Viscoelastic Structural Damping Enables Broadband Low- Frequency Sound Absorption

1 Viscoelastic Structural Damping Enables Broadband LowFrequency Sound Absorption Yanlin Zhanga,b, Junyin Lib,c, Qiongying Wud, Marco Amabilib, Diego Misseronie, and Hanqing Jiangb,c,f* aSchool of Materials Science and Engineering, Zhejiang University, Hangzhou 310027, China bSchool of Engineering, Westlake University, Hangzhou, Zhejiang 310030, China cWestlake Institute for Advanced Study, Hangzhou, Zhejiang 310024, China dZhejiang Sanlux Rubber Co. Ltd., Shaoxing, Zhejiang 312031, China eUniversity of Trento, 38123 Trento, Italy fResearch Center for Industries of the Future, Westlake University, Hangzhou, Zhejiang 310030, China *Email: [email protected] 2 Abstract Low-frequency sound absorption has traditionally relied on air-resonant structures, such as Helmholtz resonators, which are made of stiff materials that undergo negligible deformation. In these systems, energy dissipation arises primarily from air motion and thermal-viscous effects, resulting in inherently narrowband performance and bulky, complex designs for broadband absorption. Here, we presented a composite acoustic metamaterial that replaces the high-stiffness neck of a Helmholtz resonator with a soft, viscoelastic cylindrical shell. This structural modification enables material deformation and shifts the dominant energy dissipation mechanism from air resonance to intrinsic viscoelastic damping. A single unit achieves over 97% absorption across a broad low-frequency range (227–329 Hz) with deep-subwavelength thickness (λ/15 at 227 Hz). We developed a discretized impedance model that quantitatively links material properties and geometry to absorption behavior. Our results established a materials-centered design paradigm in which both material selection and geometry serve as coequal, tunable parameters for compact, broadband low-frequency sound control. Keywords: Acoustic metamaterials; Low frequency; Broadband absorption; Viscoelastic damping; Structural vibrations 3 Significance Conventional acoustic metamaterials rely on air resonance and geometrically complex, highstiffness structures to achieve low-frequency absorption, typically leading to negligible structural deformation. These approaches are inherently narrowband. We present a composite acoustic metamaterial that replaces the high-stiffness neck of a Helmholtz resonator with a soft, viscoelastic shell, shifting the dominant energy dissipation mechanism from air-based resonance to structural damping governed by material viscoelasticity. This transition enables over 97% absorption across a broad low-frequency band. A discretized impedance model links absorption performance to material and geometric parameters, establishing a predictive framework. This materials-centered approach reframes low-frequency acoustics by embedding damping into the material itself, demonstrating that both material selection and geometry serve as coequal ingredients for acoustic performance. 4 Introduction Absorbing airborne sounds, particularly at low frequencies, is essential for controlling noise pollution and improving acoustic environments (1-3). Conventional porous materials achieve efficient absorption only when their thickness is comparable to a quarter of the acoustic wavelength (4, 5), which becomes impractically large at low frequencies. In response, acoustic metamaterials have attracted considerable attention over the past decade for enabling subwavelength absorption via engineered resonant structures (6-21). Example designs of these metamaterials include Helmholtz resonators (Fig. 1A) (13, 18, 22) that rely on air-column resonance within a highstiffness neck whose deformation is negligible, Fabry-Pérot channels (15, 19, 21), and microperforated panels (20, 23, 24), which dissipate sound energy through air motion and associated thermal-viscous effects. However, these air-resonant mechanisms remain inherently effective within a narrow frequency band. To broaden the absorption bandwidth, many recent efforts have combined heterogeneous resonators of varying scales, often at the cost of added complexity and volume (15-21). A smaller number of studies have looked beyond air resonance and explored the role of flexible structures. Membrane absorbers, for example, can enhance localized energy density and improve dissipation (25, 26), but they require precise control over membrane tension, making them environmentally sensitive, prone to instability, and difficult to tune (27). Other approaches include coupling flexible materials with open-air environments (28-30) or replacing the inner walls with soft materials (3133), taking advantage of combined structural-acoustic modes to increase damping. However, these efforts often rely on parametric studies via simulations or experiments, with limited systematic theoretical frameworks. In most cases, absorption is still primarily driven by air movement rather than the material itself, necessitating complex multi-resonant assemblies for broadband performance. This research gap leads to an opportunity. Soft materials, such as elastomers and gels, offer 5 distinctive advantages for acoustic applications, including low elastic modulus (ranging from a few kPa to some MPa), high intrinsic damping (loss tangent tan δ ≈ 0.1–1.0 or higher), and large deformability (34, 35). These properties are rarely used in acoustic systems. Still, they offer a fundamentally different route to low-frequency sound absorption: dissipating energy through material deformation, rather than mainly relying on air flow. Achieving this, however, requires more than a materials swap. It calls for an advanced theoretical approach. This theory must treat material dissipation, structural dynamics, and acoustic coupling as coequal components in the absorber’s performance. In this study, we introduced a composite acoustic metamaterial that replaces the high-stiffness neck of a traditional Helmholtz resonator (Fig. 1A) with a soft, viscoelastic cylindrical shell embedded within a sealed cavity (Fig. 1F). This simple structural change fundamentally alters the energy dissipation mechanism, from air-based resonance to structural damping governed by the material’s intrinsic viscoelasticity. As a result, a single unit achieves over 97% absorption across a broad lowfrequency range (227–329 Hz), with a deep-subwavelength thickness (λ/15 at 227 Hz). Furthermore, to understand and optimize this behavior, we developed a discretized impedance model that links the material properties (i.e., elastic modulus, loss factor, density) and geometry to absorption performance. We demonstrated that this model not only captures the observed behavior but also serves as a predictive tool for tuning absorption bands through material and structural design. More broadly, our approach reframes low-frequency acoustic design, namely, rather than relying on resonance stacking or complex architectures, we turn to the material itself as the dissipative medium. By integrating viscoelastic damping into the core architecture, we unlock a new class of compact, tunable, and broadband low-frequency sound absorbers, in which material choice, not just geometry, defines performance. 6 Results and Discussion Rigid vs. Flexible-Neck Helmholtz Resonator Traditional acoustic metamaterial absorbers often assume that the structural components, particularly the resonator neck, are made of materials with sufficiently high elastic moduli, so that their deformation under acoustic excitation is negligible. We refer to this class of systems as RigidNeck Helmholtz resonators (RN-Helmholtz resonators) throughout this work. As illustrated in Fig. 1A, an RN-Helmholtz resonator features a high-stiffness neck (Young’s modulus E = 2,650 MPa, density r s = 1,150 kg/m³, loss factor h = 0.01) embedded in a sealed cavity. Incident sound waves propagate through the neck (indicated by white arrows), exciting resonance in the enclosed air column. Finite element simulations via COMSOL Multiphysics (see SI Appendix, Fig. S16 for details) were conducted to study the coupled acoustic-structural response of this system. As shown in Fig. 1B, the resulting air particle velocity distribution is concentrated within the neck region. Thus, the energy is primarily dissipated through thermal-viscous friction at the air-rigid neck boundaries, while the displacement of the neck is effectively vanishing (Fig. 1C). This mechanism confines significant absorption (defined by absorption coefficient a > 0.8) to a narrow 18 Hz bandwidth (141-159 Hz) centered at the air column’s resonance frequency (150 Hz, a = 0.999) (Fig. 1D). To broaden the absorption bandwidth, multiple RN-Helmholtz resonators of varying dimensions (e.g., different cavity volumes and neck lengths) are inevitably combined (see SI Appendix, Fig. S17 for detailed geometries). The resulting multi-resonator structure (Fig. 1E) achieves significant absorption across a 210 Hz bandwidth (300–510 Hz), featuring five distinct absorption peaks (quantitative data in SI Appendix, Fig. S17). Consequently, this multi-resonator strategy increases structural complexity and poses challenges for fabrication, tuning, and integration. In our work, we address these limitations by replacing the high-stiffness neck with a deformable, soft cylindrical shell (Young’s modulus E = 120 kPa, density r s = 1,120 kg/m3, loss factor h = 0.4) 7 (Fig. 1F). We refer to this configuration as the Flexible-Neck Helmholtz resonator (FN-Helmholtz resonator). Finite element simulations via COMSOL Multiphysics (see SI Appendix, Fig. S18 for details) were conducted to study the coupled acoustic-structural response of this system. In this configuration, incident sound waves propagate through the flexible neck region and also lead the soft shell to vibrate (indicated by blue arrows). On one hand, air-column resonance within the neck continues to contribute to energy dissipation (Fig. 1G). On the other hand, the soft shell undergoes substantial deformation (Fig. 1H), with displacement magnitudes three orders of magnitude higher than those of its high-stiffness counterpart (Fig. 1C), enabling efficient energy conversion through intrinsic viscoelastic damping. This shift in energy dissipation, from predominantly air-mediated losses to viscoelastic structural vibration, marks a fundamental departure from conventional Helmholtz resonator behavior. As a result, the FN-Helmholtz resonator achieves both broadband and high-efficiency sound absorption using a single unit, characterized by dual absorption peaks at 320 Hz ( a = 0.974) and 460 Hz ( a = 0.967) and a significant absorption with an adequate bandwidth spanning 215 Hz (294-509 Hz) (Fig. 1I). Compared to broadband absorbers constructed from arrays of RN-Helmholtz resonators (yellow line in Fig. 1J, same as in Fig. 1E), this singleunit performance (blue line in Fig. 1J, same as in Fig. 1I) achieves a slightly wider effective bandwidth (215 Hz vs. 210 Hz), while eliminating the need for complex arrays of multi-resonator systems. The reduction in structural complexity, enabled by this materials-centered design, enhances manufacturability and establishes a compact, high-performance framework for lowfrequency acoustic absorption. Experimental and Numerical Validation of Vibration-Induced Sound Absorption To experimentally verify the simulation-based findings presented in the previous section, we fabricated an FN-Helmholtz resonator, consisting of a soft cylindrical shell and a rigid outer support structure (Fig. 2A). The soft shell was made from Ecoflex-30 silicone rubber (Smooth-On Inc., Young’s modulus = 90 kPa), and the outer shell was 3D-printed using a photosensitive resin with a Young’s modulus of 2,650 MPa. The detailed geometry is provided in the caption of Fig. 8 2A, and its fabrication process is outlined in SI Appendix, Fig. S13. Dynamic mechanical analysis (DMA 850, TA Instruments) was performed in compression mode to characterize the viscoelastic behavior of Ecoflex-30. Over the testing range, its storage modulus E and loss factor h approach approximately 148.5 kPa and 0.2, respectively, at high frequency (> 181 Hz) (Fig. 2B), as supported by time-temperature superposition (TTS) experiments (36), which extrapolated the data up to 596 Hz (see SI Appendix, Fig. S7 for details). Normal-incidence sound absorption coefficients were measured using an acoustic impedance tube with an inner diameter of 29 mm (setup shown in SI Appendix, Fig. S14). The fabricated FNHelmholtz resonator exhibited broadband absorption performance, featuring two prominent peaks at 330 Hz ( a = 0.978) and 500 Hz ( a = 0.996). Except for a dip in the mid-frequency range around 400 Hz ( a = 0.677), the absorption coefficient remained above 0.8 over much of the range from 307 Hz to 535 Hz (Fig. 2C, blue). We also performed coupled acoustic–structural simulations using COMSOL Multiphysics (see SI Appendix, Fig. S19 for details) to compare with the experimental results, using the identical geometry and material properties as those of the experiment. The simulated absorption spectrum (Fig. 2C, orange) closely matched the experimental data, confirming the accuracy of the model and the broadband performance of the FN-Helmholtz resonator. To elucidate the underlying sound absorption mechanism, we measured the surface vibration displacement of the cylindrical shell using a laser Doppler vibrometer (Polytec PSV-500). Scans of displacement amplitude and phase were conducted along three axial lines located at circumferential angles of 0°, 120°, and 240° on the shell surface (see SI Appendix, Fig. S15 for details). The results revealed nearly identical displacement profiles across the three directions (SI Appendix, Figs. S15D-G), indicating that the shell vibrates in an axisymmetric mode under acoustic excitation. Averaged experimental displacement data were compared with simulation results, showing good agreement in both amplitude and phase (Figs. 2D–G). In addition, the simulations directly visualized the axisymmetric vibration modes at two representative absorption 9 peaks (330 Hz and 500 Hz) (Figs. 2H and 2I), with the displacement profiles matching well with the magnitude curves in Figs. 2D and 2E, further confirming the mechanism. Calculated distributions of dissipated power density (W/m³) at 330 Hz and 500 Hz (Figs. 2J and 2K) show that most of the energy dissipation occurs in the silicone rubber shell. We then integrated the power density over the rubber and air regions separately to obtain the total dissipated power (W) in each domain. The frequency-dependent results for both mechanisms are compared in Fig. 2L. Notably, the two prominent peaks in viscoelastic damping match well with the sound absorption peaks. This correlation confirms that the dominant energy loss in the composite absorber comes from the vibration-induced damping of the soft rubber, with a secondary contribution from air-related losses. Equivalent Circuit Analysis of Acoustic–Structural Coupling To quantitatively analyze the sound-structure interaction mechanism, we developed a discretized equivalent impedance model. As shown in Fig. 3A, this model discretizes the cylindrical shell and the enclosed air column into n equal-length segments (length = l/n) along the axial direction. Compared to the configuration in Fig. 2A, we omitted the outer shell thickness and focused only on the cavity boundary dimensions (see SI Appendix, Fig. S1 for detailed geometry). For each segment, the acoustic impedance of the air column, 𝑍!(𝑥"), and the equivalent acoustic impedance of the flexible cylindrical shell, 𝑍#(𝑥"), are connected in parallel, where 𝑥"=" $𝑙 represents the position of the ith (𝑖=1,2,,…,𝑛) segment along the shell axis. This setup captures the two primary paths of energy dissipation: sound traveling through the air column and structural vibrations within the shell. In addition, the enclosing cavity introduces a shared acoustic reactance 𝑍%, which is connected in series with the coupled impedance of the fluid-structure system. Figure 3B illustrates the complete acoustic impedance circuit diagram. The total acoustic response of the system is therefore governed by the combined effects of 𝑍!(𝑥"), 𝑍#(𝑥"), and 𝑍%. Detailed derivations of 16 peak largely unchanged (Fig. 4J). This decoupled influence allows for independent fine-tuning of the two absorption peaks. By jointly tuning material properties (E, 𝜂, 𝜌2) and geometric parameters (a, l, t), along with dynamic phase compensation through the cavity volume V, the system’s acoustic impedance can be synergistically optimized to achieve enhanced and highly customizable performance across target frequency ranges. In contrast, increasing the cavity volume V alone, without coordinated adjustments to other parameters, can shift the absorption spectrum toward lower frequencies but may degrade performance due to impedance mismatch (Fig. 4K). Likewise, simply enlarging the cross-sectional area S of the impedance tube (modulated by tube diameter d in Fig. 4L) can worsen impedance mismatch and reduce absorption efficiency, as the absorber must dissipate energy over a larger area. These results underscore the importance of system-level optimization: isolated parameter changes are insufficient for maintaining impedance balance and achieving strong, broadband absorption. Experimental Implementation: Material Optimization and Customized Absorber Designs To validate the predictive capability of the theoretical model, we experimentally optimized the viscoelastic properties of silicone rubber and adjusted geometric parameters for customized lowfrequency absorption. Specifically, we selected a softer silicone rubber (Ecoflex-20, Smooth-On Inc.) as the base material. By altering the prepolymer mixing ratio (A:B = 1:2 by mass) and introducing tungsten (W) powder (particle size <1 µm) as a density-enhancing filler, we prepared two material variants: (1) a low-modulus, high-damping rubber (Ecoflex-20, A:B = 1:2) and (2) a density-enhanced composite (Ecoflex-20 + W, A:B:W = 1:2:1.2). 17 Dynamic mechanical analysis (DMA) in compression mode showed that both materials exhibited frequency-dependent viscoelastic behavior (Figs. 5A and 5B). The storage modulus increased with frequency and stabilized at approximately 111 kPa, representing a 25.3% reduction compared to Ecoflex-30’s 148.5 kPa. Meanwhile, the loss factor rose significantly to 0.385, marking a 92.5% increase over Ecoflex-30’s value of 0.2. Importantly, the addition of W powder had minimal effect on the storage modulus and loss factor but markedly increased the material density from 1,070 kg/m3 to 1,420 kg/m3 (Fig. 5C). This ability to independently tune viscoelasticity and density underscores the flexibility of material design in achieving targeted acoustic performance. Three absorber sets were developed to exploit the improved material properties: Set 1 used Ecoflex-20 and coordinated shell dimensions (inner radius a = 4 mm, length l = 41.5 mm, thickness t = 1 mm), paired with a compensatory cavity height (h = 63 mm) to accommodate the material’s lower stiffness. Experimental measurements (Fig. 5D, blue) demonstrated broadband absorption from 260–470 Hz (defined by a > 0.83), with an average absorption coefficient of 0.912 (computed as the arithmetic mean over this range). This downward shift from the original design (307–535 Hz, Fig. 2C) validates the model’s sensitivity to modulus reduction (Fig. 4E). Set 2 adopted the density-enhanced Ecoflex-20 + W composite, while keeping the exact shell dimensions as Set 1. To compensate for the higher density, the cavity height was increased to h = 72 mm. As a result, the absorption band (defined by a > 0.9) shifted further to 248–405 Hz (Fig. 5E, blue), with an average coefficient of 0.949, aligning with the predicted density-dependent shift (Fig. 4F). Set 3 also employed the Ecoflex-20 + W composite but enlarged the shell dimensions (a = 4.5 mm, l = 50 mm, t = 1.5 mm) and expanded the cavity height to h = 100 mm. This setup achieved nearperfect absorption from 227–329 Hz (defined by a > 0.97) (Fig. 5F, blue), with an average coefficient of 0.986, while maintaining an overall thickness of just 1/15 of the wavelength at 18 227 Hz. Experimental results from all three sets showed excellent agreement with numerical simulations (Figs. 5D–F, orange curves). This consistency was further supported by the validation of normalized acoustic resistance and reactance, presented in SI Appendix, Fig. S21, reinforcing the robustness of the theoretical model in capturing the relationship between material properties, structural parameters, and acoustic performance. Together, these findings demonstrate a closedloop workflow that seamlessly integrates theoretical modeling, material formulation, structural design, and experimental validation. Discussion This work introduces a new strategy for achieving efficient low-frequency and broadband sound absorption by rethinking the role of structural materials in acoustic metamaterials. Traditional Helmholtz resonators rely on rigid components, with energy loss coming mainly from air motion and thermo-viscous effects, which limits them to narrow frequency bands. In contrast, our design replaces the rigid neck with a soft, viscoelastic cylindrical shell, allowing the structure itself to participate in the energy dissipation process through material damping and structural vibration. This shift enables a single resonator unit to achieve over 97% absorption across a tunable lowfrequency range (e.g., 227–329 Hz), with a thickness as small as 1/15 of the wavelength at the lowest frequency. The broadband performance arises from the combined effects of the shell’s viscoelastic behavior and its dynamic interaction with the surrounding air. We developed a discretized impedance model to describe this coupled system, which enables us to directly link material properties (such as modulus, damping, and density) and geometric parameters (including shell radius, length, and thickness) to acoustic performance. Notably, shell length and wall thickness offer decoupled control over the position of absorption peaks, while material properties can be independently tuned 19 using mixing ratios or fillers. This provides the system with a high degree of flexibility in meeting various design goals. The concept of embedding the damping mechanism into the structure itself, rather than relying solely on air resonance, opens the door to more compact, lightweight, efficient, and tunable absorbers. The approach also offers a practical path forward, utilizing common soft materials such as silicone rubber and straightforward fabrication methods. More broadly, it suggests that the conventional reliance on rigid structures and complex multi-resonator assemblies for broadband performance may not be necessary. Instead, a materials-centered framework, where viscoelastic properties are deliberately engineered and matched with structural geometry, can achieve highperformance absorption in a simpler, more compact, and lightweight form. Looking ahead, this design approach could be extended to systems with multiple degrees of freedom, layered or hierarchical structures, or active tunability through external stimuli like temperature or electric fields. The same principle of structure-enabled dissipation also holds promise in underwater acoustics. Furthermore, this concept also has potential applications beyond acoustic absorption, such as in vibration damping, wave filtering, or programmable mechanical metamaterials. By combining ideas from material science, mechanics, and wave physics, this work lays the groundwork for a new class of flexible, efficient sound-absorbing materials suited to the needs of dense urban areas, transportation systems, and industrial noise control. 20 Acknowledgement We thank the Research Center for Industries of the Future (RCIF) at Westlake University, Zhejiang Sanlux Rubber Co. ltd., and the Westlake Education Foundation for their support of this work. H.J. acknowledges support from the National Natural Science Foundations of China (Grants 12350003). D.M acknowledges financial support from the European Union, ERC grant HE GA 101086644 SFOAM. (Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them), and the Westlake Fellows program. We thank Zhen Yang from Instrumentation and Service Center for Physical Sciences at Westlake University for her assistance with the DMA test and data interpretation. Author contributions Y.Z., Q.W. and H.J. designed research; Y.Z. and J.L. performed research; Y.Z., J.L., D.M., and H.J. analyzed data; and Y.Z., D.M., M.A., and H.J. wrote the paper. Data availability Source data for all main and supplementary figures are provided in the accompanying Excel file. MATLAB code for calculating the sound absorption coefficient (including Full-recursive and Closed-form analytical models) is available in the GitHub repository (https://github.com/yanlinZhang865/Data-availability). 21 Materials and Methods Material Preparation and Fabrication Flexible cylindrical shells were fabricated using silicone rubbers (Ecoflex-30 and Ecoflex-20, Smooth-On Inc.). Ecoflex-30 was prepared by mixing prepolymers (Part A:B) at a 1:1 mass ratio, while Ecoflex-20 was prepared at a 1:2 ratio (Part A:B) to reduce stiffness and enhanced damping. For density modification, tungsten (W) powder (particle size <1 µm) was added to Ecoflex-20 prepolymers at a mass ratio of A:B:W = 1:2:1.2. The mixed prepolymers were then cast into highprecision CNC-machined aluminum molds (see SI Appendix, Figs. S13A-D for details) and cured at 25 ℃ for 4 hours. After curing, the silicone cylindrical shell was demolded and trimmed to its final dimensions (see SI Appendix, Fig. S13E). The fabricated shells exhibited excellent dimensional fidelity, with deviations of less than 0.05 mm from the design specifications. The high-stiffness outer shell was fabricated using 3D printing (photosensitive resin, C-UV9400A, Young’s modulus: 2,650 MPa). The soft silicone rubber shell was then embedded into the resin shell and bonded using silicone adhesive (Sil-Poxy, Smooth-On Inc.), forming a complete composite absorber (see SI Appendix, Fig. S13F). For visual clarity, in Fig. 2A, Figs. 5D-F and SI Appendix, Fig. S13F, we depict the outer shells as transparent acrylic (PMMA) to reveal the internal structure, although the actual fabricated shells were opaque. Experimental Characterization DMA. Viscoelastic properties (storage modulus and loss factor) were measured using a TA Instruments DMA 850 analyzer in compression mode (experimental setup and sample geometry shown in SI Appendix, Fig. S7). Square samples (18 × 18 × 6 mm³) were tested under frequency sweeps (1– 181 Hz, 10 Hz intervals) at 0.5% strain and 25°C. The dimensions of the square samples were chosen to minimize data fluctuations at higher frequencies ( ＞100 Hz). Comparisons with alternative testing methods (time-temperature superposition and cantilever beam resonance 22 method) are discussed in detail in SI Appendix, Section 4. Sound Absorption Measurement. Normal-incidence sound absorption coefficients were measured using an impedance tube (AWA6290T Hangzhou Aihua Instruments Co., inner diameter: d = 29 mm) following ISO 105342 standards. The experimental setup is shown in SI Appendix, Fig. S14. A 110 dB white noise signal (50-6300 Hz) served as the excitation sound source. Acoustic responses were recorded by two microphones, and absorption coefficients were calculated using the transfer function method (40). Due to the low stiffness of the flexible cylindrical shell, horizontal sample orientation induced slight gravitational sagging (see SI Appendix, Fig. S14B). To eliminate this effect, the impedance tube was reoriented vertically (SI Appendix, Fig. S14C), minimizing structural deflection during testing. Comparative analysis of absorption spectra from both horizontal and vertical configurations (SI Appendix, Fig. S14D) showed nearly identical results with overlapping curves, confirming that gravitational sagging negligibly impacted acoustic performance under the tested sample conditions. Vibration Displacement Measurement. Surface vibration displacement of the cylindrical shell under acoustic excitation at 330 Hz and 500 Hz was measured using a Polytec-PSV-500 laser Doppler vibrometer (experimental setup shown in SI Appendix, Fig. S15A). To ensure unobstructed laser access to the shell surface, the sample was enclosed in a transparent acrylic enclosure with a rectangular cross-section (SI Appendix, Fig. S15B). This design minimized spurious laser reflections from the curved cylindrical surface. The rectangular enclosed cavity volume was designed to match the cylindrical cavity volume designed in Fig. 2A. Axial vibration profiles were scanned along three axial lines at angular positions of 0°, 120°, and 240° (SI Appendix, Fig. S15C) with a spatial resolution of 0.2 mm (along x axis) to evaluate vibration symmetry. The measured displacement amplitudes and phase angles at both 23 excitation frequencies (330 Hz and 500 Hz) exhibited consistent spatial distributions across all angular orientations (SI Appendix, Figs. S15D-G), confirming the axisymmetric vibration behavior of the cylindrical shell. Numerical Simulations Coupled acoustic-structural simulations were performed in COMSOL Multiphysics (version 6.1) using the Pressure Acoustic, Thermo-viscous Acoustics and Solid Mechanics modules (SI Appendix, Fig. S19B). The resin shell was modeled as rigid, while the air domain was treated as a thermos-viscous fluid. All silicone rubbers were assumed to be nearly incompressible (Poisson’s ratio: 0.49). A perfectly matched layer (PML) boundary condition was applied at the tube outlet to eliminate wave reflections. The computational domain was discretized using hexahedral meshes, with boundary layer meshes further refined in thermo-viscous region. Boundary conditions included fixed constraints at the silicone-resin interface and acoustic-structure coupling at the solid-air interface. Viscoelastic damping in silicone rubber was quantified by integrating the dissipated power density (solid.Qh, W/m³) over the rubber domain. Similarly, thermo-viscous losses in air domains (resonator cavity and adjacent tube section) were evaluated by integrating the corresponding dissipation density (ta.diss_tot, W/m³) over fluid volumes. Stability of Sound Absorption Performance Long-term Stability. The FN-Helmholtz resonator sample (Set 3, Fig. 5F) was stored at 25°C ambient conditions and re-tested after 60 days and 80 days. The absorption coefficient, normalized acoustic resistance, and normalized acoustic reactance were measured using the same impedance tube setup. The comparative results are presented in SI Appendix, Fig. S9. Temperature Effects. 24 To evaluate temperature influence on Ecoflex-30 viscoelastic properties and acoustic performance of the FN-Helmholtz resonator, temperature-dependent data from time-temperature superposition experiments were analyzed. Coupled acoustic-structural simulations compared absorption spectra at 25°C (E = 152 kPa, 𝜂 = 0.19) and -10°C (E = 169 kPa, 𝜂 = 0.22). Results (SI Appendix, Fig. S10) show relatively stable broadband absorption across a wide temperature range. Absorption Under Oblique Incidence. To assess performance under non-normal incidence, absorption coefficients were calculated for angles of 0°, 30°, and 60° using the full recursive model with clamped-free boundaries. For a locally reacting surface, the absorption coefficient 𝛼(𝜃) is calculated by modifying the formula (Eq. 9) to account for the effective impedance projection along the direction of propagation (5): . [12] where 𝜃 is the incidence angle measured from the normal to the surface. The resulting spectra (SI Appendix, Fig. S11) show angle-dependent variations. Absorption Under High Sound Pressure. To evaluate potential influences of geometric or material nonlinearity at elevated incident sound levels, the normal-incidence sound absorption coefficient of the fabricated FN-Helmholtz resonator was measured under white-noise excitation at 90-129 dB (0.632-56.2 Pa). All curves are nearly identical across this range of incident sound pressures (SI Appendix, Fig. S12), this indicates that the absorber operates linearly even at high excitation levels. Moreover, simulations show maximal shell strain ≤1.44% at 129 dB (SI Appendix, Table S2), within Ecoflex-30's linear regime (<10% strain) (41). 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(E-J) Parametric effects on 𝛼 (other parameters fixed; 𝑉 co-adjusted per Eq.11 for optimal impedance matching): (E) Young’s modulus E; (F) Material density 𝜌&; (G) Shell radius a; (H) Loss factor 𝜂; (I) Shell length l; (J) Shell thickness t; (K-L) Parametric effects on 𝛼 with isolated tuning: (K) Cavity volume 𝑉; (L) Impedance tube diameter 𝑑. Fig. 5. Material optimization and customized absorber designs. (A) Frequency-dependent storage modulus E, (B) Frequency-dependent loss factor 𝜂 , and (C) Material density 𝜌& , for Ecoflex-30 (bronze), Ecoflex-20 (blue) and Ecoflex-20 + W (black). (D-F) Experimentally measured and simulated acoustic absorption coefficients for three resonator configurations: (D) Set 1 (Ecoflex-20; a = 4 mm, l = 41.5 mm, t = 1 mm, h = 63 mm). (E) Set 2 (Ecoflex-20 + W; a = 4 mm, l = 41.5 mm, t = 1 mm, h = 72 mm). (F) Set 3 (Ecoflex-20 + W; a = 4.5 mm, l = 50 mm, t = 1.5 mm, h = 100 mm). at V simulation experiment simulation experiment simulation experiment lh Ecoflex 30 (original) Ecoflex 20 (A:B=1:2) Ecoflex 20 (A:B=1:2 + W) 350 350 350450 450 450550 550 550250 250 250 150 150 150 0 040 4080 80120 120160 160 frequency (Hz) frequency (Hz) frequency (Hz) frequency (Hz) frequency (Hz) set #1 Ecoflex 20 (A:B=1:2), a=4 mm, l=41.5 mm, t=1 mm, h=63 mm Ecoflex 20 (A:B=1:2) + W, a=4 mm, l=41.5 mm, t=1 mm, h=72 mm Ecoflex 20 (A:B=1:2) + W, a=4.5 mm, l=50 mm, t=1.5 mm, h=100 mm set #2 set #3 0.2 0.05 0.2 600 0.1 300 0.10 0.3 900 0.4 1200 1070 1070 1420 0.15 0.5 1500 0.2 0.2 0.6 0.6 0.6 0.8 0.8 0.8 1.0 0.20 1.0 1.0 0 0 0 0 0 0 0.4 0.4 0.4 absorption, αstorage modulus, E loss factor, η density, ρ (kg/m3) absorption, α absorption, α D A B C E F Ecoflex 30 (original) Ecoflex 20 (A:B=1:2) Ecoflex 20 (A:B=1:2) + W 260 Hz 248 Hz 227 Hz 470 Hz 405 Hz 329 Hz