Active Pressure Ripple Control in Axial Piston Pumps Through High-Frequency Swash Plate Oscillations—A Theoretical Analysis

energies

Article Active Pressure Ripple Control in Axial Piston Pumps through High-Frequency Swash Plate Oscillations—A Theoretical Analysis

Paolo Casoli 1,* , Mirko Pastori 1 , Fabio Scolari 1 and Massimo Rundo 2

1 Department of Engineering and Architecture, University of Parma, 43124 Parma, Italy; [email protected] (M.P.); [email protected] (F.S.) 2 Department of Energy, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Turin, Italy; [email protected] * Correspondence: [email protected]

Received: 18 March 2019; Accepted: 4 April 2019; Published: 10 April 2019

Abstract: Pressure ripple has always been a major drawback in hydraulic circuits, since it is the main source of the overall noise emitted by the pump. This article presents a theoretical analysis on the active control of the pressure ripple in an axial piston pump by properly moving the swash plate. The reduction of the pressure oscillations is studied in an active way and for this purpose a mathematical model of the whole pump has been developed, focusing on both the ﬂuid dynamic aspects and the component dynamics. An experimental activity has been performed in order to validate the pump mathematical model. The results of the simulation, compared with the experimental data, highlight a suitable capability of the model to predict both the dynamics of the swash plate and the delivery pressure ripple. The validated model has been used for implementing an active control of the pressure ripple with the aim of properly modifying the machine displacement at high frequency in order to vary the instantaneous delivery ﬂow rate and, consequently, the outlet pressure. The control strategy is grounded on moving the swash plate for modifying the motion law of the pistons through a servo valve integrated into the displacement control system. The simulations results have demonstrated that acting on the pump displacement control is possible to considerably reduce the amplitude of the pressure oscillations.

Keywords: axial piston pump; active control; pressure ripple; swash plate

1. Introduction Variable displacement axial piston pumps are widespread in hydraulic circuits, because they are robust machines able to guarantee great reliability, considerable flexibility and to transmit high power density with also significant energy savings [1]. Like most volumetric machines, piston pumps deliver an oscillating flow rate, generating a pressure ripple that is a huge drawback in hydraulic circuits. In fact, pressure ripple produces vibrations in the pump components that are transmitted and amplified by the downstream piping (fluid borne noise); hence, the components at the delivery side of the machine are worn and stressed, compromising their functionality and durability. Many previous studies have focused on how to solve this problem, concentrating mainly on passive techniques in order to reduce the amplitude of the pressure ripple: these methodologies generally consist of geometric optimizations of some components with the aim of limiting the flow and pressure fluctuations between internal pump chambers at different pressure values. One of these optimizations concerns the design of the port plate, with particular attention paid to the geometry

Energies 2019, 12, 1377; doi:10.3390/en12071377 www.mdpi.com/journal/energies Energies 2019, 12, 1377 2 of 18 of both the inlet and the outlet relief grooves, to allow a gradual transition between the suction and delivery phases, avoiding trapped volumes and at the same time without compromising the volumetric efficiency [2]. Active control is a method for attenuating an unwanted signal by superimposing it with a second signal having the same amplitude but opposite phase: for the principle of interference, the two signals combine themselves and cancel each other out. Instead of passive techniques, an active control should be able to adapt itself to different operating settings and different fluid pressure conditions. Some researchers address the problem of active pressure ripple reduction in a hydraulic circuit by placing an actuator with a variable chamber volume, installed along the pipe, driven by a signal coming from an adaptive filter integrated with a feedforward algorithm, in the delivery hose [3]. In this work, the active reduction of the pressure ripple achieved by moving the pump swash plate is presented. The displacement control system properly changes the tilt angle of the swash plate around the equilibrium position in order to modify continuously the displacement of the machine. This technique allows the law of motion of the pistons to be varied; as a consequence, the flow rate profile changes and the pressure ripple can be reduced. In the literature there are a few articles that introduce the possibility of inducing controlled vibrations of the swash plate, in order to try to reduce the amount of the structure borne noise generated as a consequence of the interaction of the vibrations of the swash plate with the whole body of the pump. For instance, Ivantysynova et al. [4] proposed to control the swash plate through a servo valve to mitigate the swash plate vibrations and it was found that a reduction of the overall noise emitted by the pump is possible. In this work, a mathematical model of the entire pump has been developed for studying an active control solution of the pressure ripple based on swash plate controlled oscillations. The pump model concerns the fluid dynamic aspects as well as the dynamics of the components. Many researchers have analyzed in detail the typical fluid dynamic aspects of positive displacement machines: some research groups studied mechanical and volumetric losses and evaluated the friction between components in relative motion, such as pistons and cylinder block, slippers and port plate or in the axial gap of gear pumps [5–9]. Other studies evaluate experimental and theoretical pressure ripples and pump generated noise also through a three dimensional modelling approach [10–14], the estimation of forces inside pumps and the modelling of displacement control systems [15,16] or the effects of swash plate design on flow pulsations [16,17]. Further studies show the effects of fluid inertia on the delivery pressure ripple and several models of fluid characterization in order to take into account the effects of cavitation during the suction phase [18,19]. In this work, the pump has been studied adopting the filling and emptying approach, which permits us to calculate the delivery flow rate and the delivery pressure ripple. The pump dynamics model has been focused on, determining the equilibrium position of the swash plate, and for this purpose, all components inside of the pump that interact with the swash plate are studied in detail. Over the years, several studies have been conducted in order to study the dynamic behavior of the swash plate in this kind of pump. Zeiger et al. and Lin [20,21] proposed a mathematical model for the calculation of the torque acting on the swash plate; Schoenau et al. [22] developed a dynamic model of the pump in order to achieve a satisfactory correlation between experimental results and simulated transient responses; Casoli et al. [23] proposed a model for the study of the pump with its flow regulators suitable for simulating mobile applications [24–28]. The fluid dynamic model was tested by comparing the results of the simulation with the experimental data in different working conditions, in order to verify the predictive capability of the model suitable for the purposes of this research. Regarding the dynamic model, several step tests have been performed in order to verify that the transient response of the simulated pump was equivalent to the real one and a satisfying overlap of the characteristic curves had been reached. The pump model includes a sub model of a servo valve for controlling the displacement actuator, in order to obtain high dynamic performance and to allow the required movements of the swash plate. The model and the control strategy have been preliminarily applied in an ideal way (assuming pump moving parts with mass equal to zero) in order to prove a significant reduction of the amplitude Energies 2019, 12, 1377 3 of 18 of the delivery pressure oscillations could be obtained with this approach. Subsequently, the model with dynamic effects was applied and a solution for the displacement regulation system has been proposed in order to obtain better performance. Energies 2019, 12, x FOR PEER REVIEW 3 of 18 2. Fluid Model the model with dynamic effects was applied and a solution for the displacement regulation system A simple fluid model has been implemented for including the phenomenon of gaseous cavitation, has been proposed in order to obtain better performance. while vapour cavitation is not considered because of the low vapor pressure value at working ◦ temperatures2. Fluid Model (about 120 Pa at 50 C) [18]. The gaseous cavitation effects are analyzed in a simplified way, treating a gas-liquid mixture as a homogeneous fluid. Because of this assumption, it is not A simple fluid model has been implemented for including the phenomenon of gaseous possiblecavitation, to account while for vapour delays cavitation in both air is not release considered and re-dissolution because of the [29 low,30 ], vapor but for pressure the purpose value at of this studyworking the development temperatures of (about such a120 complex Pa at 50 model °C) [18 is]. notThe necessary.gaseous cavitation effects are analyzed in a Thesimplified fluid modelway, treating is based a gas on-liquid the following mixture as hypotheses: a homogeneous the fluid liquid. Because bulk modulusof this assumption, is constant, it the surfaceis not tension possible effects to account are ignored; for delays fluid in pressureboth air release and temperature and re-dissolution are the [2 same9,30], forbut bothfor the gas purpose and liquid; the fluidof this temperature study the develop is constant.ment of The such supposition a complex model of isothermal is not necessary. fluid is acceptable by the higher heat capacity ofThe the fluid liquid mode comparativelyl is based on the with following the gas. hypotheses: the liquid bulk modulus is constant, the Thesurface gaseous tension phaseeffects isare modelledignored; fluid through pressure the and state temperature equation are for the ideal same gases for both neglecting gas and any liquid; the fluid temperature is constant. The supposition of isothermal fluid is acceptable by the contribution of oil vapor and the effects of high pressure on the gas compressibility factor. The mixture higher heat capacity of the liquid comparatively with the gas. volume is the sum of the volume of each component, expressed for the unit of mass as: The gaseous phase is modelled through the state equation for ideal gases neglecting any contribution of oil vapor and the effects of high pressure on the gas compressibility factor. The v = v ·(1 − y) + v ·y (1) mixture volume is the sum of the volume of Leach componentG , expressed for the unit of mass as: and the state equation of the mixture is:푣 = 푣퐿 ∙ (1 − 푦) + 푣퐺 ∙ 푦 (1) and the state equation of the mixture is: RG·TG 1 − y v = ·y + (2) 푅퐺p∙ 푇퐺 1 − 푦p − p 푣 = ∙ 푦 + ρ 1 + 0 푝 L0 푝 − 푝0 (2) 휌퐿0 (1 + B0) 퐵0

The ﬂuidThe fluid density density versus versus pressure pressure for for different different values values ofof gasgas to liquid mass mass fraction fraction is ispresented presented in Figurein1 Figure; the gas 1; the effect gas iseffect evident is evident in the in lowerthe lower pressure pressure range. range.

FigureFigure 1. Fluid1. Fluid density density versus versus pressure. Fluid Fluid temperature temperature 320 320 K. K.

3. Fluid3. Fluid Dynamic Dynamic Model Model The fluidThe fluid flow flow is simulated is simulated by by means means ofof aa mathematical model model based based on onthe thefilling filling and andemptying emptying methodmethod (F&E (F&E). The). The axial axial piston piston pump pump has has been been divided divided inin bothboth fixedfixed and and variable variable volumes volumes in inorder order to applyto the applyF&E themodel, F&E model, as shown as shown in Figure in Figure2. 2.

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Figure 2. Volumes considered. Figure 2. Volumes considered.

The constant volumes V1A and V1M represent the internal cavities of the pump casing at the suction The constant volumes V1A and V1M represent the internal cavities of the pump casing at the and delivery side; they are connected with the external ports and with the cylinders (VCi represented suctionas variable and volumes) delivery by side means; they of are variable connected orifices. with the external ports and with the cylinders (VCi represented as variable volumes) by means of variable orifices. In order to simulate the circuit on the test bench, an additional volume, V2M, has been added to representIn order theto simulate portion ofthe pipe circuit that on connects the test thebench, delivery an additional of the pump volume, to the V2M valve, has been that simulatesadded to represent the portion of pipe that connects the delivery of the pump to the valve that simulates the the load, Ω2M. This latter is a variable orifice that permits the pressurization of delivery side of the load,pump; Ω2M downstream. This latter ofis a this variable orifice orifice there isthat the permits tank. The the generalizedpressurization Bernoulli’s of delivery equation, side of neglectingthe pump; downstreamthe gravity term, of this is: orifice there is the tank. The generalized Bernoulli’s equation, neglecting the gravity term, is: c · dc = −v · dp (3) 푐∙푑푐 = −푣∙푑푝 (3) Substituting Equation (2) in the second member of Equation (3) and integrating, the fluid velocity in a throatSubstituting is found: Equation (2) in the second member of Equation (3) and integrating, the fluid velocity in a throat is found:v u   B   u 0 · + u ρL0 pt u  B0  퐵ρ0L0  pt  c = u2· ·(1 − y)· ln ∙ 휌퐿0 + 푝푡  + RG·TG·y· ln  (4) t 퐵ρ0L0 휌B퐿00  푝p푡s  푐 = √2 ∙ [ ∙ (1 − 푦) ∙ ln ( ·ρL0 + ps) + 푅퐺 ∙ 푇퐺 ∙ 푦 ∙ ln ] (4) 휌퐿0 퐵ρL00 푝푠 ∙ 휌퐿0 + 푝푠 휌퐿0 The upstream pressure pt is the stagnation pressure, while ps is the static downstream pressure. The upstream pressure insidepressure each pt chamberis the stagnation is determined pressure through, while theps is continuity the static downstream equation; in particular,pressure. for theThe cylinder pressure volume: inside each chamber is determined through the continuity equation; in particular, for the cylinder volume: . V (z) dp dz m + ci · ci − ρ ·Ω · i = 0 (5) ∑ ci ∂p dt c ci dt i 푉푐𝑖(푧) 푑푝푐𝑖 푑푧𝑖 ∑ 푚̇ 푐𝑖 + ∂ρ ∙ − 휌푐 ∙ Ω푐𝑖 ∙ = 0 𝑖 휕푝 T0,c 푑푡 푑푡 (5) ( ⁄휕휌) where the first term is related to the mass exchange,푇0,푐 the second term to the effects of compressibility, whereunder isothermalthe first term conditions, is related andto the the mass last termexchange, considers the second the rate term of variation to the effects of the of volume compressibility, produced underby the isothermal movement conditions, of the piston, and whosethe last position term considers is defined the rate by the of variationz coordinate. of the Fixed volume volumes produced are bymodeled the movement by means of of the Equation piston (5),, whose in which position the thirdis defined term isby null. the z coordinate. Fixed volumes are modeledThe fluidby means dynamic of Eq modeluation also (5), considersin which the the third effects term of fluidis null. inertia in the delivery volume of the pumpThe body. fluid The dynamic effects ofmodel fluid also inertia considers are important the effects at the of beginningfluid inertia of in the the delivery delivery stroke, volume when of the the pumpedge of body. the cylinder The effects block of port fluid begins inertia to are uncover important the groove at the and beginning the sudden of the fluid delivery acceleration stroke, induces when theadditional edge of pressure the cylinder pulsations block port in the begins cylinder to uncover [19]. the groove and the sudden fluid acceleration inducesThe additional inertia of pressure the fluid pulsations is modeled in by the an cylinder appropriate [19]. formulation of the momentum equation, whichThe can inertia be expressed of the fluid as: is modeled by an appropriate formulation of the momentum equation, which can be expressed→ as: → . . → → → ∂ Z → Qedt − Qudt + F B − F N + F Ω dt =휕 ρ c dV dt (6) ⃗ ̇ ⃗ ̇ ∂t V 푄푒 푑푡 − 푄푢 푑푡 + [퐹퐵 − 퐹푁 + 퐹훺]푑푡 = (∫ 휌 푐 푑푉) 푑푡 (6) 휕푡 푉 The control volume considered is delimited by the port plate and the pump delivery port. The control volume considered is delimited by the port plate and the pump delivery port. The first and second terms in the first member of Equation (6) are the flux of momentum that enters and exits the volume, respectively; the term in brackets is the sum of the forces acting on the fluid in the time dt; the second member represents the change of momentum.

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The first and second terms in the first member of Equation (6) are the flux of momentum that entersEnergies and2019, exits12, x FOR the PEER volume, REVIEW respectively; the term in brackets is the sum of the forces acting on5 of the 18 fluid in the time dt; the second member represents the change of momentum. The considered forces are the surface term acting on a permeable control→ area (퐹 훺 ) and the The considered forces are the surface term acting on a permeable control area ( F Ω ) and the normal normal pressure forces acting on an impermeable surface→ (퐹푁), while the body force→ (퐹퐵) is neglectable. pressureThe forcesvector actingterms onof the an impermeableflux of momentum surface projected ( F N), while on ξ the axi bodys, represented force ( F B in) is Figure neglectable. 3, are: The vector terms of the푁 flux of momentum projected on ξ axis, represented in Figure3, are: 푄̇ = ∑ 휌 ∙ 훺 ∙ 푐 ∙ |푐 | 푄̇ = 휌 ∙ 훺 ∙ 푐 ∙ |푐 | (7) 푒 . 푐𝑖 N 푐𝑖 푐𝑖 푐𝑖 . 푢 1푀 1푀 1푀 1푀 𝑖Q=1e = ∑ ρci·Ωci·cci·|cci|Qu = ρ1M·Ω1M·c1M·|c1M| (7) The forces on permeable surfacesi=1 are:

Figure 3. Scheme of the control volume adopted to study the inertial effect. Figure 3. Scheme of the control volume adopted to study the inertial effect. The forces on permeable surfaces are: 푁

퐹Ω = ∑N 푝푐𝑖 ∙ 훺푐𝑖 − 푝2푀 ∙ 훺1푀 (8) = · − · FΩ 𝑖∑=1 pci Ωci p2M Ω1M (8) i=1 The throat area between cylinders and delivery volume 훺푐𝑖 depends on the cylinder block positionThe that throat has area been between evaluated cylinders through and the delivery kinematic volume model.Ωci depends on the cylinder block position that has been evaluated through the kinematic model. Figure 3 shows the section identified with 훺̅̅1̅̅푀̅, which is partially an impermeable surface, on whichFigure the pressure3 shows p the1M acts. section The identiﬁed forces on withimpermeableΩ1M, which surfaces is partially are expressed an impermeable by: surface, on which the pressure p acts. The forces on impermeable surfaces are expressed by: 1M ′ ′ 퐹푁 = −(∆훺 ∙ 푝1푀 + ∆훺 ∙ 푝 ) (9) 0 0 where the following terms are defined:FN = − ∆Ω·p1M + ·Ω ·p (9) ̅̅̅̅̅ 푁 ̅̅̅̅̅ ∆훺 = 훺1푀 − ∑𝑖=1 훺푐𝑖 , represents the impermeable surface portion of 훺1푀, where the pressure where the following terms are deﬁned: 1M p acts; N ′ ∆Ω∆훺 = 훺Ω11푀M−−훺̅̅1̅∑̅푀̅ Ω, ciimpermeable, represents thesurface impermeable portion due surface to the portion difference of Ω between1M, where 훺̅̅1̅ the̅푀̅ and pressure 훺1푀 , on which acts a pressurei=1 p’ that has been assumed equal to a mean value between the pressure in the p1M acts; control volume0 V1M and the pressure in the delivery one V2M. ∆Ω = Ω1M − Ω1M, impermeable surface portion due to the difference between Ω1M and Ω1M, on whichThe rate acts of a pressurechange ofp 0momentumthat has been in assumedthe control equal volume to a meanis: value between the pressure in the 1 control volume휕V1M and the pressure푑 in the delivery one푑휌V2M. 푑푝 푑푐 (∫ 휌 푐 푑푉) = (휌 ∙ 푐∗) ∙ 푉 = 푉 ( 1푀 ∙ 1푀 ∙ 푐 + 휌 ∙ 1푀) The rate of change of momentum1푀 in the control1푀 1 volume푀 is: 1푀 1푀 (10) 휕푡 푉 푑푡 푑푝 푑푡 푑푡

Z 1 The fluid∂ velocity→ c* consideredd is c1M∗ . dρ1M dp1M dc1M ρ c dV = (ρ1M·c )·V1M = V1M · ·c1M + ρ1M· (10) In conclusion,∂t V the system ofdt differential equations thatdp describesdt the fluid dynamicdt model is:

The ﬂuid velocity c* considered is c1M.

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In conclusion, the system of differential equations that describes the ﬂuid dynamic model is: 푑푝 1 푑푝 ⎧ = ∙ ∙ (푚̇ − 푚̇ )  푑푡 푉 푑휌 dp1A⎪ 1 dp . .  =푑푝 · 1 ·푑푝m1A − mCA  dt ⎪ V1A dρ ( )  ⎪ = ∙ ∙ 푚̇ − 푚̇  dp 푑푡 1 푉dp 푑휌. .  1M⎪= · ·m − m  푑푝 1 푑푝 CM 1M  dt ⎪ V1M= dρ∙ ∙ (푚̇ − 푚̇ )  푑푡 푉 푑휌  dp ⎪ 1 dp . .  2M = · · −  푑푝, 1 푑푝m1M m2M  dt V2M= dρ∙ ∙ 푚̇ − 푚̇ +휌 ∙퐴 ∙ 푧̇ (푡) 푖 = 1,2,… ,푁 (11) ⎨ 푑푡 푉 푑휌 , , , (11) dp 1 dp . . .  C,i⎪= · ·m −m + ρ ·A ·z (t) i = 1, 2, . . . , N  dt ⎪ V d CA⎡ ,i CM,i C,i cil i ⎤  Ci ρ 휌 훺 푐 |푐 | −휌 훺 푐 |푐 | +  ⎪ 푑푝   ⎢ N ⎥  푑푐 1 1 dp푑푡  ⎪ = ∙ ⎢∑ ρ Ω c |c | − ρ MΩ Mc M|c M|+ ⎥ − ∙ 1M ∙푐   ci ci ci ci 1 1 1 1  푑푝  dc1M⎪ 푑푡 1푉 ∙휌 ⎢i=1 ⎥ 휌1 dt  = · − · ·c1M  ⎪ ·  N ⎢ 푝훺 − 푝훺 + (∆Ω ∙ 푝 + ∆Ω ∙ 푝 )⎥ 푑휌dp  dt V1M ρ1M  0 0  ρ1M  ⎩ ∑ ⎣pciΩci − p2MΩ1M + (∆Ω·p1M + ∆Ω ·p ) ⎦ i=1 dρ

4.4. Dynamic Dynamic Model Model TheThe dynamic dynamic model model of of the the pump pump has has been been realized realized focusing focusing on on the the determination determination of of the the equilibriumequilibrium position position reached reached by by the the swash swash plate. plate. Preliminarily, Preliminarily, a a kinematic kinematic analysis analysis has has been been conductedconducted inin order order to to evaluate evaluate the the instantaneous instantaneous pistonpiston displacement, displacement, deﬁneddefined asas aa function function of of the the rotationrotation angleangle ofof the the cylinder cylinder block block and and of of the the swash swash plate plate angular angular position, position, through through geometrical geometrical considerations.considerations. InIn FigureFigure4 ,4, the the geometrical geometrical sketch sketch of of the the cylinder cylinder block, block, with with a a single single piston, piston, the the slipperyslippery and and the the swash swash plate, plate, is is shown. shown.

FigureFigure 4. 4.Geometrical Geometrical scheme scheme for for the the kinematic kinematic model. model.

TheThe equilibrium equilibrium position position of of the the swash swash plate plate is is determined determined by by the the equilibrium equilibrium of of the the forces forces acting acting onon it. it. In In Figure Figure5 a 5 sectionala sectional view view of the of pumpthe pump is represented is represented and theand forces the forces applied applied on the on swash the plateswash areplate highlighted. are highlighted. In detail, In the detail, forces the acting forces on acting the swash on the plate swash are the plate forces are exerted the forces by the exerted pistons by (F thep), by the cylinder block spring (F ) and by the displacement actuator (F ). pistons (Fp), by the cylinder blockk spring (Fk) and by the displacementact actuator (Fact). The rotation equilibrium equation around the axis of the oscillating plate, indicated by the spot in Figure4, is: 9 .. J·α + Ta = ∑ Tpi + Tk − Tact (12) i=1 where J represents the overall mass moment of inertia of the swash plate, slipper and slipper hold-down; Ta the friction torque; Tpi the torque exerted by the i-th piston; Tk the torque due to the spring and Tact the torque due to the control actuator.

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Figure 5. Sectional view of the pump with indicated the forces acting on the swash plate. Figure 5. Sectional view of the pump with indicated the forces acting on the swash plate. From Equation (12) it is possible to calculate the swash plate acceleration and, with a double integration,The rotation its angular equilibrium position. equation The torque around due the to theaxisi -thof the chamber oscillating can be plate, evaluated indicated from by the the force spot exertedin Figure by 4, the is: corresponding piston: Tpi = FRi·bi (13) 퐽 ∙ 훼̈ + 푇 = 푇 + 푇 − 푇 (12) The force FRi is the resultant force of the i-th piston normal to the swash plate. The force value and the moment-arm bi are defined as follows: where J represents the overall mass moment of inertia of the swash plate, slipper and slipper hold- .. . down; Ta the friction torque; TMpi thep·zi torque+ f ·zi +exertedpci·Acil by the iR-th·sin piston;ϑ − e T+k thea·sin torqueα due to the spring F = − b = i (14) and Tact the torque dueRi to the control actuator.cosα i cosα From Equation (12) it is possible to calculate the swash plate acceleration and, with a double integration,The pressure its angular trend position. inside eachThe torque cylinder due is to known the i-th fromchamber the can fluid be dynamicevaluated model, from the so force the correspondingexerted by the forcecorresponding acting on thepiston: swash plate is easily determinable. For the calculation of the term due to the viscous friction, the viscous friction coefficient f is expressed as follows [22]: 푇 = 퐹 ∙ 푏 (13) µ·π·dp· Lp + zi The force FRi is the resultant forcef of= the i-th piston normal to the swash plate. The force value(15) h and the moment-arm bi are defined as follows: p The spring inside the cylinder block generates a torque on the oscillating plate due to the 푀 ∙ 푧̈ + 푓 ∙ 푧̇ + 푝 ∙ 퐴 푅 ∙ 푠푖푛휗 − 푒 + 푎 ∙ 푠푖푛훼 퐹 = − 푏 = (14) eccentricity e, Figure 4: 푐표푠훼 푐표푠훼 Tk = Fk·e (16) The pressure trend inside each cylinder is known from the fluid dynamic model, so the wherecorrespondingFk is the springforce acting preload. on the swash plate is easily determinable. TheFor frictionthe calculation torque due of tothe the term coupling due to between the viscous the swash friction, plate the and viscous the bearings friction is coefficient studied with f is theexpressed Karnopp as frictionfollows model.[22]: This allows defining the static and viscous friction as a function of the . swash plate angular velocity. An angular velocity threshold value αth is identified: if the swash plate 휇 ∙ 휋 ∙ 푑 ∙ 퐿 + 푧 velocity is smaller than that value, the swash푓 = plate velocity is set to zero and the friction torque is(15) the ℎ minimum value between the torque acting on the swashplate (Ttot) and the maximum static friction torqueThe (Tsmax spring); conversely, inside the if the cylinder swash block plate velocity generates is greater, a torque the on friction the oscillating torque acting plate on due the swash to the plateeccentricity is defined e, Figure by Stribeck’s 4: and viscous effects. The Karnopp model mathematical formulation is the following: 푇 = 퐹 ∙ 푒 (16)  . . . ∗ where Fk is the spring α < αpreload.thTa = min (|Ttot|, Tsm)·sign(Ttot) , α = 0  The friction torque due to the couplinga between the swash plate and the bearings is studied with(17) . . . . . . ∗ .  −cv·|α| the Karnopp friction α ≥ αmodel.th Ta =ThisT allowsc + (T definingsmax − Tc )the·e static and·sign viscousα + Cfriction·α , α as= a αfunction of the swash plate angular velocity. An angular velocity threshold value 훼̇ is identified: if the swash plate velocity is smaller than that value, the swash plate velocity is set to zero and the friction torque is the minimum value between the torque acting on the swashplate (Ttot) and the maximum static friction torque (Tsmax); conversely, if the swash plate velocity is greater, the friction torque acting on the swash Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x FOR PEER REVIEW 8 of 18

plate is defined by Stribeck’s and viscous effects. The Karnopp model mathematical formulation is the following: ∗ |훼̇| < 훼̇ 푇 = 푚푖푛(|푇|, 푇) ∙ 푠푖푔푛(푇) , 훼̇ = 0 푎 (17) Energies 2019, 12, 1377 ∙|̇ | ∗ 8 of 18 |훼̇| ≥ 훼̇ 푇 = 푇 + (푇 − 푇) ∙ 푒 ∙ 푠푖푔푛(훼̇) + 퐶 ∙ 훼̇ , 훼̇ = 훼̇

The Karnopp friction coefficients have been calibrated with experimental data: different step The Karnopp friction coefficients have been calibrated with experimental data: different step tests under many working conditions have been provided in order to achieve the best possible fitting tests under many working conditions have been provided in order to achieve the best possible between the simulation results and experimental measurements. Figure 6 shows the friction torque fitting between the simulation results and experimental measurements. Figure6 shows the friction as a function of the swash plate angular speed, according to Karnopp model, adopting the optimized torque as a function of the swash plate angular speed, according to Karnopp model, adopting the coefficients. optimized coefficients.

FigureFigure 6. 6.Friction Friction torque. torque.

TheThe displacement displacement actuator actuator is is controlled controlled by by a a servo servo valve valve that that has has been been mounted mounted in in substitution substitution ofof the the original original mechanicalmechanical controlcontrol inin orderorder toto modifymodify thethe pumppump displacementdisplacement atat higherhigher frequency.frequency. TheThe force force exerted exerted by by the the actuator actuator on the on swash the swash plate is plate evaluated is evaluated from the from pressure the pressureinside the inside chamber the appliedchamber on applied the actuator on the surface; actuator the surface; pressure the is calculatedpressure is by calculated means of theby meansF&E method, of the knowingF&E method, the massknowing flow ratethe massprovided flow by rate the provided servo valve. by The the servo servo valve valve. has The the servo purpose valve of managinghas the purpose the flow of towardsmanaging the displacementthe flow towards control the actuator displacement to maintain control a fixed actuator differential to maintain pressure through a fixed the differential control orificepressure by modulatingthrough the thecontrol pump orifice displacement, by modulating allowing the the pump pump displacement, to work in a allowing load sensing the logic.pump to workThe in servoa load valvesensing is actuatedlogic. by an ideal solenoid. The mechanical aspect of the component is studiedThe as servo a second valve order is actuated system by and an the ideal instantaneous solenoid. The position mechanical of the aspect spool of is calculatedthe component using is Newton’sstudied as second a second law. order system and the instantaneous position of the spool is calculated using Newton’sThe solenoid second islaw. actuated through a signal provided by a dedicated controller, whose aim is to compareThe thesolenoid sum of is the actuated load sensing through pressure a signal (p providedLS) and the by pumpa dedicated margin controller, with the pumpwhose deliveryaim is to pressurecompare (p thed), sum in order of the to load properly sensing modulate pressure the (p displacementLS) and the pump of the margin pump with to minimize the pump the delivery error betweenpressure the (pd two), in values. order to properly modulate the displacement of the pump to minimize the error between the two values. 5. Experimental Activity 5. ExperimentalSeveral experimental Activity activities have been conducted in order to validate the mathematical model. The testsSeveral have experimentalbeen performed activities on the test have bench been in the conducted Laboratory in oforder the Engineering to validate and the Architectural mathematical Departmentmodel. The attests the have University been performed of Parma. on The the pumptest bench object in ofthe study Laboratory is the variableof the Engineering displacement and ® axialArchitectural piston pump Department Casappa atMVP60-84, the University which of is Parma. installed The in a pump Load Sensingobject of (LS) study circuit. is the The variable main characteristicsdisplacement of axial the pistonpump are pump listed Casappa in Table® 1 MVP60-84,. The pump which is also is equipped installed with in a many Load sensors Sensing with (LS) thecircuit. aim ofThe acquiring main characteristics the following of data:the pump suction are fluid listed temperature; in Table 1. The delivery pump pressure; is also equipped swash plate with angularmany sensors position; with torque the absorbed aim of acquiring by the shaft; the following shaft speed; data: delivery suction flow fluid rate temperature; and drain flow delivery rate. Thepressure; hydraulic swash scheme plate ofangular the experimental position; torque layout absorbed is reported by inthe Figure shaft;7 shaft and thespeed; transducers delivery featuresflow rate are shown in Table2. Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x FOR PEER REVIEW 9 of 18

Energiesand drain2019, flow12, 1377 rate. The hydraulic scheme of the experimental layout is reported in Figure 7 and9 of the 18 transducers features are shown in Table 2.

Table 1. Pump characteristics. Table 1. Pump characteristics. Maximum displacement 84.7 cm3/rev Maximum displacement 84.7 cm3/rev Continuous:Continuous: 250 250 MaximumMaximum outlet outlet pressure pressure Intermittent:Intermittent: 280 280 bar bar Peak:Peak: 315 315 MaximumMaximum speed speed 2500 2500 r/min r/min NumberNumber of pistons of pistons 9 9

Figure 7. Scheme of the experimental layout. Figure 7. Scheme of the experimental layout. Table 2. Transducers features. Table 2. Transducers features. Variable Sensor Main Features Variable Sensor Main Features◦ ◦ Ts Thermistor Pt100 0–100 C ± 0.2 C Ts Thermistor Pt100 0–100 °C ± 0.2 °C Ps Pressure Transducer 0–10 bar ± 0.3% FS pd, p Ps Pressure Pressure TransducerTransducer 0–10 0–400 bar bar ± 0.3%± 0.5% FS FS . LS pd, pLS Pressure Transducer 0–400 bar ± 0.5% FS Vdelivery Flow Meter 0.1–150 L/min ± 0.3% measured value . ̇ VdrainV FlowFlow MeterMeter 0.1–150 0.01–80 L/min L/min ± 0.3%± 0.3% measured measured value value α Angular sensor 0◦–360◦ ± 0.02◦ ω V̇ SpeedFlow sensorMeter 0.01–80L/min accuracy ± 0.3% measured class 0.05 value α Angular sensor 0°–360° ± 0.02° 6. Model Validation ω Speed sensor accuracy class 0.05 The fluid dynamic model, system Equation (11), and the dynamic model, Equation (12), are solved simultaneously.6. Model Validation The models have been realized in Simulink® environment (version 9.1, MathWorks Inc., Natick,The fluid MA, dynamic USA) by model, adopting system a variable-step Equation (11), solver and theode23tb dynamic, with model, a maximum Equation step (12), size are of −4 −3 1.0solved× 10 simultaneously.and with relative The and models absolute have tolerances been realized of 1.0 × in10 Simulink. ® environment (version 9.1, MathWorksBoth the Inc, fluid Natick, dynamic MA, model USA) and by adopting the dynamic a variable-step model have solver been validatedode23tb, with by comparison a maximum with step experimentalsize of 1.0× 10 data. and For with the relative fluid dynamic and absolute model, tolerances several tests of 1.0 have × 10 been. conducted under many differentBoth working the fluid conditions; dynamic model the results and the are dynamic presented model in Figure have8a,b, been where validated a single by shaftcomparison revolution with isexperimental shown. The deliverydata. For mean the fluid pressure dynamic of Figure model,8b is several higher thantests thehave mean been pressure conducted of case under reported many indifferent Figure8 workinga. The pressure conditions; has beenthe results divided are bypresented a reference in Figure value 8a,b, for confidential where a single reasons. shaft Basedrevolution on theis shown. analysis The of thesedelivery cases, mean a satisfactory pressure of agreement Figure 8b is between higher predictedthan the mean results pressure and experiments of case reported data, inin relation Figure 8a. to the The target pressure of the has model, been isdivided found. by a reference value for confidential reasons. Based on the analysis of these cases, a satisfactory agreement between predicted results and experiments data, in relation to the target of the model, is found.

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(a(a) ) (b (b) )

FigureFigure 8. 8. ( a((a)a )Delivery) Delivery Delivery pressure pressure pressure at at at1500 1500 1500 r/min, r/min, r/min, swash swash swash plate plate plate position position position 12.7°; 12.7°; 12.7 ( b(b)◦ )Delivery;( Deliveryb) Delivery pressure pressure pressure at at 2000 2000 at ◦ r/min,2000r/min, r/min, swash swash swashplate plate position plateposition position 12.8°. 12.8°. 12.8The The mean. mean The meanpressure pressure pressure of of case case of ( caseb(b) )is is (higherb higher) is higher than than thanthe the one one the of oneof case case of case( a(a).). (a).

RegardingRegarding the the dynamic dynamic model, model, several several step step tests tests have have been beenbeen performed performedperformed under under many manymany different differentdifferent workingworkingworking conditions conditions and and and some some some of of ofthe the the results results results are are arereported reported reported in in Figure Figure in Figure 9a,b, 9a,b,9 a,b,where where where the the time thetime timescale scale scalehas has been been has normalizedbeennormalized normalized for for confidential confidential for conﬁdential reasons. reasons. reasons. Both Both the the Both step-up step-up the step-up and and the the and step-down step-down the step-down phases phases phases are are simulated simulated are simulated in in an an appropriateinappropriate an appropriate manner manner manner for for each each for test. test. each In In test. general, general, In general, the the model model the modelis is able able isto to able predict predict to predict properly properly properly the the swash swash the swashplate plate position.plateposition. position.

(a(a) ) (b (b) )

FigureFigure 9. 9. ( a((a)a ))Step Step Step Test: Test: Test: 1500 1500 1500 r/min, r/min, r/min, p pd d p= = d150 150= 150bar, bar, bar, spool spool spool opening opening opening 23–48%; 23–48%; 23–48%; ( b(b) )Step (Stepb) StepTest: Test: Test: 2000 2000 2000r/min, r/min, r/min, p pd d= = 50p50d bar, =bar, 50 spool spool bar, spoolopening opening opening 28–47%. 28–47%. 28–47%. 7. Active Control of the Pressure Ripple through the Swash Plate Control 7.7. Active Active Control Control of of the the Pressure Pressure Ripple Ripple through through the the Swash Swash Plate Plate Control Control The possibility of reducing the pressure ripple by continuously changing the inclination of the TheThe possibility possibility of of reducing reducing the the pressure pressure ripple ripple by by continuously continuously changing changing the the inclination inclination of of the the swash plate is presented. The basic principle consists of changing the tilt angle of the swash plate swashswash plate plate is is presented. presented. The The basic basic principle principle consists consists of of changing changing the the tilt tilt angle angle of of the the swash swash plate plate around the equilibrium position to properly modify the machine displacement. In this way the law aroundaround the the equilibrium equilibrium position position to to properly properly modify modify the the machine machine displacement. displacement. In In this this way way the the law law of motion of the pistons and the instantaneous delivery flow rate are altered: as a consequence, ofof motion motion of of the the pistons pistons and and the the instantaneous instantaneous delivery delivery flow flow rate rate are are altered: altered: as as a a consequence, consequence, the the the delivery pressure profile is modified. In order to obtain a variation on the swash plate angular deliverydelivery pressure pressure profile profile is is modified. modified. In In order order to to obtain obtain a a variation variation on on the the swash swash plate plate angular angular position, the active control system actuates the servo valve. The control strategy consists of decreasing position,position, the the active active control control system system actuates actuates the the servo servo valve. valve. The The control control strategy strategy consists consists of of decreasing decreasing the pump displacement when the instantaneous pressure is higher than the mean value, vice versa thethe pump pump displacement displacement when when the the instantaneous instantaneous pressure pressure is is higher higher than than the the mean mean value, value, vice vice versa versa when the pressure is lower. whenwhen the the pressure pressure is is lower. lower. The servo valve is actuated by means of a square wave signal with a proper phase shift with the TheThe servo servo valve valve is is actuated actuated by by means means of of a a square square wave wave signal signal with with a a proper proper phase phase shift shift with with the the delivery pressure, in order to take into account the response time of the whole mechanic and hydraulic deliverydelivery pressure, pressure, in in order order to to take take into into account account the the response response time time of of the the whole whole mechanic mechanic and and systems. A square wave signal has been chosen because of its simplicity of implementation since hydraulichydraulic systems. systems. A A square square wave wave signal signal has has been been chosen chosen because because of of its its simplicity simplicity of of implementation implementation this is a preliminary work aimed to verify the potential of the proposed solution. In Figure 10 the sincesince this this is is a a preliminary preliminary work work aimed aimed to to verify verify the the potential potential of of the the proposed proposed solution. solution. In In Figure Figure 10 10 thethe diagram diagram representing representing the the control control strategy strategy adopted adopted to to obtain obtain the the swinging swinging of of the the oscillating oscillating plate plate

EnergiesEnergies 2019 2019, ,12 12, ,x; x; doi: doi: FOR FOR PEER PEER REVIEW REVIEW www.mdpi.com/journal/energies www.mdpi.com/journal/energies Energies 2019, 12, 1377 11 of 18 Energies 2019, 12, x FOR PEER REVIEW 11 of 18 Energies 2019, 12, x FOR PEER REVIEW 11 of 18 is presented: in detail, the blue block represents the algorithm which generates the square wave signal diagram representing the control strategy adopted to obtain the swinging of the oscillating plate is for the actuation of the servo valve, starting from the pressure at the delivery side and the mean ispresented: presented: in in detail, detail, the the blue blue block block represents represents the the algorithm algorithm which which generates generates the the square square wave wave signal signal for pressure value required. forthe the actuation actuation of the of servothe servo valve, valve, starting starting from thefrom pressure the pressure at the deliveryat the delivery side and side the meanand the pressure mean pressurevalue required. value required.

Figure 10. Scheme of the control strategy adopted for moving the pump swash plate. FigureFigure 10. SchemeScheme of of the the control control strategy strategy adopted for moving the pump swash plate. The fundamental driving frequency of the servo valve, and therefore of the signal, is expressed The fundamental driving frequency of the servo valve, and therefore of the signal, is expressed by by theThe following fundamental equation: driving frequency of the servo valve, and therefore of the signal, is expressed the following equation: by the following equation: n푛·9∙ 9 f푓== (18)(18) 푛60∙609 푓 = (18) wherewheren nis is the the shaft shaft speed speed and and 9 9 is is the the number number of of pistons pistons60 for for the the case case considered. considered. whereInIn n order isorder the to shaftto evaluate evaluate speed the andthe potentiality 9potentiality is the number of of the the of proposed pistonsproposed for solution, solution,the case the considered.the first first simulations simulations have have been been carriedcarriedIn order out out assuming assuming to evaluate an an ideal idealthe potentiality model model representing representing of the proposed a a system system solution, where where the the the masses masses first simulations of of the the pump pump have and and of beenof the the carriedservoservo valve valve out assuming components components an are idealare not not model present present representing and and the the cut-off cut-off a system frequency frequency where ofof the the servomassesservo valvevalve of the isis supposedpumpsupposed and infinite.infinite. of the servoThe valveThe servo servo components valve valve driving driving are not signal signal present is is generated andgenerated the cut-off byby a afrequency proportionalproportional of the controller, controller, servo valve which which is supposed has has the the purpose infinite.purpose ofof minimizing minimizingThe servo valve thethe errorerror driving defined defined signal as as is the the generated difference difference by between betweena proportional the the instantaneous instantaneous controller, which delivery delivery has pressure pressurethe purpose and and ofthethe minimizing mean mean desired desired the pressure: errorpressure: defined thisthis signalassignal the isdifferenceis addedadded to tobetween thethe one one the which which instantaneous allows allows the the pump deliverypump to to pressure work work in in load andload thesensingsensing mean logic, logic, desired introduced introduced pressure: in in this Section Section signal4 .4. Theis The added simulation simulation to the results oneresults which are are presented allowspresented the in inpump the the following followingto work figuresin figures load sensingwherewhere the thelogic, mass mass introduced flow flow rate, rate, thein the Section delivery delivery 4. pressure Thepressure simulation and and the the results Fast Fast FourierFourier are presented TransformTransform in (FFT)the (FFT) following amplitude amplitude figures have have wherebeenbeen divided dividedthe mass by by flow a a reference reference rate, the value deliveryvalue for for confidential pressure confidential and reasons. thereasons. Fast The Fourier The displacement displacement Transform control (FFT) control system amplitude system allows allows have the beenswashthe swashdivided plate plate to by vary ato reference itsvary tilt its angle, tiltvalue angle, around for confidentialaround the equilibrium the equilibrium reasons. position, The position, displacement of 5 degrees of 5 degrees in control amplitude, in system amplitude, as allows shown as theinshown Figure swash in 11 plateFigure: in thisto 11:vary way, in itsthis the tilt way, instantaneous angle, the around instantaneous flow the rateequilibrium flow has beenrate position, changed,has been of changed, as 5 showndegrees as in in shown Figure amplitude, 12in a.Figure It as is shownevident12a. It inis that, Figureevident with 11: that, respect in thiswith to way, therespect casethe toinstantaneous without the case the without active flow control, therate active has the been control, flow changed, rate the fluctuations flow as shown rate fluctuations havein Figure been 12a.reducedhave It beenis significantly.evident reduced that, significantly. Thiswith hasrespect a veryThis to positivethehas casea very effectwithout positive on the the effect active delivery on control,the pressure delivery the oscillations,flowpressure rate oscillations, fluctuations as shown as haveinshown Figure been in 12 Figurereducedb, where 12b, significantly. where the pressure the pressure This ripple has ripple profilea very profilepositive presents presents effect a marked on a themarked improvement. delivery improvement. pressure In particular,oscillations, In particular, it as is shownpossibleit is possible in toFigure notice to 12b,notice a reduction where a reduction the in pressure the in pressure the ripple pressure oscillation profile oscillation presents amplitude amplitude a marked of the ofimprovement. 67.5% the 67.5% compared compared In particular, to the to case the itwithoutcase is possible without active to control, activenotice control,a as reduction well as as the wellin dampingthe as pressure the of damping the oscillation oscillations of the amplitude atoscillations a higher of the frequency at 67.5% a higher compared as confirmed frequency to the by as casetheconfirmed FFT without reported by active the in FFT Figure control, reported 13 . as in well Figure as the 13. damping of the oscillations at a higher frequency as confirmed by the FFT reported in Figure 13.

FigureFigure 11. 11.Ideal Ideal model. model. Swash Swash plate plate position. position. Shaft Shaft speed speed 1000 1000 r/min. r/min. Figure 11. Ideal model. Swash plate position. Shaft speed 1000 r/min.

Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, 1377 12 of 18 EnergiesEnergies 2019 2019, 12, 12, x, xFOR FOR PEER PEER REVIEW REVIEW 12 12 of of 18 18

(a()a ) (b (b) )

FigureFigure 12. 12. (a ()a Ideal) Ideal model. model. Delivery Delivery mass mass ﬂowflow flow rate.rate. rate. Shaft Shaft speedspeed speed 10001000 1000 r/min;r/min; r/min; ( (b (b)) Ideal) Ideal Ideal model. model. Delivery Delivery Delivery pressure.pressure. Shaft Shaft speed speed 1000 1000 r/min. r/min.

Figure 13. Ideal model. Delivery pressure FFT. Shaft speed 1000 r/min. FigureFigure 13. 13. Ideal Ideal model. model. Delivery Delivery pressure pressure FFT. FFT. Shaft Shaft speed speed 1000 1000 r/min. r/min. The results based on an ideal model have successfully established the possibility of performing TheThe results results based based on on an an ideal ideal model model have have successfully successfully established established the the possibility possibility of of performing performing an active control of the pressure ripple, by the movement of the swash plate, and its effectiveness. anan active active control control of of the the pressure pressure ripple, ripple, by by the the movement movement of of the the swash swash plate, plate, and and its its effectiveness. effectiveness. Subsequently, the possibility of implementing this type of active control in a model of the pump Subsequently,Subsequently, the the possibility possibility of of implementing implementing this this type type of of active active control control in in a amodel model of of the the pump pump including the dynamic effects has been studied. It is straightforward that the type of control presented includingincluding the the dynamic dynamic effects effects has has been been studied. studied. It It is is straightforward straightforward that that the the type type of of control control presented presented in the ideal case is not physically possible. In fact, to achieve similar results, the displacement control inin the the ideal ideal case case is is not not physically physically possible. possible. In In fact, fact, to to achieve achieve similar similar results, results, the the displacement displacement control control system and the actuator itself should intervene at very high frequencies, continuously following the systemsystem and and the the actuator actuator itself itself should should intervene intervene at at very very high high frequencies, frequencies, continuously continuously following following the the signal provided by the controller. Considering the model of the real system, where the mass of the signalsignal provided provided by by the the controller. controller. Considering Considering the the model model of of the the real real system, system, where where the the mass mass of of the the components is considered in both the pump and the servo valve model, the results about the swash componentscomponents is is considered considered in in both both the the pump pump and and the the servo servo valve valve model, model, the the results results about about the the swash swash plate angular position are reported in Figure 14, without and with the control system. The displacement plateplate angular angular position position are are reported reported in in Figure Figure 14, 14, without without and and with with the the control control system. system. The The control system is able to change the amplitude of the angular position of the swash plate around its displacementdisplacement control control system system is is able able to to change change the the amplitude amplitude of of the the angular angular position position of of the the swash swash equilibrium position of only 0.7 degrees; therefore, in the simulated real case the obtainable maximum plateplate around around its its equilibrium equilibrium position position of of only only 0.7 0.7 degrees; degrees; therefore, therefore, in in the the simulated simulated real real case case the the amplitude of the swash plate oscillations is much lower with respect to the ideal case (5 degrees). obtainableobtainable maximum maximum amplitude amplitude of of the the swash swash plate plate oscillations oscillations is is much much lower lower with with respect respect to to the the The delivery flow rate is shown in Figure 15a, where only a slight reduction in the amplitude can idealideal case case (5 (5 degrees). degrees). The The delivery delivery flow flow rate rate is is shown shown in in Figure Figure 15a, 15a, where where only only a aslight slight reduction reduction in in be appreciated, but without any significant damping. The same consideration can be made for the thethe amplitude amplitude can can be be appreciated, appreciated, but but without without any any significant significant damping. damping. The The same same consideration consideration can can pressure profile presented in Figure 15b, showing a limited decrease in the amplitude of the pressure bebe made made for for the the pressure pressure profile profile presented presented in in Figure Figure 15b, 15b, showing showing a alimited limited decrease decrease in in the the amplitude amplitude oscillations around the mean value, also confirmed by the analysis of the Fourier transform visible in ofof the the pressure pressure oscillations oscillations around around the the mean mean value, value, also also confirmed confirmed by by the the analysis analysis of of the the Fourier Fourier Figure 16, but leaving almost unaltered the high frequency components. transformtransform visible visible in in Figure Figure 16, 16, but but leaving leaving almost almost unaltered unaltered the the high high frequency frequency components. components.

EnergiesEnergies 2019 2019, 12, 12, x;, x; doi: doi: FOR FOR PEER PEER REVIEW REVIEW www.mdpi.com/journal/energies www.mdpi.com/journal/energies EnergiesEnergies2019 2019, 12, 12, 1377, x FOR PEER REVIEW 13 13 of of 18 18 Energies 2019, 12, x FOR PEER REVIEW 13 of 18 Energies 2019, 12, x FOR PEER REVIEW 13 of 18

Figure 14. Swash plate position. Shaft speed 1000 r/min. Figure 14. Swash plate position. Shaft speed 1000 r/min. Figure 14. Swash plate position. Shaft speed 1000 r/min.

(a) (b) (a) (b) (a) (b) Figure 15. (a) Delivery mass flow rate. Shaft speed 1000 r/min; (b) Delivery pressure. Shaft speed 1000 Figure 15. ((aa)) Delivery Delivery mass mass flow ﬂow rate. rate. Shaft Shaft speed speed 1000 1000 r/min; r/min; (b) Delivery (b) Delivery pressure. pressure. Shaft Shaft speed speed 1000 Figurer/min. 15. (a) Delivery mass flow rate. Shaft speed 1000 r/min; (b) Delivery pressure. Shaft speed 1000 1000r/min. r/min. r/min.

FigureFigure 16. 16.Delivery Delivery pressure pressure FFT. FFT. Shaft Shaft speed speed 1000 1000 r/min. r/min. Figure 16. Delivery pressure FFT. Shaft speed 1000 r/min. Figure 16. Delivery pressure FFT. Shaft speed 1000 r/min. Therefore,Therefore, with with the the real real pump pump model, model, the the improvements improvements that that can can be be obtained obtained are are quite quite limited. limited. Therefore, with the real pump model, the improvements that can be obtained are quite limited. TheThe mainTherefore, main cause cause with is is the the the narrow narrow real pump variation variation model, of of the the improvements angular angular position position that of of can the the be plate plate obtained around around are the the quite equilibrium equilibrium limited. The main cause is the narrow variation of the angular position of the plate around the equilibrium Thepositionposition main (0.7 cause(0.7 degrees), degrees), is the narrow which which does variationdoes not not allow ofallow the obtaining angularobtaining position a a variation variation of the of of theplate the motion motion around law law the of ofequilibrium the the pistons pistons position (0.7 degrees), which does not allow obtaining a variation of the motion law of the pistons position (0.7 degrees), which does not allow obtaining a variation of the motion law of the pistons Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, 1377 14 of 18 Energies 2019, 12, x FOR PEER REVIEW 14 of 18 sufficientsufficient to to significantly significantly modify modify the the delivered delivered flow flow rate rate and and therefore therefore the the pressure pressure ripple. ripple. In In fact, fact, in in orderorder to to reduce reduce the the displacement, displacement, it is necessaryit is necessary to pressurize to pressurize the actuator the actuator chamber chamber with the with oil coming the oil fromcoming the highfrom pressure the high line. pressure On the line. contrary, On the to contrary, increase to the increase displacement, the displacement, it is essential it tois drainessential part to ofdrain the oil part to theof the tank. oil Into thethe lattertank. case,In the the latter forces case, acting the forces on the acting actuator, on whichthe actuator, define itswhich equilibrium define its position,equilibrium are given position, by the are action given of by the the pumping action of elements, the pumping due to elements, the eccentricity due to ofthe the eccentricity cylinder block of the assembly,cylinder andblock by assembly, the spring and inside by the actuator.spring inside In the the considered actuator. pumpIn the thereconsidered is a disparity pump betweenthere is a thedisparity magnitudes between of the the forces magnitudes on the of actuator the forces in the on twothe actuator cases. During in the thetwo decreasecases. During phase the the decrease forces exertedphase the by theforces fluid exerted is much by greaterthe fluid and is much this allows greater changing and this the allows position changing of the the oscillating position plateof the veryoscillating quickly. plate Conversely, very quickly. during Conversely, the phase of during increase the of phase the displacement, of increase of being the thedisplacement, force due to being the springthe force lower, due a to longer the spring time interval lower, a is longer necessary. time Thisinterval has is negative necessary. consequences This has negative on the possibilityconsequences of havingon the greater possibility ranges of ofhaving the angular greater position ranges ofof thethe swashangular plate, position even atof lowthe speedswash whereplate, theeven driving at low frequencyspeed where is lower. the driving frequency is lower. ToTo overcome overcome this this problem, problem, different different actuation actuation solutions solutions could could be be investigated, investigated, but but in in this this paper, paper, forfor brevity, brevity, the the desired desired effect effect has has been been simply simply modelled modelled by increasing by increasing the spring the spring stiffness, stiffness, omitting omitting any pumpany pump functional functional drawbacks. drawbacks. In this In way this itway is possible it is possible to obtain to obtain better better dynamics dynamics in the in phase the phase of the of displacementthe displacement increment, increment, as can as be can seen be in seen Figure in 17Figure, where 17, a where comparison a comparison between thebetween angular the position angular calculatedposition calculated with both thewith real both pump the modelreal pump and themodel modified and the spring modified stiffness spring is shown. stiffness As is proof shown. of the As bestproof dynamic of the best performance dynamic achieved,performance the curveachieved, relating the curve to the relating modified to case the hasmodified a greater case slope has a during greater theslope phase during of increment the phase of theof increment displacement. of the In displacement. this way, it is alsoIn this possible way, toit obtainis also apossible greater excursionto obtain a ingreater the angular excursion position in the of theangular swash position plate in of the the same swash time plate interval in the in same the modified time interval case within the respect modified to thecase real with case; respect as a matter to the ofreal fact case; the as oscillation a matter of amplitude fact the oscillation reaches the amplitude value of 1.1reaches degrees, the value Figure of 18 1.1, improvingdegrees, Figure the previous 18, improving result of the 0.7 degrees.previous Inresult the graphof 0.7 thedegrees. mean In value the hasgraph been the subtracted mean value from has eachbeen curve subtracted in order from to overlap each curve them. in order to overlap them.

FigureFigure 17. 17.Comparison Comparison of of the the angular angular position position of of the the swash swash plate plate between between the the real real pump pump model model and and thethe modiﬁed modified one. one.

TheThe delivery delivery flow flow rate rate shows shows aa markedmarked improvementimprovement duedue toto thethe swash plate oscillation with with a aconsiderable considerable reduction reduction in amplitude, as shown inin FigureFigure 1919a,a, andand consequently,consequently, also also the the delivery delivery pressurepressure profile profile presents presents a a significant significant decrease,decrease, equalequal to 57% compared to to the the case case in in the the absence absence of ofcontrol, control, represented represented in in Figure Figure 19b.19b. Furthermore, the higherhigher stiffnessstiffness ofof thethe swash swash plate plate actuation actuation systemsystem limits limits the the amplitudes amplitudes of of the the high high frequency frequency components components of of the the pressure pressure ripple. ripple. The The harmonic harmonic componentscomponents present present at at high high frequency frequency are are completely completely attenuated attenuated and and those those at at medium medium frequency frequency are are damped,damped, as as demonstrated demonstrated by by the the FFT FFT in in Figure Figure 20 20..

Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x FOR PEER REVIEW 15 of 18 Energies 2019, 12, x FOR PEER REVIEW 15 of 18 Energies 2019, 12, 1377 15 of 18 Energies 2019, 12, x FOR PEER REVIEW 15 of 18

Figure 18. Real pump model with higher spring stiffness. Swash plate position. Shaft speed 1000 Figure 18. Real pump model with higher spring stiffness. Swash plate position. Shaft speed 1000 r/min. Figurer/min.Figure 18. Real pumppump model model with with higher higher spring spring stiffness. stiffness. Swash Swash plate plate position. position. Shaft speedShaft 1000speed r/min. 1000 r/min.

(a) (b) (a) (b)

Figure 19. Real pump model with higher spring stiffness. (a) Delivery mass flow rate. Shaft speed Figure 19. Real pump(a) model with higherhigher springspring stiffness.stiffness. (a) DeliveryDelivery massmass (b) ﬂowflow rate.rate. Shaft speedspeed 1000 r/min; (b) Delivery pressure. Shaft speed 1000 r/min. b Figure1000 r/min; 19. Real ( (b)) Delivery Deliverypump model pressure. pressure. with Shaft Shafthigher speed speed spring 1000 1000 stiffness. r/min. r/min. (a) Delivery mass flow rate. Shaft speed 1000 r/min; (b) Delivery pressure. Shaft speed 1000 r/min.

FigureFigure 20. 20.Real Real pump pump model model with with higher higher spring spring stiffness. stiffness. Delivery Delivery pressure pressure FFT. Shaft FFT. speed Shaft 1000 speed r/min. 1000 Figure 20. Real pump model with higher spring stiffness. Delivery pressure FFT. Shaft speed 1000 r/min. Figurer/min. 20. Real pump model with higher spring stiffness. Delivery pressure FFT. Shaft speed 1000 r/min. Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies

Energies 2019, 12, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies Energies 2019, 12, 1377 16 of 18

8. Conclusions In this paper the possibility of reducing the delivery pressure ripple in piston pumps through a controlled oscillating motion of the swash plate has been theoretically investigated. The mathematical model of an axial piston pump has been presented and applied to carry out the analysis. The model has been experimentally validated in terms of delivery pressure ripple and step response. Finally, a servo valve has been modelled for implementing the high dynamic control of the pump displacement. Firstly, the validity of the working principle has been demonstrated for an ideal model, where the components have been assumed without mass. The theoretical analysis has shown interesting results with a significant reduction of flow rate oscillations and, consequently, a markable decrease of the pressure ripple. If the real masses are considered, an attenuation of the pressure oscillation is still possible, even if quite limited. The reason is the low dynamic response of the displacement control of the reference pump due to the high inertia of the components. Nevertheless, it has been demonstrated that with a quite simple modification of the displacement control, in order to reach a very short time response during both the increase and the decrease phase of the pump displacement, good results can be achieved. Hence, it is reasonable that better results could be obtained if some components of the displacement control were properly redesigned for this specific application. Overall, although the use of a servo valve implies an increment of cost, the principle of the active control of the pressure ripple through the oscillations at high frequency of the swash plate seems promising and deserves further in-depth analyses.

Author Contributions: P.C. conceived the study and wrote the draft paper, supervising the entire activity; M.P. and F.S. performed the experimental tests, applying the methodology, in order to obtain the results; M.R. analyzed the results and examined the final document. Funding: This research received no external funding. Acknowledgments: The authors would like to acknowledge the active support of this research by Casappa S.p.A., Parma, Italy. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

2 Acil Piston crown surface [m ] a Piston offset [m] α Swash plate angle [rad] B Bulk modulus [Pa] c Velocity [m/s] dp Piston diameter [m] ξ Abscissa FB Body forces [N] hp Radial clearance between piston and guide [m] Lp Length of contact between piston and chamber [m] µ Fluid dynamic viscosity [Pa·s] . m Mass flow rate [kg/s] Mp Piston mass [kg] n Shaft speed [r/min] N Number of cylinders Ω Cross sectional area [m2] p Pressure [Pa] . Q Momentum flux [kg·m/s2] ρ Density [kg/m3] R Piston-pitch radius [m] RG Specific gas constant [J/(kg·K)] t Time [s] Energies 2019, 12, 1377 17 of 18

TG Temperature [K] θ Block cylinders angular position [rad] v Volume for the unit of mass [m3/kg] V Volume [m3] Vc Cylinder volume [m3] . V Volume ﬂow rate [m3/s] y Gas to liquid mass fraction . zi Piston velocity [m/s] Abbreviation F&E Filling and emptying FFT Fast Fourier Transform Subscript

0 Reference value

References

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