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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.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 123
190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S
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
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 124
190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S
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
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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
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 126
190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S
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.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 127
190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S
Fig. 4. Physical modelling of electrohydraulic subsystem one
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Fig. 5. Physical modelling of electrohydraulic subsystem two
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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
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(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
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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
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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
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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
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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
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 135
190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S
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
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 136
190204-8686-IJMME-IJENS © August 2019 IJENS I J E N S
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