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TECHNICAL ARTICLE
Pressure Determination Approach in Speciﬁc Pressure Regions
and FEM-Based Stress Analysis of the Housing of an External
Gear Pump
R. Cinar1 , M. Ucar2 , H.K. Celik3 , M.Z. Firat4 , and A.E.W. Rennie5
1 Department of Mechanical Education, Institute of Natural and Applied Sciences, Kocaeli University, Kocaeli, Turkey
2 Department of Automotive Engineering, Technology Faculty, Kocaeli University, Kocaeli, Turkey
3 Department of Agricultural Machinery, Faculty of Agriculture, Akdeniz University, Antalya, Turkey
4 Department of Animal Science, Faculty of Agriculture, Akdeniz University, Antalya, Turkey
5 Lancaster Product Development Unit, Engineering Department, Lancaster University, Lancaster, UK
Keywords
External Gear Pump, Hydraulic Pressure
Measurements, Stress Analysis
Correspondence
H.K. Celik,
Department of Agricultural Machinery,
Faculty of Agriculture,
Akdeniz University,
Antalya
Turkey
Email: [email protected]
Received: September 3, 2013;
accepted: February 13, 2014
doi:10.1111/ext.12086
Abstract
In this study, an application algorithm has been introduced to explore structural
optimisation for gear pump housing. In the study, experimental and theoretical
(analytical and numerical) methods are utilised. A commercial external spur
gear pump which has 17.1 L min−1 volumetric flow rate with maximum
pressure capacity of 250 bar has been considered for an application case study.
In the experimental section of the study, four pressure sensors (emitters)
were placed with angle intervals of 45◦ , 90◦ , 135◦ , and 180◦ on the pump’s
housing to measure operating pressure values at specific pressure regions of the
housing from inlet to outlet. According to experimental results of the pressure
measurements, a response surface analysis (RSA) was carried out and then
an estimation model (empirical equation), which could be used to calculate
pressure values at any specific region of the housing, was obtained. According
to the RSA results gained, it appears that the estimation model has 99.9% R2
value which can be used for adequately predicting accurate pressures at any
region of the housing. Subsequently, this estimation model has been adapted for
commercial finite element method (FEM)-based engineering software and from
which stress distributions on the housing were simulated three dimensionally.
FEM-based simulation outputs showed that there were no failure signatures
on the pump housing. As the main conclusion of the study, it is seen that an
estimation model gives an adequate approach to predict pressure values at any
specific pressure region of the pump housing and, especially, stress distribution
results has highlighted that a structural optimisation study may be suitable for
the pump housing.
Introduction
One of the positive displacement types of hydraulic
pump is the external gear pump. The gear pump
is among the oldest and the most commonly used
pumps in the wide range of application areas from
agricultural machinery to heavy industrial machinery
systems.1 It has become the main choice for applicants
due to long life, minimum maintenance, high
reliability, capability in high-pressure operations, etc.
Basically, an external gear pump converts mechanical
Experimental Techniques (2014) © 2014, Society for Experimental Mechanics
power into hydraulic fluid power with high pressures.
This converting operation is described as follows.2 The
shaft of one of the gears is driven by a motor. This
gear is known as the driving gear. It engages around
the other, known as the driven gear. When the teeth
of the two gears start disengaging (near the centre
of the pump), the consequent low pressure sucks in
the fluid. Upon further rotation, the fluid is trapped
in between the void formed by this cavity and the
housing. The fluid is then carried around towards
1
Stress Analysis of the Housing of an External Gear Pump
the discharge side (outlet) of the pump with higher
pressure magnitudes. As in principle, it cannot flow
back through the very ‘‘small gap’’ between the teeth
and the housing and neither through the engaged
teeth of the gears, it is entirely ejected to the outlet.
Therefore, unlike a fan (e.g.), the flow rate is expected
not to depend on the pressure difference between the
inlet and outlet. In some of the calculations, although
it is assumed that there is no leakage from outlet to
inlet, in practice, slip of the fluid through the gap
and along the housing walls from the discharge side
(high pressure) to the suction side (low pressure) is
inevitable. Therefore, some of the drawbacks of using
gear pumps include high noise, vibration, cavitation
failure, and unavoidable outlet pressure ripple. The
ripples are the main source of the vibration, noise
wearing, and efficiency loss of gear machineries.3
During the pumps operation, cavitation and especially
alternating high-pressure spikes are very undesirable
and essentially impact on the structural stress and life
time of the housing, side plates, bearings and gears,
therefore, they may be responsible for structural
failure of these components.
In the literature, there exists a significant amount
of analytical, numerical, and experimental studies
to address the gears tooth form, cavitation, flow
characteristics, leakage from gaps, and pressure/flow
ripples of external gear pumps. In addition, the
literatures offer limited information regarding the
other structural components’ analysis details for this
type of hydraulic pump and motors with external and
internal gears.4 – 17
In this study, an application algorithm has been
developed for structural optimisation exploration of
external gear pump housing. A sample high-pressure
external spur gear pump has been considered, with
the focus on the structural analysis and structural
optimisation exploration of its housing. The study
centred on the algorithm that can be summarised as
follows: firstly, the alternating pressure distribution
issue on the gear pump from inlet to outlet and
an experimental setup are introduced with the aim
to obtain experimental data for alternating pressure
distribution at the inside pressure surfaces in the
pump housing, within defined operating conditions.
Then the experimental data is analysed utilising
response surface modelling (RSM). According to the
statistical analysis, an estimation model (empirical
equation) is derived that extracts the exposed
pressure values at any specific region of the
pump housing from inlet to outlet, numerically.
Subsequently, the pressure equation is employed
in the finite element analysis (FEA) to investigate
2
R. Cinar et al.
the structural deformation behaviour of the pump
housing. According to the simulation outputs, the
structural optimisation approach for the pump
housing is discussed.
Application Algorithm
Generally, maximum operating pressures are considered in design calculations for gear pump systems;
however, pressure distribution is not uniform in the
pressurised surface of the housing from inlet to outlet.
This phenomenon creates some difficulties in order
to reach the optimum structural design parameters
of the pump housing. At this point, it is true to
say that it is important to predict pressure values
at any specific region (between the inlet and the
outlet) of the housing in order that it is possible to
calculate stress distribution, which is one of the evaluation parameters required for the optimum structural
design process. Considering this issue, an application
algorithm that includes experimental and theoretical (analytical and numerical) methods has been
developed which can be used in pre-design decision processes and to determine the likelihood for
structural optimisation potential of the external gear
pump housing. The application steps of the algorithm
are detailed in Fig. 1.
Pressure Distribution Issue in External Gear Pumps
and Experimental Setup
In the literature, the most commonly used pressure
distribution approach for the external gear pump is a
linear approach that is defined by the gear rotation
angles and maximum operating pressure magnitudes
(Fig. 2). In this approach, no leakage is assumed
during the pumping operation and the pressure at a
specific angle’s region (Pi ) is expressed as per Eq. 1
given below.4,5
β
Pi = Pmax
(1)
π
As detailed in Fig. 2, the pressure distribution is seen
as non-uniform from inlet to outlet and the maximum
pressure is defined as Region IV in the gear pump
housing. However, this knowledge has been considered in a linear approach assuming no leakage, and
an experimental study has also been set up in order to
determine experimental pressure distribution values
with inevitable internal leakage at specific regions in
the external gear pumps. It was described as an estimation model which might be used to load the housing correctly for its structural deformation potential.
In the experimental setup, a commercial external spur
R. Cinar et al.
Figure 1 Application algorithm for exploration of structural optimisation potential.
gear pump (HEMA Industry Inc., Tekirdağ, Turkey),
which has 17.1 L min−1 volumetric flow rate and a
maximum pressure capacity of 250 bar with gears of
2.75 module and 12 teeth, was utilised. The pump was
driven at 1500 min−1 revolutions by a four pole electric motor (GAMAK™; Makina Sanayi A.Ş., İstanbul,
Turkey) which has a power capacity of 11 kW. All of
the hydraulic equipment and the electric motor have
been situated on a chassis frame. The pressure measurements were conducted by utilising a computeraided measurement system with five GEMS™ (Gems
Sensors & Controls, Plainville, MA)-3200-heavy duty
series pressure transmitters, which have a measurement range of 0–400 bar. Four of them were placed
on the pump housing with 45◦ intervals from the inlet
to the outlet lines and the fifth one was placed after
the outlet line. In order to verify this, a check valve
and an analogue pressure gage are also placed on the
output line just before the oil tank. In the experiment,
AKSOIL HYDROTRANS ISO 46 hydraulic oil was
used. The pressure measurements in the experiment
were conducted by stepping up the outlet pressure
in 10-bar interval from zero pressure to a maximum
pressure value of 250 bar (i.e. 25 steps representing
each of the recurrences). Pressure data from all of
the pressure transmitters were recorded for 30 s after
stable regime time for each of the pressure steps.
These measurement procedures had three recurrences. Placement of the pressure transmitters and
the experimental setup details can be seen in Fig. 3.
Results of the Experimental Practice
According to the data collected from the pressure
transmitters at the different regions on the pump
housing, the pressure distribution behaviour of this
gear pump housing was obtained experimentally
between the inlet and the outlet. The data was
also analysed utilising the RSM methodology which
is a useful tool in modelling curvature effects in
many scientific disciplines. Statistical analysis has also
extracted an estimation model that gives pressure (Pi )
values at any specific region depending on outlet
pressure (Pmax ) and gear rotation angle (β). The
quadratic model obtained from the response surface
analysis (RAS) is given as:
2
2
+ β4 x2i
+ β5 x1i x2i + i
yi = β0 + β1 x1i + β2 x2i + β3 x1i
(2)
3
R. Cinar et al.
Figure 2 Pressure distribution and pressure regions at the gear pump housing.
where, yi is the pressure at specific region (Pi ) (bar),
x 1i is the outlet pressure (maximum pressure) (Pmax )
(bar) and x 2i is the gear rotation angle (β) (◦ ); β 0
is a constant coefficient, β 1 , β 2 , β 3 , β 4 , and β 5 are
the interaction coefficients of linear, quadratic, and
second-order terms, respectively; and i is the error
term. The quality of the fit of the model in Eq. 2 was
evaluated using the coefficient of determination (R2 ).
The statistical significance of the polynomial model
used in this study was determined by the F-test. The
significance of the regression coefficients was tested
by Student’s t-test.
The results of the RAS are presented in Table 1 and
the predicted response is described as follows:
yi = −9.550433 + 0.511168x1i + 0.195402x2i
2
2
+ 0.000065669x1i
− 0.0008x2i
+ 0.002824x1i x2i
(3)
As can be seen from Table 1, the overall regression
F value of 15448.6 and the low probability value
(P < 0.0001) indicate that the model was significant
for the response variable pressure at a specific region.
The estimated parameters for the statistics indicated
a well-established pressure pattern which can be
described by a response surface model with linear,
quadratic, and cross-product terms [see partial Ftest and F-test for overall (total) regression results
in Table 1]. For this model, both variables x 1i and
4
x 2i and their interaction seem to be very important
parameters (t-test; in Table 1). The canonical analysis
indicates that the predicted response surface is shaped
like a saddle (Fig. 4). The eigenvalue of 13.37 shows
that the valley orientation of the saddle is less
curved than the hill orientation, with eigenvalue
of −19.39. As the canonical analysis resulted in a
saddle point, the estimated surface does not have
a unique optimum. However, the ridge analysis
indicated that maximum yields will result from higher
values of outlet pressures and gear rotation angles.
A contour plot of the predicted response surface,
shown in Fig. 4, confirms this conclusion. The
coefficient of determination (R2 = 99.9%) obtained
in this study was rather high, indicating that 99.9%
of the variability in the response could be explained
by the model (Eq. 3). On the basis of these results,
the response surface model obtained in this study
for predicting the pressure at specific region was
considered reasonable.
In addition to the contour plot in Fig. 4, correlation
between raw experimental results, estimated results,
and theoretical (From Eq. 1) results were explored
and the plots were presented for the measurement
points at 45◦ , 90◦ , 135◦ , and 180◦ on the pump’s
housing, given in Fig. 5. From initial observation
of these charts, it can be seen that the results
show a higher difference than the others at the first
measurement points and then it has an overlap at the
R. Cinar et al.
Figure 3 Experimental setup.
Table 1 Response surface analysis result
Equation parameters
β0
β1
β2
β3
β4
β5
R2 = 0.999
t value
Pr > |t|
Partial F-ratio test
1.429313
−6.68
0.013763
37.14
0.024904
7.85
0.00004785
1.37
0.00011
−7.9
0.000061541
45.88
F value = 15448.6 (total regression)
<.0001
<.0001
<.0001
0.173
<.0001
<.0001
37561.1 (0.0001)
Estimate
−9.550433
0.511168
0.195402
0.000065669
−0.00087
0.002824
SE
last measurement point, that is, the outlet line of the
pump housing. The reason for these differences can be
explained as follows: in the theoretical approach, no
leakage at the pumps gear mesh is assumed; however,
leakage is inevitable in physical practice. Therefore, a
pressure rise is seen in the charts for the experimental
results, especially at the first measurement points.
Conversely, there is good correlation between the
raw experimental and estimated results plotted in the
charts. As the charts show, linear pressure is observed
from the inlet to the outlet in the pump, producing an
uncomplicated degree of estimation. This correlation
has proved that the estimation model, which is
extracted from the statistical analysis, adequately
reflects the physical practice scenarios with this pump.
7.76 (0.0007)
2105.24 (0.0001)
Probability = < 0.001
Model Adequacy Checking
Usually, it is necessary to check the model in Eq. 3 to
ensure that it provides an adequate approximation to
the real system. Unless the model shows an adequate
fit, proceeding with the investigation and optimisation
of the likely fitted response surface will give poor
or misleading results. The residuals from the least
squares fit play an important role in judging model
adequacy. The model adequacy or the appropriateness
of the fit of the proposed model to the experimental
data can be assessed by applying the diagnostic plots,
such as a histogram of the residuals and the predicted
versus actual value plots. Figure 6 shows these two
diagnostic plots for the pressure at a specific region. A
histogram plot indicates whether the residuals follow
5
Figure 4 Contour plot of predicted response surface.
a normal distribution, in which case the data is
normally distributed. As can be seen in Fig. 6, the
predicted values of pressure obtained from the model
(Eq. 3) and the actual experimental data were in good
agreement.
FEM-Based Stress Analysis of the Pump Housing
In this step of the study, it focuses on the structural
deformation exploration of the gear pump housing
by utilising finite element method (FEM)-based
stress analysis. The results of numerical calculations
are strongly dependent on boundary conditions,
e.g. mounting, connection of the single housing
components, etc. For this reason, the boundary
conditions have to be set exactly to achieve
results comparable with the measurements.18 The
experimental study in the previous section had
extracted an estimation model which described
pressure values at any specific region depending
on outlet pressure values and gear rotation angle;
thus, in the finite element analysis (FEA), applied
pressure magnitudes in the housing separated
pressure surfaces (7 pressure sections in total) from
inlet to outlet having been calculated using this
model (Eq. 3). In the stress analysis, maximum
outlet pressure is defined as 250 bar and pressure
distribution from inlet to outlet is applied on the
pressure surface of the housing. Experimental results
also determined that there was a negative pressure
effect at the inlet of 9.55 bar (maximum). This
condition was also considered in FEA. In addition
to this, fluid pressure and bearing loads from
gear axles on the pump-bearing components were
considered. In the gear pump, pressure distribution
is non-uniform and, as is mentioned previously, the
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R. Cinar et al.
maximum pressure region is laid out between the
region of π and 3π/2 (Region IV). This phenomenon
pushes the gears from the outlet to the inlet side
which causes bearing loads. In fact, the direction of
these loads affects the bearings with a convenient
force angle from the outlet to the inlet, however, to
load the housing in a worst scenario, it is assumed
that the maximum pressure is affecting the gears at
the regions between π/2 and 3π/2 which provides
a direct force from the outlet to the inlet in the
housing for bearing loads (Fig. 7). According to the
calculations that consider the gears profile dimensions
and pressure at each gear tooth gap, the radial acting
total force on the gear was calculated as 12780 N.
Hence, each of the gears bearing forces 6390 N since
the gear bearing is symmetric. In addition to the
loading boundary condition definitions present in the
FEA setup, the side covers of the pump are assumed
to be rigid parts and the cover surfaces of the pump
have been restricted by using frictionless support
features. Bolt connection holes are restricted using
cylindrical support features in the FEM code as well.
In the analysis, three-dimensional (3D) models of
the housing and bearings were created in SolidWorks
3D parametric solid modelling design software while
Ansys Workbench commercial FEM code was utilised
for stress analysis.
Material properties of high strength 7000 series
aluminium alloy (7020-T6), stainless steel, and highleaded tin bronze (SAE 660) have been assumed for
the gear pump housing, bearing bushings, and sliding
bearings, respectively (Table 2).19 – 21 The analysis has
been set up with 3D, linear, static, and isotropic material model assumptions. The FE model of the gear
pump components (housing and bearings) used in
the FEA have been created using Ansys Workbench
meshing functions with a curvature-based meshing
approach.22 After completion of the pre-processes
in FEA, simulations were run. Von Mises equivalent stress distribution and deformation behaviour of
the housing under defined boundary conditions were
obtained. According to the simulation outputs, it is
shown that the maximum equivalent stress appeared
on the pump housing as 56.916 MPa and maximum deflection was 0.00844 mm. Mesh structure
details and related simulation outputs are presented
in Fig. 8.
Discussion for the FEA Results
As it can be seen in Fig. 8, the simulation
outputs present a good visualisation of the stress
distribution and deformation behaviour of the
R. Cinar et al.
Figure 5 Correlation between raw experimental, theoretical (Eq. 1) and estimated results (Eq. 3).
Figure 6 Diagnostic plots.
housing under defined boundary conditions. The
stress distribution is not uniform as non-uniform
pressure effects are evident at the inside surfaces
of the housing. Although, applied pressure values
increase progressively from inlet to outlet, the stress
distribution is not seen to correlate with these
progressive pressure values. Maximum stress is seen
at the outlet line of the housing as is expected;
however, the lower stress magnitude is seen at a
pressure region of P3 than P0 , which is the lowest
pressure magnitude. This can be explained as a result
of the geometric structure of the housing. Addition
to this, degree of freedom of bolt holes in the
housing was restricted in the FEA. Hence, lower
stress magnitude as a result of limited deformation
behaviour at these holes is seen at pressure region
7
R. Cinar et al.
Figure 7 Pressure and bearing loads effect in the gear pump housing.
Table 2 Material properties
Material properties used in FEA
Modulus of elasticity (GPa)
Poisson’s ratio (−)
Yield strength (MPa)
Tensile strength (MPa)
Density (kg m−3 )
Housing
Bearing
bushings
Sliding
bearings
72
0.33
280
350
2780
210
0.28
230
400
7700
100
0.32
140
240
8930
of P3 . According to the results obtained in the
simulation, safety factors (SF) have been calculated
considering the yield strength point of the housing
material (280 MPa) for seven pressure surfaces inside
the housing from inlet to outlet. In order to visualise
the change of SF at the housing pressure surfaces
from inlet to outlet, the maximum stress values for
each of the pressure regions and SF distribution chart
have been given in Fig. 9.
In Fig. 9, the non-uniform distribution of SF is
seen more clearly. The maximum SF was at the
pressure region of P3 (13.746) and the minimum SF
was at the pressure region of P6 (Outlet) (4.920). These
SF values obtained can be interpreted as quite high
due to the thickness of the housing which has been
designed with a high SF. Conversely, in the literature,
it is suggested that the gap between gear teeth and
8
inside surfaces of the housing should be between
the values of 0.005 and 0.020 mm for low operation
pressure conditions and 0.0025 and 0.010 mm for
high operation pressure conditions to avoid excessive
and undesirable fluid leakage from the gear-housing
gaps through the inlet side.5 The simulation results
determined that the maximum deflection of the
housing under defined boundary conditions was
0.00844 mm. This value stands within the suggested
values for the gear-housing operating gap. This
knowledge may allow design decisions with high SF
to be made to keep this gap between suggested values,
which might be a design constraint. However, all the
design constraints must be met in the gear pumps
total design process. Calculated high values of SF may
lead us to think about a structural optimisation to
avoid unnecessary material usage for the housing.
Concluding Remarks
As a main concluding remark, considering to the
simulation results and SF calculations, it can be
highlighted that a structural optimisation study could
be undertaken with the aim of reducing the housing
weight that would also provide a cost saving by
selecting optimum values for the housing thickness.
As such, firstly, size optimisation techniques could
be considered which may provide uniform thickness
R. Cinar et al.
Figure 8 Mesh structure details and simulation outputs: deformation behaviour and Von Mises stress distributions.
for the entire housing, although non-uniform stress
distribution may be an obstacle. As such, the outlet
side will need to be a thicker material than the
inlet side of the pump housing. Therefore, it may
be considered that a total structural optimisation
approach should be undertaken that considers
topology, shape, and size optimisation techniques in
order.
Additionally, a number of points can be summarised as follows, as final conclusion remarks of
the whole study presented in this paper:
1. An application algorithm has been introduced
for structural optimisation exploration of external
gear pump housing and it was implemented with
success within a case study for a sample external
spur gear pump.
9
R. Cinar et al.
Figure 9 Safety factor distribution chart for each of the pressure regions in the housing.
2. In the case study, an experimental study has been
set up and an estimation model has been extracted
which is adequately describes the exposed pressure
values at any specific region of the pump housing
from inlet to outlet, numerically.
3. FEM-based stress analysis has been conducted to
investigate the structural deformation behaviour
of the pump housing, which states that the
maximum stress on the housing was 56.916 MPa
and maximum deflection was 0.00844 mm. This
deflection value is within the limit gear-housing
gaps tolerance as suggested in the literature.
4. According to SF calculations considering material
yield strength point, non-uniform stress distribution is seen which is not in accordance with the
pressures applied to the pressure surfaces of the
housing.
5. SF calculation showed that the maximum SF was
13.746 and the minimum SF was 4.920 for the
pressure regions of P3 and P6 (Outlet) , respectively.
6. Final evaluation of the case study has determined
that an optimisation study could be suggested that
aims to make a reduction on the material usage
of the housing considering all design constraints
defined for the total design process of the gear
pump design. To do this, topology, shape, and
size optimisation techniques may be considered in
order to obtain optimum geometrical shape and
the size of the housing.
Acknowledgments
This study was supported financially by the Scientific
Research Fund of Kocaeli University (Project No:
2012/68) and partially supported by the Scientific
Research Projects Coordination Unit of Akdeniz
University.
Nomenclature
Pi (yi ) Pressure at specific region (bar)
10
Pmax (x 1i )
β(x 2i )
β 0–5
i
Maximum pressure (bar)
Rotation angle of the pump gears (o )
Empirical equation coefficients (−)
Equation error term (−)
Subscripts
i Index for specific rotation angle of the
pump gears
Inlet Inlet port of the gear pump housing
Outlet Outlet port of the gear pump housing
max Maximum
min Minimum
Abbreviations
FEA
FEM
RSA
RSM
SF
Finite element analysis
Finite element method
Response surface analysis
Response surface modelling
Safety factor
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MatGUID=c66e13c1d36445c29bb9f852ccf2da17
[accessed on 10 July 2013].
20. MatWeb, ‘‘Online Material Property Data:
High-Leaded Tin Bronze (SAE 660),’’ URL
http://www.matweb.com/search/DataSheet.aspx?
MatGUID=b673f55f412f40ae9ee03e9986747016
[accessed on 10 July 2013].
21. SolidWorks Product, ‘‘Material Library: Stainless
Steels,’’ Dassault Systèmes SolidWorks Corporation,
Waltham, MA (2013).
22. ANSYS Product, ‘‘Help File: Meshing. Release 14.5,’’
SAS IP, Inc., Cheyenne, WY (2012).
11

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