Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 885-894, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.885 THEORETICAL STUDY OF A TWIN-TUBE MAGNETORHEOLOGICAL DAMPER CONCEPT Janusz Gołdasz Technical Center Krakow, BWI Group, Kraków, Poland and Cracow University of Technology, Department of Control and Information Technology, Kraków, Poland e-mail: janusz.goldasz@bwigroup.com In this study, the author presents a theoretical model of a semi-active magnetorheological (MR) twin-tube damper concept. The model relies on geometric variables and material properties and can be used in engineering and research studies on damper structures. Other non-linear characteristics, namely, the fluid chamber compressibility, fluid inertia, cylinder elasticity, friction, one-way check valves are included into the model as well. The author studies the performance of the dampermodel as design variables are varied, and the results are analysed and discussed. Keywords: MRdamper, twin-tube damper concept, lumped parameter model 1. Introduction Magnetorheological (MR) fluids have always been attractive to engineers and researchers within the automotive industry. The material adapts to changing external conditions within millise- conds. Automotive (vehicle) dampers utilizing MR fluids are now found in a number of semi- -active platforms in vehicles. In the industry, the monotube damper configuration (de Carbon, 1952) is the most common structure of a flow-modeMR damper. The cylinder tube houses the floating piston (gas cup) separating the fluid from the gas-filled chamber. The piston divides the MR fluid volume into the compression chamber (fluid volume between the floating piston and the main piston assembly) and the rebound chamber (fluid volume between the rod guide and the main piston). The piston assembly contains an annular gap to permit the fluid to flow between the chambers and secondaryflowpaths (bypasses) for tuning theMRdamper low-speed performance. In a typical MR damper, the rod is attached to the vehicle body and the cylinder to thewheel hub.The relativemotion of thewheel and thebodydrives thefluidflowbetween the chambers through the annulus in the piston. The design has been a natural choice for MR ap- plications due to its simplicity, however, high operating pressures and packaging limit its scope. Moreover, manufacturing issues due to high surface finish requirements of the cylinder tube are a factor here, too. Also, gas high pressures inmonotube dampers would translate into rod guide frictionwell above that of twintube hardware.Therefore, the research on other structures ofMR dampers continues (Poynor, 2001). A standard twin-tube damper features concentric cylinder tubes. The inner cylinder houses a piston valve for controlling the flow between the adjacent fluid chambers and a base (foot) valve for regulating the flow between the fluid chamber below the piston in the inner cylinder and a reservoir (fluid volume contained between the outer tube and the inner one). The reservoir is partially filledwith oil to accommodate volume changes due to rod displacement. The dampers work at a lower gas pressure, but only upright positions are possible in vehicles, and they incorporate more valves. However, research efforts on MR twin- -tube structures have not fully succeeded. Two studies focused on a twin-tube structure of an MR damper in which the MR control valve was located in the piston inside the inner cylinder 886 J. Gołdasz (Poynor, 2001; Jensen et al., 2001). In the design of Jensen et al. (2001), a standard base valve was used for controlling the MR fluid flow into the outer reservoir. The damper structure, ho- wever, might suffer from hydraulic imbalance (a common problem affecting twintube dampers), and the range of damping forces that can be achieved with this design could be limited. The imbalance phenomenon occurs when the damper is in compression and the pressure drop across the piston is larger than the pressure drop across the base valve. As a result, most of the fluid volume is pushed through the base valve causing lags in the chamber above the piston. Another study revealed a twin-tubedamper inwhich theMRvalve regulates thefluidflow fromtheupper chamber above the piston into the reservoir volume between the cylinders (Oakley, 2006). Two one-way check valves are used for directing the flowbetween the fluid chambers.Another feature of this concept is its ability to tune its non-energized condition with passive valves. Apparently, there is no published research on the twin-tube design ofOakley (2006) related to its performan- ce. The proposed model fulfills this gap. Briefly, the generic goal of this study was to provide a lumped parameter model of a twin-tube MR damper for component as well as vehicle level analyses. The task is complicated – damper and flow channel geometry, magnetic field induced yield stress and resistance-to-flow build-up, fluid compliance, cavitation, friction, gas absorption, etc. have been among the contributors to the force output of MR dampers (Hong et al., 2006). At the same time, vehicle dampers have been a subject of intensivemodellingwork. In the past, researchers developed various models of dampers to copy their non-linear characteristics. For example, Lang (1977) as well as Segel and Lang (1981) developed a math model of a twintube automotive damper and concluded the observed hysteretic behavior was due to the compressi- bility of the fluid, cylinder tube elasticity and cavitation. Themodels of Lang (1977), Segel and Lang (1981) remain the key work on conventional dampers operating at high frequencies. Also, Lee (1997) obtained a complex model of a monotube vehicle suspension damper. The model included compressibility of fluid dampers, floating gas cup inertia and first-order heat transfer effects in addition to a deflected disc piston model. Also, Mollica (1997) proposed a non-linear model of a monotube damper using bond graph techniques. Themodel of Mollica incorporates friction elements, fluid compressibility, gas, leakage and hydraulic resistance components in the piston (Mollica, 1997). Those studies were a basis for developing the lumped parameter model described in detail below. Specifically, the goal was to obtain a dampermodel capable of copying the performance characteristics of twin-tubeMR dampers and important phenomena occurring inside the device aswell as the operational logic of the damper.Also, fluid compressibility effects andfluid inertia aremodeled, and their influence on the damping force output of anMRdamper is analysed for a selected configuration. 2. Modelling The MR twin-tube damper concept is illustrated in Fig. 1. The inner tube houses the piston separating the fluid volume into the rebound (upper) chamber volume and the compression (lower) chamber volume. The damper is driven by the displacement (velocity) input xp (vp) applied to the rod.MRvalve (1) controls the fluidflowbetween reboundand reservoir chambers. The flow rate through the MR valve is Qv,1. The flow through the piston Qv,2 is controlled by check valve (2). The valve allows flow in one direction only, from chamber (2) (compression) into chamber (1) (rebound). The flow between chambers (3) (reservoir) and (2) (compression) is controlled by one-way valve (3). This valve allows flow from chamber (3) (reservoir) into (2) (compression). Both valves are schematically shown in Fig. 1 – they may take the form of a standard deflected disc stack assembly or a preloaded spring and plate. The flow rate through check valve (3) is Qv,3. The reservoir containsMR fluid and pressurised gas. The fluid rheology in the annulus is controlled by themagnetic field H due to the current Ic in the coil of the piston Theoretical study of a twin-tube magnetorheological damper concept 887 core. The fluid is described by the yield stress τ0, viscosity µ, density ρ, and bulk modulus Bf. The MR annulus height is h, and its cross-section area Ag. Lg is the annular length, and the active section length (magnetic poles) is La (La < Lg). In rebound (see Fig. 1), the rod moves out of the damper.The flow is through valves (1) and (3), and there is no flow through valve (2); the flow through MR valve (1) is uni-directional. In compression, the rod would move into the damper. Flow through check valve (3) would be prevented, and it would occur through valves (1) and (2). In the sections that follow below, the author discusses the key phenomena occurring in the damper and outside of theMR valve. Fig. 1. MR twin-tube damper: internalMR valve 2.1. Damper model Consider the damper model in Fig. 1. With the inertia of the lumped mass of fluid in the MR valve annulus, the force balance equation is (Gołdasz and Sapiński, 2013) Q̇v,1 = Ag ρLg (Pr −Pg −∆pa −∆PH) (2.1) where ∆pa is the field-induced pressure drop along the annular gap, and ∆PH denotes losses at the holes in the inner cylinder. The term ∆pa is discussed in detail in Section 2.2. Also, fluid continuity expressions for the pressures above and below the piston are Ṗr = β(Pr) (Ap −Ar)vp − (Qv,1+Qv,2) Vr,0− (Ap −Ar)xp Ṗc = β(Pc) −Apvp +(Qv,2+Qv,3) Vc,0+Apxp (2.2) where β(P) refers to the combined bulk modulus due to fluid compressibility and cylinder compliance, whereas Vr,0 and Vc,0 are midstroke fluid chamber volumes. Gas pressure in the reservoir Pg can be expressed assuming the adiabatic process, i.e. without heat transfer between the damper and the environment Pg = Pg,0 ( Vg,0 Vg,0− ∫ (Qv,1−Qv,3) dt )n (2.3) 888 J. Gołdasz In the above equation, Pg,0 and Vg,0 are the initial gas pressure and volume, respectively, and n is the adiabatic gas constant. Also in this analysis, the effects of wall expansion with pressure are combined with the influence of fluid bulkmodulus via the relationship 1 β = 1 βf + 1 βs (2.4) where the variation of the fluid bulk modulus with pressure can be as βf(P)= β0 1+α ( Pa Pa+P )1 n 1+α P 1 n a n(Pa+P) 1+n n (2.5) Equation (2.5) reveals the bulk modulus variation with pressure of the mixture of the fluid and non-dissolved air (Manring, 2005). β0 is the pure fluid bulk modulus, Pa refers to the atmospheric (or reference) pressure, and α denotes the relative gas content. The compliance of the steel cylinder βs is (Mollica, 1997) 1 βs = 2 Es ( ν + D2o +D 2 p D2o −D2p ) (2.6) whereEs is Youngmodulus (steel), ν –Poisson’s coefficient, Do – outer diameter of the cylinder. Cavitation effects are simply modeled by imposing a constraint on the pressures Pr and Pc, Pr ­ Pv and Pc ­ Pv. Also, the pressure drop at the holes ∆PH in the inner cylinder is ∆PH = ρ Q2v,1 2(CHAH)2 (2.7) where CH is the discharge coefficient and Ao cross-sectional area of the holes. Using the one-way valve in the piston, the piston flow rate Qv,2 can be Qv,2 =    C2A2 √ 2 |Pr −Pc| ρ Pr −Pc < 0 0 Pr −Pc ­ 0 (2.8) Similarly, the flow rate Qv,3 through check valve (3) is Qv,3 =    C3A3 √ 2 |Pc −Pg| ρ Pc −Pg < 0 0 Pc −Pg ­ 0 (2.9) The check valves are assumed to open with no delay. Considering forces on the piston, the damping force Fd including friction Ff becomes Fd =(Ap −Ar)Pr −ApPc +Ff ( sgn(vp) ) (2.10) To summarize, equations from (2.1) to (2.10) form a set of expressions for simulating the output of a twin-tubeMR damper. Theoretical study of a twin-tube magnetorheological damper concept 889 2.2. MR valve model This Section shows the application of a biplastic Bingham scheme for deriving the pressure vs. flow rate characteristics of an MR valve model. TheMR valve (annulus) contains a parallel flux bypass feature. The flux bypass often takes the form of a slot feature on either surface constituting the annulus. Due to the increased (local) height of the annulus, it is characterized by a region of low flux density (yield stress) (Gołdasz and Sapiński, 2012) where the MR fluid is allowed to flow through the flux bypass section at a lower breakaway pressure drop than in the other portion of the flow channel. As a result, low forces are achieved at near-zero flow rates through the MR piston. Medium and high flow rate performance is not affected. Application of the bi-plastic scheme is based on the assumption that the dual behavior can be described with the artificial material model of parameters related to both material properties of the MR (Bingham)fluidand thepiston geometry. By expressing thepressuredrop∆pa across the control valve in terms of the dimensionless pressure number G and the plasticity S, the equation linking the term ∆pa with the flow rate through theMR valve Qv,1 is (Gołdasz and Sapiński, 2012) ∆pa = 2τ2La h G(S)+C ρQ2v,1 A2g = 2τ0La h[1−γ(1−δ)] G(S)+C ρQ2v,1 A2g G =−h∆pa 2Laτ2 S = 12µQv,1 wh2τ2 (2.11) In equation (2.11), high velocity losses are accounted for in themodel in quadratic form, and the tuning coefficient C captures the effects of the fluid entry and exit, flow development, turbulent losses, etc. The parameters γ and δ refer to the slope of the damper force (pressure) variation against velocity (flow rate) and the interception force in the pre-yield region, and τ2 is the bi-plastic material yield stress. The pre-yield viscosity (slope) µr is related to the material viscosity µ via γ = µ/µr, and the yield stress τ2 is linked to the yield stress τ0 through the equation τ0 = τ2[1−γ(1− δ)]. At γ →∞ and δ → 1, the model would reduce to that of classic Bingham’s. The bi-plastic model was studied by various authors (Gołdasz and Sapiński, 2012, 2013; Dimock et al., 2002). For example, Gołdasz and Sapiński (2012) analyzed the performance of a dual coil MR piston with the flux bypass feature and extracted non-dimensional parameters for it. The authors concluded that the non-dimensional viscosity γ was relatively invariant of the magnetic field, whereas the yield stress parameter δ varied with the current level (or flux density). Themodel allows for separating the flow regime into two distinct flow regimeswith the threshold plasticity S0 = γ(2−3δ+δ3). Briefly, the pre-yield (bypass) regime is characterized by the plasticity number S < S0 and the post-yield regime by S ­ S0. In themodel, the post-yield relationship between the pressure drop and the flow rate through the annulus for (S ­ S0 and G ­ 1) is G = 1 6 [3(1−γ(1− δ))+S] [ 2cos (1 3 arctan2(y,x) ) +1 ] (2.12) where y =12 √ −81b2+12ba3 x =−108b+8a3 a = 3 2 (1−γ(1− δ))+ 1 2 S b = 1 2 (1−γ(1− δ3)) (2.13) In the pre-yield flow regime, S < S0, thematerial behavior is governed by themodifiedBingham plastic formula G = δ 1 6 ( S δγ +3 )[ 2cos (1 3 arctan2(y′,x′) ) +1 ] (2.14) 890 J. Gołdasz where x′ =−27+27 S γδ +9 ( S γδ )2 + ( S γδ )3 y′ =6 √ 3 √ 27 S γδ +9 ( S γδ )2 + ( S γδ )3 (2.15) To summarize, equation (2.11) accompanied by equations (2.14) and (2.15) allow for calculation of the pressure drop ∆pa across the energized annulus. 3. Simulations Thesimulations involved theMRtwin-tubedampermodel subjected toadisplacementwaveform at the rod as in Fig. 1 and used the data in Table 1. The friction estimate Ff of 70N has been obtained from a real damper; the gas pressure Pg,0 is equal to 0.8MPa, and the adiabatic constant 1.4. TheMR fluid bulkmodulus βf is 1500MPa, the density ρ is 2.68g/cc, and its air contents α equal to 0.001. The viscosity of the fluid µ is 62cP at the temperature Ta of 30◦C – see Fig. 2. The steel modulus of elasticity Es is 2.1 ·105MPa, and the Poisson coefficient equals to 0.29. Table 1. Twin-tube dampermodel inputs Symbol Description Value Lr,0 Initial rebound chamber length, [mm] 150 Lc,0 Initial compression chamber length, [mm] 150 Aeff = Ap−Ar Upper chamber cross-section area, [mm2] 683.48 Ap Cylinder cross-section area, [mm2] 804.24 Vr,0 Initial rebound chamber volume, [mm3] 1.206 ·105 Vc,0 Initial compression chamber volume, [mm3] 1.025 ·105 Vg,0 Initial gas chamber volume, [mm3] 0.861 ·105 A2, A3 Check valve flow areas, [mm2] 220 C2, C3, CH Discharge coefficients, [–] 0.7 AH Cylinder holes area, [mm2] 301 tw Cylinder wall thickness, [mm] 1.8 La Active length, [mm] 25.8 L Annulus length, [mm] 37 h Annulus height, [mm] 0.89 w Mean circumferential width, [mm] 88.60 C Flow coefficient, [–] 0.1 The piston parameters, the yield stress ratio and the viscosity ratio variation with current, respectively, copy the dual-coil assembly byGołdasz and Sapiński (2012). In the study, the two parameters γ and δ are identified from real piston performance data. The identified viscosity ratio γ varied from 0.0175 at the coil current Ic of 1A through 0.0167 at 3A to 0.0149 at the maximum coil current level of 5A. The yield stress ratio varied from 0.179 (Ic = 1A) through 0.363 (Ic = 3A) to 0.492 (Ic = 5A). Here, the MR piston is simply described by the steady-state pressure vs. flow rate characteristics in Fig. 3. The ∆pa −Qv,1 characteristics in Fig. 3 are based on the geometry and material properties, and then input into the Simulink model. The fluid data are in Fig. 2; B – magnetic flux density, H – field strength. The results given by equations (2.2) through (2.10) are presented in Figs. 4 through 7. Briefly, the model Theoretical study of a twin-tube magnetorheological damper concept 891 Fig. 2. MR fluid characteristics: B-H, τ0-B (Gołdasz and Sapiński, 2012) Fig. 3. MR piston steady-state characteristics: ∆pa vs. Qv,1 Fig. 4. Influence of rod size on the damping force; Xp =30mm, Vp =1024mm/s 892 J. Gołdasz Fig. 5. Graphs of force-displacement and force-velocity; Xp =30mm, Ic =5A Fig. 6. Graphs of pressure-displacement and pressure-velocity; Xp =30mm, Ic =5A is subjected to the displacement xp(t) = Xp sinωt applied to the rod. The results are shown as force-velocity and force-displacement loops. In the simulations, the effects of velocity, coil current and rod size on the damping force output are examined. Specifically, Fig. 4 shows the impact the rod diameter (area) has on the damper force. As seen in Figs. 4a through 4d, smaller rod sizes (Dp = 12.4mm) contribute to major asymmetry in the damping force. The rebound- -to-compression ratio (asymmetry ratio) for thedamping force is above 5:1 at thepeakvelocity of 1024mm/s. In the cases shown, the rebound forces decreasedwhen the pistondiameter increased Theoretical study of a twin-tube magnetorheological damper concept 893 Fig. 7. Influence of the frequency; Vp =382mm/s up to 22mm.The asymmetry decreased at the expense of rebound forces. It can be shown that as the piston rod is in compression, check valve (2) in the piston is opened, and check valve (3) in the base valve is closed, so that the annular flow rate is related to the rod area Ar. Smaller rod sizes develop larger force output asymmetry. Increasing the rod size impacts the hysteresis between the force and velocity (see Figs. 4a and 4c and 5) and rotates the damping force ellipses into the first quadrant of the force-displacement plane due to the gas force. The hysteresis is larger when in compression than in rebound. Also, it can be shown that the gas force change magnitude is directly related to the rod area. Next, Fig. 6 reveals the pressures in each chamber of the damper vs. piston displacement and velocity. Note that the rebound chamber pressure dominates regardless of the damper operating conditions, i.e. it is clear that when the damper is in rebound the pressure in the lower chamber drops below gas pressure. Check valve (3) in the base valve opens, and there is flow through check valve (3) from the reservoir and into the compression chamber. In compression, the check valve in the piston opens and there is flow from the compression chamber into the rebound one. The effect of frequency manifested by an increase in the hysteresis in the force-velocity loops and force oscillations are shown in Fig. 7. 4. Conclusions The author has analysed a novel model of a twin-tube MR damper concept. The study shows numerical results, however, the MR valve model is based on a verified bi-plastic theory and against real datawhich allows one to analyze the resultswith confidence (Gołdasz and Sapiński, 2012, 2013).Apart fromtheMRvalve, thedamperutilizes twoone-way checkvalves in thepiston and the base valve, respectively. The check valves offer extrameans of tuning the output force in off-state conditions; this aspect of the concept is beyond the scope of this paper.Additionally, by using the check valves at the piston and the base of the damper, the flow through the piston is always in the samedirection.To theauthor’s knowledgeno suchmodel hasbeendeveloped so far. As opposed to presentMR structures, this configuration is asymmetric rebound-to-compression; the asymmetry is related to the rod size. To conclude, larger rod sizes minimize the asymmetry at the cost of rebound forces. The damper is more complex than single-tube structures but any performance and cost benefits, namely, lower friction, less stringent cylinder surface finish, may favour its applications. The damper works at a lower gas pressure than other MR damper structures, too. The twin-tube dampermodel can be a useful tool in various studies. Themodel relies on the information extracted mainly from engineering drawings and fluid data, which makes it suitable for fast sizing studies early in the design development stage. Transient studies through the B-τ0 coupling are possible, too. 894 J. Gołdasz References 1. de Carbon Ch., 1952, Shock absorbers, US Patent No. 2774446 2. Dimock G.A., Yoo J.-H., Wereley N.M., 2002, Quasi-steady Bingham biplastic analysis of electrorheological and magnetorheological dampers, Journal of Intelligent Material Systems and Structures, 13, 9, 549-559 3. Gołdasz J., Sapiński, B., 2012, Nondimensional characterization of flow-mode magnetorheolo- gical fluid dampers, Journal of Intelligent Material Systems and Structures, 23, 14, 1545-1562 4. Gołdasz J., Sapiński B., 2013, Verification of magnetorheological shock absorber models with various piston configurations, Journal of Intelligent Material Systems and Structures, 24, 15, 1846-1864 5. 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Poynor J.C., 2001, Innovative designs for magnetorheological dampers, M.Sc. Thesis, Virginia Polytechnic Institute and State University, US 13. Segel L., Lang H., 1981, The mechanics of automotive hydraulic dampers at high stroking frequencies,Vehicle System Dynamics, 10, 2, 82-85 Manuscript received September 23, 2014; accepted for print May 8, 2015