonators are then used to produce a multi-modal locally resonant add-on. In this configuration, the ring frequency is targeted as well as the anti-resonance produced by F1 resonators, a well-known phenomenon for tuned mass dampers.¹² Note that although the number of resonators remains unchanged for the three configurations, the added mass is slightly decreased in the multi-modal configuration, as F2 and F3 resonators are lighter than F1 resonators (see Table 1). The effect of the sound absorbing treatment is finally tested alone and in combination with the D3 distribution of resonators in single mode (F1).

Results obtained are reported in Figs. 3(a)–3(d). The reference STL curve for the bare panel is systematically included in Figs. 3(a)–3(c) in order to ease comparison between test cases described in Table 1. In Fig. 3(d), the effect of four key configurations is also plotted in terms of insertion loss so as to highlight STL improvement brought by each setup.

A theoretical ring frequency f_r of approximately 634 Hz is calculated using the simplified relation for a non-stiffened thin shell² $fr=(2πr)−1E/ρ(1−ν2)$⁠, with E the Young's modulus (=70 GPa), ρ the mass density (=2700 kg/m³), ν the Poisson's ratio (=0.3), and r the curvature radius. Since only the panel skin is clamped into the mounting frame, this commonly used approximation^1,2 seems reasonable. The estimated f_r is indicated by a vertical arrow in Fig. 3(a), and the largest decrease of measured STL is obtained at the 630 and 670 Hz 1/12 octave bands. Note that tests previously performed on the same panel identified the ring frequency in the 630 Hz third octave band,¹⁵ in agreement with current tests and with the calculated f_r value. Note that the F1 frequency of the resonators is tuned for enlarging the STL in the frequency band where it is found to be the lowest (the 670 Hz 12th octave band).

The STL of the fuselage is measured before and after placing the resonators tuned on a single mode (F1, 670 Hz) with distributions D1, D2, and D3 (see Table 1). Corresponding results are provided in Fig. 3(a). For the three distributions, the largest STL enhancements are not obtained in the 670 Hz 12th octave band but in the preceding one, centered on a frequency of 630 Hz. Compared with the bare panel, the obtained STL gains for D1, D2, and D3 distributions are 6.6, 10.3, and 11.8 dB, respectively. The more concentrated D1 distribution produces a lower and smoother STL peak than when the resonators are more uniformly spread (i.e., D2 and D3 distributions). This result might indicate that local sound radiation is predominant around the ring mode's frequency bandwidth and that a broader spatial distribution of the resonators improves the sound insulation properties in the tuned frequency's bandwidth. This could also be a consequence of the spatial averaging done when measuring radiated sound intensity, which includes untreated bays in the case of the D1 distribution. Sound intensity mapping using discrete measurement points instead of continuous scanning could be considered in order to clarify specific bay or part contributions. The STL reduction observed between 800 and 900 Hz in Fig. 3(a) is a known consequence of the tuned mass dampers' anti-resonance. The dip nevertheless appears at different center frequencies depending on the considered D1, D2, or D3 distribution, with reduction of 5.1, 2.9, and 2.1 dB at 1/12 octave band frequencies of 900, 850, and 800 Hz, respectively (with the bare panel result as a reference). Note that with the exception of this STL reduction, no other alteration of the sound insulation occurs in the 100–5000 Hz frequency range [see the result given in terms of insertion loss in Fig. 3(d)].

In summary, the results presented in Fig. 3(a) indicate that the most significant STL boost near the ring frequency (and the lowest alteration above) is achieved with the D3 distribution. Nevertheless, the D2 distribution provides a good compromise in terms of overall STL improvement in the 500–800 Hz frequency range and is therefore chosen for studying the effect of different resonator frequencies. Combinations of F1, F2, and F3 resonators are used with the D2 distribution and the results obtained are provided in Fig. 3(b). Results show that the 2-mode and 3-mode configurations help limit the negative effect of the anti-resonance and increase the STL in the 800–950 Hz bandwidth. Compared with a single frequency configuration, the three mode configuration leads to an average STL improvement of 3.3 dB per 1/12th octave with a maximum of 5.7 dB at 800 Hz. This multi-resonant solution even reduces the add-on weight from 5.7% to 4.9% (see Table 1). A multi-modal configuration of the resonators clearly increases the upper limit of the effective bandwidth from 750 to 900 Hz. Interestingly, changing the frequency distribution within a fixed number of resonators does not modify the STL results in the rest of the frequency spectrum [see the result given in terms of insertion loss in Fig. 3(d)].

In Fig. 3(c), the results for the D3-F1 configuration is recalled [see Fig. 3(a)], and compared to the results obtained with the melamine foam alone and combined with the D3-F1 configuration. The addition of the foam treatment leads to an increase in the measured sound insulation starting at 250 Hz but nevertheless fails to fully compensate the STL loss resulting from the panel's ring mode as a result of its broadband behaviour (limited to a masking effect). When combined with the D3-F1 resonator configuration, the broadband effect of the melamine is nearly always additive to the narrow-band effect of the resonators. Compared with the foam alone case, the obtained STL result is boosted at all frequencies (at the exception of two STL reductions that can be found at 400 and 800 Hz). It is noteworthy that the additive effect is only limited to the peak enhancement brought by the resonators [see the result given in terms of insertion loss in Fig. 3(d)]. The maximum STL increase is found at the 630 Hz frequency band, with values of 11.8 and 13.5 dB for the foam alone and combined with the resonators, respectively.

The performance of a locally resonant solution for improving sound transmission loss of curved panels at ring frequency is demonstrated on an aircraft sidewall panel under a diffuse acoustic field excitation. The STL reduction occurring at and near the ring mode frequency is entirely suppressed with up to 10 dB gain using resonators tuned at a single frequency. The use of a multi-resonant configuration proves to be a solution for both enlarging the efficiency bandwidth and limiting the anti-resonance negative effect. It is finally demonstrated that the proposed concept can be combined with a broadband sound insulation treatment, the resonators' narrow-band and foam's broadband effects being nearly always additive in terms of STL benefits. Based on this first proof-of-concept, the on-going steps of this work are (1) a numerical model guiding an optimization of both the frequency tuning and number of resonators so as to be able to draw a “cost function-like” map to propose reachable dB enhancements vs a target added mass, and (2) the design of a solution for plane panels so as to solve ring and coincidence frequency STL reductions. Finally, this compact and low-cost solution of 3D-printed small-scale cantilever beams could also be tested on arbitrarily shaped complex industrial-scale structures for solving other vibro-acoustic issues.

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 675441. The help of Patrick Levesque at Université de Sherbrooke for the transmission loss experimental setup was greatly appreciated. The authors would also like to acknowledge Benoit Minard and Navdeep Sharma at Ecole Centrale de Lyon for their support in designing and manufacturing the resonators, and Patrick O'Donoughue at Université de Shebrooke for kindly proofreading the final manuscript.