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Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights Author's personal copy Applied Acoustics 77 (2014) 99–111 Contents lists available at ScienceDirect Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust Modelling dynamic behaviour and noise generation in gear pumps: Procedure and validation Emiliano Mucchi a,⇑, Alessandro Rivola b, Giorgio Dalpiaz a a b Engineering Department in Ferrara, Università degli Studi di Ferrara, Via Saragat 1, I-44122 Ferrara, Italy DIEM – Department of Mechanical Engineering, University of Bologna, Viale del Risorgimento 2, I-40136 Bologna, Italy a r t i c l e i n f o Article history: Received 19 November 2012 Received in revised form 5 October 2013 Accepted 21 October 2013 Keywords: External gear pumps Vibro-acoustic analysis Experimental validation Fluid–structure interaction a b s t r a c t The paper presents a methodology for noise and vibration analysis of gear pumps and its application to an external gear pump for automotive applications. The methodology addresses the use of a combined numerical model and experimental analyses. The combined model includes a lumped-parameter model, a finite-element model and a boundary-element model. The lumped-parameter (LP) model regards the interior parts of the pump (bearing blocks and gears loaded by the pressure distribution and the driving torque), the finite element (FE) model regards the external parts of the pump (casing and end plates), while the boundary element (BE) model enables the estimation of the emitted noise in operational conditions. Based on experimental evidences, attention has been devoted to the modelling of the pump lubricant oil: the fluid–structure interaction between the oil and pump casing was taken into account. In the case of gear pumps all these important effects have to be considered in the same model in order to take their interactions into account. The model has been assessed using experiments: the experimental accelerations and acoustic pressure measured in operational conditions have been compared with the simulated data coming from the combined LP/FE/BE model. The combined model can be considered a very useful tool for design optimisation. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Noise, vibration and harshness (NVH) is an important attribute in vehicles. It is usually among the top five attributes in terms of its priority in the design of any vehicle type [1]. Like other attributes of safety, performance, dynamics and fuel economy, this attribute has to be considered closely in the design process. The attempts to reduce vibrations and radiated noise, while improving system performance, have become of increasing interest for the automotive industry in order to achieve high level of comfort in vehicles. Each vehicle component should produce low level of noise and vibration. Thus, a proper design of gear pumps used in the vehicle steering is crucial to control the emitted noise and vibration in operational conditions, maintaining high overall pump performance. Gear pumps use a very simple mechanism to generate flow and consist of a low number of parts. They combine good performance and low costs. The simplicity of their design translates into higher reliability as compared to other positive displacement pumps using more complex designs. However, gear pumps are often accompanied with noise levels that are generally higher than other types of pumps. Such noise levels are the consequence of ⇑ Corresponding author. Tel.: +39 0532 974913; fax: +39 0532 974870. E-mail address: emiliano.mucchi@unife.it (E. Mucchi). 0003-682X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2013.10.007 the dynamic forces within the system, related with the flow and pressure ripple as well as the variable meshing stiffness and gear errors. Thus, in the hydraulic systems using this pump as a power source, taking countermeasures for noise and vibration reduction is one of the key points. In this scenario, numerical or analytical analyses can be useful to study the dynamics of gear pumps. In the literature, several authors have addressed efforts to study gear pump dynamics. Refs. [2–9] contain a good description of the lumped parameter (LP) models used to simulate the pressure distribution and variable pressure forces acting on gears. In [10–14], the computational fluid dynamics (CFD) has been used for the development of numerical models of gear pumps in order to estimate the pressure distribution around the gear in operational conditions; a few of these works use structured mesh in each time step for the simulation, others dynamic non-structured mesh. With this latter technique, the mesh is adapted, through a spring-based algorithm, to the new geometry in each time step. In order to keep the mesh quality above a certain limit, cells are created and agglomerated where necessary. Li et al. in [15] have developed a dynamic model of gears which takes into account the contribution of the trapped oil in order to estimate the gears acceleration in an external spur-gear pump. In such a work the contribution of the journal bearing is not considered. Refs. [16,17] contain a good description of the mathematical models used to simulate the meshing stiffness Author's personal copy 100 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 Nomenclature Latin symbols Boil oil bulk modulus torsional viscous damping coefficient of the driving CT shaft h, l, w height, length, width of a generic fluid film element fbxk, fbyk bearing reaction applied to gear k in directions X and Y, respectively fmg meshing force fpxk, fpyk pressure force applied to gear k in direction X and Y, respectively F ecX 0 ; F ecY 0 external forces applied to pump casing and plate in direction X 01 and Y 01 , respectively. F gcX 0 ; F gcY 0 forces applied to pump casing and plate from gears in direction X 01 and Y 01 , respectively F icX 0 ; F icY 0 inertia forces concerning pump casing and plate in direction X 01 and Y 01 , respectively F igX 0 ; F igY 0 inertia forces concerning gears in direction X 01 and Y 01 , respectively Jk moment of inertia of gear k KT torsional stiffness of the driving shaft M ecO1 external moment applied to pump casing and plate about point O1 M gcO1 moment applied to pump casing and plate from gears about point O1 inertia moment concerning pump casing and plate M icO1 about point O1 phenomena in gear coupling; the majority of the simulations to date are based on rigid-body lumped-parameter models with the meshing teeth assimilated to a lumped variable stiffness. The simulation of the dynamic behaviour of a speed-increasing gearbox was also carried out using finite element (FE) methods in [18], where a 3D-contact FE model is used to model the time variable meshing stiffness of the gears, while the gearbox housing is modelled using tetrahedral solid elements. The combined analyses of gears and oil bearings have been developed in the literature using several oil bearing formulations. In particular, in Ref. [19] the dynamics of a spur gear pair supported by journal bearings was studied using the theory proposed in [20]. Gearbox vibrations have been widely studied in the literature by using several methods; in [21] by using a torsional vibration model, in [22,18] by using an FE model, in [23,24] by using a multibody model. Moreover, the noise and vibration behaviour of a gearbox has been modelled by using FE methods in [22]. The emitted noise of gear pump has also received attention in [25–27]. Thus, works that specifically deal with the dynamic phenomena occurring in gear pump were found in the literature, as mentioned above; however such effects have been shown and discussed separately; on the contrary, in the case of gear pumps all these important effects have to be considered in the same model in order to take their interactions into account. In fact, it has to be underlined that, gear accelerations, which can be estimated by LP models, are not directly related to noise and vibration emitted, due to the presence of the pump casing dynamics and noise propagation issues. Moreover, the flow pulsation and the variable pressure field around the gears, which can be foreseen by CFD or LP models, are not the only source of fluid borne noise. Several studies such as [28] show that there is a relation between flow ripple and air borne noise, at least in many operating conditions. It is also true that other phenomena related to the meshing process, bearing dynamics or casing resonances also contribute to noise emissions. For this reason, a complete methodology for noise and vibration M igO1 mk Mm Mpk N Pb pi Vi t T (xk, yk) inertia moment concerning gears about point O1 mass of gear k motor driving torque pressure torque applied to gear k maximum number of isolated tooth spaces base pitch pressure in control volume i volume of control volume i periodic time (0 6 t < T) meshing period (T ¼ 60=nz, n is the rotational speed in rpm) coordinates of the centre of gear k in reference frame OkXkYk Greek symbols aw pressure angle in operational conditions Dp pressure drop between adjacent control volumes DQ difference between the volumetric flow rate, coming into control volume i and coming out h angular coordinate hp angular pitch l oil dynamic viscosity x angular speed Subscripts i = 1 . . . N denotes isolated tooth space volumes (control volumes) k = 1, 2 denotes gears analysis of gear pumps including the dynamic behaviour of gear pairs, the pressure evolution in a gear tooth space during the pump rotation, the journal bearing behaviour, the dynamic response of the external casing, the noise propagation as well as experimental verification has been developed, since it cannot be found in the literature. In this work an external gear pump for vehicle steering is studied. The most usual configuration has two twin gears (see Fig. 1), which are assembled by a couple of lateral floating bearing blocks that act as seals for the lateral ends. Gears and floating bearing blocks are jointly packed inside a casing that encloses all the components and defines the isolated spaces that carry the fluid from the low to the high pressure chamber. The bearing blocks act as supports for the gear shafts by means of two hydrodynamic bearings, which are hydraulically balanced in order to avoid misalignments between gear shaft and journal bearing. Power is applied to the shaft of one gear (gear 1) and transmitted to the driven gear (gear 2) through their meshing. The driving shaft is connected by an Oldham coupling with an electrical drive. This pump works with a pressure ranging from 3.5 to 100 bar and angular speed ranging from 1500 to 3400 rpm. These authors have already developed and experimentally assessed a lumped-parameter model of such a pump [29–34]; the lumped-parameter (LP) model regards the internal parts as gears and bearing blocks. The LP model is a non-linear kineto–elastodynamic model and includes the most important phenomena involved in the pump operation, as timevarying oil pressure distribution on gears, time-varying meshing stiffness, gear errors and hydrodynamic journal bearing reactions [29–31]. The LP model was used in order to analyse the influence of a few design and operational parameters on the pump dynamic behaviour. In particular, the effect of operational pressure and speed, the influence of the clearance in the journal bearing and between tooth tip and pump casing, and the effect of the dimension of the relief grooves in the bearing blocks have been thoroughly discussed [32]. As model results, this analysis gives the gear Author's personal copy E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 (BE) model for the estimation of the radiated noise. The combined model has been assessed using experiments: the experimental accelerations on pump surface and the acoustical pressure measured in operational condition have been compared with the numerical simulations. θ1 Y’1 First isolated tooth space Y1 Last isolated tooth space γ O1 2. Method X’1 δ X1 Inlet Outlet Y2 Y’2 X’2 O2 101 θ2 (a) X2 The methodology proposed in this research for the vibro-acoustic analysis of gear pumps takes advantage of a combined model which integrates a lumped-parameter model, a finite-element model and a boundary-element model of a gear pump. The combined model aims at evaluating the acceleration on the exterior parts of the gear pump (external surface of the casing, flange, cover, see Fig. 1b) as well as the acoustical pressure during operational conditions. The used methodology is as follows. The lumpedparameter (LP) model of the moving components gives as output the dynamic forces and torques acting on the gears; in particular, the forces exchanged between the gears and the casing are the pressure forces and bearing reactions. These are the forces exciting the casing vibrations. Then, a finite-element (FE) model of the pump casing (also including the experimental apparatus) was developed and used in order to estimate the casing vibrations caused by the exciting forces applied by the gears and obtained by the LP model. The effect of the lubricant oil inside the gear pump is considered, based on experimental frequency response function measurements and numerical fluid–structure interaction models. Finally, the casing vibrations are used as input data in an indirect boundary element (BE) model for the estimation of the acoustic pressure in different operational conditions. In this sense the model is globally a combined LP/FE/BE model. For the development of the combined model it is assumed that the casing vibration produces negligible influences on the moving part dynamics behaviour. The presented model has been experimentally assessed by a number of experiments: modal analysis, frequency response function measurements, acceleration, force and acoustic pressure measurements. Hereafter, an overview of the three models (LP, FE, BE models) and the relative validation procedures are shown. 3. Lumped parameter model of the rotating components and experimental validation (b) Fig. 1. (a) Schematic of the gear pump and reference frames and (b) exploded view. accelerations as a function of the above-mentioned design and operational parameters. The manufacturing companies and the customers are more interested in the evaluation of the accelerations on the external surface of gear pumps as well as on emitted noise during operational conditions. Note that the vibrations on the external surfaces are directly related to the vibration transfer to the neighbouring structures (e.g. car chassis) and together with the pump emitted noise contribute to low comfort level in the passenger compartment. In this context a combined lumped-parameter finite-element boundary-element model of an external gear pump for automotive applications is presented and experimentally assessed. The FE model makes it possible to carry out a forced vibration analysis for the evaluation of the acceleration levels produced on the external surfaces of the casing by the excitation forces acting on the internal surface (bearing reactions and pressure forces), previously estimated using the LP model. Finally, the casing vibrations are used as input data in a boundary element 3.1. LP model: equations of motion A non-linear lumped-parameter kineto–elastodynamic model was built in order to evaluate the dynamic behaviour of the internal components of the gear pump. The model was already developed and experimentally verified in previous works of the authors [29– 34]. Hereafter e brief description of the model is given. It is a non-linear planar model with 6 degrees of freedom, taking into account the main phenomena involved in the operation of this kind of devices such as meshing effects, variable pressure distribution around the gears and the non-linear behaviour of the journal bearings. Two different reference frames for each gear are used (see Fig. 1a), both having their origins coinciding with the centres of the gears. In reference frames O1X1Y1 and O2X2Y2, the X-axis and the Y-axis are perpendicular and parallel to the Direct Line of Action (DLA), respectively. On the other hand, in reference frames O1 X 01 Y 01 and O2 X 02 Y 02 , the Y0 -axis is along the line connecting the centres of the gears and the X0 -axis is orthogonal. For each gear, the degrees of freedom are the displacements along Xk and Yk directions and the angular displacement: coordinates x1, y1, and h1 are relative to gear 1 (driving gear), while coordinates x2, y2, and h2 are relative to gear 2 (driven gear). Coordinates h1 and h2 are taken as positive in the direction of rotation of gears 1 and 2, Author's personal copy 102 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 respectively, that is to say, clockwise for h1 and anticlockwise for h2. The known input of the model is coordinate h0, representing the angular displacement of the electrical drive, assumed to rotate at constant speed. Coordinate h0 is connected to h1 by a torsional spring-damper element (KT, CT) that represents the dynamic behaviour of the shaft (see Fig. 1b). The force components acting on each gear are resultant forces fpxk, fpyk and torque Mpk of the pressure distribution, meshing force fmg, bearing reactions fbxk, fbyk and motor driving torque Mm. Thus, in the case of meshing contact along the DLA, the equations of motion in reference frames OkXkYk, are: 8 m1 €x1 ¼ fbx1 þ fpx1 > > > > > €1 ¼ fby1 þ fpy1 þ fmg m1 y > > > > < J €h ¼ r f  M þ M p1 m b mg 1 1 > m2 €x2 ¼ fbx2 þ fpx2 > > > > > m2 y €2 ¼ fby2 þ fpy2  fmg > > > : € J h2 ¼ rb fmg  M p2 ð1-6Þ 2 The list of symbols is given in the Nomenclature and in Table 1. The solution of Eqs. (1–6) leads to obtained the translational gear vibration along Xk and Yk coordinate axes as well as the torsional gear vibration around hk coordinates: the vibratory behaviour of the gears in the XkYk plane can therefore be estimated. In the following, the formulations for the estimation of the pressure forces and torques, journal bearing reactions and meshing forces are briefly outlined; the complete formulation can be found in Refs. [29,30]. 3.2. LP model: pressure phenomena Pressure forces and torques (fpxk, fpyk, Mpk of Eqs. (1–6)) are calculated as the resultant forces and moment of the pressure distribution around the gears; this pressure distribution as well as the resultant pressure forces and torques are time-varying during gear rotation. In particular, in order to study the pressure distribution, the pump is divided into several ‘‘control volumes’’, i.e. the ‘‘sealed’’ spaces between teeth, bearing blocks and casing (the control volume are identified with subscript i). They are the isolated tooth space volumes depicted in Fig. 1a, from the ‘‘first isolated tooth space’’ to the ‘‘last isolated tooth space’’. Each control volume Table 1 Dimensions and properties of gears and operational oil. Value for gear 1 b1 = 12.1 mm Value for gear 2 Centre distance of gear pair Gear face width E = 210109 Pa m = 0.3 Young’s modulus Poisson’s ratio J1 = 4.0714107 kg m2 KT = 8.053102 Nm/rad J2 = 3.9564107 kg m2 – m1 = 0.0333 kg m2 = 0.0216 kg ^ ¼ 1:150 mm m rb1 = 6.484 mm rb2 = rb1 ^ ^ x1 = x2 ¼ ^ x1 z1 = 12 Description a = 14.65 mm b2 = b1 z2 = z1 a = 20 deg aw = 27.727 deg Boil = 1400 MPa l = 14 mPa s q = 800 kg/m3 d = 139.36 deg c = 220.64 deg Gear moment of inertia Torsional stiffness of the driving shaft Gear mass Gear module Base radius Addendum modification coefficient ( confidential) Number of teeth Pressure angle Pressure angle in operational condition Oil Bulk modulus Oil viscosity Oil density is treated as an open thermodynamic system with mass transfer with its surroundings. The rate of change in fluid pressure induced by the mass transfer and volume variation, being the fluid characterised by its isothermal bulk modulus Boil, can be conveniently expressed in terms of gear angular coordinate h [2,3]:   dpi Boil dV i ¼ DQ i  x dh V i x dh ð7Þ By using this relation, it is possible to determine the pressure variation inside control volume i in the angular region defined by angle c (Fig. 1a), caused by the flow rate gain DQi and by the volume vari iation of dV . The flow rate gain DQi is the balance between the flows dh coming in and out from control volume i by the clearance between the tooth tip and casing and between the tooth lateral flanks and bearing blocks; furthermore, the drainage flow has also taken into account. For the calculation of flow rate gain DQi in Eq. (7), the contribution of the pressure drop between adjacent volumes and the entrained flow have been taken into account. It is well-known that for a fluid film element with height h, length l and width w, the volumetric flow rate due to the pressure drop Dp can be calculated using Poiseuille’s equation that supposes laminar flow [3]: 3 Qp ¼ wh Dp 12l l ð8Þ On the other hand, the volumetric flow rate due to the entrained flow has a linear distribution from zero to u, where u is the relative velocity of the upper part of the fluid film element with respect to the bottom part. It can be calculated using the well-known relation [3]: Qu ¼ wuh 2 ð9Þ The formulation of the meatus height also takes into account the modification of the casing internal profile due to the running in process carried out in order to increase the volumetric efficiency [33]. Applying continuity Eq. (7) to each control volume (the ‘‘sealed’’ spaces between teeth, bearing blocks and casing), the pressure inside such control volumes is computed by means of an iterative solution process. It can be noted that the number of considered control volumes will be variable during the gear rotation because of the variation in the number of isolated tooth spaces between the gears and the casing. The initial position in the integration procedure is chosen as the position when a tooth space becomes isolated at the inlet side, as shown in Fig. 1a for gear 1: in such a condition the number of control volumes is the maximum (N). Then, when the last isolated tooth space starts the communication with the outlet chamber, the number of control volumes is reduced to N  1. Therefore, the number of equations must be modified. Finally, the starting situation is repeated after one angular pitch when a new tooth tip arrives at the initial position and a new tooth space is isolated from the inlet chamber. The number of control volumes becomes maximum again. For the pump being studied N is equal to 7. In order to define the pressure around the gears in the remaining part (identified by angle d of Fig. 1a), the pressure in the inlet and outlet chamber is assumed as constant and equal to the atmospheric pressure and the output pressure, respectively. In the gear meshing area, when two tooth pairs come into contact, a trapped volume could arise and undergo a sudden volume reduction leading to a violent change in its pressure. To avoid this, the trapped volume is always connected with the high or low pressure chamber. This role is performed by the relief grooves milled in the internal face of the lateral bearing blocks, whose dimension is very important in the resulting dynamic behaviour (as demonstrated in [32]). In particular, in the pump Author's personal copy E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 being studied, the inlet and outlet chambers are in contemporaneous communication with the trapped volume for the interval equal to 3.5% of the meshing period, due to the dimension of the relief grooves. During this time interval a linear decrease of the pressure in the trapped volume from the outlet to the inlet value has been assumed. The latter assumption has an experimental origin. Experimental measurements of the pressure evolution in a tooth space for a complete gear rotation performed in external gear pumps [3] depict an almost linear pressure transient in the meshing area. However, in some experimental measurements reported in [3] a pressure peak occurs in the meshing area when the rotational speed highly increases. This phenomenon is not taken into account in the current model, but it has been verified that the error introduced in the estimation of the dynamic pressure forces is negligible. 3.3. LP model: gear meshing and bearing reactions Gear meshing phenomena have received particular attention [30,31]: in particular the time-varying meshing stiffness, the tooth profile errors, the lubricant squeeze and the effects of the backlash between meshing teeth have been included in the model. The gears being studied are low contact ratio gears, equal to 1.35, so in each line of action (direct line of action and inverse line of action) there should be one or two meshing tooth pairs. To consider this situation, each one of the possible tooth pairs in contact is associated with a stiffness and a viscous damper. This damper takes the structural damping as well as other damping effects into account. The damping coefficient is taken proportionally to the corresponding stiffness if teeth are in contact; when tooth separation occurs, the damping coefficient is computed in order to represent the lubricant squeeze effect. The tooth profile errors represent the relative gear errors of the meshing teeth; when two pairs of teeth come into contact there will be two separate error functions, each acting on a different spring and damper. In this paper, the error functions are defined in order to only represent the tooth profile errors evaluated by means of metrological measurements; other types of gear errors are neglected. Moreover, the backlash along the line of action is considered: it is obtained by Wildhaber measurements. Taking the contribution of such effects, the formulation of the meshing forces (fmg) becomes highly non- linear (Eqs. (1–6)). One of the particular features of gear pump design is the use of hydrodynamic journal bearings for gear shaft support. The non-linear behaviour of this kind of bearings has been modelled using Childs’ theory called ‘‘finite impedance formulation’’ [19,20]. In brief, this formulation consists in taking into account and composing the results from ‘‘short bearing’’ and ‘‘long bearing’’ theories. This formulation leads to obtain bearing reactions (fbxk, fbyk) of Eqs. (1–6). Such a formulation states that the magnitude of the reaction forces in the journal bearings increases as rotational speed, lubricant oil dynamic viscosity, eccentricity ratio of the journal axis and its derivatives increase and as the ratio between radial clearance and journal radius decreases. 3.4. LP model: ‘‘static’’ equilibrium position of gear axes The equations of motion (1–6) are numerically integrated in Simulink environment [35] by using a variable step integration method. The geometry is hp periodic, where hp = (2p)/z is the angular pitch. Therefore, in steady-state operational conditions, the excitation components (pressure forces and torques, meshing forces) have periodicity equal to the angular pitch and consequently to meshing period T. As a consequence, each gear axis has a periodic motion; in particular, the gear centreline trajectory is an orbit around an eccentric position with respect to the casing. There is a certain interest in studying this eccentric position: it is 103 not the actual gear axis position, but it can be considered as a reference position during the axis orbital motion. This position can be obtained by the ‘static’ equilibrium of gears. It is the equilibrium in the ideal steady-state condition in which the periodic variation in forces and torques is neglected and they are taken as a constant value, equal to their mean value. This eccentric position is hereafter referred to as ‘static’ equilibrium position (SEP). The SEP is obtained by an iterative procedure that finds this equilibrium position by taking into account the average value of the pressure forces and torques over one gear pitch, the mean meshing forces and the bearing reactions (more details are given in [29]). 3.5. LP model: experimental validation Hereafter, a brief description of the validation procedure will be given and some comparison results between experimental and simulation data will be shown; all the details about the validation methodology and the comparison results can be found in [34]. A non-conventional validation procedure has been developed in order to assess the dynamic model. It is a vibration-based validation that enables to quantify the precision in the estimation of the sum of the dynamic forces of Eqs. (1–6) and thus of gear accelerations. Since we are interesting in the global dynamic behaviour of the gears, this kind of validation approach, which aims at verifying gear vibrations should be considered satisfactory. A specific experimental set-up and procedure were developed in order to validate the model. Tests were carried out on a test bench in which the pump is fastened on an ergal plate and it is driven by an electrical motor. Model validation is not a simple task in complex systems like gear pumps where it is not easy to directly obtain vibration data concerning rotating components. Thus, it is necessary to acquire secondary measurements that, after proper processing, provide quantities that can be compared with simulation results. In particular, D’Alembert’s principle is applied to the whole system constituted by the pump and the ergal plate: some terms are obtained by means of measurements and others are simulation results. D’Alembert’s equations are hereafter expressed in reference frame O1 X 01 Y 01 ; the reference point about which the moments are taken is origin O1 and the sense of the moments is in accordance with coordinate h1 (see Fig. 1a), thus: P P P ð F ecX 0 þ F icX 0 ÞT þ ð F igX 0 ÞS ¼ 0 P P P ð F ecY 0 þ F icY 0 ÞT þ ð F igY 0 ÞS ¼ 0
P P P MecO1 þ MicO1 T þ ð MigO1 þ M m ÞS ¼ 0 ð10-12Þ where the external forces and moments acting on the ergal plate and pump casing ðF ecX 0 ; F ecY 0 ; MecO1 Þ are measured by means of four piezoelectric triaxial force sensors, the inertia forces and moments of the plate and casing ðF icX 0 ; F icY 0 ; MicO1 Þ are measured by piezoelectric accelerometers mounted on the casing external surface and on the plate, while the inertia forces and moments acting on gears ðF igX 0 ; F igY 0 ; MigO1 Þ and the external torque due to the driving shaft (Mm) is given by the elastodynamic model. In D’Alembert’s equations, the sum of the terms obtained by simulations (identified with subscript S in Eqs. (10–12)) are compared with the sum of the terms obtained by means of experimental tests (identified with subscript T in Eqs. (10–12)). The validation is carried out both in time and frequency domain. In particular, hereafter the results of the validation in time domain are presented; they concern output pressure of 34 and 90 bar and rotational speed of 2000, 3000 and 3350 rpm. These are the standard operational conditions for this type of pump. Model results have been obtained by using gear and pump parameters of Table 1. Fig. 2 shows the comparison between the experimental RMS values of the synchronous average (SA) of the above-mentioned components and the simulation ones (Eqs. (10–12)). Author's personal copy 104 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 with 0 6 # < H 3 2 1 0 2000 rpm-34 3350 rpm-34 2000 rpm-90 3000 rpm-90 bar bar bar bar Force Y'1-dir (RMS Values) (b) 3.6. LP model: evaluation of the variable loads exciting the casing The evaluation of the variable loads exciting the casing is important for the calculation of the emitted noise and vibration. To this aim, it is interesting to note that in the equilibrium of Eqs. (10–12), the inertia and external components directly acting on gears (identified with subscript S) are transmitted through the casing. They load the pump casing and the plate and produce the external and inertia components acting on pump casing and plate (identified with subscript T). The same components are transmitted to the casing in operational conditions as well; thus these are the variable forces and moments applied to the pump casing and exciting casing vibration. These components can be evaluated in terms of gear accelerations by using Eqs. of motion (1–6); in reference frame O1 X 01 Y 01 they become: 1 4 ð13Þ If sufficient averages are taken, the SA closely approximates a truly periodic signal with periodicity corresponding to one revolution of the selected rotational element. This process strongly reduces the effect of components non-synchronous with the reference. Moreover, measurement noise is reduced too. In the present case, the SA have been performed in post-processing with a linear interpolation. In the calculation of the moment components obtained by simulations (identified with subscript S), the mean value of the driving shaft torque, Mm, was cancelled in order to compare these quantities with the experimental quantities. The agreement between the RMS values is not satisfactory at 3000 rpm to 90 bar in X 01 -direction and at 3350 rpm to 34 bar in Y 01 -direction, but in all other operational conditions and directions the agreement is rather good. This indicates that the model is able to give a satisfactory evaluation of the dynamic forces and thus of the gear accelerations. P €1 þ m2 y€2 Þ cos aw F gcX0 ¼ ð F igX 0 ÞS ¼ ðm1 €x1 þ m2 €x2 Þ sin aw  ðm1 y P F gcY 0 ¼ ð F igY 0 ÞS ¼ ðm1 €x1 þ m2 €x2 Þ cos aw  ðm1 y€1 þ m2 y€2 Þ sin aw P €2 cos aw Þ MgcO1 ¼ ð MigO1 þ Mm ÞS ¼ ðaÞðm2 €x2 sin aw  m2 y J h€1 þ J h€2 þ Mm Simulation 5 ð14-16Þ 4.5 Experimental Data 4 Simulations 3.5 3 [N] X 1 M1 pð# þ lHÞ; M l¼0 Experimental Data 6 2.5 2 1.5 1 0.5 0 2000 rpm-34 bar (c) 3350 rpm-34 bar 2000 rpm-90 bar 3000 rpm-90 bar Moment (RMS Values) 0.25 Experimental Data Simulations 0.2 [Nm] mp ð#Þ ¼ Force X'1 -dir (RMS Values) (a) 7 [N] Synchronous Averaging [36,37] is a common method used to process the signal in presence of rotational element. It consists in the synchronisation of the sampling for the measured signal with the rotational element of interest, and the evaluation of the cyclic average over many revolutions with the start of each frame at the same angular position. In this way a signal called Synchronous Averaging (SA) is obtained, which in practice contains only the components synchronised with the rotational element in question. The SA mp(#) of the measured signal p(#), synchronised with the rotor shaft in the angle domain #, is evaluated over a number of rotations M, each corresponding to one angular period H, as follows: 0.15 0.1 0.05 0 2000 rpm-34 bar 3350 rpm-34 bar 2000 rpm-90 bar 3000 rpm-90 bar Fig. 2. RMS values of the SA relative to one meshing period of (a) force in X 01 direction, (b) force in Y 01 -direction, and (c) moment about point O1. used as input data in the FE analysis described in the following Section. 4. Finite element model of the casing and experimental validation 2 4.1. Structural FE model: description Fig. 3 presents the effect of output pressure variation on the variable loads exciting the casing ðF gcX 0 ; F gcY 0 ; M gcO1 Þ. These variable loads are noticeably affected by the output pressure, since they are obtained as the combination of gear accelerations (see Eqs. (13–15)). In particular, their peaks significantly increase as the output pressure increases. Furthermore, it is worth noting that moment MgcO1 Þ in Fig. 3 clearly shows damped waves in the second half of the meshing period, following the impulsive events. The frequency of these damped waves corresponds to the first torsional natural frequency of the system, evaluated through an undamped linearised model (4130 Hz, see [30]). These variable loads will be The structural FE model regards the gear pump and also the plate to which the pump is fastened during the experimental tests. The model must be validated using experiments and therefore it has been modelled the same apparatus used for the experimental measurements, i.e. the pump, the ergal plate and the four force sensors supporting the plate as shown in Fig. 4a. Obviously, once the model has been assessed, it should be possible to model the real boundary conditions as in the actual location in vehicles. It can be noted that the test bench used in the validation of the LP/ FE model is the same test bench used for the validation of the LP Author's personal copy 105 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 this effect, rigid spider connections (Card RBE2 in Nastran) are used in order to connect the casing surface with the end-plate surfaces as shown in Fig. 5a. Furthermore, the presence of bearing blocks, oil and relief valve is taken into account. Each of these components is modelled by means of a concentrated mass and an inertia momentum (Card CONM2 in Nastran) located in its centre of mass and connected to the surrounding mesh by means of interpolation spiders (Card RBE3 in Nastran). The ergal plate of the test bench is modelled using tetrahedral elements. The ergal plate is connected to the pump by means of two aluminium screws (Figs. 4 and 5a). These screws are modelled using beam elements – in the same way as for the steel screws – and are connected to the ergal plate and to the pump by means of interpolation spiders. Moreover, rigid spiders are used between the end-plate lower surface and the ergal plate surface for modelling the connection between the two surfaces and in order to avoid penetration between the surfaces themselves. Finally, the ergal plate is connected to ground by four triaxial force sensors located under the ergal plate. These sensors are modelled by means of spring elements having the nominal stiffness of each sensor as collected in Table 3; the spring elements are joined to the plate mesh by means of rigid spiders as depicted in Fig. 5b. model. In more details, the structural FE model includes the casing, the two end-plates (the lower one is indicated as flange and the upper one as cover), as well as the ergal plate to which the pump is fastened on the test bench and the four force sensors supporting the plate. The casing and the two end-plates are meshed using tetrahedral elements as indicated in Table 2. The casing and the end-plates have different Young Modulus and density: the casing is in aluminium whereas the end-plates are made of steel. In the actual pump, the three components are fastened together by means of two steel screws (M6, length 65 mm) as shown in Fig. 4. The two screws, with proper tightening torque, guarantee the connection between the casing and the end-plates. The two screws are modelled as beam elements having the same cross section area and inertia properties of the actual screws joined to the surrounding mesh of the end-plate by means of interpolation spiders (Card RBE3 in MSC.Nastran [38]) as shown in Fig. 5a. This way, the motion of the two end-points of the beam element is defined as the weighted average of the motions of the surrounding mesh. The screw tightening torque is not applied to the screws in the model because it represents a static torque and therefore it gives no contribution in a dynamic analysis. Moreover, the tightening torque produces the effect of joining the casing surface to the end-plate surfaces. Therefore in order to model FgcX' F 60 40 gcY' 3000 rpm - 90 bar 3000 rpm - 90 bar 40 [N] [N] 20 0 20 -20 -40 0 0 20 40 60 80 -20 100 10 0 20 40 60 80 100 40 3000 rpm - 20 bar 20 -10 3000 rpm - 20 bar [N] [N] 0 0 -20 -30 0 20 40 60 80 -20 100 0 20 40 %T 60 80 100 %T M 3 gcO1 [Nm] 2.5 2 3000 rpm - 90 bar 1.5 1 0 20 40 60 80 100 1.5 [Nm] 3000 rpm - 20 bar 1 0.5 0 0 20 40 60 80 100 %T Fig. 3. Forces and moments exciting the pump casing over one meshing period, expressed in reference frame O1 X 01 Y 01 at different output pressure values and rotational speed of 3000 rpm. Author's personal copy 106 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 Steel screws Y1’ Z 1’ X1’ Aluminium screws Ergal plate (a) (b) Fig. 4. (a) Pump in the test bench with transducers and (b) relative FE model with directions of reference frame X 01 Y 01 Z 01 . 4.2. Structural FE model: experimental validation 4.3. Fluid-structural FE model: theoretical overview Experimental tests have been carried out in order to validate the structural FE model. In particular, frequency response function (FRF) measurements have been acquired in the pump fastened to the test bench exciting point C in direction X 01 (Fig. 6a) by an hammer and measuring the acceleration response along the same direction by an accelerometer (PCB 353B18, frequency range 1 Hz to 10 kHz) located in point B (Fig. 6a). The FRF test has been performed in two different conditions: without oil inside the pump and with the presence of oil at 23 bar. The 23 bar pressure has been obtained by an external equipment. Fig. 6b shows the comparison of the experimental FRFs at the two mentioned operational conditions, in the frequency range till 5 kHz, where the main pump modes occur. The figure clearly shows two peaks at about 2300 Hz and 2700 Hz corresponding to the first two natural frequencies of the pump. It is interesting to note that the peaks referring to the condition ‘‘without oil’’ occur at major frequencies than the peaks referring to the condition ‘‘with oil at 23 bar’’. This is due to the mass effect that the oil introduces in the system under study. This phenomenon can be captured by the developed structural FE model, which includes the effect of the oil as concentrated mass. The figure also highlights that the amplitude of the FRF is higher in the case ‘‘with oil at 23 bar’’, due to the contribution of the oil in pressure [39]. This phenomenon cannot be accounted by the structural FE model presented above. However, it is very important since the FRF amplitude is increased of about 4 times in correspondence of the peak at 2700 Hz (from 6 to 24 kg1), see Fig. 6b. Hereafter, a brief overview of the coupled fluid-structural problem solved by the finite element scheme is presented. The equation of motion regarding the structural part considers the external structural loading conditions and the coupled fluid loading. On the other hand, the fluid response in the cavity is caused by external fluid excitation and by structural vibration on the boundary. The coupled dynamic equation of the FE model for the fluid, can be written as follow [40]: € þ C f p_ þ K f p ¼ AT u € þ g fluid Mf p ð17Þ where Mf, Cf and Kf are the global inertia, damping and stiffness matrices resulting by an indirect formulation of the fluid problem, gfluid is the fluid load, AT is the term which considers the effect of the structural deformation in fluid problem, p is the fluid pressure and u is the structural displacement. The equation of motion for the solid linearised dynamics is given as: € þ C s u_ þ K s u ¼ Ap þ g ext Ms u s s ð18Þ s where M , C and K are the standard mass, damping and stiffness matrices of the solid structure, gext is the mechanical external load on the structure, Ap represents the loads applied by the fluid on the structure. Thus, Eqs. (17) and (18) can be combined as follows:  Ms AT    s € u C þ f € p M 0 0    s u_ K þ f p_ 0 C 0 A Kf   u p ¼ ( g ext g fluid ) ð19Þ Table 2 Properties of the system components modelled by the FE method. System components # Nodes # elements Element type Casing 46,024 27,398 Flange 38,554 22,192 Cover 37,605 22,120 Pump screws Ergal plate 2 39,478 1 24,740 Plate screws Oil 2 10,521 1 7910 Tetrahedral, 10 nodes(card CTETRA) Tetrahedral, 10 nodes (card CTETRA) Tetrahedral, 10 nodes (card CTETRA) Beam(card CBEAM) Tetrahedral, 10 nodes (card CTETRA) Beam(card CBEAM) Hexahedral, 8 nodes (card CHEXA) The differential equations system (19) has to be solved in order to obtain the fluid pressure and the structural deformation in a coupled fluid-structural problem. 4.4. Fluid-structural FE model: description and experimental validation The structural FE model was improved and a coupled fluidstructural problem has been solved simultaneously to include the mutual coupling interaction between the fluid pressure and the structural deformation. It is a matter of fact that in the real system the oil in pressure can transfer vibration from the interior to the exterior of the pump. Thus, the oil has to be accounted as a mechanical system with distributed mass and stiffness characteristics coupled with the structural mesh parts. Author's personal copy 107 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 Interpolation spider Aluminium screws Rigid spider Steel screws (a) (b) Fig. 5. (a) Modelling of screws (beam elements in blue and yellow) and rigid spiders (in red) between case and cover and (b) rigid spiders (in red) connecting the ergal plate mesh and spring elements (not visible in the figure), modelling the force sensors between ground and the ergal plate. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 3 Stiffness of the spring elements in the lower surface of the ergal plate (values taken from sensor data). Stiffness (N/m) X 01 -direction Y 01 -direction Z 01 -direction 1E9 1E9 2.6E9 modes occur. The amplitude of the fluid-structural FRF is similar to the amplitude of the experimental FRF. On the other hand, the structural FRF presents amplitude extremely smaller than the experimental one, as already discussed in Section 4.1. Concerning the frequency contents, Table 4 resumes the first two natural frequencies for the same three scenarios of Fig. 8. The numerical natural frequencies are obtained by a modal analysis, SOL 103 in MSC.Nastran driven from LMS Virtual.Lab. The experimental natural frequencies are obtained by an experimental modal analysis performed on the pump. The results concerning the structural simulation and the fluid-structural simulation are the same in terms of natural frequencies (Table 4). This is due to the fact that both the structural FE model and the fluid-structural FE model take into account the oil mass, the first as concentrated mass, while in the second by the mesh. The difference in percentage between the experimental frequencies and the numerical ones is 13% and 5% for the first and second natural frequency, respectively. This difference in also confirmed by the peak location along the horizontal axis in the FRFs of Fig. 8. Moreover, on the fluid-structural FE model, a forced response analysis has been carried out (SOL 111 in MSC.Nastran) with the aim at obtaining the vibration level on the entire pump in operational conditions. The excitations are the pressure forces and moments and the bearing reactions exchanged between the gears and the casing (see Section 3.3 and Fig. 3) as well as the variable Fig. 7 depicts the cavity mesh developed for the pump being studied, which has been coupled to the structural one for the solution of the fluid-structural problem. The properties of the cavity mesh are reported in Table 2. The cavity mesh has the fluid properties of the pumping oil (see Table 1), in particular the oil density and the oil bulk modulus have been introduced in the model in order to take into account the distributed mass and stiffness effects, respectively. Moreover the continuity of the mesh between the structural and fluid part has been imposed. In order to verify the effectiveness of the coupled fluid-structural FE model with respect to the genuine structural FE model described above, a numerical FRF analysis (SOL 111 in MSC.Nastran driven by LMS Virtual.Lab [38,41]) is carried out with excitation point in C and response point in B, along X 01 direction (see Fig. 6a). The numerical FRFs obtained by the fluid-structural FE model and the structural FE model are compared with the experimental FRF obtained by hammer and accelerometers (see Section 2.2). Fig. 8 depicts such a comparison in the frequency range of interest (1500–3500 Hz), where normal Y1’ X1’ B A C FRF Amplitude [1/kg] 25 without oil with oil at 23 bar 20 15 10 5 0 0 1000 2000 3000 4000 5000 Frequency [Hz] (a) (b) Fig. 6. (a) Input and output points in the FRF analysis and (b) experimental FRF amplitude for tests without oil and with oil at 23 bar with excitation in C and response in B, along the X 01 direction. Author's personal copy E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 output pressure of the oil exerted to the pump casing, flange and cover. In the fluid-structural FE model, such forces have been located in the centre of the cavity where the gears are housed. The LP model can also estimates the variable output pressure. This pressure has been applied to the areas depicted in Fig. 9 where actually this pressure acts in operational conditions. The forced analysis has been carried out using excitations estimated by the LP model at the operational condition of 3000 rpm and 90 bar. It is worth noting that the frequency-dependent damping introduced in the numerical FRF and forced response analyses described above, is evaluated by means of an experimental modal analysis (EMA) performed on the pump at the condition ‘‘with oil at 23 bar’’, see Table 5. In particular, six PCB piezoelectric accelerometers (frequency range 1–10,000 Hz) are mounted on the pump casing and on the ergal plate in order to measure the response in terms of acceleration. Moreover, a hammer PCB 068C04 is used to excite the system in 24 measurement points both in direction X 01 and Y 01 . The procedure used in order to perform the EMA is the conventional procedure in which both excitation and response are measured simultaneously for obtaining the so called Inertance, i.e. the FRF between acceleration and force. The response points are maintained fixed during the test, while the excitation moves from one measurement point to another in order to obtain the FRF among all the considered points. The signals are acquired using a sample frequency of 40,960 Hz and frequency resolution of 1.25 Hz; furthermore the FRFs are calculated by using the H1 estimator available in LMS Test.Lab [42]. Curve-fitting methods are used to calculate the modal parameters: as well known, they can operate both on the response characteristics in the frequency domain, i.e. on the frequency response functions themselves, and in the time domain as well; the latter method considers that the Inverse Fourire Transform of the FRF is itself another characteristic function of the system that represents the response of the system to a single unit impulse as excitation. In particular, two modal parameter extraction methods are used in this work: the LSCE method (Least Square Complex Exponential) operating in the time domain and the PolyMAX method operating in the frequency domain, both available in LMS Test.Lab. The complete formulation of such methods can be found in [43,44]; once the stable poles are chosen in the stabilization diagram, the modal damping can be evaluated by averaging the modal damping obtained by the LSCE method and the PolyMAX method (Table 5). Fig. 10a shows the comparison between the experimental and numerical acceleration spectra of the casing surface in correspondence of point B in X 01 direction (Fig. 6b). The experimental acceleration has been measured by means of piezoelectric accelerometer mounted on point B at the operational condition of 3000 rpm and 90 bar, while the numerical one has been ob- 2 10 1 10 FRF Amplitude [1/kg] 108 0 10 Fluid-Structural FRF Structural FRF Experimental FRF -1 10 -2 10 -3 10 1500 2000 2500 3000 3500 Frequency [Hz] Fig. 8. Amplitude of FRFs with excitation in C and response in B, along the X 01 direction. Table 4 First two natural frequencies obtained from experimental modal analysis, structural FE simulation and fluid-structural FE simulation. Mode Experimental, fn (Hz) Structural, fn (Hz) Fluid-structural, fn (Hz) 1 2 2325 2651 2013 2518 2013 2518 tained by the forced analysis performed on the fluid-structural FE model. The curves show peaks in correspondence of the meshing frequency and relative harmonics as expected. The peaks are amplified by the resonances of the casing, in particular at the frequency around 2400 Hz. The comparison between the numerical and experimental curves highlights that the developed model is able to capture the trend of the experimental curves, even if differences occur, in particular at the meshing frequencies of 2400 Hz (3rd harmonics) and 3000 Hz (4th harmonics). These differences are due to the overestimated input force in X 01 direction (see Fig. 2). It has to be underlined that the fluid-structural FE forced analysis has been carried out at the output pressure of 90 bar, while damping has been introduced as modal damping estimated by an experimental modal analysis (EMA). This EMA was performed on the pump at the condition ‘‘with oil at 23 bar’’, in which the Cavity mesh Fig. 7. Cavity mesh. Author's personal copy 109 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 23 bar pressure has been obtained by an external equipment, as stated above. It has been demonstrated that the modal damping is very sensitive to the output pressure. Table 5 collects the modal damping values obtained by an experimental modal analysis at two different conditions: ‘‘without oil’’ and ‘‘with oil at 23 bar’’. The difference is large in particular for the 2nd mode, as also depicted in Fig. 6b. Therefore, it is expected that the modal damping could be different for a pressure of 90 bar. This can be a source of discrepancy between numerical and experimental results. 5. Boundary element model simulating noise generation and experimental validation 5.1. BE model: description The vibration level on the pump surface obtained by the forced fluid-structural FE analysis has been used as input data for the acoustic simulation in order to determine the noise emitted by the pump in operational conditions. The developed acoustic model should reproduce the acoustic behaviour of test bench in which the pump has been experimentally tested, in order to enable the comparison between experimental and numerical data. The same test bench used for the validation of the LP and FE model has been used hereafter for the acoustic characterisation (Fig. 4a). Two microphones have been used in order to measure the acoustic pressure in operational conditions, located at a distance of 20 cm from the pump. From an acoustic point of view, the test bench represents a chamber with walls of unknown acoustic impedance. The chamber’s walls are sandwich panels of steel and mineral wool. The boundary element (be) method [45,46] has been used in order to model such a system. Moreover, since we are interested in the acoustic pressure field in the exterior of the pump and in the interior of the test bench, the indirect variational boundary element scheme has been selected [47–50]. By means of the indirect BE method the interior and exterior problems are solved simultaneously. Two output points, B and C (on the same side as points B and C used during the vibration analysis, respectively) have been located in the same position as the microphones in order to estimate the acoustic pressure to be compared with experiments (see Fig. 11a). The test bench walls have been modelled as a box of the same dimension as the real one (see Fig. 11b) and characterised by the complex acoustic impedance of the mineral wood (variable in the frequency domain). The acceleration level on the pump casing surface obtained by the previous fluid-structural FE analysis (Section 4.2) has been used as input data for the determination of the acoustic pressure inside the box. 5.2. BE model: experimental validation Fig. 10b shows the comparison between the experimental and numerical spectra of the acoustic pressure in correspondence of field point B. The figure shows the first ten harmonics of the meshing frequency, between 600 Hz (1st harmonics) and 6000 Hz (10th harmonics). Differences between numerical and experimental spectra occur in the amplitude of several harmonics (1st, 2nd, 4th, 6th,7th) while for others (3rd, 5th,10th) the correspondence is good. In between the harmonics, the spectra are largely different: this is due to the excitation forces estimated by the LP model, which exist only in correspondence of the meshing frequency and relative harmonics, while they are zero elsewhere. However, this difference does not influence the global acoustic behaviour that is dominated by the highest harmonics peaks. Globally, the pressure level estimated by test is 73.8 dB and it is 74.4 dB when estimated by the model. It is interesting to note that the agreement between numerical and experimental data is better for sound (Fig. 10b) than for acceleration (Fig. 10a). This is due to the fact that the acceleration displayed in Fig. 10a mainly depends on the input force along the X 01 direction, which is overestimated with respect to the experimental data (Fig. 2). On the other hand, the calculation of the simulated acoustical pressure uses as input the vibration level obtained by the fluid-structural FE model, in all the directions. Since the agreement in the Y 01 and rotational direction was good at input force level (Fig. 2), globally the predicted sound agrees with the experimental data. The good agreement between the numerical and experimental results confirms that the approximation in using in the model, the modal damping estimated with 23 bar of pressure inside the pump, can be considered acceptable. Thus, this model is able to capture the global vibro-acoustic behaviour of the real pump and it can be used for design optimisation. 6. Concluding remarks This work addresses the development of a combined LP/FE/BE model for the vibro-acoustic analysis of gear pumps for automotive applications. The lumped-parameter part of the model aims at obtaining the gear accelerations and the forces between the moving parts and the casing, the FE model estimates the external casing accelerations using the excitation forces coming from the LP model; the BE model estimates the noise emitted by the pump in operational conditions. Particular attention has been paid in the inclusion of the oil effect inside the pump casing: the fluid–structure interaction between oil and pump casing has been taken into account. In the literature, such effects have been shown and discussed separately, on the contrary, in the case of gear pumps all these important effects have to be considered in the same model in order to take their interactions into account. The LP model has been experimentally verified by a non-conventional validation Table 5 Experimental modal damping obtained by modal analysis in different scenarios. Scenario Modal damping (%) for the 1st mode Modal damping (%) for the 2nd mode Without oil With oil at 23 bar 6 5 4 0.87 Fig. 9. Areas of application of the variable output pressure. Author's personal copy 110 E. Mucchi et al. / Applied Acoustics 77 (2014) 99–111 80 10 0 60 10 10 Pressure [dB] Acceleration [m/s2] 40 -2 Experimental Fluid-structural -4 20 0 -20 -40 -60 Experimental Fluid-structural -80 10 -6 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 -100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] Frequency [Hz] (a) (b) Fig. 10. (a) Acceleration spectra in point B at 3000 rpm and 90 bar and (b) acoustic pressure spectra at location B: experimental and simulation results. C* B* 55 cm 20 cm 20 cm 71 cm 76 cm (b) (a) Fig. 11. (a) BE model and relative output points (B and C) and (b) dimension of the box representing the test bench. procedure. It is a vibration-based validation that enables to quantify the precision in the estimation of the gear accelerations. Validation results can be considered rather satisfactory. The FE model has been validated by a two-step procedure. Firstly, several experimental and numerical frequency response functions have been compared in order to verify the accuracy of the modal behaviour of the model. Secondly, the numerical and experimental casing accelerations have been compared leading to a good agreement. Eventually, the BE model has been verified by comparison between the numerical and experimental acoustical pressure in operational conditions; the pressure level estimated by the model agrees with the experimental data. Thus, the validation results show that the model is able to estimate the amplitude of gear accelerations, the vibrations on the external surfaces of the pump as a function of working conditions as well as the emitted noise. Furthermore, the model can identify the system resonances and capture the global vibro-acoustic behaviour of the real pump. The combined model can be considered a very useful and effective tool: for the evaluation of the contribution of the pump casing that acts as a flexible body amplifying several harmonics of the meshing frequency; in prototype development to identify the origin of unwanted dynamic effects; in design optimisation in the early phase of prototype development as well as in the following phase of design improvement and solution of functional problems. Acknowledgements This work has been developed within the Advanced Mechanics Laboratory (MechLav) of Ferrara Technopole, realized through the contribution of Regione Emilia-Romagna – Assessorato Attività Produttive, Sviluppo Economico, Piano telematico – POR-FESR 2007-2013, Activity I.1.1. References [1] Qatu MS, Abdelhamid MK, Pang J, Sheng G. Overview of automotive noise and vibration. Int J Veh Noise Vib 2009;5(1/2):1–35. [2] Mancò S, Nervegna N. Modello matematico di pompe oleodinamiche a ingranaggi esterni. Oleodinamica-Pneumatica 1987;1. [3] Mancò S, Nervegna N. Pressure transient in an external gear hydraulic pump. In: Maeda T, editor. Proceedings of the second JHPS international symposium on fluid power, 14 October, 1993, Tokyo; 1993. [4] Paltrinieri F, Dilani M, Borghi M. Modelling and simulating hydraulically balance external gear pumps. In: Proceedings of the 2nd international FPNI Ph.D. symposium on fluid power, 3–6 July, 2002, Modena, Italy; 2002. 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