Physical Modelling, Simulation and Experimental Analysis

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 122

190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S

Physical Modelling, Simulation and Experimental

Analysis for Synchronizing Multiple Hydraulic

Actuators Olayinka Mohammed Olabanji, Khumbulani Mpofu

Abstract-- This paper focuses on theoretical design and

experimental investigation of multiple hydraulic actuators used

for precise positioning of elements in a reconfigurable assembly

fixture. Design analysis of the hydraulic system is achieved by

deriving mathematical models of the components. A physical

model of the hydraulic system was developed in SimHydraulics

using Matlab-Simulink. The developed model was

parameterized using the result obtained from the design

analysis. The responses of the actuators were obtained for

different input signals at the directional control valves.

Experiment was carried out on an electrohydraulic test bench

in order to observe the performance of the system, and confirm

the synchronized extension and retraction of the actuators.

Results obtained from the simulation and experiment are

presented graphically and discussed extensively.

Index Term-- Hydraulic actuators, Position synchronization,

Electrohydraulic System, Physical Modelling and Simulation

1. INTRODUCTION

Industrial machineries such as reconfigurable assembly

fixtures (RAFs), manufacturing and laboratory test

equipment, robots, automobile and aeronautical equipment

requires the use of hydraulic actuators. The core reason for

the application of hydraulic actuator is the production of high

force compare to the overall weight of the system [1].

Another advantage is high precision control of variable

speed, force and displacement [2, 3]. Hydraulic actuators play an important role in an electrohydraulic system [4, 5]. They

convert the fluid energy obtained from the pumping system

through the electrohydraulic valve, and allow flexibilities in

operations [6, 7]. The electrohydraulic valve can be a

proportional control valve or a directional control valve. Most

applications involving multiple actuators requires effective

synchronization [8]. Basically, there are three approaches to

achieve synchronization of multiple hydraulic actuators.

These approaches are, the use of flow dividers circuit,

mechanical connection of the actuators by linkage design and

electrohydraulic synchronization [9]. In literature,

electrohydraulic systems comprising of multiple hydraulic actuators uses several proportional control valves or flow

dividers to ensure that the displacement of the actuators are

synchronized [10, 11]. The use of several proportional

control valves is not cost effective and it is tedious during the

automation and valve sequencing. Despite the fact that the

electrohydraulic system produce high force, controllable

speeds, force and displacement, they still have some intrinsic

non-linear effects which make it difficult to control them.

They are properly modelled to ensure that the output response

is similar to the actual input or desired action [12, 13]. In

essence the research on synchronization of multiple hydraulic actuators has focused on the use of several flow dividers and

multiple proportional control valves for each of the actuators

in the system (Vasiliu et al., 2004). In this paper, we will

consider the synchronization of multiple hydraulic actuators

using a directional control valve, three way flow control

valves, and pressure reducing valves. The rest of the paper is

structured as follows. In the next section, a brief description

of the RAF using the electrohydraulic system is presented. In

section three, model equations for the components of the

hydraulic system is obtained. In section four, a physical

model of the electrohydraulic system is developed in Simulink (Simscape-SimHydraulics) and the simulation

results are presented graphically in section 5. In section six

the experimental set up for the electrohydraulic system is

presented with the results obtained from the experiment.

Finally conclusions are made based on the results obtained.

An Introduction should provide a review of the recent

literature on the topic and sufficient background information

to allow the results of the article to be understood and

evaluated.

2. DESCRIPTION OF THE RAF A computer aided design of the reconfigurable assembly

fixture (RAF) is shown in Fig. 1 [14]. RAFs are required to

assemble products with varying dimensions. They are

precision equipment used to completely immobilize varying

dimensions of a product within the reconfigurable limit [1].

The electrohydraulic system of the RAF is divided into two

subsystems. Subsystem one has one directional control valve,

eight pressure reducing valves, eight flow control valves and

four hydraulic actuators (called the finger cylinders FC). The

finger cylinders are controlled synchronously to position the

fingers so as to fit the dimensions of the workpiece.

Subsystem two also has one directional control valve, four pressure reducing valves, four flow control valves and two

hydraulic actuators (called the movable frame cylinders

MFC). The movable frame cylinders are also controlled

synchronously to move the movable jaw of the fixture. The

directional control valves are driven by solenoids. The

solenoids operating the first hydraulic subsystem is A and C

for extension and retraction respectively, while the solenoids

operating the second hydraulic subsystem is B and D for

extension and retraction respectively, as presented in Fig 2.



Fig. 1. Pictorial View of the RAF

3. MODEL EQUATIONS FOR COMPONENTS OF THE

HYDRAULIC SYSTEM

In order to obtain a physical model of the electrohydraulic

system it is necessary to derive the mathematical model of

the components in the system. The parameters of the system

considered in the formulation are: the pressure in each hydraulic subsystem, volumetric flow rate of pump, valve

spool equation, valve coefficients and gains and the damping

force due to transient flow. Other parameters considered are:

the net force on the piston, the diameter of the actuator and piston rod, the resistive force and leakage flow coefficient,

the elastic load stiffness and coefficient of viscous friction.

Also, the mean flow rate of hydraulic fluid into the actuators

and the acceleration of the piston were also obtained.

The directional control valves are sized in order to obtain

effective performance. The model equations are used

separately for each subsystem. However the values of the

parameters in the model differs for each subsystem. The

pressure in each subsystem is a function of the total dynamic

column, and the inlet pressures of the actuators. The total

dynamic column can be obtained from the static discharge column of the piping system, discharge column diameter,

head loss and velocity head in the discharge column [15-18].

The pressures in subsystems one and two can be deduced in

equations 1 and 2 respectively.

1

n=41

1 nn=1

PI FC

dc

s

c f dc c

ρgS +1

P =

d 4C S + d P - P

(1)

2

i=22

2 ii=1

PI MFC

dc

s

c f dc c

ρgS +1

P =d 4C S + d P - P

(2)

The volumetric flow rate of hydraulic fluid from the pump to

all the actuators in the system can be deduced in equation 3.

It depends on the power input to the pump and drive

efficiency. It is also a function of the total dynamic column

in each subsystem and the specific gravity of the hydraulic

fluid [19-21].

1 2

d in

g

p

tc tc

0.1γ PQ =

D + D S

(3)

The displacement of the spool is initiated by an

electromagnetic force created by the solenoid [22, 23]. This electromagnetic force depends on rate of change of

inductance with respect to spool displacement and current

through the solenoid. The current flowing through the coil is

also a function of the applied voltage. Furthermore, the

current is a function of resistance and inductance of the coil.

The inductance depends on the spool position of the valve.

The spool of the valve is driven by the plunger of the

solenoid. This force must be able to overcome the mass of the

spool, force spring rate of the spool and the damping

coefficient due to transient flow force. If subsystems one and

two are denoted with subscript 1 and 2, then the equation of

motion of the valve spool force can be expressed from Newton’s second law as; [24-26].

s s

ss=1,2 s s s

2

sp sp ct sr2

d U dUF = M + d + U F

dtdt (4)

The damping coefficient due to transient flow force is also

deduced in terms of the pressure requirements of the actuators

in the subsystems and the valve symmetry as shown in Fig 3.

0.5

l 1 2

n=41

1 nn=1

s s

ct 2 1 d

s FC

ρ P + P -

d = L - L C w

P - P

(5)

0.5

l 1 2

i=22

2 ii=1

s s

ct 2 1 d

s MFC

ρ P + P -

d = L - L C w

P - P

(6)

As the spool is displaced, a net pressure is created. This net pressure must overcome the inertia of the pistons in each

actuators of the subsystems. In order to obtain a synchronized

displacement the effect of this net pressure is uniformly

distributed by the three way flow control valves and pressure

reducing valve which is proposed in this article [26, 27].

The directional control valves are considered as a matched

and symmetrical valve as shown in Fig 3. In order to supply

hydraulic fluid to the actuators, it is necessary to determine

the valve flow coefficient (VFC), valve gain (Vg) and

pressure coefficient (PC) in terms of the actuator

requirements and the flow rate in each subsystem [7]. Using

subscripts one and two for subsystems one and two respectively, the valve flow coefficient, valve gain and

pressure coefficient of the directional control valves can be

deduced in equations 7-12.

FC Fingers

MFC



Fig. 2. Schematic Diagram of the RAF showing all Hydraulic components

0.5-1

n=41

1 1 nn=1

p

r

FC g s FC

F

V = S P - PNF

(7)

0.5-1

i=22

2 2 ii=1

p

r

FC g s MFC

F

V = S P - PNF

(8)

*

0.5

n=4

1 1 2 1 nn=1

l

d

s s s FCg

πLC 1V = P + P - P - P

6 ρ (9)

*

0.5

i=2

2 1 2 2 ii=1

l

d

s s s MFCg

πLC 1V = P + P - P - P

6 ρ (10)

*

-0.5n=4

21 1 2 1 nn=1

C FC

l

s s s

πLCdP = P + P - P - P

12ρ

(11)

*

-0.5i=2

22 1 2 2 ii=1

C MFC

l

s s s

πLCdP = P + P - P - P

12ρ (12)

The diameter of the actuator is a function of the external load

pushed, the pressure in the system and the pressure developed

in the actuator due to the movement of the piston. It is

expected that the pressure in each subsystem will be

uniformly distributed into all identical actuator in the system.

This will be achieved by the pressure reducing valves

controlling each of the actuator. Again, if subscript one and

two is used to represent subsystems one and two, then the diameter of each actuator in subsystems one and two can be

expressed by equation 13;

0.5

s

s s=1,2s s

actact

act s

4Fd =

π P + P (13)

Furthermore, the net force on the piston is a summation of

acceleration force, stiffness force and force due to viscous

friction. The acceleration force on the piston is a function of

Pressure

reducing valve 3 way Flow

control valve

DCV (A)

DCV (B)

SOL D

SOL

B

SOL

C SOL

A

Finger

Cylinders Movable

Frame Cylinder

Subsystem 2

Subsystem 1

Pumping

Unit

Fingers

Extension

line

Return

Line



the external load pushed. In order to obtain perfect

synchronization, the load pushed by identical actuators in any

subsystem must be equal. The diameter of the piston rod is a

function of the external load pushed, diameter of piston, and

the free buckling length of the piston rod at full extension

considering the type of mounting arrangement. In essence the piston rod diameter of the actuators in the subsystems is

deduced in equation 14;

-12

s

ss s=1,2s

fbcpr s act

act

Ld = 4F F πσ 1- a

d (14)

The leakage flow coefficient is deduced from the volumetric

flow rate of hydraulic fluid entering the actuator and the

pressure in the subsystem. The volumetric flow rate of

hydraulic fluid entering and leaving each identical actuator is

controlled by the three way flow control valves in order to obtain synchronization. Equations 15 and 16 presents the

leakage flow coefficient for each hydraulic actuator in

subsystems one and two respectively [28].

Fig. 3. Schematic diagram of the directional control valves

FC 1

1

1

1 P e

Lf

s FC

Q - A V

C =P - P

MFC 2

2

2

2 P e

Lf

s MFC

Q - A V

C =P - P

The resistive force due to viscous friction is a function of the

viscous friction coefficient and the velocity of the piston. It

is linearly proportional to the velocity and contact area of the piston, and inversely proportional to the clearance between

the piston and the cylinder wallcl

y . The force produced by

the piston must overcome this resistive force in order to

achieve motion in the actuator [29]. This resistive force can

be represented by equation 17. The total coefficient of

viscous friction of each actuator can also be obtained from

equation 18; [30].

l l ss

s s=1,2

s

act e

res

cl

υ ρ A V

F =y

FressVes

l l

s

s s=1,2

s

act

v

cl

υ ρ A

b =y

The elastic load stiffness of the actuator is an important factor

in the dynamic performance of the actuator. It depends on the

volume of hydraulic oil in the hoses connected to the piston

n

n=4 n

n=1n 1

n 1

FC

FC

1

O np

T n p

X

A +Pt

Q =V P

+ K P2β t

(20)

i

i=2 i

i=1i 2

i 2

MFC

MFC

2

O np

T np

X

A +Pt

Q =V P

+ K P2β t

(21)

Furthermore, since it is expected that the dimensions (such as

area, volume and travel length), displacement, viscous

friction and acceleration force of all identical actuators are

uniform, then the total supply from the directional control

valve will be an equal distributive portion of individual

actuator in a particular subsystem through the three way flow

control valve and pressure reducing valves. In order for the piston of the actuators to translate, the net force on it must be

greater than the summation of the acceleration force, viscous

friction and the stiffness force. This net force is a function of

the piston area and the differential pressure across the ports

of the actuators. Since the aim of this modelling is to

synchronize the position of the actuators, then the

acceleration of the pistons will be obtained from the net force

acting on the piston as presented in the dynamic equations 22

and 23 [32].

12n

n

2n

n=1-4 1 n

npFC

FC

FCF

le FC

A P -PX 1

=X

Mtb - K Xv

t

(22)

L

Spool

Displacement

initiated by a

solenoid Port 3

Port 4

Port 1

Port

2



22i

i

2i

i=1-2 2 i

npMFC

MFC

MFCMF

le MFC

A P -PX 1

=X

Mtb - K Xv

t

(23)

4 PHYSICAL MODELLING AND SIMULATION OF

THE HYDRAULIC SYSTEM

A physical model of the two subsystems is developed using SimHydraulics in the Matlab-Simulink environment, as

presented in Figs 4 and 5. The aim of the simulation is to

obtain the response of the actuators and synchronize their

outputs. Synchronization is achieved when all identical

actuators have equal output response using different source

signals at the directional control valves. The parameters of

the components obtained from design analysis and model

equations was inputted into the SIMSCAPE model in order

to obtain the response of the actuators. The parameters used

for simulation is presented in Table I. Furthermore, the

simulation will assist in judging the synchronization of

identical actuators, since the retraction and extension of the actuators is a function of the valve spool displacement [33,

34]. In essence, the solenoids of the directional control valves

will be subjected to various source inputs such as step, sine

and ramp. It is expected that the scope of the actuators will

be in form of the input at the directional control valves [35,

36]. Considering Fig 4, the fluid leaving the directional

control valves enters into the flow control valves. The flow

control valves regulates the amount of fluid entering the

piston side and annulus side of the actuators. The pressure

reducing valves ensures that the induced differential pressure

from the directional control valves is uniformly distributed among identical actuators in each subsystem. This implies

that the flow and pressure of the fluid entering and leaving

each identical actuator are supplied by several ports that are

controlled to operate at equal state thereby giving a perfect

synchronization [9, 10]. The 3-way flow control valve

ensures that the same amount of fluid enters and leaves each

identical actuator. Each of the 3-way flow control valve uses

one pressure reducing valve. These pressure reducing valves

also regulate the pressure of the hydraulic fluid entering and

leaving each identical actuator. This is necessary in order to

obtain a synchronized clamping force. Furthermore, the function of the 3 way flow control valve in

the hydraulic system is to ensure that the flow rate of

hydraulic fluid supplied to the actuators is constant, and equal

to the flow supplied to other identical actuator in the same

subsystem. As the fluid leaves the flow control valves, its

pressure is also kept constant by the pressure regulating

valves. This is necessary because, it will ensure that the

actuators are filled with hydraulic oil at the same rate thereby

producing synchronization and equivalent gripping force of

all identical actuators. The same situation holds for the

movable frame cylinders in subsystem two as shown in Fig

5.

5 SIMULATION RESULTS AND DISCUSSION

The results obtained from the simulation are presented in this

section of the article. The results of the simulation are the

responses of the actuators to different inputs at the directional control valves in each of the subsystem. The response of the

actuators in subsystem one to ramp, step, and sine inputs are

presented in Fig 6. Similarly, the response of the actuators in

subsystem two to sine, step, and ramp inputs are presented in

Fig. 7. Considering Figs. 6 and 7, it is evident that the

response of identical actuators in each subsystem are

equivalent irrespective of the type of source input at the

directional control valve compared to the response of the

actuators obtained from the use of flow dividers. The output

response of the actuators are not perfectly synchronized when

flow dividers are used particularly for a step response and

large number of actuators [1]. Further, the simulation approach used in this paper yield responses for individual

actuators in the subsystem compared to the use of FluidSim

simulation [10]. The exactness of the graphical response of

all identical actuators denotes that the actuators are

synchronized [11]. Hence, it can be hypothetically stated that

synchronization is achieved with the use of three way flow

control valve and pressure reducing valve. The implication of

subjecting the system to a ramp input is to obtain the linear

response of the actuators with respect to a linear displacement

of the spool in the directional control valve. It is expected that

there should be a linear relationship between the spool displacement of the directional control valve and the

synchronized extension of all the identical actuators in a

subsystem.

In Fig 6, subsystem one is subjected to a unit ramp, and the

actuators responds linearly to the input at the rate of 2.5*10-

14m in 10s. This implies that the actuators will extend or

retract synchronously and linearly within ten seconds of

applying the input. The actuators in subsystem two responds

linearly to a unit ramp at the rate of 4.9*10-16m in 10s. This

implies that subsystem one attains linear state faster than

subsystem two, because the displacement covered in 10s is

greater than that of subsystem two. More also, the implication of subjecting the system to a step input is to obtain the

response of the actuators to a change of state in the spool

position of the directional control valves. It is anticipated that,

there will be a change of state of the actuators for an

infinitesimal displacement of the spool. Considering Fig 6,

when subsystem one is subjected to a unit step input, there is

an indication that the actuators will change state (start to

extend or retract) linearly for a displacement of 2.25*10-14m

of the spool. Actuators in subsystem two change state linearly

for a displacement of 4.5*10-16m by the spool. Additionally,

the system is subjected to a sine input in order to obtain the displacement and time required by the actuators in each

subsystem to response to the spool displacement. It is desired

to obtain the time required by the actuators to respond to a

sinusoidal input at the directional control valves.



Fig. 4. Physical modelling of electrohydraulic subsystem one



Fig. 5. Physical modelling of electrohydraulic subsystem two



Table I

System Parameters used for modelling

(a) Step input

S/N Parameters Symbols Units Subsystem 1 Subsystem 2

1 Area of actuator actA

2m

0.023 0.075

2 Area of piston rod prA

2m

0.0095 0.038

3 Travel length of actuator actL

m 0.75 2.3

4 Coefficient of viscous

friction vb

2/Ns m

669551 9461050

5 Elastic load stiffness leK

5 /m Ns

38535828 81038249

6 Resistive force due to

viscous friction resF

N 33867 171

7 Mass of load pushed /FC MFM M

kg

12806 50218

8 Coefficient of discharge dC

0.7 0.7

9 Valve flow coefficient FCV

0.0002 0.002

10 Critical Reynolds number eR

12 12

11 Valve maximum opening opV

m 0.038 0.045

12 Pump displacement pQ

3 /m s

0.0132 0.0132

13 System pressure sP 2/N m

3887159 1549572

14 Volumetric efficiency of the

pump epV

0.92 0.92

15 Pump efficiency p

0.8 0.8

16 Valve damping coefficient ctd

Nm 31147 42536



(b) Sine input

c) Ramp input

Fig. 6. Actuator responses of hydraulic subsystem one to sine, step and ramp input at the directional control valve



Fig. 7. Actuator responses of hydraulic subsystem two to sine, step and ramp input at the directional control valve

Also, it is clear from Fig 6 that, when the directional control

valve in subsystem one is subjected to a sine input, it is

observed that the actuators took 3secs to extend to a

displacement of 2*10-11m. This implies that the actuators will

respond to the spool displacement of the valve after 3secs,

thus extending or retracting synchronously. Also, when the

directional control valve in subsystem two is subjected to a

sine input (Fig 7), it is observed that the actuators took

3.5secs to extend to a displacement of 1.18*10-11m. This

implies that the actuators will respond to the spool

displacement of the valve after 3.5secs, thus extending or

retracting synchronously.

6 EXPERIMENTAL INVESTIGATION, RESULTS AND

DISCUSSION

In order to observe and investigate the synchronized

extension and retraction of the hydraulic actuators, the

hydraulic subsystems were set up experimentally in the

mechatronics laboratory using FESTO didactic test bench as

presented in Fig 8. In the experimental set up, each actuator

uses two flow control vales to accurately synchronize the



position of the actuators. The experiment was initially carried

out in order to ensure that the flow control valves produces

synchronized extension and retraction of the actuators. At

complete extensions of the actuators the piston rod was

cleaned in order to create notable spots (to represents actuator

positions 50mm, 100mm, 150mm 200mm). The hose supplying fluid to the piston and annulus ports of the

actuators was disengaged in order to force the retraction of

the piston rod into the actuators, and ensure that there is no

fluid in the actuators. The hydraulic system was reconnected

and the pressure readings of the flow distributor in the

annulus and piston sides of the system during extension and

retraction strokes were noted. The pressure readings of the

flow distributor in the annulus and piston sides of the system

during extension and retraction stroke are presented and

discussed in this section. As stated earlier, the piston rods of

the actuators are pushed back into the actuator after

confirming their synchronized extensions and retractions, hence there will be no fluid in the actuators before the first

extension stroke and as such the pressure readings of the flow

distributors are all zero. It is clear from Fig. 9 to Fig. 12 that

the maximum pressure occurs at the maximum extension of

the actuators. Furthermore, the pressure readings of the flow

distributors at maximum extension is equal to that of the

minimum retractions because retraction begins at maximum

extension in the experiment. The pressure readings of the

flow distributor during the extension and retraction

of the finger actuators differs from that of the movable frame

actuators as presented clearly in Fig. 9 to Fig. 12. The

difference in pressure is because the number of actuators

controlled in subsystem one are more than the number of

actuators controlled in subsystem two. It is expected that the

three way flow control valves will ensure that an equal amount of fluid enters and leaves all identical actuator

thereby making it possible to achieve synchronization.

Furthermore, the differences in response time and

displacement of the actuators observed in the simulation for

the two subsystems is because the volume of oil under

compression in the piston sides of the actuators in each

subsystems are different. This difference can also be linked

to the pressure readings of the two subsystems in the piston

and annulus sides of the hydraulic system as presented in

Table II. It is evident that the subsystem with high pressure

responds faster by covering a notable distance in a small time.

The pressure in the piston side of the system is lesser than the annulus side during the retraction stroke and higher during

the extension stroke due to the varying volume of oil under

compression in the ports as seen in Figs 9 to 12. In addition,

it can also be observed that the pressure reading of subsystem

one is higher than that of subsystem two due to large number

of ports under compression in the subsystem. In view of this,

it can be hypothetically stated that larger volume of oil under

compression can be attributed to high pressure in the system.

Fig. 8. Experimental set up of the Hydraulic system



Table II

Pressure readings of flow distributors in the hydraulic system

Subsystems

Actuator

Positions (mm)

Pressure Readings of Piston and Annulus sides of the Hydraulic

Subsystems during extension and retraction strokes 2 3( / )*10N m

Extension stroke Retraction Stroke

Piston Side Annulus Side Piston Side Annulus Side

Subsystem

one

0 0 0 38790 50952

50 32852 28951 41580 48954

100 39491 34742 44357 47218

150 46713 39965 46456 45321

200 50005 44820 50005 44820

Subsystem

two

0 0 0 25320 39682

50 20510 17895 28988 38521

100 25300 21422 29320 37224

150 29210 25071 31854 34318

200 36088 33213 36088 33213

Fig. 9. Pressure readings of piston and annulus sides of the hydraulic Subsystem one during extension

Fig. 10. Pressure readings of piston and annulus sides of the hydraulic Subsystem one during retraction

Fig. 11. Pressure readings of piston and annulus sides of the hydraulic Subsystem two during extension



Fig. 12. Pressure readings of piston and annulus sides of the hydraulic Subsystem two during retraction

7 CONCLUSION

The performance of hydraulic actuators as clamping

elements for a fixturing device depends on effective

synchronization. An innovative approach to position

synchronization of multiple hydraulic actuator have been

achieved using directional control valves, pressure reducing valves and flow control valves. The approach have been

tested by simulation and experiment. The model for the

electrohydraulic subsystems are simulated using Simscape

hydraulics. The results of the simulation have been

confirmed with the experimental test carried out on the

FESTO didatic hydraulic test bench. The exactness of the

graphical results obtained for all identical actuators in the

same hydraulic subsystem also depicts that the actuators are

perfectly synchronized. However future work needs to be

considered in the aspect of synchronizing the clamping

force and designing an effective controller for the

electrohydraulic systems. This is necessary, in order to

obtain uniform gripping force of the reconfigurable

assembly fixture. To this end, it can be stated that the

contributions of this article can be seen from the Synchronization of multiple hydraulic actuator using a

directional control valve, three way flow control valves and

pressure reducing valves. Experimental validation of the

designed electrohydraulic system on an electrohydraulic test

bench and physical modelling and simulation of the

electrohydraulic system in Simscape SimHydraulics in

order to obtain the system response to various inputs at the

directional control valve.

8 NOMENCLATURES

1 2P and Ps s [N/m2] pressures in the subsystems for subsystems one and two respectively

P and PFC MFC [N/m2] inlet pressures of the actuators for subsystems one and two respectively

1 2S and Sdc dc [m] static discharge column for subsystems one and two respectively

cd [m] discharge column diameter

fC coefficient of friction of the column wall

PIP [N/m2] pressure of the hydraulic fluid at the pump inlet

n and i number of actuators in subsystem one and two respectively.

Q p [m3/s] volumetric flow rate of hydraulic fluid from the pump

Sg specific gravity of the hydraulic fluid

Pin [W] power input to the pump

d drive efficiency

1 2 and tc tcD D [m] total dynamic column of subsystem one and two respectively

sU [m] displacement of spool

spM [kg] mass of spool

srF [N/m] foce spring rate

spF [N] valve spool force

ctsd Damping coefficient due to trasient flow force



dC coefficient of discharge

w [m2] area gradient of spool

l [kg/m3] density of the hydraulic fluid

N numerical constant based on unit factor

pF numerical constant based on pipe geometry factor

r1 r2F and F [m3/s] volumetric flow rate of hydraulic fluid in subsystems one and two respectively

actd [m] diameter of the actuator

actF [N] external load pushed by the actuator

Ps [N/m2] pressure in the hydraulic system

actP [N/m2] pressure developed in the actuator

c [N/ m2] crushing stress a Rankine’s constant

sF factor of safety

prd [m] diameter of the piston rod.

fbL [m] free buckling length of the piston rod

PFC PMFCA and A [m2] areas of the pistons in the finger and movable frame cylinders respectively

e1 e2V and V [m/s] velocities of the piston in the finger and movable frame cylinders during extension respectively

Lf1 Lf2C and C leakage flow coefficients for identical actuators in subsystems one and two respectively

Q [m3/s ] volumetric flow rate of hydraulic fluid entering the actuator.

resF [N] resistive force due to viscous friction

vb viscous friction coefficient

esV [m/s] velocity of the piston in each subsystems

l [m2/s] kinematic viscosity of the hydraulic fluid

actA [m2] area of the actuator

cly [m] clearance between the piston and cylinder wall

lek [N] elastic load stiffness of the actuator

nA [m2] annulus area of the actuator

[N/m2] bulk modulus of the oil

L1 L2V and V [m3] volume of hydraulic oil in the hoses connected to the piston and annulus side respectively

1 2V and V [m3] total volumes of hydraulic oil in the actuators before displacement.

On OiV and V [m3] initial volumes of hydraulic oil in the actuators in subsystems one and two respectively

np1 np2P and P [N/m2] differential pressures across the ports of the valves in subsystem one and two respectively.

Tn TiK and K [m3] total leakage flows for each actuators in subsystem one and two respectively.

1 2Q and Q [m3/s] mean flow rate of subsystem one and two respectively

FCn MFCiX and X [m] displacements of the actuators in subsystem one and two respectively

F MFM and M [N] external loads pushed by the actuators in subsystems one and two



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Documents

Physical Modelling, Simulation and Experimental Analysis