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2014, Applied Acoustics

2015 •

Archives of Acoustics

Vibroacoustic Measurements and Simulations Applied to External Gear Pumps. An Integrated Simplified Approach2016 •

This paper describes the development phases of a numerical-experimental integrated approach aimed at obtaining sufficiently accurate predictions of the noise field emitted by an external gear pump by means of some vibration measurements on its external casing. Harmonic response methods and vibroacoustic analyses were considered as the main tools of this methodology. FFT acceleration spectra were experimentally acquired only in some positions of a 8.5 cc/rev external gear pump casing for some working conditions and considered as external excitation boundary conditions for a FE quite simplified vibroacoustic model. The emitted noise field was computed considering the pump as a ‘black box’, without taking into account the complex dynamics of the gear tooth meshing process and the consequent fluid pressure and load distribution. Sound power tests, based on sound intensity measurements, as well as sound pressure measurements in some positions around the pump casing were performed for val...

In previous works, the authors have presented a numerical model of the dynamic behaviour of an external gear pump for automotive applications. The model takes the most important phenomena involved in the operation of this kind of machine into account. The simulation results have shown that the variable meshing stiffness has a notable contribution on the dynamic behaviour of the pump but this is not as important as the pressure phenomena. As a consequence, the original model was modified with the aim of improving the calculation of pressure forces and torques in order to achieve a result that was closer to measured values. The new pressure formulation includes several phenomena not considered in the previous one, such as the pressure variations at input and output ports, as well as an accurate description of the trapped volume and its connections with high and low pressure chambers. In particular in this paper the relative effect of each considered pressure phenomenon will be discussed in detail, highlighting which are the most important ones, in order to develop a simple but accurate dynamic gear pump model. So, the improved model could be used in order to analyse the dynamic behaviour of the pump and to identify noise and vibration sources.

Energy Procedia

Experimental and Numerical Validation of an Automotive Subsystem through the Employment of FEM/BEM Approaches2015 •

In this paper, to study vibrations of the main components of the engine due to combustion, a dynamic model is presented based on Lagrange's equation. The effects of both reciprocal and rotational components of the engine in producing sound and vibration were accurately considered in the model. This model includes a mass matrix and a velocity matrix whose components were obtained as periodic functions with respect to time. Applying the model to data received from OM-355 engine manufactured by Idem industry, it is possible to analyse noise and vibration due to resonance phenomenon in the engine body. It is concluded that noise propagation resulting from the sixth vibration mode is drastically increased from the engine block, especially from crankcase. The results obtained from this research can be used for design of engine.

2004 •

A clear trend in recent years is increased legal and customer demands for lower noise levels for construction machinery.Even if the gear whine type of noise is not the loudest source, its pure tone ...

6th International Power Transmission and Gearing Conference: Advancing Power Transmission Into the 21st Century

Comparison of Analysis and Experiment for Gearbox Noise1992 •

Low-contact-ratio spur gears were tested in the NASA gear-noise rig to study the noise radiated from the top of the gearbox. Experimental results were compared with a NASA acoustics code to validate the code for predicting transmission noise. The analytical code is based on the boundary element method (BEM) which models the gearbox top as a plate in an infinite baffle. Narrow-band vibration spectra measured at 63 nodes on the gearbox top were used to produce input data for the BEM model. The BEM code predicted the total sound power based on this measured vibration. The measured sound power was obtained from an acoustic intensity scan taken near the surface of the gearbox at the same 63 nodes used for vibration measurements. Analytical and experimental results were compared at four different speeds for sound power at each of the narrow-band frequencies over the range of 400 to 3200 Hz. Results are also compared for the sound power level at meshing frequency plus three sideband pairs ...

Sustainability

Parameterization, Modeling, and Validation in Real Conditions of an External Gear Pump2021 •

This article presents a methodology for predicting the fluid dynamic behavior of a gear pump over its operating range. Complete pump parameterization was carried out through standard tests, and these parameters were used to create a bond graph model to simulate the behavior of the unit. This model was experimentally validated under working conditions in field tests. To carry this out, the pump was used to drive the auxiliary movements of a drilling machine, and the experimental data were compared with a simulation of the volumetric behavior under the same conditions. This paper aims to describe a method for characterizing any hydrostatic pump as a “black box” model predicting its behavior in any operating condition. The novelty of this method is based on the correspondence between the variation of the parameters and the internal changes of the unit when working in real conditions, that is, outside a test bench.

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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
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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
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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,
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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
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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)).
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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
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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.
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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.
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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.
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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.
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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.
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