dissertation ezeldin
TRANSCRIPT
Technisch-Naturwissenschaftliche Fakultät
Investigations on Fast Hydraulic Accumulators for Hydraulic Switching Control
DISSERTATION
zur Erlangung des akademischen Grades
Doktor im Doktoratsstudium der TECHNISCHEN WISSENSCHAFTEN
Eingereicht von: MSc. Eng. Mohamed Mohamed Ez ElDin
Angefertigt am:
Institut für Maschinenlehre und hydraulische Antriebstechnik
Erster Beurteiler: O.Univ.-Prof. Dipl.-Ing. Dr. Rudolf Scheidl
Zweiter Beurteiler:
O.Univ.-Prof. Dipl.-Ing. Dr. Hans Irschik
Linz, März 2011
___________________________________________________________________________
Johannes Kepler Universität Linz, Altenberger Straße 69, 4040 Linz, Österreich, www.jku.at
II
Acknowledgement
This thesis would not have been possible unless Allah gave me the power and patience.
I owe my deepest gratitude to o.Univ.-Prof. Dipl.-Ing. Dr. Rudolf Scheidl for his advices,
guidance; I learned a lot from his infinity experience.
I am grateful to DI. Dr. Florian Maier, DI. Andreas Tairych, DI Rainer Haas, Ing. Siegfried
Grammer and DI. Dr. Markus Resch for their kind help in the hydraulic laboratory of the
Johannes Kepler University. It is an honor for me to thank o.Univ.-Prof. Dipl.-Ing. Dr.Hans
Irschik for reviewing my thesis.
I would like to thank a.Univ.-Prof. Dipl.-Ing. Dr. Bernhard Manhartsgruber for the test rig
idea.
I express my gratitude to my parents and my professors of my home university in Egypt. It
is a pleasure to thank Habibi for the moral support.
I offer my regards and blessings to OeAD organisation (Österreichischer Austauschdienst)
who supported me financially during the completion of my PhD; and at the end Elhamd Ellah.
III
Abstract
Hydraulic switching technology which is based on fast switching valves requires an
innovative fast hydraulic accumulator concept to attenuate high frequency pulsations in the
some hundred hertz range. The new accumulator concept should be simple in construction,
allow low cost production, should be easy to integrate into the hydraulic system, should have
high fatigue life and no need to recharging gas, and, if needed, be applicable also in some
corrosive conditions of pure water hydraulics.
The main trend to solve the compressed gas diffusion in the accumulator oil chamber
which is the essential problem of the conventional accumulator is to replace the elastomer
diaphragm with either a multilayer or micro-thickness metal diaphragm. Many investigations
presented that the multilayer diaphragm has problems of layers separation due to the variation
of materials properties. This lowers the accumulator working life and reduces diaphragm
flexibility which affects the accumulator volume capacity.
Recent developments of ultimately thin, high strength steel strips created the possibility to
think of realizing new types of metal diaphragm accumulators.
It is worth mentioning that the accumulator is always embedded in some hydraulic
transmission system and that its attenuation performance couples strongly with the dynamics
of that system. For this reason, distributed and 2DOF discrete parameter pipe models are
developed and used to investigate the transmission system’s influence on the attenuation
performance of the accumulator system.
In general, the one dimensional transmission line model of Leonard gives good dynamic
response results compared with the 3D FE acoustic model results in the investigated
frequency range. Only at junctions or at bent zones the results differ due to the radial velocity
components which are not represented in the Leonard model.
The new accumulator design, called “Diaphragm cap accumulator”, is numerically
simulated using FEM to obtain a feasible realization being able to cope with the extreme
requirements. The simulations happened as series of sequentially developed trials to optimise
the design of the new accumulator.
IV
The stress-displacement analyses of “Diaphragm cap accumulator” demonstrate that the
finally selected thin steel diaphragm shape performs axisymmetric deformations without any
instability problem with stresses below the material yield strength.
Both stamping and hydroforming simulations were performed to determine the most
economic and accurate manufacturing process to form the thin steel diaphragm of 50 μm
thickness, to compute the right clamping force, and to estimate the springback effect for the
high strength steel material.
The stamping simulation results present the ability of the stamping process to form a
complex geometry. The hydrofroming simulations confirm the precise hydroforming pressure
used to form the cap diaphragm.
A test rig is designed to prove the feasibility of the new accumulator design. The
experiments show the new accumulator behaves axisymmetrically and performs normally as
conventional accumulator. Chips or solid particles contained into the hydraulic oil can lower
the working life of the cap diaphragm.
Another innovative hydraulic accumulator is investigated in the thesis “the combined
round-weld bellows type accumulator” which has the advantage of low stress values, high
hydraulic capacity and long fatigue life. It could be used instead of the weld bellows type in
the bellows accumulator to avoid the strength reducing effects of welding.
Zusammenfassung
Um die hydraulische Schnellschalttechnologie, welche wesentlich auf den hydraulischen
Schnellschaltventilen basiert, voranzutreiben, bedarf es neuer schneller Hydrospeicher,
welche bis zu einer Frequenz von mehreren hundert Hertz betrieben werden können. Ein
derartiger Hydrospeicher sollte eine einfache Konstruktion und hohe Lebensdauer aufweisen,
billig herzustellen und einfach in ein Hydrauliksystem zu integrieren sein. Des Weiteren sollte
ein neuer Hydrospeicher auch in einer korrosiven Umgebung (z.B. Wasserhydraulik)
einsetzbar sein.
Um das Problem der Gasdiffusion in die Ölkammer in einem konventionellen Elastomer-
Blasenspeicher zu lösen, wird die Verwendung einer sehr dünnen Metallmembran oder
Mehrschichtmembran anstatt der Elastomerblase vorgeschlagen. Untersuchungen haben
bezüglich der Mehrschichtmembranen gezeigt, dass es hier häufig zu Problemen kommt, da
sich die Schichten voneinander ablösen können. Dies wiederum reduziert die Lebensdauer
von solchen Speichern.
Auf Grund neuer Erkenntnisse in der Materialtechnologie von sehr dünnen
Metallmembranen mit hoher Festigkeit, eröffnet sich die Möglichkeit neue Hydrospeicher mit
dünnen Metallmembranen zu realisieren.
Es muss erwähnt werden, dass ein Hydrospeicher immer in einem Hydrauliksystem
eingebettet ist. Somit koppeln die Eigenschaften des Speichers mit den Eigenschaften des
Hydraulikkreises. Aus diesem Grund wurden sowohl verteilt-parametrische als auch diskrete
Rohrleitungsmodelle (mit 2 Freiheitsgraden) entwickelt und verwendet um die Einflüsse der
Hydraulikleitungen auf die Speicherperformance beurteilen zu können.
Generell liefert das eindimensionale Leitungsmodell von Leonhard im untersuchten
Frequenzbereich gute Ergebnisse im Vergleich zu einem dreidimensionalen FE
Akustikmodell. Nur bei Abzweigungen oder gebogenen Stücken kommt es zu Abweichungen,
da die Radialgeschwindigkeiten im Leonardmodell nicht berücksichtigt werden.
Das neue Hydrospeicherkonzept „Membran-Kappen-Speicher“ wird numerisch mittels
Finiter Elemente simuliert um eine realisierbare Lösung zu erhalten, welche den sehr hohen
Anforderungen gerecht wird. Die Berechnung erfolgt in einer Serie von sequentiellen
Analysen um das Konzept zu optimieren.
V
Die Spannungs- und Verschiebungsanalysen des „Membran-Kappen-Speichers“
demonstrieren, dass eine dünne Stahlmembran achssymmetrische Deformationen aufweist,
ohne dass ein Stabilitätsproblem auftritt. Die mechanischen Spannungen bleiben dabei unter
der Materialstreckgrenze.
Sowohl Stanz- als auch Hydroumformsimulationen wurden durchgeführt um eine
effiziente Herstellbarkeit der Membranen von 50ym Dicke sicherzustellen. Die
Stanzsimulationen zeigen, dass komplexe Geometrien mit dem Stanzvorgang hergestellt
werden können. Die Hydroumformsimulationen liefern das Druckprofil, welches für den
Umformprozess verwendet werden muss.
Ein Prüfstand wurde aufgebaut, um die Funktion des Speichers zu zeigen. Die Experimente
zeigen, dass sich die Membran achssymmetrisch verhält und der Hydrospeicher wie ein
konventioneller Speicher funktioniert. Grobe Partikel im Hydrauliköl führten zu einer stark
reduzierten Lebensdauer des Speichers.
Ein weiterer innovativer Hydrospeicher ist ebenfalls in der vorliegenden Arbeit angeführt.
Dieser hat den Vorteil kleiner mechanischer Spannungen, hoher hydraulischer Kapazitäten
und einer hohen Lebensdauer. Ein solches Konzept könnte statt geschweißten Faltenbalgen
verwendet werden, da die Festigkeitswerte nicht durch Schweißung beeinflusst werden.
VI
List of Notations
Symbol Description Unitα The strain hardening exponent [1]
A the cross sectional area of a fluid volume [m2]
accA , 4A The surface area of the separator element of the hydro-pneumatic
accumulator.
[m2]
2vv A,A The cross sectional areas of the output throttle valve and of the
throttle valve located at the entrance of the accumulator
respectively
[m2]
c Speed of sound in the fluid [m/s]
dC The coefficient of discharge flow rate [1]
fc The fluid particle (element) damping coefficient [N.s/m]
HC The hydraulic capacitance [m3/Pa]
4H3H
2HHi
C,C,C,C
The hydraulic capacitance of several pipes and of a cavity (see.
Section 3.2.2.7)
[m3/ Pa]
p̂Δ The pressure variation in frequency domain [Pa]
K,GdP The change of the gas pressure in the accumulator [Pa]
GdV The differential of the gas pressure in the accumulator [m3]
effE The effective Bulk modulus [Pa]
flE The fluid Bulk modulus [Pa]
The plastic strain [1]
γ The propagation operator [1]
1γ , 2γ ,
3γ , 4γ
The propagation operator of different pipes and of an oil chamber
(see Section 3.3.2)
[1]
i The current [A]
Ci The current passing through the capacitor [A]
j The imaginary unit )1j( −= [1]
0J , 2J The Bessel functions of the first kind and order 0 and 2 [1]
VII
K
The strength coefficient [Pa]
fk The fluid particle (element) stiffness [N/m]
L The pipe length [m]
HL The electric coil impedance or the hydraulic inductance [kg/m4]
4H3H
2HHi
L,L,L,L
The hydraulic inductance several pipes and an oil chamber,
respectively (see. Section 3.2.2.7)
[kg/m4]
diaphm = m The mass of the separator element of the hydro-pneumatic
accumulator
[kg]
μ Poisson 's ratio [1]
n The polytropic exponent [1]
p The acoustic pressure [Pa]
p,p &&& The time derivatives of the acoustic pressure. [Pa/s] and
[Pa/s2]
1AP , 1AP)
The input pressure to the first horizontal (upstream) pipe in time
and frequency domain
[Pa]
2AP , 2AP)
The input pressure to the second horizontal (upstream) pipe in
time and frequency domain
[Pa]
4AP , 4AP)
The input pressure of the accumulator oil chamber in time and
frequency domain
[Pa]
2avav P,P The average pressures at the output throttle valve and the throttling
at the entrance of the accumulator respectively.
[Pa]
1EP , 1EP)
The output pressure from the first horizontal (upstream) pipe in
time and frequency domain
[Pa]
2EP , 2EP)
The output pressure from the second horizontal (upstream) pipe in
time and frequency domain
[Pa]
3EP , 3EP)
The output pressure rate of the pipe connecting the accumulator in
time and frequency domain
[Pa]
4EP , 4EP) The output pressure of the accumulator oil chamber in time and
frequency domain
[Pa]
GP The actual gas pressure in the accumulator gas chamber [Pa]
GP& The time derivative of GP [Pa/s]
0GP The gas initial pressure in the accumulator gas chamber [Pa]
VIII
inP , inP̂ The input pressure to the transmission line in time and frequency
domain
[Pa]
KP The gas pressure at certain point [Pa]
oilP , oilP̂ The oil pressure inside the accumulator oil chamber in time and
the frequency domain respectively.
[Pa]
outP , outP̂ The output pressure from the transmission line in time and the
frequency domain respectively.
[Pa]
1outP̂ , 2outP̂ The outlet pressure of the first and second pipe in frequency
domain respectively.
[Pa]
1AQ , 1AQ)
The input discharge flow rate to the first horizontal (upstream)
pipe in time and frequency domain respectively.
[m3/s]
2AQ , 2AQ)
The input discharge flow rate to the second horizontal (upstream)
pipe in time and frequency domain respectively.
[m3/s]
3AQ , 3AQ)
The input discharge flow rate to the pipe connecting the
accumulator in time and frequency domain
[m3/s]
1EQ , 1EQ)
The output discharge flow rate from the first horizontal (upstream)
pipe in time and frequency domain
[m3/s]
2EQ , 2EQ)
The output discharge flow rate from the second horizontal
(upstream) pipe in time and frequency domain
[m3/s]
3EQ , 3EQ)
The output flow rate of the pipe connecting the accumulator [m3/s]
4EQ , 4EQ)
The output pressure and flow rate of the accumulator oil chamber
in time and frequency domain
[m3/s]
inQ , inQ̂ The input flow rate in time and frequency domain. [m3/s]
oilQ , oilQ̂ The oil flow rate of the accumulator in time and the frequency
domain
[m3/s]
outQ , outQ̂ The output flow rate from the transmission line in time and the
frequency domain
[ m3/s]
1outQ̂ , 2outQ̂ The outlet flow rate of the first and the second pipe in frequency
domain
[m3/s]
vQ , vQ̂ The discharge flow rate of the throttle valve in time and frequency
domain respectively.
[m3/s]
ρ The fluid density [kg/m3]
r The radius of the pipe [m]
IX
HR The hydraulic resistance [Pa/m3/s]
4H3H
2HHi
R,R,R,R
The hydraulic resistance of the first, second horizontal pipe, the
pipe connecting the accumulator and the oil chamber
[Pa/ m3/s]
The applied stress on the material, [Pa]
s Laplace operator
t Physical time [s]
u The fluid velocity in the axial direction [m/s]
fu The fluid particle displacement; [m]
ff vu =& The fluid particle velocity; [m/s]
fu&& The fluid particle acceleration; [m/s2]
v The fluid velocity in the radial direction [m/s]
ν Kinematic viscosity of the fluid [m2/s]
)t(vC The capacitor voltage [Volt]
GV& The time derivative of GV [m3/s]
0GV The gas initial volume of the accumulator gas chamber [m3]
GV The actual gas volume of the accumulator gas chamber [m3]
)t(vL The electrical coil voltage [Volt]
)t(vin The input voltage to the RLC circuit [Volt]
)t(vin& The time derivate of the input voltage [Volt]
)t(vR The electrical resistance voltage [Volt]
ϖ The Laplace frequency or the angular velocity. [rad/s] y The displacement of the separator element of the hydro-pneumatic
accumulator
[m]
y&& The acceleration of the separator element of the hydro-pneumatic
accumulator
[m/s2]
cz The characteristic impedance [Pa/ m3/s]
1cz , 2cz ,
3cz , 4cz
The characteristic impedance for the upstream pipe, downstream
pipe, the vertical pipe and the oil chamber respectively(see Section
3.3.2)
[Pa/ m3/s]
X
Table of contents
Acknowledgement
II
Abstract
III
Zusammenfassung
V
List of Notations
VI
Table of contents
XI
1 1 2 4 5 6 6 6 7 9
10
11
11
12
13
13
1 Introduction
1.1 Overview and problem setting
1.2 Importance of fast response accumulators
1.3 Functions of accumulators
1.3.1 Hydro-pneumatic type accumulator
1.3.2 Diaphragm accumulator
1.3.2.1 Weld type diaphragm accumulator
1.3.2.2 Screw type diaphragm accumulator
1.3.3 Bladder diaphragm accumulator
1.3.4 Piston accumulator
1.3.5 Novel accumulator concepts
1.3.5.1 Metal strands type
1.3.5.2 Solid elastomer type
1.3.5.3 Metal bellows accumulator
1.3.6 Spring loaded accumulator type
1.3.7 Dead weight accumulator type
XI
16
16
16
30
2. New accumulator concepts – overview of the state of the art
2.1 Introduction
2. 2 Literature review
2.3 Conclusions
31
31
32
32
34
34
37
38
39
44
45
48
56
56
58
61
75
77
77
79
3. Hydraulic system dynamic response modelling
3.1 Introduction
3.2 Discrete parameter models
3.2.1 Case of study
3.2.2 Elements of the hydraulic system model
3.2.2.1 Linear hydraulic capacitance
3.2.2.2 Hydraulic inductance
3.2.2.3 Laminar, steady state flow resistance in a circular cross
section pipe
3.2.2.4 Discrete transmission line model
3.2.2.5 Hydraulic throttle
3.2.2.6 The Hydro-pneumatic accumulator
3.2.2.7 Discrete parameter SDOF model of the case of study.
3.3 Distributed parameter models
3.3.1 Transmission line model
3.3.2 Case of study
3.4 Results and discussions
3.5 3D Finite Element acoustic models with frequency dependent friction
3.5.1 Acoustic finite element models of some hydraulic systems
3.5.1.1 Test case straight pipe with pressure excitation
3.5.1.2 Accumulator in a transmission line with pressure rate
excitation
XII
3.6 Conclusions
85
4. Theoretical investigations of alternative accumulator concepts
4.1. ’Diaphragm cap’ accumulator
4.1.1. Diaphragm concept and design
4.1.2. Nonlinear FE model of diaphragm deformation and stress state
4.1.3. Diaphragm cap accumulator simulation results
4.1.4. Dynamical response behaviour
4.1.4.1. FE acoustic model and simulation results
4.2. Bellow type accumulator
4.2.1. Bellow type accumulator simulation results
4.3. Conclusions
86
86
86
88
90
104
104
111
111
115
116
116
117
119
119
120
121
121
121
123
123
127
129
132
5. Experimental investigation of diaphragm cap accumulator
5.1. Design of a prototype
5.2. Material selection for the diaphragm
5.3. Diaphragm forming processes
5.3.1 Stamping
5.3.2 Hydroforming
5.4. Simulations of the diaphragm forming processes
5.4.1 Simulation of stamping process
5.4.1.1 Finite element model of the stamping process
5.4.1.2 Finite element model of the hydroforming process
5.4.1.3 Results of the stamping and hydroforming simulations
5.5. Manufacturing of diaphragm cap (hydroforming)
5.6. Test set-up
5.6.1 Test results
XIII
5.6.1.1 Static test
5.6.1.2 Fatigue test
5.7. Conclusions
133
134
136
References
138
Eidesstattliche Erklärung
142
Curriculum vitae
143
Annex 1 Prototype engineering drawings
A1. The diaphragm cap
A2. The intermediate part
A3. The upper housing
A4. The lower housing
145
145
146
147
148
149
149
149
150
153
155
157
159
162 168
Annex 2
A.2.1 Abaqus input files
A.2.1.1 Abaqus input file for acoustic analysis of closed straight pipe
model
A.2.1.2 Acoustic analysis of ideal accumulator connected with
transmission line
A.2.1.3 Acoustic analysis of the ideal diaphragm cap accumulator
A.2.1.4 Nonlinear behaviour of the Diaphragm cap with dimples
A.2.1.5 Nonlinear behaviour of the contact interaction between the
Diaphragm cap and the lower accumulator housing
A.2.1.6 Hydroforming a flat membrane
A.2.1.7 Stamping a flat membrane
A.2.1.8 Nonlinear behaviour of the accumulator bellows
XIV
A.2.2 Matlab files
A.2.2.1 The distributed and the 2DOF discrete parameter models of
transmission line connecting with a hydro-pneumatic accumulator.
A.2.2.1.1 The distributed parameter model of transmission line
connecting with an accumulator
A.2.2.1.2 The 2DOF discrete parameter model of transmission line
connecting with an accumulator.
171
171
171
172
XV
1 Introduction
1.1 Overview and problem setting
In high pressure hydraulic power systems, hydraulic accumulators serve as energy storage
devices, provide emergency or standby power, compensate for leakage loss, dampen
pulsations and shocks of periodic excitation sources, such as hydraulic pumps, hydraulic
motors, switching valves, etc., and constitute auxiliary energy sources. The hydro-pneumatic
accumulators are designed to store the hydraulic working fluid from the system under
pressure by moving a rigid or flexible separator against highly compressed inert gas acting
like a spring.
Various types of gas charged accumulators are known, such as diaphragm, bladder, piston
and bellow-type accumulators. In diaphragm hydro-pneumatic accumulators, a flexible
diaphragm, such as rubber, resign, metallic material or combination of these materials is fixed
within a metallic shell and subdivides the interior space of the metallic shell into two fluid-
tight pressure chambers on opposite sides of the diaphragm. One chamber is exposed to the
hydraulic system. The hydraulic fluid flows into and out of this chamber depending on the
pressure situation. The other chamber is charged with an inert gas (which can not react with
the hydraulic fluid), for example, nitrogen gas, under high pressure to act as a spring and in
this way also as an energy storage medium. The working hydraulic fluid enters the hydraulic
accumulator when the hydraulic fluid pressure exceeds the pressure of the compressed gas;
the diaphragm is elastically deformed and moves against the compressed gas on the other
side. The gas pressure expels the fluid out of the fluid chamber into the hydraulic fluid system
when the pressure of the hydraulic liquid falls below the gas pressure. The change in gas
pressure and volume determine the hydraulic capacity of the accumulator or the amount of
liquid that can be added to or withdrawn from the accumulator. However, unlike mechanical
springs1, compressing a gas tends to heat it and expanding a gas tends to cool it. Either of
these effects can substantially affect accumulator capacity. Expansion or compression of a gas
1 Apart from the comparatively very small thermo-elastic effect!
1
resulting in a change of gas temperature produces polytropic state change which depends on
how fast expansion or compression occurs. During the polytropic compression process, heat
energy transfers from the hot gas to the accumulator walls and finally to the surrounding air.
A major problem of conventional accumulator designs is their inability to fully prevent the
diffusion of the compressed gas of the gas chamber to the hydraulic fluid chamber through the
elastomer diaphragm. Consequently, the hydro-pneumatic accumulators tend to gradually lose
their charge and require periodic recharging or replacement; also the gas dissolves in the
hydraulic fluid and increases the saturated gas levels in the hydraulic fluid and, thus, increases
fluid compressibility considerably, particularly at lower pressures. That reduces reliability and
the performance of the hydraulic system and increases service frequencies.
1.2 Importance of fast response accumulators
Nowadays, there is a trend in the hydraulic drive technology to imitate electrical switching
converters such as the buck converter. Such hydraulic converters preferably are run at
comparatively high switching frequencies of about 100 Hz. In combination with broad band
frequency excitation due to switching frequencies in the order of up to some Kilohertz may
occur. Filtering of such high frequency pulsation requires adequately fast response
accumulators.
Switching converter technology is just one branch of hydraulic switching technology some
other applications are described in the sequel:
• Anti Lock Braking system [ABS] which prevents the wheels from locking and
maintains directional stability by using a electronic control module (ECU) which
act via fast switching valves on the hydraulic brake at each wheel (see Fig.1. 1).
The ECU in this system monitors the number of rotations per minute of each wheel
by collecting the signals transmitted by each speed sensor and observes the values
of the wheel velocity or deceleration and compares with the vehicle velocity,
derives from these values the slip percentage which is kept in a safe range by
controlling the brake pressure. An ABS controller is capable of modulating the
brake pressure at a given wheel up to15 Hz; so, if the vehicle has four tyres,
pulsations in the hydraulic brake system may go up to 60 Hz and even beyond,
when the higher order frequencies are considered.
2
Fig.1. 1 the hydraulic circuit of Anti Brake System (ABS) ([LEXUS 2009])
• Hydraulic switching control is applicable also in agricultural machines like ploughs or
harvesters. In a research project of LCM (Linz Center of Mechatronics) [Winkler
2002] a harvester pick-up was developed that is actively controlled, among other
purposes to avoid crashes with obstacles. The pick-up is guided at certain distance
above the ground by some automatic level control with some fast response positioning
hydraulic drive. Other material on hydraulic switching control in agricultural
machinery can be found in [Scheidl 2000].
• Another application is the hydraulic buck converter (see Fig.1. 2) which is similar to
electrical buck converter. It consists of a switching valve, a check valve, the
inductance realized by a pipe, and a hydraulic accumulator. When the switching valve
opens, the fluid in the pipe inductance is accelerated or flows to charge the
accumulator and at the same time enters to the consumer line. After the valve closes,
the kinetic energy of the fluid in the pipe generates a suction of oil from the tank line.
This suction effect is responsible for achieving a higher efficiency than proportional
hydraulics. The fluctuations of pressure generated by the switching process with
frequencies in range of 50 up to a few hundred Hertz are attenuated by a fast hydraulic
accumulator. This switching causes strong pressure fluctuations in the system and that
may deteriorate machine performance. The hydraulic accumulators needed to filter
3
this fluctuation have to be to be quite fast since switching provokes many higher order
pulsation frequencies. As an estimate a few kHz might be given as limit frequency up
to which the accumulator should respond nearly immediately [Scheidl 2008].
Fig.1. 2 The hydraulic Buck-Converter ([Scheidl 2008])
1.3 Functions of accumulators
Several hydraulic accumulator types are used in hydraulic power systems, the optimal
selection of which depends on their performance characteristics relative to the performance
requirements. As mentioned before, the main functions of hydraulic accumulators can be:
1. To attenuate pressure pulsation from an excitation source in a hydraulic system. Such
sources are: pumps and motors; pulsating oil consumption by periodic motion of the
drive at high frequencies like in modern punching machines; switching in a hydraulic
switching systems like the buck converter; pulsating fluid consumption in fluid
systems like a modern Common Rail Fluid Injection System.
2. To constitute an energy storage device. An actual application of this function are
hydraulic hybrid drives for vehicles in which the accumulator stores energy during the
4
vehicle deceleration (braking) and covers peak power to keep the combustion engine
smaller than usual.
3. To provide auxiliary energy like in electro-hydraulic power steering systems to
prevent the pump from delivering pressure all time and waste energy in this way.
4. To have an energy source in case of emergency like for a winch or a hydraulic lift,
where the prime mover is typically an internal combustion engine, to carry out
emergency operation at failure of the prime mover.
5. To compensate leakage loss or thermal expansion volume for instance in clamping
units of machine tools.
Attenuation of pulsation needs a fast response accumulator having small hydraulic capacity
and small hydraulic inertia to keep the limit frequency high enough. Inertia stems from the
hydraulic fluid in narrow passages or from accumulator movable solid components, like the
piston of a piston accumulator.
1.3.1 Hydro-pneumatic type accumulator
The hydro-pneumatic accumulator is the most commonly used accumulator in the
industrial domain. It comprises a sealed hollow housing defining a pressure vessel, a working
fluid inlet port connectable to a high pressure fluid system, and a movable part which is
mostly a flexible element like a diaphragm or bladder from elastomer or a composite material.
Recently and also in this work, a very flexible thin metallic material is used or proposed,
respectively, as such a flexible element. Also combinations of elastomers and metals have
been addressed in the patent literature. A different principle of separating gas from the
hydraulic fluid is a rigid piston mostly of metal. The gas chamber is charged with gas through
a gas fill port which is sealed with a gas valve. The main function of the separator element is
to avoid a mixing of gas with the hydraulic fluid. It is the critical element and decisive for
reliability and performance of the accumulator.
A further important and critical function is the protection of flexible separating elements
from gas pressure when the hydraulic pressure falls below the gas pressure which is the case
when the accumulator is emptied.
For safety reasons dry nitrogen is used since air forms with oil an explosive air-oil vapour.
The different types of hydro-pneumatic accumulators currently used are:
5
1.3.2 Diaphragm accumulator
It consists of a lower and an upper metallic housing which are separated by a flexible
diaphragm. The upper and lower housing are connected by welding or by a screw joint. The
diaphragm material should be a very flexible material such as an elastomer or a flexible
composite. But also thin metallic sheets – as dealt with in this thesis - or combinations of the
mentioned materials in form of a multilayer composite diaphragm are possible. The peripheral
edge of the diaphragm is clamped to the housing for providing a fluid tight seal between the
two chambers.
The advantages of the diaphragm accumulator are:
It is simple in construction, cheap in cost and has long working life. It does not need to
charge gas in case of using metallic or multilayer composite diaphragm with an impermeable
gas layer. But, no product of the latter type is marketed today.
The disadvantages of this accumulator type are:
It has small capacity compared to piston and bladder types. There is some small gas
permeation into the hydraulic fluid in case of the usual rubber diaphragms. A means is
required to protect the diaphragm against gas pressure in absence of a sufficient oil pressure.
This is mostly realized by an inlay of some stiffer plastic part in the centre of the diaphragm
in combination with a rather small inlet bore. This small bore prevents high flow rates and
hence a fast impact of the inlay part with the housing. But this causes a considerable dynamic
performance loss of the accumulator.
The diaphragm accumulator is realized in two different versions:
1.3.2.1 Weld type diaphragm accumulator
In this type, the upper and lower housings are joined together by welding their ends
employing welding technologies with little heat input, like electron beam welding. Care is
taken, that heat generated by the welding is not transferred to the elastic diaphragm (see Fig.1.
3).
1.3.2.2 Screw type diaphragm accumulator
In this type, the upper and lower housings are strongly fitted together with a thread or
sometimes by using an outer ring which is fixed to the upper housing and screwed with the
6
lower one (see Fig.1. 4 ). No heat problem of welding arises. The main advantage of this type
is that diaphragms can be changed in case of failure. The production costs, however, are
significantly higher than of the welded type such that only for multiple diaphragm changes
(~ 6÷8 times) total costs become lower.
Fig.1. 3 Diaphragm accumulator-weld type [Smsproducts 2011].
1.3.3 Bladder diaphragm accumulator
It consists of a synthetic polymer rubber bladder like chloroprene, nitrile, or any other
rubber material inside a metallic shell in cylindrical shape. The bladder is filled with
compressed inert gas such as nitrogen (see Fig.1. 5). When the hydraulic fluid enters from the
hydraulic port to the hydraulic chamber with high pressure and forces the diaphragm to move
upward, the diaphragm compresses the inert gas into the gas chamber and hence, the
accumulator can store hydraulic energy from the hydraulic system or achieve its functions as
mentioned before. The bladder responds quickly to receiving or expelling flow of the working
hydraulic fluid. The bladder has to perform strong deformation which increases the chance of
bladder failure.
7
Fig.1. 4 Diaphragm accumulator-screw type ([Narender 2011])
The advantages of the bladder hydraulic accumulator are:
It has larger hydraulic capacity than the diaphragm type, is simpler in design and less
expensive than the piston type. The shell is only one part which avoids any of the mentioned
problems of joining the two housing parts of the diaphragm accumulator.
The disadvantages are:
For medium or large hydraulic capacity, the bladder should be made from a very flexible
material such as rubber. These materials are not fully gas tight. Gas may permeate into the
hydraulic fluid and the accumulator fails after certain working life. Like the diaphragm
accumulator, there is a valve at the hydraulic port to secure the bladder to not excessively
deform. This valve may close if fast outflow occurs due to flow forces. This limits the
dynamical performance of the accumulator.
8
Fig.1. 5 Bladder accumulator [Tobul 2011]
1.3.4 Piston accumulator
It consists of a cylindrical metal shell mostly fabricated from steel alloys material to resist
corrosion and withstand high hydraulic fluid pressure. Upper and lower caps are joined with
the cylindrical shell by a thread. A movable piston separates gas from oil. It is made of steel
or light metal such as aluminium, titanium or any other light material (see Fig.1. 6).
For providing a fluid tight seal between the cylindrical shell and the piston, there is more
than one sealing element used for this purpose. Of course, this seal is the critical element and
has to trade-off friction and tightness.
The advantages of the piston accumulator are:
It has a large hydraulic capacity compared to the diaphragm and the bladder types; it is safe
to use it in high pressure hydraulic systems because the piston withstands cyclic operation in
contrast to the bladder or the flexible diaphragm which, in turn, means long working life;
there is no need to use a valve at the hydraulic port to avoid the exit of the piston like in
9
diaphragm and the bladder types. This makes it a favourable candidate if excellent high
frequency response is required.
The disadvantages of this type are:
It needs regular maintenance for changing the sealing elements of the piston and also
charging the hydraulic accumulator with inert gas; it is more complicated in design and has
higher costs than the diaphragm type.
Fig.1. 6 Piston accumulator [Tobul 2011]
1.3.5 Novel accumulator concepts
In literature several trials for innovative accumulator solutions are reported heading
typically for a simple accumulator design with a lesser number of components, new materials
to make it less weight and more efficient, higher reliability or safety, and reduced cost.
10
1.3.5.1 Metal strands type
This type was published in [Yeapple 1996]. It is basically a piston accumulator. It consists
of a cylindrical metal shell with a piston moving inside along its axis (see Fig.1. 7). This
piston divides the cylindrical shell into a gas and a hydraulic chamber. The actually new thing
is metal particles added to this gas chamber. These metal particles increase the energy storage
capacity because of their higher thermal capacity which brings the accumulator close to an
isothermal behaviour. The latter also avoids problems with the significant temperature rise in
close to adiabatic compression which may endanger seals. The metal particles cause
additional weight. The main challenge is to assure a homogenous distribution of the particles
in the gas –a prerequisite for a proper functioning.
Fig.1. 7 Metal strands accumulator [Yeapple Fluid power 1996]
1.3.5.2 Solid elastomer type
It consists of a cylindrical metal shell including an elastomer part like silicon. This
elastomer acts as a spring element instead of the compressed gas of the hydro-pneumatic
accumulator and stores the hydraulic energy. There is no need for a separator to divide the
accumulator into two chambers (see Fig.1. 8). The hydraulic fluid enters from the hydraulic
port to the hydraulic chamber and pressurises the elastomer against the end wall of the
cylindrical shell [Yeapple 1996].
11
The advantages of the solid elastomer are:
Simple in construction since there is no need for a separating element; less number of
parts; cheap; using elastomer is safer because it will not explode; needs not to be charged with
gas and no gas leakage, hence has a long working life; no tight seal between fluid and gas is
required; operates from zero pressure on, in contrast to gas filled accumulators which start
working for pressure above the gas filling pressure; the expansion of the elastomer due to
high temperature is less than of the compressed gas in hydro-pneumatic accumulators.
The disadvantages of this type are:
It has less hydraulic capacity than the hydro-pneumatic type because the elastomer material
has a limit deformation (The elastomer takes more place in the compression stroke than the
compressed gas), higher weight than the hydro-pneumatic accumulator because the inert gas
is almost weightless.
Fig.1. 8 Solid elastomer accumulator [Yeapple 1996]
1.3.5.3 Metal bellows accumulator
The meta1 bellows accumulator consists of a cylindrical metal shell including a metallic
bellow; one end of the metallic bellows unit is fixed to one end wall of the cylindrical shell.
The metallic bellows divides the hydraulic accumulator into an outer chamber containing the
hydraulic fluid connected to the hydraulic port and an inner chamber containing the inert gas.
The metal bellows accumulator should have a stopper like a rubber bush for preventing the
bellows from the collapse with the end wall of the cylindrical shell in the unladen case (see
12
Fig.1. 9). When the hydraulic fluid enters from the hydraulic port into the hydraulic
chamber it forces the metallic bellow to move mainly in axial direction to the cylindrical
shell. The metallic bellow pressurises the inert gas into the gas chamber and hence the
accumulator can store energy from the hydraulic system [Hydac 2009].
The advantages of the metal bellows accumulator are:
Gas tight, hence no leakage of nitrogen gas; that means long working life and no
maintenance; it has medium hydraulic capacity. Metals have a significant durable strength;
thus, this type has an infinite lifetime in contrast to elastomers.
The disadvantages of this type are:
It is complicated in design and more expensive and has a smaller capacity to volume ratio
than piston, diaphragm or bladder accumulators.
1.3.6 Spring loaded accumulator type
This accumulator is used in some hydraulic systems. A single spring or multiple springs
act(s) against a hydraulic piston forcing the hydraulic fluid to flow out of the accumulator. It
consists of a cylinder body, a movable piston, and a coil spring. The pressure in the
accumulator is determined by the stiffness rate of the spring (see Fig.1. 10). [Wiki 2010]
The advantages of the spring loaded accumulator are:
These accumulators are usually smaller and less expensive than the dead weight type;
mounting is easy; there is no need for charging the hydraulic accumulator with inert gas and it
needs rarely a maintenance unless the spring stiffness has to be changed or the oil seal around
the piston is damaged after a long working life; since there is no compressed gas no problems
with a gas temperature rise occur.
The disadvantages of this type are:
It has a rather small hydraulic capacity unless in case of very small operating pressures.
1.3.7 Dead weight accumulator type
The dead weight accumulator consists of a piston loaded with a dead weight and moving
within a large metallic cylinder that exerts pressure on the hydraulic fluid. The dead weight
may be of some heavy material such as iron or concrete (see Fig.1. 11).
The advantages of the dead weight accumulator are:
13
The pressure remains constant for the full stroke, it provides a huge capacity and; they are
most often used in huge central hydraulic systems; there is no need to charging the
accumulator with inert gas [Wiki 2010].
The disadvantages of this type are:
The dead weight accumulator has a very large size which makes it inapplicable for most
hydraulic applications.
Fig.1. 9 Quarter section view of the bellows accumulator.
14
Fig.1. 10 Spring accumulator [Tobul 2011]
Fig.1. 11 Dead weight accumulator [Tobul 2011]
15
2. New accumulator concepts – overview of the state of the art
2.1 Introduction
For obvious reasons, the majority of hydraulic accumulators consist of liquid and gas parts,
with a bladder, piston or diaphragm as separating element. The liquid filling the hydraulic
chamber is connected to the hydraulic circuit. When the liquid pressure rises the gas is
compressed and stores some energy and when the pressure falls the compressed gas expands
again and forces the accumulated liquid into the hydraulic circuit.
The task of this thesis is not only to study new accumulator concepts of this principle but
also to improve diaphragm design and to obtain an excellent dynamic performance
characteristic. Such an accumulator should have the following further properties:
• simple in construction
• essentially impervious to gas diffusion and leak-tight
• highly flexible to provide large volumetric displacement capacity over a wide range of
operating temperatures
• able to withstand repeated displacement without degrading
• reliable, durable and having a long service life without requiring recharging or
frequent servicing.
2. 2 Literature review
This section reviews the published hydraulic accumulator literature with emphasis on the
diaphragm accumulator type because it is the field of study of this thesis. It shows different
innovative concepts either in construction or in material used for the diaphragm.
Axinte Ionita [Ionita 2001] investigated an accumulator to absorb the hydraulic pulsation
in the hydraulic system with numerous small holes on one-side of the containment which
16
allows hydraulic fluid to enter into or exit from the accumulator without much impedance.
The diaphragm is made of rubber (see Fig.2. 1).
Fig.2. 1 Axinete model of diaphragm accumulator.
The author analyses the deformation of the diaphragm to find an explanation for an
observed crack formation caused by non-symmetric radial deformation. The results indicate
that the deformation pattern depends on the shape of the diaphragm rather than on the
magnitude of the applied load. The author did not concern other effects such as thermal or
chemical effects which may accelerate the creation of the cracks or might be responsible for
the crack initiation. The assessment of this design is, that it needs frequent service to
discharge gas into the gas chamber due to permeability of the diaphragm rubber material and
that the elastomer diaphragm will sharply deform through the holes when a high gas pressure
applies; hence, it has a short working life.
17
Kenji Hattori et al [Hattori 1993] investigate a new hydro-pneumatic diaphragm
accumulator used in antilock brake and traction control systems in the automotive field. In
order to prevent gas from permeating into the liquid chamber from the gas chamber, the
diaphragm is constructed of multiple layers (laminated fabric material); the middle layer is
formed of a thin sheet element of metal or resinous material having a small gas-permeability.
The upper and lower layers are made of an elastomer (see Fig.2. 2). With this construction the
permeation of gas is inhibited by the thin sheet element of the metal or resinous material and
the strength of the thin sheet – element is reinforced by the laminated fabric material. But the
thin sheet element if made of metal has low admissible elongation. Hence, fatigue is induced
by repeated displacement of the diaphragm. Even though cracking of the diaphragm as a
whole can be inhibited by the laminated fabric material, gas permeation through the
diaphragm occurs if the gas tight metal layer gets cracked. If the thin sheet element is made of
polyvinylidene-fluoride or -chloride similar durability problems arise since these material
have low elongation at low temperatures. Polyvinylidene-fluoride or -chloride have a high
resistance to solvents and, therefore, it is difficult to improve the cold temperature resistance
of the material by the addition of a plasticizer. Moreover, the elongation of the laminated
fabric material is extremely small and, hence, the displacement of the diaphragm is restrained
by the laminated fabric material. That limits the hydraulic capacity of the accumulator. The
author found that to solve such a problem, a material exhibiting a large elongation at low
temperatures should be used. For example, if the gas-impervious member is formed of
polyvinyl alcohol with some glycerine, that could be achieved.
Another problem addressed by the author is the influence of some hydraulic fluid
constituents such as Ethylene glycol alkyl ether on diaphragm materials. Ethylene glycol alkyl
ether (used as brake fluid) has a hardening effect on polyvinyl with glycerine which in turn
tends to crack particularly at low temperatures since the glycerine is extracted from the layer.
A solution tested to solve this problem is to affix a second layer of synthetic resin at one
surface of the middle layer. But repeated deformation of the diaphragm tends to separate the
layers which in turn leads to gas permeation again.
Assessments of the design are that it is simple in construction, and may have longer
working life than conventional elastomer diaphragms. However, the production of such
layered diaphragm material may be costly and there might be problems with the robustness of
such compound materials.
18
Fig.2. 2 Hattori et al model of the diaphragm accumulator.
Alan R. Larsen [Larsen 2000] presented a high pressure accumulator design with a
composite flexible diaphragm (see Fig.2. 3). It is made of only two layers; the first layer
serves as the main elastic layer and is composed of an elastomer, such as nitrile rubber, butyl
rubber, styrene rubber, chloroprene rubber, or ethylene-propylenediene (EPDM) rubber.
EPDM rubber is preferred. The rubber layer has typically a thickness between 1.2 and about
2.5 mm. The second layer serves as the gas-impermeable layer and is affixed to the surface of
the first layer on the side facing the gas chamber. It is composed of a gas-impermeable
material, such as a thin metal layer vapour deposited to the first layer. The metal film may
consist of aluminum, titanium, antimonytin, or other metals or alloys. The metal layer
typically has a thickness between about 2 µm and about 8 µm. The composite diaphragm still
has excellent elastic flexibility as a whole, even with a metal layer being affixed to the rubber
layer.
19
Fig.2. 3 Larsen model of the diaphragm accumulator.
John H. Crankshaw [Crankshaw 1974] invented a new accumulator design which
consists of two or more concentric steel cylinders with at least one cylinder sleeve of
neoprene acting as a spring between them. The contact surfaces of the steel and the neoprene
are bonded together and have shear strength equal to the shear strength of the neoprene itself
(see Fig.2. 4). The inner cylinder is connected to a piston rod and in turn to a piston. The fluid
pressure acts on this piston against the sleeve of the neoprene. The outer cover encloses the
assembly. This design is simple in construction, efficient to use and does not need gas
charging. After a certain working life the sleeve of neoprene gets damaged leading to a
sudden failure. Thus, this design would need additional elements to avoid this malfunction.
20
Fig.2. 4 Crankshaw model of the piston accumulator.
James R. Mayer [Mayer 1976] created a new flexible diaphragm integrating a metal disc
into an elastomer diaphragm. The diaphragm contains also a plurality of concentric circular
ridges for engagement with the wall surface of the accumulator lower housing which contains
concentric circular rows of fluid discharge openings (see Fig.2. 5). This prevents an extrusion
of the diaphragm through the discharge port and reduced the tendency of the diaphragm to
trap hydraulic fluid between the diaphragm and the walls of the accumulator housing which
provided maximum flow area.
This design is assessed as follows: it is a little bit complicated in construction and more
expensive than conventional diaphragm accumulators, but needs no recharge the accumulator
21
Fig.2. 5 Mayer model of the diaphragm accumulator.
Kip R. Steveley [Steveley 1988] discusses the utilization of an internal spring having at its
end an ellipsoidal cap in a normal hydraulic accumulator. This cap shapes the flexible
membrane of the accumulator when pressurized fluid enters the accumulator, thereby
minimizing the tensional forces within the membrane (see Fig.2. 6Fig.2. 6).
This design is relatively simple, although more complex than a conventional diaphragm
accumulator; it is probably not suitable for very high operating frequencies due to the small
inlet bore. The claimed service life extending effect of the elliptical cap is hard to asses
without experiments.
22
Fig.2. 6 Steveley model of the diaphragm accumulator.
Onishi [Onishi 2000] proposes a new design for the lower housing of a high-pressure
accumulator which defines the limit of deformation for a flexible disk-shaped metal
diaphragm. Excessive concentrations of stresses in the diaphragm are prevented by the
curvature of the contact surface of the lower housing (see Fig.2. 7). To obtain the right
curvature the author divided the diaphragm surface into two zones: the first on its
circumference which is subjected to a uniformly distributed load; and the other one is the
central portion of the diaphragm which is defined by a large deflection. This design is simple
in construction, cheap in cost and does not need gas charging. Excessive stresses occur in a
plate made of rather stiff metal if deformations are large. This is the main shortcoming of this
design.
23
Fig.2. 7 Onishi model of the diaphragm accumulator.
Takamatsu et al [Takamatsu 2000] present an accumulator with a diaphragm comprising
a resin intermediate layer for gas shielding, an upper rubber layer adjacent to the gas chamber
made of three layers of different material. The upper layer adjacent to the gas is made of butyl
rubber, the intermediate layer is from EPDM (Ethylene-Propylene-Diene Terpolymer), and
another rubber layer works as the lower layer adjacent to the oil chamber. When the
accumulator is working the rubber is repetitively compressed and strained by elastic
deformation (see Fig.2. 8). The rubber is rubbed and worn, resulting in cracking of the rubber
layers but after a longer working life than the normal diaphragm construction. This diaphragm
design is expected to be more complicated in manufacturing, hence more expensive, and has a
tendency of a separation of the diaphragm layers since the resign and the rubber materials
have different modulus of elasticity.
24
Fig.2. 8 Takamatsu et al model of the diaphragm accumulator.
Sasaki et al [Sasaki 2006] investigate a hydraulic accumulator with an elastic composite
diaphragm which consists of the intermediate layer (gas shielding layer) made of ethylene-
vinyl alcohol copolymer, the inner elastic layers made of the polyamide resin, and the outer
elastic layers made of ester based elastic plastic. All these layers are included in an outer
rubber layer (see Fig.2. 9). This new composite diaphragm prevents gas permeation for a long
time. Furthermore, the resistance to the low temperature operation under -40 ° C is improved.
This design is assessed to be more costly but having longer working life than conventional
accumulators; a diaphragm with several layers has lower flexibility than single layer designs
which affects the accumulator hydraulic capacity negatively.
25
Fig.2. 9 Sasaki et al model of the diaphragm accumulator.
Nakamura et al [Nakamura 2004] study a bellow type accumulator comprising a
cylindrical shell including a metallic bellow for partitioning the interior of the shell into a
hydraulic chamber and a gas chamber (see Fig.2. 10). Operation of the accumulator over long
time can be ensured. The design is available for medium capacity accumulators. It depends on
the maximum height of the bellow. The assessment of the design is, that this accumulator type
promises adequate hydraulic capacity, is gas tight but costly in manufacturing.
Suzuki et al [Suzuki 2004] propose a new way to efficiently bleed air out of the hydraulic
accumulator. This is accomplished by a specific liquid chamber and a separate inflow –
outflow channel pair. During an air bleeding operation to be carried out when the hydraulic
accumulator is attached to the support member, the operating liquid flows into the liquid
chamber from the liquid inflow port via the inflow passageway (see Fig.2. 11). Fluid is
progressively accumulated in the liquid chamber until the liquid level reaches the liquid-
chamber-side end of the outflow passageway.
26
Fig.2. 10 Nakamura et al model of the bellows accumulator.
Air within the liquid chamber is forced out toward the liquid outflow port via the outflow
passageway. Further, air remaining in the upper portion within the liquid chamber is mixed in
form of bubbles into the operating liquid flowing into the liquid chamber via the inflow
passageway, and these bubbles, together with the operating liquid, flow out toward the liquid
outflow port. Therefore, by the air bleeding operation in which an operating liquid is
progressively supplied to the liquid inflow port of the hydraulic accumulator, air within the
liquid chamber can be discharged to the outside of the liquid chamber, thus achieving the
intended excellent air removal.
The author of the thesis doubts the need for such a rather complicated air removal system.
Under high pressure conditions and if several rapid fluid charging and discharging cycles
occur air is removed quite quickly.
27
Fig.2. 11 Suzuki et al model of the bellows accumulator.
Robert Mutschler [Mutschler1999] carries out analytical and experimental studies of
metallic bellow accumulators. Arguing the well known shortcoming of the elastomer elements
of conventional accumulators, namely gas diffusion and limited service life; metallic bellows
are investigated as an alternative separator element (see Fig.2. 12).
[Senior Aerospace 2011] an accumulator manufacturing company - reports about their
development of bellow accumulators. They study different types of bellow materials such as
stainless steel, different metal alloys as titanium, and a carbon fibre material for high strength
light weight, e.g. aerospace, applications.
28
Fig.2. 12 Mutschler model of the diaphragm accumulator.
The product is completely maintenance and service free (absolutely no loss of gas charge),
has higher reliability than bladder or piston types, long predictable life under crude operating
conditions, has high corrosive resistance for chemical fluids, saves weight and cost by
eliminating charging lines and valves, and has extreme temperature capabilities (see Fig.2.
13).
Fig.2. 13 Bellows accumulator of Senior Space company.
29
2.3 Conclusions
All these researches address either new designs or new material or material compounds.
The well known problem of gas diffusion with elastomer material is tried to be overcome by
the use of metallic diaphragms or bellows, or special multilayered membranes combining
different plastic or metal material. Multi-layers diaphragms promise gas tightness and can be
integrated in the accumulator in a traditional way, but there is the risk of layer separation
which tends to reduce its working life time. Metallic bellows type has a moderate capacity, is
maintenance free, can be used with some chemical products if appropriately resistant steel
grades are employed and has long life time; a major drawback of this type of separator
element is the cost. In a few cases low temperature and resistance against wear or against
some chemical fluids are addressed.
30
31
3. Hydraulic system dynamic response modelling
3.1 Introduction
As already outlined in Section1.2, hydraulic systems may exhibit significant flow and
pressure pulsation, the attenuation of which is one of the important functions of
accumulators. The characteristic of such pulsation is strongly influenced by the dynamic
properties of the hydraulic system. The accumulator’s dynamic properties do not define its
attenuation performance in a specific application but it is always necessary to consider the
whole system’s dynamics. Hydraulic systems such as fuel pipelines of internal combustion
engines, hydroelectric power plants, petroleum transmission lines, motivated researchers
since long time to study fluid transmission line dynamic behaviour. One important aspects of
this behaviour is known as water hammer, which is a significant pressure rise if a flow is
strongly decelerated. Mathematical modelling and simulation are inevitable tools for a
systematic analysis of a hydraulic system’s dynamic behaviour. Of course, this is also true
for the study of the accumulator dynamic performance. Different model types and model
granularity can be selected for these purposes. Simple models have the advantage to need
less information and less mathematical effort for their solution than more complex models
which potentially can yield more accurate results, provided adequately precise system data
are fed into these models.
The purpose of modelling a hydraulic system in the context of accumulator design is to
quickly assess new designs of the hydraulic accumulators with respect to the required
performance, and to optimize design parameter configurations.
What are the main performance criteria that are addressed by this design? The new
accumulator design should be simple, need no maintenance, is as cheap as possible, is easy to
integrate into the hydraulic system and, of course, has to dampen pulsation to a necessary
extent.
The accumulator is somehow connected to the source of excitations (e.g., a hydraulic
pump, switching valves, …). This connector may influence the effect of the accumulator
significantly.
32
In this chapter, two types of hydraulic models are presented:
discrete parameter models and
distributed parameter models.
To study how accumulator dynamics interacts with the transmission line dynamics a
benchmark problem is defined.
3.2 Discrete parameter models
Discrete parameter models have finite dimensional states, typically combined to a state
vector x. The higher the number of states of such models the better is the achievable
accuracy. Discrete parameter models lead to a system of ordinary differential equations
(ODE). This is an advantage over the distributed parameter models which lead to partial
differential equations (PDE) which need a lot more effort for solving. Due to the nonlinear
characteristics of some of the hydraulic elements, these ODE are typically nonlinear. To
solve such systems in frequency domain linearization of these equations is necessary.
Before investigating the hydraulic modelling of each element, the relevant hydraulic
systems – the cases of studies - are characterized.
3.2.1 Case of study
An accumulator is nearly always embedded in a hydraulic transmission line. Thus,
typically the situation of a T-junction arises, as shown in Fig.3. 1. The fluid enters from the
left port of the left pipe in the system of Fig.3. 1controlled by a (fast) switching valve; but
this is just one possibility how pulsation is excited. At the T-junction the flow is divided, one
part going into the accumulator, one into the right pipe at which’s right port it leaves the
system. The entering flow carries some pulsations, either of the flow rate or of the pressure.
In case of a switching actuation as indicated in Fig.3. 1, these pulsations will be quite
extreme and will provoke very high frequencies. The accumulator should dampen these
pulsations such that at the output port a smaller pulsation level occurs. The load at this
system’s end is just a throttle. Of course, this is only the simplest case which can simulate a
real hydraulic load which is often much more complicated and exhibits typically a more
complex response dynamics. The vertical pipe is used to connect the hydraulic accumulator
with the hydraulic system in practice. It is often not possible to connect the accumulator
directly. It is clear that this connection line may have significant influence on the attenuation
performance of the accumulator system and that is desirable keeping this connection line
very short for high operation frequencies. Incorporating this element in the model should
reveal its influence. A possibility to connect the hydraulic accumulator to the hydraulic
system is to use a hydraulic block manifold. This block could also contain the valve (the
source of excitations). Such design not only rules out the vertical pipe but also the left pipe,
hence is an optimal design with respect to the accumulator performance. But some sort of
small flow channel is even present in the classical accumulator designs and also in the
valves. Playing with the parameters of these pipe elements in the mathematical model can
show how different designs influence the dynamics.
The hydraulic accumulator model of this analysis allows addressing the intended new
accumulator design with a metallic separator membrane and a special charging port realized
by several small holes [Fig.3. 2].
Fig.3. 1 Case of study.
33
Fig.3. 2 The new diaphragm accumulator with a multi-hole charging port.
For the purpose of modelling this system’s dynamic performance the following hydraulic
elements are required.
3.2.2 Elements of the hydraulic system model
3.2.2.1 Linear hydraulic capacitance
A linear hydraulic capacity is given by a fluid filled cavity due to fluid compressibility
and/or a flexible boundary. From the definition of the bulk modulus E of a fluid which is
based on the assumption of a constant mass (M) system with initial (or reference volume) V0
which changes its volume by ΔV and its pressure by Δp, the following compressibility
relation results, see Fig.3.3. [Esposito 1988]
o
fl
fl VV
pEΔ
Δ−=
eq.3. 1
The bulk modulus is a material property characterizing the compressibility of a fluid.
It can be largely decreased by entrapped air bubbles.
flE
Assuming a state change with a constant mass M and the basic linear compressibility
definition of eq. 3.1 the compressibility law relating density and pressure can be derived:
34
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛+≈
−=−=
−=⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟
⎠⎞
⎜⎝⎛ −=
−⇔−=
fl0
fl
0
fl
0
fl
00
000
fl
0fl
Ep1
Ep1
;E
p1
Ep1
MV
MV
MVV
EVpV
ΔρΔρρΔ
ρρ
ΔρρρρΔΔ
eq.3. 2
In case of a constant control volume of size V0 to which fluid flows in or out with a flow
rate Q (positive inward) the compressibility law leads to a relation as shown below.
flHflfl
fl
CQp
EVp
EVpQ
QEpVVM
=⇒≈=
===
&&&
&&&
000
000
ρρ
ρρρ
eq.3. 3
Where pM &&& ,,ρ are the time derivatives of the mass, the density and the
pressure of the fluid.
The formal equivalence with the state law of an electric capacitance motivates the
definition of a hydraulic capacity flHC
eq.3. 4
fl
0H
E
VCfl=
One may work with the effective bulk modulus of the fluid which is a superposition of the
compressibility of the hydraulic oil, air or gas bubbles and the pipe’s flexibility. If the
applied hydraulic oil pressure is exceeds 100 bar air bubbles influence is negligible for usual
air contents of hydraulic fluids.
Fluid pressure excites a radial force on the pipe wall which due to pipe flexibility has
effect on the effective bulk modulus of the system.
For a thick wall pipe the volume change due to pipe expansion is [Murrenhoff 2005]
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−++Δ−=Δ
1
)21(3)1(22
2_0
pipe
pipe
el
pipepipe E
VpV
β
μμβ
Where: i
opipe d
d=β and μ is the Poisson’s ratio
eq.3. 5
flpipeltotal VVV ΔΔΔ += eq.3. 6
35
Taking into consideration that the nominal volume of the fluid is equal to the pipe inner
volume and that the total volume change pipe0 VV = totalVΔ considers fluid compressibility
and pipe extension the corresponding effective bulk modulus can be written as effE
Fig.3. 3 The linear hydraulic capacitance model.
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−−++
+Δ−=Δ1
)21(3)1(2112
2
0pipe
pipe
elfltotal EE
VpVβ
μμβ
eq.3. 7
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−−++
+=1
)21(3)1(21112
2
pipe
pipe
elfleff EEE βμμβ
eq.3. 8
and the corresponding linear hydraulic capacity can be written as
eff
0H
E
VC =
eq.3. 9
36
3.2.2.2 Hydraulic inductance
When a mass of fluid flows in a conductor with a variable velocity, i.e. that mass is
accelerated, the fluid inertia opposes the velocity change (see Fig.3.4). The acceleration term
in the Eulerian coordinate system is divided into local and convective acceleration [Prieve
2000]. The local acceleration is the rate of change of velocity with respect to time at an
arbitrary point in the flow while the convective acceleration is the rate of change of velocity
due to the change of position of fluid particles with respect to the Elerian reference frame
[Cohen 2002].
The fluid acceleration reads:
xvv
tv
DtDva
∂∂
+∂∂
== eq.3. 10
For typical flow situations in hydraulic transmission lines, the so called convective
acceleration term can be neglected. A justification for this can be found in [Murrenhoff
2005].
With this cancelling of the convective term eq.3.10 simplifies to
dtdQ
A1
tv
DtDva =
∂∂
== eq.3. 11
Due to Newton’s second law, the total pressure force applied on a fluid volume is equal to
the inertia force
aLAFaVamF ρρ =⇒== eq.3. 12
By substituting the acceleration term from eq.3. 11 into eq.3. 12 one gets [Furesz 1988]
dt
dQLFdt
dQ
A
1LAF ρρ =⇒⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=
eq.3. 13
The hydraulic force is the difference between the applied pressures pΔ on the surface of a
fluid volume
A
FppAF =⇒= ΔΔ eq.3. 14
37
Fig.3. 4 The hydraulic inductance mathematical model.
By substituting eq.3. 13 into eq.3. 14
QA
Lp &ρΔ = eq.3. 15
Again, by analogy to an inductance of electrical engineering, the hydraulic inductance is
defined
A
LLHρ
= eq.3. 16
which brings eq.3.16 into the following form
HLpQ Δ
=& eq.3. 17
3.2.2.3 Laminar, steady state flow resistance in a circular cross section pipe
When a flow passes through a pipe of circular cross section, there is a frictional (shear)
force which resists the fluid motion. This action is dissipative and transfers hydraulic power
to thermal energy. The shear force depends on the physical properties of the fluid and the
size and the shape of the pipe [Murrenhoff 2005].
Fig.3. 5 The pipe flow resistance model.
38
In steady state, no fluid acceleration occurs, hence inertia forces are zero. The shear forces
have to be balanced by pressure forces resulting from a pressure difference ( pΔ ), see
Fig.3. 5.
2221 ypy)pp(Ly2 πΔπτπ =−=
eq.3. 18
For a Newtonian fluid the shear stress between fluid layers is proportional to the velocity
gradient in the direction perpendicular to the layers
μμμμτ
Lrpvdy
Lypv
Lyp
dydv
dydv r
422
2
max0
Δ=⇒
Δ=⇒
Δ−=⇒−= ∫
eq.3. 19
In hydraulics, the flow rate Q is preferred over the flow velocity. Writing the friction
pressure loss relation in terms of the flow rate reads
LrpvdyyQ
r
μππ8
24
0
Δ== ∫ eq.3. 20
By defining the resistance as
4H
r
L8Rπ
μ=
eq.3. 21
transfers eq.3. 22into
QRp H=Δ eq.3. 23
This relation is well known as Poiseuille’s law which applies only to Newtonian fluids
and steady state flow.
3.2.2.4 Discrete transmission line model
Transmission lines such as pipes and hoses have capacitive, resistive and inertial
impedances along the pipe length. Discrete parameter models contain one or more capacitor,
inductance and resistance as in the modelling of an electric circuit to obtain solutions with
suitable accuracy. The discrete transmission line is modelled as a series connection of
resistance due to fluid friction, the fluid inertia (hydraulic inductivity) and fluid
compressibility (hydraulic capacity), see Fig.3. 6.
39
Pipe wall flexibility as well as nonlinear fluid compressibility due to entrapped gas
bubbles are neglected. This model is an RLC circuit as it is well known in electrical
engineering. An electrical RLC circuit is depicted in Fig.3. 7. The direct analogy to its
hydraulic pendant is shown in Fig.3. 6.
Electric-hydraulic analogy:
The electric-hydraulic analogy has been addressed already in the derivation of the
hydraulic capacitance and inductance. It will be employed also here to derive the state
equations for a system comprising a combination of such elementary elements. This is done
for the analogue electrical system first and can then be translated into the hydraulic notation.
The electrical resistance, inductance, and capacitance equations are
)()( tiRtv RHR = Electrical resistance eq.3. 24
dttdiLtv L
HL)()( = Electrical inductance
eq.3. 25
HCC C)t(v)t(i &= Electrical capacitance eq.3. 26
vR,L,C are the voltages at the resistance, the inductance, and at the capacitance, and iR,L,C
the respective currents.
The combinations of the hydraulic and electric elements, respectively, to form the most
elementary pipe element featuring resistive, inductive, and capacitive modelled are shown in
Fig. 3.7 and Fig.3.8.
Fig.3. 6 The Hydraulic model for SDOF of the transmission line.
40
Fig.3. 7 The electrical model for SDOF of the transmission line.
From Kirchhoff's law, the sum of all the voltages around the closed loop is equal to zero.
)t(v)t(v)t(v)t(v CLRin ++= eq.3. 27
By substituting eq.3. 24, eq.3. 25 and eq.3. 26 in eq.3. 27, one gets
∫++= dt)t(iC1
dt)t(diL)t(iR)t(v
HHHin eq.3. 28
And differentiating eq.3. 28 to eliminate the integral yields
HH2
2
Hin
C)t(i
dt)t(diR
dt)t(idL
dt)t(dv
++=
And eq.3. 29 can be further modified to
)t(i)t(iCR)t(iCL)t(vC)t(vC HHHHoutHinH ++=− &&&&&
Where: )t(i)t(vC)t(vC CCHoutH −== &&
eq.3. 29
)t(i)t(iCR)t(iCL)t(i)t(vC HHHHCinH ++=+ &&&& eq.3. 30
The electrical model (eq.3. 30) can be directly transferred into its hydraulic pendant
)t(Qdt
)t(dQCR
dt
)t(QdCL
dt
)t(dpC)t(Q in
inHH
in2
HHin
Hout +++−=
eq.3. 31
41
The following equivalences have been applied in this transfer
)t(pˆ)t(v);t(Qˆ)t(i);t(Qˆ)t(i inininoutC === eq.3. 32
The time domain model can be transformed to the frequency domain using Laplace
transform [Tuma 1979]:
(0)F-sF(0)-f(s)s(t)}F{LF(0),-sf(s)(t)}FL{ 2 ′=′′=′
The transformation of eq.3. 31 into the frequency domain by Laplace transform gives
( ) inHHHH2
inHout Q̂1CRjCLP̂CjQ̂ ++−+−= ϖϖϖ
Where:
ϖ : the Laplace frequency or the angular velocity
j : the imaginary number )1( −=j
outQ̂ and inQ̂ : the output and the input flow rates in the frequency domain
inP̂ : the input pressure to the transmission line in the frequency domain
eq.3. 33
The inductance and the resistance relations of the hydraulic model in time domain are
)t(QR)t(p)t(p)t(QR)t(p)t(p inHinRinHRin −=⇒=− eq.3. 34
H
outRin L
)t(p)t(p)t(Q −=&
eq.3. 35
)(t)QR(t)p(t)p(L1(t)Q inHoutinH
in −−=& eq.3. 36
⎟⎠
⎞⎜⎝
⎛ +−= (t)QRdt
(t)dQL(t)p(t)p inH
inHinout
eq.3. 37
The equations enable to obtain the following frequency domain equation, to describe the
output pressure’s (pout) Laplace transform as function of the input quantities . outP̂ inin QP ˆ,ˆ
( ) inHHinout Q̂LjRP̂P̂ ϖ+−= eq.3. 38
Both eq.3. 33 and eq.3. 38 could be combined to one matrix equation, representing a
SDOF (Single Degree Of Freedom) discrete frictional pipe model
42
eq.3. 39
( ) ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++−−
+−=⎥
⎦
⎤⎢⎣
⎡
in
in
HHHH2
H
HH
out
out
Q̂P̂
1CRjCLCj)LjR(1
Q̂P̂
ϖϖϖϖ
For modelling a transmission line by two identical RLC elements connected in series, as
Fig.3. 8 shows for the electric case, the LCR circuit must be applied twice and the
corresponding matrix of the transmission line will be just the square of the matrix of one
element:
First element transmission matrix
( ) ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++−−
+−=⎥
⎦
⎤⎢⎣
⎡
in
in
HHHH2
H
HH
1out
1out
Q̂P̂
1CRjCLCj)LjR(1
Q̂P̂
ϖϖϖϖ
eq.3. 40
Second element transmission matrix
( ) ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++−−
+−=⎥
⎦
⎤⎢⎣
⎡
1out
1out
HHHH2
H
HH
2out
2out
Q̂P̂
1CRjCLCj)LjR(1
Q̂P̂
ϖϖϖϖ eq.3. 41
Combining both by substituting eq.3. 40 into eq.3. 41 gives
( ) ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++−−
+−=⎥
⎦
⎤⎢⎣
⎡
in
in2
HHHH2
H
HH
2out
2out
Q̂P̂
1CRjCLCj)LjR(1
Q̂P̂
ϖϖϖϖ
eq.3. 42
Where: 1out1out Q̂,P̂ : The Laplace transform of the outlet pressure and flow rate of the first
pipe which are also the inlet values of the second pipe respectively.
2out2out Q̂,P̂ : The Laplace transform of the outlet pressure and flow rate from the second
pipe, respectively.
Of course, the whole pipe can be constituted of N identical RLC circuits (see Fig.3. 8),
which results in the following transfer matrix
( ) ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++−−
+−=⎥
⎦
⎤⎢⎣
⎡
in
inN
HHHH2
H
HH
out
out
Q̂P̂
1CRjCLCj)LjR(1
Q̂P̂
ϖϖϖϖ
eq.3. 43
where N is the number of discrete pipe elements.
43
Fig.3. 8 The multi degrees of freedom discrete model of the transmission line represented by its
equivalent electrical circuit
3.2.2.5 Hydraulic throttle
A hydraulic throttle is typically realized as a thin disc with a central hole. Its hydraulic
behaviour is modelled by the orifice equation. [Sullivan 1998]
ρ))t(p)t(p(2AdC)t(Q 21
vv−
=
Where:
dC : the coefficient of discharge
vQ : the discharge flow rate of the throttle valve
eq.3. 44
ip , : the fluid pressure at both ports ,...2,1i =
To get a frequency domain model of the throttle the orifice equation must be linearized at
a certain working point, i.e. a certain pressure , see avP Fig.3. 9
The resulting frequency domain equation reads
ρρ
Δ
av
vdv P
)p̂(2AC
2
1Q̂ = eq.3. 45
44
Fig.3. 9 The linearization of the pressure-flow rate relation of a hydraulic throttle.
3.2.2.6 The Hydro-pneumatic accumulator
As mentioned previously, the hydro-pneumatic accumulator under study consists of liquid
and gas chambers, separated by a diaphragm. The liquid filling the hydraulic chamber is
connected to the hydraulic circuit. When the liquid pressure rises the gas is compressed
adiabatically or polytropically. When the oil pressure decreases the compressed gas expands
again and forces the accumulated liquid into the hydraulic circuit.
Gas chamber
From the polytropic equation [Kokmaz 1982]
n
G
GGG V
VPP ⎟⎟⎠
⎞⎜⎜⎝
⎛= 0
0
eq.3. 46
and by linearization at a certain pressure Pk, see Fig.3. 10, one gets the linearized state
equation of the gas spring:
GGo
n1
Gon1n
KK,G dV
VPPndP
−+
−= eq.3. 47
The gas volume change is related to the displacement of the separator element, which for
modelling purposes is assumed to be a rigid disc. Its actual axial displacement is denoted by
45
y (see Fig.3. 11). If flexible membranes are employed the corresponding relations are more
complex. Since this model should just be a first order approximation of the separator
element’s influence on the accumulator dynamic response dynamics, this simple model of a
rigid disk may be acceptable.
yAdV accG = eq.3. 48
yAdVdtQ accGoil ==∫ eq.3. 49
acc
oil
A
dtQy ∫=
eq.3. 50
∫−+
−= dtQV
PPndP oil
0G
n
1
0Gn
1n
KK,G
eq.3. 51
Separator element with inertia
The separator element between the gas and the oil chambers (see Fig.3. 11) has some
inertia which might have influence on the response dynamics, hence it is taken into
consideration.
The load of excitation is . accoil AP
The inertia resistance of the metallic diaphragm is ymdiaph &&
The equation of motion reads:
accK)Goildiaph A)dPP(ym −=&& eq.3. 52
Combining equations eq.3. 50 and eq.3. 51 with eq.3. 52 results to
)dtQ(V
PPndt
)dtQ(d
A
mP oil
0G
n1
0Gn1n
K2oil
2
2acc
diaphoil ∫∫
−+
−=
eq.3. 53
the frequency domain version of which reads
46
oil0G
n1
0Gn1n
K22
acc
diaphoil Q̂
VPPn
A
mjP̂
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+=
−+
ϖϖ
eq.3. 54
Fig.3. 10 The linearization method for the relation pressure-volume of gas inside the accumulator.
GV : is the actual gas volume of the accumulator gas chamber
GV& : is the time derivative of GV
n : is the polytropic exponent
y : is the displacement of the separator element of the hydro-pneumatic accumulator
y&& : is the acceleration of the separator element
accA : is the surface area of the separator element of the hydro-pneumatic accumulator
diaphm : is the mass of the separator element of the hydro-pneumatic accumulator
The complete hydraulic system is composed by connecting these hydraulic model
elements. This leads to a matrix type equation of the complete system.
47
Fig.3. 11 The separator element model inside the accumulator.
3.2.2.7 Discrete parameter SDOF model of the case of study.
The discrete parameter model of the case of study (see section 3.2.1) is represented as a
collection of pipes where each pipe is modelled as SDOF transmission line model, throttles
and a gas chamber (see Fig.3. 12); its frequency domain representation can be condensed into
one matrix equation:
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡++−−
+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
0000000000
Q)1CRjCL(Q)RLj(
.
4D3D2D1D4C3C2C1C4B3B2B1B4A3A2A1A
yQPPQPQPQQPP
1A1H1H1H1H2
1A1H1H
4E
4E
4A
3E
3E
2E
2E
2A
1E
1E
1A
ϖϖϖ
eq.3. 55
48
The coefficients A1 to D4 of the matrix equation (eq.3. 55) are
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
0101-0Cj-
01-11A H1ϖ , , , ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
01-Lj-R-000000
2A
H2H2 ϖ ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
3A⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
4A
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=
33
H2
100000Cj-0
1
HH RLjB
ϖ
ϖ ,
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
−=
00LjR
12
10
1-01+CRj+CL-
2
H3H3
H3H3H3H32
ϖρ
ϖϖ
av
vd
PAC
B ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
001000000
3B , , ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
4B
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=000000
1+C.Rj+CL-Cj-01C
H3H3H3H32
H3 ϖϖϖ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=000000001)+CRj+CL(--
2H3H3H3H3
2 ϖϖC ,
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
=
1Lj-R-02
112
1010
3
H4H4
2
2
2
2
ϖρρ av
vd
av
vd
PAC
PAC
C
, , , ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
1D⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
2D⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=000000Cj-1+CRj+CL-0
3H4H4H4H4H4
2 ϖϖϖD
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
=
−
1-V m
APPnj-0m
A-
1-Aj-0
0104D
0
24
n1
0n
1n+
K2
4
4
ωϖ
ϖ
Where:
1AP , are the input pressure and the input flow rate amplitude of the first 1AQ
horizontal (upstream) pipe.
1EP , are the output pressure and flow rate of the first horizontal (upstream) 1EQ
pipe.
2AQ is the input flow rate of the second horizontal (downstream) pipe.
49
2EP , are the output pressure and flow rate of the second horizontal 2EQ
(downstream) pipe.
3EP , are the output pressure and flow rate of the vertical pipe or the pipe 3EQ
connecting the accumulator.
4AP is the input pressure of the accumulator oil chamber.
4EP , are the output pressure and flow rate of the accumulator oil chamber. 4EQ
y is the diaphragm displacement inside the hydraulic accumulator.
4H3H2HHi R,R,R,R are the hydraulic resistances of the first, second horizontal pipe,
the pipe connecting the accumulator and the oil chamber
respectively.
4H3H2HHi C,C,C,C are the hydraulic capacitances of the first, second horizontal
pipe, the pipe connecting the accumulator and the oil chamber
respectively.
4H3H2HHi L,L,L,L are the hydraulic inductances of the first, second horizontal
pipe, the pipe connecting the accumulator and the oil chamber
respectively.
2, vv AA are the cross sectional areas of the output throttle valve and of the
throttle valve located at the entrance of the accumulator.
is the surface area of the separator element of the hydro-pneumatic 4A
accumulator.
m is the diaphragm mass
ϖ is the angular velocity
To demonstrate the effect of the accumulator on the system’s pulsation dynamics also a
model without an accumulator is established to compare the results of both cases. This
50
accumulator free model is just a series connection of two SDOF pipe elements as derived
above. It is depicted in Fig.3. 13.
By comparison with more refined continuous parameter models (see Section3.4) it turned
out that a two SDOF pipe model, in other words a two degree of freedom model (2DOF),
gives reasonable results in the frequency range of interest. The SDOF model, however, gives
quite unsatisfactory results in the parameter range of interest.
The discrete parameter model of the case of study is represented by a matrix equation:
⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎦
⎤⎢⎣
⎡=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
000
)QA11)+CRj+CL(-+)Lj-(-RC(-j-1))QA1+CRj+CL)(-Lj-(-R+Lj-(-R-
.4A3A2A1A
QPQPP
2H1H1H1H1
2H1H1H1
H1H1H1H12
H1H1H1H1
2E
2E
1E
1E
1A
ϖϖϖϖϖϖϖϖ
eq.3. 56
The coefficients of the matrix equation (eq.3. 56) are
⎥⎦
⎤⎢⎣
⎡=
01)+CRj+CL.(-Cj-Cj-1-)Lj-(-RCj-1
1AH1H1H1H1
2H1H1
H1H1H1
ϖϖϖϖϖϖ , ,
⎥⎦
⎤⎢⎣
⎡−
=001000
2A
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
001)+CRj+CL(-Cj-Cj-0
)Lj-(-RCj-103A H2H2H2H2
2H2H2
H2H2H2
ϖϖϖϖϖϖ
,
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−+=
1-P
AC2
10
101)+CRj+CL(-)Lj-.(-RCj-01-1)+CRj+CL).(-Lj-(-R+Lj-R-
4A
av
vd
2H2H2H2H2
2H2H2H2
H2H2H2H22
H2H2H2H2
ρ
ϖϖϖϖϖϖϖϖ
To obtain more accurate solutions in the case of study a 2DOF discrete parameter model
is used instead of the SDOF model.
The 2DOF discrete parameter model can be modelled in the following matrix in the
frequency domain as:
51
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
0000000000
)QA11)+CRj+C(-L+)R-LCH1(-j(-j-1))QA1+CR j+C)(-LR-L(-j+R-L(-j-
.
4D3D2D1D4C3C2C1C4B3B2B1B4A3A2A1A
yQPPQPQPQQPP
2H1H1
2H1H1H1H1
H1H12
H1H1H1H1H1H1
4E
4E
4A
3E
3E
2E
2E
2A
1E
1E
1A
ϖϖϖϖϖϖϖϖ
eq.3. 57
or in simplified form , where is the variables vector, is the parametric
matrix and is the input vector.
uAx 1−= x A
u
The coefficients of the matrix equation (eq.3. 57) are
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
0)R-L.(-jCj-101-01)+CRj+C.(-LCj-Cj-
01-)R-L.(-jCj-11A
H2H2H2
H1H12
H1H1H1H1
H1H1H1
ϖϖϖϖϖϖ
ϖϖ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
01-1)+CRj+CL)(-Lj-(-R+Lj-R-000000
2A
H2H2H2H22
H2H2H2H2 ϖϖϖϖ ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
3A , ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
4A⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
1H3H3H3
1
)R-L(-jCj-1000000
1
B
B
f
eB
ϖϖ
Where: 1)+CRj+C.(-LCj-C-je H2H22
H2H2H2H2B1 ϖϖϖϖ=
1)+CRj+CL)(-Lj-(-R+Lj--Rf H3H3H3H32
H3H3H3H3B1 ϖϖϖϖ=
52
Fig.3. 12 The discrete parameter SDOF model of the case of study.
53
Fig.3. 13 The discrete parameter model of the case of study without the hydraulic accumulator in an
equivalent electric depiction.
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++
−=
00)1CRj+CL)(-Lj-(-R-LjR
1P
AC2
10
1-01)+CRj+CL(-+)Lj-(-RCj-
2B
H3H3H3H32
H3H3H3H3
av
vd
2H3H3H3H3
2H2H2H2
ϖϖϖϖρ
ϖϖϖϖ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
010000000
3B
, ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
4B
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=000000
1)+CRj+CL(-+)Lj-(-RCj-e01C
2H3H3H3H3
2H3H3H31C ϖϖϖϖ
Where: 1)+CRj+C(-LCj-C-je H3H32
H3H3H3H31C ϖϖϖϖ=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=000000001)+CRj+CL(--)Lj-(-RCj
2C
2H3H3H3H3
2H3H3H3 ϖϖϖϖ
54
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
=
)R-L(-jCj-1f0P
AC2
11P
AC2
1010
3C
H4H4H43C
2av
2vd
2av
2vd
ϖϖρρ
Where: 1)+CRj+CL)(-Lj-(-R+Lj--Rf H4H4H4H42
H4H4H4H43C ϖϖϖϖ=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
001000000
4C
,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
1D
, ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
2D
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=000000
1)+CRj+C(-LCj-Cj-e03D
H4H42
H4H4H4H43D ϖϖϖϖ
Where: 2H4H4H4H4
2H4H4H43D 1)+CRj+CL(--)Lj-(-RCje ϖϖϖϖ=
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
=
4D2H4H4H4
H4H4H4
f0)(m
))Lj-(-RCj-(1-
1-))Lj-(-RCj-(1
j-0010
4D
ϖϖϖ
ϖϖϖ
Where: 1-V m
))Lj-(-RCj-(1PPnj-f
0
2H4H4H4
n1
0n
1+n
K4D ω
ϖϖ−
=
55
3.3 Distributed parameter models
3.3.1 Transmission line model
The real study of the propagation of the pressure wave in the fluid transmission can be
based on the early work of Navier and Stokes. They derived the fundamental equations for
the flow of a fluid, known as Navier-Stokes equations.
For the purpose of transmission line modelling Navier-Stokes equations are preferably
expressed in cylindrical coordinates as [D'Souza 1964]:
The momentum equation along the pipe axial direction is:
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +∂∂
∂∂
+∂∂
+∂∂
+∂∂
+∂∂
−=⎥⎦⎤
⎢⎣⎡
∂∂
+∂∂
+∂∂
rv
rv
x31
ru
r1
ru
xu
34
xp
ruv
xuu
tu
2
2
2
2μρ
eq.3. 58
and the momentum equation in the radial direction:
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
∂∂
+−∂∂
+∂∂
+∂∂
−=⎥⎦⎤
⎢⎣⎡
∂∂
+∂∂
+∂∂
xv
ru
31
xrv
34
rv
r1
34
rv
34
rp
rvv
xvu
tv
22
2μρ
eq.3. 59
The continuity equation reads:
0r
vx
urv
rv
xu
t=
∂∂
+∂∂
+⎥⎦⎤
⎢⎣⎡ +
∂∂
+∂∂
+∂∂ ρρρρ
eq.3. 60
The above equations are non linear partial differential equations (PDE) for which only in
special cases analytical solutions can be found. Therefore, several techniques for an
approximate computation of the transient flow in the hydraulic system have been developed.
D'Souza and Oldenburger [D'Souza 1964] derived transmission line models which
considered also the viscosity and the compressibility of fluid passing through a circular
cross-sectional pipe in the laminar case neglecting the elasticity of the hydraulic pipe walls.
In 1972, Leonard and Goodson [Leonard 1972] developed a distributed parameter model
in frequency domain. This viscous laminar model which gives sufficiently accurate results
has a very compact representation in frequency domain [Manhartsgruber 2000] which in a
matrix form representation reads:
56
⎥⎦
⎤⎢⎣
⎡⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−
−=⎥
⎦
⎤⎢⎣
⎡)(ˆ)(ˆ
))(cosh()(
))(sinh())(sinh()())(cosh(
)(ˆ)(ˆ
ωω
ωγωωγ
ωγωωγ
ωω
jQjp
jjZ
jjjZj
jQjp
in
in
c
c
out
out
eq.3. 61
The meaning of the different variables and parameters are:
The characteristic impedance
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
r.ijJ
rijJ
r
E)j(Z
2
0
2eff
c
νω
νω
π
ρω
eq.3. 62
Where , are the Bessel functions of the first kind and 0J 2J r is the radius of the pipe.
The characteristic impedance Zc according to eq.3. 62 is not a function of the pipe length
as it was the case for the discrete parameter model, but it represents local impedance
combined of a capacitance, inductance, and resistance of the pipe cross section.
The propagation operator γ defined as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
rjjJ
rjjJ
c
Lj)i(
2
0
ν
ω
ν
ω
ωωγ eq.3. 63
accounts for the propagation of the input pressure through the transmission line while the
characteristic impedance controls the fluid flow [King 2006].
The two-port model eq.3. 61 can be brought into impedance, admittance, or mixed form,
depending on the requirements.
1- The impedance form:
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=⎥
⎦
⎤⎢⎣
⎡
)(ˆ)(ˆ
))(sinh())(cosh()(
))(sinh()(
))(sinh()(
))(sinh())(cosh()(
)(ˆ)(ˆ
ωω
ωγωγω
ωγω
ωγω
ωγωγω
ωω
jQjQ
jjjZ
jjZ
jjZ
jjjZ
jpjp
out
in
cc
cc
out
in eq.3. 64
57
2- The admittance form:
⎥⎦
⎤⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
)(ˆ)(ˆ
))(sinh()())(cosh(
))(sinh()(1
))(sinh()(1
))(sinh()())(cosh(
)(ˆ)(ˆ
ωω
ωγωωγ
ωγω
ωγωωγωωγ
ωω
jpjp
jjZi
jjZ
jjZjjZj
jQjQ
out
in
cc
cc
out
in
eq.3. 65
3- The first mixed form:
eq.3. 66
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=⎥
⎦
⎤⎢⎣
⎡
)(ˆ)(ˆ
))(cosh()())(sinh(
))(cosh(1
))(cosh(1
))(cosh())(sinh()(
)(ˆ)(ˆ
ωω
ωγωωγ
ωγ
ωγωγωγω
ωω
jpjQ
jjZj
j
jjjjZ
jQjp
out
in
c
c
out
in
4- The second mixed form:
⎥⎦
⎤⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
)(ˆ)(ˆ
))(cosh())(sinh()(
))(cosh(1
))(cosh(1
))(cosh()())(sinh(
)(ˆ)(ˆ
ωω
ωγωγω
ωγ
ωγωγωωγ
ωω
jQjp
jjjZ
j
jjjZj
jpjQ
out
in
c
c
out
in
eq.3. 67
These transfer function matrices of the transmission line method allow efficient
computations or rather accurate solutions of pressure and the flow rate whether from the
input or output port of the transmission line. It can be easily integrated to a large model of a
hydraulic system, provided the other elements are linear and are transferred into frequency
domain.
Since this transmission line model is of distributed parameter type it is no surprise that it
has infinitely many oscillations or modes.
3.3.2 Case of study
The model problem is modelled by the discrete parameter model as well as by a
distributed model and both models will be compared. The distributed parameter model of the
case of study is represented as a collection of pipes – modelled in a distributed parameter
model fashion - and throttles and a gas chamber being just discrete parameter models (see
Fig.3. 14).
58
A matrix equation representation reads:
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
0000000000
)cosh(Q)sinh(Qz
.
4D3D2D1D4C3C2C1C4B3B2B1B4A3A2A1A
yQPPQPQPQQPP
11A
11A1c
4E
4E
4A
3E
3E
2E
2E
2A
1E
1E
1A
γγ
or in simplified form uAx 1−=
eq.3. 68
The coefficients of the matrix equation (eq.3. 68) are
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
0)cosh(0
1-0z
)sinh(01-)cosh(
1A
2
1c
1
1
γ
γγ
, , ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
01-)sinh(z000000
2A
22c γ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
3A ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
4A ,
59
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
−
=)sinh(z)cosh(0
000
0z
)sinh(0
1B
33c3
2c
2
γγ
γ
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
00)sinh(z
1P
AC2
10
1-0)cosh(
2B
33c
av
vd
2
γρ
γ
,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
001000000
3B ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
4B ,
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡ −
=000000
)cosh(z
)sinh(0
1C
33c
3 γγ
,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
00000000)cosh(-
2C3γ
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−
−−
=
4c
44
444c
z)sinh()cosh(0
)cosh()sinh(z0010
3Cγ
γ
γγ ,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=
010001000
4C , , ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
1D
60
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
2D ,
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
ρρ 2av
2vd
2av
2vd
PAC
21-1-
PAC
21
000000
3D ,
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡ −
=
−
000
1-Vm
APPni-0
mA
-
Ai10
4D0
24
n1
0n
1+n
K2
4
4
ωω
ϖ
3.4 Results and discussions
As mentioned in Chapter 2, the main purpose of this thesis is to create a new design for an
accumulator employing an alternative metallic diaphragm instead of the usual elastomer
diaphragm.
Before presenting the results the purpose of these dynamic models should be stated:
An ideal accumulator which is based on a gas spring is just guided by the relations of the
gas chamber (eq.3. 51). In case of small amplitudes the pressure pulsation or flow rate
pulsations follow the equation of a capacitance; the only parameter is the hydraulic capacity
CH.
From the dynamic perspective the new accumulator should come reasonably close to this
ideal situation. The purpose of this section and the models developed is to quantify the role
of the other effects that may deteriorate the dynamic performance, in particular also the role
of the main design parameters. These other effects can be called parasitic, since they are
unwanted. They slip in due to design-, manufacturing-, material-, and assembly constraints.
The accumulator design has to keep them as small as possible or, at least, within tolerable
limits.
The diverse results constitute a parameter study. A reference case is varied with respect to
most of its parameters to analyze the sensitivity of the system with respect to these
parameters. Hence, it also could be seen a sensitivity analysis.
61
Fig.3. 14 The distributed parameter model of the case of study.
62
63
The nominal conditions are:
The excitation frequency (f) 100 Hz
Bulk modulus of the hydraulic oil (Efl) 1.6x109
N/m2
The kinematic oil viscosity at 40°C (ν) 46x10-6 m2/s
The all pipes diameters (d) 15 mm
The diameter of the accumulator (D) 52 mm
The length of the upstream pipe (the first horizontal tube)
(l1) 170 mm
The length of the downstream pipe (the second horizontal
tube) (l2) 1000 mm
The connecting accumulator pipe (the vertical pipe)
length (l3) 50 mm
The nominal accumulator volume (V0) 0.25 l
The sound velocity in the hydraulic oil (c) 1372 m/s
The input flow rate excitation (QA1) 60 l/min
The adiabatic exponent (n) 1.4
The linearized pressure of the output throttle valve (Pav) 100 bar
The linearized pressure of the throttling at the
accumulator entrance (Pav2) 30 bar
The initial gas pressure of the accumulator gas chamber
(P0) 5 bar
The linearized gas pressure (PNenn) 50 bar
The oil density (ρ) 850 kg/m3
The steel density (ρst) 7800 kg/m3
The thickness of the separator 50x10-3 mm
The opening area of the output throttle valve (Av) in % of
the downstream pipe 50%
The opening area of the output throttle valve (Av2) in % of 70%
the connecting pipe to the accumulator
the gas volume (Vg) in % of the total accumulator volume 80%
Table 3. 1 the case of study parameters
As can be seen from several results below, the discrete parameter SDOF pipe model often
has too low accuracy. Therefore, in addition, a two degrees of freedom discrete parameter
pipe model is used for the four pipe elements in this parameter study. They give results quite
similar to the distributed parameter model concerning maximum amplitudes but, of course,
higher eigenfrequencies deviate significantly. The purposes of presenting results from four
different pipe models is to find out of which order the significant pipe dynamics for the
behaviour of the whole system with respect to the accumulator effect is; in other words, how
many oscillation modes are significantly involved.
The minimum dimension models reduce computational costs. This is particularly
important, if such pipe models are part of a hydraulic system model with nonlinear effects,
since then time domain modelling is required and the computational effort becomes
significantly higher than for the frequency domain models of this analysis.
The system behaviour and the accumulator dynamic performance, respectively, are
assessed from the frequency response of pressure and flow rate signals at certain points.
These are: the outlet pressure of the second horizontal (downstream) pipe ( ) which, of
course, is the main performance criterion together with the flow rate (Q
2EP
E2) there.
See Fig.3. 12 and Fig.3. 14 for a schematic of the hydraulic system.
64
65
Influence of membrane inertia
This part of the parameter study should reveal the role of diaphragm inertia, or more
generally, of the separating elements inertia on the dynamic performance of the accumulator.
This should also provide an information basis for the design of the metal diaphragm by
clarifying if its inertia is an issue at all and has to be considered as a design constraint or
optimization criterion.
In the results presented in Fig.3. 15 the metallic diaphragm with thickness 50µm is
compared with the normal elastomer diaphragm with 4 mm thickness to compare their
dynamic responses.
The pressure-frequency and flow rate-frequency responses of the distributed and the
SDOF discrete parameter transmission line models for the normal and metallic diaphragm
did not match properly because of the low accuracy as mentioned before in Section 3.2.3.
Both, the distributed and the 2DOF discrete models have approximately the same amplitudes
of pressures and flow rates at the first resonance. It can be concluded that the difference
between the dynamic responses of a steel diaphragm with thickness of 50µm and a NBR
rubber diaphragm with 4mm thickness is negligible.
Influence of changing the accumulator charge gas pressure
The accumulator charge gas pressure plays an important role in the hydraulic circuit. The
standard system configuration with a small capacity accumulator of 0.25 l nominal volume
and nominal charge gas pressure 5 bar is compared with the same accumulator volume with
charge gas pressure 90 bar to study the accumulator effect on the hydraulic system
performance. The results of both, the distributed and the 2DOF discrete parameter models,
showed that the accumulator can well dampen the pressure pulsations, especially at higher
frequencies (see Fig.3. 16). In this context it must be noted that gas filled accumulators have
nonlinear characteristics; hence the eigenfrequencies influenced by the accumulator will shift
with the mean gas pressure.
Discrete parameter model Distributed parameter model
Fig.3. 15 The comparison of the exit ports pressure and flow rate transfer function amplitudes for the
normal elastomer diaphragm with thickness 4mm and the metallic one with thickness 50µm.
66
Discrete parameter model Distributed parameter model
Fig.3. 16 The influence of changing the accumulator charge gas pressure.
Influence of inlet throttling
One of the common problems of an accumulator for its high frequency attenuation
performance is the inlet throttling which resists the oil flow in and out the accumulator. This
throttling problem is modelled as a linearized throttle valve. Throttling is parameterized by
67
the throttle cross section. In the subsequent figures this is quantified by a percentage of the
inlet pipes cross sectional area.
Discrete parameter model Distributed parameter model
Fig.3. 17 The influence of the throttling on the new accumulator design (“A=70%” means the orifice
cross section is equal 70% of the connecting pipe cross sectional area, “no orifice” means there is no
throttling at the entrance of the accumulator, “A=0%” means no accumulator connected to the hydraulic
circuit)
68
Fig.3. 17 and Fig.3. 18 show the influence of throttling the entrance of the accumulator. A
throttling reduces the attenuation effect of the accumulator off the resonance areas but
dampens the pulsation in the resonance zones. From these results of the distributed parameter
model it can be concluded that a moderate throttling is not deteriorating the accumulator
performance and is even helpful to reduce the resonance peaks.
The results show also that the insertion of an accumulator with this configuration would
probably improve the hydraulic system performance at high frequencies.
Discrete parameter model (continue) Distributed parameter model (continue)
Fig.3. 18 The influence of the throttling on pressure and flow rate amplitudes at the accumulator input
port.
69
70
In the discrete parameter model with higher degrees of freedom the pressure and flow
rates amplitudes are almost similar to the distributed parameter model at the first resonance
frequency.
In the discrete parameter model, there are significant variations in the pressure and flow
rate amplitudes and that may result from the Poiseuille resistance model which accounts only
for steady state friction. It yields lower resistance values than the distributed parameter
model even at moderate frequencies and the more the higher the frequency. That can be
clearly seen in the resonance peak height.
Influence of the hydraulic system on the accumulator’s attenuation performance
studied by variations of lengths and diameters of the pipes of the hydraulic model.
Variation of upstream pipe
As shown in and Fig.3. 19 and Fig.3. 20, the upstream and downstream pipes lengths have
a significant effect on the dynamic behaviour of the hydraulic system. Since transmission
lines have their own dynamics with mostly little or moderate damping they may couple
heavily with the dynamics of the other system elements. The important conclusion for system
design aiming at low pulsation is that the system dynamics must be properly known, the
accumulator must be included in the system model, there is hardly a chance to reduce
pulsation amplitudes in a wide frequency range, and, hence, system tuning is a complex task.
Already this simple system model shows, why the often primary expectations of some
engineers in the industrial practice concerning the action of system modifications targeting
pulsation attenuation, may be difficult to fulfil.
Discrete parameter model Distributed parameter model
Fig.3. 19 The influence of the first horizontal (upstream) pipe length on the hydraulic system.
71
Variation of the downstream pipe
Discrete parameter model Distributed parameter model
Fig.3. 20 The influence of the second (downstream) horizontal pipe length on the hydraulic system.
72
The influence of the connection pipe length on the attenuation performance
In Fig.3. 21, the discrete parameter models show relatively low influence of the
connecting pipe length on the system behaviour in the low frequency range. However, the
distributed parameter model clearly indicated that a longer connection line deteriorates the
accumulator performance at higher frequencies. This difference between the discrete and
distributed model results from the missing higher eigenfrequencies of the discrete models.
Discrete parameter model Distributed parameter model
Fig.3. 21 The influence of the vertical pipe length on the hydraulic system.
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The influence of the pipe diameter
In this analysis, the pipes diameters are the same for all pipes and are varied in steps as
indicated in the figures. As shown in Fig.3. 22, the pressure and flow rate amplitudes of the
distributed and the discrete are quite similar.
By increasing the pipe diameter the pipe impedance 2eff r/EZ πρ= decreases which
intrinsically reduces the pressure pulsation. If a system is excited by a given flow rate
pulsation, reducing this impedance will reduce pressure fluctuation throughout the system.
The improvement goes inversely proportional with the pipe cross section area ( ). 2rπ
Discrete parameter model Distributed parameter model
Fig.3. 22 The influence of the pipe diameter on the hydraulic system..
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3.5 3D Finite Element acoustic models with frequency dependent friction
Fast changes of flow or pressure create waves in hydraulic systems. For small amplitudes
these waves follow the linear wave equation [Beranek 1996]. In Section 3.3 wave
propagation was limited to transmission lines exploiting spatially one dimensional models. In
this section a theory presented in [Scheidl 2009] is briefly recalled and applied to some
typical hydraulic configurations with relevance for fast response hydraulic accumulators.
This theory adopts acoustic finite elements and the frequency dependent boundary
impedance elements as available in several advanced finite element codes to model the effect
of fluid viscosity on wave propagation. This method operates in frequency domain. For the
evaluation and application of this theory the finite element code Abaqus [Abaqus 2009] was
applied. All subsequent statements related to the finite element part of modelling and
computations are referred to Abaqus as well.
Acoustic fields are strongly dependent on the conditions at the boundary of the acoustic
medium. The reactive acoustic boundary is represented as a thin layer of material placed
between acoustic media and rigid stationary wall, whose own motions are very small. This
thin layer of material provides a “reactive surface,” or impedance boundary condition, to the
acoustic medium. This boundary impedance specifies the relationship between the pressure
of an acoustic medium and the normal motion at the boundary. The reactive or the
impedance boundary condition at any point along the acoustic medium surface is governed
by
eq.3. 69 p
cp
kvu
ffff
11+== &&
eq.3. 69 can be written in Frequency domain:
eq.3. 70 p
ki
cv
fff ˆ)1(ˆ ϖ
+=
Where:
ϖ : The angular velocity.
fk
1 : The proportional coefficient between the pressure and the displacement normal to the
surface.
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fc1 : The proportional coefficient between the pressure and the velocity normal to the
surface.
These reactive acoustic boundaries can have a significant effect on the pressure
distribution in the acoustic medium. The coefficients and can be evaluated to [Scheidl
2009].
fk fc
eq.3. 71
flfflf E2k1and
E2c1
ϖνϖν
==
If no impedance, loads, or fluid-solid coupling are specified on the surface of an acoustic
mesh, the acceleration of that surface is assumed to be zero and this is equivalent to the
presence of a rigid wall at that boundary.
The acoustic medium itself is considered as a non-viscous fluid and it has only friction
(distributed impedance) at the dynamic boundary layer [Scheidl 2009], see Fig.3. 23.
Where:
p The acoustic pressure
p,p &&& The time derivatives of the acoustic pressure
fu The fluid particle displacement
ff vu =& The fluid particle velocity
fu&& The fluid particle acceleration
fc The fluid particle (element) damping coefficient
fk The fluid particle (element) stiffness
flE The bulk modulus of the acoustic medium
ν The kinematic viscosity of the acoustic medium
ϖ The angular velocity
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3.5.1 Acoustic finite element models of some hydraulic systems
In the sequel several cases are studied employing the viscid acoustic modelling technique
outlined above. Comparison is made with models employing the frequency domain
transmission line model as described in Section 3.2.
3.5.1.1 Test case straight pipe with pressure excitation
This case just demonstrates the equality of the finite element modelling technique with the
one dimensional transmission line model of Leonard [Leonard 1972], already used in
Section 3.2.
Test case:
A fluid filled pipe of diameter 52mm and length 180mm is excited with pulsating pressure
(amplitude 50 bar) at its inlet port and its other end is closed (flow rate Q≡0).
Abaqus acoustic model:
The modelling data are given in Annex 2. The pressure amplitude plot for a frequency of
101 Hz can be seen in Fig.3. 24. In Fig.3. 25 the finite element model results are compared
with the transmission line model in terms of the pressure amplitude at the pipe’s closed end.
There is a very good agreement of both models
Fig.3. 23 The dynamic boundary layer model of the acoustic medium.
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Fig.3. 24 The pressure contour plot for the closed pipe model with constant cross section.
Fig.3. 25 The pressure at the end of the closed pipe model– frequency relation
78
79
3.5.1.2 Accumulator in a transmission line with pressure rate excitation
The results showed that the resonance frequency for both 3D FE model and the one
dimensional transmission line model are identical in the low frequency range while at high
frequencies they are slightly different.
The pressure and the flow rate amplitudes for the 3D FE model in the first resonant
frequency are lower than in the one dimensional model due to the higher fluid damping
coefficient of the impedance at low frequency [eq.3. 71], see Fig.3. 31- Fig.3. 35.
In the FE model the radial velocity components at the transition zone of the resonator
(from the inductance to the capacitance pipe) constitute an extra portion of kinetic energy
that is missing in the one dimensional models. This explains why the input flow rate to the
downstream pipe in the one dimensional model is different to the 3D finite acoustic model
(see Fig.3. 32). In the high frequency range, less flow goes to the accumulator than in the one
dimensional transmission line model due to the fluid inertia which forces the flow to
continue in the axial direction and not making a strictly sharp bent, see Fig.3. 29, Fig.3. 30,
and Fig.3. 32. The 3D FE model shows that the first resonance frequency of the hydraulic
system is in the low frequency range (12 Hz). To shift up the lowest natural frequency, the
system should have small hydraulic inductivity by lowering the upstream pipe length or
increasing the pipe diameter assuming the hydraulic capacity is kept constant.
The results of the one dimensional distributed parameter and 3D finite acoustic models of
the input flow rate to the downstream pipe (see Fig.3. 32) are different because the mesh in
the transitional zone is non-symmetric which makes it difficult to obtain the average
magnitude acoustic velocity at this section of the downstream pipe.
Fig.3. 26 Mesh of the acoustic model.
Fig.3. 27 The pressure contour plot of the case of study at frequency 1 Hz.
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Fig.3. 28 The acoustic velocities contour plot of the case of study at frequency 1 Hz.
Fig.3. 29 Half section model for the acoustic velocities contour plot and the velocity resultant vector at
frequency 101 Hz
81
Fig.3. 30 Half section model for the acoustic velocities contour plot and the velocity resultant vector at
frequency 1000 Hz
Fig.3. 31 The output flow rate of the first horizontal (upstream) pipe – frequency relation.
82
Fig.3. 32 The input flow rate of the second horizontal (downstream) pipe – frequency relation.
Fig.3. 33 The output pressure of the second horizontal (downstream) pipe – frequency relation.
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Fig.3. 34 The input flow rate to the oil chamber– frequency relation.
Fig.3. 35 The pressure at the end of the gas chamber– frequency relation.
84
85
3.6 Conclusions
The dynamic response results demonstrate that the two degrees of freedom (2DOF)
discrete parameter pipe model gives reasonable results and quite similar to the distributed
parameter model concerning maximum amplitudes and eigenfrequencies in the investigated
frequency range.
The 2DOF discrete models results do not match with the distributed parameter model at
the high frequencies because of their low order that cannot represent higher order oscillation
modes.
The dimensions of the connection line have significant influence on the attenuation
performance of the accumulator system.
The influence of the diaphragm inertia is negligible compared with the oil inductivity in
the hydraulic system.
The distributed parameter model shows that throttling the accumulator entrance more than
50% relative to the reference case resists the flow to enter in or withdraw from the
accumulator and creates a slight pressure drop.
The one dimensional transmission line models of Leonard compare very well with the 3D
FE modelling techniques in the investigated frequency range.
4. Theoretical investigations of alternative accumulator concepts
In this Section different new accumulator concepts are evaluated with respect to their
dynamical and strength performance employing Finite Element techniques.
4.1. ’Diaphragm cap’ accumulator
The cap accumulator is a new hydraulic accumulator concept promising fast response. It is
similar to the normal diaphragm accumulator concerning its basic design. This accumulator
consists of oil and gas chambers separated with a metallic diaphragm. The upper and the
lower housings have special shapes to reduce the stresses acting on the metallic diaphragm to
achieve long working life, see Fig.4. 1. The lower part has central cylindrical bores instead of
a normal hydraulic inlet port (similar to the concepts proposed by [Mayer 1976] and [Onishi
2000]) in the diaphragm accumulator to let the fluid enter or leave the accumulator; the reason
of these many small bores is to prevent the metallic diaphragm from excessive load in
absence of a hydraulic pressure. The cap accumulator design is numerically simulated using
FEM models to obtain the hydraulic performance and the diaphragm stress-displacement
behaviour.
4.1.1. Diaphragm concept and design
The main object of the new accumulator design is to achieve:
• a simple geometric shape,
• maximum capacity,
• fast response,
• stresses below the fatigue limit to obtain long work life.
This subsection focuses on the metallic diaphragm to obtain its optimal shape. This shape
defines also the upper and lower inner contours of the housing.
The reference load acting on the diaphragm surface is a pressure with 10 bar. The outer
end of the diaphragm is kept fixed in the radial and axial directions. This is accomplished by
clamping the diaphragm between the two housing parts or by welding it to one or both parts.
The outer diameter is 60mm. A piston is connected to the lower part of the diaphragm and its
86
function is to prevent the diaphragm to withdraw outside the accumulator from the hydraulic
port when the oil pressure is eliminated (see Fig.4. 4).
The study concentrates on ultimately thin (thicknesses in the range 20-50µm) membranes
of high strength steel. Such material is offered by special steel manufacturers.
Fig.4. 1 A-Exploded view of the cap accumulator; B - sectional view of the cap accumulator assembly.
Sandvik offers its 11R51, EN 1.4310 austenitic stainless steel with down to 20 µm
thickness. It has high fatigue strength properties and corrosion resistance, (ultimate strength:
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2050 MPa, yield strength: 1975 MPa). These properties are bound to cold rolled condition
with only 0.5% elongation: Of course, in this state it can hardly be formed to the desired
shape. Sandvik experts recommended 12R11, an AISI (301) Austenitic stainless steel material
which is in soft state with a minimum ultimate strength of 800 MPa, good corrosion resistance
and good spring properties. The minimum available thickness of this material is 50µm.
Simulations of the diaphragm are run with thicknesses 20µm and 50µm.
The optimality criterion of is to realize a diaphragm which can resist highest deformation
(displacement volume) without exceeding allowable stress limits.
4.1.2. Nonlinear FE model of diaphragm deformation and stress state
Shell structures are applied in several engineering fields, especially in aerospace and
automotive applications, due to their excellent mechanical characteristics [Falzon 2008]. The
thickness of a shell is small compared with its other dimensions and may be the same
everywhere or it may vary from point to point. The shell of revolution - which is considered
in this work - is already applied in hydraulic accumulator technology as a vessel and as
separator element. Since in the latter application the bending stiffness of the shell is an
unwanted property such separator shells are often referred to as membranes.
This subsection reports about the development of the optimal shape for the thin metallic
diaphragm of the new design cap accumulator employing FE models for the stress and
displacement studies.
The diaphragm is simulated as an axisymmetric shell with a special contour. Its thickness
is constant. This, of course, neglects thickness variations due to the diaphragm manufacturing
by forming techniques, for instance hydroforming.
To allow for relatively large deformations, an ultimately thin shell is used. Such thin shells
(or membranes) have a very small bending stiffness and come close to pure membrane
behaviour1. In non stretched state unloaded membranes have infinitely many equilibrium
states. They tend to wrinkle, like can be observed for instance with thin aluminium foils for
household use. The only stabilizing parameter avoiding such wrinkling is bending stiffness.
Furthermore, relatively large deformations occur and need to be simulated. This brings
2 A membrane is a shell with a vanishing bending stiffness!
88
buckling phenomena into play. Both effects, the low bending stiffness and the tendency to
buckling, make the numerical simulation of large deformations of such membrane like
structures a formidable task.
To avoid numerical problems with buckling Abaqus offers an arc-length method (see
Fig.4. 3), designated ‘Riks-Wempner’ method. This method helps to overcome limit points,
i.e. local extremes of the deformation load path as shown in Fig.4.2. [ANSYS 2009].
Fig.4. 2 The load-displacement nonlinear behaviour.
Fig.4. 3 The arc-Length Convergence or Riks-Wempner method. [Wiki 2011]
But of course, it cannot avoid convergence problems resulting from a real bifurcation point
where more than one solution paths meet at one point in the load – deformation space.
89
Technically, the occurrence of wrinkled states is strongly undesirable because such states
cause strong bending stresses which most likely deteriorates lifetime significantly.
Unfortunately, many diaphragm variants seemingly show such buckling behaviour. In
these cases Abaqus was unable to compute a full deflection path, but stopped at some point
with the message that negative eigenvalues occurred. Even though such an error message is
no full proof of such bifurcation, it is at least a strong hint. Technically, such bifurcation to
some wrinkled state is absolutely undesirable. Hence, the occurrence of such a computational
problem in Abaqus was considered a reason to disqualify a certain diaphragm variant.
4.1.3. Diaphragm cap accumulator simulation results
The first version of the diaphragm has a cosine shape with an inverted cap in the centre and
a depth of 10mm. It is shown in Fig.4. 4. Abaqus aborted to complete the computations of
large deformation because it did not converge to a fully displaced solution. Abaqus stopped at
deformation = 2.6 mm, see Fig.4. 6, which is even far down the middle position. The max.
stress2 was 1150 MPa (see Fig.4. 5) at diaphragm side surface wrinkles. The deformation of
the cosine diaphragm shape is not axisymmetric due to using a non-symmetric mesh. The
results from this cosine shape computation give a real vision for the instability (the buckling
problem) of a very thin shell and show the challenge of computing such very flexible
structures with very large deformations.
The problems with the deep cosine shaped diaphragm led to think about a shallower shell
(a lower depth). The shallow shell diaphragm’s meridian curve is composed of an arc with
radius 175mm, bent downwards and an inverted cap in the centre, see Fig.4. 7. The total depth
is 4.5 mm. Abaqus was able to compute the full deflection of this shallow diaphragm up to 8
mm vertical deflection. Furthermore, this design shows acceptable von Mises stress of 1549
MPa. The highly reduced depth reduces also the hydraulic capacity of the accumulator. Fig.4.
8 and Fig.4. 9 show the van Mises stresses and the deformation in the final position.
2 Max. stress means maximum von Mises equivalent stress throughout the whole Section
90
Fig.4. 4 The construction of the diaphragm with cosine shape connected with piston
Fig.4. 5 The stress contour plot of the diaphragm with cosine shape.
91
Fig.4. 6 The displacement contour plot of the diaphragm with cosine shape.
Fig.4. 7 A section of the diaphragm with positive curvature shape.
92
Fig.4. 8 The stress contour plot of the diaphragm with positive curvature shape.
Fig.4. 9 The displacement contour plot of the diaphragm with positive curvature shape.
93
Alternatively, a shallow diaphragm with same depth but negative curvature of same radius
as before was investigated (see Fig.4. 10). The idea was to obtain higher accumulator
capacity. Results are shown in Fig.4. 11 and Fig.4. 12. The deformation of the diaphragm is
able to attend 10mm height. This design has a higher volumetric capacity but has also higher
stresses which exceed even the ultimate strength of the ultimate material (Sandvik’s 11R51).
Fig.4. 10 A section of the diaphragm with negative curvature shape.
Fig.4. 11 The stress contour plot of the diaphragm with negative curvature shape.
94
Fig.4. 12 The displacement contour plot of the diaphragm with negative curvature shape.
The next attempt to achieve higher accumulator capacity with lower stress values is to
combine positive and negative curvature zones with a cone according to Fig.4. 13. The depth
is 11 mm and the positive and negative radii are 95mm each. The results of the Abaqus
computation as given for an intermediate deformation are shown in Fig.4. 15 and Fig.4. 16,
show a kink in the cone shape with excessive stresses. Throughout the whole deformation
history the deformed diaphragm is composed of an outer inverted part and an inner
undeformed part both connected by the sharp kink mentioned before with the very high
bending curvature leading to the high bending stresses. This kink starts at the outermost
radius and progresses inwards with ongoing deformation. Fig.4. 14 sketches this deformation
history. This design is unfeasible because of the much too high stresses.
95
Fig.4. 13 The diaphragm with positive-cone-negative curvature shape.
Fig.4. 14 The deformation history of the positive-cone-negative shape.
96
Fig.4. 15 The stress contour plot of the diaphragm with positive-cone-negative curvature shape.
Fig.4. 16 The displacement contour plot of the diaphragm with positive-cone-negative curvature shape.
97
A next version directly connects a positively and negatively curved shell. The positive
curvature radius is 110mm. The negative one was optimized by several simulation runs
aiming to minimise max stress. The optimal ratio is found⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛== %8
R
Rr
positive
negative . The max depth of
the design is 4.1 mm (see Fig.4. 17). The results show the stress level is well below the yield
strength of the propsed material (Sandvik 11R51) without any kink. But the volumetric
capacity of this design is low. Abaqus results for von Mises stresses and vertical deformation
are given by Fig.4. 18 and Fig.4. 19.
Fig.4. 17 The shallow diaphragm shape with positive -negative curvature.
Fig.4. 18The stress contour plot of the shallow diaphragm shape with positive -negative curvature.
98
Fig.4. 19 The displacement contour plot of the shallow diaphragm shape with positive -negative
curvature.
A fourth diaphragm variant is equipped with three circumferential grooves with 2 mm
radius and 0.1 mm depth. These grooves are expected to add flexibility to allow for high
deflection without excessive stresses. The diaphragm has total depth of 4 mm, see Fig.4. 20.
The FE computation of this design encountered stability problems (Abaqus reported negative
eigenvalues). It is unlikely that the solutions bifurcating from the axisymmetric solution
constitute some wrinkled state since the grooves tend to enforce an axisymmetric state.
However, a different axisymmetric solution might bifurcate. The computed stresses are
moderate (see Fig.4. 21). But since such transition to a different state which cannot be
assessed properly due to computational problems constitutes an uncertainty this solution is not
considered further.
99
Fig.4. 20 The shallow diaphragm with grooves.
Fig.4. 21 The stress contour plot of the shallow diaphragm with grooves
100
Fig.4. 22 The displacement contour plot of the shallow diaphragm with grooves.
A further design aiming at fewer parts replaces the piston which supports the diaphragm at
the inlet port against the gas filling pressure when no oil is present. The lower housing’s port
consists of several small bores (2 mm diameter). The diaphragm cap is strengthened with
dimples placed at bores. In this way the thin diaphragm can resist the gas filling pressure. The
basic diaphragm shape is similar to the positive-negative curvature with the radius
ratio ⎟⎟⎠
⎞⎜⎜⎝
⎛== %8
positive
negative
RR
r , see Fig.4. 23.
The computation of the deformation with Abaqus worked without any problems. The
diaphragm deforms axisymmetrically until it reaches its inverse shape at the upper limit
without kinks. No instability was encountered (see Fig.4. 25). The stress levels are well below
the material yield strength (see Fig.4. 24). The nonlinearity behaviour of the diaphragm cap
due to the large deformation using the arc length method is shown in Fig.4. 26. The load
displacement curve in Fig.4. 26 shows that the maximum difference pressure which the
diaphragm needs to get deformed is only 4% of the nominal pressure which was set in the
Abaqus FE model. Since this nominal pressure was 10 bar, the required pressure is only 0.4
bar.
101
Fig.4. 23 The final shape of the diaphragm and the accumulator lower housing.
Fig.4. 24 The stress contour plot of the final shape of the diaphragm using Arc length method.
102
Fig.4. 25 The displacement contour plot of the final shape of the diaphragm using Arc length method.
Fig.4. 26 The load proportionality factor-displacement ratio relation of the final shape of the
diaphragm.
103
Resistance to gas pressure:
The dimples in the cap have to resist the gas filling pressure to exit from the oil bores of
the lower housing when no oil pressure is present. The corresponding strength problem is also
investigated by an Abaqus model. The applied gas pressure is 20 bar. This is modelled by
Abaqus contact elements without friction. This non-linear contact problem is simulated using
the arc-length method to avoid convergence problems.
The results showed that the stress levels in the diaphragm’s dimples are less than the yield
stress of the proposed high strength material.
The dimples strengthen the diaphragm. The regions around the outer dimples show higher
stresses (see Fig.4. 27) due to some complex deformation: these zones are pulled into the
direction of the dimples caused by the membrane stresses in the dimples.
4.1.4. Dynamical response behaviour
4.1.4.1. FE acoustic model and simulation results
Two FE simulated models were set-up and simulated: the first is an ideal cap accumulator
in which only gas and oil determine the dynamical properties; see Fig.4. 29. The second
model is adding the metallic diaphragm identical to the first model, to study its effect on the
acoustic results (see Fig.4. 28). The exciting ‘load’ is the oil pressure with 30 bar amplitude.
The gas chamber is filled with nitrogen at a pressure of 20 bar, and a temperature of 40°C.
These are the reference states for the linearization of the nonlinear state equations of the gas
for the acoustic simulation in the FE models.
The accumulator’s maximum displacement volume (nominal volume) is 6 cm3, with
corresponding diaphragm displacement of 8 mm. All cap accumulator dimensions are shown
in Annex 1.
104
Fig.4. 27 The stress contour plot of the cap accumulator.
105
Fig.4. 28 The cap accumulator model with the metallic diaphragm.
Fig.4. 29 The ideal cap accumulator model
As shown in Fig.4. 30 and Fig.4. 31, the FE acoustic results of both, the ideal cap
accumulator and the cap accumulator with the metallic diaphragm, have similar values of gas
pressure and oil flow rate responses.
106
Fig.4. 30the output flow rate of the oil chamber cap accumulator – frequency relation.
Fig.4. 31the gas pressure of the cap accumulator – frequency relation.
107
The coupled acoustic structural model has slightly more damping at the resonance
frequency than the ideal model and there is also a small frequency shift between the two
models due to the diaphragm’s inertia and elastic forces.
The pressure and acoustic velocity amplitudes of the gas and oil chamber of the ideal cap
accumulator are shown in Fig.4. 32 to Fig.4. 35 for a frequency of 101 Hz. This frequency
was selected because significant pressure variations in the nitrogen volume (gas chamber)
occur. The oil and the gas pressure amplitudes at the contact surface have the same values at
the oil chamber and gas chamber side, see Fig.4. 34 and Fig.4. 32. This physically obvious
fact just indicates that the modelling technique in Abaqus employing master and slave nodes
worked properly. The resultant acoustic velocities in the oil bores have slightly different
amplitudes and different directions where the bores enter the accumulator oil chamber (see
Fig.4. 36). The reason of this is the different geometrical situations of the oil and gas chamber
zones above each bore. The outer bores’ flows have also to supply the small wedge shaped oil
and gas areas in the outer radial areas, the centre bore’s flow is heading mainly in axial
direction.
Fig.4. 32 The pressure contour plot of the ideal cap accumulator gas chamber at frequency 101Hz.
108
Fig.4. 33 The acoustic velocity magnitude contour plot plot of the ideal cap accumulator in the gas
chamber at 101Hz.
Fig.4. 34 The pressure contour plot plot of the ideal cap accumulator in the oil chamber at 101Hz.
109
Fig.4. 35 The acoustic velocity magnitude contour plot plot of the ideal cap accumulator in the oil
chamber at 101Hz.
Fig.4. 36 The acoustic velocity resultant vector of the whole accumulator half section at 101 Hz.
110
From resonance peaks shown in some of the previous results one can conclude, that the
diaphragm cap accumulator has Helmholtz resonator properties with a resonance frequency of
425 Hz. To raise this resonance frequency the diameter of the oil bores or their number should
be enlarged to reduce their total hydraulic inductance but that makes the diaphragm stiffer and
decreases its life time. The diaphragm inertia has no measurable influence on the accumulator
performance.
4.2. Bellow type accumulator
Metal bellows are cylindrical vessels which consist of numerous annular shells with special
contour. The lower end of the metallic bellows is fixed with the accumulator lower housing,
while its upper end is attached to the piston, see Fig.4. 37. When the oil enters the oil chamber
it forces the piston to move upwards against the gas chamber. The metal bellows accumulator
has the advantages of being absolutely gas tight and of avoiding friction between its working
parts. The metal bellows are formed by pressing or hydro-forming processes.
Metal bellows have a low spring rate compared with the gas stiffness. The metallic bellows
can be produced from different kind of alloys depending on the hydraulic applications, for
instance, of stainless steel or brass, the latter having higher chemical resistance.
4.2.1. Bellow type accumulator simulation results
Different bellows constructions are simulated in this section to obtain the optimum bellows
design for long working life. The round, the weld and combined round-weld bellows types
with different diameters are simulated for a certain displacement (10 mm) to analyze the
stresses. All the simulated bellows types have the same surface area of approximately 0.1 m2,
and 400µm thickness. The applied load is a pressure of 30 bar acting on the inner side of the
bellow and on the lower side of the piston.
111
Fig.4. 37 The bellows accumulator type
• The round bellow type
Fig.4. 38 The stress contour plot of the round bellows type.
112
• The weld bellow type
Fig.4. 39 The stress contour plot of the weld bellows type.
• Combined round-weld bellow type, version I
Fig.4. 40 The stress contour plot plot of the first round-weld bellows type.
113
• Combined round-weld bellow type, version II
Fig.4. 41 The stress contour plot of the second round-weld bellows type.
The results showed that the round-weld bellow type, version II has lowest stresses. It must
be noted that the simulation did not consider the stress raising (residual stresses) and strength
reducing effects of welding. Both would shorten its fatigue life. Therefore, the combined
round-weld bellows types are proposed to avoid the last problem. Furthermore, round–weld
bellow types have smaller stresses than the weld bellow type (see Fig.4. 38 until Fig.4. 41).
They also might have the advantage of much lower production costs, since the welding
process is definitely very costly.
The author performed some acoustic analysis of the bellows accumulator. But the results
are not presented in this thesis because this accumulator was not considered for further
development since the industrial company HYDAC has already developed such an
accumulator.
114
4.3. Conclusions
The acoustic and the dynamic response results of a steel diaphragm with thickness of
50µm prove that the influence of the diaphragm inertia is negligible. To increase the first
resonance frequency (425 Hz), the oil bores diameters should be enlarged but then the
diaphragm becomes stiffer and the diaphragm’s working life time is most likely reduced.
The acoustic analysis of the cap accumulator showed also the resultant velocity vectors
coming into the oil chamber are quite the same for all the oil bores even though the inner
accumulator geometry is complex.
The stress and displacement analysis of the diaphragm cap accumulator using Abaqus CAE
showed that the diaphragm performs axisymmetric deformations until it reaches its inverse
shape at the upper limit without any kink or instability problem and that the stresses are below
the material yield strength.
The FE analysis of the diaphragm dimples - lower housing contact region showed that the
stress levels in the diaphragm’s dimples are less than the yield strength of the proposed high
strength material.
The FE results of the combined round-weld bellows types have the advantage of low stress
values. It could be used instead of the weld bellows type in the bellows accumulator to avoid
the strength reducing effects of welding.
115
5. Experimental investigation of diaphragm cap accumulator
5.1. Design of a prototype
In this subsection, the mechanical design of the diaphragm cap accumulator as derived
from the extensive simulations in section 4.2.1 is presented. Each of its components is shown.
The corresponding technical drawings are placed in Annex 1.
The main purpose of this prototype is to analyze
• if the proposed manufacturing of the diaphragm by a hydroforming process works
properly
• to run tests for an experimental verification or falsification of the basic design
The prototypal design is not adequate for series production. In the experimental testing
various trials and modifications had to be expected to finally achieve a reasonable realization.
Of course, for the cap accumulator the diaphragm is the critical component, because of its
ultimate thinness, the high stresses, the high strength material properties and the
hydroforming process. Particularly the latter was expected to be a particular challenge, since
experts of two companies (Kleiner Stanztechnik in Pforzheim, Mr. Großkopf, visited at 4th
April 2010, and Haerter in Koenigsbach-Stein (Baden-Wuerttemberg), Mr. Kaupert, visited at
20th April 2010) with a high expertise in stamping technology judged hydroforming of this
diaphragm to be not promising.
To enable frequent replacements of the diaphragm a detachable connection of upper and
lower housing is necessary. It was decided to select a screw fastening since this allows
creating high clamping forces to the diaphragm. To save costs, the upper and lower housing
parts should also serve the hydroforming process to a large extent.
The lower housing is divided into two parts: the intermediate part and lower housing. The
reason of this arrangement is to avoid twisting the diaphragm during the accumulator
assembly when both parts are screwed together. The intermediate part is fixed with 4 pins to
prevent the rotary motion of the part during the accumulator assembly and to maintain the
diaphragm in the right position (see Fig.5. 1). The lower housing has the same contour as the
diaphragm while the upper housing has the inverse shape of the diaphragm to limit its
deformation when high hydraulic pressures are occurring.
116
The lower housing has conical inlet bore to reduce flow losses. The upper housing can be
provided with an outer thread to insert it into a hydraulic block.
Fig.5. 1 The diaphragm cap accumulator assembly
5.2. Material selection for the diaphragm
The basic requirement on the diaphragm material is
• to enable high deformation
• to have high fatigue limit
• to be gas tight
• enable cheap manufacturing in high lot size production
From the material selections viewpoint the selected material must allow to be produced as
such very thin sheet (some tenths of microns) in an economic way. According to the today’s
available manufacturing technologies this means that
• this material must be formable, since only forming techniques allow such a production
at low cost.
117
• stainless steel material with excellent spring properties that in most cases fulfill
demands on corrosion resistance, mechanical strength and fatigue resistance [Sandvik
2011].
Other important requirements are:
• allow heat treatment for increasing its strength, if forming has to be done at soft
(annealed) state.
• be weldable to fix the diaphragm with the accumulator housing and, possibly, to make
it gas proof.
Not many vendors provide such material currently. One is Sandvik with its 12R11 (DIN
1.4310 or AISI 301). This is an austenitic stainless steel material which gets closest to the
above material requirements. It has excellent spring properties and it is available in
thicknesses down to 20 μm. 12R11 is actually a base grade for a whole family of ultimate
strength steels. The other grades of this family can be obtained by further treatment of 12R11.
Treatment reaches from annealing, over hardening to cold rolling. 12R11 has relatively low
ultimate strength of 800 MPa, but has sufficient elongation for the forming of the required
diaphragm shape. With annealing ultimate strength can be increased up to 1900 MPa. The
yield strength of this material family is generally 85% of the ultimate stress. Such yield
stresses are higher than the maximum stresses found in the stress analysis of the diaphragm in
Section 4.2.1. Unfortunately, nothing can be found on the endurance limit for 50µm
thickness.
The highest ultimate strength (2350 MPa) of this family has 11R51. This steel grade is
basically obtained from 12R11 by cold rolling and tempering at 425°C /4 h. 11R51 has also
superior qualities with respect to fatigue resistance and corrosion resistance. The tensile
strength of both steel grades is shown in Table 5.1. The usage of 11R51 for the diaphragm
requires a complex forming process. 11R51 in its final state has too low ductility to allow a
hydroforming or stamping. On the other hand, the already hydroformed diaphragm of soft
11R51 cannot be further formed to obtain the high strength properties since this would change
its geometry. Thus, the only way to get a high strength 11R51 final diaphragm is to apply a
very sophisticated forming process which provides both, the right amount of forming to get
the wanted strength properties and the right shape.
118
Sandvik grade Tensile strength [MPa]
12R11 800–1900
11R51 1700–2350
Table 5. 1 The tensile strength of Sandvik 12R11 and 11R51.
5.3. Diaphragm forming processes
5.3.1 Stamping
Stamping is a plastic deformation process in which the metal sheet blank is plastically
deformed between the tools (the punch and the die) to obtain the desired configuration.
Depending on the complexity of the formed part, one or more stages of forming processes are
required to form the sheet to desired final shape.
In a stamping operation, the sheet metal is formed against the die by the press or the punch
while the blank holder applies a predefined force to control the material flow into the die.
The blank holder force (BHF) when designed correctly can avoid failure by tearing and
wrinkling in the formed part.
Stamping operation is performed using either a single action or multi action press. In single
action press stamping (see Fig.5. 2) the sheet metal is initially clamped between the upper and
lower blank holders, a movement of the press attached to the top ram draws the sheet against
the die in presence of BHF acting on the sheet. [Palaniswamy 2007]
Fig.5. 2 The stamping process.
119
5.3.2 Hydroforming
Sheet hydroforming is widely applied in the field of sheet metal forming. The
hydroforming process is an alternative to the stamping process where either punch or the die
is replaced by hydraulic medium, which generates the pressure and forms the required part
[Palaniswamy 2007]. Sheet hydroforming is classified into two types SHF-P and SHF-D: In
Sheet Hydroforming with Punch (SHF-P), the hydraulic fluid replaces the punch while in the
Sheet Hydroforming with Die (SHF-D), the hydraulic fluid replaces the die (see Fig.5. 3).
In SHF-P and SHF-D, the quality of a formed part is determined by the amount of material
drawn into the die cavity during the forming process which is controlled by the applied blank
holder force. Absence of either punch or the die in the hydroforming process reduces the
tooling cost. Similar to the multiple stages stamping process, the formed part in SHF-P and
SHF-D process is subject to sequence forming operations to obtain the final part geometry.
Typical tools for sheet hydroforming consist of a punch or die, blank holders and a pressure
chamber [Palaniswamy 2007].
Sheet hydroforming has also other advantages as increased drawing ratio, better surface
quality, less spring back, minimizing thickness variations of the products, and reduced tooling
costs, especially for nonsymmetrical components that lead to possibility of manufacturing
complex sheet metal shape with low tooling effort.
Fig.5. 3 The hydroforming process.
120
5.4. Simulations of the diaphragm forming processes
In this section, FE simulations of the diaphragm forming process are reported. The aim is
to get quantitative information of the forming and to assess in this way its feasibility for the
production of the specially shaped diaphragm. This `virtual tryout` before tool construction
helps to avoid unpromising designs, in particular also tool designs.
Stamping and Hydroforming are two possible technologies to form a thin sheet metal. Both
of these forming technologies are numerically simulated here. The FE program Abaqus
which has been employed for stress before has also the capability to simulate large
deformation and is frequently applied for the simulation of forming processes.
The deformation of the tool (die or punch) is not taken into account because of the ultimate
thinness of the diaphragm sheet material with correspondingly low required stamping forces
or hydroforming pressures.
5.4.1 Simulation of stamping process
5.4.1.1 Finite element model of the stamping process
The full model of the diaphragm and the stamping tools are simulated using Abaqus CAE.
The finite element models for punch, die, and blank holder are constructed as deformable
models and can be also simulated as rigid models to save calculation time.
The simulation process runs in sequential steps to study the springback effect and the
residual stress in the formed diaphragm.
• Starting with the clamping step where the blank is clamped with the upper and lower
holder blanks,
• the stamping step where the punch forces the blank to form into the stamping die; in
this step the Riks method with nonlinear behaviour is applied,
• removing the punch step where the punch is returned to its initial position,
• and finally with the de-clamping step where the clamping force is released.
The high strength stainless steel material properties (Sandvik 12R11) with 50µm
thickness are used in Abaqus material definition module.
121
Poisson ratio 0.3
Modulus of elasticity 210 GPa
Density 7800 kg/m3
Table 5. 2 Sandvik 12R11 properties in elastic zone
In the material plastic region, the work hardening phenomenon is described by Hollomon's
equation [Wiki2 2011] αεσ pK=
Where:
σ : the applied stress on the material,
K : the strength coefficient is calculated with value of 456 MPa,
: the plastic strain, pε
α : the strain hardening exponent α =0.44 [Wiki1 2011].
Stress (MPa) Plastic strain (mm/mm)
1275 (yield strength) 0
1397.3 0.05
1440.9 0.1
1473.2 0.15
1500 (the ultimate strength) 0.2
Table 5. 3 Sandvik 12R11 properties in plastic zone.
The stress value of the proportional limit is set as 1275 MPa and the ultimate strength is
1500 MPa at a strain of up to 20%.
The blank upper part is fixed with the clamping holders in all spatial directions.
The blank is modeled by shell elements type S3 and S4R, in total 2759 elements. This is a
relatively coarse mesh for this analysis. However, since the main interest in this case is to
study the possibility to form the diaphragm in principle and to obtain the necessary clamping
to prevent the membrane from slip during the forming process such a coarse mesh is
acceptable.
122
In the simulation of the stamping, the mechanical interactions between surfaces of the
punch and the blank, and the surfaces of the blank and the die are modeled as frictional
contact with coefficient 0.25.
5.4.1.2 Finite element model of the hydroforming process
In the hydroforming, also the full model of the blank and the hydroforming die are
simulated.
The material properties and the mesh elements are the same as used in the simulation of the
stamping. The hydroforming is performed in one step in which the pressure is applied on the
upper blank surface as a ramp function in range from 0 to 200 bar in 100 seconds to form the
diaphragm in the desired shape and the upper part of the blank is fixed in the global
coordinates. The interaction between the blank and the hydroformed die is modeled as
frictional contact interaction with 0.25 friction coefficient.
5.4.1.3 Results of the stamping and hydroforming simulations
The results of the simulations of the stamping (see Fig.5. 4) showed that the stamping
process gives the desired diaphragm geometry. In the hydroforming process (see Fig.5. 5) the
main shape of the diaphragm can be produced but the outer dimples cannot be formed even
though the pressure reaches 200 bar (the pressure limit of the used pump in the hydraulic
laboratory of the Johannes Kepler University).
The maximum stress amplitude acting on the diaphragm in the stamping and the
hydroforming simulation are the same. The high stress levels are at the center and the middle
dimples region due to the complicated geometry at this zone.
The diagram pressure-diaphragm over nodal displacement showed that there is no need to
apply more than approximately 140 bar to fully hydroform the diaphragm’s main shape (see
Fig.5. 6). As shown in Fig.5. 6 the center dimple is getting in contact with the die at a pressure
of 15 bar and the formation of the middle dimples needs about 40 bar pressure. However, with
the indicated hydroforming pressures the actual nodal displacements of some characteristic
parts of the diaphragm did not fully reach the desired shape. This was also noticed in the
practically hydroformed diaphragm.
123
Fig.5. 4 The stress and the displacement contour plots of the stamping simulation.
124
Fig.5. 5 The stress and the displacement contour plots of the hydroforming simulation.
125
Fig.5. 6 The pressure-node displacement of the hydroformed diaphragm.
Springback effect
Springback is a defect occurring in nearly all sheet-metal forming processes; the material
has a tendency to partially return to its original shape because of the elastic recovery of the
material. This effect is influenced not only by the yield strength and the hardening, but also by
thickness, bend radius and bend angle. [Spring 2011]
In the application of the new accumulator design, the diaphragm material is of high
strength steel alloy which has high springback after the forming process. FE simulations are
used to predict the springback in the diaphragm after forming to investigate the possibility to
compensate the springback in the die design.
In Fig.5. 7, the simulation results show that the springback reaches 28% at the complex
geometry of the dimples’ area. The residual stresses after the forming process in the
diaphragm material reach 437 MPa at the edges of the dimples. This may be less a problem in
reality, since the dimple boundaries are filleted which reduces the strains there. In the
simulation it was not possible to model these fillets because this would have increased the
mesh size to levels beyond the capacity of the used computer.
126
5.5. Manufacturing of diaphragm cap (hydroforming)
Sheet hydroforming with a die (SHF-D) was selected as forming method to manufacture
the diaphragm. This selection was mainly motivated by cost reasons to save the punch. A
further advantage of hydroforming is the option to form the diaphragm directly in the
assembled accumulator, by clamping the blank between upper and lower housing parts and
hydroform it from the gas chamber side with an appropriate pressure. This would be the last
but one manufacturing step just before charging the nitrogen into the gas chamber and sealing
it.
For the prototypal manufacturing hydroforming can be realized with the housing
components of the accumulator prototype. Only an extra die (see Fig.5. 10) had to be
designed and manufactured to protect the diaphragm in the dimples’ zones against too high
pressure since the lower housing’s bores cannot provide such support. This die has basically
the same shape as the lower housing’s intermediate part but is deeper such that after
springback the diaphragm has the same contour as the lower housing. All the hydroforming
parts including the formed diaphragm have been manufactured in house.
The hydroforming circuit consists of a hydraulic pump which can provide pressures
beyond 200 bar, a proportional control valve which directs the fluid flow to/from the
hydroforming die and controls the fluid pressure to form the diaphragm in the desired shape,
see Fig.5. 8. A control system (type dSpace 1104) is used to control the displacement of
hydraulic pump and to record the results. The fluid pressure is applied as a ramp function
from 0 to 150 bar. The actual pressure is fed back to the controller by a pressure sensor
mounted at the high pressure line coming from the variable displacement pump.
The diaphragm should be handled carefully due to its ultimate thinness of on only 50 μm.
As shown in Fig.5. 9, the formed diaphragm does not need any further modification after
hydroforming. As already found in the hydroforming simulation, only the central and the
middle raw dimples are formed. The outer row is influenced by the strong radial curvature of
the diaphragm which stiffens considerably and prevents the desired forming of dimples.
An important matter in this hydroforming process is a sufficient clamping force to avoid
the diaphragm slipping into the die during the hydroforming which in turn most likely leads to
wrinkles.
127
The radial force at each node of the upper end of the diaphragm is presented in Fig.5. 11.
The simulation of the diaphragm forming was used to calculate the required clamping force. It
was magnified by some safety factor to compensate for inaccuracies of the friction modulus
and some small unevenness of the components due to manufacturing tolerances.
Fig.5. 7 Springback effect of the diaphragm cap
128
Fig.5. 8 The hydraulic circuit of the hydroforming process.
5.6. Test set-up
For the testing of some performance criteria of the new accumulator design a test rig had to
be established.
The test rig which is shown in Fig.5. 12 consists of:
- Variable displacement pump which is working in a pressure control mode; the desired
pressure could be remotely adjusted
- Directional control valve (Hoerbiger MSV 32) is a direct solenoid actuated 3/2 poppet
valve. It has a flow capacity of 10 l/min at pressure loss of 23 bar and a maximum
operating pressure of 350 bar.
- The oil used in the test rig is a Shell Tellus Oil S 32. This is a highly refined oil with a
density of 872 kg/m3 at 15°C and a kinematic viscosity of 32 mm2/s at 40°C.
- Oil pressure sensor which is mounted at the entrance of the accumulator oil chamber
to monitor the oil pressure
- Gas pressure sensor which is fixed at the gas charging point to indicate the gas
pressure in the accumulator gas chamber. Both oil and gas pressure sensors are PDCR
129
4060. PDCR 4060 pressure sensor is based on micro-machined silicon diaphragm
technology with sensitivity ±0.027 MPa and with accuracy 0.04%, the sensor pressure
range is from 70 mbar to 70 bar.
Fig.5. 9 The real parts of the accumulator after manufacturing.
130
Fig.5. 10 The hydroforming die
Fig.5. 11 The radial force at each node of the diaphragm upper part.
- Graduated glass with sub-scale 1 cm3 to measure the quantity of the oil stored in the
accumulator.
The power supply can deliver hydraulic oil within the pressure range 20 to more than 200
bar and to obtain fluid pressure less than the minimum pressure limit a throttle valve is used
in the hydraulic circuit.
5.6.1 Test results
Two types of tests were performed:
131
1. static tests of the accumulator volumetric stiffness (capacitance) and
2. fatigue tests
Before starting a test, the hydraulic circuit is flushed to expel most of the air bubbles from
the hydraulic fluid in order not to falsify the results by this entrapped air. The accumulator is
pre-charged with nitrogen at 20 bar pressure. All the experimental works was done using the
diaphragm made of soft state material (Sandvik 12R11).
Fig.5. 12 The accumulator test rig
5.6.1.1 Static test
132
In this test, the accumulator is evaluated with respect to its static capacity. The capacity
stems from gas compressibility and is represented by pressure volume diagram. The
behaviour of gas spring accumulators is usually limited by the two extremes: isothermal and
adiabatic behaviour.
The whole test is carried out in several cycles. When the directional control valve (7) is
switched on and the handle valve (6) is closed, the fluid flows from the pump to the
accumulator (2) until the pressure in the system reaches the set values of the pump pressure
control system. When the directional control valve is switched off and the handle valve (6) is
opened the oil stored in the accumulator in the previous filling phase is expelled to the
graduated glass. Here the amount of oil stored by the accumulator for the given system
pressure can be measured. This cycle is repeated for different system pressure values. Each
pressure – amount of stored oil volume is entered in a pressure volume diagram. To average
out some measurement errors 10 measurement cycles are done for each system pressure level.
The obtained values in the gas volume pressure diagram are shown in Fig.5. 13. They are
between the calculated adiabatic and isothermal curves of an ideal accumulator of this volume
size.
The values are obviously considerably scattered. This has the following reasons:
1. The pump pressure control was not very accurate. The final pressure was corrected by
adjusting the set point of the pump pressure control system until the oil pressure
measured in the experimental set-up at the pressure sensor (3) was fluctuating around
the desired value. This actual fluctuation causes some scattering.
2. The measurement of the stored amount of oil is also subject to some errors. The oil
surface exhibits a considerable meniscus which makes an exact reading of the
effective height of the fluid in the graduated glass impossible. The sub-scale of the
graduation was 1 cm3 and this can be also guessed as accuracy limit of the reading. A
further inaccuracy may result from the emptying of the glass after each measurement
series. The high oil viscosity makes this a very retarded process which does not come
to a well defined end in reasonable time. Thus, the amount of remaining liquid at the
glass bottom may have differed from case to case.
133
Fig.5. 13 The gas volume – gas pressure relation.
5.6.1.2 Fatigue test
The purpose of the fatigue test is to get some first experimental assessment of the
diaphragm’s strength properties. In this test, the directional control valve is switched on/off
with a trigger box to apply a cyclic load with certain pressure amplitude and frequency until
the diaphragm is damaged. The operation frequency was 4 Hz. A schematic of this test series
is shown in Fig.5. 14.
After the damage the accumulator was disassembled to visually check the diaphragm
condition. The measured pressure-time curves are given by Fig.5. 15.
These curves show that the new diaphragm accumulator has roughly the expected
performance. There is some small phase shift between the oil and the gas pressure due to the
throttling effect of the oil bores in the intermediate part see Fig.5. 9.
After 11000 cycles the diaphragm was damaged by a small crack. This was observed from
the pressure signals since then the nitrogen diffused quickly into the oil and the accumulator
lost its hydraulic capacitance. The diaphragm showed a plastic deformation (see Fig.5. 16)
which had to be expected because the material is in the soft state [12R11] and the calculated
elastic stresses exceed the yield strength. Thus, considerable low cycle fatigue must be
expected.
134
Fig.5. 14 The hydraulic circuit of the fatigue test.
Fig.5. 15 The pressure – time relation.
135
Fig.5. 16 The diaphragm shape before and after the fatigue test.
5.7. Conclusions
The experimental work of the cap accumulator brought up a further problem of such
ultimately thin diaphragms: they are very sensitive to chips or solid particles when such are
placed between lower housing and the diaphragm when there is no oil pressure. These
particles are impressed into the diaphragm by the gas pressure. The resulting dimples reduce
the diaphragm flexibility which in turn means higher stresses when the diaphragm gets
deformed. Therefore, before assembling the accumulator every part should be cleaned to
prevent any such particles and the oil should be well filtered.
The hydroforming process has low accuracy to form a complicated geometry and needs
high pressures to form the sharp corners (see the relation between the hydroform pressure and
the nodal displacement at some points of the diaphragm Fig.5. 6). The most important
measure for a proper hydroforming process is a sufficient clamping force to avoid the
diaphragm slipping into the die.
The springback of the high strength steel material must be taken into account. It can be
investigated by FE simulations prior to manufacturing to save costly experimental trial and
136
error procedures. At some regions of the diaphragm under study the computed springback
values reach 28% as presented in section 5.4.1.3.
The material selection and its heat treatment possibility are the basic steps before FE
investigation and accumulator manufacturing.
Sandvik 12R11 has good ability for forming but it has only moderate strength. Therefore, it
cannot replace Sandvik 11R51 for this extreme application.
Sandvik 11R51 could be the right material to be utilized as a diaphragm due to its fatigue
properties but it has only 0.5% elongation limit which does not allow its forming.
Unfortunately, there is no way to upgrade Sandvik 12R11 after forming the requested
shape to obtain 11R51, since this is only possible with a cold forming process which
inevitably changes shape. The only way would be, to use one cold forming process to get
both, the intended final shape and the required stretching of the material to get 11R51 strength
properties from the 12R11 blank.
In the tested design no O-ring between diaphragm and upper housing was placed.
Nevertheless, the gas chamber was tight. This is a promising result indicating the possibility
of achieving a fully gas proof accumulator just with metal components and without any
elastomer sealing element.
The throttling region in the accumulator intermediate part is not deteriorating the
accumulator performance as expected before the manufacturing process.
137
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141
142
Eidesstattliche Erklärung
Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig
und ohne fremde Hilfe verfasst, andere als die angegebenen
Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen
Stellen als solche kenntlich gemacht habe.
Author’s signature: Mohamed Ez ElDin
Linz, März 2011
143
Curriculum vitae
PERSONAL DATA
Name: Mohamed Mohamed Ez ElDin
Date of Birth: 02. September 1977
Born in Cairo, Egypt
Email: [email protected]
EDUCATION
- 1996-2000 BSc in Automotive and Tractors Department in the Faculty of Engineering
at Helwan University, Cairo, Egypt
- 1996 Mechanical drawing course
- 1997 Production processes course
2001-2005 MSc in Automotive and Tractors Department at the same university
- 1998 -2000 Auto service and repair course in BMW and Nil of automotive service
company, Cairo, Egypt
- 2001-2002 MSc preparatory courses at the same university
- 2006 PhD preparatory courses at the same university
- 2007-2009 PhD study at Johannes Kepler University Linz, Austria.
EMPLOYMENT
- October 2000- Mai 2005 Demonstrator in Automotive and Tractors
Department in the Faculty of Engineering at Helwan
University, Cairo, Egypt
- September 2005 – October 2007 Assistant lecturer at the same university
- 2000-2007 Assistant-Supervisor for several bachelor graduated
projects in automotive field such as hybrid and
electric vehicles, suspension, front and 4 wheels
steering, engine dynamic performance, brake systems
and automatic transmissions, electronic ignition.
- 2004-2005 Teaching several courses in automotive field for
144
technical schools
PUBLICATIONS
- Rudolf Scheidl, Bernhard Manhartsgruber, Mohamed Mohamed Ez El Din, "Finite
Element Analysis of 3D Viscid Periodic Wave Propagation in Hydraulic Systems",
International Journal of Fluid Power, Vol. 10, Nummer 1, 3-2009, ISSN: 1439-9776.
Annex 1
Prototype engineering drawings
A1. The diaphragm cap
The diaphragm cap construction.
145
A2. The intermediate part
The intermediate part construction.
146
A3. The upper housing
The upper housing construction.
147
A4. The lower housing
The lower housing construction.
148
Annex 2
A.2.1 Abaqus input files
A.2.1.1 Abaqus input file for acoustic analysis of closed straight pipe model
*Heading ** Job name: Acoustic analysis of closed straight pipe model with 180mm length and 52mm diameter of Section 3.5.1.1 ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** PARTS *Part, name=oil **Define the nodes coordinates of the mesh *Node 1, 0.0251140706, -0.0140000004, 0.00672929501 2, 0.0225166604, -0.0140000004, 0.0130000003 3, 0.018384777, -0.0140000004, 0.018384777 ... 2239, 0.00719966646, 0.165999994, -0.0015414753 2240, -0.00177428278, 0.165999994, -0.000468688464 2241, -0.00143424107, 0.165999994, 0.00376663706 **Define the mesh element type: Acoustic 3D element with 8 nodes used to model wave propagation *Element, type=AC3D8 1, 7, 6, 28, 71, 90, 89, 111, 154 2, 26, 5, 4, 25, 109, 88, 87, 108 1818, 2151, 2154, 2126, 2123, 2234, 2237, 2209, 2206 1819, 2135, 2130, 2129, 2153, 2218, 2213, 2212, 2236 1820, 2157, 2138, 2137, 2158, 2240, 2221, 2220, 2241 **Define node set and element sets *Nset, nset=Set2, internal, generate 1, 2241, 1 *Elset, elset=Set2, internal, generate 1, 1820, 1 *Nset, nset=pressure_inlet, generate 1, 83, 1 *Elset, elset=pressure_inlet, generate 1, 70, 1 **Define section properties as a homogeneous solid section and its material properties ** Section: oil *Solid Section, elset=Set2, material=oil *End Part ** ASSEMBLY *Assembly, name=Assembly *Instance, name=oil-1, part=oil *End Instance **Define the node and element sets of the boundary conditions and the viscous boundary layers *Nset, nset=Set10, internal, instance=oil-1, generate 1, 83, 1 *Elset, elset=Set10, internal, instance=oil-1, generate 1, 70, 1 *Surface, type=ELEMENT, name=_PickedSurf8, internal __PickedSurf8_S2, S2 __PickedSurf8_S3, S3 __PickedSurf8_S4, S4
149
__PickedSurf8_S6, S6 __PickedSurf8_S5, S5 *End Assembly ** MATERIALS **Define the hydraulic oil properties as its density and Bulk modulus *Material, name=oil *Acoustic Medium 1.6e+09, *Density 850., ** INTERACTION PROPERTIES **Define the acoustic boundary conditions by which the viscous boundary layer is modelled ** State the oil admittance values **The first column is the values of the real part 1/cf **The second column is the values of the imaginary part 1/kf
**The third column is the frequency ω *Impedance Property, name=admittance_oil 1.2e-12, 7.51e-12, 1. 1.69e-13, 5.31e-11, 50. 2.42e-14, 3.72e-10, 2450. 2.39e-14, 3.76e-10, 2500. ** PHYSICAL CONSTANTS *Acoustic Wave Formulation ** STEP: acoustic *Step, name=acoustic, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 1., 2500., 250, 1. ** BOUNDARY CONDITIONS **Define the oil pressure amplitude 50 bar (the real part) ** Name: p_in Type: Acoustic pressure *Boundary, load case=1 Set10, 8, 8, 5e+06 *Boundary, load case=2 Set10, 8, 8 ** INTERACTIONS ** Interaction: oil *Simpedance, property=admittance_oil _PickedSurf8 ** OUTPUT REQUESTS ** FIELD OUTPUT: F-Output-1 **Define the desired output: the pressure And the acoustic velocity *Output, field *Node Output POR, *Element Output, directions=YES ACV, ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step
A.2.1.2 Acoustic analysis of ideal accumulator connected with transmission line
*Heading ** Job name: Acoustic analysis of ideal accumulator connected with transmission line, see Section 3.5.1.2. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** PARTS *Part, name=Nitrogen gas
150
*Node 1, 0.0253160819, 0., 0.00592419133 2, 0.02234881, 0., 0.0132864853 ... 35531, 0.00720497966, 0.0920000002, -0.0017008516 35532, 0.00240557059, 0.0920000002, 0.00116190163 *Element, type=AC3D8 1, 145, 97, 36, 35, 901, 853, 792, 791 2, 99, 642, 39, 38, 855, 1398, 795, 794 ... 32843, 34613, 34605, 34628, 34596, 35369, 35361, 35384, 35352 32844, 34539, 34774, 34770, 34769, 35295, 35530, 35526, 35525 *Nset, nset=Set6, internal, generate 1, 35532, 1 *Elset, elset=Set6, internal, generate 1, 32844, 1 ** Section: Nitrogen gas *Solid Section, elset=Set6, material=gas *End Part *Part, name=oil *Node 1, 0.407499999, 0.0149999997, 0. 2, 0.407499999, 0.0679999962, 0. ... 60622, 0.414498657, -0.00492892321, 0.00289387745 60623, 0.398189098, -0.00194143027, 0.00588307763 *Element, type=AC3D8 1, 1000, 1, 12, 961, 17650, 35, 1108, 17611 2, 34, 1, 1000, 959, 1130, 35, 17650, 17609 3, 961, 12, 13, 962, 17611, 1108, 1109, 17612 57132, 60439, 59949, 17410, 60405 57133, 59757, 59944, 60489, 60574 *Nset, nset=Set12, internal, generate 1, 60623, 1 *Elset, elset=Set12, internal, generate 1... 49955 ** Section: oil *Solid Section, elset=Set12, material=oil *End Part ** ASSEMBLY *Assembly, name=Assembly *Instance, name= Nitrogen gas-1, part=Nitrogen gas 0.4, 0.091, 0. *End Instance *Instance, name=oil_system-1, part=oil *End Instance *Elset, elset=__T0_oil_system-1_M_S4, internal, instance=oil_system-1 52165, 54150, 54861, 54974, 55439, 56337, 56355 *Surface, type=ELEMENT, name=_T0_oil_system-1_S, internal __T0_oil_system-1_S_S1, S1 *Tie, name=_T0_oil_system-1, position tolerance=0.0212132 _T0_oil_system-1_S, _T0_oil_system-1_M *Surface, type=ELEMENT, name=_T1_oil_system-1_S, internal __T1_oil_system-1_S_S1, S1 *Tie, name=_T1_oil_system-1, position tolerance=0.0212132 _T1_oil_system-1_S, _T1_oil_system-1_M *Surface, type=ELEMENT, name=_T2_oil_system-1_S, internal __T2_oil_system-1_S_S1, S1 *Tie, name=_T2_oil_system-1, position tolerance=0.0212132 _T2_oil_system-1_S, _T2_oil_system-1_M
151
** Constraint: oil - gas_accumulator *Tie, name="oil - gas_accumulator", adjust=yes, type=SURFACE TO SURFACE, no thickness accumulator_gas-1."tie gas - oil_accumulator", oil_system-1.oil_gas_tie *End Assembly ** MATERIALS *Material, name=gas *Acoustic Medium 7e+06, *Density 55.66, *Material, name=oil *Acoustic Medium 1.6e+09, *Density 850., ** INTERACTION PROPERTIES *Impedance Property, name=admittance_oil 1.196e-12, 7.5e-12, 1. 8.46e-13, 1.06e-11, 2. ... 3.8e-14, 2.375e-10, 999. 3.8e-14, 2.376e-10, 1000. *Impedance Property, name=gas_N2 2.416e-11, 1.52e-10, 1. 1.708e-11, 2.15e-10, 2. ... 7.6e-13, 4.797e-09, 999. 7.6e-13, 4.8e-09, 1000. ** PHYSICAL CONSTANTS *Acoustic Wave Formulation ** STEP: acoustic *Step, name=acoustic, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 1., 1000., 200, 1. ** BOUNDARY CONDITIONS ** Name: oil_pressure Type: Acoustic pressure *Boundary, load case=1 oil_system-1.pressure_in, 8, 8, 5e+06 *Boundary, load case=2 oil_system-1.pressure_in, 8, 8 ** INTERACTIONS ** Interaction: admittance_gas *Simpedance, property=gas_N2 accumulator_gas-1."interaction accumul_gas" ** Interaction: oil_system *Simpedance, property=admittance_oil oil_system-1.oil_impedance ** OUTPUT REQUESTS ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output POR, *Element Output, directions=YES ACV, ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step
152
A.2.1.3 Acoustic analysis of the ideal diaphragm cap accumulator
*Heading ** Job name: Acoustic analysis of the ideal “diaphragm cap accumulator” see Section 4.3.2. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** PARTS **Define the nodes coordinates of Nitrogen gas in the accumulator gas chamber *Part, name=N2 *Node 1, -6.05403613e-07, -0.00136087986, -0.000582084875 2, -6.05403613e-07, -0.00143569161, 0. ... 4090, -0.00644935062, 0.00247915299, -0.00425915467 4091, -0.00738840736, 0.00241698255, -0.00304517127 ** Define the element type of Nitrogen gas: Acoustic 3D element with 8 nodes *Element, type=AC3D8 1, 38, 464, 36, 1, 677, 685, 129, 128 2, 36, 464, 37, 2, 129, 685, 102, 14 ... 2031, 443, 444, 4091, 4090, 901, 902, 1136, 1135 2032, 444, 34, 3908, 4091, 902, 106, 953, 1136 *Nset, nset=Set9, internal, generate 1, 4091, 1 *Elset, elset=Set9, internal, generate 1, 2032, 1 ** Section: N2 *Solid Section, elset=Set9, material=N2 *End Part ** Define the nodes coordinates of hydraulic oil in the accumulator oil chamber *Part, name=oil_total *Node 1, -0.00185871043, -0.00449999981, -0.000660450722 2, -0.00185871043, -0.00836099964, -0.000660450722 ... 9386, -0.0027775683, -0.00230968976, 0.0104190316 9387, -0.00419776607, -0.00258428906, -0.00861875061 ** Define the element type of oil: Acoustic 3D element with 6 and 4 nodes respectively *Element, type=AC3D6 1, 7424, 1474, 1475, 1430, 110, 111 2, 7423, 1476, 1477, 1429, 112, 113 ... 1531, 2697, 505, 506, 2698, 8185, 2738, 2739, 8186 1532, 508, 2690, 2700, 507, 2741, 8178, 8188, 2740 *Element, type=AC3D4 1533, 8190, 8191, 8192, 8193 1534, 8190, 8194, 2421, 8195 .. 25982, 6884, 6296, 6885, 6900 25983, 7262, 7107, 6532, 6822 *Nset, nset=oil_pressure 2, 6, 8, 10, 14, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42 ... 2699, 2700, 2701 *Elset, elset=oil_pressure 7, 8, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88 ... 1530, 1531, 1532 ** Section: oil_total *Solid Section, elset=Set5, material=oil *End Part ** ASSEMBLY
153
*Assembly, name=Assembly *Instance, name=N2-1, part=N2 *End Instance *Instance, name=oil_total-1, part=oil_total *End Instance **Define the constraints between the nodes of the gas and the hydraulic oil in the accumulator *Tie, name=_T20_oil_total-1, position tolerance=0.00282674 _T20_oil_total-1_S, _T20_oil_total-1_M ** Constraint: oil-N2-tie *Tie, name=oil-N2-tie, adjust=yes, type=SURFACE TO SURFACE, no thickness N2-1.N2-oil-tie, oil_total-1.oil-N2-tie *End Assembly ** MATERIALS **Define the Nitrogen properties as its density and Bulk modulus at temperature 40°C and pressure 20 bar *Material, name=N2 *Acoustic Medium 2.8e+06, *Density 22.264, **Define the hydraulic oil properties as its density and Bulk modulus *Material, name=oil *Acoustic Medium 1.6e+09, *Density 850., ** INTERACTION PROPERTIES *Impedance Property, name=N2 9.154e-11, 5.8e-10, 1. 3.737e-11, 1.41e-09, 6. ... 2.91e-12, 1.811e-08, 991. 2.9e-12, 1.815e-08, 996. *Impedance Property, name=oil 1.196e-12, 7.5e-12, 1. 4.88e-13, 1.84e-11, 6. ... 3.8e-14, 2.365e-10, 991. 3.8e-14, 2.371e-10, 996. ** PHYSICAL CONSTANTS *Acoustic Wave Formulation ** STEP: accumulator *Step, name=accumulator, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 1., 1000., 200, 1. ** BOUNDARY CONDITIONS ** Name: oil_pressure Type: Acoustic pressure *Boundary, load case=1 oil_total-1.oil_pressure, 8, 8, 3e+06 *Boundary, load case=2 oil_total-1.oil_pressure, 8, 8 ** INTERACTIONS ** Interaction: N2_impedance *Simpedance, property=N2 N2-1.N2_impedance ** Interaction: oil_impedance *Simpedance, property=oil oil_total-1.oil_impedance ** OUTPUT REQUESTS ** FIELD OUTPUT: F-Output-1 *Output, field
154
*Node Output POR, *Element Output, directions=YES ACV, ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step
A.2.1.4 Nonlinear behaviour of the Diaphragm cap with dimples
*Heading ** Job name: non linear behaviour of the “Diaphragm-cap with dimples” using Riks method, see Section 4.2.1. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** PARTS *Part, name=central dimple *Node 1, 0., -0.00325000007, 0. 2, -0.000103742408, -0.00325190858, 0. ... 759, -0.00100310007, -0.0030465608, 9.15962664e-05 760, -0.00109544257, -0.00300000003, 0.000100028352 **the dimple element type: Shell element with 4 nodes *Element, type=S4R 1, 2, 3, 14, 13 2, 3, 4, 15, 14 ... 758, 1, 739, 750 759, 1, 750, 2 *Nset, nset=Set8, internal, generate 1, 760, 1 *Elset, elset=Set8, internal, generate 1, 759, 1 **Define the Diaphragm-cap section with 50x10-6m thickness ** Section: central dimple *Shell Section, elset=Set8, material=steel 5e-05, 5 *End Part *Part, name= dimple_first_row *Node 1, 0., -0.00325000007, 0. 2, -0.000102999009, -0.00326028909, 0. ... 1066, -0.00120461872, -0.00304775732, 9.24840351e-05 1067, -0.00129618554, -0.00300000003, 9.9514029e-05 *Element, type=S4R 1, 2, 3, 16, 15 2, 3, 4, 17, 16 ... 1065, 1, 1042, 1055 1066, 1, 1055, 2 *Nset, nset=Set9, internal, generate 1, 1067, 1 *Elset, elset=Set9, internal, generate 1, 1066, 1 ** Section: dimple_first_row *Shell Section, elset=Set9, material=steel 5e-05, 5 *End Part
155
*Part, name=diaphragm *Node 1, -0.00601515174, -0.00387023459, 0. 2, -0.00601542229, -0.00387027697, 1.90076062e-05 ... 3568, -0.00852377992, -0.00336264702, 0.000406037667 3569, -0.0075850077, -0.00365063944, 0.000361318438 **the Diaphragm-cap element type: Shell element with 3 nodes *Element, type=S3 1, 862, 61, 753 2, 754, 60, 8 ... 3823, 36, 724, 26, 35 3824, 35, 26, 25, 7 *Nset, nset=BC_diaphragm_side 8,..., 703 704, 705, 706, 707 *Elset, elset=BC_diaphragm_side 1298, 1299, ..., 3805 ** Section: diaphragm *Shell Section, elset=50, material=steel 5e-05, 5 *End Part ** ASSEMBLY *Assembly, name=Assembly *Instance, name=diaph-1, part=diaphragm *End Instance **Define the constraint of the dimples nodes with the Diaphragm-cap nodes ** Constraint: dimple_diaph *Tie, name=dimple_diaph, adjust=yes, position tolerance=0.0001, type=SURFACE TO SURFACE dimples_diaph_tie, diaph-1.diaph_tie *End Assembly ** MATERIALS **Define the material properties as density and the modulus of elasticity *Material, name=steel *Density 7800., *Elastic 2.1e+11, 0.3 ** STEP: accumulator *Step, name=accumulator, nlgeom=YES, extrapolation=NO, convert sdi=YES *Static, riks 0.001, 1., 1e-08, , , ** BOUNDARY CONDITIONS **The Diaphragm-cap upper end is fixed in the three transitional and rotational coordinates ** Name: end Type: Displacement/Rotation *Boundary Set452, 1, 1 ... Set452, 6, 6 ** LOADS **Define the pressure load acting on the lower surface of the Diaphragm-cap with value of 10 bar. ** Name: pressure Type: Pressure *Dsload pressure_surface, P, 1e+06 ** OUTPUT REQUESTS *Restart, write, frequency=0 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output
156
U *Element Output, directions=YES S ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step
A.2.1.5 Nonlinear behaviour of the contact interaction between the Diaphragm cap and the lower accumulator housing
*Heading ** Job name= non-linear behaviour of the contact interaction between the Diaphragm-cap and the lower accumulator housing, see Section 4.2.1. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** PARTS *Part, name=dimple *Node 1, 0., -0.00325000007, 0. 2, -0.000103742408, -0.00325190858, 0. ... 759, -0.00100310007, -0.0030465608, 9.15962664e-05 760, -0.00109544257, -0.00300000003, 0.000100028352 *Element, type=S4R 1, 2, 3, 14, 13 2, 3, 4, 15, 14 ... 758, 1, 739, 750 759, 1, 750, 2 *Nset, nset=Set8, internal, generate 1, 760, 1 *Elset, elset=Set8, internal, generate 1, 759, 1 ** Section: dimple *Shell Section, elset=Set8, material=steel 5e-05, 5 *End Part *Part, name=dimple_first_row *Node 1, 0., -0.00325000007, 0. 2, -0.000100176308, -0.00325027714, 0. ... 477, -0.000760549447, -0.00305446866, 9.05884226e-05 478, -0.000844033959, -0.00300000003, 0.000100532197 *Element, type=S4R 1, 2, 3, 12, 11 2, 3, 4, 13, 12 ... 423, 476, 477, 9, 8 424, 477, 478, 10, 9 *Element, type=S3 425, 1, 2, 11 426, 1, 11, 20 ... 476, 1, 461, 470 477, 1, 470, 2 *Nset, nset=Set9, internal, generate 1, 478, 1 *Elset, elset=Set9, internal, generate
157
1, 477, 1 ** Section: dimple_first_row *Shell Section, elset=Set9, material=steel 5e-05, 5 *End Part *Part, name=Diaphragm-cap *Node 1, -0.00601515174, -0.00387023459, 0. 2, -0.00359691377, -0.00335424743, 0. ... 414, -0.000982530997, -0.00253854366, -0.000915984041 415, -0.000524001778, -0.00256696111, 0.00132726424 *Element, type=S3 1, 254, 168, 17 2, 255, 20, 238 ... 657, 415, 223, 224 658, 415, 388, 389 *Nset, nset=Set50, internal, generate 1, 415, 1 *Elset, elset=Set50, internal, generate 1, 658, 1 ** Section: Diaphragm-cap *Shell Section, elset=Set50, material=steel 5e-05, 5 *End Part *Part, name=lower_housing *Node 1, -0.00401571486, -0.00346932001, 0. 2, -0.00601542229, -0.00387027697, 1.90076062e-05 ... 4934, -0.000458726339, -0.00838949438, 0.00375138083 4935, -0.00717415474, -0.0085634971, -0.000585289148 *Element, type=C3D4 1, 2632, 2633, 2634, 2635 2, 2636, 2637, 2638, 2639 ... 22163, 3296, 1796, 1789, 1788 22164, 4576, 3301, 1109, 1804 *Nset, nset=Set4, internal, generate 1, 4935, 1 *Elset, elset=Set4, internal, generate 1, 22164, 1 ... ** Section: lower_housing *Solid Section, elset=Set4, material=steel *End Part ** ASSEMBLY *Assembly, name=Assembly *Instance, name=Diaphragm-cap-1, part=Diaphragm-cap *End Instance *Elset, elset=_pressure_surface_SPOS, internal, instance=dimple_first_row-1, generate 1, 477, 1 *Surface, type=ELEMENT, name=diaphragm_interaction _diaphragm_interaction_SPOS, SPOS ** Constraint: dimple_Diaphragm-cap *Tie, name=dimple_Diaphragm-cap, adjust=yes, position tolerance=0.0002, type=SURFACE TO SURFACE dimples_tie, Diaphragm-cap-1.Diaphragm-cap_tie *End Assembly ** MATERIALS
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*Material, name=steel *Density 7800., *Elastic 2.1e+11, 0.3 ** INTERACTION PROPERTIES *Surface Interaction, name=Diaphragm-cap_lower-housing 1., *Friction 0., ** INTERACTIONS ** Interaction: Diaphragm-cap - lower_housing *Contact Pair, interaction=Diaphragm-cap_lower-housing, type=SURFACE TO SURFACE, adjust=0.0 diaphragm_interaction, lower_housing-1.interaction ** STEP: accumulator ** *Step, name=accumulator, nlgeom=YES, convert sdi=YES *Static, riks 0.001, 1., 1e-05, , , ** BOUNDARY CONDITIONS **The Diaphragm-cap upper end is fixed in the three transitional coordinates ** Name: end Type: Displacement/Rotation *Boundary Diaphragm-cap-1.BC_Diaphragm-cap_end, 1, 1 Diaphragm-cap-1.BC_Diaphragm-cap_end, 2, 2 Diaphragm-cap-1.BC_Diaphragm-cap_end, 3, 3 ** Name: lower_housing Type: Displacement/Rotation *Boundary lower_housing-1.lower_housing, 1, 1 lower_housing-1.lower_housing, 2, 2 lower_housing-1.lower_housing, 3, 3 ** LOADS ** Name: pressure Type: Pressure *Dsload Surf468, P, 2e+06 ** Interaction: diaphragm - lower_housing ** INTERACTIONS *Contact Interference diaphragm_interaction, lower_housing-1.interaction, 0.0002, ** OUTPUT REQUESTS *Restart, write, frequency=0 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output U *Element Output, directions=YES S ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step
A.2.1.6 Hydroforming a flat membrane
*Heading ** Job name: hydroforming flat membrane with pressure 200 bar, see Section 5.4.1.3. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO
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** PARTS *Part, name=flat_membrane *Node 1, 0., 0., 0. 2, 0.000250000012, 0., 0. ... 2700, 0.0287713259, 0., -0.00363466376 2701, 0.0297634415, 0., -0.00375999697 *Element, type=S4R 1, 2, 3, 57, 56 2, 3, 4, 58, 57 ... 2699, 1, 2594, 2648 2700, 1, 2648, 2 *Nset, nset=_Set2, internal, generate 1, 2701, 1 *Elset, elset=_Set2, internal, generate 1, 2700, 1 ** Section: membrane_flat *Shell Section, elset=_Set2, material=steel 5e-05, 5 *End Part *Part, name=Hydroforming die *Node 1, 0.00495344866, -0.00433870777, -0.0035988912 2, 0.00358935189, -0.00417631166, -0.00362968515 ... 2329, 0.0189014431, -0.00240271329, 0.0205324143 2330, 0.0141664669, -0.00445721112, 0.0127947684 *Element, type=C3D4 1, 1263, 1264, 828, 1265 2, 1263, 1264, 252, 256 ... 10120, 2143, 1220, 1856, 2142 10121, 2206, 1820, 1203, 404 *Nset, nset=_Set2, internal, generate 1, 2330, 1 *Elset, elset=_Set2, internal, generate 1, 10121, 1 ** Section: Hydroforming die *Solid Section, elset=_Set2, material=steel_die_presse *End Part ** ASSEMBLY *Assembly, name=Assembly ** *Instance, name=flat_membrane-1, part=flat_membrane 0., 0.0001, 0. *End Instance *Instance, name= Hydroforming die -1, part= Hydroforming die *End Instance *End Assembly ** MATERIALS *Material, name=steel *Density 7800., *Elastic 2.1e+11, 0.3 *Plastic 1275 *106, 0.
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1397.3 *106, 0.05 1440.9 *106, 0.1 1473.2 *106, 0.15 1500 *106, 0.2 *Material, name=steel_die_presse *Density 7800., *Elastic 2.1e+11, 0.3 ** INTERACTION PROPERTIES *Surface Interaction, name=contact 1., *Friction, slip tolerance=0.005 0.25, ** BOUNDARY CONDITIONS ** Name: membrane Type: Displacement/Rotation *Boundary flat_membrane-1.BC, 1, 1 flat_membrane-1.BC, 2, 2 flat_membrane-1.BC, 3, 3 ** INTERACTIONS ** Interaction: lower_part *Contact Pair, interaction=contact, type=SURFACE TO SURFACE, adjust=0.0 flat_membrane-1.lower_part, lower_part_final-1.contact ** STEP: stamping *Step, name=stamping, nlgeom=YES, unsymm=YES, convert sdi=YES *Static, stabilize=0.002, 0.01, 100, 1e-06, 100. ** BOUNDARY CONDITIONS **The hydroforming die is fixed in the three transitional coordinates ** Name: Hydroforming die Type: Displacement/Rotation *Boundary Hydroforming die -1.BC, 1, 1 Hydroforming die -1.BC, 2, 2 Hydroforming die -1.BC, 3, 3 **The flat membrane is fixed in the three transitional coordinates ** Name: membrane Type: Displacement/Rotation *Boundary flat_membrane-1.BC, 1, 1 flat_membrane-1.BC, 2, 2 flat_membrane-1.BC, 3, 3 ** LOADS ** Name: pressure_hydroforming Type: Pressure *Dsload flat_membrane-1.pressure, P, 2.0e+07 ** Interaction: Hydroforming die ** INTERACTIONS *Contact Interference flat_membrane-1. Hydroforming die, Hydroforming die -1.contact, 0.0001, ** OUTPUT REQUESTS *Restart, write, frequency=0 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output RF, RM, U, UR, UT *Element Output, directions=YES S *Contact Output CDISP, CFORCE, CSTRESS ** HISTORY OUTPUT: H-Output-1
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*Output, history *Energy Output ETOTAL *End Step
A.2.1.7 Stamping a flat membrane
*Heading ** Job name: Stamping flat membrane, see Section 5.4.1.3. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=central_dimple *Node 1, 0., -0.00400000019, 0. 2, 0., -0.00319999992, 0. ... 1487, 1862, 1867, 1877, 1868, 124, 129, 139, 130 1488, 1867, 820, 821, 1877, 129, 19, 18, 139 *Element, type=C3D6 337, 2, 10, 257, 9, 121, 913 338, 9, 121, 913, 8, 122, 914 ... 1535, 8, 1860, 122, 7, 1861, 123 1536, 7, 1861, 123, 1, 814, 25 *Nset, nset=Set2, internal, generate 1, 1877, 1 *Elset, elset=Set2, internal, generate 1, 1536, 1 ** Section: cent_dimple_4mm *Solid Section, elset=Set2, material=steel_die_presse *End Part ** *Part, name=dimple_mittle_raw_4mm *Node 1, 0.00164143497, -0.0031426684, 0. 2, 0., -0.0031426684, 0. ... 1980, 0.000202069394, -0.00355437351, -2.45356659e-05 1981, 0.000199114074, -0.00334759126, -2.41768248e-05 *Element, type=C3D8R 1, 7, 8, 132, 131, 272, 273, 984, 983 2, 8, 9, 133, 132, 273, 274, 985, 984 ... 1559, 1962, 1961, 1981, 1980, 123, 122, 142, 141 1560, 1961, 972, 973, 1981, 122, 12, 13, 142 *Element, type=C3D6 352, 16, 140, 992, 3, 17, 164 353, 15, 141, 993, 16, 140, 992 ... 1611, 14, 1981, 142, 15, 1980, 141 1612, 2, 973, 13, 14, 1981, 142 *Nset, nset=Set2, internal, generate 1, 1981, 1 *Elset, elset=Set2, internal, generate 1, 1612, 1 ** Section: dimple_mittle_raw_4mm
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*Solid Section, elset=Set2, material=steel_die_presse *End Part ** *Part, name=flat_membrane *Node 1, 0., 0., 0. 2, 0.00039999999, 0., 0. ... 2759, 0.0319202878, 0., 0.00225724676 2760, 0.0339153074, 0., 0.00239832466 *Element, type=S4R 1, 2, 3, 34, 33 2, 3, 4, 35, 34 ... 2669, 2758, 2759, 31, 30 2670, 2759, 2760, 32, 31 *Element, type=S3 2671, 1, 2, 33 2672, 1, 33, 64 ... 2758, 1, 2699, 2730 2759, 1, 2730, 2 *Nset, nset=Set2, internal, generate 1, 2760, 1 *Elset, elset=Set2, internal, generate 1, 2759, 1 ** Section: membrane_flat *Shell Section, elset=Set2, material=steel 5e-05, 5 *End Part ** *Part, name=die *Node 1, 0.00202427106, -0.00346453628, 0.000879591913 2, 0.00146207574, -0.00346453628, 0.00165338733 ... 3175, -0.0068435641, -0.00800000038, 0.0264390595 3176, -0.00444200169, -0.00800000038, 0.027096374 *Element, type=C3D8R 1, 807, 791, 145, 144, 1112, 1096, 286, 287 2, 806, 788, 791, 807, 1111, 1093, 1096, 1112 ... 1464, 699, 700, 2858, 2857, 2879, 2880, 3176, 3175 1465, 700, 12, 2789, 2858, 2880, 258, 3107, 3176 *Element, type=C3D6 360, 960, 961, 956, 1265, 1266, 1261 361, 17, 204, 206, 302, 1092, 295 ... 1470, 95, 54, 55, 727, 508, 69 1471, 2850, 2851, 2835, 3168, 3169, 3153 *Nset, nset=Set12, internal, generate 1, 3176, 1 *Elset, elset=Set12, internal, generate 1, 1471, 1 ** Section: die *Solid Section, elset=Set12, material=steel_die_presse *End Part ** *Part, name=presse
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*Node 1, 0.00220711459, -0.00346453628, 0. 2, 0., 0.0179999992, 0. ... 14811, 0.00542244362, -0.00344255567, 0.000956123113 14812, 0.0054769665, -0.00393000618, 0.000965736981 *Element, type=C3D8R 1, 67, 699, 685, 1, 1821, 4459, 4445, 182 2, 699, 700, 686, 685, 4459, 4460, 4446, 4445 ... 13751, 14732, 14801, 14807, 928, 997, 1003 13752, 14778, 14805, 14810, 974, 1001, 1006 *Nset, nset=Set2, internal, generate 1, 14812, 1 *Elset, elset=Set2, internal, generate 1, 13752, 1 ** Section: presse_new *Solid Section, elset=Set2, material=steel_die_presse *End Part ** *Part, name=dimple_outer_raw_4mm *Node 1, 0.00168721389, -0.00307392236, 0. 2, 0., -0.00307392236, 0. ... 1980, 0.000217379406, -0.00351735111, -2.63946386e-05 1981, 0.000213614025, -0.0032945876, -2.59374392e-05 *Element, type=C3D8R 1, 7, 8, 132, 131, 272, 273, 984, 983 2, 8, 9, 133, 132, 273, 274, 985, 984 ... 1559, 1962, 1961, 1981, 1980, 123, 122, 142, 141 1560, 1961, 972, 973, 1981, 122, 12, 13, 142 *Element, type=C3D6 352, 16, 140, 992, 3, 17, 164 353, 15, 141, 993, 16, 140, 992 ... 1611, 14, 1981, 142, 15, 1980, 141 1612, 2, 973, 13, 14, 1981, 142 *Nset, nset=Set2, internal, generate 1, 1981, 1 *Elset, elset=Set2, internal, generate 1, 1612, 1 ** Section: dimple_outer_raw *Solid Section, elset=Set2, material=steel_die_presse *End Part ** ASSEMBLY *Assembly, name=Assembly *Instance, name=central_dimple_4mm-1, part=central_dimple_4mm 0., 0.007679, 0. *End Instance *Instance, name=dimple_mittle_raw_4mm-1, part=dimple_mittle_raw_4mm 0.00407598754916549, 0.00657324025946356, 0. 0.00407598754916549, 0.00657324025946356, 0., 0.00407598754916549, 0.00657324025946356, -1., 23.5000012074692 *End Instance *Instance, name=presse-1, part=presse 0., 0.007, 0.
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*End Instance *Instance, name=dimple_outer_raw_4mm-1, part=dimple_outer_raw_4mm 0.00576794210196859, 0.00593601963025582, 0. 0.00576794210196859, 0.00593601963025582, 0., 0.00576794210196859, 0.00593601963025582, -1., 10.4999999167598 *End Instance *Instance, name=die-1, part=die *End Instance *Instance, name=flat_membrane-1, part=flat_membrane 0., 0.0001, 0. *End Instance *Nset, nset=BC_presse_dimples, instance=central_dimple_4mm-1 1, ..., 821, ... *Elset, elset=_presse_dimple_total_S3, internal, instance=outer_raw_4mm-1 1,..., 1556 *Surface, type=ELEMENT, name=presse_dimple_total _presse_dimple_total_S4, S4 _presse_dimple_total_S5, S5 _presse_dimple_total_S2, S2 _presse_dimple_total_S6, S6 _presse_dimple_total_S1, S1 _presse_dimple_total_S3, S3 ** Constraint: presse *Tie, name=presse, adjust=yes, no thickness dimples_presse_tie, presse-1.presse_tie *End Assembly ** MATERIALS *Material, name=steel *Density 7800., *Elastic 2.1e+11, 0.3 *Plastic 1275 *106, 0. 1397.3 *106, 0.05 1440.9 *106, 0.1 1473.2 *106, 0.15 1500 *106, 0.2 *Material, name=steel_die_presse *Density 7800., *Elastic 2.1e+11, 0.3 ** INTERACTION PROPERTIES *Surface Interaction, name=contact 1., *Friction, slip tolerance=0.005 0.25, ** BOUNDARY CONDITIONS ** Name: die Type: Displacement/Rotation *Boundary die-1.BC_die, 1, 1 die-1.BC_die, 2, 2 die-1.BC_die, 3, 3 ** Name: membrane Type: Displacement/Rotation *Boundary flat_membrane-1.BC, 1, 1 flat_membrane-1.BC, 2, 2 flat_membrane-1.BC, 3, 3
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** Name: presse Type: Displacement/Rotation *Boundary BC_presse_dimples, 1, 1 BC_presse_dimples, 2, 2 BC_presse_dimples, 3, 3 ** INTERACTIONS ** Interaction: die *Contact Pair, interaction=contact, type=SURFACE TO SURFACE, tracking=STATE, adjust=0.0 flat_membrane-1.die, die-1.die_contact ** Interaction: presse *Contact Pair, interaction=contact, type=SURFACE TO SURFACE, tracking=STATE, adjust=0.0 presse_dimple_total, flat_membrane-1.presse ** STEP: fixiation *Step, name=fixiation, nlgeom=YES, amplitude=STEP, inc=2 *Static 1., 1., 0.001, 1. ** INTERACTIONS ** Interaction: die *Model Change, type=CONTACT PAIR, remove flat_membrane-1.die, die-1.die_contact ** Interaction: presse *Model Change, type=CONTACT PAIR, remove presse_dimple_total, flat_membrane-1.presse ** OUTPUT REQUESTS *Restart, write, frequency=0 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output RF, U *Element Output, directions=YES S ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step ** STEP: stamping *Step, name=stamping, nlgeom=YES, unsymm=YES, convert sdi=YES *Static, stabilize=0.002, 0.01, 3., 1e-06, 3. ** BOUNDARY CONDITIONS ** Name: membrane Type: Displacement/Rotation *Boundary flat_membrane-1.BC, 1, 1 flat_membrane-1.BC, 2, 2 flat_membrane-1.BC, 3, 3 ** Name: presse Type: Displacement/Rotation *Boundary BC_presse_dimples, 2, 2, -0.00672 ** INTERACTIONS ** Interaction: die *Model Change, type=CONTACT PAIR, add flat_membrane-1.die, die-1.die_contact *Contact Interference flat_membrane-1.die, die-1.die_contact, 0.0001, ** Interaction: presse *Model Change, type=CONTACT PAIR, add presse_dimple_total, flat_membrane-1.presse *Contact Interference presse_dimple_total, flat_membrane-1.presse, 0.0001, ** OUTPUT REQUESTS *Restart, write, frequency=1
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** FIELD OUTPUT: F-Output-1 *Output, field *Node Output RF, U *Element Output, directions=YES LE, PE, S ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step ** STEP: presse_upward *Step, name=presse_upward, nlgeom=YES, amplitude=STEP, inc=10 *Static, stabilize=0.0002 1., 1., 0.001, 1. ** BOUNDARY CONDITIONS ** Name: membrane Type: Displacement/Rotation *Boundary flat_membrane-1.BC, 1, 1 flat_membrane-1.BC, 3, 3 ** Name: presse Type: Displacement/Rotation *Boundary BC_presse_dimples, 2, 2, 0.007 ** INTERACTIONS ** Interaction: die *Model Change, type=CONTACT PAIR, remove flat_membrane-1.die, die-1.die_contact ** Interaction: presse *Model Change, type=CONTACT PAIR, remove presse_dimple_total, flat_membrane-1.presse ** OUTPUT REQUESTS *Restart, write, frequency=1 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output RF, U *Element Output, directions=YES LE, PE, S ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step ** STEP: end_process *Step, name=end_process, nlgeom=YES, amplitude=STEP, inc=10 *Static 1., 1., 0.001, 1. ** BOUNDARY CONDITIONS ** Name: presse Type: Displacement/Rotation *Boundary BC_presse_dimples, 2, 2 ** OUTPUT REQUESTS *Restart, write, frequency=1 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output RF, U *Element Output, directions=YES LE, PE, PEEQ, PEMAG, S ** HISTORY OUTPUT: H-Output-1 *Output, history, variable=PRESELECT *End Step
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A.2.1.8 Nonlinear behaviour of the accumulator bellows
*Heading ** Job name: non-linear behaviour of the accumulator bellows 400µm thickness using Riks method, see Section 4.4.1. ** Generated by: Abaqus/CAE Version 6.8-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** PARTS *Part, name=bellows *Node 1, 0.0448008329, 0.0545638837, 0. 2, 0.0466426089, 0.0531210341, 0. ... 29399, 0.0474205427, 0.00780243892, 0.00198750966 29400, 0.0454875715, 0.00765754795, 0.00190649414 *Element, type=S4R 1, 1, 85, 11074, 385 2, 85, 2, 11075, 11074 ... 29249, 29399, 29400, 10924, 10923 29250, 29400, 10925, 84, 10924 *Nset, nset=lower_bellows 84, ..., 11073 *Elset, elset=lower_bellows, generate 28654, 29250, 4 ** Section: bellows *Shell Section, elset=Set52, material=steel 0.0004, 5 *End Part *Part, name=piston_new *Node 1, 0., 0.0603206195, 0. 2, 0., 0.070363678, 0. ... 6843, -0.0262842029, 0.0613464303, 0.00276258122 6844, -0.0217496827, 0.0632079393, 0.00228598388 *Element, type=C3D8R 1, 37, 659, 657, 38, 1049, 3319, 3317, 1048 2, 659, 660, 658, 657, 3319, 3320, 3318, 3317 ... 5399, 6828, 6841, 6792, 6837, 703, 716, 667, 712 5400, 6830, 6844, 6787, 6839, 705, 719, 662, 714 *Element, type=C3D6 1261, 2, 37, 1049, 36, 659, 3319 1262, 36, 659, 3319, 35, 660, 3320 ... 5519, 6824, 6792, 6841, 699, 667, 716 5520, 6802, 6834, 6838, 677, 709, 713 *Nset, nset=Set2, internal, generate 1, 6844, 1 *Elset, elset=Set2, internal, generate 1, 5520, 1 ** Section: all_piston *Solid Section, elset=Set2, material=steel 1., *End Part ** ASSEMBLY *Assembly, name=Assembly *Instance, name=bellows-1, part=bellows *End Instance
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*Instance, name=piston_new-1, part=piston_new *End Instance *Elset, elset=_pressure_load_SNEG, internal, instance=bellows-1, generate 1, 29250, 1 ... *Surface, type=ELEMENT, name=pressure_load _pressure_load_SNEG, SNEG _pressure_load_S4, S4 _pressure_load_S6, S6 _pressure_load_S2, S2 _pressure_load_S5, S5 _pressure_load_S3, S3 ** Constraint: bellows-piston *Tie, name=bellows-piston, adjust=yes, type=SURFACE TO SURFACE bellows-1.upper_bellows-piston, piston_new-1.piston_bellows_tie *End Assembly ** MATERIALS *Material, name=steel *Density 7800., *Elastic 2.1e+11, 0.3 ** INTERACTION PROPERTIES *Surface Interaction, name=piston-upper_housing 1., *Friction 0., ** PHYSICAL CONSTANTS ** STEP: deformation *Step, name=acoustic, nlgeom=YES, extrapolation=PARABOLIC, convert sdi=YES *Static, riks 0.001, 1., 1e-08, , , *Solution Technique, type=CONTACT ITERATIONS 1, 100 ** BOUNDARY CONDITIONS ** Name: lower_bellows Type: Displacement/Rotation *Boundary bellows-1.lower_bellows, 1, 1 bellows-1.lower_bellows, 2, 2 bellows-1.lower_bellows, 3, 3 ** Name: piston Type: Displacement/Rotation *Boundary piston_new-1.piston_BC, 1, 1 piston_new-1.piston_BC, 2, 2, 0.04 piston_new-1.piston_BC, 3, 3 ** LOADS ** Name: pressure Type: Pressure *Dsload pressure_load, P, 3e+06 ** OUTPUT REQUESTS *Restart, write, frequency=0 ** FIELD OUTPUT: F-Output-1 *Output, field *Node Output RF, UT *Element Output, directions=YES EE, S ** HISTORY OUTPUT: H-Output-1 *Output, history *Energy Output
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ETOTAL *End Step
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A.2.2 Matlab files
A.2.2.1 The distributed and the 2DOF discrete parameter models of transmission line connecting with a hydro-pneumatic accumulator.
The simulations of the influence of changing the model parameters on the model performance are similar in programming and the author preferred to present one Matlab file such as the influence of changing the pipes diameters to list the used commands (codes)
A.2.2.1.1 The distributed parameter model of transmission line connecting with an accumulator %===============================in frequency domain============ %===============================with discharge flow rate excitation ===== %===============================Leonard viscous model==================== %=================at each case all pipes diameters are the same===== % The nominal conditions are stated in Section 3.4 % The first case diameter pipe is [d=15 mm] j=0; for we=1*2*pi:20*2*pi:2500*2*pi % we is the angular velocity (rad/s) Rf=i*sqrt(i*we/nu)*R; Rf4=i*sqrt(i*we/nu)*R4; % propagation operator gamma1=i*we*l1/c*sqrt(-besselj(0,Rf,1)/besselj(2,Rf,1)); ... gamma4=i*we*H/c*sqrt(-besselj(0,Rf4,1)/besselj(2,Rf4,1)); % the pipes impedance zf1=z1*sqrt(-besselj(0,Rf,1)/besselj(2,Rf,1)); ... zf4=z4*sqrt(-besselj(0,Rf4,1)/besselj(2,Rf4,1)); % The equations of the model are presented in a matrix, see eq.3.68: X=inv(A)*u; % where x is the model variables vector % the Variables = [PA1, PE1, QE1, QA2, PE2, QE2, PE3, QE3, PA4, PE4, QE4, y, dPG] % A is the parametric matrix A= [cosh(gamma1) -1 0 0 0 0 0 0 0 0 0 0; ...; 0 0 0 0 0 0 Cd * Av2 * sqrt(0.2e1) * (Pav / rho) ^ (-0.1e1/ 0.2e1)/ rho / 0.2e1 -1 -Cd * Av2 * sqrt(0.2e1) * (Pav / rho) ^ (-0.1e1 / 0.2e1) / rho / 0.2e1 0 0 0]; % u is the input vector u=[zf1*QA1*sinh(gamma1); -QA1*cosh(gamma1); ...; 0; 0]; % By solving eq.3.68 one can obtain the model solution j=j+1; PA1(j)=X(1);...;dPG(j)=X(12); We(j) =we; end % the second and the third case [d=6 mm and d=30 mm respectively] are similar as the first one
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... % Plot the variables in the frequency domain figure() plot(We/(2*pi),abs(PE2)*10^-5,...); title('f & PE2') xlabel('f [Hz]') ylabel('abs(PE2) [bar]') legend('d=15mm','d=6mm','d=30mm') grid on ...
A.2.2.1.2 The 2DOF discrete parameter model of transmission line connecting with an accumulator. %===============================in frequency domain============ %===============================with discharge flow rate excitation ===== %=================at each case all pipes diameters are the same===== % The nominal conditions are stated in Section 3.4 % The first case diameter pipe is [d=15 mm] j=0; for we=1*2*pi:1*2*pi:1000*2*pi % we is the angular velocity (rad/s) % The equations of the model are presented in a matrix, see eq.3.57: X=inv(A)*u; % where x is the model variables vector % the Variables = [PA1, PE1, QE1, QA2, PE2, QE2, PE3, QE3, PA4, PE4, QE4, y, dPG] % A_2DOF is the parametric matrix A_2DOF = [1 - i * (-i * LH1 * we - RH1) * we * CH1 -1 0 0 0 0 0 0 0 0 0 0; ...; 0 0 0 0 0 0 0 0 0 -(1 - i * (-RH4 - i * we * LH4) * we * CH4)/ we ^ 2 / m 0 -i * n * pNenn ^ ((n + 1) / n) * p0 ^ (-1 / n)/ V0 * (1 - i * (-RH4 - i * we * LH4) * we * CH4) ^ 2 / we / m – 1]; % u is the input vector U_2DOF=[-(-i*LH1*we-RH1+(-i*LH1*we-RH1)*(-LH1*CH1*we^2+i*RH1*CH1*we+1))*QA1; -(-i*(-i*LH1*we-RH1)*we*CH1+(-LH1*CH1*we^2+i*RH1*CH1*we+1)^2)*QA1; ...; 0; 0]; % By solving eq.3.57 one can obtain the model solution j=j+1; PA1(j)=X(1);...;dPG(j)=X(12); We(j) =we; end % the second and the third case [d=6 mm and d=30 mm respectively] are similar as the first one ... % Plot the variables in the frequency domain figure() plot(We/(2*pi),abs(PE2)*10^-5,...); title('f & PE2') xlabel('f [Hz]') ylabel('abs(PE2) [bar]') legend('d=15mm','d=6mm','d=30mm') grid on ...
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