Performance Analysis of a Magnetorheological Shock Absorber Prototype Designed According to a Quasi-Static No-Slip Model

actuators

Article Performance Analysis of a Magnetorheological Shock Absorber Prototype Designed According to a Quasi-Static No-Slip Model

Danilo D’Andrea 1, Giacomo Risitano 1 and Lorenzo Scappaticci 2,*

1 Department of Engineering, University of Messina, Contrada di Dio (Sant'Agata), 98166 Messina, Italy; [email protected] (D.D.); [email protected] (G.R.) 2 Sustainability Engineering Department, Guglielmo Marconi University, via Plinio 44, 00193 Rome, Italy * Correspondence: [email protected];

Abstract: The aim of this paper is to investigate the limits of the quasi-static, no-slip approach when modeling the activation of magnetorheological (MR) devices. A quasi-static model is implemented to define the hydraulic and magnetic characteristics of an MR damper prototype. Then, an FE (Finite Element) magnetic simulation activity is carried out to validate theoretical findings, and an optimization procedure is carried out to adjust nominal geometry to actual application. Furthermore, a prototype is realized re-using the maximum number of components that constitute the existing conventional shock absorber. Finally, experimental tests at bench stands are performed. The predictable results demonstrate that neglecting the transient slipping effects, the Force–Velocity performance of the device is correlated with the model findings only for low current intensities acting in the magnetic circuit.

Keywords: magnetorheological damper; smart materials; shock absorber; FE simulation

1. Introduction Citation: D’Andrea, D.; Risitano, G.; In a motorcycle, the suspensions are the set of parts that connect the sprung and Scappaticci, L. Performance Analysis of a Magnetorheological Shock unsprung masses, allowing relative motion. The unsprung masses are all those elements Absorber Prototype Designed connected to the ground—tires, rims, the brake system, and the parts of the suspension According to a Quasi-Static No-Slip integral with them. Instead, all the components above the suspension belong to the sprung Model. Actuators 2021, 10, 13. masses, such as the chassis, engine, fairing, and driver [1]. The role of the suspension https://doi.org/10.3390/act10010013 system is crucial in the vehicle design, as it performs different tasks, and its performance requirements are often conflicting: it guarantees the contact between the vehicle tires and Received: 15 December 2020 road while isolating the vehicle body from the vibrations due to the road roughness [2]. Accepted: 8 January 2021 Mechanically, a conventional shock absorber is composed of a spring and a central Published: 11 January 2021 body that is designed as a hydraulic circuit to ensure a certain damping effect; the dynamic driving behavior can be customized by adjusting the fluid passage sections when the Publisher’s Note: MDPI stays neu- vehicle is stationary. When setting the suspension of a vehicle, it is necessary to satisfy tral with regard to jurisdictional clai- different requirements such as driving comfort and good tire grip [1]. Using traditional ms in published maps and institutio- hydraulic shock absorbers, a compromise choice must be found between comfort and nal affiliations. safety needs; comfort riding often requires a plush, underdamped shock absorbers setting, while the search for grip and traction mostly suggests overdamped characteristics. Magnetorheological (MR) fluids devices applied in the field of vehicle suspensions

Copyright: © 2021 by the authors. Li- allow dynamically adjusting damping characteristics, exploiting the physical characteristics censee MDPI, Basel, Switzerland. of the MR fluids [3–5] so as to overcome the constraint of a compromise setting. Discovered This article is an open access article by Jacob Rabinow [6], MR fluids consist of micron-sized, magnetically polarizable ferrous distributed under the terms and con- particles suspended in a carrier fluid such as oil. MR fluids are smart materials, whose ditions of the Creative Commons At- chemical–physical characteristics can vary in a controlled way, depending on the intensity tribution (CC BY) license (https:// of the magnetic field. Different design solutions of an MR damper can be chosen, according creativecommons.org/licenses/by/ to the “fluid activation” strategy (direct shear—see Figure1—or pressure-driven flow). 4.0/).

Actuators 2021, 10, 13. https://doi.org/10.3390/act10010013 https://www.mdpi.com/journal/actuators Actuators 2021, 10, x FOR PEER REVIEW 2 of 19

Actuators 2021, 10, 13 2 of 18 according to the “fluid activation” strategy (direct shear—see Figure 1—or pressure- driven flow).

FigureFigure 1.1.The The DirectDirect ShearShear activationactivation schemescheme forfor anan applicationapplication ofof magnetorheologicalmagnetorheological (MR) ﬂuidsfluids forfor automotive automotive shock shock absorbers. absorbers.

TheThe MRMR devicesdevices are mainly used used in in the the civil civil field field [7–9] [7–9 ]for for the the control control of ofstructural structural vi- vibrations,brations, in in the the automotive automotive field field [10–12], [10–12], as as well well as as some uses in the industrialindustrial fieldfield [[13,14].13,14]. LiLi etet al.al. [15[15]] developeddeveloped andand testedtested aa steeringsteering damperdamper forfor motorcyclemotorcycle use,use, alsoalso taking taking into into considerationconsideration thethe easeease ofof assembly.assembly. TheThe devicedevice byby ZhangZhang etet al.al. [[16]16] isis equippedequipped withwith aa pistonpiston with with an an internal internal bypass bypass valve valve and and a a permanentpermanent magnet. magnet. The The valve valve serves serves to toprevent prevent thethe blockageblockage ofof thethe fluidfluid inin casecase of of high high currents currents applied, applied, while while the the magnet magnet guarantees guarantees minimumminimum safetysafety magnetizationmagnetization andand aa widewide dynamic dynamic range. range. AnAn innovationinnovation in in the the motorcycle motorcycle field field is reportedis reported by Wangby Wang et al. et [17 al.] which [17] which introduced intro- aduced winding a winding integral integral with the with shock the absorber shock abso cylinderrber cylinder for automatic for automatic detection detection of the piston of the positionpiston position (IRDS). This(IRDS). system This issystem interesting, is intere as thesting, piston as the winding piston simultaneouslywinding simultaneously activates theactivates MR fluid the andMR actsfluid as and a displacement acts as a displacement sensor. sensor. Again,Again, with with regard regard to to control control systems, systems, Metered Metered et al. et[ 18al.] [18] developed developed a neural a neural network- net- basedwork-based one. These one. These allow allow a low a energylow energy demand, demand, long long service service life, life, and and a minimal a minimal use use of theof the sensors, sensors, making making it theit the most most convenient convenient system system in in economic economic terms. terms. Spelta Spelta et et al. al. [ 12[12]] reportedreported thethe studystudy of a control control system system for for se semi-activemi-active suspension suspension applied applied to toa dirt a dirt motor- mo- torcycle. Regarding shock absorbers with permanent magnets, Du et al. [19] carried out cycle. Regarding shock absorbers with permanent magnets, Du et al. [19] carried out stud- studies on the flow losses in the magnetic circuit, indicating significant differences between ies on the flow losses in the magnetic circuit, indicating significant differences between experimental results and modeling and specifying the appropriate corrective factors for experimental results and modeling and specifying the appropriate corrective factors for the case under consideration. Chen et al. [20] published studies on an MR shock absorber the case under consideration. Chen et al. [20] published studies on an MR shock absorber for vehicles, which were equipped with self-sensing speed and the ability to recover the for vehicles, which were equipped with self-sensing speed and the ability to recover the mechanical energy of vibration with transformation into electricity. The main advantages mechanical energy of vibration with transformation into electricity. The main advantages are savings in terms of costs, weight, dimensions, and, of course, energy. are savings in terms of costs, weight, dimensions, and, of course, energy. In this paper, a quick technical approach was used to define geometrical dimensions In this paper, a quick technical approach was used to define geometrical dimensions of the MR shock absorber, through which the minimum and maximum damping forces of the MR shock absorber, through which the minimum and maximum damping forces were determined. Subsequently, the hydraulic and magnetic circuits were dimensioned towere guarantee determined. the above Subsequently, forces. As the a performancehydraulic and requirement, magnetic circuits it was were established dimensioned that ato semi-active guarantee damperthe above should forces. perform As a performa a wide rangence requirement, “Force vs. Velocity”it was established (F-V) diagram that a thatsemi-active is superimposable damper should to that perform of a conventional a wide range high-end “Force vs. SuperSport Velocity’’ shock (F₋V) absorber, diagram i.e.,that equippedis superimposable with high to and that low of speed a conventional adjustment, high-end both for compressionSuperSport shock and rebound. absorber, This i.e., functionalequipped with prerogative high and represents low speed the adjustment most challenging, both for objectivecompression since, and within rebound. the fullThis rangefunctional of hydraulic prerogative adjustments represents available, the most the ch traditionalallenging shockobjective absorbers since, forwithin SuperSport the full applicationsrange of hydraulic show performances adjustments which,available, to date, the traditional are considered shock as stateabsorbers of the for art. SuperSport Moreover, weapplications also decided show to performances keep the overall which, dimensions to date, are of considered the MR prototype as state unchangedof the art. Moreo- with referencever, we also to the decided traditional to keep device. the overall The prototype dimensions building of the approach MR prototype was based unchanged on the ideawith ofreference re-using to the the maximum traditional number device. of The components prototype that building constitute approach the existing was based traditional on the passiveidea of device.re-using In the this maximum way, any implementation number of comp ononents the motorcycle that constitute could takethe placeexisting without tradi- anytional mechanical passive device. intervention In this onway, the any frame implem andentation suspension, on the at amotorcycle minimal cost. could take place withoutThe any aim mechanical of this paper intervention is to investigate on the the frame limits and of thesuspension, quasi-static, at a no-slipminimal approach cost. when modeling the activation of the MR fluid of devices that must guarantee high performance for a wide spectrum of acting forces, such as in the case of shock absorbers for motorcycle use. The quasi-static approach to modeling the activation of a MR fluid, if the slip effect is neglected, is lean and essentially requires the design of the induced magnetic

Actuators 2021, 10, 13 3 of 18

field, but what is the downside? The experiment described in the paper demonstrates that the range of use in which the device offers the required damping forces is predictable for low current intensities acting in the magnetic circuit. The results could be useful to understand the accuracy of the required mathematical model as a function of the desired F-V response, and this is a novelty in the field. Therefore, the present paper is basically an overview on the whole activity, i.e., from mathematical models to the experimental comparative testing at bench stands, highlighting the limits of the quasi-static approach with no slip. First, a quasi-static axisymmetric model is implemented in Matlab to obtain geometrical characteristics of the gap and to design the magnetic circuit. Therefore, a magnetic simulation activity is carried out to validate theoretical findings. Furthermore, an optimization procedure is carried on adjusting nominal geometry to actual application. Finally, experimental tests are proposed.

2. Materials and Methods As hinted in Section1, the philosophy of the present work is proposing a global view approach for MR dampers design: from the performance requirements to a prototype built by re-engineering a passive shock absorber, from the numerical modeling of the magnetic field to the choice of the material, until operational validation. The main activities can be summarized as following: • Determination of the gap geometry: By means of geometric parameters, estimate the force from the device as a function of the fluid volume stimulated by the magnetic field. • Calculation of the total reluctance of the equivalent magnetic circuit: By means of the definition of the equivalent RL circuit, this activity returns an estimate of the number of windings of the wire. • Calculation of the winding configuration: According to time response issues, define the configuration of the winding. • Design optimization by means magnetic simulation: Based on the theoretical findings, Finite Element Analysis (FEA) is employed for numerical modeling of magnetic field and stresses distribution according to the chosen geometry. Geometry itself is optimized to ensure functionality • Manufacturing of the prototype: the MR prototype is manufactured starting from components of a traditional, passive, shock absorber. A new piston is realized, and the main cylinder housing in which the piston is engaged is replaced to ensure the closure of the magnetic field. Once the MR device is assembled, it is tested under typical working conditions. • Final operational validation: bench tests are performed at different current intensities (including challenging limits) acting in the RL circuit, and a comparative analysis with the passive traditional shock absorber is proposed.

2.1. Design of the MR Damper The functional characteristics of a hydraulic shock absorber for high-end SuperSport purpose, such as the F-V diagram were used as a reference. So, a characterization of a traditional shock absorber available on the market (Matris M46R) was ﬁrst performed on the test bench (Figure2). The reference to a speciﬁc product does not make the work lose generality, since the top-of-the-range products of each manufacturer show similar characteristics. Moreover, with a view to re-engineering an MR shock absorber exploiting a high number of existing components taken from a traditional device, the geometric dimensions related to the cylinder housing and the useful stroke were considered settled. Actuators 2021, 10, x FOR PEER REVIEW 4 of 19

Actuators 2021, 10, 13 4 of 18 number of existing components taken from a traditional device, the geometric dimensions related to the cylinder housing and the useful stroke were considered settled.

Figure 2. “Force“Force vs. vs. Velocity’’ Velocity” (F (F-V)₋V) characteristic characteristic curves curves of the of Matris the Matris M46R M46R in compression. in compression. The clicks The clicksare an areindex an of index the hydraulic of the hydraulic brake, and brake, considering and considering a single piston a single speed, piston it is possible speed, itto isobserve possible the to observeincrease in the the increase damping in force the damping and therefore force in and the thereforedamping incoefficient, the damping which coefficient, occurs with which the passage occurs withfrom thethe passagehydraulic from open the condition hydraulic (24 openclicks) condition to the condition (24 clicks) of closed to the hydraulics condition (six of closed clicks). hydraulics (six clicks). The nominal performance forces for the MR prototype were established as follows. ReferringThe nominal to the F₋V performance curves of the forces conventional for the MR shock prototype absorber were (see establishedFigure 2), at asmaximum follows. Referringspeed, in the to the condition F-V curves of fluid of the activated conventional with maximum shock absorber current (see intensity, Figure2), the at maximum MR shock speed,absorber in should the condition offer a of force fluid equal activated to the with traditional maximum shock current absorber intensity, with thethe MRhydraulic shock absorberclosed. Instead, should the offer force a force required equal toin thethe traditionalcondition of shock deactivated absorber fluid with (zero the hydraulic current) closed.should Instead,be equal the to that force of required the conventional in the condition device of with deactivated open hydraulic fluid (zero condition. current) Assum- should being equal a maximum to that ofpiston the conventionalvelocity equal device to VMAX with = 0.5 open m/s, hydraulic the force that condition. the magnetic Assuming shock a maximumabsorber should piston exert velocity in the equal hydraulic to VMAX closed= 0.5 condition m/s, the is force FON Max that the= 4431 magnetic N and shockin the absorbercondition should of hydraulic exert inopen, the hydraulicit is FOFF Min closed = 3235 condition N. is FON Max = 4431 N and in the conditionOnce ofthe hydraulic performance open, requirements it is FOFF Min are= esta 3235blished, N. a Design Algorithm (DA) was im- plementedOnce the in Matlab performance [21] exploiting requirements the findin are established,gs of a quasi-static a Design axisymmetric Algorithm (DA) model was [22]. im- plementedEssentially, in Matlab that [DA21] exploitingrepresents thethe findingssizing of of the a quasi-static magnetic circuit axisymmetric as a function model of [ 22the]. desiredEssentially, output forces that DA and represents of the physical the sizing characteristics of the magnetic of the materials. circuit as aThe function DA worked of the desiredwith “if” output and “while” forces and loops of the(see physical Figure characteristics3) and proposed of thethe materials. geometry The of the DA gap worked as a withfunction “if” of and the “while” maximum loops force (see required Figure3 and) and minimum proposed volume the geometry of activated of the fluid, gap and as a it functioniterates the of process the maximum for the quantities force required definition and minimum to obtain the volume shortest of activated magnetization fluid, time and it iterates the process for the quantities definition to obtain the shortest magnetization of the winding. The DA received nine constants as input and three parameters to be iter- time of the winding. The DA received nine constants as input and three parameters to be ated and returned seven values that completely define the gap, piston geometry, and iterated and returned seven values that completely define the gap, piston geometry, and winding characteristics as well as the F₋V characteristics at maximum current intensity winding characteristics as well as the F-V characteristics at maximum current intensity and and zero current. zero current. Other input data were the magnetic permeability values of the MR fluid and the steel, which were necessary to characterize the magnetic circuit, the maximum current intensity applicable to the winding, the resistivity of the copper, and the curves B = B (H) and τ0 = τ0 (H) related to the fluid. In addition, other parameters were used such as the diameter of the copper wire of the winding, the number of wire passes, and the internal diameter of the piston.

Actuators 2021, 10, 13 5 of 18 Actuators 2021, 10, x FOR PEER REVIEW 5 of 19

Figure 3. Block diagram of the Design Algorithm (DA). Figure 3. Block diagram of the Design Algorithm (DA). The geometry of the gap depends on the piston geometry, magnetic properties of the materials,The geometry and geometry of the gap of the depends gap. All on thethe inputspiston andgeometry, outputs magnetic are shown properties in Table 1of. the materials, and geometry of the gap. All the inputs and outputs are shown in Table 1. Table 1. Input and output data. Table 1. Input and output data. Name Symbol Unit Name Symbol Unit Inputs Inputs Internal diameter of the cylinder Dcint mm Internal diameter of the cylinder Dcint mm External diameter of the cylinder Dcest mm External diameterPiston of strokethe cylinder Dcest c mm mm RelativePiston magnetic stroke permeability of steel c µr fe mm / RelativeAbsolute magnetic magnetic permeability permeability of of steel the fluid μr fe µa fl / H/m Magnetic permeability in vacuum µ H/m Absolute magnetic permeability of the fluid μa fl 0 H/m Resistivity of copper ρ Ω * m Magnetic permeability in vacuum μ0 H/m Constant for minimum volume activated α / ρ ResistivityMaximum of current copper intensity Imax Ω*m A Constant for minimuSolenoidm volume diameter activated α Dsol / mm Maximum Wirecurrent diameter intensity Imax Φwire A mm Number of wire passes Npass / Solenoid diameter Dsol mm Wire diameterOutputs Φwire mm 3 NumberVolume of MRwire fluid passes activated Npass Vmin / mm OutputsLength of gap g mm Gap thickness h mm Volume of MR fluid activated Vmin mm3 Gap width w mm Length of gap g mm Piston length Lpist mm GapPiston thickness diameter h Dp mm mm GapNumber width of turns w Nsp mm / Piston length Lpist mm In orderPiston to obtain diameter a short time magnetic response Dp in the gapmm [23], an economic and easily workableNumber ferromagnetic of turns material (AISI 1018) N wassp chosen for/ the piston prototype. According to the experimental B(H) curve of the AISI 1018 steel, it was set as the limit value forOther the magnetic input data induction were theBsat magnetic Fe = 1.5 T,permeabi which islity the values magnetic of the field MR saturation. fluid and the steel, which ThewereB necessarysat Fe value to was characterize a constraint the conditionmagnetic incircuit, the DA the andmaximum acted as current a control intensity parameter ap- plicableto check to thethe pistonwinding, geometry. the resistivity Whenever of the thecopper, design and flow the curves did not B lead = B (H) to a and geometry τ0 = τ0 (H) that

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related to the fluid. In addition, other parameters were used such as the diameter of the copper wire of the winding, the number of wire passes, and the internal diameter of the piston. In order to obtain a short time magnetic response in the gap [23], an economic and easily workable ferromagnetic material (AISI 1018) was chosen for the piston prototype. According to the experimental B(H) curve of the AISI 1018 steel, it was set as the limit Actuators 2021, 10, 13 value for the magnetic induction Bsat Fe = 1.5 T, which is the magnetic field saturation.6 of 18 The Bsat Fe value was a constraint condition in the DA and acted as a control parameter to check the piston geometry. Whenever the design flow did not lead to a geometry that allows compliance with the saturation limits in the gap, the tangential stress value τ0, that allows compliance with the saturation limits in the gap, the tangential stress value τ , that is the nominal τ obtainable by magnetizing the MR fluid in the gap, was decreased.0 is the nominal τ obtainable by magnetizing the MR fluid in the gap, was decreased. In the block called “Calculation of the gap geometry”, the direct shear (see Figure 1) In the block called “Calculation of the gap geometry”, the direct shear (see Figure1) activation mode for MR fluids was implemented within a while loop; exiting the while activation mode for MR fluids was implemented within a while loop; exiting the while loop was allowed if the damping force in the activated fluid condition FoutON exceeded the loop was allowed if the damping force in the activated fluid condition FoutON exceeded FONMax, which is the maximum exerted damping force for the traditional device experi- the FONMax, which is the maximum exerted damping force for the traditional device mentally observed. To obtain this result, the height of the gap g was increased in steps of experimentally observed. To obtain this result, the height of the gap g was increased in stepsa tenth of of a tentha millimeter. of a millimeter. The calculation of the minimum volume of activated fluid Vmin occurred with follow- The calculation of the minimum volume of activated fluid Vmin occurred with follow- inging equationequation [[24]:24]: ! ! η 𝜂 F𝐹on Vmin𝑉= α=𝛼∗∗ ∗∗ ∗𝐹∗ Fon ∗∗𝑉Vmax (1) τ2 F (1) y 𝜏 𝐹o f f

where the Vmin becomes a reference value (Vmin ref), which is used when changing the τ0 to where the Vmin becomes a reference value (Vmin ref), which is used when changing the τ0 to calculate the initial Vmin according to the following equation: calculate the initial Vmin according to the following equation: 2 ! τ 0max Vmin𝑉 start = V=min 𝑉re f ∗ ∗ .. (2) 2 (2) τ0

In thethe firstfirst geometrygeometry definitiondefinition attempt,attempt, allall thethe valuesvalues werewere calculatedcalculated accordingaccording toto thethe Vmin startingstarting point. point. Subsequently, Subsequently, having having a a fi firstrst attempt attempt geometry of the gap available, at the end of each cycle, VVmin waswas recalculated recalculated according according to (1) to be reused in the next step, as long as Foutout ON ON remainedremained below below the the FFONON Max Max. . Having to vary the response ofof the shock absorberabsorber inin relation toto thethe speed withwith whichwhich itit isis compressed,compressed, itit isis importantimportant thatthat thethe fluidfluid magnetizesmagnetizes toto thethe desireddesired valuevalue inin aa shortshort time.time. Since the fluid fluid needs a few tens of milliseconds milliseconds to react to the applied applied magnetic field, field, itit followsfollows thatthat itsits responseresponse timestimes dependsdepends onon thethe timetime constantconstant ((ττmgmg) ofof thethe equivalentequivalent RLRL electric circuit (see(see FigureFigure4 4).).

Figure 4. Magnetic circuit: (a) control sections,sections, ((b)) equivalentequivalent circuit.circuit.

2.2. Determination the τmg 2.2. Determination the τmg Inside the block “Choice of the configuration of the parameters to be iterated with Inside the block “Choice of the configuration of the parameters to be iterated with the criterion of the minor τmg“, the magnetic circuit was modeled. In compliance with the criterion of the minor τmg“, the magnetic circuit was modeled. In compliance with the the tangential stress required in the fluid (τ0) and the necessary Bfl value, and given the geometry of the gap, the magnetic flow φ that must take place in the magnetic circuit defined by the piston, gap, and cylinder housing is univocally defined. By the reluctance

method, one can deﬁne the number of turns Nsp:

Nsp ∗ Imax = Rtot ∗ φ (3)

where Imax is the maximum intensity current and Rtot is the total reluctance. Actuators 2021, 10, 13 7 of 18

The schematic representation of the magnetic circuit (see Figure4), which is necessary to deﬁne the reluctances of the actual application, was also revised considering the observations in [25]. Reluctance R1 and section S1 relate to the central body of the piston, while R2 and sections S2 and S3 relate to the piston wings. Rtr and Str relate to the MR ﬂuid, and R4 and S4 refer to the cylinder housing. π S = ∗ D2 (4) 1 4 sol

S2 = π ∗ Dsol ∗ g (5)

S3 = 2π ∗ Rpist ∗ g (6)

Str = 1.05 ∗ S3 (7) π S = D2 − D2 (8) 4 4 cest cint

To calculate the reluctance of the piston wings that have a variable section from S2 to S3, the reluctance expressed in inﬁnitesimal terms (9) was integrated by Dsol/2 to Rpist, which is based on the heat conduction model through the cylindrical wall.

dr Z Re dr dR = => R = (9) µa f l ∗ 2π ∗ r ∗ g Ri µa f l ∗ 2π ∗ r ∗ g

N sp ∗ φ + g Npass wire R1 = (10) µaFe ∗ S1

1 2Rpist R2 = ∗ ln (11) µa f l ∗ 2π ∗ g Dsol h Rtr = (12) µa f l ∗ Str N sp ∗ φ + g Npass wire R4 = (13) µaFe ∗ S4

As shown, Nsp appears in (10). Total reluctance depends on physical constants (absolute magnetic permeability of steel and MR ﬂuid), and the geometric values (lengths l and areas of sections S of the circuit) are all known or calculable, except for the number of turns of the winding; therefore, Equation (3) can be rewritten by specifying Nsp. Considering: Rtot = R1 + 2R2 + 2Rtr + R4 (14) One can obtain: φ(g ∗ A + B) Nsp = (15) (I − φ) ∗ φ ∗ A wire Npass where A and B are equal, respectively: ! 1 1 A = 4 + (16) ∗ ∗ 2 2 2 µaFe π Dsol µaFe ∗ π ∗ (Dcest − Dcint) ln 2Rpist/Dsol h B = + . (17) µaFe ∗ π ∗ g 1.05 ∗ µa f l ∗ π ∗ Rpist ∗ g Once the winding magnetization time was determined, an RL equivalent circuit was resolved to determine the length of the wire: Lwire = π ∗ Nsp ∗ Dsol + Npass ∗ φwire . (18) Actuators 2021, 10, x FOR PEER REVIEW 8 of 19

𝑅 =𝑅 +2𝑅 +2𝑅 +𝑅 (14) One can obtain:

𝜙(𝑔∗𝐴 +𝐵 𝑁 = 𝐴 (15) (𝐼−𝜙 ∗𝜙 ∗ 𝑁

where A and B are equal, respectively: 1 1 𝐴 =4 + (16) 𝜇 ∗𝜋∗𝐷 𝜇 ∗𝜋∗(𝐷 −𝐷

/ 𝐵= + . (17) ∗∗ .∗∗∗∗ Once the winding magnetization time was determined, an RL equivalent circuit was Actuators 2021, 10, 13 resolved to determine the length of the wire: 8 of 18

𝐿 = 𝜋∗ 𝑁 ∗ 𝐷 +𝑁 ∗𝜙. (18)

SinceSince the piston geometrygeometry andand the the winding winding characteristics characteristics are are all all determined, determined, therefore, there- Fe 1 2 4 fore,BFe was B was calculated calculated in the in control the control sections sectionsS1, S 2S, and, S , andS4. In S addition,. In addition, since since the magneticthe magnetic ﬂux fluxis constant, is constant, in correspondence in correspondence of section of sectionS3, the S3 maximum, the maximum magnetic magnetic induction induction value value of the offerromagnet the ferromagnet was obtained. was obtained. If its value If its wasvalue less was than less the thanBsat the Fe limit, Bsat Fe the limit, design the design process pro- was cessﬁnished; was finished; otherwise, otherwise, the algorithm the algorithm reduced the reducedτ0 and the restarted τ0 and fromrestarted the determination from the deter- of minationthe gap geometries. of the gap geometries.

2.3.2.3. FEA FEA and and Optimization Optimization ToTo validate the theoretical findings, findings, a a ma magneticgnetic FE FE analysis analysis was was performed. performed. A A two- two- dimensionaldimensional axisymmetric axisymmetric model model was was implemented implemented in in Ansys. According According to to the the Direct ShearShear activation activation (Figure (Figure 22),), anan FEFE model (Figure(Figure5 5)) was was designed designed to to evaluate evaluate the the magnetic magnetic fieldfield of of the the MR MR fluid fluid in in the the gap. gap. To To model model the the magnetic magnetic properties properties of of the the various various materials, materials, whichwhich were were subsequently subsequently assigned assigned to to the the appr appropriateopriate areas areas of of the model, the B(H) curve ofof the the MR MR fluid fluid was was introduced, introduced, while while for for coppe copper,r, insulation resin, resin, and steel, the relative magneticmagnetic permeability permeability μµ waswas considered.

FigureFigure 5. 5. ((aa)) Two-dimensional Two-dimensional (2D) (2D) axisymmetric axisymmetric Finite Finite Element Element (FE) (FE) model, model, (b ()b Particular) Particular of ofthe the piston,piston, ( (cc)) A A detail detail of of the the mesh. mesh.

The magnetization curve of AISI 1018 steel (cylinder housing), Fe 360 steel (piston), and MR fluid were implemented by points in the pre-processing phase of the analysis. The model consisted of about 91,000 nodes and 35,000 elements (PLANE53 of Ansys library). Boundary conditions were set as amagnetic for the outer regions. Simulation activity allowed us also to estimate the shear stress τ to be used in the direct shear model for the calculation of the force applied and to check that the magnetic induction values in the steel parts did not exceed the imposed limit. The DA of the MR shock absorber was performed by acting on the parameters to be iterated (Table1) and on the Bsat Fe magnetic induction limit. A first analysis was carried out by setting Bsat Fe = 1.4 T, keeping its constraint control limit equal to 1.5 T. The first run of the DA with the input data values shown in Table1 did not produce satisfying results; the magnetic induction values in the section S4, i.e., the cross-section of the shock absorber cylinder housing, exceeded the control limits. Since S4 is constant and univocally determined by the internal and external diameters of the cylinder, the input data were varied, increasing the Dcest diameter. By performing the DA, the solution shown in Table2 was obtained. Actuators 2021, 10, 13 9 of 18

Table 2. Program output data.

Parameter Value Unit

τ0 35.7 kPa Dsol 24 mm g 4.8 mm Npass 3 / h 0.3 mm φwire 0.57 mm

However, with these geometrical parameters, the total length of the piston exceeded the axial overall dimension available, which is equal to 25 mm. Therefore, some input parameters were varied, as shown in Table3.

Table 3. New input values for sizing.

Parameter Value Unit

Dsol 26-27-28 mm Dcest 51 mm Bsat fe 1.4 T Npass 5-8-10 / φwire 0.57 mm

To limit the piston axial length, both the number of passages (Npass) and the diameter of the solenoid were considered. The diameter of the wire has a signiﬁcant inﬂuence only on the resistance, so it was set equal to φ wire = 0.57 mm. The magnetic analysis solution showed high induction values in the control section S2 Actuators 2021, 10, x FOR PEER REVIEW 10 of 19 and in some parts of the cylinder housing (see Figure6); moreover, we noticed a difference of about 0.25 T with respect to the average values calculated by the DA.

FigureFigure 6.6. FEFE analysisanalysis ofof aa solutionsolution thatthat exceedsexceeds thethe inductioninduction limits.limits.

TheThe variationvariation ofof thethe magneticmagnetic inductioninduction waswas furtherfurther investigatedinvestigated byby varyingvarying onlyonly oneone parameterparameter atat aa time.time. MagneticMagnetic inductioninduction derivingderiving from thethe dimensionsdimensions chartchart inin TableTable3 3 waswas explored,explored, consideringconsideringN Npasspass valuesvalues ofof 10,10, 12,12, andand 1515 andand findingfinding aa slightslight deteriorationdeterioration ofof thethe magneticmagnetic fieldfield andand aa muchmuch moremore markedmarked increaseincrease inin termsterms ofof magnetizationmagnetization time.time. TheseThese trendstrends suggestedsuggested keeping keeping lower lower the the number number of of passages passagesN Npasspass.. ByBy fixingfixing thethe numbernumber ofof passagespassages NNpasspass == 8, 8, BBfefe waswas analyzed analyzed by by varying varying DDsolsol. As. As DsolD solincreased,increased, the the magnetic magnetic in- inductionduction BBfe feestimatedestimated by by DA DA decreased decreased in in the the control control sections S11 and SS44, increasedincreased inin SS33,, andand remainedremained approximatelyapproximately unchangedunchanged inin SS22.. TheThe samesame trendstrends werewere observedobserved fromfrom thethe results of the magnetic analysis (see Figure 7), which stands at average values greater than 0.2–0.3 T. Therefore, Bsat limit fe = 1.2 T was set.

Figure 7. Comparison between FE analysis with different solenoid diameters (Dsol): (a) 26 [mm]; (b) 30 [mm]; (c) 34 [mm].

The trends of some fundamental geometric dimensions and physical characteristics as a function of the Npass and Dsol parameters can be deduced from the data listed in Table 4.

Table 4. Output data as Npass and Dsol vary.

Npass h [mm] g [mm] τ0 [kPa] Lpist [mm] Nsp τmg [s] 5 0.3 4.2 38.3 22.1 120 0.47 8 0.3 4.2 38.3 16.4 112 0.58 10 0.3 4.2 38.3 14.7 110 0.86 Dsol [mm] 26 0.3 4.2 38.3 16.4 112 0.58 30 0.2 3.1 41.7 11.9 80 0.67 34 0.2 2.3 43.5 10.3 80 0.78

Actuators 2021, 10, x FOR PEER REVIEW 10 of 19

Figure 6. FE analysis of a solution that exceeds the induction limits.

The variation of the magnetic induction was further investigated by varying only one parameter at a time. Magnetic induction deriving from the dimensions chart in Table 3 was explored, considering Npass values of 10, 12, and 15 and finding a slight deterioration of the magnetic field and a much more marked increase in terms of magnetization time. Actuators 2021, 10, 13 These trends suggested keeping lower the number of passages Npass. By fixing the number10 of 18 of passages Npass = 8, Bfe was analyzed by varying Dsol. As Dsol increased, the magnetic induction Bfe estimated by DA decreased in the control sections S1 and S4, increased in S3, and remained approximately unchanged in S2. The same trends were observed from the resultsresults ofof thethe magneticmagnetic analysisanalysis (see(see FigureFigure7 ),7), which which stands stands at at average average values values greater greater than than 0.2–0.3 T. Therefore, Bsat limit fe = 1.2 T was set. 0.2–0.3 T. Therefore, Bsat limit fe = 1.2 T was set.

Figure 7. Comparison between FE analysis with different solenoid diameters (Dsol): (a) 26 [mm]; Figure 7. Comparison between FE analysis with different solenoid diameters (Dsol): (a) 26 [mm]; ((bb)) 30 30 [mm]; [mm]; ( c(c)) 34 34 [mm]. [mm].

TheThe trendstrends of of some some fundamental fundamental geometric geometric dimensions dimensions and and physical physical characteristics characteristics as aas function a function of theof theNpass Npassand andD Dsolsolparameters parameters can be deduced from from the the data data listed listed in in Table Table 4.4 .

Table 4. Output data as Npass and Dsol vary. Table 4. Output data as Npass and Dsol vary.

Npass h [mm] g [mm] τ0 [kPa] Lpist [mm] Nsp τmg [s] Npass h [mm] g [mm] τ0 [kPa] Lpist [mm] Nsp τmg [s] 5 0.3 4.2 38.3 22.1 120 0.47 5 0.3 4.2 38.3 22.1 120 0.47 88 0.30.3 4.2 38.3 38.3 16.4 16.4 112 1120.58 0.58 1010 0.30.3 4.2 38.3 38.3 14.7 14.7 110 1100.86 0.86 DsolDsol[mm] [mm] 2626 0.30.3 4.2 38.3 38.3 16.4 16.4 112 1120.58 0.58 Actuators 2021, 10, x FOR PEER REVIEW 30 0.2 3.1 41.7 11.9 80 0.6711 of 19

3430 0.20.2 2.33.1 41.7 43.5 11.9 10.3 80 800.67 0.78 34 0.2 2.3 43.5 10.3 80 0.78

TheThe solenoidsolenoid diameterdiameterD Dsolsol exhibited thethe greatestgreatest inﬂuenceinfluence onon thethe magneticmagnetic inductioninduction values,values, while the effect effect of of changing changing the the numb numberer of of passes passes mainly mainly influenced inﬂuenced the the magnetiza- magne- tizationtion time. time. By examining By examining the possible the possible combinations combinations between between Dsol variablesDsol variables from 30 from to 36 30 mm to 36and mm Npass and fromNpass 2 tofrom 5, a solution 2 to 5, a was solution obtained was only obtained by considering only by considering Dsol ≥ 32 andD Nsolpass≥ ≥ 323. Since and Na passleast≥ number3. Since aof least wire number passes guarantees of wire passes lower guarantees τmg, numerical lower analysesτmg, numerical were performed analyses werekeeping performed this constraint, keeping fully this exploiting constraint, the fully available exploiting space thein the available axial direction space in LpistMAX the axial = 25 directionmm. LpistMAX = 25 mm. ThreeThree furtherfurther casescases ( D(Dsolsol == 32,32, 34,34, 3636 mm,mm,N Npasspass = 3) were investigated,investigated, andand magneticmagnetic inductioninduction valuesvalues werewere alwaysalways lowerlower thanthan thethe limitlimit ofof 1.51.5 T,T, exceptexcept forfor thethe internalinternal cornerscorners ofof thethe pistonpiston wherewhere locallocal peakspeaks ofof aboutabout 1.81.8 TT werewere noted.noted. GivenGiven thethe locallocal charactercharacter ofof thethe phenomenonphenomenon (see(see FigureFigure8 ),8), the the results results have have been been considered considered acceptable. acceptable.

Figure 8. FE analysis with the parameter Npass = 3, which respects the induction limits: (a) Dsol = 32 Figure 8. FE analysis with the parameter Npass = 3, which respects the induction limits: (a) Dsol = 32 [mm]; ([mm];b) 34 [mm]; (b) 34 ([mm];c) 36 [mm]. (c) 36 [mm].

AccordingAccording toto thethe quasi-staticquasi-static parallelparallel plateplate model,model, forfor eacheach ofof thesethese designdesign options,options, thethe MRMR shockshock absorberabsorber showedshowed oversizedoversized F-VF-V curvescurves ifif comparedcompared toto thethe hydraulichydraulic shockshock absorberabsorber reference.reference. TheThe mostmost criticalcritical datadata werewere thethe performanceperformance atat zerozero currentcurrent intensityintensity (FMR OFF), since it did not allow the device to fully cover the speed field below 0.25 m/s, as shown in Figure 9.

Figure 9. Comparison of the F₋V curves by varying Dsol with respect to the performance of the hydraulic shock absorber.

In order to further optimize the model, the clearance between the piston and cylinder was increased, without excessively deteriorating the performance at maximum current intensity (FMR ON). In this contest, geometric solutions with Dsol = 33, 35, and 36 mm were considered, also varying the gap height in steps of 0.3 mm, which is approximately 10% of the initial value, and the gap thickness of 0.05 mm, corresponding to a variation of 0.1 mm in the piston diameter. The h-g combinations that guaranteed F-V curves with the best compromise between covering the area at low speed and achieving adequate force in the activated fluid condition were obtained with h = 0.3 mm and g = 3.4 mm (Figure 10).

Actuators 2021, 10, x FOR PEER REVIEW 11 of 19

The solenoid diameter Dsol exhibited the greatest influence on the magnetic induction values, while the effect of changing the number of passes mainly influenced the magnetization time. By examining the possible combinations between Dsol variables from 30 to 36 mm and Npass from 2 to 5, a solution was obtained only by considering Dsol ≥ 32 and Npass ≥ 3. Since a least number of wire passes guarantees lower τmg, numerical analyses were performed keeping this constraint, fully exploiting the available space in the axial direction LpistMAX = 25 mm. Three further cases (Dsol = 32, 34, 36 mm, Npass = 3) were investigated, and magnetic induction values were always lower than the limit of 1.5 T, except for the internal corners of the piston where local peaks of about 1.8 T were noted. Given the local character of the phenomenon (see Figure 8), the results have been considered acceptable.

Figure 8. FE analysis with the parameter Npass = 3, which respects the induction limits: (a) Dsol = 32 [mm]; (b) 34 [mm]; (c) 36 [mm]. Actuators 2021, 10, 13 11 of 18 According to the quasi-static parallel plate model, for each of these design options, the MR shock absorber showed oversized F-V curves if compared to the hydraulic shock absorber reference. The most critical data were the performance at zero current intensity ((FFMRMR OFF ),), since since it it did did not not allow allow the the device device to fullyfully cover thethe speedspeed ﬁeldfield belowbelow 0.25 0.25 m/s, m/s, asas shownshown inin FigureFigure9 9..

sol FigureFigure 9. ComparisonComparison of of the the F₋ F-VV curves curves by by varying varying D D withsol with respect respect to the to performance the performance of the of hy- the hydraulicdraulic shock shock absorber. absorber.

InIn orderorder toto furtherfurther optimizeoptimize thethe model,model, thethe clearance betweenbetween thethe pistonpiston andand cylindercylinder waswas increased,increased, withoutwithout excessivelyexcessively deterioratingdeteriorating thethe performanceperformance atat maximummaximum currentcurrent intensityintensity ((FFMRMR ON ).). In In this this contest, contest, ge geometricometric solutions with Dsolsol = 33, 33, 35, and 36 mm werewere considered,considered, alsoalso varyingvarying the the gap gap height height in in steps steps of of 0.3 0.3 mm, mm, which which is approximatelyis approximately 10% 10% of theof the initial initial value, value, and and the the gap gap thickness thickness of 0.05 of 0.05 mm, mm, corresponding corresponding to a variationto a variation of 0.1 of mm 0.1 inmm the in piston the piston diameter. diameter. The h-gThe combinations h-g combinations that guaranteedthat guaranteed F-V curvesF-V curves with with the best the Actuators 2021, 10, x FOR PEER REVIEW 12 of 19 compromisebest compromise between between covering covering the area the area at low at speedlow speed and and achieving achieving adequate adequate force force in the in activatedthe activated ﬂuid fluid condition condition were were obtained obtained with with h = 0.3h = mm0.3 mm and and g = 3.4g = mm3.4 mm (Figure (Figure 10). 10).

FigureFigure 10.10. F–VF–V curvescurves whenwhen parametersparameters hh andand gg vary.vary.

Finally,Finally, DDsolsol = 33 mm was consideredconsidered thethe bestbest compromise,compromise, sincesince itit allowedallowed holdingholding thethe magneticmagnetic inductioninduction limits,limits, especiallyespecially inin thethe cylindercylinder housinghousing sectionsection (see(see FigureFigure 11 11).).

Figure 11. Result of the FE analysis with Dsol = 35 [mm]; h = 0.3 [mm]; g = 3.4 [mm].

3. Manufacturing the MR Prototype For the prototyping of the magnetorheological shock absorber, a hydraulic shock absorber was used, keeping unchanged most of the overall dimensions and damping characteristics. The MR shock absorber was realized maintaining the same piston rod, piston guide, and compensation tank. The main cylinder housing was replaced by a ferromagnetic one. As for the piston, the lamellar pack was removed, keeping the disc perforated to ensure the coaxiality of the new piston with the cylinder: this feature guarantees a space through which the fluid can flow and can be magnetized. The material used for the piston is a Fe 360 steel, which was chosen for the good compromise it offers between magnetic characteristics and workability. To avoid MR fluid contact-winding and MR fluid-upper and lower surfaces of the piston, the epoxy resin produced by Dolph’s was interposed as an insulator, mixing hardener, and resin with a 10:1 ratio. The new MR piston was built in three phases: • Realization of the piston and protections (caps); • Realization of the wire winding; and • Application of the resin in parallel with the assembly of the pieces.

Actuators 2021, 10, x FOR PEER REVIEW 12 of 19

Figure 10. F–V curves when parameters h and g vary. Actuators 2021, 10, 13 12 of 18 Finally, Dsol = 33 mm was considered the best compromise, since it allowed holding the magnetic induction limits, especially in the cylinder housing section (see Figure 11).

FigureFigure 11.11. Result of the FE analysisanalysis withwith DDsolsol = = 3535 [mm];[mm]; hh == 0.30.3 [mm];[mm]; gg == 3.4 [mm].

3.3. Manufacturing the MR Prototype ForFor the prototyping prototyping of of the the magnetorheologic magnetorheologicalal shock shock absorber, absorber, a hydraulic a hydraulic shock shock ab- absorbersorber was was used, used, keeping keeping unchanged unchanged most most of of the the overall overall dimensions dimensions and and damping damping characteristics.acteristics. The MR shock absorber was realizedrealized maintainingmaintaining thethe samesame pistonpiston rod,rod, pistonpiston guide,guide, andand compensationcompensation tank. tank. The The main main cylinder cylinder housing housing was was replaced replaced by aby ferromagnetic a ferromag- one.netic Asone. for As the for piston, the piston, the lamellarthe lamellar pack pa wasck was removed, removed, keeping keeping the the disc disc perforated perforated to ensureto ensure the the coaxiality coaxiality of of the the new new piston piston with with the the cylinder: cylinder: this this feature feature guarantees guaranteesa a spacespace throughthrough whichwhich thethe fluidfluid cancan flowflow andand cancan bebe magnetized.magnetized. TheThe materialmaterial used forfor thethe pistonpiston isis aa FeFe 360360 steel,steel, whichwhich waswas chosenchosen forfor thethe goodgood compromisecompromise itit offersoffers betweenbetween magneticmagnetic characteristicscharacteristics andand workability.workability. ToTo avoidavoid MRMR fluidfluid contact-windingcontact-winding and MR fluid-upperfluid-upper and lowerlower surfacessurfaces ofof thethe piston,piston, thethe epoxyepoxy resinresin producedproduced byby Dolph’sDolph’s waswas interposedinterposed as anan insulator,insulator, mixingmixing hardener,hardener, andand resinresin withwith aa 10:1 ratio. 10:1 ratio. The new MR piston was built in three phases: The new MR piston was built in three phases: Actuators 2021, 10, x FOR PEER REVIEW 13 of 19 • Realization of the piston and protections (caps); • • RealizationRealization ofof thethe wirepiston winding; and protections and (caps); • ApplicationRealization of of the the wire resin winding; in parallel and with the assembly of the pieces. • OnOnApplication the the upper upper of surface, surface,the resin two twoin holes parallel holes were werewith made the made forassembly the for winding the of winding the wires.pieces. wires. Finally, Finally, the insu- the latinginsulating resin resin was wasapplied, applied, and and the thefollowing following were were installed: installed: internal internal spring, spring, seal, seal, O-ring, O-ring, cylindercylinder cap, cap, and and lower lower uniball uniball (Figure (Figure 12). 12). Figure Figure 13 13 shows shows the the fundamental fundamental dimensional dimensional parametersparameters of of the the MR MR shock shock absorber. absorber.

FigureFigure 12. 12. (a(a) )Piston Piston and and piston piston rod rod assembly assembly of of the the MR MR prototype prototype damper, ((b)) TheThe prototypeprototype ofof the theMR MR shock shock absorber. absorber.

Figure 13. Dimensional parameters of the MR shock absorber.

4. Experimental Testing To validate the F-V curves of the model, tests at the bench stands were performed at 300 mm/s. The test bench is a rod–crank type with adjustable stroke up to 120 mm and equipped with a data logger. The test bench is equipped with a load cell with 1000 kg of full scale, a magnetoresistive position transducer with a stroke of 200 mm, a contact thermocouple with a measuring range from −200 to +400 °C, and accuracy of 1 °C.

4.1. Zero Current Intensity Tests At a first instance, three tests with zero current intensity for the MR shock absorber were performed, with a central time of 15 s and sinusoidal forcing at 1, 2, 3, and 4 Hz were considered (Figure 14).

Actuators 2021, 10, x FOR PEER REVIEW 13 of 19

On the upper surface, two holes were made for the winding wires. Finally, the insu- lating resin was applied, and the following were installed: internal spring, seal, O-ring, cylinder cap, and lower uniball (Figure 12). Figure 13 shows the fundamental dimensional parameters of the MR shock absorber.

Actuators 2021, 10, 13 13 of 18 Figure 12. (a) Piston and piston rod assembly of the MR prototype damper, (b) The prototype of the MR shock absorber.

FigureFigure 13. 13. DimensionalDimensional parameters parameters of of the the MR MR shock shock absorber. absorber.

4.4. Experimental Experimental Testing Testing ToTo validate validate the the F-V F-V curves curves of of the the model, model, te testssts at at the the bench bench stands stands were were performed performed at at 300300 mm/s. mm/s. TheThe test test bench bench is is a arod–crank rod–crank type type with with adjustable adjustable stroke stroke up up to to 120 120 mm mm and and equipped equipped withwith a a data data logger. logger. The The test test bench bench is isequipped equipped with with a load a load cell cell with with 1000 1000 kg kgof offull full scale, scale, a magnetoresistivea magnetoresistive position position transducer transducer with with a stroke a stroke of of 200 200 mm, mm, a acontact contact thermocouple thermocouple ◦ ◦ withwith a a measuring measuring range range from from −−200200 to to +400 +400 °C,C, and and accuracy accuracy of of 1 1°C.C. 4.1. Zero Current Intensity Tests 4.1. Zero Current Intensity Tests At a ﬁrst instance, three tests with zero current intensity for the MR shock absorber At a first instance, three tests with zero current intensity for the MR shock absorber Actuators 2021, 10, x FOR PEER REVIEWwere performed, with a central time of 15 s and sinusoidal forcing at 1, 2, 3, and 4 Hz14 were of 19 were performed, with a central time of 15 s and sinusoidal forcing at 1, 2, 3, and 4 Hz were considered (Figure 14). considered (Figure 14).

FigureFigure 14.14. F-SF₋S plot ofof thethe MRMR shockshock absorberabsorber atat 00 [A],[A], withwith differentdifferent valuesvalues ofof forcingforcing frequency.frequency.

4.2.4.2. Increasing CurrentCurrent TestsTests InIn aa secondsecond instance, instance, tests tests at at 0.5 0.5 Hz Hz were were performed performed with with the the current current intensity intensity varying vary- froming from 0 to 1.80 to A 1.8 with A anwith increase an increase of 0.3 A. of Passing0.3 A. Passing from 0 to from 0.3 A,0 to it was0.3 A, possible it was to possible highlight to highlight an increase in the force exerted by the MR shock absorber, while the graphs maintained the overall shape (Figure 15).

Figure 15. Comparison between run at 0 [A] (blue) and 0.3 [A] (red); (a) F₋S; (b) F₋V.

5. Results and Discussion As it is known from the literature, such MR devices often show a non-linear behavior: in our activity, when the power supply for the magnetic circuit exceeded 0.3 A, the trends in the F₋S plane took on a pipe shape that was accentuated more and more as the current intensity increased (Figure 16). The divergence of the F₋S of the MR damper from a traditional device can be explained as follows [26]. Within a certain displacement range, the ferrite particle chains act as springs that stick both to the piston and the inner housing surface, because all particle chains along the entire radius have not yet reached the sliding angle γ [27]. The force growth due to the slip effect occurs between pre-yield and post-yield regions when ferrite chains start to slide relative to the peripheral area of the piston and relative to the outer surface of the cylinder, generating the friction force (Section C–A in Figure 16).

Actuators 2021, 10, x FOR PEER REVIEW 14 of 19

Figure 14. F₋S plot of the MR shock absorber at 0 [A], with different values of forcing frequency.

Actuators 2021, 10, 13 14 of 18 4.2. Increasing Current Tests In a second instance, tests at 0.5 Hz were performed with the current intensity varying from 0 to 1.8 A with an increase of 0.3 A. Passing from 0 to 0.3 A, it was possible to highlightan increase an in increase the force in exerted the force by theexerted MR shock by the absorber, MR shock while absorber, the graphs while maintained the graphs the maintainedoverall shape the (Figure overall 15 shape). (Figure 15).

FigureFigure 15. ComparisonComparison between between run run at 0 [A [A]] (blue) and 0.3 [A] (red); ( a) F F-S;₋S; ( b) F F-V.₋V.

5. Results and Discussion 5. Results and Discussion As it is known from the literature, such MR devices often show a non-linear behavior: As it is known from the literature, such MR devices often show a non-linear behavior: in our activity, when the power supply for the magnetic circuit exceeded 0.3 A, the trends Actuators 2021, 10, x FOR PEER REVIEWin our activity, when the power supply for the magnetic circuit exceeded 0.3 A, the trends15 of 19 in the F-S plane took on a pipe shape that was accentuated more and more as the current inintensity the F₋S increasedplane took (Figure on a pipe 16). shape that was accentuated more and more as the current intensity increased (Figure 16). The divergence of the F₋S of the MR damper from a traditional device can be explained as follows [26]. Within a certain displacement range, the ferrite particle chains act as springs that stick both to the piston and the inner housing surface, because all particle chains along the entire radius have not yet reached the sliding angle γ [27]. The force growth due to the slip effect occurs between pre-yield and post-yield regions when ferrite chains start to slide relative to the peripheral area of the piston and relative to the outer surface of the cylinder, generating the friction force (Section C–A in Figure 16).

FigureFigure 16.16. PipePipe shapedshaped F-SF₋S chart.chart.

TheThe divergencefriction force of thesteadily F-S of increases the MR damper to leve froml B (Section a traditional A–B in device Figure can 16) be within explained the astransition follows [zone26]. due to the increasing sliding area on the piston with increasing stroke. WhenWithin all particle a certain chains displacement along the entire range, radius the ferrite are in particle the sliding chains regime act as and springs the migration that stick bothprocess to the of pistonparticles and is thefinished, inner housing the friction surface, force because (Section all B–D particle in Figure chains 16) along is generated the entire radiusby the haveMR damper not yet reached[27,28]. the sliding angle γ [27]. TheThis force physical growth interpretation due to the slip of the effect observed occurs betweenforce response pre-yield was and confirmed post-yield by regionsrheom- wheneter tests ferrite on chainsMR fluid start samples to slide relative[29]. Compressi to the peripheralon behavior area is of much the piston less sensitive and relative to the to theslip outer effect, surface since the of thecompensation cylinder, generating circuit in thethis friction phase is force not (Sectionsubject to C–A cavitation. in Figure 16). TheAs the friction current force intensity steadily increased, increases the to levelcycle B became (Section increasingly A–B in Figure deformed 16) within (Figure the transition17a,b). Analyzing zone due the to F₋ theS plot increasing obtained sliding at 0.5 Hz area (Figure on the 17a), piston it is withnoted increasing that the response stroke. of the MR device changed when the power supply for the magnetic circuit exceeded 0.3 A; in the last part of the rebound stroke, damping forces seem to agree with the F₋S curve of a standard device. As a result, the increase in the supply current of the magnetic circuit causes, during the extension phase, the displacement of point B (Figure 15) toward the reversal stroke point; when this current is 1.8 A, the point of damping force restoring, in essence, coincides with the beginning of the compression phase.

Actuators 2021, 10, 13 15 of 18

When all particle chains along the entire radius are in the sliding regime and the migration process of particles is finished, the friction force (Section B–D in Figure 16) is generated by the MR damper [27,28]. This physical interpretation of the observed force response was confirmed by rheome- ter tests on MR fluid samples [29]. Compression behavior is much less sensitive to the slip effect, since the compensation circuit in this phase is not subject to cavitation. As the current intensity increased, the cycle became increasingly deformed (Figure 17a,b). Analyzing the F-S plot obtained at 0.5 Hz (Figure 17a), it is noted that the response of the MR device changed when the power supply for the magnetic circuit exceeded 0.3 A; in the last part of the rebound stroke, damping forces seem to agree with the F-S curve of a standard device. As a result, the increase in the supply current of the magnetic circuit causes, during the extension phase, the displacement of point B (Figure 15) toward the Actuators 2021, 10, x FOR PEER REVIEW 16 of 19 reversal stroke point; when this current is 1.8 A, the point of damping force restoring, in essence, coincides with the beginning of the compression phase.

FigureFigure 17. 17.MR MR shockshock absorber absorber behavior behavior at at 0.5 0.5 [Hz], [Hz], varying varying current current intensity intensity ( a(a)) F-S; F₋S; ( b(b)) F-V. F₋V.

A differentA shape different from shape the usual from one the of usual a traditional one of a traditionalshock absorber shock one absorber can also one be can also be noted in thenoted F₋V curve in the (Figure F-V curve 17b), (Figure on which 17b), the on same which reference the same points reference used can points still usedbe can still traced. It isbe well traced. known It is that well the known maxima that of the damping maxima force, of damping both in force,compression both in and compression re- and bound, consequentlyrebound, consequently tends to move tends from to the move extremes from theof the extremes velocity of range the velocity (midpoints range of (midpoints the stroke) ofto thethe stroke)center (stroke to the center reversal (stroke points). reversal points). If one considersIf one considersthe damping the damping force value force in value the invelocity the velocity range range under under exam exam (+32 (+32 mm/s/ mm/s/−30.5− mm/s),30.5 mm/s), when whenvarying varying the current the current intensit intensity,y, the values the values of force of exerted force exerted increase increase by a by a variablevariable quantity quantity from from300 to 300 500 to N, 500 unlike N, unlike the theplate-parallel plate-parallel no noslip slip model. model. This This implies implies thatthat the themodel model underestimates underestimates the theexperi experimentalmental data data for currents for currents from from 0.3 to 0.3 1.8 to A, 1.8 A, unlike unlike whatwhat happens happens at the at extremes. the extremes. Therefore, Therefore, it is only it is from only 0 from to 0.3 0 A to of 0.3 input A of current input current that that the the quasi-staticquasi-static parallel no-slip parallel model no-slip ca modeln be considered can be considered descriptive descriptive (Figure 17a,b). (Figure 17a,b). The experimentalThe experimental tests show how tests the show hysteretic how the behavior hysteretic is behaviorhighly frequency-dependent: is highly frequency-dependent: F₋V performanceF-V performance correlation between correlation the model between and the the modelexperimental and the tests experimental (Figure 18) testswhen (Figure 18) varying thewhen forcing varying frequency the forcing becomes frequency much more becomes evident. much more evident.

Figure 18. Graph F₋V: comparison between experimental data and model at 0 [A].

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Figure 17. MR shock absorber behavior at 0.5 [Hz], varying current intensity (a) F₋S; (b) F₋V.

A different shape from the usual one of a traditional shock absorber one can also be noted in the F₋V curve (Figure 17b), on which the same reference points used can still be traced. It is well known that the maxima of damping force, both in compression and rebound, consequently tends to move from the extremes of the velocity range (midpoints of the stroke) to the center (stroke reversal points). If one considers the damping force value in the velocity range under exam (+32 mm/s/−30.5 mm/s), when varying the current intensity, the values of force exerted increase by a variable quantity from 300 to 500 N, unlike the plate-parallel no slip model. This implies that the model underestimates the experimental data for currents from 0.3 to 1.8 A, unlike what happens at the extremes. Therefore, it is only from 0 to 0.3 A of input current that the quasi-static parallel no-slip model can be considered descriptive (Figure 17a,b). Actuators 2021, 10, 13 The experimental tests show how the hysteretic behavior is highly frequency-dependent:16 of 18 F₋V performance correlation between the model and the experimental tests (Figure 18) when varying the forcing frequency becomes much more evident.

FigureFigure 18.18. GraphGraph F-V:F₋V: comparisoncomparison betweenbetween experimentalexperimental datadata andand modelmodel atat 00 [A].[A].

Since the plate-parallel model is quasi-static, it is possible to compare only the extreme points of the experimental curves, which correspond to quasi-static behaviors. In fact, the velocity at extremes is closest to the quasi-static condition, being a = dv/dt = 0. It is evident that a device with the aforementioned dynamic characteristics essentially becomes unusable for vehicle applications; in this regard, it is sufﬁcient to observe that, passing for example from 0.3 to 0.6 A in the graph F-V (Figure 17a), for a stroke equal to, say, −10 mm, the damping force changes sign. This proves that the quasi-static parallel model in which the slip prediction is not included proves to be predictive only for low supply currents. The restoring force (point B) can be modeled by a modiﬁed Bouc–Wen– Baber–Nory model [28,29], which considers the slip effect. That given, our intent is to implement a reviewed model to redesign the magnetic circuit of the prototype.

6. Conclusions In order to overcome the limits of the traditional hydraulic shock absorber, a magnetorheological shock absorber has been designed and built, trying to replicate the performance of the reference traditional device. A DA was implemented based on the quasi-static no slip mathematical model of the parallel plate geometry, which is capable of providing the characteristic dimensions of the damper as output. Subsequently, a prototype of a magnetorheological damper has been realized, using a traditional shock absorber as the starting point. The comparison between the experimental and theoretical damping characteristics, with current intensity equal to zero, shows an excellent correspondence. Good correlation is maintained for low current intensities tests. In fact, the current prototype shows good performances only in the context of a reduced stroke, say from −3 to + 5mm, from 0 to 0.6 A of supply current. As the current intensity increases, the mathematical model tends to overestimate the rebound damping force while underestimating the compression one. Moreover, correlation is strictly dependent on forcing frequency. As the testing velocity and intensity of the current grown up, both the experimental F-S and F-V of the MR device show damping force lacks during both compression and rebound phases. Based on these observations, a review of the DA is in progress, implementing in the calculation both the effect of the slip, according to the Bouc–Wen model [30,31], and the determination of the contribution due to the pressure acting in the compensation tank [32] Furthermore, since it is challenging to match F-D and F-V of the traditional model for a wide range of current intensities, it is planned to develop an algorithm-based non-linear control strategy [33] exploiting the feedback loop. On balance, the no-slip parallel plate model is slender, since it is almost based on the search for the maximum and minimum damping force obtainable from τ0, neglecting the transient effects. On the other hand, it is not suitable for demanding dynamical applications, Actuators 2021, 10, 13 17 of 18

making it useful mostly for applications requiring minor adjustments of damping performance. The results of this research could be a useful reference for designers to understand the complexity of the mathematical model to adopt according to the expected performance.

Author Contributions: Formal analysis, D.D.; Methodology, D.D. and L.S.; data curation, D.D. and L.S.; writing–original Draft, D.D.; writing–review & Editing, D.D. and L.S.; validation, L.S. and G.R.; conceptualization, G.R.; supervision, G.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conﬂicts of Interest: The authors declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article.

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