Eddy current damping is commonly modeled using linear viscous-equivalent formulations. However, for finite-width conductor plates, relative motion between permanent magnets and the conductor causes the electromagnetic interaction region to approach the plate edges, where the effective conducting area decreases, giving rise to a displacement-dependent nonlinear damping mechanism. This paper investigates edge-induced nonlinear eddy-current damping and its influence on dynamic behavior, using a bilateral-plate eddy-current tuned mass damper (BP-ECTMD) as a representative configuration. An analytical model is developed by explicitly accounting for the translating integration bounds associated with finite conductor plates in a magnet-fixed coordinate system. The resulting damping force exhibits a pronounced nonlinear dependence on relative displacement, characterized by a two-stage, S-shaped decay. This behavior is efficiently approximated by a compact logistic-type function suitable for nonlinear dynamic analysis. The analytical predictions are validated against three-dimensional finite-element simulations for different air gap thicknesses, with errors generally below 10%. Nonlinear dynamic analyses show that linearized damping models are adequate only for small vibration amplitudes. At larger displacements, edge-induced effects significantly reduce the effective damping and alter vibration response characteristics. In the numerical example, if the nonlinear BP-ECTMD is designed using the optimal damping coefficient of the linear TMD, its peak structural response reaches 1.655 times that of the linear TMD; after re-optimization, the best nonlinear design is obtained at 1.5 times this damping amplitude, reducing the peak response to 0.968 times that of the linear TMD. These results demonstrate that edge-induced nonlinear damping is an intrinsic feature of plate-type eddy-current dampers and must be explicitly considered in nonlinear mechanical modeling.
AkounGYonnetJ-P (1984) 3D analytical calculation of the forces exerted between two cuboidal magnets. IEEE Transactions on Magnetics20(5): 1962–1964. https://doi.org/10.1109/tmag.1984.1063554
2.
BaeJ-SHwangJ-HRohJ-H, et al. (2012) Vibration suppression of a cantilever beam using magnetically tuned-mass-damper. Journal of Sound and Vibration331(26): 5669–5684. https://doi.org/10.1016/j.jsv.2012.07.020
3.
BaeJ-SParkJ-SHwangJ-H, et al. (2018) Vibration suppression of a cantilever plate using magnetically multimode tuned mass dampers. Shock and Vibration Bhatti AQ2018(1): 3463528. https://doi.org/10.1155/2018/3463528
4.
BauerJGronebergDA (2016) Measuring spatial accessibility of health care providers – introduction of a variable distance decay function within the floating catchment area (FCA) method. PLOS ONE Deribe K11(7): e0159148. https://doi.org/10.1371/journal.pone.0159148
5.
BourquinFCarusoGPeigneyM, et al. (2014) Magnetically tuned mass dampers for optimal vibration damping of large structures. Smart Materials and Structures23(8): 085009. https://doi.org/10.1088/0964-1726/23/8/085009
6.
CadwellLH (1996) Magnetic damping: analysis of an eddy current brake using an airtrack. American Journal of Physics64(7): 917–923. https://doi.org/10.1119/1.18122
DuJDuJBaoC, et al. (2025) Theoretical model and shaking table experiment of eddy current–enhanced friction pendulum tuned mass damper. The Structural Design of Tall and Special Buildings34(2): e2211. https://doi.org/10.1002/tal.2211
9.
EbrahimiBKhameseeMBGolnaraghiMF (2008) Design and modeling of a magnetic shock absorber based on eddy current damping effect. Journal of Sound and Vibration315(4–5): 875–889. https://doi.org/10.1016/j.jsv.2008.02.022
10.
EbrahimiBKhameseeMBGolnaraghiF (2009) Eddy current damper feasibility in automobile suspension: modeling, simulation and testing. Smart Materials and Structures18(1): 015017. https://doi.org/10.1088/0964-1726/18/1/015017
11.
FengZDengFLiangL, et al. (2025) Bilateral-plate eddy current damping: Paradigm shift for enhancing energy dissipation in vibration control. Journal of Engineering Mechanics151(5): 04025008. https://doi.org/10.1061/jenmdt.emeng-7735
12.
GaloshejerdiLHAssadiA (2025) Design and optimization of a nonlinear tuned mass damper with eddy current damping. International Journal of Structural Stability and Dynamics: 2650397. https://doi.org/10.1142/s0219455426503979
13.
HeWZhouYZengW, et al. (2025) Shaking table test and performance evaluation of spring eddy-current tuned mass damper. Structures76: 109053. https://doi.org/10.1016/j.istruc.2025.109053
14.
HealdMA (1988) Magnetic braking: improved theory. American Journal of Physics56(6): 521–522. https://doi.org/10.1119/1.15570
15.
HuangZWHuaXGChenZQ, et al. (2018) Modeling, testing, and validation of an eddy current damper for structural vibration control. Journal of Aerospace Engineering31(5): 04018063. https://doi.org/10.1061/(asce)as.1943-5525.0000891
16.
Kwan CheahSSodanoHA (2008) Novel eddy current damping mechanism for passive magnetic bearings. Journal of Vibration and Control14(11): 1749–1766. https://doi.org/10.1177/1077546308091219
17.
LavanO (2015) Optimal design of viscous dampers and their supporting members for the seismic retrofitting of 3D irregular frame structures. Journal of Structural Engineering141(11): 04015026. https://doi.org/10.1061/(asce)st.1943-541x.0001261
18.
LeiXShenLNiuH, et al. (2024) Development and application of eddy current damping-based tuned mass damper for wind-induced vibration control of transmission tower. International Journal of Structural Stability and Dynamics24(05): 2450053. https://doi.org/10.1142/s0219455424500536
19.
LiLLiB (2024) Simplified damping model of the eddy current damper with a double-annular-plate configuration. Journal of Sound and Vibration570: 118102. https://doi.org/10.1016/j.jsv.2023.118102
20.
LiSLiYMaoW, et al. (2022) A novel 500 kN axial eddy current damper using rack and gear mechanism: design, testing, and evaluation. Structural Control and Health Monitoring29(10): 3022. https://doi.org/10.1002/stc.3022
21.
LiSLiYWangJ, et al. (2022) Theoretical investigations on the linear and nonlinear damping force for an eddy current damper combining with rack and gear. Journal of Vibration and Control28(9–10): 1035–1044. https://doi.org/10.1177/1077546320987787
22.
LiuMLiSWuT, et al. (2021) Eddy-current tuned mass dampers for mitigation of wind-induced response of the noor III solar tower: design, installation, and validation. Journal of Structural Engineering147(12): 05021009. https://doi.org/10.1061/(asce)st.1943-541x.0003180
23.
LiuZWangCZhangD (2025) Study on vibration control of wind turbine with an optimised eddy current tuned rolling cylinder damper. Structural Control and Health Monitoring Rakicevic Z2025(1): 6726023. https://doi.org/10.1155/stc/6726023
24.
LuXZhangQWengD, et al. (2017) Improving performance of a super tall building using a new eddy-current tuned mass damper: improving performance of a super tall building using eddy-current TMD. Structural Control and Health Monitoring24(3): e1882. https://doi.org/10.1002/stc.1882
25.
MuallaIHBelevB (2002) Performance of steel frames with a new friction damper device under earthquake excitation. Engineering Structures24(3): 365–371. https://doi.org/10.1016/s0141-0296(01)00102-x
26.
NasarSAXiongG (1988) Determination of the field of a permanent-magnet disk machine using the concept of magnetic charge. IEEE Transactions on Magnetics24(3): 2038–2044. https://doi.org/10.1109/20.3398
27.
PuYHuangZZhangH, et al. (2023) Shock vibration control of SDOF systems with tubular linear eddy current dampers. Applied Sciences13(4): 2226. https://doi.org/10.3390/app13042226
28.
ShanJLiuJLoongCN, et al. (2021) Design and analysis of plate-type eddy-current damper with high energy-dissipation capability. Smart Structures and Systems27(5): 769–781.
SodanoHABaeJ-SInmanDJ, et al. (2005) Concept and model of eddy current damper for vibration suppression of a beam. Journal of Sound and Vibration288(4–5): 1177–1196. https://doi.org/10.1016/j.jsv.2005.01.016
31.
SodanoHABaeJ-SInmanDJ, et al. (2006) Improved concept and model of eddy current damper. Journal of Vibration and Acoustics128(3): 294–302. https://doi.org/10.1115/1.2172256
32.
SoongTTDargushGF (1997) Passive Energy Dissipation Systems in Structural Engineering. Chichester Weinheim: Wiley.
33.
VarelaWDBattistaRC (2011) Control of vibrations induced by people walking on large span composite floor decks. Engineering Structures33(9): 2485–2494. https://doi.org/10.1016/j.engstruct.2011.04.021
34.
VargasRBruneauM (2007) Effect of supplemental viscous damping on the seismic response of structural systems with metallic dampers. Journal of Structural Engineering133(10): 1434–1444. https://doi.org/10.1061/(asce)0733-9445(2007)133:10(1434)
35.
VermaMFriedlMAFinziA, et al. (2016) Multi-criteria evaluation of the suitability of growth functions for modeling remotely sensed phenology. Ecological Modelling323: 123–132. https://doi.org/10.1016/j.ecolmodel.2015.12.005
36.
WangZChenZWangJ (2012) Feasibility study of a large-scale tuned mass damper with eddy current damping mechanism. Earthquake Engineering and Engineering Vibration11(3): 391–401. https://doi.org/10.1007/s11803-012-0129-x
37.
WiederickHDGauthierNCampbellDA, et al. (1987) Magnetic braking: simple theory and experiment. American Journal of Physics55(6): 500–503. https://doi.org/10.1119/1.15103
38.
ZhangHYChenZQHuaXG, et al. (2020) Design and dynamic characterization of a large-scale eddy current damper with enhanced performance for vibration control. Mechanical Systems and Signal Processing145: 106879. https://doi.org/10.1016/j.ymssp.2020.106879
39.
ZhangJLiuYXueS (2025) Enhancing structural seismic performance with a novel variable damping inertial eddy current damper. Structures71: 108129. https://doi.org/10.1016/j.istruc.2024.108129
40.
ZhongTFengXZhangY, et al. (2022) Experimental study on the effect of EC-TMD on the vibration control of plant structure of PSPPs. Smart Structures and Systems29(3): 457–473.
