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)

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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.

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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.

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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

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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

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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:

87

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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[Scheidl 2000] M. Garsternauer, B. Manhartsgruber, R. Scheidl, " Switching Type

Control of Hydraulic Drives – A Promising, Perspective for

Advanced Actuation in Agricultural Machinery", New Fluid Power

Applications and Components, Society of Automotive Engineers,

Warrendale, pp 37-47, 9-2000, ISBN: 0-7680-0650-3.

[Scheidl 2008] Kogler, H., Scheidl, R. 2008. Two Basic Concepts of Hydraulic

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3rd October 2008, Tampere, Finland.

[Scheidl 2009] Rudolf Scheidl, Bernhard Manhartsgruber, Mohamed Mohamed Ez

El Din, "Finite Element Analysis of 3D Viscid Periodic Wave

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International Journal of Fluid Power, Vol. 10, Nummer 1, 3-

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[11-03-2011]

[Spring 2011] http://metals.about.com/library/bldef-Springback.htm [5. Jan. 2011]

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Kip R. Steveley "Apparatus and Method of an Accumulator with

Rigid Secondary Diaphragm", United States patents number

4777983, 18.Oct.1988.

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[Suzuki 2004]

Kenichi Suzuki, Yuichiro Skakibara, Yutaka Yamashita, "Hydraulic

Accumulator", "Hydraulic Accumulator", United States patents

number US 2004/0231738 A1, 25.Nov.2004.

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Shigeaki Takamatsu, Hiroaki Nagaoka, "Hydraulic Accumulator",

United States patents number 6092552, 25.Jul.2000.

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New York, 1979.

[Winkler 2002] Winkler B., Scheidl R.; Automatic Level Control in Agricultural

Machinery. In Proceedings of the 2nd International FPNI PhD

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[Yadav 2008] Ajay D. Yadav, „Process Analysis and Design in Stamping and

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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: mohamedezeldin@hotmail.com

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

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__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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