On the assessment of lumped parameter models for gear pump performance prediction

Simulation Modelling Practice and Theory 99 (2020) 102008

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On the assessment of lumped parameter models for gear pump performance prediction

T

Mattia Battarra , Emiliano Mucchi ⁎

University of Ferrara, Engineering Dept. Via G. Saragat, 1 - 44122 Ferrara, Italy

ARTICLE INFO

ABSTRACT

Keywords: Gear pumps Efficiency Validation Trace-driven simulation,

The present work describes a statistical approach for the assessment of discrete models for gear pump efficiency prediction. A critical discussion is performed on the input data assumptions that are commonly adopted to carry out the analysis, with particular regards to the actual bearing clearances and casing radial clearances. The proposed model adopts well-established techniques for simulating the pump fluid-dynamics in association with a novel approach, which allows us to take into account the effects produced by the gearpair micromotions. Moreover, the possibility to study both spur and helical gears, as well as non-unitary transmission ratio gearpairs, has been included, in order to ensure the wide applicability of the model in modern design solutions. Measured data obtained from an extended experimental campaign, involving 20 nominally identical samples of the same pump design, are used to establish the assessment procedure. Each sample is geometrically characterized by measuring the actual clearances at the end of the production process and then tested at 14 different working conditions, leading to 280 tests. The entire set of test conditions is then adopted to carry out a trace-driven simulation analysis, showing that the lumped parameter approach may reach different levels of accuracy depending on both the analyzed working conditions and the simulated pump samples. The results underline that reliability and accuracy of this kind of model should be evaluated with respect to a population of pumps, defined on the basis of a statistical approach, since referring to a single pump sample may easily lead to an over/under-estimate of the quality of the proposed model. In addition, they also demonstrate that real clearance values need to be included in the model to obtain high fidelity estimations.

1. Introduction Fluid dynamic performance and dynamic behavior of gear pumps are related to each other and strictly connected to several factors, e.g. clearances between components, gear shape, gear meshing, pressure ripple. For this reason, the definition of empirical correlations between dynamics, fluid dynamics and design choices would require extremely demanding experimental campaigns, both in terms of time and costs. In order to overcome this practical limitation, starting from the 80s until the present day, numerical models based on Lumped Parameter (LP) approaches have been proposed to study performance of gear pumps by focusing the attention on volumetric efficiency, pressure distribution around the gears and outlet pressure ripple. In the last decade, the remarkable spread of Computational Fluid Dynamic (CFD) models may have made zero-dimensional models appear to be obsolete, but in the field of volumetric machines the practical application of CFD models is still limited by the considerable computational effort

⁎

Corresponding author. E-mail address: [email protected] (M. Battarra).

https://doi.org/10.1016/j.simpat.2019.102008 Received 30 January 2019; Received in revised form 6 September 2019; Accepted 15 October 2019 Available online 15 October 2019 1569-190X/ © 2019 Elsevier B.V. All rights reserved.

Simulation Modelling Practice and Theory 99 (2020) 102008

M. Battarra and E. Mucchi

Nomenclature m τ J Fb(i) F p(i) x y

T p(i) r Fm Td(i) Tshaft θ β ω ηHM p B t V w n

h L b μ ρ Qin Qout μ μ0 U Q A ηV Vm mn Sd z d α Toil pout Qsim Qmeas 1,2

gear mass transmission ratio gear polar mass journal bearing reaction applied to gear i pressure load applied to gear i translational motion of the gear, normal to the gear mesh line of action translational motion of the gear, along the gear mesh line of action pressure torque applied to gear i base radius gear mesh force friction torque related to gear i driving torque rotational motion of the gear gear helix angle angular velocity in rad/s hydro-mechanical efficiency oil pressure Bulk’s modulus time volume of the generic gear pocket channel width pump angular velocity in rpm

channel height channel length gear facewidth oil dynamic viscosity oil density volumetric flow rate entering the control volume volumetric flow rate leaving the control volume oil dynamic viscosity true mean mean oil velocity within the channel volumetric flow rate porting area between control volumes volumetric efficiency gear pump theoretical displacement gear normal module standard deviation tooth number model performance parameter significance level oil temperature pump delivery pressure simulated delivered volumetric flow rate simulated delivered volumetric flow rate subscripts and superscripts representing driving and driven gear, respectively

required. For this reason, LP methodologies continue to be improved and numerical results are compared to measured data in order to give evidence of the reliability of the proposed approaches. The very first works on this subject pertain to Foster K., Mancò S. and Bonacini C. In particular, Foster K. et al. in [1] proposed a mathematical model to predict pressure evolution in gear pockets by using a numerical approach. The fluid was considered to be theoretically incompressible and compressibility effects were taken into account by means of an iterative procedure. The trapped volume was assumed ideally relieved in order to neglect overshooting pressure phenomena as well as cavitation. Mancò S. et al. in [2] introduced a different methodology, based on an eulerian approach, to determine the oil pressure inside gear pockets by solving a continuity equation where compressibility effects were accounted by the Bulk’s modulus. Theoretical demonstration of the adopted continuity equation was provided in the appendix of the same work. A similar approach was later proposed by Bonacini C. et al. in [3], but the study was particularly focused on the leakage discretization scheme adopted to compute flow rates exchanged between consecutive gear pockets in the pressurizing zone. Since these studies represented the first attempts to calculate the pressure distribution around gears with discrete dynamic models, the described works mainly concentrated on enlightening the face validity of the proposed approaches, by giving explanation for the so called structural assumptions of the model, i.e. assumptions related to how the system behaves (see ref. [4], Chapter 10). These first works concentrated on addressing the face validity of the overall modeling approach, with particular reference to the theoretical bases of the adopted equations and the numerical values of some crucial parameters such as the discharge flow coefficients related to leakage definition. Subsequent studies, on the contrary, limited the face validity assessment on the original aspects introduced in each work and put much more effort in validating the input-output transformations of the models, i.e. in validating the results by means of experimental comparisons. In particular, Mancò S. et al. in [5] provided a detailed explanation of the model structure and a comparison between numerical results and measured data referring to suction and delivery pressure ripple. Later, the physical behavior of the oil pressure evolution around gears was addressed in [6] by means of a dedicated experimental campaign. An analogous approach was then followed by Casoli P. et al. in [7], where the authors compared the predicted outlet pressure ripple with experimental data referring to a single pump prototype at a single working condition. Similar limitations characterize the validation of further approaches proposed with the purpose to take into account various physical aspects. Within this framework, Mucchi et al. in [8,9] introduced a mathematical model to predict the Static Equilibrium Position (SEP) of the gear pair with respect to the pump body. The method was further developed by Vacca et al. in [10], which actually seems to be the one proposing the most complete validation process. In particular, the authors in [10] proposed a graphical evaluation of the model capability to predict the steady state operating condition of both pumps and motors, as well as delivery pressure oscillations and casing radial wear due to break-in processes. The assessment takes into account a large variety of working conditions and two different machine designs, however, it is still limited by the fact that measurements are referred to a single pump sample. Other relevant works focused on evaluating the presence of cavitation [11,12] and the use of helical gears and tandem pump configurations [13–15]. Eaton et al. [16] analyzed the pressure inside the trapping volume, with particular reference to aero-engine fuel pumps, underlining the lack of investigations on high-speed working conditions. Concurrently, other studies focused on 2

Simulation Modelling Practice and Theory 99 (2020) 102008

M. Battarra and E. Mucchi

lubrication phenomena [17,18] and gear balancing, in reference to the presence of journal bearings and lateral bearing blocks [19–23]. Despite the large amount and variety of experimental campaigns carried out throughout these studies, no objective validation tests have been presented to address the statistical accuracy and reliability of the proposed approaches. Validation is always performed by graphical comparison, independently on the reference parameter adopted to assess the model, which may vary from the delivery pressure ripple, to the suction pressure ripple and the volumetric efficiency. The described scenario enlightens the considerable effort dedicated to demonstrate the face validity of the introduced LP models and their overall validation based on graphical comparisons; however, it also underlines the lack of quantitative estimations. The zero-dimensional modeling of gear pumps is often evaluated by juxtaposing trends and time series that give an idea of the quality of the estimation, but a quantitative evaluation of the model accuracy and result reliability based on statistical tests is not present. Within this framework, it is the authors’ belief that further evaluations should also be performed to address the model data assumptions referring to the pump geometry values adopted as input parameters for these models. In this context, the present work attempts to demonstrate the necessity of a statistical approach for the quantitative assessment of the accuracy of this kind of model, by showing that, when the comparison is restricted on a single pump sample, results may easily lead to an over/under-estimate of the quality of the proposed method. This approach represents an original path with respect to the classical one, which has characterized the specialized literature since the first works till the latest one. In addition, the proposed validation method is used to assess the model data assumptions related to the pump radial clearances. According to Refs. [1,8–10], casing radial clearances and bearing clearances play a fundamental role in defining the pressure evolution around gears and the volumetric efficiency. These parameters, which are defined by design, are usually of the order of some tens of micrometers, but the accuracy limits of modern production processes require the practical definition of tolerance intervals that let them vary within the range of some micrometers or more. Moreover, the tolerance chain between casing radial clearances and bearing clearances may further worsen the phenomenon, with the practical result of producing pump samples that present non-negligible differences in terms of their micro-geometry. In order to assess the influence of these input parameters and evaluate their effect on the pump performance, a dedicated experimental campaign is described in the present work, involving tests on 20 nominally identical samples of a gear pump with helical teeth and nonunitary transmission ratio. Each sample is geometrically characterized by measuring the mean actual casing radial clearances and bearing clearances on both gears; data are statistically analyzed and their influence on the machine performance is addressed by testing each pump sample at 14 different working points. Experimental results enlighten the influence of slight variations of the radial clearance values on the pump performance, even if such values fall within the acceptable tolerance interval defined by design. This outcome constitutes a preliminary demonstration that the pump clearance variability should be taken into account in the process of effectively validating this kind of models. In order to further address this aspect, the entire experimental campaign, involving 280 working condition points, has been numerically reproduced with a dedicated LP model, which couples the gearpair radial movements to the pump fluid-dynamic behavior for predicting its volumetric efficiency. Within this framework, a careful attention is devoted to the description of the original aspects characterizing the model, i.e. the estimation of the gearpair micromotions and the iterative scheme proposed to solve them in conjunction with the pump fluid-dynamics. Numerical and experimental data are graphically compared, showing that the model accuracy may significantly depend on the analyzed sample. A statistical procedure based on confidence intervals is then proposed to quantify the model accuracy with respect to each analyzed working condition. Results demonstrate that a trustworthy validation approach requires to be based on the accurate reproduction of the pump radial clearances and extended to a population of pump samples, in contrast with the classical trend followed in the specialized literature, where validation is limited to the comparison with measurements taken from a single pump sample. The following Section describes the mathematical bases of the overall modeling approach. In particular, the first Subsection regards the mathematical model for calculating gearpair radial movements that has been coupled to the LP fluid dynamic model, while the subsequent one proposes the mathematical model adopted to calculate the pressure distribution around the gears. Third

Fig. 1. Dynamic model (damping associated with the shaft not shown). 3

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Subsection describes the whole model structure together with its iterative solving work-flow. The following Sections concern the experimental studies and the assessment of the simulation results, respectively. Finally, last Section is devoted to concluding remarks. 2. Overview of the model The present Section provides a brief description of the mathematical approach and the physical assumptions constituting the bases of the proposed model. Attention is particularly focused on the definition of the equation system adopted to model the gearpair micromotions, which assumes a key role in the accurate estimation of the pump volumetric efficiency. The entire model is based on the hypothesis of lumped element system, assuming that the behavior of the distributed system is well represented by a finite number of discrete entities. Finally, last Subsection focuses the attention on the assessment of the iteration scheme adopted to calculate the pressure field with respect to the instantaneous gearpair micromotions. 2.1. Gearpair radial movements The present Section introduces the modeling procedure defined to calculate gearpair radial movements with respect to the variable loads applied. The mechanical system has been discretized as shown in Fig. 1. Gear pumps for automotive applications are constituted by two gears, usually of the same tooth number, and their respective shafts, each one supported by a pair of journal bearings. An electric motor/internal combustion engine usually drives the pump by means of a flexible joint. Since the inertia of the pump is considerably smaller than that of the motor, the latter can be replaced with a fixed frame connected to the driving gear by a flexible shaft. This modeling solution is also allowed by the fact that the frequency range typically influenced by the first harmonics of the crankshaft angular speed is considerably lower than the frequency range that characterizes the first harmonics of the gear pump mesh frequency. Each gear i (i=1,2) is allowed to vibrate along the transverse plane of the gears and vibrate torsionally about its nominal rigid rotation i = i t where ωi is the nominal rotational speed of gear i. The non-linear dynamic model described in Fig. 1 leads to the following 6 DOF equation system: (1) (1) m1 x¨1 + Fbx + F px =0 (1) (1) m1 y¨1 + Fby + F py + Fm = 0

J1 ¨1 + Tshaft m2 x¨2 +

(2) Fbx

r1 Fm + Td(1) +

(2) F px

T p(1) = 0

=0

(2) (2) m2 y¨2 + Fby + F py

Fm = 0

J2 ¨2

T p(2) = 0

r2 Fm +

Td(2)

(1)

where x1 and x2 are the translational motions of gears normal to the gear mesh line of action, y1 and y2 are translational motions along the line of action, and θ1 and θ2 are the rotational vibrations about the nominal rotational angles ϑ1 and ϑ2, respectively. Terms (i ) (i ) , Fby mi, Ji and ri denote mass, polar mass moment of inertia and the base radius of gear i, respectively. Terms Fbx are the components (i ) (i ) (i ) of the journal bearing’s reaction supporting gear i, while F px , F py and T p are the variable pressure loads applied to gear i. Finally, Fm

represents the mesh stiffness, Tshaft is the driving torque and Td(i) is the friction torque referred to gear i. As already demonstrated in [8], for the purposes of the present model, the inertia terms in Eq. 1 can be neglected. This assumption does not affect the accuracy of the overall modeling approach since the gearpair torsional dynamics is highly damped by the oil surrounding the mechanism. The resulting low amplitude oscillations do not affect the oil pressure field and the connected loads applied on the pump. For this reason, the dynamic system described in Eq. 1 can be simplified as follows: (1) (1) Fbx + F px =0 (1) (1) Fby + F py + Fm = 0

Tshaft

r1 Fm + Td(1)

T p(1) = 0

(2) (2) Fbx + F px =0 (2) (2) Fby + F py

r2 Fm +

Td(2)

Fm = 0 T p(2) = 0

(2)

where the first two equations describe the radial motion of the driving gear and the fourth and fifth equations describe the radial motion of the driven one. The hypothesis adopted in Eq. 2 reduces the mathematical model describing the gearpair centers position to a 4 DOF equation system, since all the terms left depend on the gear nominal rigid rotation i = i t . In this context, third and sixth equations can be used to estimate the meshing force, while the remaining equations are effectively required to calculate the gearpair radial motion: 4

Simulation Modelling Practice and Theory 99 (2020) 102008

M. Battarra and E. Mucchi (1) (1) Fbx + F px =0 (1) (1) Fby + F py + Fm = 0 (2) (2) Fbx + F px =0 (2) (2) Fby + F py

Fm = 0

(3)

which substantially represents the force balance both for contributions along the line of action and normal to it. Moreover, the two equations describing the torque balance on the two gears can be used to estimate the meshing force Fm without necessarily estimating the mesh stiffness and damping coefficients. In particular, starting from the equation system reported hereafter:

r1 Fm = Tshaft + Td(1) r2 Fm = T p(2)

T p(1)

Td(2)

(4)

The case where driving and driven gear are identical, i.e. r1 = r2 = r, has been already discussed by Mucchi et al. in [8] and it leads to the following expression defining meshing force Fm:

Fm =

T p(2)

(5)

r

On the contrary, in the case where driving and driven gear are not identical, i.e. in general r1 ≠ r2 and transmission ration τ ≠ 1, by subtracting term by term the two equations in Eq. 4, meshing force Fm is defined as follows:

Fm =

Tshaft

Td(1)

Td(2) r1

T p(1)

T p(2) (6)

r2

Moreover, taking into account that the total friction torque can be defined with respect to the hydro-mechanical efficiency ηHM of the pump:

Tdtot = Td(1) + Td(2) = Tshaft (1

(7)

HM )

meshing force Fm can be determined without directly calculating the friction terms:

Fm =

HM Tshaft

r1

T p(1)

T p(2) (8)

r2

It is worth noting that Eq. 8 does not apply for pumps made by two identical gears since the denominator would be equal to zero. The presented approach substantially differs from the methodology introduced in [8], as well as the one proposed in [10], since in the present dissertation the estimation of the meshing force takes into account the contribution of the pressure torque applied on both gears together with the contribution of the driving and friction torques. On the contrary, the method proposed in [8] estimates meshing force Fm by assuming the absence of friction. This approach requires no extra-data regarding the gear pump under study, i.e. hydromechanical efficiency ηHM and driving torque Tshaft, but it is based on more limiting hypotheses. Concurrently, the method proposed in [10] does not consider the presence of friction torque by assuming HM = 1. In the same work, meshing force Fm is assumed as a constant along the entire revolution and no information are provided about its estimation. For the sake of clarity, it has to be noticed that the proposed model it is not focused on evaluating the gearpair dynamics and, therefore, it can be considered as a reduced version with respect to the state of the art dynamic modeling of gears [24], where all the terms that have a limited effect on the pump performance have been neglected. Once meshing force Fm has been defined, the determination of the gear center position requires the calculation of the con(i ) (i ) (i ) (i ) , F py , T p(i) and bearing reactions Fbx tributions provided by variable pressure loads F px and Fby . In particular, the formers are directly calculated from the pressure distribution determined by solving the relative lumped parameter model. The analytical procedure followed for their determination is fully described in Ref. [15] and it is based on the integration of the pressure distribution with (i ) (i ) respect to the pocket surface. Regarding the journal bearing reactions, components Fbx and Fby are estimated by following the analytical approach described in [25]. The methodology is based on the correction of the formulae obtained under the half Sommerfeld conditions for infinitely long/short bearings [26,27], by means of polynomial correction functions that depend on the bearing aspect ratio L/D. 2.2. Gear pump lumped parameter modeling The present Subsection provides a brief description of the mathematical approach used to determine the pressure variation around gear pumps for a complete gear rotation; the model concurrently estimates flow rates, pressure ripple, pressure course around gears and volumetric efficiency. The entire pump and the relative outlet piping system are discretized with a constant number of control volumes, in which all the fluid properties are considered as a constant. On the basis of this assumption, cavitation, as well as overheating phenomena, are not taken into account in the proposed model. As described in Section Introduction, several authors [5,8,10,28] have already applied a similar approach, each of them proposing a different subdivision of the pump volumes, in particular in the meshing zone. The present 5

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model takes advantage of the meshing zone discretization scheme proposed in [15], with each pocket defined by the Continuity Equation linking flow rates to the pressure difference between control volumes:

dp B = dt V

Qin

Q out

dV dt

dV var dt

(9) in

where B is the Bulk modulus, V and dV are the volume of a generic gear pocket and its derivative, respectively. Q is the generic volumetric flowrate entering the control volume, while Qout is the volumetric flowrate leaving it. Term Vvar is a variable volume alternatively added to the inlet or outlet chamber, according to the definition proposed in Ref. [10]. Flow rates connecting control volumes can be defined on the basis of two different approaches. In the hypothesis of low Reynolds numbers #Re, the Couette-Poiseuille equation may be a satisfactory representation of the phenomenon:

Q=

wh3 p whU + 12µ L 2

(10)

where U represents the tangential velocity, while w, h and L are the channel width, height and length, respectively. A classical application of Eq. 10 refers to the modeling of tooth tip leakages: channel width w is given by the gear facewidth, channel length L is represented by the tooth tip width while channel height h, which is the most influencing parameter of Eq. 10, is given by the radial distance between the pump casing and the tooth tip. This latter value, in particular, is calculated by taking into account the actual tooth tip radial clearance in reference to the radial position of the gear center, which is given by Eq. 3. It is worth mentioning that Eq. 10 is also adopted to define axial and bearing leakages. A detailed explanation of the calculation workflow is given in the following Subsection. When the hypothesis of low Reynolds numbers does not match the actual flow conditions, flow rates may be defined with the help of the discharge coefficients Cd, by using Eq. 11, which is obtained from the application of the Bernoulli’s equation under specific hypotheses:

Q = Cd A

2 p

sign ( p)

(11)

where A represents the porting area between two control volumes. According to the different links between flow and pressure drop defined by Eqs. 10 and 11, it is important to correctly characterize the several flows describing the interconnections between volumes. This classification is typically performed on the basis of experience and trial and error processes, since, a priori, no objective information is available regarding the nature of the various flow rates exchanged between control volumes. However, in the specific field of gear pumps modeling, the discretization of these flow rates have been extensively assessed by classifying them in reference to the clearance sizes and the angular position of the reference control volume. The complete description of the modeling approach chosen for each flow rate is avoided due to the large amount of literature on this subject, such as the detailed dissertations in refs. [3,7–10,14–16,21]. 2.3. Model implementation and workflow The present Subsection describes the assessment of the iteration scheme adopted to calculate the pressure field with respect to the gearpair translational micro-motions, which are calculated by using the approach proposed in the first Subsection. Despite the described mathematical approach may seem to require a small set of geometrical parameters referring to the analyzed gear pump, as it can be appreciated from Eqs. 9, 10 and 11, the evolution of several geometrical parameters, i.e. tooth pocket volume and its derivative, length, width and height of each flow channel, must be firstly calculated and stored along a complete revolution for each frame of calculus. If the analysis is focused on a spur gear pump, the calculation is entirely performed in a 2D domain, by assuming no dependence from the axial direction. On the contrary, if the analysis is focused on a helical gear pump, influence of the helix must be taken into account with a dedicated procedure. In particular, both helical gears are sliced into an arbitrary number of cross sections, obtained by sectioning them along the face width. Hence, the parameters affected by the helix angle are calculated for each cross section and then a numerical integration is used to achieve the results referred to the helical gears. Further details regarding the adopted procedure are given in refs. [14,15]. Once the entire set of parameters has been obtained for both gears, look-up tables are generated and stored by timing the data with respect to the starting reference position of the gearpair. This last step produces all the required geometrical data to run the dynamic simulations. Moreover, in the case where τ ≠ 1, then Fm is obtained from Eq. 8 and therefore both driving torque Tshaft and hydro-mechanical efficiency ηHM must be defined. Such data may be, in general, obtained from direct measurements [29] or, in case no prototypes are available, by estimating them with specific models [30,31]. Before introducing the iterative scheme adopted to reach the solution, it is worth clarifying the physical phenomena correlating the oil pressure distribution with the gearpair radial movements. As underlined in the first Subsection, the main goal of the model is the determination of the pump volumetric efficiency through the estimation of the pressure distribution within the pump, thanks to the fluid-dynamic system defined by Eq. 9. As stated by Eq. 9, since the volume variation term is non-zero exclusively inside the meshing zone, the oil pressure around the gears is calculated on the basis of the algebraic sum between the exchanged flow rates. Such flow rates are leakages calculated with Eq. 10, in which the governing parameter is the channel height h, i.e. the actual radial clearance between the tooth tip and the pump casing. It is therefore straightforward to understand that an accurate estimation of the 6

Simulation Modelling Practice and Theory 99 (2020) 102008

M. Battarra and E. Mucchi

pressure distribution comes from an accurate determination of the pump radial clearances. In addition, as stated by Eq. 3, such clearances depend on the pressure distribution, since gearpair micro-motions are given by the balancing between pressure loads and bearing reaction. In order to take into account the reciprocal correlation between the pressure field and gearpair micromotions, an iterative solution scheme has been designed. In particular, Fig. 2 describes the solution workflow for a generic angular position of the gearpair i. At the first iteration (u = 1 in Fig. 2), the code uses the solution of the previous angular position as starting data to get the first estimation of the gearpair center position (step 1), also known as Stationary Equilibrium Position (SEP), since it is obtained from Eq. 3, which neglects the inertia terms. Consequently, the code calculate all the influenced geometrical parameters (step 2), i.e. radial clearances at each tooth-tip and backlash clearance, and collects the new look-up tables (step 3). Detailed description on the determination of the radial clearances in reference to the actual gearpair eccentricity may be found in Ref. [8]. Since all the position dependent data are now available, the fluid-dynamic system is solved by taking the pressure field of the previous angular position i 1 as initial condition (step 4). The outcome of step 4 is the pressure field at the angular position i, which is used to calculate the pressure loads applied to the gearpair (step 5). After this first iteration, the code directly goes for the second turn u = 2, by using the pressure field calculated at iteration u = 1 to obtain a new SEP. Once the calculus is entirely repeated for the second time, then pressure force, torque and gearpair center position estimated by two consecutive iterations are compared. If the relative tolerance of each term is below the threshold value, then the solution achieved during the last iteration is stored and the code moves to the next angular position. Otherwise, in case at least one term shows a relative tolerance above the threshold value, the code moves to another iteration u = 3 referred to the same angular position i. The procedure is repeated until the tolerance condition is satisfied or the iteration number reaches a limit value. The convergence of the calculus is usually pretty fast and three-four iterations are sufficient to satisfy a tolerance condition with a threshold value equal to 10 3 . However, on the basis of the author’s experience, the maximum number of iterations should always be fixed, since in case of low speed or light loads, the code might tend to fall in a local minimum. 3. Experimental study A dedicated experimental campaign has been carried out in order to address the correct data assumptions and achieve a statistical validation for the proposed modeling approach. An external gear pump has been specifically designed and manufactured for the tests; the main design parameters are reported in Table 1. As it can be noted, the gearpair is constituted by two helical gears with different tooth number, so that τ ≠ 1. Despite this choice might be considered unusual, studies as the one reported in [32] are making it more and more common. The characteristic τ ≠ 1 requires a more time-consuming design procedure since all the grooves are asimmetrical and their timing is different between the two gears. On the other hand, this design feature allows to considerably reduce the pump size without affecting the machine displacement, which is governed by the number of teeth of the driving gear, as demonstrated in ref. [32]. In addition, it is worth specifying that the adopted gearpair is a single flank contact design with backlash to normal module ratio higher than 0.1. Fig. 3 shows the mechanical setup of the pump; the helical gearpair is located inside a cast iron casing, therefore radial clearances on both driving and driven side are designed to not require any running in process, since contact friction between

Fig. 2. Iterative solution scheme for a given gearpair angular position i, where u stands for the iteration number. 7

Simulation Modelling Practice and Theory 99 (2020) 102008

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Table 1 Main design parameters of the designed gear pump. Tooth geometry

helical

z1 z2 Displacement Vm

14 9

14.65·10 6m3/rev

Facewidth b

26·10 3m

Normal module mn

2.25·10 3m 6∘

Helix angle β

Fig. 3. Mechanical components of a tested pump sample.

tooth tips and casing cannot take place. Gears are then covered by a thrustplate and the entire machine is packed by a coverplate, which is clumped to the casing with four screws. A rubber seal is placed between the casing and the coverplate, in order to reduce leakage. Journal bearings are directly obtained from the casing and the coverplate, while grooves are milled on the thrustplate and the casing. As it can be observed from Fig. 3, dedicated by-pass grooves are designed on the casing, the thrustplate and the coverplate in order to guarantee the oil supply from the suction chamber to each journal bearing. The pump is designed to work in submerged conditions, as usually required for pumps working as oil supplier in automatic transmissions. The designed pump has been manufactured in 20 nominally identical samples, in order to perform a statistical characterization of the pump behavior. As a matter of fact, despite each sample is based on the same design that should lead to 20 identical pumps, design tolerances, together with the accuracy of the production process, necessarily cause the presence of minor discrepancies. As a result, the 20 samples will globally show the same macro-behavior, but each pump will also have its own characteristic curve. It is therefore clear that, to proper address the accuracy of the proposed model, a detailed characterization of the samples is mandatory as a first step. On the basis of this consideration, the mean actual value of the tooth-tip/casing clearances and bearing radial clearances has been measured on both gears for each pump, at the end of the production process. The former, in particular, represents the mean actual sealing gap between the tooth tips and pump casing, when the gear center coincides with the journal bearing center, accounting for manufacturing errors on both the gear and the pump casing. The value is obtained by calculating the difference between the mean measured radius of the pump casing surface and the mean measured tip radius of the gear. Similarly, bearing radial clearances represent the mean gap between the housing hole radius and the gear shaft radius. Due to the high accuracy required, measurements have been performed by private laboratory specifically qualified in performing non-destructive dimensional testing. Measurements related to the actual clearances of the grooves milled inside the pressurizing zones as well as the actual axial 8

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clearances have not been included in the experimental study. This choice is based on the fact that pressurizing grooves are obtained with design tolerances that are three orders of magnitude smaller than their groove global dimensions. Similarly, the tolerance chain has been designed to obtain nominal axial clearances that are half of the tooth tip/casing clearances. This choice, together with the thinnes of the tooth tip and the high number of teeth for a gear pump, leads to demonstrate that axial leakages are two orders of magnitudes less relevant than the radial clearances. By considering that both radial and axial leakages are modeled by Eq. 10, the following correlations are recognized for the pump under study:

wa = 2.25mn wr = zmn Lr 0.1La

(12)

where wa and La represent the channel width and length for axial leakages, while wr and Lr represent the channel width and length for radial leakages. Since ha ≃ hr/2, axial leakage flow rate Qa and radial clearance flow rate Qr related to two consecutive tooth pockets become:

Qa

2.25mn hr3 p 960µ Lr

Qr

zmn hr3 p 12µ Lr

+

+

2.25mn hr Ua 4

zmn hr Ur 2

(13)

where Ua is the mean tangential velocity of the lateral tooth flank while Ur is the tangential velocity of the tooth tip and therefore Ur > Ua. Eq. 13 demonstrates that in the present study axial leakage flow rate are negligible with respect of radial leakages by 2 order of magnitudes. On the basis of these observations, despite axial clearances and pressurizing grooves are recognized to play a key role in the definition of the pump performance, their variability within the design tolerance interval may be considered of minor

Fig. 4. Measured tooth tip clearances deviation from the design value. Dash line represents the normal distribution with sample mean x¯ and sample standard deviation S.

9

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relevance. For this reason, both axial clearances and pressurizing grooves are assumed to be equal to their nominal values defined by design. The charts in Figs. 4.a and 5.a report the measured tooth tip and journal bearing radial clearances (for both gears), respectively. For confidentiality reasons, measured data xm have been normalized with respect to the reference design value xd as shown in Eq. 14:

x m* =

xm

xd

(14)

xd

It is worth specifying that, for each geometrical quantity considered, parameter xd is the same throughout the entire set of samples. As it can be noted, both tooth tip/case and journal bearing clearances may deviate consistently from the design value. In particular, for tooth tip radial clearances (Fig. 4), the mean difference between the measured values and the design one x¯ is around 0.025 for the driving gear (Fig. 4.b) and 0.035 for the driven one (Fig. 4.c). Moreover, such a deviation may occasionally reach also 0.2, leading to a standard deviation S around 0.06 for both gears. The measured values reported in Fig. 4 demonstrate that the obtained manufacturing tolerances on tooth tip radial clearances are one to two orders of magnitude smaller with respect to the design value. A similar situation, but amplified, is observed in the case of journal bearing radial clearances: mean difference between the measured values and the design one x¯ is around 0.1 for the driving gear (Fig. 5.b) and reaches almost the 0.17 for the driven one (Fig. 5.c). Standard deviation S in both gears is similar to the one observed in case of tooth tip clearances. Normality of the measured data has been previously verified by means of a chi-squared normality test [4], which is used to test the hypothesis that the measured set of samples follows the normal distribution. It is worth underlining that the large discrepancy between the values of x¯ referred to bearing clearances (Fig. 5.b) and the one referred to casing clearances (Fig. 4.b) depends on the different design value xd used for their calculation. The quality of the production process is comparable for the two clearance types, as demonstrated by the estimated standard deviation, which is similar for both bearing and casing clearance. As a matter of fact, the designed clearance in bearings is smaller than the designed clearance in the casing, but the tolerance interval is defined in terms of absolute values rather than in terms of relative values, leading to an apparently different mean accuracy of the production process. The results of this analysis demonstrate

Fig. 5. Measured journal bearing clearances deviation from the design value. Dash line represents the normal distribution with sample mean x¯ and sample standard deviation S. 10

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that actual clearances may deviate consistently (but still within the design tolerances of some micrometers) from the design values, however, in order to evaluate their actual effect on the pump behavior, attention must be focused also on the measured pump performance. The test rig adopted to carry out the experimental analysis is shown in Fig. 6. The gear pump is installed inside a sealing box that makes it working submerged and driven by a brushless AC motor with speed controller. A torque meter is located along the shaft connecting the pump with the electric motor. The pipeline for the oil supply system is constituted by two branches: one connecting the tank to the sealing box and the other one connecting the pump outlet chamber to the tank. A proportional pressure relief valve located on the latter branch of the pipeline allows for the regulation of the oil delivery pressure. The oil temperature is regulated with an additional pipeline connected to the tank and controlled by a dedicated system, namely the temperature controller, while the pump temperature is regulated by a climatic chamber. The overall system guarantees an accuracy in controlling the temperature equal to ± 0.5oC. Both pipeline systems described above are equipped with a drainage system, various valves and filters in order to allow for a safe and easy management of the test rig during the measurement procedures. The description of the auxiliary parts of the test rig is neglected being out of the scope of the present work. As shown in Fig. 6, the set of transducers adopted for the tests is constituted by two digital pressure gauges placed on the suction and delivery ports, respectively, and a Kracht Gear Type Flow Meter (model VC 5 P3 PS/124, with ± 0.5% accuracy within the measuring range), placed on the delivery pipeline, in order to measure the outlet flow rate and therefore determine the volumetric efficiency. The instantaneous angular speed can be directly acquired from the tachometer connected to the speed controller of the electric motor. The set of sensors is directly acquired by the data acquisition system integrated in the test bench. With the aim to characterize each pump sample throughout a wide working condition range, two different tests have been conducted. The first testing procedure consists of evaluating the pump performance at four different operating speed values, while delivery pressure is kept as a constant. On the contrary, the second testing procedure consists of evaluating the pump performance at different delivery pressure values, while the working speed is kept as a constant. Both testing procedures have been repeated at two different oil temperatures. Table 2 summarizes the fourteen working conditions tested; the entire set of tests has been repeated for each pump sample, leading to 280 different tests. Fig. 7, from (a) to (e), shows the measured volumetric efficiency values with respect to delivery pressure variation for the entire set of 20 samples; data have been collected at a working speed equal to 1000rpm and oil temperature equal to 60∘C. Volumetric efficiency has been determined by measuring the mean delivered flow rate Qmeas and then by applying Eq. 15: v

=

Qmeas Vm· n/60

(15)

M

where Vm is the pump theoretical displacement, which has been calculated with the formulation proposed by Manring N. et al. in [32] for gear pumps with r1 ≠ r2. It is worth specifying that Vm is calculated with respect to the nominal dimensions of the gearpair and the obtained value is considered as a constant for all the samples. The actual value of Vm would change from sample to sample, however its exact estimation cannot be straightforwardly performed since it depends on both production tolerances and load condition. As also explained in [29], a conservative value for the uncertainty introduced by the adopted assumption may be taken as equal to ± 0.5% of the measured efficiency; this leads to an expected global uncertainty on the efficiency equal to ± 1% of the measured value, which is always lower than the scatter obtained on the measured efficiency throughout the set of samples. As it can be appreciated from Fig. 7.a, efficiency tends to decrease as the delivery pressure increases, as expected. However, what is more interesting relates to how the measured values are distributed: in particular, it can be noted that, as the delivery pressure increases, the standard deviation tends to increase as well. This phenomenon is due to the scatter of the measured radial clearances: as the delivery pressure is increased, leakages become more relevant and, step by step, the scatter of the measured radial clearances amplifies the scatter of the measured efficiency values. A similar trend is also appreciated in Fig. 7 from (f) to (l), which shows the measured efficiency values with respect to delivery pressure variation, collected at a working speed equal to 1000rpm and oil temperature equal to 120∘C. Measured data referring to volumetric efficiency with respect to speed variation have been post-processed with the same approach (see Fig. 8). In particular, from Fig. 8.a to 8.e, measured efficiency values for the 20 samples at a delivery pressure pout = 30bar and

Fig. 6. ISO schematic of the test rig configuration and sensor disposition adopted for the experimental study. 11

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Table 2 Test conditions adopted for the experimental campaign. Outlet pressure (bar)

Oil temperature ∘ C

30

60

120

10 20 30 40 10 20 30 40

60

Working speed (rpm) 500 1000 2000 4000 500 1000 2000 4000 1000

120

Toil = 60 C are shown together with the estimated mean value and standard deviation for the four different working speed analyzed. As it can be appreciated, the speed increasing causes the volumetric efficiency to increase as well, while the standard deviation tends to decrease. Again, the phenomenon is related to the scatter of the measured radial clearances: as the speed is reduced, leakages become progressively more relevant with respect to the delivery flow rate and, step by step, the scatter of the measured radial clearances amplifies the scatter of the measured efficiency values. Such a trend is confirmed also for an oil temperature equal to 120∘C. The torque required to operate the pump has been also recorded, as it can be appreciated from Fig. 9 depicting the measured data with respect to delivery pressure and Fig. 10 depicting the measured data with respect to working speed. Results obtained from the experimental campaign demonstrate that radial clearances measured at the end of the production process may show high value of the estimated standard deviation, even if the recorded values stand within the designed tolerance interval. This result should be considered as the normal outcome of modern production processes. What is more important is that, although design limitations are satisfied, slight modifications of the radial clearances may have non-negligible effects on the pump performance, both in terms of volumetric efficiency and required torque. In addition, such a phenomenon is clearly amplified as the temperature increases. Experimental results suggest that data assumptions regarding the input geometry parameters of LP models for gear pump performance prediction should be carefully assessed in order to achieve a proper validation. 4. Model results and assessment In the present Section, results obtained from the numerical model described in the second Section are introduced and discussed. Attention is particularly focused on the comparison with data obtained from the experimental study, with the aim to precisely address the accuracy of the model with respect to the volumetric efficiency estimation. In presence of a set of samples with measured input parameters, validation is performed by using a trace-driven simulation approach [33]. Trace-driven simulation is a technique which uses the actual measured data as model input. The method provides an alternative approach with respect to self-driven simulation methods, in which the input data sets are generated by random numbers sampled from distributions or stochastic processes. The chosen technique allows to compare each measured output with the relative simulated result based on the same input parameters, providing a statistical characterization of the model accuracy. However, it is worth noting that the computational and experimental effort requested by this validation process tends to overflow as the number of samples increases. For this reason, the definition of a reasonable compromise between number of tested samples and computational time required is compulsory. In order to perform an accurate evaluation of the model reliability, each pump sample has been reproduced and their behavior throughout the 14 different working conditions defined in Table 2 has been numerically simulated. Figs. 11 and 12 report the graphical comparison between the simulated and measured characteristic curve of each pump sample, obtained at fixed working speed (1000rpm) and four different delivery pressure values. In particular, simulated efficiency is calculated with Eq. 16: v

=

Qsim Vm· n/60

(16)

where Qsim represents the mean delivery flow rate estimated by the model. It is worth specifying that, since the model provides the instantaneous delivery flow rate, its mean value calculated on a complete mesh period gives Qsim. From Fig. 11 and 12 two relevant aspects may be observed. Firstly, the model is sensitive to the variability introduced by taking into account the effect of the production tolerances. Similarly to the measured efficiency, the simulated one differs from one sample to the other due to the inclusion of this phenomenon. With the same relevance, it is also observed that the model shows a different accuracy depending on the pump sample examined. It appears to be clear that, if the validation would have been performed by considering measured data taken from a single sample, there would have been the risk to achieve misleading conclusions. In order to clarify this aspect, let us assume the case 12

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Fig. 7. Measured efficiency ηv with respect to pressure variation, for the 20 samples. From (a) to (e), test conditions are: n = 1000rpm and Toil = 60 C, while from (f) to (l) test conditions are: n = 1000rpm and Toil = 120 C. Data are normally distributed for each analyzed working condition.

in which the model assessment is performed exclusively on pump sample N20. The comparison would lead to the conclusion that the model is representative of the pump behavior and, therefore, it is taking into account all the relevant phenomena. Similarly, let us now consider the case in which the model assessment is performed exclusively on pump sample N18. In this scenario, the conclusion would have been the opposite, since on the basis of the graphical comparison discrepancies between measured and simulated efficiency appear to be consistent. The same model would have been declared not representative of the pump behavior. These results may be caused by the temporary and spatially localized phenomena, e.g. axial micro-movements and micro-tilt of the balance plate, 13

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Fig. 8. Measured efficiency ηv with respect to speed variation, for the 20 samples. From (a) to (e), test conditions are: pout = 30bar and Toil = 60 C, while from (f) to (l) refer to the same case with Toil = 120 C. Data are normally distributed for each analyzed working condition.

localized temperature variation, micro-wear. Such phenomena produce statistical fluctuations affecting the measurement, regardless of the high level accuracy reached in controlling the test conditions. The proposed considerations tend to confirm that reliability and accuracy of this kind of models should be evaluated with respect to a population of pumps, since referring to a single pump sample may easily lead to over/under-estimate the quality of the proposed approach. Similar considerations arise by comparing simulated and measured volumetric efficiency of each pump sample, evaluated at a fixed delivery pressure equal to 30bar and 4 different 14

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Fig. 9. Torque Tshaft required to operate the pump with respect to pressure variation, for the 20 samples. From (a) to (e), test conditions are: n = 1000rpm and Toil = 60 C, while from (f) to (l) test conditions are: n = 1000rpm and Toil = 120 C. Data are normally distributed for each analyzed working condition.

working speed values (see Figs. 13 and 14). As previously noted, the model reaches different levels of accuracy depending on the analyzed sample. Concurrently, the graphical comparison already demonstrate that slight modifications of the input geometry parameters (within the design tolerance interval) affect the results of the LP approach, but a more detailed analysis is necessary to obtain quantitative outcomes. 15

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Fig. 10. Torque Tshaft required to operate the pump with respect to speed variation, for the 20 samples. From (a) to (e), test conditions are: pout = 30bar and Toil = 60 C, while from (f) to (l) refer to the same case with Toil = 120 C. Data are normally distributed for each analyzed working condition.

In order to give a quantitative estimation of the obtained accuracy, the LP model is considered as an Input-Output Transformation, where the inputs are represented by the bearing and casing radial clearances, together with the pump working conditions, while the output is represented by the simulated volumetric efficiency. The hypothesis of normally distributed data regarding simulated volumetric efficiency has been verified by applying the chi-square normality test, for each working condition analyzed, as done also for the measured values (see previous Section). The hypothesis of input data independence allows us to apply a statistical approach based 16

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Fig. 11. Comparison between simulated (dashed line) and measured (solid line) volumetric efficiency, with n = 1000rpm and four different delivery pressure values, for the pump samples from N1 to N10. Black and gray lines refer to oil temperature values equal to 60∘C and 120∘C, respectively.

on confidence intervals calculated on a performance parameter d defined as:

d=

m v

s v

(17)

where vs and vm represent simulated and measured volumetric efficiency, respectively. If the model would exactly reproduce the reality, then the sample average E (d ) = d¯ would be expected to coincide with the true mean µ 0 = 0 . For this reason, the accuracy of the numerical model may be checked with a validation test based on the null hypothesis H0 : E (d ) = 0 versus H1: E(d) ≠ 0. Chosen the level of significance α (typically the 5%), the test attempts to verify that the null hypothesis cannot be discarded and that there is no 17

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Fig. 12. Comparison between simulated (dashed line) and measured (solid line) volumetric efficiency, with n = 1000rpm and four different delivery pressure values, for the pump samples from N11 to N20. Black and gray lines refer to oil temperature values equal to 60∘C and 120∘C, respectively.

reason to consider the model invalid. Here, the level of significance represents the probability to reject the model when it is actually valid. However, rather than validating the proposed model on the basis of a strict valid/invalid proposition, the present work has the purpose to statistically assess the level of the LP method accuracy with respect to the data assumptions at its basis. For this reason, it is the authors’ belief that the most interesting aspect is to evaluate which is the minimum accuracy level that the model statistically guarantees with a specified value of α. In order to carry out the analysis, a statistic based on the Student’s t-distribution can be applied, where the true variance σ2 is 18

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Fig. 13. Comparison between simulated (dashed line) and measured (solid line) volumetric efficiency, with pout = 30bar and four different working speed values, for the pump samples from N1 to N10. Black and gray lines refer to oil temperature values equal to 60∘C and 120∘C, respectively.

substituted with the sample variance Sd2 :

t0 =

d¯

µ0

(18)

Sd/ ns

where ns represents the number of observations, i.e. the number of pump samples, and Sd is the sample standard deviation of d. By defining a level of significance α, the confidence interval can be calculated with the following equation: 19

Simulation Modelling Practice and Theory 99 (2020) 102008

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Fig. 14. Comparison between simulated (dashed line) and measured (solid line) volumetric efficiency, with pout = 30bar and four different working speed values, for the pump samples from N11 to N20. Black and gray lines refer to oil temperature values equal to 60∘C and 120∘C, respectively.

[dmin, dmax ] = [d¯

t

, ns 1 Sd /

ns , d¯ + t

, ns 1 Sd /

(19)

ns ]

Once the confidence interval has been calculated for a given working condition, as also clarified in [4], it is possible to define the model closeness-to-reality Δν as the maximum distance between the bounds of the confidence interval and the true mean:

= µ0

(20)

max (dmin, dmax )

The estimated values of d¯, Sd and Δν for

= 0.05 are reported in Table 3 with respect to the 14 working conditions analyzed. It is 20

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Table 3 Estimated values of sample mean d¯, sample standard deviation Sd and model closeness-to-reality Δη for each working condition analyzed. Pout

Speed

d¯

Sd

Δη

(bar)

(rpm)

60∘C

120∘C

60∘C

120∘C

60∘C

120∘C

10 20 30 40 30 30 30

1000 1000 1000 1000 500 2000 4000

–0.001 –0.019 –0.027 –0.028 –0.058 –0.003 0.007

–0.028 –0.015 0.001 0.015 0.002 0.005 0.006

0.022 0.019 0.012 0.013 0.022 0.012 0.013

0.016 0.021 0.023 0.023 0.021 0.016 0.009

0.006 0.026 0.036 0.038 0.069 0.009 0.013

0.036 0.025 0.011 0.026 0.012 0.013 0.011

worth noting that sample mean d¯ may give a rough measure of the model accuracy and Sd represents a rough measure of the model precision. In this context, parameter d¯ helps the analysis to recognize the potential presence of a relevant bias error. In this kind of models, where some parameters need to be set as constant values for all the working conditions and for all the pump samples (e.g. the discharge coefficients), such an error may become not negligible in magnitude. The results in Table 3 shows that an evident and repeatable bias between the values of d¯ and the expected value (i.e. zero) is not recognizable. Despite this important link between d¯, Sd and the model reliability, Δν is the parameter that actually defines the performance of the model. The results underline that the model reaches different level of accuracy depending on the working condition analyzed, but Δν is always below the 0.04 limit, except for the 500rpm/30bar condition at Toil = 60 C , where Δν reaches 0.069. This result indicates that the variability of the main radial clearances, within common production tolerance limits, affects the efficiency estimation in agreement with the measured results. The proposed approach allows us to give a quantitative estimation of the model accuracy, however, the authors are aware that many other approaches could be followed to validate trace-driven simulations, as the one defined by Kleijnen et al. in [34], which is based on regression analysis and it concurrently assesses both accuracy and precision of the simulations. In this context, it must be taken into account that their practical applicability in case of gear pump efficiency simulations is limited by the fact that such approaches require a large number of observations, both in terms of measured data and simulated results, in order to obtain a statistical test with a high value of power, i.e. with a low risk to accept an invalid model. Within this framework, it is worth underlining that the number of observations cannot be incremented by simply increasing the number of working conditions tested. The reason resides in the main hypothesis at the basis of these statistical tests, which requires the input data samples to be independent and identically distributed. As demonstrated by the measured data, this fundamental hypothesis appears to be verified when data refer to different pumps tested at the same working condition, while it is rejected when data refer to the same pump at different working conditions. 5. Concluding remarks The present work describes a statistical procedure for the assessment of LP models for gear pump performance prediction based on a trace-driven simulation approach, with the attempt to demonstrate two fundamental aspects. The first purpose is to demonstrate that the validation of this kind of models needs to be performed with a statistical approach, based on measurements obtained from a population of nominally identical pumps. The second purpose is to assess the influence of the data assumptions regarding the pump radial clearances on the model accuracy. The validation procedure is carried out on a novel LP model, which couples the effects produced by the fluid-dynamic field with the gearpair micromotions. The physical phenomena are reciprocally solved, at each timestep, to obtain a precise estimation of the machine behavior. The mathematical model adopted to solve the gearpair radial movements is based on a detailed set of equations of motion, which is shown to give an accurate definition of the different loads applied to the gears, by taking into account multiple load sources such as the presence of speed dependent friction torque. In order to perform the statistical assessment, an extended experimental campaign has been carried out on a gear pump designed for automotive applications. The pump design has been reproduced in 20 nominally identical samples, each one marked by measuring its mean casing radial clearances and journal bearing radial clearances. Results demonstrate that radial clearances measured at the end of the production process may show high value of the estimated standard deviation, even if such values stand within the design tolerance interval. The obtained data provided a statistical characterization of the main geometry parameters and allowed us to experimentally evaluate their own influence on the pump performance. Within this framework, each sample has been tested on a dedicated test rig, collecting both mean delivery flow rate and driving torque at 14 different working conditions resulted from the combination of 4 delivery pressure values, 4 working speed values and 2 oil temperatures. The complete experimental campaign was therefore constituted by 280 tests. Results underline that, although design limitations are satisfied for each sample, slight modifications of the radial clearances may affect the pump performances significantly. Such a phenomenon is amplified as the temperature increases. This experimental outcome is a preliminary demonstration that the variability of the pump clearances requires a dedicated assessment before the validation of the model could be approached. Experimental data have been finally adopted to set up the LP model and perform its validation. Each test condition has been simulated for each pump sample, obtaining so 280 different points of comparison, where the input data are not simply based on the working condition, but also on the probability distribution of the main pump clearances. The graphical comparison based on the volumetric efficiency has shown that the model cannot guarantee a constant level of accuracy, but it may show different accuracy 21

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levels depending on the analyzed working condition. Concurrently, such a comparison has enlightened that the model shows a different relative error value depending on the pump sample examined. This result tends to confirm that reliability and accuracy of this kind of models should be evaluated by means of a statistical approach, since referring to a single pump sample may easily lead to an over/under-estimate of the quality of the proposed approach. In this context, it underlines the necessity to overcome the basic validation procedures usually adopted as common practice in this field, where the reference benchmark is constituted by measurements taken from a single pump sample. In addition, the analysis corroborates the idea that these kind of models requires a dedicated assessment of the data assumptions regarding the pump radial clearances, since their variability plays a non-negligible role in efficiency performance predictions. The reason for this result is given by the fact that slight deviations of the main clearances from the design values can effectively affect the simulation outputs. In order to achieve a quantitative overview of the model reliability, a statistical comparison based on confidence intervals has been carried out. The proposed approach is devoted to quantify the accuracy that the LP method can reach with a specific level of significance 1 , rather than checking the validation on the basis of an either/or statement. The analysis has underlined that the predicted volumetric efficiency is expected to show a maximum deviation below 0.04 from the real value. Such a result is confirmed for all the analyzed working conditions, except for the 500rpm/30bar condition at Toil = 60 C , where the relative deviation may reach a magnitude equal to 0.069. For the sake of completeness, it is worth underlining that, in order to achieve this level of accuracy, the LP approach requires a careful definition of the machine geometry and a number of input data that may be experimentally defined, e.g. the driving torque. In case no measured data are available, these input parameters need to be assumed, with a potentially negative effect on the model accuracy. References [1] K. Foster, R. Taylor, M. Bidhendi, Computer prediction of cyclic excitation sources for an external gear pump, Proc. Instit. Mech. Eng. Part B 199 (3) (1985) 175–180. [2] S. Mancó, N. Nervegna, Modello matematico di pompe oleodinamiche a ingranaggi esterni, Oleodinamica Pneumatica (1987). [3] C. Bonacini, M. Borghi, Calcolo delle pressioni nei vani fra i denti di una macchina oleodinamica ad ingranaggi esterni, Oleodinamica Pneumatica 11 (1990) 128–134. [4] J. Banks, B.L. Nelson, J.S. Carson, D.M. Nicol, Discrete-Event System Simulation, 4th, Pearson, 2010. [5] S. Mancó, N. Nervegna, Simulation of an external gear pump and experimental verification, JHPS. International symposium on fluid power, Tokyo, (1989). [6] S. Mancó, N. Nervegna, Pressure transients in an external gear hydraulic pump, Fluid Power (1993). [7] P. Casoli, A. Vacca, G. Franzoni, A numerical model for the simulation of external gear pumps, 6th JFPS International symposium on Fluid Power, (2005), pp. 705–710. [8] E. Mucchi, G. Dalpiaz, A. Fernandez Del Rincon, Elastodynamic analysis of a gear pump. part i: pressure distribution and gear eccentricity, Mech. Syst. Signal Process. 24 (7) (2010) 2160–2179. [9] E. Mucchi, G. Dalpiaz, A. Rivola, Elastodynamic analysis of a gear pump. part II: meshing phenomena and simulation results, Mech. Syst. Signal Process. 24 (7) (2010) 2180–2197. [10] A. Vacca, M. Guidetti, Modelling and experimental validation of external spur gear machines for fluid power applications, Simul. Model. Practice Theory 19 (9) (2011) 2007–2031. [11] M. Borghi, M. Milani, F. Paltrinieri, B. Zardin, The influence of cavitation and aeration on gear pumps and motors meshing volumes pressures, ASME IMECE 2006, Chicago, (2006), pp. 1–10. [12] J. Zhou, A. Vacca, P. Casoli, A novel approach for predicting the operation of external gear pumps under cavitating conditions, Simul. Modell. Practice Theory 45 (2014) 35–49. [13] E. Mucchi, G. Dalpiaz, A. Fernandez Del Rincon, Elasto-dynamic analysis of a gear pump. part IV: improvement in the pressure distribution modelling, Mech. Syst. Signal Process. (2014) 1–21. [14] M. Battarra, E. Mucchi, G. Dalpiaz, A model for the estimation of pressure ripple in tandem gear pumps, ASME IDETC/CIE, (2015). V010T11A018; 9 pages [15] M. Battarra, E. Mucchi, A method for variable pressure load estimation in spur and helical gear pumps, Mech. Syst. Signal Process. 76–77 (2016) 265–282. [16] M. Eaton, P.S. Keogh, K.A. Edge, The modelling, prediction, and experimental evaluation of gear pump meshing pressures with particular reference to aeroengine fuel pumps, Proc. Instit. Mech. Eng., Part I 220 (2006) 365–379. [17] R.H. Frith, W. Scott, Comparison of an external gear pump wear model with test data, Wear 196 (1996) 64–71. [18] E. Koç, A.O. Kurban, C.J. Hooke, An analysis of the lubrication mechanisms of the bush-type bearings in high pressure pumps, Tribol. Int. 30 (8) (1997) 553–560. [19] E. Koç, Bearing misalignment effects on the hydrostatic and hydrodynamic behaviour of gears in fixed clearance end plates, Wear 73 (1/2) (1994) 199–206. [20] E. Koç, C.J. Hooke, An experimental investigation into the design and performance of hydrostatically loaded floating wear plates in gear pumps, Wear 209 (1997) 184–192. [21] F. Paltrinieri, M. Milani, M. Borghi, Modelling and simulating hydraulically balanced external gear pumps, 2nd International FPNI Ph.D. Symposium on Fluid Power, (2002). [22] M. Borghi, M. Milani, F. Paltrinieri, B. Zardin, Studying the axial balance of external gear pumps, SAE Int. (2005), https://doi.org/10.4271/2005-01-3634 SAE Technical Paper 2005-01-3634, 19 pages. [23] S. Dhar, A. Vacca, A novel CFD – Axial motion coupled model for the axial balance of lateral bushings in external gear machines, Simul. Modell. Pract. Theory 26 (2012) 60–76. [24] X. Liang, M.J. Zuo, Z. Feng, Dynamic modeling of gearbox faults: a review, Mech. Syst. Signal Process. 98 (2018) 852–876. [25] Y. Bastani, M. de Queiroz, A new analytic approximation for the hydrodynamic forces in finite-length journal bearings, J. Tribol. 132 (1) (2009). 014502 [1]–014502 [9] [26] B.J. Hamrock, Fundamentals of Fluid Film Lubrication, McGraw-Hill, New York, NY, 1994. [27] A. Harnoy, Bearing Design in Machinery: Engineering Tribology and Lubrication, CRC Press, New York, NY, 2003. [28] B. Zardin, F. Paltrinieri, M. Borghi, M. Milani, About the prediction of pressure variation in the inter-teeth volumes of external gear pumps, Proceedings of the 3rd FPNI - PhD Symposium on Fluid Power, Terrassa, Spain, (2004). [29] N.D. Manring, Measuring pump efficiency: uncertainty considerations, J. Energy Resour. Technol. 127 (4) (2005) 280. [30] G.L. Zarotti, N. Nervegna, Pump efficiencies approximation and modelling, 6th International Fluid Power Symposium, Hannover, (1981), pp. 145–164. [31] D. McCandlish, R.E. Dorey, The mathematical modelling of hydrostatic pumps and motors, Proc. Instit. Mech. Eng., Part B 198 (3) (1984) 165–174. [32] N.D. Manring, S.B. Kasaragadda, The theoretical flow ripple of an external gear pump, J. Dyn. Syst., Measure. Control 125 (3) (2003) 396–404. [33] O. Balci, R.G. Sargent, Some examples of simulation model validation using hypothesis testing, The Winter Simulation Conference, (1982), pp. 621–629. [34] J.P.C. Kleijnen, B. Bettonvil, W. Van Groenendaal, Validation of trace-Driven simulation models: a novel regression test, Manage. Sci. 44 (6) (1998) 812–819.

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