Simkin FinalReport

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Analysis of Accumulator Response to an

External Force acting on a Hydraulic Actuator

by

Leonid Simkin

An Engineering Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING

Major Subject: Mechanical Engineering

Approved:

_________________________________________Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute

Hartford, Connecticut

(For Graduation May 2012)


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Copyright 2012

by

Leonid Simkin

All Rights Reserved


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CONTENTS

Numerical Method Analysis of the Response of an Accumulator to an Impulse Force in a

Hydraulic System .......................................................................................................... i

LIST OF TABLES ............................................................................................................. v

LIST OF FIGURES .......................................................................................................... vi

LIST OF SYMBOLS ....................................................................................................... vii

ACKNOWLEDGMENT ................................................................................................. viii

ABSTRACT ...................................................................................................................... ix

1. Introduction .................................................................................................................. 1

2. Methodology ................................................................................................................ 3

2.1 Assumptions ....................................................................................................... 4

2.2 Accumulator Behavior Modeling ....................................................................... 4

2.2.1 Accumulator Initial Conditions .............................................................. 5

2.2.2 Accumulator Behavior Equations .......................................................... 5

2.3 Actuator Behavior Modeling .............................................................................. 7

2.3.1 Actuator Initial Conditions ..................................................................... 8

2.3.2

Actuator Behavior Equations ................................................................. 8

2.4 Complete Hydraulic System Behavior Modeling .............................................. 9

2.4.1 Hydraulic System Initial Conditions ...................................................... 9

2.4.2 Hydraulic System Behavior Equations ................................................. 10

2.5 Euler Method .................................................................................................... 11

2.6 Runge-Kutta Method ........................................................................................ 11

2.7 FLOWMASTER Method ................................................................................. 11

3. Results and Discussion............................................................................................... 14

3.1 Comparison of Accumulator Behavior ............................................................. 14

3.1.1 Model Comparisons for = 0.05 .......................................................... 153.1.2 Model Comparisons for = 0.005 ........................................................ 15


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3.1.3 Model Comparisons for = 0.0005 ...................................................... 173.1.4 Model Comparisons for = 0.00005 .................................................... 183.1.5 Discussion of Accumulator Behavior Results ...................................... 20

3.2 Comparison of Complete Hydraulic System Behavior .................................... 20

3.2.1 Model Comparisons for = 0.05 .......................................................... 213.2.2 Model Comparisons for = 0.005 ........................................................ 223.2.3 Model Comparisons for = 0.0005 ...................................................... 243.2.4 Model Comparisons for = 0.00005 .................................................... 253.2.5 Discussion of Hydraulic System Behavior Results .............................. 27

4. Conclusion ................................................................................................................. 28

4.1 Suggestions for Further Research ..................................................................... 29

5. References .................................................................................................................. 30

Appendices ....................................................................................................................... 32

A. Input Variables for Accumulator Simulation ............................................................. 32

B. Differential Equations for Accumulator Behavior ..................................................... 33

C. Accumulator Behavior Simulation using Euler Method ............................................ 34

D. Accumulator Behavior Simulation using Runge-Kutta Method ................................ 34

E. Accumulator Behavior Simulation using FLOWMASTER ...................................... 36

F. Input Variables for Hydraulic System Simulation ..................................................... 37

G. Differential Equations for Hydraulic System Behavior ............................................. 37

H. Hydraulic System Behavior Simulation using Euler Method .................................... 38

I. Hydraulic System Behavior Simulation using Runge-Kutta Method ........................ 39

J. Hydraulic System Behavior Simulation using FLOWMASTER .............................. 41


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LIST OF TABLES

Table A-1: Input Variables for the Accumulator Simulation ......................................... 32

Table E-1: FLOWMASTER Accumulator Simulation Component 1: Air Vessel ......... 36

Table E-2: FLOWMASTER Accumulator Simulation Component 2: Rigid Pipe ......... 36Table F-1: Input Variables for the Hydraulic System Simulation .................................. 37

Table J-1: FLOWMASTER Hydraulic System Simulation Component 1: Single-Rod

Double-Acting Cylinder ................................................................................................. 42

Table J-2: FLOWMASTER Hydraulic System Component 2: Rigid Pipe .................... 42

Table J-3: FLOWMASTER Hydraulic System Component 4: Air Vessel .................... 42


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LIST OF FIGURES

Figure 2-1: Hydraulically Driven Suspension System with Accumulators [6] ................. 3

Figure 2-2: Typical Piston-Type Accumulator [9] ............................................................ 5

Figure 2-3: Piston-Type Accumulator Schematic .............................................................. 5

Figure 2-4: Hydraulic Actuator Schematic ........................................................................ 8

Figure 2-5: Hydraulic System Schematic .......................................................................... 9

Figure 2-6: FLOWMASTER Accumulator Simulation Graphic ..................................... 13

Figure 2-7: FLOWMASTER Hydraulic System Simulation Graphic ............................. 13

Figure 3-1: Accumulator Piston Behavior Comparison h=0.005 ................................. 16

Figure 3-2: Accumulator Gas Pressure Comparison h=0.005 ...................................... 16

Figure 3-3: Accumulator Piston Behavior Comparison h=0.0005 ............................... 17

Figure 3-4: Accumulator Gas Pressure Comparison h=0.0005 .................................... 18

Figure 3-5: Accumulator Piston Behavior Comparison h=0.00005 ............................. 19

Figure 3-6: Accumulator Gas Pressure Comparison h=0.00005 .................................. 19

Figure 3-7: Hydraulic System: Accumulator Piston Comparison h=0.05 .................... 21

Figure 3-8: Hydraulic System: Gas Pressure Comparison h=0.05 ............................... 22

Figure 3-9: Hydraulic System: Accumulator Piston Comparison h=0.005 .................. 23

Figure 3-10: Hydraulic System: Gas Pressure Comparison h=0.005 ........................... 23

Figure 3-11: Hydraulic System: Accumulator Piston Comparison h=0.0005 .............. 24

Figure 3-12: Hydraulic System: Gas Pressure Comparison h=0.0005 ......................... 25

Figure 3-13: Hydraulic System: Accumulator Piston Comparison h=0.00005 ............ 26

Figure 3-14: Hydraulic System: Gas Pressure Comparison h=0.00005 ....................... 26

Figure A-1: Bore Size 125 mm, Model AP-10 Dimension (Reference [14]) ................. 33

Figure C-1: Print Screen of Spread Sheet using Euler Method with h=0.005................. 34

Figure D-1: Print Screen of Spread Sheet using Runge-Kutta Method with h=0.005 ... 36

Figure H-1: Print Screen of Spread Sheet using Euler Method with h=0.005 ............... 39

Figure I-1: Print Screen of Spread Sheet using Runge-Kutta Method with h=0.005 ..... 41


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LIST OF SYMBOLS

... Area of the hydraulic actuator piston (m ). Area of the accumulator piston (m ).. Area of the hydraulic actuator rod (m ) Time step interval (s) Length/ height of the accumulator (m).. Mass of actuator piston and attached component (kg) Mass of the accumulator piston (kg). Atmospheric Pressure (Pa).. Pressure of the gas in the accumulator (Pa) Pressure of the liquid (Pa)...... Flow rate into the accumulator (l/s)...... Flow rate out of the actuator (l/s) Length/stroke of the linear actuator (m) Time (s).. Volume of gas in the accumulator (l).. Change in volume of gas in the accumulator (l/s) Position of the actuator piston (m).. Velocity of the actuator piston (m/s). Acceleration of the actuator piston (m/s ) Position of the accumulator piston (m).. Velocity of the accumulator piston (m/s). Acceleration of the accumulator piston (m/s ) Ratio of specific heat(l-liter, m-meter, s-second)


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ACKNOWLEDGMENT

I would like to thank my advisor, Professor Ernesto Gutierrez-Miravete, whose constant

reminders and strict schedule helped me complete my project. I would like to thank my

family and friends whose constant questioning motivated me to continue working. Iwould also like to thank my supervisor, who allowed me the flexibility to change my

work schedule so that I can finish my project.


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ABSTRACT

Pressure transient can occur in piping system due to many different abnormal system

conditions. In a hydraulic system, one such condition that can cause a pressure transient

is the action of an external force on an actuator in the system. An example where thistype of transient can occur is in mechanical shocks of hydraulically operated suspension

system for a dump truck. A proven method to suppress this type of pressure transient is

to incorporate an accumulator into the system near the actuator. This project outlined a

numerical approach for analyzing the response of such an accumulator affected by an

external force acting on a hydraulic actuator. The numerical approach was based on the

law of conservation of mass, Newtons second law, and the ideal gas law. The numerical

approach was solved using the Euler method and the Runge-Kutta method. The methods

were implemented using Microsoft Excel 2007 edition. A simulation using

FLOWMASTER, a one-dimensional computational fluid dynamics software, was also

created to compare with the numerical approach. All three methods were used to analyze

the response of two separate models: a standalone accumulator with a pressure transient

input and a complete hydraulic system with an impulse force input. The outputs for the

behavior of the accumulator in each of the models were compared for all three methods.

The Euler method and the Runge-Kutta method were easily implemented and

manipulated in a Microsoft Excel 2007 spreadsheet and showed similar outputs for both

models. However, when compared to the FLOWMASTER simulation, the outputs for

both models did not agree with the numerical approach. Although the outputs for the

numerical methods did not match the FLOWMASTER simulation, the output from the

FLOWMASTER simulation were unexpected for the behavior of the hydraulic system

while the output using the numerical approach produced expected results. Further

comparison between the outlined numerical approach and other computational fluid

dynamic software or experimental results need to be accomplished to determine the

accuracy of the outlined numerical approach.


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1. IntroductionMany factors have to be taken into account in the design of a piping system. Depending

on the type of piping system being designed and its application, several normal and

abnormal conditions have to be considered in the design process. One such factor thatneeds to be accounted for during abnormal conditions is the occurrence of pressure

transients in the piping system.

Pressure transients in piping systems can occur due to many different abnormal

system conditions. Most common pressure transients studied today occur due to one of

two conditions. Condition one is the stoppage of a moving fluid due to the instantaneous

closure of a valve. Just before valve closure, the pipe is full of fluid moving at a certain

velocity. If the valve is suddenly closed, the instantaneous stoppage of the moving fluid

converts the kinetic energy, i.e. velocity, of the moving fluid to potential energy by

increasing the pressure of the fluid at the point it was stopped. However, the fluid farther

upstream of the valve continues to move. The increase in the pressure of the fluid at the

point of stoppage causes the fluid element directly upstream of it to come to rest and

then begins to increase the pressure of that element of fluid [1]. This causes a pressure

wave to develop. This type of pressure transient is also referred to as a water hammer.

Water hammer is usually caused by quick shutting valves within the piping system or the

stoppage of a power element in the system [2]. Condition two is the changing of an

operating condition of a turbomachine which results in unsteady flow in the attached

piping system. This may be the startup or stoppage of centrifugal pump, or a load

adjustment at a motor on an attached pump [3]. Sudden changes to the operating

conditions of the pressure source (i.e. pump) of a piping system will cause pressure

transients in the attached system.

A third abnormal condition in a piping system that can cause pressure transients

is the action of an external force on an actuator in the system. The external force moves

the actuator, which causes movement of the liquid inside the actuator and the associated

piping system. The acceleration of the liquid within the piping system due to the external

force causes a pressure transient and therefore a pressure increase. Examples where this

type of transient can occur are hydraulically operated mill rolls if a large piece of foreign

matter enters the mill and drives the roll up thereby driving up the actuators [4], rock


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drills which oscillate and back drive the actuator when operating [5] and mechanical

shocks in hydraulically driven equipments such as suspension systems for fork-lifts,

mobile cranes and dump trucks [6].

All the above mentioned transient conditions create pressure increases in the

associated piping systems which can lead to pipe ruptures and component damage. One

possible solution is to beef up components so they can withstand the pressure increase.

However, this solution can be very expensive depending on the size and complexity of

the system [7]. A better and proven method to suppress pressure transients is to

incorporate an air chamber or accumulator into the system near the component where the

pressure transient can potentially occur.

This project outlines a numerical approach to analyze the behavior of such an

accumulator that can be easily implemented in a spreadsheet or other engineering

calculation software. The most available software for calculations is currently Microsoft

Excel, and therefore Microsoft Excel will be used to model the accumulator. The

determination whether or not the developed approach is easily implemented will be if it

can be easily modeled and manipulated in the spreadsheet. If the approach cannot be

easily implemented and manipulated, then for the purposes of this project it is

impracticable.


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2. MethodologyThe numerical approach used in this project to analyze the response of an accumulator to

pressure transients in a hydraulic system is developed specifically for the accumulators

affected by impulse forces acting on hydraulic actuators. An example of a hydraulicsystem with accumulators that are used to attenuate this type of impulse force is shown

in Figure 2-1. Figure 2-1 shows a simplified schematic of a hydraulically operated

suspension system for a dump truck. In this system, an impulse force would act on the

actuators thereby driving liquid into the system and increasing the pressure. A set of

equations is obtained to model the behavior of the major components of interest. As seen

in Figure 2-1, the actuators and accumulators are isolated from any other hydraulic

system components, i.e. pump, motor, tank, by the control valve being in the center

blocked position. Therefore, for the purpose of the model only the behavior of one

accumulator and actuator shall be modeled.

Figure 2-1: Hydraulically Driven Suspension System with Accumulators [6]

The set of equations to model the response of the hydraulic system will be based

on the law of conservation of mass, Newtons second law, and the ideal gas law. The

obtained set of differential equations will be modeled using the Euler method and the

Runge-Kutta method to approximate the behavior of the hydraulic system. The behavior

of the hydraulic system (i.e. pressure, flow rate, piston velocity) using the numerical


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approach will then be compared to the results of the behavior of the same system as

simulated by FLOWMASTER, a simulation program used to model thermodynamic and

fluid systems [8]. The Euler method, Runge-Kutta method and the FLOWMASTER

simulations were first used to model a standalone accumulator with a transient pressure

input and then a complete hydraulic system with a transient force input.

2.1 AssumptionsIn developing the model in this project several whole system and component specific

assumptions were used.

1. The accumulators are piston-type [9]. The accumulators initially contain a givenvolume of gas, , at a given pressure, .

2. When the system is working normally (i.e. no impulse force), the accumulator isat its lowest pressure and the accumulator piston is at its lowest point of travel.

3. Pressure drop through piping is negligible. Since the model is based on a dumptruck suspension system, the actuator and accumulator will be in close proximity

to one another and the length of pipe between them shall be minimal.

4. The liquid in the hydraulic system is incompressible. Comparing thecompressibility of the liquid to the compressibility of the gas in the system, it is

reasonable to consider the liquid to be incompressible.

5. There is no leakage in the system. Any external or internal leakage should beminimal in a hydraulic system.

6. The air in the accumulator is an ideal gas and the change in air states in theaccumulator is an adiabatic process and is governed by [10].An adiabatic process can be assumed due to the very short duration of the

pressure transient.

2.2 Accumulator Behavior ModelingA schematic representation of the effects of a pressure transient in a typical piston-type

accumulator is shown in Figure 2-2. Figure 2-2 shows the movement of the piston in the

accumulator when the pressure of the liquid in the system temporarily exceeds the

pressure of the gas in the accumulator.


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Figure 2-2: Typical Piston-Type Accumulator [9]

Figure 2-3 shows a simplified schematic representation of the change in the accumulator

when the pressure of the liquid in the system, , increases above the initial pressure ofthe gas in the accumulator, . For now the time dependent input variable into thesystem is the pressure of the liquid,

. The flow rate of the liquid,

, is related to

.

However, the relationship between and will be discussed further whenexamining the behavior of the hydraulic system as a whole.

Figure 2-3: Piston-Type Accumulator Schematic

2.2.1 Accumulator Initial ConditionsUsing the previous assumptions listed in section 2.1, two initial conditions were assumed

for the behavior of the accumulator.

1. The initial position of the accumulator piston,, is zero2. The initial velocity of the accumulator piston,

, is zero

2.2.2 Accumulator Behavior EquationsUsing Figure 2-3, the equations for the behavior of the accumulator can be obtained with

changing . The behavior of the accumulator is characterized by the position, ,


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velocity, , and acceleration, , of the accumulator piston. The equations governingthe accumulator behavior are obtained as follows.

Using the initial condition that the initial position of the accumulator piston is

zero and knowing the accumulator length,

, and accumulator piston area,

, the

volume of gas in the accumulator, , can be obtained using Equation (1). The volume ofthe gas is in liters.

(1)The pressure of the gas in the accumulator can be obtained using assumption six

listed in section 2.1 and is given by Equation (2).

(2)Also, using the initial condition that the velocity of the accumulator piston is

zero, the time rate of change of the accumulator gas volume, , can be obtained usingEquation (3).

(3)Since the accumulator piston has a lower position limit, Equation (3) is only valid if the

position of the piston is above zero or the velocity of the piston is positive. Otherwise,

.The acceleration of the accumulator piston can be obtained by using Newtons

second law and balancing the forces acting on the accumulator piston as shown in

Equation (4).

(4)The forces acting on the accumulator piston include the pressure of the gas, , on theupper surface of the piston and the pressure of the liquid, , on the lower surface of thepiston as shown in Figure 2-3. Again, since the accumulator piston has a lower position

limit, Equation (4) is only valid if the position of the piston is above zero or the pressure

of the liquid is greater or equal to the pressure of the gas.

The piston velocity and position is obtained using Equations (5) and (6)

respectively.


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(5) (6)The volume of the accumulator is obtained using Equation (7).

(7)Additionally, the flow rate of liquid into or out of the accumulator, , is equal

to the change in volume of gas in the accumulator and is given by Equation (8).

(8)2.3 Actuator Behavior ModelingAs shown in Figure 2-1, the type of actuator which will be modeled is a single acting

linear hydraulic cylinder. The terms actuator and cylinder will be used interchangeably

for the remainder of this report. The term used will depend on the context of the

statement and the source of the information. Single acting cylinders are operated through

a hydraulic port only at one end of the cylinder. When liquid enters that port, the piston

extends out of the actuator. External forces acting on the piston would return it to its

normal position, and force the liquid out of the actuator port. Figure 2-4 shows a

schematic of a single acting linear hydraulic actuator. The blind side of the actuator

piston is in contact with the liquid. The rod side of the piston is in contact withatmosphere. Therefore the force acting on the blind side area, , of the piston is thepressure of the liquid, . The rod side of the piston, whose area is minus the area ofthe rod, , is open to atmospheric pressure, . In addition, an external impulseforce, , is assumed to be acting on the piston. The mass of the piston, , will includethe mass of the piston and the mass of the component or devise being moved. This

definition for is used to establish that in hydraulic system the impulse force neveracts directly on the piston but acts on the component being driven. Therefore, the massthat is accelerated by the impulse force is not just the mass of the piston but also the

mass of the attached component.


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Figure 2-4: Hydraulic Actuator Schematic

2.3.1 Actuator Initial ConditionsTo simplify the model for the behavior of the actuator, several initial conditions for the

actuator were assumed.

1. The initial position of the actuator piston, , is equal to stroke of the actuator, 2. The initial velocity of the actuator piston, , is zero2.3.2 Actuator Behavior EquationsUnlike the behavior of the accumulator, which can be modeled as a standalone

component using the equations in section 2.2.2 and a given changing value, thebehavior of the actuator cannot be modeled as a standalone component. When

considering the actuator as a standalone component, the number unknown variables is

greater than the number of equations. Therefore the following equations cannot be used

to model the behavior of an actuator alone, but will be used when modeling the behavior

of the entire hydraulic system. Therefore, assuming is known for the system and usingFigure 2-4, the acceleration of the actuator piston,

, can be obtained for a givenfrom a force balance equation given in Equation (9). (9)

The actuator piston velocity and position can be obtained from the acceleration and

velocity respectively from Equations (10) and (11).

(10) (11)


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Additionally, the flow rate out of the actuator, , can be obtained using Equation (12). (12)

2.4 Complete Hydraulic System Behavior ModelingFor the situation shown in Figure 2-1, the only components whose behavior will be

modeled will be the accumulator and actuator. A simplified schematic of one actuator

and accumulator from Figure 2-1 is shown in Figure 2-5. The components downstream

of the control valve and the control valve itself are not considered in the model because

the control valve is in the shut position and separates these components from any

upstream pressure changes.

Figure 2-5: Hydraulic System Schematic

Based on the schematic shown in Figure 2-5, the movement of the actuator piston due to

the applied impulse force, , causes an increase of the fluid pressure, , in the systemand flow rate out of the actuator, as explained in section 2.3. The fluid flowing outof the actuator then causes a flow rate into the accumulator, . The pressure of theliquid, , is assumed to be the same at every point in the system.2.4.1 Hydraulic System Initial ConditionsThe initial conditions of the system shown in Figure 2-5 were assumed to be the

following.1. The initial position of the accumulator piston,, is zero2. The initial velocity of the accumulator piston, , is zero3. The initial position of the actuator piston, , is equal to stroke of the actuator, 4. The initial velocity of the actuator piston, , is zero


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5. The system is initially at equilibrium. This means that a balance of forces existsbetween the impulse force, , the atmospheric pressure, , the liquid pressure,, and the gas pressure, .

6. The acceleration of the accumulator piston,

, is zero7. The acceleration of the actuator piston, , is zero2.4.2 Hydraulic System Behavior EquationsThe equations that model the behavior of the hydraulic system shown in Figure 2-5, are

a combination of the equations that model the behavior of the accumulator and actuator

in section 2.2.2 and section 2.3.2 respectively. There exists more than one approach to

combining the equations of the previous sections to obtain the equations that model the

behavior of the hydraulic system. This section will only describe one possible

combination.

Using Figure 2-5, the equations for the behavior of the system can be obtained

with a given function for . The behavior of the accumulator will be characterized by theposition, , velocity, , and acceleration, , of the accumulator piston, and thebehavior of the actuator will be characterized by the position, , velocity, , andacceleration, , of the actuator piston. The first values obtained are the volume, , andpressure,, of the gas in the accumulator, using Equations (1) and (2) respectively.Using the pressure of the gas in the accumulator, the liquid pressure, , of thesystem can be obtained by rearranging Equation (4) as shown in Equation (13).

(13)Using , obtained from Equation (13), the acceleration of the actuator piston,, can be obtained from Equation (9). The actuator piston velocity and position can

then be obtained using Equations (10) and (11) respectively.

Since the hydraulic system being evaluated is an isolated system, the flow rateout of the actuator, , is equal in magnitude but opposite in direction to the flow rateinto the accumulator, . Using the relationship between the accumulator pistonvelocity and flow rate in Equation (8) and the actuator piston velocity and flow rate in

Equation (12), Equation (14) is used to obtain the accumulator piston velocity.


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(14)Using the velocity of the accumulator piston, the position and acceleration of the

accumulator piston can be obtained using Equations (6) and (5) respectively.

2.5 Euler MethodThe Euler method is a first order accurate, single-step numerical procedure useful for

solving initial value problems for ordinary differential equations. Using the initial

conditions given in section 2.2.1 and section 2.4.1 for the accumulator and the complete

hydraulic system respectively, the Euler method was used as one numerical approach to

obtain the behavior of the two models. The equations used with the Euler method to

model the accumulator and the hydraulic system are shown in Appendices C and H

respectively.

2.6 Runge-Kutta MethodThe Runge-Kutta method is a fourth order accurate, single-step numerical procedure

useful for solving initial value problems for ordinary differential equations. Similar to

the Euler method, using the initial conditions for the accumulator and the complete

hydraulic system, the fourth order Runge-Kutta method was used as a second numerical

approach to obtain the behavior of the two models. The equations used with the Runge-

Kutta method to model the accumulator and the hydraulic system are shown in

Appendices D and I respectively.

2.7 FLOWMASTER MethodFLOWMASTER is a one-dimensional computational fluid dynamics software that

allows the modeling and analysis of complex piping systems. The software can calculate

pressure transients and flow rates in piping networks. It can also be used to conduct

pressure drop calculations, heat transfer analysis and analyze system operations such as

load inputs and pump curves. The piping networks are drawn onscreen using a list of

piping components, such as pipes, actuators, pumps, orifices and pipe fittings, and

boundary devices, such as pumps, tanks or pressure/ flow inputs. More information on

this software can be found in Reference [11].


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FLOWMASTER is capable of modeling compressible and incompressible piping

systems. To model the accumulator in Figure 2-3 and the hydraulic system in Figure 2-5,

the incompressible license of the software was used. This was done for two reasons. The

first reason is that according to assumption four in section 2.1, the behavior of the liquid

in the hydraulic system being modeled is considered incompressible. Having the

FLOWMASTER simulation and the numerical models use the same assumption will

allows for better comparison between results. The second reason is that although the

accumulator in the hydraulic system being modeled contains a compressible gas, in

FLOWMASTER, the behavior of the gas in the accumulator is built into the components

and not dependent on the simulation license being used. Additionally, several of the

accumulators simulated in FLOWMASTER are required to be used with an

incompressible license.

To simulate the accumulator system shown in Figure 2-3, three components were

used. The accumulator itself was simulated using a FLOWMASTER component called

an air vessel. Several different components are available in the FLOWMASTER

components menu under the accumulator category. However, none of the components

exactly matched the features of a piston-type accumulator. The air vessel components

comes the closest to matching the properties and behavior of a piston-type accumulator.

However, the air vessel component lacks a piston. The air vessel component does not

model weight and friction due to piston inside an accumulator. Due to the large pressures

which will be simulated, it is expected that the accumulator piston weight or friction in

the model would have a negligible effect on the results.

To simulate the pressure input into the system, a component called a pressure

source was used. This component allows a user to set the pressure input into the system

and changes the flow rate into the system to match the required pressure input.

Additionally, it is necessary to simulate a hydraulic pipe between the pressure source

and the air vessel. Pressure nodes, which connect components in FLOWMASTER, store

and transfer pressure data between components during a simulation. These nodes do not

transfer flow rates. Therefore it is necessary to simulate a pipe between any two

components in piping network in order for the simulation to work properly. The


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FLOWMASTER model for the simulation of an accumulator is shown in Figure 2-6.

The properties for each component in Figure 2-6 are listed in Appendix E.

Figure 2-6: FLOWMASTER Accumulator Simulation Graphic

To simulate the hydraulic system shown in Figure 2-5, the accumulator was

simulated using an air vessel component as explained above. The single acting linear

hydraulic actuator was simulated using a single-rod double-acting cylinder component.

A pressure-source component was attached to the rod side of the cylinder and was set to

provide a constant pressure equal to atmospheric pressure. This combination of

components was used to ensure that the rod side of the actuator was properly simulated.

The blind side of the cylinder component and the air vessel component were connected

by a pipe due to the built in functionality of FLOWMASTER as explained above. To

simulate the impulse force acting on the actuator a cylinder load curve controller

component was used. This component connects to the cylinder component and can be

used to exert an external force on the cylinder. The FLOWMASTER model for the

simulation of the hydraulic system is shown in Figure 2-7. The properties for each

component in Figure 2-7 are listed in Appendix J.

Figure 2-7: FLOWMASTER Hydraulic System Simulation Graphic


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3. Results and DiscussionThe following is the comparison of the modeled behavior of the accumulator and the

complete hydraulic system using the three methods discussed in section 2. The modeling

equations for the Euler and Runge-Kutta numerical methods were programmed intoMicrosoft Excel spreadsheets.

For modeling the behavior of an accumulator, the pressure transient of the liquid

was characterized by Equation (15).

(15)This pressure transient was obtained from an example evaluation in Reference [12]. The

duration of the pressure transient was chosen such that an adiabatic condition can be

assumed .Other properties used to model the system are in outlined Appendix A.For modeling the behavior of a potential hydraulic system used in a dump truck

suspension system, the impulse force acting on the actuator was characterized by

Equation (16).

(16)

The minimum and maximum values of the of the impulse force were assumed to be theempty and loaded conditions for the rear axle of a Caterpillar 740 articulated dump truck

[13]. The force was assumed to cycle to create a situation where the accumulator

behavior can be modeled. The duration of the impulse force was chosen to be equal to

the duration used for accumulator behavior modeling for consistency and so that an

adiabatic condition can be assumed.

3.1 Comparison of Accumulator BehaviorEach of the three methods used to model the accumulator behavior were attempted using

four time steps, = 0.05, 0.005, 0.0005 and 0.00005. The largest time step, = 0.05, waschosen based on Reference [3] which states that the most reliable simulations require at

least eight distance intervals per wavelength or eight time intervals per period

(Reference [3], page 262). Since the period, or total time, of the pressure increase was


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0.5 seconds, the largest time step chosen was . Smaller time steps werethen chosen to see if the output of each consecutive model changed. Each consecutive

time step used is an order of magnitude smaller than the previous one. A time step

smaller than

= 0.00005 was not used because it would require more than 32,000 data

points which is the maximum number of data points which can be graphed in Microsoft

Excel 2007 edition. If the numerical method cannot be easily modeled and manipulated

in this software, then for the purposes of this project it is impracticable. The accumulator

behavior was modeled using the following attributes: accumulator piston position,

accumulator piston velocity, and gas pressure.

3.1.1 Model Comparisons for = 0.05The Euler method, using the = 0.05 time step, became unstable and began to outputinvalid numerical values at of the model. The Runge-Kutta method alsobecame unstable and began to output invalid numerical values at of themodel. The FLOWMASTER simulation showed no change in the piston position over

the entire simulation and was also considered invalid.

3.1.2 Model Comparisons for = 0.005The Euler Method, using the

= 0.005 time step, became unstable and began to output

invalid numerical values at of the model. The Runge-Kutta method andthe FLOWMASTER simulation outputted stable results using this time step.

Figure 3-1 shows the accumulator piston position and velocity for the Runge-

Kutta method and the FLOWMASTER simulation.

Figure 3-2 shows the pressure of the gas in the accumulator during the pressure

transient for the Runge-Kutta method and the FLOWMASTER simulation.


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Figure 3-1: Accumulator Piston Behavior Comparison h=0.005

Figure 3-2: Accumulator Gas Pressure Comparison h=0.005


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3.1.3 Model Comparisons for = 0.0005Using the = 0.0005 time step, all three modeling methods outputted stable results.Figure 3-3 shows the accumulator piston position and velocity for the each of the three

methods. Figure 3-4 shows the pressure of the gas in the accumulator during the pressuretransient for the each of the three methods.



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methods. Figure 3-6 shows the pressure of the gas in the accumulator during the pressure

transient for the each of the three methods.


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3.1.5 Discussion of Accumulator Behavior ResultsTo compare the three separate methods to each other and to compare each method to

itself at each of the time steps, the accumulator piston position was first used. The

accumulator velocity and the gas pressure were then used to confirm the comparison.

Comparing the accumulator piston position results in Figure 3-1, Figure 3-3 and

Figure 3-5, several observations can be made. The first observation is that the Runge-

Kutta method output similar piston position graphs for each of the three time steps. The

second observation is that the piston position at each time step does not match for the

Euler method. However, it can be observed, from Figure 3-3 and Figure 3-5, that the

Euler method results overlap the Runge-Kutta method results as the time step was

reduced.

For the FLOWMASTER simulation, the accumulator piston position results

shown for each of the three time steps do not overlap well. The average values of the

piston positions and the overall vector of the graphs overlap, however the actual do not

overlap. As the time step decreases, the accumulator piston position graphs diverge from

one another. It is also observed that the overall vectors of the accumulator piston

position results for the two numerical methods match the FLOWMASTER simulation.

However, the results for the Euler method and the Runge-Kutta method do not overlap

the FLOWMASTER simulation results.

The observations outlined above are the same for the accumulator piston

position, accumulator piston velocity and the gas pressure.

3.2 Comparison of Complete Hydraulic System BehaviorThe hydraulic system described in section 2.4 was modeled using the Euler method, the

Runge-Kutta method, and a FLOWMASTER simulation. Four time steps were used for

each of the methods. The time step used were,

= 0.05, 0.005, 0.0005 and 0.00005. The

reasoning behind the use of these time steps to model the complete hydraulic system is

identical to the reasoning described for the accumulator behavior in section 3.1. The

hydraulic system behavior was modeled using the following attributes: accumulator

piston position, accumulator piston velocity, and gas pressure. Other behaviors, such as


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the actuator piston position and velocity were analyzed and will be discussed; however

those attributes were not graphed.

3.2.1 Model Comparisons for = 0.05Using the = 0.05 time step, all three modeling methods outputted stable results. Figure3-7 shows the accumulator piston position and velocity for the each of the threemethods. Figure 3-8 shows the pressure of the gas in the accumulator during the impulse

force for the each of the three methods. Figure 3-8 also shows the liquid pressure in the

hydraulic system assuming the liquid is incompressible and an accumulator is not

installed. This information was included in the graph to show the benefit of including an

accumulator in the system only.

Figure 3-7: Hydraulic System: Accumulator Piston Comparison h=0.05


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Figure 3-8: Hydraulic System: Gas Pressure Comparison h=0.05

3.2.2 Model Comparisons for = 0.005Using the = 0.005 time step, all three modeling methods outputted stable results. Figure3-9 shows the accumulator piston position and velocity for the each of the three

methods. Figure 3-10 shows the pressure of the gas in the accumulator during the

impulse force for the each of the three methods. Figure 3-10 also shows the liquid

pressure in the hydraulic system assuming the liquid is incompressible and an

accumulator is not installed.


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methods. Figure 3-12 shows the pressure of the gas in the accumulator during theimpulse force for the each of the three methods. Figure 3-12 also shows the liquid





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methods. Figure 3-14 shows the pressure of the gas in the accumulator during the

impulse force for the each of the three methods. Figure 3-14 also shows the liquid




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3.2.5 Discussion of Hydraulic System Behavior ResultsTo compare the three separate methods to each other and to compare each method to

itself at each time step, the accumulator piston position was first used. The accumulator

velocity and the gas pressure were then used to confirm the comparison. Additionally,

the pressure of the gas in the accumulator and the pressure of the liquid in the system

were within less than 0.3% of each other in value in all simulations.

Comparing the accumulator piston position results in Figure 3-7, Figure 3-9,

Figure 3-11, and Figure 3-13, several observations can be made. The first observation is

that both the Euler method and the Runge-Kutta method output similar piston position

results for at each of the four time steps. For time steps = 0.05 and 0.005, the results ofthe numerical methods do not overlap and both results change with decreasing time step.

For the following two time steps used, the numerical method results overlap at each time

step and the results at each of the time steps overlap each other.

For the FLOWMASTER simulation, the change in the accumulator piston

position decreases with each decrease in time step. When the FLOWMASTER

simulation is run at smaller time steps than the ones discussed, the accumulator piston is

observed to be stationary. The FLOWMASTER results closely match the numerical

methods results for = 0.005, but diverge as the time step is reduced.The observations outlined above are the same for the accumulator piston

position, accumulator piston velocity and the gas pressure.


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FLOWMASTER simulation, it is inconclusive whether the numerical approach is

accurate. However, the numerical approach does produce an expected output for the

behavior of the hydraulic system, while the FLOWMASTER simulation produces an

output which shows no effect from the impulse force as the time step size is reduced.

Therefore, it would be premature to argue that the numerical method outlined in this

project does not produce an accurate output for the behavior of an accumulator.

4.1 Suggestions for Further ResearchDue to the disagreement between the outlined numerical method and the

FLOWMASTER simulation results, further research needs to be conducted to determine

the whether the numerical approach or the FLOWMASTER simulation is inaccurate.

Further research can be conducted in determining the accuracy of the numericalapproach by either comparing the results to the results of different computational fluid

dynamic software, or comparing it to experimentally obtained values.

Currently little information exists on the subject of simulating the behavior of

accumulators for any application. If subsequent computational fluid dynamic software

confirms the accuracy of the outlined numerical approach, then suggested further

research would include determining how much of an effect the assumption used in

section 2.1 have on the results. The research can concentrate on how the output would be

affected if the liquid in the hydraulic system was considered compressible, the air in the

accumulator was treated as a real gas, or the compression of the gas was not considered

an adiabatic process.


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5. References[1] Massey BS. Mechanics of Fluids 6thedition. Padstow, Cornwall: T.J. Press Ltd;

1989. 599 p.

[2]

Merritt HE. Hydraulic Control Systems. New York; John Wiley and Sons, Inc.;1967. 358 p.

[3] Wylie EB and Streeter VL. Fluid Transients in Systems. Upper Saddle River, NJ:Prentice Hall; 1993. 463 p.

[4] Accumulator Circuits. Hydraulics and Pneumatics. 2012. Available fromhttp://hydraulicspneumatics.com/200/techzone/accumulators/article/false/6468/te

chzone-accumulators. Accessed 2012 February 15.

[5] Ijas, M. Damping of Low Frequency Pressure Oscillation. Tampere University ofTechnology. 2007.

[6] Weemac Brochure. WEEMAC Oy. 2011. Available fromhttp://www.weemac.fi/media/weemacbroschure9_11.pdf. Accessed 2012 April 7.

[7] Zahid Z. Using Accumulators to Smooth Hydraulic Transients. Machine Design.1977; 16:75-77.

[8] Ghidaoui MS, Zhao M, McInnis DA and Axworthy DH. A Review of WaterHammer Theory and Practice. Applied Mechanics Reviews. 2005; 58:49-76.

[9] Accumulators Part 1, Accumulators can increase efficiency, provide smoother,more reliable operation, and store emergency power in case of electrical failure.

Hydraulics and Pneumatics. 2012. Available from

http://hydraulicspneumatics.com/200/TechZone/Accumulators/Article/False/644

6/TechZone-Accumulators. Accessed 2012 February 15.

[10] Fox RW, McDonald AT, and Pritchard PJ. Introduction to Fluid Mechanics 6 thedition. New Jersey; John Wiley and Sons, Inc.; 2004. 787 p.

[11] Welcome to FLOWMASTER. FLOWMASTER Group. Available fromhttp://www.flowmaster.com/index.html. Accessed 2012 March 25.

[12] Accumulators: Application and Selection. EPE Process Filters and AccumulatorsPVT LTD. 2012. Available from

http://www.accumulatorsandfilters.com/accumulators/Appln.pdf. Accessed 2012,

April 7.


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[13] RitchieSpecs: Caterpillar 740 Articulated Dump Truck. RichieSpecs EquipmentSpecifications Richie Bros. Auctioneers. 2012. Available from

http://www.ritchiespecs.com/specification?type=&category=Articulated+Dump+

Truck&make=Caterpillar&model=740&modelid=91910. Accessed 2012 April 7.

[14] Piston Type Accumulators: Type-AP. EPE Process Filters and AccumulatorsPVT LTD. 2012. Available from

http://www.accumulatorsandfilters.com/accumulators/Piston.pdf. Accessed 2012

March 10.

[15] Hydraulic Cylinders: WW: Specification. WEEMAC Oy. 2011.Available fromhttp://www.weemac.fi/pages/products/hydraulic-cylinders/ww/specification.php.

Accessed 2012, March 25.


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Figure A-1: Bore Size 125 mm, Model AP-10 Dimension [14]

B. Differential Equations for Accumulator BehaviorThe following three equations make up the differential equations which govern the

behavior of the system. Equation (17) is the combination of Equations (2), (4), and (5).

Equation (18) is a combination of Equations (3) and (7).

(17) (18) (6)These equations will obtain the volume of the gas in the accumulator and the

accumulator piston position and velocity. Any other variables, such , can be solved forby converting from another variable.


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C. Accumulator Behavior Simulation using Euler MethodThe following are initial conditions which were assumed for simulating the accumulator

behavior.

Using the above initial conditions and the input variables in Appendix A, the following

equations were solved for in the order listed below.1. (2)2. (3)3. (4)4. (5)5. (6)6. (7)7.

Figure C-1: Print Screen of Spread Sheet using Euler Method with h=0.005

D. Accumulator Behavior Simulation using Runge-Kutta MethodThe following are initial conditions which were assumed for simulating the accumulator

behavior.


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The system of equations which were solved using the Runge-Kutta method was made up

of the following two equations. The Equation (19) was obtained by combining Equations

(1), (2), and (4).

(19) (5)

Using the above initial conditions and the input variables in Appendix A, the following

equations were solved at for in the order listed below.1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. (3)13. (2)


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Figure D-1: Print Screen of Spread Sheet using Runge-Kutta Method with h=0.005

E. Accumulator Behavior Simulation using FLOWMASTERFigure 2-6 shows the FLOWMASTER model that was created to simulate the

accumulator. The model was made using three components and two nodes. Component

one is an air vessel, which was used to simulate the accumulator. The properties of the

air vessel are listed in Table E-1. Component two is a rigid pipe whose properties are

listed in Table E-2. Component three is a pressure source with a pressure versus time

output equal to the Equation (15). The reason for choosing these components to simulate

the accumulator is explained in section 2.7.

Table E-1: FLOWMASTER Accumulator Simulation Component 1: Air Vessel

Feature Value RemarkGas Pressure 350,000 Pa Precharge pressure

Liquid Level 0 m Initial condition for simulation

AccumulatorHeight

0.815 m Equal to - value in Table A-1Diameter of Inlet 0.1 m Chosen to minimize friction and pressure drop

Base Level above

Reference0 m Chosen to eliminate/ eliminate gravitational effects

Outflow Loss 0 Chosen to eliminate friction and pressure drop

Inflow Loss 0 Chosen to eliminate friction and pressure drop

Liquid Type Water

Polytropic Index 1.4 Equal to - value in Table A-1Horizontal cross-sectional area

0.0123 m2 Equal to - value in Table A-1

Table E-2: FLOWMASTER Accumulator Simulation Component 2: Rigid Pipe

Feature Value Remark

Absolute

Roughness1E-12 mm Chosen to eliminate friction and pressure drop


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Friction Factor 0.02 Chosen to eliminate friction and pressure drop

Diameter 1 m Chosen to eliminate friction and pressure drop

Length 1 m

F. Input Variables for Hydraulic System SimulationThe following table lists the input variables needed to use the numerical method to

model the behavior of an accumulator and the values used.

Table F-1: Input Variables for the Hydraulic System Simulation

Variable Value Source

0.0165 m2 Based Cylinder Dimensions 125/70, Model WW[15] 0.0491 m2 Based Accumulator bore size 250 mm, ModelAP-100 [14]

0.0038 m2 Based Cylinder Dimensions 125/70, Model WW[15]

0.05 s0.005 s

0.0005 s

0.00005 s

Several time steps were chosen to see the effect of

time step size to modeled behavior. See section3.1 for discussion.

2,440,509,370 Pa*liter1.4 , where 2.037 m 6380 kg Mass acting on the center axle of a Caterpillar 740Articulated Dump Truck [13] 57.9 kg

Based on Accumulator bore size 250mm Model

AP-100, aluminum density of 2810 kg/m3andFigure A-1 scaled dimension [14] 101,353 Pa Standard Atmospheric Pressure

3 m Stroke chosen for the actuator to make sure theactuator piston does not reach the end of theactuator during the impulse force.

100 liters Based Accumulator bore size 250 mm, ModelAP-100 [14] 1.4 Ratio of Specific Heat for AirG. Differential Equations for Hydraulic System BehaviorThe following five equations make up the differential equations which govern the

behavior of the hydraulic system. Equation (20) is the combination of Equations (2), (9),

(10) and (13).


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(20)

(5) (6) (14) (11)

These equations will obtain the position and velocity of the actuator piston and the

position, velocity, and acceleration of the accumulator piston. Any other variables, such

, , or flow rate, can be solved for by converting from another variable.H. Hydraulic System Behavior Simulation using Euler MethodThe following are initial conditions which were assumed for simulating the behavior of

the hydraulic system.

Using the above initial conditions and the input variables in Appendix F, the following

equations were solved for in the order listed below.1. (1)2. (2)3.

(13)

4. (9)5. (14)6. (5)


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7. (6)8. (10)9. (11)

Additionally, for comparison purposes the pressure of the liquid was calculated

assuming no accumulator in the system. In this case, the pressure of the liquid would be

equal to the sum of the force as shown in Equation (21).

10. (21)

Figure H-1: Print Screen of Spread Sheet using Euler Method with h=0.005

I. Hydraulic System Behavior Simulation using Runge-Kutta MethodThe following are the initial conditions for the Runge-Kutta method were assumed for

simulating the behavior of the hydraulic system.


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The system of equations which were solved using the Runge-Kutta method was made up

of the following three equations.

(20) (14) (10)

Using the above initial conditions and the input variables in Appendix F, the following

equations were solved at for in the order listed below.1.

2. 3. 4.

5. 6. 7.

8. 9.


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10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Figure I-1: Print Screen of Spread Sheet using Runge-Kutta Method with h=0.005

J. Hydraulic System Behavior Simulation using FLOWMASTERFigure 2-7 shows the FLOWMASTER model that was created to simulate the hydraulic

system. The model was created using five components and three nodes. The

components shown are as follows (1) single-rod double acting cylinder, (2) rigid pipe,

(3) pressure source, (4) air vessel, and (5) cylinder load curve controller. Properties for


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components one, two and four are listed in Table Table J-1, Table J-2, and Table J-3

respectively. Component three is used to simulate a constant atmospheric pressure, value

given in Table F-1, on the rod side of the actuator. Component five is used to simulate a

force versus time output equal to Equation (19) onto the actuator cylinder. The

properties of these components are listed in the tables below. The reason for choosing

each of these components to simulate the system is explained in section 2.7.

Table J-1: FLOWMASTER Hydraulic System Simulation Component 1: Single-

Rod Double-Acting Cylinder


Port 1 Diameter 1 m Chosen to eliminate friction and pressure drop

Port 2 Diameter 1 m Chosen to eliminate friction and pressure drop

Rod Diameter 0.07 m Diameter equivalent to

in Table F-1

Cylinder Diameter 0.145 m Diameter equivalent to in Table F-1Max Rod Travel 3 m Equal to - value in Table F-1Initial Rod Position 3 m Initial condition used for Hydraulic System

Mass of Piston 6380 kg Equal to - value in Table F-1Table J-2: FLOWMASTER Hydraulic System Component 2: Rigid Pipe


Absolute

Roughness1E-12 mm Chosen to eliminate friction and pressure drop

Friction Factor 0.02 Chosen to eliminate friction and pressure drop

Diameter 1 m Chosen to eliminate friction and pressure dropLength 1 m

Table J-3: FLOWMASTER Hydraulic System Component 4: Air Vessel


Gas Pressure 3,867,950 Pa Precharge pressure

Liquid Level 0 m Initial condition for simulation

Accumulator

Height2.03718 m Equal to - value in Table F-1

Diameter of Inlet 1 m Chosen to minimize friction and pressure drop

Base Level above

Reference

0 m Chosen to eliminate/ eliminate gravitational effects

Outflow Loss 0 Chosen to eliminate friction and pressure drop

Inflow Loss 0 Chosen to eliminate friction and pressure drop

Liquid Type Water

Polytropic Index 1.4 Equal to - value in Table F-1

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