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Learn more: PMC Disclaimer | PMC Copyright Notice Commun Eng . 2026 Mar 5;5:69. doi: 10.1038/s44172-026-00630-7 Search in PMC Search in PubMed View in NLM Catalog Add to search Non-contact electroelastic modulation of conventional media leveraging two-way electromagnetic induction Joshua Dupont Joshua Dupont 1 School of Mechanical, Aerospace, and Manufacturing Engineering, University of Connecticut, Storrs, CT USA Find articles by Joshua Dupont 1 , Richard Christenson Richard Christenson 2 School of Civil and Environmental Engineering, University of Connecticut, Storrs, CT USA Find articles by Richard Christenson 2 , Jiong Tang Jiong Tang 1 School of Mechanical, Aerospace, and Manufacturing Engineering, University of Connecticut, Storrs, CT USA Find articles by Jiong Tang 1, ✉ Author information Article notes Copyright and License information 1 School of Mechanical, Aerospace, and Manufacturing Engineering, University of Connecticut, Storrs, CT USA 2 School of Civil and Environmental Engineering, University of Connecticut, Storrs, CT USA ✉ Corresponding author. Received 2025 Sep 22; Accepted 2026 Feb 19; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13076841 PMID: 41786912 Abstract This research reports the synthesis of a wave altering non-contact design. Resonators enable vibration and wave modulation, but typically require mechanical attachments that modify host properties and limit retrofit. Inspired by remote field technologies, we introduce a tunable non-contact resonator that traps propagating wavefronts and suppresses targeted modes in elastic structures. Eddy-current interactions provide remote, bidirectional coupling between structural vibrations and coil voltage. Analog impedance converters tune the shunt impedance, establishing LC-resonance at select frequencies while compensating for dissipative loss. In this manner, local resonance is induced on electrically conductive media, facilitating tunable dispersion and modal suppression. Intrinsic host properties are preserved, facilitating non-intrusive elastodynamic control for in-service retrofit, delicate structures, and challenging environments. Analytical modeling predicts underlying electromagnetics and structural dynamics, informing electrical tunings while highlighting functional dependencies. Experiments demonstrate tunable suppression and wave-blocking, unveiling constraints and improvement paths. The results establish a foundation for adaptive, non-contact elastic metamaterials. Subject terms: Electrical and electronic engineering, Mechanical engineering Joshua Dupont and colleagues report a tunable electromagnetic unit cell to remotely impose local resonance in conductive media. Analytical and experimental studies of the coupled system show selective wave blocking and modal vibration suppression Introduction Built-up periodic structures, commonly termed metamaterials or metasurfaces, leverage coupled interactions and/or tunable features to extend elastodynamic control over conventional media. Through engineered unit cells, auxiliary dynamics and energy flow paths may be integrated into elastic structures, facilitating spatially localized tunings with macroscale influence 1 . These attributes have established metamaterials in emergent fields of energy harvesting 2 – 5 , wave modulation 6 , and vibration control 7 , with exotic characteristic features 8 – 14 . Recent progress emphasizes reconfigurability 15 , 16 and tunable control across a diverse application space, including soft actuators 17 , 18 , tactile interfaces 19 , 20 , and transportation systems 21 – 23 , where compact and adaptable solutions are increasingly needed. However, most metamaterials depend on bonded substructures or geometric alterations/inclusions to modify wave propagation and dynamic responses. Phononic crystals 24 , local resonators 25 , and multifunctional systems 26 are typically designed a priori with some level of direct mechanical integration. Such implementations can alter fundamental properties of the host structure, limiting applicability for in-service retrofit or deployment under constrained and challenging environments 27 , 28 . For instance, piezoelectric metamaterials 29 offer adaptable electrical tunability but require surface bonding with soldered interconnections, which may be unsuitable for sensitive or delicate structures. Consequently, there is a need to develop fully non-contact and detachable mechanisms to extend tunable control over existing platforms 30 . To address this gap, this work investigates a non-contact electromagnetic transducer harnessing bidirectional Eddy-current coupling to remotely impose local resonance on conductive media. This mechanism draws inspiration from foundational work in remote field technologies 31 and electromagnetic shunt dampers 32 , 33 . By placing a permanent magnet near the vibrating surface, steady-state oscillations drive continual changes in magnetic flux, generating corresponding Eddy currents and Lorentz forces 34 , 35 . Electroelastic coupling is then established through strategic alignment of a secondary coil, exploiting electromagnetic induction 36 to facilitate energy transfer between the mechanical and electrical domains. Despite the non-contact potential, challenges with coupling efficiency and electrical resistivity have historically confined the approach to applied sensing 37 – 40 . Metamaterial configurations have been explored, but primarily through augmentation of traditional elastic couplings with electromagnetic enhancements. Representative designs include permanent magnets serving as conventional resonator masses that oscillate relative to fixed coils 41 , whose impedance is electronically compensated to provide a non-contact stiffness tuning. Other configurations have leveraged relative motion between magnet and nearby conductor, introducing tunable Eddy-current damping to inertial dampers 42 or periodic resonator arrays 43 . These magneto-mechanical metastructures introduce dynamic reconfigurability via externally applied fields, facilitating elastic wave modulation and control 43 – 45 . Related efforts have also tailored the global response of vibratory structures 46 , but still rely on mechanical bonding to achieve adequate performance 47 . Distinct from previous work, this research presents a wave-altering non-contact design, WAND, to remotely impose locally resonant dynamics on conductive structures using fully non-contact and mechanically detached unit cells. The transducer assembly consists of a magnet and an impedance-compensated coil, facilitating a bidirectional Eddy current coupling between structural motion and an analog shunt. This coupling mechanism prioritizes deployability over efficiency, enabling detached, electrically tunable resonators to be temporarily applied to in-service metal structures and thin, delicate panels without permanent modification. Related studies have successfully applied this fully non-contact architecture to active modal suppression 48 , 49 , relying on real-time feedback to appropriately modulate coil current. By contrast, this work builds upon progress in piezoelectric metamaterials, which has established reliable semi-passive methods to compensate for resistive damping 50 and enhance coupling efficiency 29 , 51 . Leveraging mechatronic synthesis, the transducers are formed into tunable LC-oscillators coupled to elastic vibrations via electromagnetic induction. Negative impedance converters 52 , 53 and capacitance multipliers 54 provide analog tunability while compensating for dissipative loss, strengthening coupling at select frequencies without active control. Among periodic architectures, local resonance is particularly well-suited to remote Eddy current coupling. Tuning the LC-resonators modulates elastic wave propagation, enabling the versatile realizations of transmission band gaps 55 , waveguiding 56 , sensing enhancement 57 , wavefield imaging 58 , mode conversions 59 , and vibration suppression 60 – 62 . In this research, a lumped parameter model is developed to capture underlying physics and inform electrical calibration. Two representative case studies are then examined, including a wave-blocking metamaterial and an electrically tunable beam damper. The analyses highlight key features, functional constraints, and preferred electrical tunings of the proposed WAND, while experimental testing confirms the feasibility of the model and approach. Through this mechanism, remote imposition of locally resonant dispersion and modal suppression is achieved, offering a versatile framework for continuing progress in smart structures, wavefield modulation, and vibration control. Results Governing dynamics of Eddy-current resonators Resonant substructures impose forces to passively counteract local mechanical oscillations at a given frequency. In elastic wave propagation, these resonators absorb portions of the incident energy which are subsequently released out of phase, such that tuning their natural frequency relative to the incident frequency ω , tailors the superimposed wavefield to delay, steer, or block transmission. This capability benefits waveguiding and vibration control, but requires bidirectional energy transfer between the elastic medium and the local resonator. To achieve this coupling through an air gap, the WAND leverages electromagnetic Eddy-current interactions and Lorentz forces driven by mechanical oscillations in the host structure. The WAND transducer in Fig. 1a comprises a permanent magnet fixed within a copper coil. This assembly is centered above a conductive host structure with a lift-off distance h . The coil inductance L s and capacitive load C s form a passive electrical LC-oscillator with natural frequency f e in Hz. f e = 1 2 π L s C s 1 Fig. 1. Metamaterial system description. Open in a new tab a Non-contact electromagnetic unit cell with resonant LC shunt, highlighting key parameters. By adjusting C s and R n , the frequency and damping are electrically tuned. b Wave-altering non-contact design (WAND) metamaterial applied to locally resonant band gaps and elastic wave-blocking. Tuning the shunt capacitance adjusts this LC-oscillation, which is converted to elastic local resonance as detailed by the dynamic interactions in Fig. 2a . Wave-induced deformation results in a time-varying displacement d dt q ( t ) = q ° of the host structure within the permanent magnet’s static field B p , generating a changing magnetic flux Φ ° t through the underlying cylindrical volume. By Faraday’s Law 34 , this drives differential Eddy current rings in the highlighted regions of Figs. 1 and 2b . Mechanical oscillations modulate the Eddy-current strength, producing a secondary time-varying magnetic field in accordance with the Biot-Savart law 36 . This electromagnetic induction then generates a corresponding electromotive force in the coil, constituting the forward path of energy transfer. In the reverse path, when the mechanical oscillation approaches the LC resonance, coil current is maximized 180° out of phase with the motion. Through mutual inductance, this coil current induces additional Eddy currents in the elastic medium, which interact with the permanent magnet to generate Lorentz forces 34 that oppose the motion. Through these interactions, the WAND establishes a bidirectional, non-contact, and electrically tunable coupling between the resonator and host structure, facilitating elastic wave modulation and modal suppression. Fig. 2. Bidirectional Eddy current coupling. Open in a new tab a Block diagram of bidirectional coupling between elastic vibrations in the conductive host and LC-resonator current; P 1 , P 2 , k 1 , and k 2 label the lumped terms in Eqs. ( 2 – 5 ), associated with each path. b Electromagnetic unit cell with neodymium core field lines, showing motion-induced differential Eddy-current rings in the conductive host. c Equivalent circuit model of electrical coupling between Eddy currents and coil current via mutual inductance. To quantify this magnetic coupling, a Lagrangian formulation based on the extended Hamilton’s principle and Euler-Bernoulli beam theory is employed 63 , 64 . Electromagnetic interactions are represented by a generalized Lorentz force 64 F L , expressed as a volume integral over the highlighted conduction zones. F L = ∫ ∫ ∫ J i × B p d V = − k 1 I ° ( t ) − k 2 q ° ( t ) 2 where J i represents the density of the i t h Eddy-current ring and q t is the generalized mechanical displacement. I t = Q ° is the LC-oscillator current, or equivalently, the time derivative of the shunt’s charge displacement Q ( t ) . Formulated in cylindrical space, this integral aggregates contributions of differential Eddy current rings (see Fig. 2b ) to the Lorentz force exerted on the host. The lumped parameter k 1 scales this force based on the changing coil (LC-oscillator) current and represents the inductive coupling. Meanwhile, k 2 represents additional viscous damping generated by motion through the magnet’s static field B p , inducing opposing Eddy-currents that dissipate energy in the resistive substrate. These coefficients are derived from electromagnetic first principles, incorporating coil and beam geometry, boundary conditions, and modal or traveling-wave solutions. The resulting closed-form integral expressions, provided in Supplementary Eqs. ( S1 – S15 ), are numerically evaluated via Gaussian quadrature as detailed in Methods. On the electrical side, Fig. 2c defines an equivalent circuit through which Eddy currents are modeled as individual loops coupled to the LC oscillator through mutual inductances L m i 40 . The electromotive force V b i follows Faraday’s law, with a path resistance R b i originating from the conductivity of the elastic medium. To analytically resolve the distributed current density J i of each differential ring, the Eddy current self-inductance L b i is neglected, an approximation valid at mid-to-low frequencies where ω L b i < R b i 39 , such that Eddy-current impedance remains resistance dominated. In Supplementary Fig. S1 , this margin is investigated further 65 and estimated below 1200 Hz for the geometry considered here. Below this frequency, the total electromotive force induced in the LC oscillator by the aggregated mutual inductance L m is compactly defined as L m J ° = ∬ L M i J i ° d s = − P 1 Q ⋅ ⋅ ⋅ ( t ) − P 2 q ¨ ( t ) 3 where ds = 2 π r i dr is the differential area of each Eddy-current ring. The coefficient P 2 defines an electromechanical inerter 32 , relating the Eddy-current induced coil voltage to the host acceleration q ¨ ( t ) . The parameter P 1 arises from higher-order inductive feedback, in which coil currents generate self-interacting Eddy currents that inductively reflect onto the electromagnetic shunt. The overall term P 1 Q ... perturbs the shunt dynamics as a high-order resistance, but is negligible for the geometry considered here. However, since it scales in accordance with electrical jerk Q ⋅ ⋅ ⋅ ( t ) , its contribution may become more substantial at high frequencies. Incorporated into Hamilton’s principle, these non-contact electromagnetic interactions yield a coupled set of governing expressions, which together describe both the elastic Eq. ( 4 ) and electric Eq. ( 5 ) reduced order dynamics of the system as m q ¨ + ( c + k 2 ) q ° + k q + k 1 Q ¨ = F ext 4 P 1 Q ⋅ ⋅ ⋅ + L s Q ¨ + R s Q ° + 1 C s Q + P 2 q ¨ = 0 5 where the lumped parameters m , c , and k are the effective mass, damping, and stiffness from Euler–Bernoulli beam theory, and F ext is an externally applied mechanical excitation. In practice, the total shunt resistance R s varies between transducers due to parasitic impedances and frequency dependence, is typically large ( k Ω range), and increases with frequency from coil skin/proximity effects. Aside from the electric circuit parameters L s , C s , and R s , the remaining lumped parameters are case-specific and spatially dependent (e.g., elastic wavenumber k x or mode shape ϕ ) functions of the geometry, material properties, and boundary conditions. For reproducibility, closed-form expressions for each lumped parameter are given in Supplementary Eqs. ( S1 – S15) , while the derivation is adapted from methods in the supporting literature 34 , 36 – 40 , 63 , 64 . In this formulation, we also note that the lift-off h is considered constant, a small-signal approximation that eliminates non-linear amplitude dependence in the lumped-parameter terms. Dispersion relation and locally resonant band gaps To assess the capability of this non-contact electromagnetic resonator for elastic wave modulation, the lumped formulation considers a periodic WAND array as depicted in Fig. 1b . This magneto-mechanical metamaterial is governed by the cumulative effect of individually local Eddy current couplings. Mechanical and electrical state variables are expressed as traveling wave solutions within the Lagrangian framework, while Bloch–Floquet boundary conditions impart wavenumber dependence on the lumped parameters in Eqs. ( 4 ) and ( 5 ). Performing an eigenvalue analysis on the coupled system yields a characteristic equation, resolving wave frequencies ω available to each elastic wavenumber k x . For the magneto-mechanical metamaterial, this dispersion relation may be expressed as 1 − ω 2 ω m 2 + j ω ω m 2 c + k 2 1 − ω 2 ω e 2 + j ω ω e 2 R s − ω 2 P 1 − ω 4 k 1 P 2 C s k = 0 6 where j = − 1 is the imaginary unit, ω m = k / m denotes the mechanical resonance, and ω e = 2 π f e is the radial LC-oscillation frequency. The relation can be viewed as a dynamic equivalence between the decoupled system dynamics (bracketed terms) and a magnetic coupling term ( ω 4 k 1 P 2 C s / k ) . Together, these terms reflect how strongly the inductive Eddy-current coupling modifies the otherwise independent mechanical and electrical dynamics of the system. Equation ( 6 ) deviates from other canonical forms of metamaterials because Eddy-current coupling is rate-driven, resulting in the high-order coupling and electrical jerk in the governing expressions. In the dispersion relation, this results in a frequency-dependent effective coupling and dissipative contribution, reflecting the nature of electromagnetic induction. The locally resonant characteristics of the WAND metamaterial are revealed in Fig. 3a , numerically resolving the dispersion relation by neglecting imaginary terms and plotting frequency against the real component of normalized wavenumber. The remaining panels in Fig. 3 (as well as Supplementary Figs. S2 and S3 ) are derived from this underlying solution set. A locally resonant bandgap emerges near the tuned frequency f e of the electromagnetic resonator, enabling electrically tunable wave suppression without the need for direct physical contact or geometric alterations. Fig. 3. Analytical metamaterial model. Open in a new tab a Elastic dispersion for two capacitive shunt impedances, demonstrating bandgap tunability. b Dispersion for C s = 43 nF with increasing lift-off h , where narrowing band gaps reflect losses in magnetic coupling. c Magnitude of magnetic coupling term versus wavenumber k x for several beam materials, showing the influence of conductivity, stiffness, and density. d Magnetic coupling magnitude versus frequency, including ±10% neodymium magnet remanence tolerance to illustrate sensitivity and bounds due to manufacturing uncertainty. The overall strength of magnetic coupling is tied directly to the unbracketed term in Eq. ( 6 ), exhibiting parametric sensitivity to geometry, material properties, and excitation frequency. The effect of varying lift-off h is studied in Fig. 3b , illustrating the 681 Hz bandgap progressively narrowing as this air gap increases. This reduced bandwidth reflects a corresponding decrease in the magnetic coupling term (detailed in Supplementary Fig. S3 ), arising from weakened Eddy-current generation and mutual inductance as the transducer is moved away from the conductive surface. In Fig. 3c , the wavenumber-frequency solution set corresponding to the lower f e = 681 Hz dispersion band is applied to this magnetic coupling, plotted against wavenumber for several host material selections. Note that all cases follow the geometry of the experimental setup, while varying the host’s Young’s modulus E b , mass density ρ b , and electrical conductivity σ (see explicit values in Supplementary Tables S1 and S2 ). In general, the materials best suited to this magnetic coupling are lightweight and compliant, or those demonstrating high electrical conductivity. In contrast, heavy and stiff materials (e.g., gold and brass) exhibit weak magnetic coupling, as these properties limit the rapid deformations necessary for Eddy-current generation. Aluminum offers the best balance overall, with copper and magnesium performing reasonably well in comparison. These trends support the versatility of the unit-cell concept, particularly with regard to thin and compliant hosts. Since electromagnetic induction is driven by changes in magnetic flux, coupling effectiveness increases strongly with frequency, as illustrated in Fig. 3d . Coupling also scales with the static magnetic field strength B p , such that stronger magnets or superconducting coils may enhance coupling and suppression bandwidth. While this analysis offers insight into maximizing performance, several trade-offs and challenges must be considered. Close proximity enhances coupling but may introduce modeling uncertainty from lift-off variations in the permanent-magnet’s nonlinear field 39 . Additionally, predictable performance is constrained to frequency regimes high enough for adequate electromagnetic coupling, but low enough to avoid frequency-dependent parasitic impedance, coil-resistance, and higher-order effects. Nevertheless, passive local resonance remains feasible through careful design and calibration, accounting for frequency dependence while leveraging synthetic hardware to mitigate dissipative loss. Challenges and analog impedance compensation A primary challenge in extracting meaningful performance from the electromagnetic resonators is dealing with Eddy-current damping and frequency-dependent shunt resistance, arising from coil losses, Eddy-current damping, and non-ideal analog hardware. These mechanisms reduce the local resonant quality factor, limiting the current available for magnetic coupling. The shunt resistance R s of the LC-oscillator can be decomposed into individual contributions, R s ω = R DC + Δ R skin ( ω ) + Δ R prox ( ω ) + R parasitic ( ω ) − R n 7 where each term and its effect are detailed in Supplementary Table S3 . The DC coil resistance R DC sets a baseline ohmic loss across all frequencies. As excitation frequency increases, skin-depth and proximity effects ( Δ R skin , Δ R prox ) confine current within localized regions of each conductor, increasing shunt resistance 66 . Additionally, parasitic resistances R parasitic arise from op-amp output impedance, trace and terminal resistance, and other analog hardware non-idealities. Separately, inductive Eddy-current coupling between the coil and metallic host contributes additional resistive and viscous damping, as captured in the governing expressions. Without compensation, these dissipative factors dampen the LC-oscillation, impeding current flow and weakening magnetic coupling, diminishing the control authority over wave propagation at targeted frequencies. To overcome this challenge, previous work has leveraged active control 48 , 49 to amplify current and strengthen Lorentz forces. Others have reduced complexity by eliminating intermediary steps 41 , 43 , enhancing efficiency by directly attaching coils or magnets to the host structure. In contrast, this work retains passive tunability and fully non-contact operation via analog impedance converters, reducing the shunt resistance to strengthen magnetic coupling. The mechatronic synthesis combines a capacitance multiplier 54 , facilitating frequency adjustments of the locally resonant bandgap, with a tunable negative resistance 50 to offset parasitic losses. Figure 4c summarizes this tunable hardware, while more detailed schematics are provided in Supplementary Fig. S4 . Each circuit offers semi-passive control, whereby DC power is fed to operational amplifiers, and reconfiguration is achieved via continuous potentiometer adjustments. In this way, digital controllers and real-time feedback are avoided, preserving the reliability and scalability of the non-contact approach. Fig. 4. Experimental setup and calibration. Open in a new tab a Locally resonant magneto-mechanical metamaterial, featuring a 2 m aluminum beam and 7 non-contact unit cells with 1.5 m lift-off. b RLC calibration performed with a signal analyzer to measure electrical frequency responses. c Shunt circuit diagram for the RLC configuration, where analog impedance converters adjust C s and R n . Details in Supplementary Fig. S1 . d RLC response measurements demonstrating tunability of the electrical LC-oscillator. With this augmentation, remote local resonance depends on accurate and consistent electrical tuning across the metamaterial array. Variations in shunt impedance can lead to mismatched LC-resonances, diminishing the cumulative bandgap strength. Leveraging the analog impedance converters, uniformity across the LC-oscillators is achieved via electrical calibrations. The electromagnetic transducer and shunt are arranged as an RLC circuit with a test resistor and positioned near the elastic host, capturing any coil impedance shifts due to Eddy-current interactions. An Agilent 35670 A signal analyzer then drives the circuit and records its electrical frequency response, revealing the realized LC resonances. By tuning the capacitance multiplier, each unit cell resonance is adjusted until all peak responses align at a common frequency. Figure 4d shows a demonstrative set of responses, demonstrating frequency tunability as potentiometer R 5 is decreased from 5.86 k Ω to 3.86 k Ω . Similarly, the negative resistance branch adjusts the peak LC-oscillation magnitudes via R 3 . Due to parasitic effects and parametric uncertainty, the resistive load of each electromagnetic shunt varies considerably. The dispersion curves in Fig. 2 reflect near-zero resistance and damping, as the real component of eigenvalues is characterized by reactive energy-storing dynamics, not viscous loss. Thus, experimentally, the resistance of each unit cell at a particular target frequency should be minimized, whilst remaining positive R s ω > 0 Ω . This condition provides an approximate stability margin, since instability arises when the combined dissipative terms in Eqs. ( 4 ) and ( 5 ) become net negative, precipitating rapid op-amp saturation and flattening of the measured LC-resonance. Increasing the negative resistance R n reduces the net shunt resistance R s , strengthening the LC resonances as shown in Fig. 4 d. Rather than specifying R s for calibration, which is challenging due to frequency-dependent uncertainties, the peak response is incrementally raised until op-amp saturation is observed (flattened resonant peak) and then backed off. During calibration, a slight tilt of the LC resonances is also observed. This behavior appears consistent with weak non-linear softening, likely attributed to small lift-off variations 37 – 40 as Lorentz forces move the host within the neodymium magnet’s cubic-decaying magnetic field B p ∝ r − 3 . The effect is minor over the tested band and does not impede the calibration process. In this manner, the effective impedance of each resonant shunt is tailored to enhance magnetic coupling while maintaining stability. Tunable Wave Modulation via WAND through Resonator Array To translate the conceptual WAND to a practical demonstration, a one-dimensional magneto-mechanical metamaterial is synthesized to address key challenges under realistic constraints. The core setup, Fig. 4a , consists of a thin 1.8-m 6061-O aluminum beam excited by a piezoelectric actuator swept over 500–700 Hz. The beam length is selected to capture a well-developed wavefield across the incident and transmitted regions, ensuring that at least elastic wavelengths are measured to accurately determine the transmission ratio while minimizing spatial aliasing and near-field artifacts. The beam ends are clamped between layers of 70-00 durometer sorbothane, a viscoelastic rubber with high hysteretic damping in the acoustic range 25 . These boundary conditions emulate infinite periodic boundaries by dissipating energy and minimizing elastic-wave reflections. Meanwhile, aluminum is selected for its compliance, low density, and electrical conductivity, facilitating strong elastodynamic interactions between Eddy-currents and Lorentz forces. The metastructure comprises seven evenly spaced resonators, illustrated in Fig. 4b , with a 1.5 mm lift-off above the elastic substrate. Each electromagnetic transducer consists of a copper coil and a neodymium magnet embedded in hardened epoxy, restricting relative motion while allowing precise standalone placement. These self-contained, remotely coupled units provide practical benefits, including durability in harsh environments 27 , and ease of replacement or repositioning. The potential for the magneto-mechanical metamaterial to modulate elastic waves via local resonance is experimentally verified in Fig. 5 . A PolyTec PSV-500-M scanning laser vibrometer measures out-of-plane velocity at 255 scan points spanning the incident and transmitted regions to map the wavefield over the excitation range. Two electrical calibrations are tested, with the LC oscillators tuned first to 527 Hz and then to 681 Hz. A representative wavefield measurement is shown in Fig. 5a , illustrating wave propagation within the 681 Hz bandgap. Incident waves travel from left to right through the magnetically coupled region, where Eddy currents and Lorentz forces counteract mechanical oscillations. The transmitted field on the right exhibits a 9 dB amplitude reduction relative to an otherwise identical configuration with Eddy-current damping and disconnected coils. Figure 5b casts this measurement into a transmission ratio, highlighting narrowband suppression at each tuned frequency. The observed band gaps agree with the analytical prediction of Eq. ( 6 ), confirming elastic wave modulation via remotely imposed local resonance. By adjusting the capacitance multiplier, these band gaps may be shifted to arbitrary frequencies within the operational domain of the coupling mechanism and electrical hardware. Fig. 5. Magneto-mechanical metamaterial experiment. Open in a new tab a Spatial distribution of out-of-plane vibration velocity magnitude for a 681 Hz incident wave, measured using a PolyTec PSV-500-M scanning laser vibrometer. b Corresponding transmission ratio in decibels, demonstrating elastic band gaps (wave-blocking) at 527 and 681 Hz resonant tunings. The experimental realization of electrically tunable, remotely imposed resonant band gaps confirms the underlying coupling mechanism and associated physics, marking a step forward in resonant elastic metastructures. Table 1 compares this performance against associated electroelastic metamaterials and absorbers explored in recent literature. Overall, these electromagnetic resonator band gaps remain narrow and moderately deep, reflecting ongoing challenges in coupling efficiency, mitigating parasitic losses, and broadband performance. Many of these factors are inherent to Eddy-current coupling, which exhibits frequency dependence and weak nonlinearity. Mitigating these challenges offers opportunities for future research and advancement, which may enhance fundamental properties like magnetic field strength and material conductivity, or augmentation through mechanical and circuit-level adaptations. In this context, enhancing eddy current generation while addressing functional constraints is key to improved performance at larger lift-off distances. Performance-based optimization could also be leveraged to expand and tailor these attenuation characteristics. For example, performance-based optimization using graded or multi-resonant spatial distributions of C s and R n could expand and tailor attenuation by merging individual band gaps for broadband suppression. Alternatively, geometric optimization may further maximize coupling to targeted modes under practical lift-offs. This may include optimized transducer placement near high-acceleration regions or tailoring the host geometry to localize Eddy-current generation and enhance magnetic induction. Nevertheless, these experiments demonstrate electrically tunable local resonances via a fully non-contact mechanism, establishing a reconfigurable method to control delicate structures, perform in-service retrofits, and deploy to constrained environments. Table 1. Summary of tunable contact and non-contact metamaterial architectures Type Architecture Tuning/control mechanism Freq. range Achieved attenuation Bandgap width Contact Piezoelectric metamaterial with LC-Resonant Shunts 50 Tunable potentiometers, analog impedance circuits featuring negative resistance & negative capacitance 4–16 kHz Up to 28 dB Up to 1.2 kHz Contact Temporally modulated metamaterial via piezo LC-resonators 69 Voltage-controlled resistors modulated via a time-varying reference control signal 7–15 kHz 4–8 dB Multiple bands Up to 100 Hz Contact Spatiotemporally modulated metamaterial via piezo LC-resonators 70 Synthetic inductors augmented with digitally programmed time-varying potentiometers 2–8 kHz Up to 13 dB Multiple bands Up to 1 kHz Partially Non-Contact Spring/mass metamaterial featuring tunable non-contact Eddy current damping 43 Bandgap tuning via spring stiffness and absorber mass. Non-contact dissipative tuning via magnet position 35–60 Hz Up to 40 dB Up to 66 Hz Partially non-contact Spring-mass metamaterial featuring electromagnetically coupled LC-oscillators 41 Combined mechanical & electrical tuning via stiffness/mass and impedance compensated coil 30–200 Hz Up to 80 dB (simulated) Up to 60 Hz Fully non-contact Eddy current coupled beam absorber with active control system 49 Velocity feedback drives an electromagnet to induce active Eddy current damping 30–100 Hz Up to 17 dB N/A Fully non-contact (present work) WAND metamaterial featuring semi-passive Eddy current coupled LC-oscillators Tuned potentiometers, impedance compensated coil with capacitance multiplier & negative resistance 500–700 Hz Up to 10 dB Up to 5 Hz Open in a new tab Non-contact modal suppression of a cantilever beam To further demonstrate versatility and effectiveness, the non-contact coupling and model are extended to modal suppression of a cantilever beam. Contrasting with wavefront modulation, this configuration forgoes periodicity and treats the augmented structure as an enclosed system, electrically tuned to moderate a targeted mechanical resonance. The fundamental physics remain consistent with the bidirectional Eddy current coupling previously discussed. Figure 6a details the corresponding experimental setup in Fig. 6b , consisting of a 220 mm, 6061-T6 aluminum beam with rigidly clamped boundaries. A single electromagnetic resonator, identical to those used earlier, is placed near the beam’s free end at a vibrational antinode. This placement maximizes induced Eddy currents, enhancing coupling efficiency and suppression performance. To determine the lumped parameters of Eqs. ( 4 ) and ( 5 ) for this configuration, the assumed-mode method based on Hamilton’s principle 67 is employed to discretize the continuous beam dynamics. This method leverages separation of variables, casting bending deformations u x , t into a known cantilever mode shape ϕ x multiplied by a generalized displacement q ( t ) . In lieu of the piezoelectric actuator, the model applies a vertical external force F ext ( t ) at the free end of the cantilever. With relevant assumptions and some rearrangement, the closed-form transmissibility of the magnetically coupled tuned beam damper is obtained as q ¯ f M ¯ = 1 C s − ω 2 L s + j ω R s − ω 2 P 1 k − m ω 2 + j ω c + k 2 1 C s − ω 2 L s + j ω R s − ω 2 P 1 − ω 4 P 2 k 1 8 where q ¯ and f M ¯ denote steady-state amplitudes of the beam displacement and forcing input, respectively. The remaining parameters are analogous to their metamaterial counterparts but obtained via cumulative integration under the assumed-mode method. Final closed-form expressions for these lumped terms are provided in Supplementary Eqs. ( S7 – S15) , for reproducibility of the model and analytical results. Fig. 6. Non-contact modal suppression. Open in a new tab a Electromagnetic resonator configured as a tuned damper for targeted vibrational modes. b Close-up of the non-contact resonator assembly on an optics post mount. c Analytical displacement–force response near the 2nd bending mode for varying shunt resistance R s . d Measured velocity-voltage response for standalone beam, Eddy current damping, and impedance compensated configurations. Echoing bandgap formation, the cumulative shunt impedance strongly influences the electromagnetic coupling and performance of the tuned damper. When the LC oscillator is tuned to the beam’s 2nd natural frequency, the resonant terms in Eq. ( 8 ) cancel, reducing the peak response amplitude. This behavior is shown in Fig. 6c , comparing the predicted response of the standalone beam to that with resonant electromagnetic coupling. The simple introduction of an open-circuit transducer ( R s = ∞ ) adds viscous damping, reducing response amplitude as Eddy currents dissipate energy in the resistive host. With the LC-oscillator active, minimal improvement is observed due to the large inherent coil resistance ( R s ≈ 2 kΩ ), motivating negative resistance compensation. Decreasing R s lowers corresponding terms in Eq. ( 8 ), enhancing electromagnetic coupling and modal suppression. As this continues, a near-optimal combination of C s and R s is reached, indicated by the flattened frequency response ( R s ≈ 330 Ω ). Although the exact optimum is non-trivial, an approximate solution follows from an adapted invariant-point method 68 . Assumptions and details are discussed further in the Methods, with the corresponding expressions provided in the Supplementary Eqs. ( S16 – S24) . These trends are experimentally replicated in Fig. 6d , which reports analogous transmissibility curves from piezoelectric excitation near the beam’s 2nd vibrational mode. Incrementally adjusting the negative resistance yields a flattened modal response exceeding the performance of Eddy-current damping, further demonstrating the electromagnetic resonator’s potential for semi-passive, electrically tunable vibration control. Discussion In summary, we introduce and experimentally validate an electrically tunable electromagnetic resonator that remotely imposes local resonance in conductive media through an air gap. The motivation and contributions lie in strict adherence to a non-contact architecture, facilitating elastic wave modulation and vibration suppression without physical attachments, inclusions, or other permanent modifications. This wave-altering non-contact design, WAND, provides the adaptability of electromechanical smart structures while preserving intrinsic dynamical properties of the host, which is particularly advantageous for in-service retrofit, sensitive and delicate structures, or deployment in challenging environments. Leveraging mechatronic synthesis, analog impedance converters semi-passively counteract coil resistance at target frequencies, providing electrical tunability without real-time feedback control. This compensation enhances the strength of eddy-current coupling to facilitate bidirectional energy transfer and improve effectiveness in elastic control. These principles are successfully confirmed through experimental demonstrations of wave-blocking and modal vibration suppression, highlighting versatility across different scenarios. Limitations of the proposed method are discussed, particularly regarding narrow operational bandwidths and functional constraints, which motivate further investigations and study. Future work will examine the functional dependencies and modeling framework in greater detail, emphasizing direct quantification of electromagnetic coupling and its frequency dependence, as well as solidifying optimal tuning criteria. Detailed finite-element simulations will be developed to investigate nonlinear influences and statistically quantify parametric uncertainties. Further efforts may also target methods to enhance two-way magnetic coupling and broaden band gaps at greater lift-off distances. Performance-based geometric optimizations offer additional opportunities, including dispersive tailoring, improved coupling efficiency, and broadband attenuation. Continuing foundational advancements could advance next-generation, remotely coupled adaptive metastructures and reconfigurable systems, with multifunctional applications in integrated sensing, energy harvesting, and vibration control. Methods Wave dispersion analysis For the periodic formulation the coupled governing equations of motion, Eqs. ( 4 ) and ( 5 ), emerge from the extended Hamilton’s Principle ∫ t 1 t 2 ( δ T − δ U + δ W ) dt = 0 . The procedure draws upon contributions and derivation strategies from existing literature 34 , 37 – 40 , 63 , 67 , synthesized and adapted to the context of the non-contact electromagnetic resonator (see Fig. 1a ). The kinetic energy variation δ T is defined and evaluated as a volume integral over the unit cell length, while the potential energy variation δ U is extracted from the strain-energy equation, substituting in the linear stress–strain relation and Euler–Bernoulli definition of strain 64 . The variational virtual work term δ W = ∫ − L b / 2 L b / 2 F L x , t δ w dx , where w is the elastic deformation, encapsulates the cumulative Lorentz force exerted on the local elastic medium. Importantly and uniquely, the Lorentz force is treated as a distributed load across the cylindrical Eddy-current conduction zone. This is applied via a Heaviside function H ( x ) in the definition F L ( x , t ) = − H x + W b 2 − H x − W b 2 ∭ J i × B p dV , which ensures the Lorentz force maintains spatial dependency on the elastic wavenumber k x . Deformation of the elastic substrate is assumed to be a traveling wave in the form w x , t = w 0 e j k x x e − j ω t = q t e j k x x , with no applied external forces F ext = 0 . In conjunction with the cited literature, these principles form the basis of the analytical derivations. Closed-form, final solutions to the lumped parameters of the magneto-mechanical metamaterial are detailed in the Supplementary Eqs. ( S1 – S6) , for reproducibility of the analytical results. Cantilever beam transmissibility analysis For the transmissibility analysis of the electromagnetically coupled cantilever beam, the lumped-parameter model is derived using the assumed mode method 40 , 63 , 67 . This approach separates the transverse displacement of the beam into independent spatial and temporal components w x , t = ϕ x q ( t ) . Where ϕ x is the cantilevered mode shape of an Euler–Bernoulli beam as detailed in Supplementary Eqs. ( S14 ) and ( S15) . The generalized displacement t = q ¯ e j ω t and external modal forcing function F t = ∫ 0 L b δ x − L b ⋅ f ¯ e j ω t ϕ ( x ) dx are treated as harmonic functions, where δ x − L b is the Dirac-delta function constraining the applied force to the beam’s free end. Unlike the dispersion analysis, the Lorentz force is treated as a point load applied along the center axis of the electromagnetic coil. The resulting expression F L ( x , t ) = − δ ( x − x m ) ∭ J i × B p dV simplifies the analytical model for low-order bending modes. In conjunction with the cited literature, these principles form the basis of the analytical derivations. Closed-form, final solutions to the lumped parameters of the tuned beam damper are detailed in the Supplementary Eqs. ( S7 – S15) , for reproducibility of the analytical results. Optimal shunt tuning approximation for modal suppression To obtain the near-optimal electrical tuning criteria for the flattened transmissibility response in Fig. 6c , the classical invariant point method 68 for spring-mass vibration absorbers is adapted to a non-contact electromagnetic beam absorber. Starting from the displacement force transmissibility in Eq. ( 8 ), the targeted mechanical mode is assumed to be sufficiently narrowband such that the frequency-dependent resistive term R s − ω 2 P 1 = > R s − ω m 2 P 1 and coupling term ω 4 P 2 k 1 = > ω m 4 P 2 k 1 may be evaluated at the beam resonance ω m . Following Den Hartog’s invariant point criterion 68 , the electrical resonance is first tuned to the targeted mechanical mode C s = 1 / L s ω m 2 with an optimal shunt circuit resistance R s = ω m 2 P 1 + 2 k 1 P 2 / m C s . Here, m is the effective modal mass, k 1 is the inertial Eddy current coupling, and P 1 and P 2 are the higher-order electromagnetic feedback and LC coupling constants for the tuned damper configuration. These lumped parameters are obtained from the electromagnetic modeling and assumed mode formulation 40 , 63 , 67 ; provided in Supplementary Eqs. ( S7 – S13) together with numerical values in Supplementary Tables S1 and S2 . Numerical computation of nested integrations The governing equations are expressed in terms of the lumped parameters m for effective mass, c for mechanical damping, k for elastic stiffness, and the electromagnetic coupling terms k 1 , k 2 , P 1 , and P 2 . As shown in Supplementary Eqs. ( S1 − 17) , many of these quantities are defined by lengthy nested integrations that arise from the electromagnetic field analysis. These formulations are required to represent the interaction between Eddy current fields, host structure vibrations, and the electrical shunt circuitry in a consistent way. To evaluate these parameters, numerical integration is performed via Gauss–Legendre quadrature in the spatial coordinates. The relevant expressions are detailed in Supplementary Eqs. ( S10 – S13) . In this approach, each definite integral is approximated by a weighted sum of the integrand evaluated at a finite number of quadrature nodes, referred to as Gaussian integration points. To validate convergence and select an appropriate number of integration points, a numerical study is performed where k 1 , k 2 , P 1 , and P 2 are computed as the number of integration points (per nested integration) is increased. The details of this study are summarized in Supplementary Fig. S5 . Based on this study, all reported simulations use 5 Gauss–Legendre integration points per dimension, with a maximum simulated relative error of 0.3%. This selection provided an appropriate balance between numerical accuracy and computational cost for the purposes of this electromagnetic coupling model. Experimental fabrication details The electromagnetic transducers are assembled from a prefabricated APW FC-6490 electrical coil and a 25.4 × 19 mm cylindrical neodymium magnet purchased from McMaster-Carr 5862K335. A customized cylindrical spacer is 3D-printed to secure the magnet coaxially within the APW coil, followed by immersion in liquid–resin epoxy with a cylindrical silicone mold. Once hardened, the transducer is sanded flat to expose the magnet surface. The finished transducer is then fixed to a 3D printed mounting bracket, secured by a second magnet as illustrated in Fig. 4b and Fig. 6b before being wired to the corresponding shunt circuitry. The analog electromagnetic shunt circuits are constructed from commercial off-the-shelf electronic components on a solderless breadboard for experimental validation and tuning. A concise overview of the shunt architecture is provided in Fig. 4c , which leverages OPA445 operational amplifiers powered by a dual-rail ± 18-V DC power supply. This configuration synthesizes an electrically tunable shunt capacitance (via potentiometer R 5 ) and negative resistance (potentiometer R 3 ). A detailed schematic of the experimental circuit hardware with nominal component values and tolerances is provided in Supplementary Fig. S4 . For the magneto-mechanical metamaterial, the locally resonant unit cells are calibrated as described in the main text. For tuning modal suppression, the PolyTec PSV-500-M scanning laser vibrometer is first used to experimentally identify the 2nd mode of the cantilevered beam. The electromagnetic resonator is then fixed in place with a 2 mm lift-off, before proceeding with RLC calibration. Once the resonances of the LC-oscillator and the beam are aligned, the negative resistance is incrementally tuned to flatten the measured transmissibility. Measurement and characterization procedures Elastic wave transmission through the metamaterial is characterized using a PolyTec PSV-400 scanning laser doppler vibrometer (SLDV). The host structure is excited via a surface-bonded piezoelectric actuator driven by a 1 V swept-sine input across the frequency range of interest. To quantify the incident wave amplitude, out-of-plane velocity measurements are acquired over a grid of 255 scan points positioned upstream of the metamaterial array (Fig. 5a ). At each scan location, ×20 signal averaging is applied to reduce the influence of ambient noise and disturbances. The resulting frequency response functions (FRFs) are spatially averaged to establish a baseline incident wave amplitude. This procedure is then repeated using an identical downstream grid to capture the transmitted wave field. The transmission coefficient, expressed in decibels, is computed as the ratio of spatially averaged transmitted to incident response magnitudes using: T dB = 20 log 10 FR F transmitted FR F incident . To adjust the bandgap frequency, the individual shunt circuits are electrically recalibrated to the new resonant frequency, and the measurements are repeated. The experimental velocity-voltage transmissibility of the cantilever beam is captured via the same SLDV system. The integrated vibrometer controller is used to drive the piezoelectric patch actuator with a 1-V frequency sweeping sine signal across the frequency range of interest. By taking the ratio of the output velocity at the beam tip to the input voltage supplied to the PZT actuator, an experimental FRF analogous to the analytical model is obtained. Use of AI large language models To improve language quality, the authors used an AI tool (OpenAI ChatGPT) to edit the author-written text for clarity, grammar, and spelling. All AI-assisted edits were reviewed by the authors to ensure that the original meaning was not altered through this refinement. Supplementary information Transparent Peer Review file (3.3MB, pdf) Supplemental material file (1.3MB, pdf) Acknowledgements The authors gratefully acknowledge support in part by the U.S. National Science Foundation (NSF) under GRFP grant 2136520 and in part by NSF contract number 1825324. Author contributions J.T. and J.D. conceptualized the research. J.D. derived the analytical and numerical models. J.D. designed the hardware and experimental approach. J.D. conducted the experiments and collected the data. J.D. and J.T. analyzed the data and interpreted the results. J.D. prepared figures and visual representations. J.D. and J.T. wrote the initial draft of the paper. R.C. and J.T. supervised the project and provided critical feedback. R.C. and J.T. secured funding for the project. All authors reviewed and edited the paper. Peer review Peer review information Communications Engineering thanks Ting-Wei Wang and Kaijun Yi for their contribution to the peer review of this work. Primary Handling Editors: [Philip Coatsworth]. A peer review file is available. Data availability The data supporting the conclusions of this study are either included within the article and the supplementary information or are available upon reasonable request. 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