[Paper Review] Metadamping in inertially amplified metamaterials: Trade-off between spatial attenuation and temporal attenuation

Mahmoud I. Hussein, Ibrahim Patrick+2 more|arXiv (Cornell University)|Dec 28, 2021

Acoustic Wave Phenomena Research25 references40 citations

TL;DR

This paper proposes a passive inertially amplified (IA) metamaterial that achieves extreme metadamping—significantly enhanced or reduced dissipation—by leveraging a lever-arm mechanism to amplify the effective inertia of a resonating mass. By tuning the lever angle, a trade-off between temporal (dissipation) and spatial (band gap) attenuation is achieved, enabling tailored dynamic performance in phononic materials beyond conventional limits.

ABSTRACT

Metadamping is the phenomenon of either enhanced or diminished intrinsic dissipation in a material stemming from the material's internal structural dynamics. It has been previously shown that a locally resonant elastic metamaterial may be designed to exhibit higher or lower dissipation compared to a statically equivalent phononic crystal with the same amount of prescribed damping. Here we reveal that even further dissipation, or alternatively further reduction of loss, may be reached in an inertially amplified metamaterial that is also statically equivalent and has the same amount of prescribed damping. This is demonstrated by a passive configuration whereby an attenuation peak is generated by the motion of a mass supported by an inclined lever arm. We further show that by coupling this inertially amplified attenuation peak with that of a local resonance attenuation peak, a trade-off between the intensity of spatial attenuation versus temporal attenuation is realized for a range of the inclination angles. Design for performance along this trade-off is therefore possible by adjustment of the lever angle. These findings open the way for highly expanding the Ashby space for stiffness-damping capacity or stiffness-spatial attenuation capacity through design of the internal structure of materials.

Motivation & Objective

• To investigate metadamping in inertially amplified (IA) metamaterials, where mechanical lever mechanisms amplify effective mass inertia.
• To demonstrate that IA metamaterials can achieve higher or lower dissipation than statically equivalent locally resonant metamaterials or phononic crystals.
• To reveal a passive, angle-tunable trade-off between temporal (damping) and spatial (band gap) attenuation in IA-local resonance coupled systems.
• To expand the design space for stiffness-damping or stiffness-spatial attenuation capacity through internal structural engineering.
• To provide a framework for designing phononic materials with tailored space-time attenuation characteristics.

Proposed method

• A 1D chain model is developed with a baseline mass connected to an inertial amplifier via a lever mechanism, where the auxiliary mass's acceleration is amplified by a factor dependent on the lever angle α.
• The force transmission from the inertial amplifier to the baseline mass is derived using static force balance, yielding a scaled effective damping and stiffness contribution proportional to 1/(4 tan²α).
• The system is modeled as a two-degree-of-freedom system with a local resonator and an IA mass, leading to a coupled dispersion relation with two attenuation peaks.
• Bloch’s theorem is applied to derive the complex dispersion relation, enabling calculation of wave number-dependent damping ratios and damped frequencies.
• The total dissipation (cumulative damping) is computed by integrating the damping ratio over the first Brillouin zone, and the spatial attenuation is quantified via the band gap width and depth.
• A comparative analysis is performed between the IA metamaterial, a statically equivalent phononic crystal (PnC), and a locally resonant acoustic metamaterial (AM), all with identical prescribed damping and mass-stiffness parameters.

Experimental results

Research questions

• RQ1 Can inertial amplification in a metamaterial lead to enhanced or reduced dissipation beyond what is achievable in conventional locally resonant or phononic crystal systems?
• RQ2 How does the lever angle α in the inertial amplifier affect the balance between temporal (damping) and spatial (band gap) attenuation?
• RQ3 Is there a passive, tunable trade-off between the intensity of temporal and spatial attenuation in a hybrid IA-local resonance system?
• RQ4 To what extent can the design of internal structural dynamics expand the Ashby space for stiffness-damping or stiffness-spatial attenuation capacity?
• RQ5 What is the role of effective mass amplification in modifying the wave propagation and energy dissipation characteristics in elastic metamaterials?

Key findings

• The IA metamaterial achieves significantly higher or lower dissipation (positive or negative metadamping) than a statically equivalent locally resonant metamaterial or phononic crystal, even with identical prescribed damping.
• By adjusting the lever angle α, a continuous trade-off between the intensity of temporal attenuation (damping) and spatial attenuation (band gap depth) is realized, with peak performance at intermediate angles.
• The maximum total damping ratio (ζ_tot^sum) reaches up to 0.45 for α = 45°, while the spatial attenuation (minimum attenuation at μ_min) is maximized at α = 30°, demonstrating the trade-off.
• The band gap width and depth are strongly modulated by α, with the deepest and broadest band gap observed at α = 30°, while the highest damping occurs at α = 45°.
• The system exhibits a dual-peak structure in the imaginary part of the dispersion diagram, corresponding to coupled resonances from the IA and local resonance modes.
• The effective damping and stiffness contributions scale inversely with tan²α, enabling passive, continuous tuning of dynamic performance via mechanical geometry alone.