Modelling and Test of an Integrated Magnetic Spring-Eddy Current Damper for Space Applications

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Article Modelling and Test of an Integrated Magnetic Spring-Eddy Current Damper for Space Applications

Efren Diez-Jimenez 1,* , Cristina Alén-Cordero 1 , Roberto Alcover-Sánchez 1 and Eduardo Corral-Abad 2

1 Mechanical Engineering Area, Signal Theory and Communications Department, Universidad de Alcalá, Ctra. Madrid-Barcelona, Km 33,66, 28805 Alcalá de Henares, Spain; [email protected] (C.A.-C.); [email protected] (R.A.-S.) 2 Department of Mechanical Engineering, Universidad Carlos III de Madrid, Av de la Universidad 30, 28911 Leganés, Spain; [email protected] * Correspondence: [email protected]

Abstract: We present the design, manufacturing, and dynamical characterization of a mechanical suspension made by a passive magnetic spring and an eddy current damper integrated into a single device. Three configurations with 2, 3, and 4 permanent magnets axially distributed with opposite polarizations are designed, simulated, manufactured, and tested. Stiffness of 2410, 2050, 2090 N/m and damping coefficient of 5.45, 10.52 and 17.25 Ns/m are measured for the 2-, 3-, and 4-magnets configurations, respectively. The magnetic suspension provides good mechanical properties combined with excellent cleanness and high reliability, which is very desirable in mechanical systems for space applications.

Keywords: vibration damping; mechanical design; magnetic spring; eddy current damper

1. Introduction Citation: Diez-Jimenez, E.; Mechanisms for space applications typically need to fulﬁl additional requirements due Alén-Cordero, C.; Alcover-Sánchez, R.; to the extreme operational environment. Deployment mechanisms for antenna [1], solar Corral-Abad, E. Modelling and Test of panels [2], or docking elements have to be light and elastic [3] but at the same time they an Integrated Magnetic Spring-Eddy need to damp vibrations. In space, any vibrating structure needs an additional damping Current Damper for Space mechanism in order to mitigate vibrations after deployments or operations. Conventional Applications. Actuators 2021, 10, 8. damping elements as friction or viscous dampers [4] can hardly be applied in space systems https://doi.org/10.3390/act10010008 since oils, metallic chips, and other type of dust particles can damage precision optical components and delicate sensors. Therefore, other types of mechanical solutions providing Received: 14 November 2020 Accepted: 28 December 2020 spring-damping behaviour must be explored. Similar cleanliness requirements can be Published: 2 January 2021 found in mechanical systems for surgical applications [5]. Although metal springs are extensively applied in mechanical systems, their vibration Publisher’s Note: MDPI stays neu- damping capacity is limited. Moreover, they consume metal and cause vibration and noise, tral with regard to jurisdictional clai- resulting in negative effects on the reliability and stability of the system. Even more, metal ms in published maps and institutio- springs are liable to suffer from fatigue and permanent plastic deformation, reducing the nal afﬁliations. reliability of the system. Besides conventional springs, metal springs are normally con- structed as a bunch of metal wires winded up to create a multilayer coaxial strand following the helical direction. Stranded-wire helical springs show higher damping capacity and longer lifetime under certain dynamic applications [6]. However, this damping capacity Copyright: © 2021 by the authors. Li- comes from the fretting of wires among themselves, which generates chips and wear [7]. censee MDPI, Basel, Switzerland. Thus, these type of metal springs, even in dampers, can generate issues for optical space This article is an open access article instruments. Therefore, it can be worthwhile to explore more exotic solutions in order to distributed under the terms and con- provide mechanical components with elastic response and damping capacity adequate for ditions of the Creative Commons At- application in space mechanisms deployment. tribution (CC BY) license (https:// creativecommons.org/licenses/by/ Innovative solutions based on magnets can be an option for space mechanisms applica- 4.0/). tions. High performant magnetic systems appeared after the development of high magnetic

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product magnets based on Nd-Fe-B alloys [8,9]. This type of new magneto-mechanisms have in common some unique features: cleanness, oil-free, reliability, low wear, no power consumption, long life, and low noise [10–20]. Among these new magneto-mechanisms, magnetic springs are widely used as an auxiliary element of energy harvesting systems [21–23]. In magnetic springs, the damping is negligible and the magnetic spring only acts as a stiffness element. Other applications of magnetic springs are in combination with active magnetic suspension [24–28]. In these cases, they provide contactless support of at least one degree of freedom and again without damping. A magnetic spring has been also been used in negative stiffness vibration dampers [29,30]. However, these solutions are not valid for space applications where cleanness is required. Another option are magnetorheological dampers, but although they offer very good mechanical performance in terms of damping and reliability [31,32], they still require oils for their operations and thus, are not very convenient for space applications. Eddy current dampers are a good solution as they are clean and efficient dampers. Eddy current dampers were successfully used in space mechanisms [33,34]. In these cases, the damping element is combined with other elastic components such as a metal spring or the structure itself. Eddy current damping has also been used in other industrial applications like in rotary sensors or shock absorbers [35–37]. The most performant eddy current damper ever done is the device published in [12,13]. This outstanding eddy current damper includes a magnetic linear gear to amplify the input vibration, which maximizes, through impedance matching, the effectiveness of the damper. The damping density of this device is the largest ever achieved for an eddy current damper demonstrating a value of 8.4 MN s m−4, even operating at high temperatures. In the present work, we propose a simpler device that integrates a passive magnetic spring and an eddy current damper in a single device. The key point is to get the benefit of the moving magnets used in the magnetic spring to couple their magnetic field with a conductive element to provide damping capacity. More specifically, we propose to use ring-shaped NdFeB magnets inserted on an aluminium rod, which acts as both an eddy current dissipater and plain bearing guide. In this way, the whole device can be more compact, lighter, oil-free and reliable, therefore very adequate for space applications. In this article, the mechanic and magnetic design of the device is described in detail. A dynamic model considering eddy current and Coulomb damping is presented and analyzed. A prototype of the magnetic parts of the devices was manufactured and assembled. The experimental static and dynamic characterization provide a validation of the model used for the dynamic response and behaviour of the device. This information can be useful for considering this device in future space and other type of mechanical applications.

2. Mechanical and Electromagnetic Design and Analysis 2.1. Mechanical Design The design of the device is simple and straightforward, as shown in Figure1. Two spherical ends, (1) and (5), which are used for external mechanical connections, enclose the magnetic spring and the eddy current damper. An inner cylindrical aluminum rod (3) serves as guide for the magnets’ motion as well as a conductive element for the eddy current dissipation. A set of ring-shaped magnets (2) are distributed along this aluminum rod. The number of magnets can be easily changed from 4, to 3 and 2 within the same device, depending on the damping and stiffness requirement. The top end is connected to the top magnet through a hollowed cylindrical linking part (4). The hollowed cylinder is needed to allow the motion of the magnets from the magnetic spring and to host the conductive layer for extreme spring operation positions. Any external mass or element can be connected to the two spherical ends through two corresponding bolts or axles. Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 3 of 18 Actuators 2021, 10, 8 3 of 18

FigureFigure 1. 1. 3D3D CAD CAD model model of of the the device device in in four four magnets magnets configuration. conﬁguration. (1) (1)—top—top spherical spherical end, end, (2) (2)NdFeB NdFeB N38H N38H magnets, magnets, (3) conductive(3) conductive aluminum aluminum inner inner rod, rod, (4)linking (4) linking hollowed hollowed rod, rod and, (5)and bottom (5)spherical bottom end. spherical end.

TheThe whole device waswas designeddesigned to to be be as as light light and and compact compact as possibleas possible and and to maximize to max- imizethe damping the damping capacity. capacity. The design The design allows allows three three magnetic magnetic topologies topologies with with two, tw three,o, three and, andfour four magnets magnets aligned aligned along along the the conductive conductive rod rod in in alternantalternant magnetization directions directions.. EmptyEmpty axial axial space space in in the the linking linking rod rod permits permits the the full full compression compression of of the the magnetic magnetic spring. spring. DifferentDifferent operation points points and and preload preload conditions conditions could could be defined be defined by using by using limiting limiting bulks bulksin the in linking the linking rod. Therod. inner The inner cylindrical cylindrical rod is rod also is hollowedalso hollowed to alleviate to alleviat mass,e mass, but with but a thickness large enough to generate all the tangential eddy currents that maximize the with a thickness large enough to generate all the tangential eddy currents that maximize damping; see Figure1. the damping; see Figure 1. The device has a total size of 204 mm length and 51.4 mm diameter, including top The device has a total size of 204 mm length and 51.4 mm diameter, including top and bottom ends. Dimensions are shown in Figure2 for the four magnets configuration. and bottom ends. Dimensions are shown in Figure 2 for the four magnets configuration. The main material in the device is aluminum 7075. It permits a reduced total mass of only The main material in the device is aluminum 7075. It permits a reduced total mass of only 517 g in case of four magnets. Permanent Magnets are made of NdFeB with quality grade 517 grams in case of four magnets. Permanent Magnets are made of NdFeB with quality of N38H with axial magnetization direction. According to the manufacturer, the quality grade of N38H with axial magnetization direction. According to the manufacturer, the assures a magnetic remanence of 1.23–1.26 T and a coercivity of 899 kA/m at a maximum quality assures a magnetic remanence of 1.23–1.26 T and a coercivity of 899 kA/m at a operational temperature of 120 ◦C (limit for most space applications). It is important to maximum operational temperature of 120 °C (limit for most space applications). It is im- note the 0.05 mm diametrical clearance between magnets and aluminum inner rod that portant to note the 0.05 mm diametrical clearance between magnets and aluminum inner assures the correct linear guiding while preventing the rotation of magnets; see Figure2. rodDetailed that assures list of materials, the correct weight, linear andguiding dimensions while preventing are given inthe Table rotation1. of magnets; see FigureBesides 2. Detailed themechanical list of materials, and geometricalweight, and design,dimensions specific are given detailed in Table analysis 1. linking magnetic behavior with dynamical parameters are done. Stiffness, eddy current damping, and friction Coulomb damping were estimated during the design phase.

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FigureFigure 2. 2.CrossCross-section-section view view of ofthe the devices devices including including main main dimensions. dimensions.

TableTable 1. 1.PartPart list. list. Part Main Dimensions (mm) Material Weight (g) Part Main Dimensions (mm) Material Weight (g) Top and bottom ends Aluminum 7075 107 Top and bottom ends Aluminum 7075 107 PermanentPermanent MagnetMagnet NdFeBNdFeB N38H N38H 49 InnerInner conductive conductive rod rod AluminumAluminum 7075 7075 47 LinkingLinking hollowed rod rod AluminumAluminum 7075 7075 60 TOTALTOTAL for for 2 2 magnets magnets configurationconfiguration 419 TOTALTOTAL for for 3 3 magnets magnets configurationconfiguration 468 TOTALTOTAL for for 4 4 magnets magnets configurationconfiguration 517

Besides the mechanical and geometrical design, specific detailed analysis linking 2.2. Simulation of Stiffness—k magnetic behavior with dynamical parameters are done. Stiffness, eddy current damping, and frictionSelection Coulomb of the damping magnets were and conductive estimated during rod element the design was donephase. after magnetic simulation. The objective of the design was to maximize the stiffness and damping capacity 2.2.of Simulation the system of while Stiffness keeping—k a low total weight. Moreover, other considerations such as manufacturability and simplicity in assembly were taken into account. Selection of the magnets and conductive rod element was done after magnetic simu- All the simulations were performed using ANSYS Electromagnetic Desktop software lation. The objective of the design was to maximize the stiffness and damping capacity of v2019 R2. This is a Finite Element Model (FEM) software for electromagnetics solutions. the system while keeping a low total weight. Moreover, other considerations such as man- It includes a magnetostatics and transient solver. For simulation of the stiffness, the ufacturability and simplicity in assembly were taken into account. magnetostatics solver is used. The magnetostatics solution is achieved following two All the simulations were performed using ANSYS Electromagnetic Desktop software Maxwell’s equations: v2019 R2. This→ is→ a Finite E→lement Model (FEM) software for electromagnetics solutions. It includes∇ × aH magnetostatics= J and ∇·B and= 0 transient with the followingsolver. For relationship simulation forof the each stiffness, material: the magnetostatics solver is used. The→ magnetostatic→ → s solution→ is achieved→ following two Max- B = µ H + M = µ ·µ ·H + µ ·M , (1) well’s equations:   0 0 r 0 p  H = J  B = 0 where B is the magnetic and field density, with theH is following the magnetic relationship field intensity, for eachJ is thematerial: current density, M µ µ p is the permanent magnetization퐵⃗ = 휇0(퐻⃗⃗ + 푀⃗⃗0 )is= a휇0 constant⋅ 휇푟 ⋅ 퐻⃗⃗ + of휇0 permeability⋅ 푀⃗⃗ 푝, of vacuum and(1) r corresponds to the relative permeability. The solver obtains the magnetic field distribution whereproduced B is the by magnetic a spatial distributionfield density, of H objects is the withmagnetic permanent field intensity, magnetization J is the [ 38current] and a density,combination Mp is the of DC permanent current densities.magnetizationµ All the0 simulationsis a constant were of permeability run on a computer of vacuum with andIntel µ r correspo Core i4-4690nds to and the 8 relative Gb of RAM permeabili memory.ty. The solver obtains the magnetic field dis- tributionThe produced FEM model by a is spatial based ondistribution 2D axisymmetric of objects geometric with permanent as the symmetry magnetization of the device[38] allows to do. By using 2D models, the computation time is drastically reduced. Static

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Actuators 2021, 10, 8 and a combination of DC current densities. All the simulations were run on a comput5 of 18er with Intel Core i4-4690 and 8 Gb of RAM memory. The FEM model is based on 2D axisymmetric geometric as the symmetry of the device allows to do. By using 2D models, the computation time is drastically reduced. Static simulationssimulations of of two, two, three, three and, and four four ring-shaped ring-shaped magnets magnets were were done done in order in order to simulate to simulate the magnetic field, as shown in Figure3. The magnetic field generated has minimum values the magnetic field, as shown in Figure 3. The magnetic field generated has minimum val- in the middle of the separation distance of the magnets because magnets are oriented in ues in the middle of the separation distance of the magnets because magnets are oriented opposite poles configuration. The repulsion of the magnets by pairs is what generates the in opposite poles configuration. The repulsion of the magnets by pairs is what generates stiffness of the system. When there is no load, magnets tend to be in the largest separation the stiffness of the system. When there is no load, magnets tend to be in the largest sepa- distance, limited by bulks limits of the mechanical structure. ration distance, limited by bulks limits of the mechanical structure.

FigureFigure 3. 3.2D 2D axisymmetric axisymmetric FEM FEM results results for for the the magnetic magnetic ﬁeld field in in four four magnets magnets conﬁguration configuration (50 (50 mm mm outer diameter). outer diameter).

OneOne ofof thethe main targets targets of of any any mechanical mechanical system system for for spaces spaces is to is optimize to optimize the com- the compactnesspactness of ofthe the system, system, i.e. i.e.,, to to achieve achieve the the desired performance performance with with the the minimum minimum size/volume.size/volume. WeWe used used a a parameter parameter 2D 2D axisymmetric axisymmetric model model to obtainto obtain the the Force Force and and magnet mag- volumenet volume ratio ratio as a as function a function of the of the outer outer diameter diameter (see (see Figure Figure4). 4 From). From this this analysis, analysis we, we determineddetermined that that an an optimal optimal outer outer diameter diameter for for the the magnet magnet would would be be around around 50 50 mm. mm. From the magnetic ﬁeld simulation, in a post-processing step, forces between the different magnets for different separation distances are calculated by the virtual forces method. The most useful value is the force acting on the magnet at the top of the stack since this is the force that the system could exert against an external load. From these force calculation results, the stiffness of the system as a function of the distance between magnets poles can be derived. Next, Figure5 presents the relationship between repulsion force and separation distance of magnets faces for the three conﬁgurations: 2, 3, and 4 magnets. It is not possible to just simulate a pair of magnets and extrapolate since linear superposition Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 6 of 18

0.004

) 3 0.003

Actuators 2021, 10, 8 6 of 18 0.002

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principle, widely used in mechanics, is not applicable to magnetic ﬁelds. The results for Force / Volume (N/mm 0.001 the stiffness are also shown in Figure5.

0.004 50 100 150 200 250 300 350 Magnet outer diameter (mm)

) 3 Figure0.003 4. Simulation results of force/volume ratio for different magnet outer diameters (inner diameter and magnet thickness were fixed parameters).

From the magnetic field simulation, in a post-processing step, forces between the dif- 0.002 ferent magnets for different separation distances are calculated by the virtual forces method. The most useful value is the force acting on the magnet at the top of the stack since this is the force that the system could exert against an external load. From these force Force / Volume (N/mm 0.001 calculation results, the stiffness of the system as a function of the distance between magnets poles can be derived. Next, Figure 5 presents the relationship between repulsion force and separation50 distance100 of magnets150 faces200 for250 the three300 configuration350 s: 2, 3, and 4 magnets. It is not possible to just simulateMagnet outer a pair diameter of magnets (mm) and extrapolate since linear superposi-

tion principle, widely used in mechanics, is not applicable to magnetic fields. The results Figure 4. Simulation results of force/volume ratio for different magnet outer diameters (inner Figurefor the 4. stiffness Simulation are results also shown of force in/volume Figure ratio 5. for different magnet outer diameters (inner di- ameterdiameter and and magnet magnet thickness thickness were were fixed ﬁxed parameters). parameters).

From the magnetic field simulation, in a post-processing step, forces between the different magnets for different separation distances are calculated by the virtual forces method. The most useful value is the force acting on the magnet at the top of the stack since this is the force that the system could exert against an external load. From these force calculation results, the stiffness of the system as a function of the distance between magnets poles can be derived. Next, Figure 5 presents the relationship between repulsion force and separation distance of magnets faces for the three configurations: 2, 3, and 4 magnets. It is not possible to just simulate a pair of magnets and extrapolate since linear superposition principle, widely used in mechanics, is not applicable to magnetic fields. The results for the stiffness are also shown in Figure 5.

Figure 5.5. ((a)a) Load capacity and (b) (b) stiffness stiffness calculation calculation for for different different magnets magnets separation separation distances distances andand differentdifferent magnetsmagnets conﬁgurations.configurations.

It is signiﬁcant that the stiffness of the magnetic spring is not linear, but it progressively increases with decrease of distance between the magnets. This can be explained since magnetic force interaction of two dipoles depend inversely on the distance to the fourth power. It should be noted that stiffness and maximum load is quite similar for all three cases. Intermediate magnets, as they all have the same quality, just provide a continuity of the force interaction between the top and bottom magnets. From Figure5, we can determine that a static force of 19.36 N must be applied in order to keep a distance of 10 mm with four magnets conﬁguration as intended from design. This leads to a stiffness of 1936 N/m in the surroundings of 10 mm separation distance. Figure 5. (a) Load capacity and (b) stiffness calculation for different magnets separation distances and different magnets configurations.

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It is significant that the stiffness of the magnetic spring is not linear, but it progressively increases with decrease of distance between the magnets. This can be explained since magnetic force interaction of two dipoles depend inversely on the distance to the fourth power. It should be noted that stiffness and maximum load is quite similar for all three cases. Intermediate magnets, as they all have the same quality, just provide a continuity of the force interaction between the top and bottom magnets. Actuators 2021, 10, 8 From Figure 5, we can determine that a static force of 19.36 N must be applied in order7 of 18 to keep a distance of 10 mm with four magnets configuration as intended from design. This leads to a stiffness of 1936 N/m in the surroundings of 10 mm separation distance. AsAs an an additional additional result result of of the the simulation, simulation, a a larger larger number number of of magnets magnets was was considered. considered. MaximumMaximum expected force waswas simulatedsimulated for for up up to to eight eight magnets magnets in in opposite opposite magnetization magnetiza- tiondirections. directions The. The behavior behavior is shown is shown in Figurein Figure6. The6. The maximum maximum force force oscillates oscillates for for 2 to2 to 6 6magnets magnets and and then, then, it it gets gets constant constant around around 180 180 N N for for a larger a larger number number of magnets.of magnets. While While the thenumber number of magnetsof magnets is reduced is reduced (2, 3(2, and 3 and 4), there4), there is a is direct a direct inﬂuence influence between between the topthe andtop andbottom bottom magnets magnets among among themselves. themselves However,. However, when the when number the ofnumber magnets of increases, magnets thein- creases,inﬂuence the between influence the between top and bottomthe top magnetsand bottom is reduced, magnets affecting is reduced only, affecting the more only adjacent the moreones. adjacent ones.

220

210

200

190

180

Maximum Force (N) 170

160

2 4 6 8 Number of magnets

Figure 6. Maximum force simulation for different numbers of magnets with alternative magnetization Figure 6. Maximum force simulation for different numbers of magnets with alternative magnetiza- tiondirections. directions.

2.3. Simulation of Eddy Current Damping—ced 2.3. Simulation of Eddy Current Damping—ced When a magnetic field varies inside an electrical conductor, the motion generates eddy currentsWhen in a the magnetic electrical field conductor. varies inside These currentsan electrical rotates conductor, in such a the path motion that an generates opposing eddymagnetic current fields in is the created, electrical appearing conductor. the so Th calledese currents Lorentz rotates forces. in These such forcesa path resultthat an in opposingdamping magnetic effect of the field magnets is created motion, appearing and produces the so heatcalled in Lorentz the electrical forces conductor.. These forces The resulttotal energyin damping transferred effect toof thethe conductormagnets motion is equal and to theproduces variation heat of in kinetic the electrical energy of con- the ductormagnets.. The Power total energy losses pertransferred unit mass to canthe becondu calculated,ctor is equal under to assumptions the variation of of uniform kinetic energymaterial, of uniformthe magnets. magnetic Power field, losses and per null unit skin mass effect; can as be proportional calculated, u to:nder assumptions of uniform material, uniform magnetic field, and null skin effect; as proportional to: P ∝ a· f 2 (2) 푃 ∝ 푎 ∙ 푓2 (2) wwherehere PP isis the the power power lost lost per unit mass (W/kg), ff isis the the frequency frequency (Hz) (Hz) of of the the oscillation, oscillation, oror variation, variation, of of the the magnetic magnetic field field applied applied and and aa isis a a constant constant depending depending on on the the material material conductivity,conductivity, magnetic magnetic field field of the magnet and geometrical dimensions. EquationEquation (2) (2) is is valid valid on onlyly in inthe the quasi quasi-static-static conditions, conditions, where where the thefrequency frequency of the of magnetthe magnet movement movement is low is enough low enough not to notgenerate to generate skin effect; skin in effect; such ina way such that a way electro- that magneticelectromagnetic waves wavesfully penetrates fully penetrates the material. the material. For the For case the of case the ofproposed the proposed device, device, skin depthskin depth is estimated is estimated in the range in the of range 10 mm, of10 much mm, larger much than larger the thanarea thewhere area eddy where currents eddy currents will be generated. Therefore, the assumption of Equation (2) is correct. Power losses of a viscous damper can be described as:

P = F ·v (3) L

where FL is damping force (Lorentz force), and v is the velocity of the moving suspended mass. By linking mechanical losses and eddy current losses, we can demonstrate that:

2 a· f = FL·v (4) Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 8 of 18

will be generated. Therefore, the assumption of Equation (2) is correct. Power losses of a viscous damper can be described as:

푃 = 퐹퐿 ∙ 푣 (3)

where FL is damping force (Lorentz force), and v is the velocity of the moving suspended mass. By linking mechanical losses and eddy current losses, we can demonstrate that: Actuators 2021, 10, 8 8 of 18 2 푎 ∙ 푓 = 퐹퐿 ∙ 푣 (4) Moreover, as oscillatory motion frequency is directly proportional to the amplitude of the linearMoreover, speed asas oscillatory푣 = 퐴 ∙ 2 ∙ motion휋 ∙ 푓, we frequency can determine is directly that proportionalthe damping toforce the- amplitudespeed ra- of tio isthe a linearconstan speedt value, as vced=, dependingA·2·π· f , we on can the determine electromagnetic that the behavior damping of force-speed the magnet: ratio is a constant value, ced, depending on the electromagnetic behavior of the magnet: 퐹퐿 푐푒푑 = (5) 푣FL ced = (5) Dynamic simulations of the magnet along thev conductive rods were done using a transientDynamic 2D FEM simulationssolver. Different of the speeds magnet were along simulated. the conductive For the dynamic rods were simulations, done using a justtransient a single magnet 2D FEM along solver. the Different aluminum speeds rod is were included simulated. since Foreddy the currents dynamic are simulations, signifi- cantjust only a single in the magnet surroundings along the of aluminumthe magnet, rod thus is included magnetic since interaction eddy currents of adjacent are significant mag- netsonly is negligible. in the surroundings Figure 7 presents of the magnet, the Lorentz thus force magnetic and linear interaction speed of for adjacent a specific magnets case is as annegligible. example. Figure With 7the presents asymptotic the Lorentz value of force the andforce, linear we can speed determine for a specific the damping case as an coefficient.example. By With simulating the asymptotic Lorentz value force of at the different force, we speeds, can determine we have thefound damping that for coefficient. all of them,By the simulating damping Lorentz force-speed force ratio at different is constant speeds, with wea total have value found of 5.15 that Ns/m. for all This of them, value the appliesdamping for the force-speed two magnets ratio case. is constant For three with magnets a total valueconfiguration of 5.15 Ns/m., the damping This value coeffi- applies cientfor is the 10.30 two Ns/m magnets and case.for the For four three magnets magnets configur configuration,ation, the thedamping damping coefficient coefficient is is 15.4510.30 Ns/m. Ns/m Thus, and it foris expected the four magnetsthat the prototype configuration, with the four damping magnets coefficient will dissipate is 15.45 faster Ns/m. thanThus, the prototype it is expected with thattwo theor three prototype magnets with since four a magnetslarger amount will dissipate of eddy currents faster than are the generatedprototype in total. with two or three magnets since a larger amount of eddy currents are generated in total.

0.50 0.10 5.4 0.45 0.09 5.2 0.40 0.08 5.0 0.35 0.07 4.8 0.30 0.06 4.6 0.25 0.05 4.4 0.20 0.04

Speed (m/s) Speed 4.2

Lorentz Force (N) Force Lorentz 0.15 0.03 Lorentz Force (N) 4.0 0.10 0.02 Speed (m/s) 3.8 DampingCoefficient (Ns/m) 0.05 Damping Coefficient (Ns/m) 0.01 3.6 0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.05 time (s)

FigureFigure 7. Results 7. Results of the of tran the transientsient FEM FEM model model for the for thedamping damping force force and and damping damping ratio ratio calculation. calculation.

2.4. Estimation of Coulomb Friction Damping Coefﬁcient—cCou 2.4. Estimation of Coulomb Friction Damping Coefficient—cCou Inner conductive aluminum rods serve as an eddy current dissipater but also as a Inner conductive aluminum rods serve as an eddy current dissipater but also as a mechanical guide for the magnets’ linear motion. Linear guiding unavoidably generates mechanical guide for the magnets’ linear motion. Linear guiding unavoidably generates friction forces when moving. Thus, it is necessary to analyze the expected friction forces friction forces when moving. Thus, it is necessary to analyze the expected friction forces during the damping of the magnet motion. during the damping of the magnet motion. The resistance friction force, Ff, dissipates energy as WCou = 4 ·Ff ·X, where X is the displacementThe resistance of thefriction moving force, element. Ff, dissipates The energy energy dissipated as 푊퐶표푢 by = the4 ∙ friction퐹푓 ∙ 푋, dampingwhere X forceis the displacement of the moving element. The energy dissipated by the friction damping can be linked with an equivalent viscous damping coefﬁcient for Coulomb friction, cCou, forceusing can thebe linked oscillatory with angularan equivalent speed ωviscousas [39 ]:damping coefficient for Coulomb friction, cCou, using the oscillatory angular speed ω as [39]: 4 ·Ff c = (6) Cou π ·ω·X

For the free vibration test, oscillatory angular speed ω will correspond to natural radian frequency and displacement will be considered as ±2 mm. Thus, in order to estimate damping coefﬁcient cCou, we need to estimate friction force. Friction force only appears if magnets frets against the inner rod. In an ideal situation, where magnets are perfectly aligned axially and radially, the force acting on the magnets in the radial direction would be zero, leading to a null normal force, thus no friction. Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 9 of 18

4 ∙ 퐹푓 푐 = (6) 퐶표푢 휋 ∙ 휔 ∙ 푋 For the free vibration test, oscillatory angular speed ω will correspond to natural radian frequency and displacement will be considered as ±2 mm. Thus, in order to estimate damping coefficient cCou, we need to estimate friction force. Actuators 2021, 10, 8 Friction force only appears if magnets frets against the inner rod. In an ideal situation,9 of 18 where magnets are perfectly aligned axially and radially, the force acting on the magnets in the radial direction would be zero, leading to a null normal force, thus no friction. How- However,ever, as there as there is a small is a small 0.05 0.05mm mmdiametric diametric clearance clearance between between the theinner inner conductive conductive rod rodouter outer radius radius and andthe themagnet magnet inner inner radius, radius, magnetic magnetic torque torque between between magnets magnets tends tends to tilt to tiltmagnets magnets creating creating a normal a normal force force between between the the magnet magnet and and the the inner inner conductive conductive rod. rod. By simplesimple geometrics,geometrics, it cancan bebe demonstrateddemonstrated thatthat aa diametricdiametric clearanceclearance ofof 0.050.05 mmmm allows the magnetsmagnets to rotaterotate anan angleangle ofof maximummaximum 0.550.55◦° inin aa directiondirection perpendicularperpendicular toto thethe revolutionrevolution axis.axis. Therefore,Therefore, magnetostaticsmagnetostatics simulations of the torquetorque actingacting onon eacheach moving magnet for a 0.550.55◦° rotationrotation werewere done,done, asas shownshown inin FigureFigure8 8.. These These torques torques are are transformedtransformed to to normal normal forces forces between between the the magnets magnets and and inner inner conductive conductive rod rod by byusing using the therelation relation of 푇 of=T퐹=푁 ∙F푑N푓· d(pairf (pair of offorces). forces). These These normal normal forces forces multiplied multiplied by by an estimated estimated frictionfriction coefficientcoefficient betweenbetween aluminumaluminum andand NdFeBNdFeB inin solidsolid lubricatedlubricated conditionsconditions givegive thethe totaltotal frictionfriction forcesforces actingacting onon thethe device.device. TheThe simulationsimulation waswas donedone forfor eacheach configurationconfiguration (2,(2, 3,3, andand 44 magnets).magnets). TheThe 3D3D simulationsimulation modelmodel usedused forfor torquetorque calculationcalculation inin 44 magnetsmagnets configurationconfiguration inin shownshown in in Figure Figure8. 8 Results. Results of of the the calculations calculations of Coulombof Coulomb friction friction damping damp- coefficientsing coefficients are summarizedare summarized in Table in Table2. 2.

Figure 8.8. 4 magnets 3D staticstatic model forfor simulationssimulations torquetorque aroundaround X.X.

TableTable 2.2. EstimationEstimation ofof CoulombCoulomb frictionfriction damping.damping.

CommonCommon ParametersParameters Rotation angle 0.55° Friction Coeff. 0.05 Rotation angle 0.55◦ Friction Coeff. 0.05 Pair of forces arm 5 mm X displacement ±2 mm Pair of forces arm 5 mm X displacement ±2 mm Configuration 2 Magnets 3 Magnets 4 Magnets TorqueConﬁguration magnet 1 (mNm) 2 Magnets3.6 3 Magnets3.5 4 Magnets5.8 TorqueTorque magnet magnet 1 (mNm) 2 (mNm) 3.6- 3.52.9 5.82.2 TorqueTorque magnet magnet 2 (mNm) 3 (mNm) -- 2.9- 2.27.5 Torque magnet 3 (mNm) - - 7.5 TotalTotal Normal Normal force force (N) (N) 0.720.72 1.281.28 3.13.1 TotalTotal axial axial Friction Friction force (N)force (N) 0.0500.050 0.080.08 0.2170.217 NaturalNatural radian radian frequency frequency (rad/s) (rad/s) 31.3031.30 31.3031.30 31.3031.30 Damping coefﬁcient (Ns/m) 1.02 1.821 4.41

By using Equation (4), the total damping coefﬁcients are estimated for each case. We have analyzed all the static and dynamic parameters determining the expected performance of the suspension. A summary of the expected dynamical parameters is presented in Table3 for each conﬁguration operating at 10 mm distance magnet separation. Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 10 of 18

Damping coefficient (Ns/m) 1.02 1.821 4.41

By using Equation (4), the total damping coefficients are estimated for each case. We have analyzed all the static and dynamic parameters determining the expected performance of the suspension. A summary of the expected dynamical parameters is presented Actuators 2021, 10, 8 10 of 18 in Table 3 for each configuration operating at 10 mm distance magnet separation.

Table 3. Simulated dynamics parameter summary. Table 3. Simulated dynamics parameter summary. Load k —Stiffness ced—Eddy Current Damping ccou—Coulomb Friction Damping Configuration/Topology (N)Load (N/m)k—Stiffness c —Eddy(Ns/m) Current Damping c —Coulomb(Ns/m) Friction Damping Configuration/Topology ed cou 2 Magnets Configuration 22.36(N) 2236(N/m) 5.15 (Ns/m) 1.02(Ns/m) 3 Magnets2 Magnets Configuration Configuration 18.51 22.36 1851 223610.30 5.151.82 1.02 4 Magnets3 Magnets Configuration Configuration 19.46 18.51 1946 185115.45 10.304.41 1.82 4 Magnets Configuration 19.46 1946 15.45 4.41 2.5. Dynamical Model 2.5.The Dynamical dynamical Model model of the device follows the conventional damped harmonic oscillator equTheations dynamical as applicable model for of theany devicespring- followsviscous thedamping conventional suspension. damped It include harmonics stiffnessoscillator provided equations by the as magnetic applicable spring, for any eddy spring-viscous current viscous damping damping suspension. given Itby includes the magnetstiffness-conductive provided rod by magnetic the magnetic interaction spring, and eddy equivalent current viscous viscous damping Coulomb given friction by the dampingmagnet-conductive due to mechanical rod fretting magnetic of interactionthe magnet andagainst equivalent the conductive viscous rod. Coulomb It has been friction considereddamping that due oscillation to mechanical amplitude fretting is ofvery the limited magnet so against that stiffness the conductive can be rod.considered It has been constantconsidered within that the oscillation motion range. amplitude Therefore, is very the limited linear so model that isstiffness valid. can Moreover, be considered the modconstantel includes within the corresponding the motion range. suspended Therefore, mass them. As linear the system model has is valid. only one Moreover, Degree the Of Freedommodel includes (DoF), x the axis corresponding—vertical direction, suspended the model mass m. is Assimple, the system and it can has be only described one Degree as shownOf Freedom in Figure (DoF), 9. x axis—vertical direction, the model is simple, and it can be described as shown in Figure9.

Figure 9. 1 DoF damped harmonic oscillator model. Figure 9. 1 DoF damped harmonic oscillator model. The balance of forces for the damped harmonic oscillator model is then:

The balance of forces for the damped harmonic dxoscillatord2 modelx is then: F = F − kx − c = m (7) ∑ d 푑푥dt 푑2dt푥 2 ∑ 퐹 = 퐹 − 푘푥 − 푐 = 푚 (7) 푑 푑푡 푑푡2 In the case of a sinusoidal driving force, Fd, this can be rewritten into the form: In the case of a sinusoidal driving force, 퐹푑, this can be rewritten into the form: d2x dx 1 2 + + 2 = ( ) 푑 푥 2 2ξω푑푥 ω0 x 1 F0 sin ωt (8) dt dt 2 m (8) 2 + 2휉휔 + 휔0 푥 = 퐹0sin (휔푡) q 푑푡 푑푡 푚 where ω = k is the undamped angular frequency of the oscillator or natural radian 0 푘 m wherefrequency, 휔0 = √ξ = is the√c undampedis the damping angular ratio, frequencyF is the of amplitude the oscillator of the or driving natural motion, radian and 푚 · 0 푐 2 m k frequency,ω is the 휉 frequency= is of the the damping sinusoidal ratio, driving F0 is motion.the amplitude of the driving motion, and 2√푚∙푘 ω is the frequencyA general of solution the sinusoidal is the sum driving of a motion steady. state part that is not depending on the initial conditions and a transient part that depends on initial conditions. The general solutions depend only on the driving amplitude F0, driving frequency ω, undamped angular frequency ω, and the damping ratio ξ. The steady-state solution returns as proportional to the driving force with phase shift:

F x(t) = 0 sin(ωt + ϕ) (9) mZmω Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 11 of 18

A general solution is the sum of a steady state part that is not depending on the initial conditions and a transient part that depends on initial conditions. The general solutions depend only on the driving amplitude F0, driving frequency ω, undamped angular frequency ω, and the damping ratio 휉. Actuators 2021, 10, 8 The steady-state solution returns as proportional to the driving force with phase11 shift of 18 :

퐹0 푥(푡) = sin(휔푡 + 휑) (9) 푚푍푚휔 q 2 1 2 Z ( ) + 1 ( 2 − 2) where m = 2ω0ζ 2 ω2 ω20 ω2 2 is the absolute value of the impedance or linear where 푍푚 = √(2휔0휁) + (휔 − 휔 ) is the absolute value of the impedance or linear 휔2 0 = 2ωω0ζ + response function, and ϕ arctan 22휔휔20휁 nπ is the phase of the oscillation relative to ω −ω0 response function, and 휑 = 푎푟푐푡푎푛 ( 2 2) + 푛휋 is the phase of the oscillation relative to the driving force. 휔 −휔0 the driving force. Transmissibility represents the ratio of the amplitude of the force transmitted to the Transmissibility represents the ratio of the amplitude of the force transmitted to the supporting structure to that of the exciting force. From previous equations, the vibration transmissibilitysupporting structure from topto that to bottom of the endexciting can beforce. described From previous as: equations, the vibration transmissibility from top to bottom end can be described as: ! 2ωω2휔0ζ휔0휁 ϕ =휑 arctan= 푎푟푐푡푎푛 ( +) +nπ푛휋 (10)(10) 2 −2 2 2 ω 휔 ω−0 휔0

ByBy usingusing thisthis model,model, accelerationacceleration measurementsmeasurements andand aa knownknown suspendedsuspended mass; mass; we we cancan determinedetermine dampingdamping ratioratio ofof thethe systemsystem and and its its natural natural frequency frequency experimentally. experimentally.

3.3. PrototypePrototype ManufacturingManufacturing andand TestbenchTestbench Set-Up Set-Up 3.1.3.1. PrototypePrototype ManufacturingManufacturing andand AssemblyAssembly AA simpliﬁedsimplified prototypeprototype was was manufactured manufactured and and assembled. assembled. Permanent Permanent magnets magnets made made ofof NdFeBNdFeB N38HN38H werewere purchasedpurchased from from HKCM HKCM Company Company (Germany), (Germany), while while the the conductive conductive innerinner aluminumaluminum rodrod andand baseplatebaseplate werewere manufacturedmanufactured atat UniversidadUniversidad dede AlcalAlcaláá mechan-mechan- icalical workshop.workshop. Figure Figure 10 10 shows shows a aphotograph photograph of ofthe the different different parts parts in the in theassembled assembled con- conﬁguration.figuration. Accurate Accurate tolerance tolerance adjustments adjustments between between the magnets the magnets’’ inner inner radius radius and inner and innercylinder cylinder were were done done in order in order to to provide provide smooth smooth displacement, displacement, reducing reducing friction while while avoidingavoiding wedge wedge blockings. blockings. A A clearanceclearance of of just just 0.050.05 mmmm betweenbetween the themagnet’s magnet’s inner inner radius radius andand rodrod outer outer radius radius is is achieved. achieved. Adapted Adapted top top and and bottom bottom ends ends were were 3D 3D printed printed including includ- threadeding threaded holes holes to attach to attach the accelerometer the accelerometer and theand suspended the suspended mass. mass.

FigureFigure 10.10.Parts Parts andand assembledassembled prototype. prototype. (1)—adapted (1)—adapted top top end, end, (2) (2) NdFeB NdFeB N38H N38H magnets, magnets, (3) (3) con- ductiveconductive aluminium aluminium inner inner rod, rod, (4) threaded (4) threaded M4holes M4 holes for sensors for sensors and (5)and bottom (5) bottom end. end.

acquisition system, and a software to register and analyze the measurements. The test prototype was attached through the bottom end to a wider rigid baseplate. An uniaxial accelerometer was connected to the top end of the prototype. Above the top end, a suspended mass m of 2 kg was placed. The accelerometer was powered through a coaxial wire connected to the current power source. Then, by using BNC connectors, the Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 12 of 18

3.2. Measurement System Set-Up and Data Analysis Procedure The measurement test bench was composed of a baseplate, a suspended mass, an accelerometer, a current source and signal conditioning for the accelerometer, a data acquisition system, and a software to register and analyze the measurements. Actuators 2021, 10, 8 The test prototype was attached through the bottom end to a wider rigid baseplate.12 of An 18 uniaxial accelerometer was connected to the top end of the prototype. Above the top end, a suspended mass m of 2 kg was placed. The accelerometer was powered through a coaxial wire connected to the current power source. Then, by using BNC connectors, the output outputvoltage voltage of the accelerometers of the accelerometers was read was by read a data by aacq datauisition acquisition card and card the and data the were data wereregis- registeredtered and treated and treated in a personal in a personal computer. computer. All the All equipment the equipment are shown are shown in Fig inure Figure 11. 11.

FigureFigure 11.11. TestTest bench: bench: (1)—prototype(1)—prototype in in 4 4 magnets magnets conﬁguration, configuration, (2) (2) baseplate, baseplate, (3) (3) vertical vertical direction direction accelerometer, (4) current source, (5) data acquisition card, (6) personal computer and (7) 2 kg accelerometer, (4) current source, (5) data acquisition card, (6) personal computer and (7) 2 kg load load mass. mass.

TheThe accelerometeraccelerometer usedused waswas anan IEPEIEPE uniaxialuniaxial 3035B3035B modelmodel fromfrom DYTRANDYTRAN withwith aa sensitivitysensitivity of of 100 100 mV/g, mV/g, frequency frequency response response of of 0.5 0. to5 to 10,000 10,000 Hz, Hz, and and 2.5 g2.5 of grams mass. Currentof mass. sourceCurrent was source also fromwas DYTRAN,also from DYTRAN, model E4114B1, model with E4114B1, an adjustable with an output adjustable current output between cur- 2rent and between 20 mA. 2 Measurements and 20 mA. Measurements were taken usingwere taken a DAQ using system a DAQ USB-6211 system fromUSB-6211 National from InstrumentsNational Instruments and registered and registered by means by ofmeans a personal of a personal computer computer and Labview and Labview software. soft- Measurementware. Measurement of the of voltage the voltage provided provided by the by the accelerometers accelerometers were were done done using using a a fixed fixed numbernumber ofof 10,00010,000 samplessamples atat aa samplingsampling frequencyfrequency raterate ofof 11 kHz,kHz, i.e.,i.e.,10 10 ss of of continuous continuous registering,registering, FigureFigure 11 11.. StaticStatic teststests were were used used in in order order to to determine determine the the stiffness stiffness at differentat different magnet magnet separation separa- distance.tion distance. The procedure The procedure was straightforward: was straightforward: addition addition of different of different masses masses above the above device the anddevice by and measuring by measuring the separation the separation distance, distance, load andload stiffnessand stiffness were were obtained. obtained. A similar A sim- procedureilar procedu wasre usedwas used for the for three-magnets the three-magnets configuration. configuration. TheThe dynamicdynamic teststests werewere donedone inin freefree vibration vibration mode. mode. AnAn initialinitial manualmanual excitationexcitation waswas appliedapplied inin thethe mass,mass, pushingpushing thethe massmass fromfrom itsits initialinitial positionposition toto aa lowerlower position,position, aa separationseparation distancedistance ofof 88 mmmm betweenbetweenmagnets. magnets. Then,Then, atat thisthis position,position, thethe registrationregistration waswas started,started, andand thethe massmass releasedreleased toto vibratevibrate freely.freely. TheThe samesame testtest procedureprocedurewas was donedone forfor allall thethe configurations.configurations. FreeFree vibrationsvibrations can can be be used used to to measure measure properties properties of of a dynamic a dynamic system. system. After After applying apply- aing certain a certain impulse impulse excitation, excitation, by using by using a hammer a hammer or manually, or manually, natural nat frequencyural frequency mode mode was activated.was activated. Then, Then, by means by means of the of accelerometer,the accelerometer, we canwe determinecan determine the displacement,the displacement, by integratingby integrating twice, twice, from from the pointthe point where where the structurethe structure was was excited. excited. TheThe resultsresults areare aa graphgraph similarsimilar toto thethe illustrativeillustrativegraph graph shown shown in inFigure Figure 12 12.. From From this this measurement,measurement, twotwo importantimportant quantitiesquantities areare derived.derived. TheThe firstfirst importantimportant elementelement isis thethe periodperiod ofof oscillation, oscillation which, which is is the the time time between between two two peaks. peaks. As As the the signal signal is supposed is supposed to be to periodic,be periodic, it is it often is often best best to estimate to estimateT as: T as: t − t T = n 0 (11) n

where tn is the time at which the nth peak occurs, as shown in Figure 12. Secondly, logarithmic decrement. This can be deﬁned as follows: x(t ) δ = log n (12) x(tn+1) Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 13 of 18

푡푛−푡0 푇 = (11) 푛 Actuators 2021, 10, 8 13 of 18 where 푡푛 is the time at which the nth peak occurs, as shown in Figure 12. Secondly, logarithmic decrement. This can be defined as follows:

푥(푡푛) where x(tn) is a displacement located훿 = at푙표푔 the( nth peak.) In principle, by selecting any(12) two 푥(푡 ) close peaks, and calculating δ, we can get the same푛+1 answer, whichever peaks you choose. However,where 푥(푡 it푛) is is often a displacement more precise located to obtain at theδ nthusing peak the. In next principle, formula: by selecting any two close peaks, and calculating δ, we can get the same answer, whichever peaks you choose. However, it is often more precise to obtain1 훿 usingx(t0 the) next formula: δ = log (13) n x(tn) 1 푥(푡0) 훿 = 푙표푔 ( ) (13) 푛 푥(푡 ) From T and δ, we can deduce ωn and ξ as follows:푛

From T and δ, we can deduce 휔푛 and 휉 as follows√ : δ 4π2 + δ2 ξ = √ ωn = 2 2 (14) 2훿 2 √4휋 T+ 훿 휉 = 4π + δ 휔푛 = (14) √4휋2 + 훿2 푇 From Equation (6) definitions, we can thus obtain stiffness k and damping coefficient c. From Equation (6) definitions, we can thus obtain stiffness k and damping coefficient c. All the conﬁgurations (with 2, 3, and 4 magnets) were mounted and tested in order to All the configurations (with 2, 3, and 4 magnets) were mounted and tested in order characterize dynamically their behavior. to characterize dynamically their behavior. Displacement x(t) x(t) x(t) x(t) time

t t1 t2 t3 t4

T FigureFigure 12.12. DisplacementDisplacement versus versus time time in in free free vibration vibration damped damped oscillations. oscillations.

4.4. TestTest ResultsResults 4.1.4.1. StiffnessStiffness Static Test Results StiffnessStiffness for each configurationconﬁguration was was obtained obtained from from the the load load vs. vs. separation separation distance distance curve. As stated, masses were progressively added against the set of magnets up to the Actuators 2021, 10, x. https://doi.org/10.3390/xxxxxcurve. As stated, masses were progressively added against the set of magnets up to the limit14 of 18 limitof 0 mm of 0 distance, mm distance, Figure Figure 13, where 13, wherethe magn theets magnets’’ poles encounter poles encounter their corresponding their corresponding ones. ones.

FigureFigure 13.13. Four-magnetFour-magnet conﬁgurationconfiguration underunder aa staticstatic verticalvertical loadloadof of240 240N. N.

Figure 14 presents the results of the load vs. separation distance measurements. A maximum admissible load of 240 N can be achieved by the two-magnet configuration. As expected from simulation results, maximum load for 2 and 4 magnets configuration are the highest, while maximum load for 3 magnets configuration is slightly lower, 180 N. Deviations from simulation results are not larger than 8%, which can be explained by dif- ferences between magnetic properties of permanent magnets in reality versus ideal properties in the simulation. Load measured at 10 mm separation distance is 29.5 N for 2 magnets configuration and 22.5 N for 3 and 4 magnets configuration. Based on this 10 mm separation distance measurement, we selected the load mass for dynamic tests. Moreover, from these measurements, we can retrieve the stiffness, summarized in Table 4.

2 magnets EXPERIMENTAL 200 3 magnets EXPERIMENTAL 4 magnets EXPERIMENTAL 150

100 Load (N)

0 0 2 4 6 8 10 12 14 16 18 20 distance (mm)

Figure 14. Load versus separation distance for 2, 3, and 4 magnets configuration.

4.2. Damping Dynamic Free Vibration Test Results Damping coefficients were obtained from dynamic free vibration measurements. Ac- celeration measurements after several dynamic excitations for 4 magnets configuration are presented in Figure 15a. Raw acceleration measurements were numerically filtered by applying a low pass filter with a cut-off frequency of 100 Hz, Figure 15b. Both curves are

Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 14 of 18

Actuators 2021, 10, 8 14 of 18 Figure 13. Four-magnet configuration under a static vertical load of 240 N.

Figure 14 presents the results of the load vs. separation distance measurements. A Figure 14 presents the results of the load vs. separation distance measurements. A maximum admissible load of 240 N can be achieved by the two-magnet configuration. As maximum admissible load of 240 N can be achieved by the two-magnet configuration. expected from simulation results, maximum load for 2 and 4 magnets configuration are As expected from simulation results, maximum load for 2 and 4 magnets configuration theare highest the highest,, while while maximum maximum load load for 3 for magnets 3 magnets configuration configuration is slightly is slightly lower, lower, 180 180N. DeviationsN. Deviations from from simulation simulation results results are not are notlarger larger than than 8%, 8%,which which can be can explained be explained by dif- by ferencesdifferences between between magnetic magnetic properties properties of permanent of permanent magnets magnets in reality in reality versus versus ideal prop- ideal ertiesproperties in the in simulation. the simulation. Load measured Load measured at 10 mm at 10 separation mm separation distance distance is 29.5 N is for 29.5 2 Nmag- for nets2 magnets configuration configuration and 22.5 and N 22.5for 3 N and for 4 3 magnets and 4 magnets configuration. configuration. Based on Based this 10 on mm this separation10 mm separation distance distancemeasurement, measurement, we selected we the selected load mass the loadfor dynamic mass for tests. dynamic Moreover, tests. fromMoreover, these frommeasurements these measurements,, we can retrieve we can the retrieve stiffness, the summarized stiffness, summarized in Table 4. in Table4.

2 magnets EXPERIMENTAL 200 3 magnets EXPERIMENTAL 4 magnets EXPERIMENTAL 150

100 Load (N)

0 0 2 4 6 8 10 12 14 16 18 20 distance (mm)

FigureFigure 14. 14. LoadLoad versus versus separation separation distance distance for for 2, 2, 3 3,, and and 4 4 magnets magnets configuration. configuration. Table 4. Measured dynamics parameters summary versus simulated values. 4.2. Damping Dynamic Free Vibration Test Results c—Damping Load at 10 mmDamping Distance coefficients were obtainedk—Stiffness from dynamic free vibration measurements. Ac- Configuration Coefficient celeration(N) measurements after several(N/m) dynamic excitations for 4 magnets configuration (Ns/m) are presented in Figure 15a. Raw acceleration measurements were numerically filtered by Measure.applying a Simul. low pass filter Measure.with a cut-off frequency Simul. of 100 Hz, Measure. Figure 15b. Both Simul. curves are 2 Magnets 29.5 22.36 2410 2236 5.45 6.17 3 Magnets 22.5 18.51 2050 1851 10.52 12.12 4 Magnets 22.9 19.46 2090 1946 17.25 19.86

4.2. Damping Dynamic Free Vibration Test Results Damping coefficients were obtained from dynamic free vibration measurements. Acceleration measurements after several dynamic excitations for 4 magnets configuration are presented in Figure 15a. Raw acceleration measurements were numerically filtered by applying a low pass filter with a cut-off frequency of 100 Hz, Figure 15b. Both curves are shifted to +1.6 m/s2 due to a voltage bias error. Once the measurement signals were filtered, they were integrated twice in order to obtain displacement versus time measurements. From the displacement curve, logarithmic decay and oscillation period can be derived. Using logarithmic decay and oscillation period, damping ratio, natural frequency can be calculated and therefore, we obtain stiffness and total damping coefficient (eddy current damping plus Coulomb friction damping). A similar procedure was applied for the 2- and 3-magnets configurations. The results are listed in Table4. Actuators 2021, 10, x. https://doi.org/10.3390/xxxxx 15 of 18

shifted to +1.6 m/s2 due to a voltage bias error. Once the measurement signals were filtered, they were integrated twice in order to obtain displacement versus time measurements. From the displacement curve, logarithmic decay and oscillation period can be derived. Using logarithmic decay and oscillation period, damping ratio, natural frequency

Actuators 2021, 10, 8 can be calculated and therefore, we obtain stiffness and total damping coefficient 15(eddy of 18 current damping plus Coulomb friction damping). A similar procedure was applied for the 2- and 3-magnets configurations. The results are listed in Table 4.

FigureFigure 15 15.. AccelerationAcceleration measurements measurements treatment treatment procedure procedure.. (a ()a R)aw Raw dir directect acceleration acceleration measure- measure- mentsments;; ( bb)) L Lowow pass filtered ﬁltered signal.

AsAs shown shown in in Table Table 44,, thethe measurementmeasurement resultsresults areare inin agreementagreement withwith simulations,simulations, with deviations lower lower than than 8% for stiffness and deviations lower than 14% for damping Actuators 2021, 10, x. https://doi.org/10.3390/xxxxxcoefficient measurements. 16 of 18 coefficient measurements. 4.3. Transmissibility Curve Table 4. Measured dynamics parameters summary versus simulated values. a separationThe transmissibility distance of 10 curvemm. Damping for each configurationratios are different was calculateddepending usingon the Equation configu- (11) and presented in Figure 16. The measured natural frequency ω is 4.98c—Damping Hz (31.4 rad/s) ration.Load Damping at 10 mm ratios Distance, 휉, are calculated ask 0.03,—Stiffness 0.08, and 0.12 for0 2, 3, and 4 magnets Configuration common for all configurations. This can be explained since load mass wasCoefficient adjusted to set a configuration,(N) respectively. (N/m) separationThe four distance-magnets of 10configuration mm. Damping has a ratios larger are damping different ratio depending and therefore, on(Ns/m) the resonance configuration. peakDampingMeasure. is lower ratios, thanξ, for are the calculatedSimul. rest of configura as 0.03,Measure. 0.08,tions. and Transmission 0.12Simul. for 2, 3, of and vibration 4Measure. magnets decays configuration, expo-Simul. 2 Magnets nentiallyrespectively.29.5 above natural frequency.22.36 2410 2236 5.45 6.17 3 Magnets 22.5 18.51 2050 1851 10.52 12.12 4 Magnets 22.9 19.46 2090 1946 17.25 19.86

4.3. Transmissibility Curve The transmissibility curve for each configuration was calculated using Equation (11) and presented in Figure 16. The measured natural frequency ω0 is 4.98 Hz (31.4 rad/s) common for all configurations. This can be explained since load mass was adjusted to set

Figure 16. 16. TransmissibilityTransmissibility curve curve from from measured measured dynamic dynamic parameters. parameters.

5. Conclusions It can be worthwhile to explore more exotic solutions in order to provide mechanical components with elastic response and damping capacity, but at the same time clean and reliable for space mechanisms applications. In this work, we propose to integrate a passive magnetic spring and an eddy current damper into a single device. The key point is to get the benefit of the moving magnets used in the magnetic spring to couple their magnetic field with a conductive element to provide damping capacity. This way, the device can be compact, lighter, and suitable for space applications. Three configurations with 2, 3, and 4 magnets axially distributed and in opposite polarizations were designed, simulated, manufactured, and tested. A stiffness of 2410, 2050, 2090 N/m and damping coefficient of 5.45, 10.52 and 17.25 Ns/m was measured for the 2- , 3- and 4-magnets configurations, respectively. This type of suspension demonstrates a specific stiffness performance of the same order of magnitude as those of mechanical conventional suspensions. On the other hand, specific measured damping is an order of magnitude lower. In any case, this magnetic suspension provides good mechanical behavior combined with cleanness and reliability, which are properties highly sought after in mechanical engineering for space developments.

Author Contributions: Conceptualization, E.D.-J.; methodology, E.D.-J. and C.A.C.; state of the art, R.A.-S., mechanical design, E.D.-J. and R.A.-S.; manufacturing and assembly, C.A.C. and E.D.- J.; simulation, E.D.-J.; validation test, E.D.-J. and R.A.-S.; formal analysis, E.C.-A.; literature review, R.A.-S.; data curation, R.A.-S. and E.C.-A.; writing—original draft, E.D.-J. and C.A.C.; writing— review and editing, C.A.C. and R.A.-S. supervision, E.D.-J.; project administration, E.D.-J.; funding acquisition, E.D.-J. All authors have read and agreed to the published version of the manuscript. Funding: The research leading to these results has received funding from the Spanish Ministerio de Economía y Competitividad under the Plan Estatal de I+D+I 2013–2016, grant agreement n° ESP2015-72458-EXP. Data Availability Statement: Data sharing not applicable Acknowledgments: The authors recognize the work of Alba Martínez Pérez in preparation of the figures.

Actuators 2021, 10, 8 16 of 18

The four-magnets conﬁguration has a larger damping ratio and therefore, resonance peak is lower than for the rest of conﬁgurations. Transmission of vibration decays expo- nentially above natural frequency.

5. Conclusions It can be worthwhile to explore more exotic solutions in order to provide mechanical components with elastic response and damping capacity, but at the same time clean and reliable for space mechanisms applications. In this work, we propose to integrate a passive magnetic spring and an eddy current damper into a single device. The key point is to get the benefit of the moving magnets used in the magnetic spring to couple their magnetic field with a conductive element to provide damping capacity. This way, the device can be compact, lighter, and suitable for space applications. Three configurations with 2, 3, and 4 magnets axially distributed and in opposite polarizations were designed, simulated, manufactured, and tested. A stiffness of 2410, 2050, 2090 N/m and damping coefficient of 5.45, 10.52 and 17.25 Ns/m was measured for the 2-, 3- and 4-magnets configurations, respectively. This type of suspension demonstrates a specific stiffness performance of the same order of magnitude as those of mechanical conventional suspensions. On the other hand, specific measured damping is an order of magnitude lower. In any case, this magnetic suspension provides good mechanical behavior combined with cleanness and reliability, which are properties highly sought after in mechanical engineering for space developments.

Author Contributions: Conceptualization, E.D.-J.; methodology, E.D.-J. and C.A.-C.; state of the art, R.A.-S., mechanical design, E.D.-J. and R.A.-S.; manufacturing and assembly, C.A.-C. and E.D.-J.; simulation, E.D.-J.; validation test, E.D.-J. and R.A.-S.; formal analysis, E.C.-A.; literature review, R.A.-S.; data curation, R.A.-S. and E.C.-A.; writing—original draft, E.D.-J. and C.A.-C.; writing— review and editing, C.A.-C. and R.A.-S. supervision, E.D.-J.; project administration, E.D.-J.; funding acquisition, E.D.-J. All authors have read and agreed to the published version of the manuscript. Funding: The research leading to these results has received funding from the Spanish Ministerio de Economía y Competitividad under the Plan Estatal de I+D+I 2013–2016, grant agreement n◦ ESP2015-72458-EXP. Data Availability Statement: Data sharing not applicable. Acknowledgments: The authors recognize the work of Alba Martínez Pérez in preparation of the figures. Conflicts of Interest: The authors declare that there is no conflict of interest regarding the publication of this paper. All results data are available from the corresponding author upon request.

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