Exploring the dynamics of hourglass shaped lattice metastructures

Metastructures, inspired by metamaterials, derive their properties from geometry rather than composition and can be tuned via local resonances. Early work on electromagnetic metamaterials (negative permittivity and permeability) motivated extensions to acoustic/elastic metamaterials and phononic crystals exhibiting periodicity. While traditional phononic crystals have fixed bandgaps once patterned, metamaterials enable tunability but many operate over narrow bands, motivating tunable designs for applications such as acoustic imaging, cloaking, and seismic protection. Metastructures demonstrate excellent wave absorption and high stiffness-to-weight ratios, and tailoring the geometry of building blocks can tune wave behavior. However, many prior metastructure unit cells (e.g., lumped masses, cycloidal or cylindrical resonators) have limited tunability due to simple geometries. Architected lattice metastructures offer enhanced static properties and wave control, with 3D printing enabling diverse topologies (lattice, woven, hierarchical, honeycomb, foams). Dome-based lattice structures have been rarely explored in metastructure design despite their potential to tune stiffness and Poisson’s ratio. This work expands the design space by integrating lattice geometry with an hourglass shape composed of two oppositely oriented coaxial domes. Auxetic (re-entrant) lattices, with negative Poisson’s ratio, deform synclastically into domes and can reduce snap-through compared to conventional lattices. The objective is to develop the foundation of hourglass-shaped latticed metastructures, addressing homogeneous (symmetric lattices) and non-homogeneous (unsymmetrical lattices) unit cells. The study presents theoretical and experimental models of deformation mechanics and behavior, finite element modal analyses of six hourglass structures, vibration attenuation via local resonators, tunable stiffness and dispersion curves of periodic hourglass oscillators, and a concept of prestressing (e.g., via piezoelectrics) for remotely controlled bandgaps.

The paper situates the work within metamaterials and metastructures research. Key themes include: (1) electromagnetic metamaterials exhibiting negative constitutive parameters, extended to acoustic/elastic metamaterials and phononic crystals; (2) tunability via local resonances to overcome narrow operating bands, relevant to imaging, cloaking, and seismic protection; (3) metastructure designs ranging from 1D lumped mass models to arrays of resonators, which often have limited tunability due to simple unit-cell geometry; (4) architected lattices (honeycombs, auxetics, chiral, hierarchical) offering improved static properties and wave control, enabled by additive manufacturing; (5) limited exploration of dome-based lattice metastructures, despite literature showing auxetic materials’ negative Poisson’s ratio, synclastic dome formation, and reduced snap-through instabilities. This motivates an hourglass, double-dome, lattice-based unit cell for increased tunability of stiffness, damping, and dynamic response.

Overview: The authors combine analytical modeling, finite element simulations, additive manufacturing, and experiments to investigate six hourglass metastructure classes: homogeneous (solid shell; honeycomb; auxetic) and non-homogeneous (solid-honeycomb; solid-auxetic; auxetic-honeycomb). Analytical model: The hourglass (two coaxial domes) is modeled as two coned-disc (Belleville) springs in series, yielding a cubic load–deflection relation P = K1 δ + K2 δ^2 + K3 δ^3 for a single dome, and equivalent series expressions for homogeneous and non-homogeneous combinations. The formulation relates stiffness coefficients to geometric and material parameters, including shell thickness t, free height h, lattice cell angle θ (re-entrant negative to positive honeycomb), cell length, a/b radius ratio, and material properties (E, ν). Small angular deflection and uniform loading/support around the circumference are assumed. Non-dimensional parametric studies examine the influence of h/t and lattice angle on stiffness, nonlinearity, and potential negative stiffness regions. Fabrication: Samples were 3D-printed as single bodies (two domes joined by a smooth spline) on an Ultimaker 3 Extended multi-material printer using PCTPE (plasticized copolyamide thermoplastic elastomer) for the metastructure and PLA for fixtures. Dome sphere radii: 50 mm; dome height h = 24 mm (hourglass undeformed height H = 48 mm); radial shell thickness t = 4 mm; base radius 42 mm; lattice beam thickness 1 mm. Printing used 0.15 mm layer thickness, 100% infill (triangular pattern), rectilinear/zig-zag supports (≥45° overhang), with consistent build orientation across samples. Material properties (PCTPE): E ≈ 75 MPa, ν = 0.28, ρ ≈ 0.96 g/cm^3. Aluminum plates (radius 50 mm, thickness 8 mm, density 2.7 g/cm^3; 200 g each) served as base/top masses. Static testing: Quasi-static compression tests (Instron UTM-1195, 2 kN load cell) at 0.5 mm/min were conducted to 35% compression of initial height. Load–deflection data were analyzed for nonlinearity and fitted with third-/fourth-degree polynomials; comparison to analytical predictions assessed model validity. Dynamic testing: Displacement transmissibility was measured using a 3D Laser Doppler Vibrometer (Polytec PSV-400) under base excitation (electrodynamic shaker LDS V780) with pseudo-random input (1600 FFT lines) and sine sweeps up to 500 Hz. NI DAQ handled acquisition; Savitzky–Golay filtering (order 1, frame 31) reduced noise. Samples were sandwiched between aluminum plates and clamped with 16 N·m torque for repeatability. Natural frequencies were identified from transmissibility peaks; damping ratios were computed using the half-power bandwidth method. Finite element simulations: Modal analyses (ANSYS Mechanical APDL, tetrahedral mesh, subspace solver) computed the first ten eigenmodes for all six samples. Base rib had zero displacement along the hourglass axis (z). The sixth mode, aligned with the axis, corresponded to the fundamental axial mode observed experimentally. Dispersion analysis: A 1D diatomic, locally resonant metamaterial analog was constructed with an hourglass-based oscillator (internal resonating mass coupled to an outer ring), using experimentally inferred linear equivalent stiffness for small deflections: k_eq ≈ 50 N/mm (auxetic) and 266 N/mm (honeycomb), and mass ratio m_r = 0.125. Dispersion relationships were solved to obtain real/imaginary parts of the wavenumber and illustrate tunable attenuation bands. A prestressing concept is proposed to tune nonlinearity (and bandgaps), including potential piezoelectric actuation.

• Analytical–experimental agreement: The cubic analytical model for a single dome and its series combination for the hourglass predicts load–deflection curves in good agreement with experiments.
• Static behavior: Load–deflection is linear up to ~3 mm axial deflection, then becomes nonlinear (third-degree). Hysteresis during unloading indicates energy absorption. Nonlinearity is highly sensitive to geometry; h/t strongly controls stiffness profiles. A wide zero-stiffness (“zero-rate”) region can be generated near a critical h/t, and negative stiffness arises for sufficiently large h/t (> ~3), with magnitude influenced by lattice topology (auxetic vs honeycomb).
• Lattice angle effect: Varying the lattice cell angle from re-entrant (auxetic, negative) to positive (honeycomb) significantly alters stiffness. Honeycomb regions exhibit larger stiffness variation and more snap-through tendency; auxetic lattices stabilize response and reduce snap-through buckling.
• Dynamics (transmissibility and damping): Natural frequencies are lattice-dependent. Lowest f_n observed for homogeneous auxetic hourglass (~67.1 Hz) and highest for homogeneous solid shell (~250 Hz). Damping ratios (half-power bandwidth): highest for homogeneous auxetic (ζ ≈ 0.125); lowest for non-homogeneous solid-shell with honeycomb lattice (ζ ≈ 0.0997). Others: non-homogeneous honeycomb–auxetic (ζ ≈ 0.1057), non-homogeneous solid-shell–auxetic (ζ ≈ 0.1042), homogeneous solid shell (ζ ≈ 0.1033). Homogeneous auxetic shows ~21% higher damping than homogeneous solid shell under comparable conditions.
• FE–experiment correlation: Modal analysis (first ten modes) matches experimental transmissibility; the sixth eigenmode aligns with the hourglass axis and corresponds to the measured natural frequency peak.
• Dispersion and bandgap tuning: Using linearized stiffness values k_eq ≈ 50 N/mm (auxetic) and 266 N/mm (honeycomb) with mass ratio m_r = 0.125, dispersion curves show auxetic-based hourglass oscillators favor low-frequency attenuation, while honeycomb-based favor higher frequencies. Bandgaps are tunable via h/t and lattice angle θ; prestress can further adjust nonlinearity and attenuation bandwidths.
• Application implications: Nonlinear stiffness profiles (including zero-rate and negative-stiffness regions) are promising for variable-load isolators and widening attenuation bands in dynamic vibration neutralizers.

The study addresses the need for highly tunable metastructure unit cells by introducing an hourglass geometry that integrates two domed lattices. Findings demonstrate that geometric parameters (notably h/t and lattice cell angle) can systematically tailor stiffness nonlinearity, damping, and natural frequencies. Auxetic lattices in dome form mitigate snap-through instabilities and enhance damping relative to honeycomb and solid shells, enabling better energy absorption and low-frequency attenuation. The close agreement between analytical predictions, FE modal analysis, and transmissibility experiments validates the modeling approach and underscores the feasibility of designing hourglass-based oscillators as locally resonant elements. The tunability of bandgaps via lattice topology and potential prestressing (including smart material actuation) suggests the hourglass unit cell can serve as a versatile building block for broadband vibration suppression, wave attenuation, and adaptive devices (e.g., tunable filters, waveguides, resonators, isolators, acoustic diodes).

Six hourglass-shaped lattice metastructures (three homogeneous and three non-homogeneous combinations of solid, honeycomb, and auxetic domes) were designed, fabricated, modeled, and tested. The hourglass architecture provides customizable nonlinear stiffness profiles and lattice-dependent damping and natural frequencies. Auxetic domes yield lower stiffness and higher damping than honeycomb or solid shells, favoring low-frequency attenuation and energy absorption. Analytical cubic load–deflection models, FE modal analyses, and experimental transmissibility measurements are in close agreement. A diatomic locally resonant model with hourglass oscillators shows tunable dispersion/bandgaps, with stiffness governed by lattice topology and h/t ratio, and potential for further tuning via prestress. Future work should explore broader geometric variations (dome shapes, lattice parameters), incorporate full nonlinear dispersion analyses (beyond linearization), and realize active tuning via piezoelectric/magnetostrictive coupling to electrically switch bandgaps. The hourglass lattice metastructure is a promising unit cell for tunable metamaterials and vibration isolation platforms.

• Analytical simplifications: Small angular deflection and uniform circumferential loading/support are assumed; hourglass is modeled as two Belleville-type springs in series, which approximates but may not capture all local lattice effects.
• Linearization in dispersion: To avoid complexity, dispersion analysis uses experimentally derived linear stiffness for small deflections; full nonlinear wave analyses (where Bloch’s theorem is not directly applicable) are not implemented here.
• Manufacturing constraints: 3D printing layer orientation and support strategies can influence mechanical properties; samples were printed with a single orientation but variability may persist.
• Material and test scope: Tests use PCTPE elastomer and aluminum plates, with frequency response limited to 0–500 Hz and a specific undeformed height H = 48 mm for main samples; generalization to other materials/geometries and higher frequencies requires further study.
• Parameter coverage: While h/t and lattice angle θ were varied analytically, experimental validation across the full parameter space (e.g., extreme h/t leading to negative stiffness) is limited and deferred to future work.