theoretical simulation of static and dynamic behavior

Scientific Bulletin of the
Politehnica University of Timisoara
Transactions on Mechanics Special issue
ABSTRACT
The 6 th International Conference on
Hydraulic Machinery and Hydrodynamics
Timisoara, Romania, October 21 - 22, 2004
THEORETICAL SIMULATION OF STATIC AND DYNAMIC
BEHAVIOR OF ELECTRO-HYDRAULIC SERVO VALVES
Victor BALASOIU, Prof.*
Mircea Octavian POPOVICIU, Prof.
Department of Hydraulic Machinery
Department of Hydraulic Machinery
“Politehnica” University of Timisoara
“Politehnica” University of Timisoara
Ilare BORDEASU, Assoc. Prof.
Department of Hydraulic Machinery
“Politehnica” University of Timisoara
*: Bv Mihai Viteazu 1, 300222, Timisoara, Romania, Tel.: (+40) 256 403681,
Fax: (+40) 256 403700, (+) 256 4030682, Email: balasoiu@mec.utt.ro, vbalasoiu@online.ro
The electro-hydraulic servo-valves (EHSV) as interface
in automatic hydraulic systems are in essence
hydroelectric directional control valves with cylindrical
spool, with integral reaction. The output quantity
(flow rate, pressure) is modified proportional with
the control signal (voltage, current) together link
(electrical, hydraulic or mechanical). Both the manner
in which the reaction signal is produced and the
position of the information circuit where it is applied,
characterises the type but also static and dynamic
behavior of the servo valve. The servo valves currently
used for automatic hydraulic systems are those with
two stages able to develop great output hydraulic
powers for small input electrical signals.
In the paper, on the basis of electro-hydraulic
analogy it is developed the mathematical model able
to give theoretical analyses of the static and dynamic
behavior for the main stage, the valve with the cylindrical
spool. After the definition of the basic equations
of the aggregate spool-distributor for zero clearances
and overlaps it was analyzed the directional spool
valve with radial clearances and zero overlap. Finally
the generalized equation of the adjustment characteristic
for flow rate [QM = f(yS)∆pM=0], pressure
[∆pM = f(yS)QM = 0] and load [QM = f(∆pMyS)] for
transition and laminar flow, with non-dimensional was
plotted both for linear and non linear zones, taking
into account the annular clearance J and the overlap
range Yoi ≠ 0. For the pressure valve taken into con-
sideration there have been deter-mined and plotted the
adjustment characteristics [ QMA, B = f ( ∆p
MA,
B , YS
) ],
for the flowing directions A and B, for which there
have been emphasized the influence of the overlap
range Yoi and the clearance J over the linearity degree
and the magnitude of the adjusted flow.
The model of the dynamic equilibrium of the spool
valve is defined starting with the computation of forces
acting on the spool. Introducing the concept of interaction
between the aggregate components nozzle-spooldistributor
it was defined the mathematical model for
the directional spool valve as a whole.
After defining the transfer function, both analyze
and syntheses of performances were possible for the
considered servo valve, in terms of time or frequency.
The frequency characteristics and the transfer function
hodograph were plotted, putting into evidence the overlap
degree Yoi, the input pressure at constant control
electric current ∆ic or the control electric current for
different constant input pressure. The analyze of the
dynamic behavior in time had in view the computation
of the response at a unitary step input by determining
the behavior of the output magnitude and the position
of the cylindrical spool YS(t). The variations of the
output magnitude represent the solution of the linear
differential equation that is describing the running of
the servo valve by an III degree transfer function,
which was obtained in the present work.
In the framework of the paper was worked out a
unitary mathematical model for the servo-valve with
two stage having pressure reaction permitting the study
of the influence of geometrical and functional parameters
upon the behavior of static and transient regimes.
KEYWORDS
Electro-hydraulic servo valves, spool-distributor,
adjustment characteristics transfer function, mathematical
model
303

1. INTRODUCTION
Electro hydraulic servo valves (EHSV) in current
use for the automatic hydraulic systems are those with
two stages able to develop great hydraulic power, for
small input electric signals. Running with constant
pressure, regardless of the constructive solution, they
have a functional dependence between the flow and
the applied command current Q = f(∆iC), ensuring both
a good stability and linearity. With the view to establish
the dynamic and static characteristics, which are determined
by a great number of physical, geometrical,
electrical and mechanical parameters, there are taken
into account the fundamental laws of mechanics and
hydraulics, completed with the automatic system theory.
2. THEORETICAL ANALYSES OF THE
DISTRIBUTION STAGE WITH LINEAR
CYLINDRICAL SPOOL VALVE
Appealing to the model introduced by W Backe [5]
the stage spool valve-distributor body can be reduced
to a scheme tip A + A, as is shown in Fig.1. For a
permanent motion the continuity equation becomes:
2.
∆pi,
e
Q = αD.
π.
DS.
(1)
ρ
304
Utilizing the modified flow equation both for the
α D = f ( ∆p,
ν,
Re , and the transition
laminar ( )
Q10
= K Dj
Q20
= K Dj
Q30
= K Dj
Q40
= K Dj
Qio
i=1,
2,3,
4
⎡
( Y +
2
+
2 ⎢
S Y01)
J
⎢
⎢⎣
⎡
( Y −
2
+
2 ⎢
S Y02
) J
⎢
⎢⎣
⎡
( Y −
2
+
2 ⎢
S Y03)
J
⎢
⎢⎣
⎡
( Y +
2
+
2 ⎢
S Y04
) J
⎢
⎢⎣
( Y + Y )
S
( Y − Y )
S
( Y − Y )
S
K
2
T
2
+
2
01 J
( Y + Y )
S
K
2
T
2
+
2
01 J
K
2
T
2
+
2
03 J
K
2
T
2
+
2
04 J
+
+
+
+
α D ,(Re=100…2500), introduced
by H. Weule and H. J. Feigel [1], the equation (1)
written for the directional control valve became:
motion ( = f ( ∆p)
( p − p )
MA
( p − p )
03
( p − p )
MB
( p − p )
03
Fig.1.
Q = KD
⎡
2
Yj
+ 1⎢
⎢
⎣
2
K K
⎤
t t − ⎥
2 2
J j + 1 Yj
+ 1⎥
⎦
(2)
04
MA
03
MB
−
−
−
−
⎤
KT
⎥
⎥
( Y +
2
+
2
S Y01)
J ⎥⎦
⎤
KT
⎥
⎥
( Y −
2
+
2
S Y02
) J ⎥⎦
⎤
KT
⎥
⎥
( Y −
2
+
2
S Y03)
J ⎥⎦
⎤
KT
⎥
⎥
( Y +
2
+
2
S Y04
) J ⎥⎦
non zero overlaps linear zone non linear zone (4)
Y < Y < Y
Y S YSN
YS <
−YSN
(3)

305
Q10
( ) ( )
( ) ( ) ⎥ ⎥⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
−
−
−
+
+
+
+
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
−
−
−
+
+
+
+
+
+
2
2
01
S
T
04
MA
04
MA
2
2
01
S
2
T
2
2
01
Y
S
DJ
2
2
01
S
T
04
MA
04
MA
2
2
01
S
2
T
2
2
01
Y
S
DJ
J
Y
Y
K
)
p
p
(
sign
p
p
J
Y
Y
K
J
)
Y
(
K
-
J
Y
Y
K
)
p
p
(
sign
p
p
J
Y
Y
K
J
)
Y
(
K
0
Q20
( ) ( )
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
+
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
−
−
−
−
+
+
−
+
−
J
K
-
)
p
p
(
sign
p
p
J
K
J
.
K
-
J
Y
Y
K
)
p
p
(
sign
p
p
J
Y
Y
K
J
)
Y
Y
(
K
T
MA
03
MA
T
T
03
MA
T
03
2
2
DJ
2
2
02
S
MA
03
2
2
02
S
2
2
2
02
S
DJ
Q30
( )
( ) ( ) ⎥ ⎥⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
−
−
−
+
+
+
+
−
2
2
03
S
2
2
03
S
2
2
2
03
S
DJ
J
Y
Y
K
)
p
p
(
sign
p
p
J
Y
Y
K
J
Y
Y
K
T
MB
03
MB
03
T
–
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
−
+
J
K
)
p
p
(
sign
p
p
J
K
K
T
2
2
T
DJ MB
03
MB
03
Q40
( ) ( )
( ) ( ) ⎥ ⎥⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
−
−
−
+
+
+
+
+
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
−
−
−
+
+
+
+
+
2
2
04
S
MB
2
2
04
S
2
2
2
04
S
DJ
2
2
04
S
MB
2
2
04
S
2
2
2
04
S
DJ
J
Y
Y
K
)
p
p
(
sign
p
p
J
Y
Y
K
J
)
Y
Y
(
K
-
J
Y
Y
K
)
p
p
(
sign
p
p
J
Y
Y
K
J
)
Y
Y
(
K
0
T
04
04
MB
T
T
04
04
MB
T
(5)
( )
( )
SN
0i
2
2
01
S
T
MA
03
MA
03
2
2
01
S
2
T
2
01
J
T
MA
03
MA
03
2
2
T
2
SN
03
MN
MA
Y
Y
Y
pentru
J
)
Y
Y
(
K
)
p
p
(
sign
.
.
p
p
J
)
Y
Y
(
K
p
Y
Y
J
K
)
p
p
(
sign
.
p
p
J
K
1
Y
p
1
Q
Q
<
<
⎥
⎥
⎥
⎦
⎤
+
+
−
−
⎢
⎢
⎣
⎡
+
+
+
+
+
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
+
+
+
=
( )
( )
SN
0i
2
2
04
S
T
2
2
04
S
2
T
2
03
2
04
J
T
2
2
T
2
SN
03
MN
MB
Y
Y
Y
pentru
J
)
Y
Y
(
K
)
p
p
(
sign
.
p
p
J
)
Y
Y
(
K
)
1
Y
(
p
Y
Y
J
K
)
p
p
(
sign
.
p
p
J
K
1
Y
p
1
Q
Q
MB
03
MB
03
SN
04
MB
04
MB
<
<
⎥
⎥
⎦
⎤
+
+
−
−
⎢
⎢
⎣
⎡
−
+
+
+
+
+
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
−
+
+
=
were:
•
ρ
2
)
J
D
(
π
.
C
K S
D
+
= -is a geometric constant, •
ρ
ν
=
2
a
.
4
.
C
K
2
T
- is the coefficient for laminar flow.

The relation (2) valid for laminar and transition flow
will be used for the theoretical analysis of the distribution
stages.
The directional control valve with negative overlap
constructively differ from the ideal distributor by the
fact that all four throttle openings are unsealed for
YS = 0. For a spool valve stroke YS (depending on the
value YS±Y0i) there is realized a throttle opening for
which the flow capacities given by (1) become:
Appealing to the computing model introduced by
F. Klinger [11] and utilizing for the reference flow
capacity the value 2 2
Q MN = K DJ YSN
+ J p the 03
relation (4 ), valid in the sections Q10, Q20, Q30, Q40,
is obtained. For the directional control valve with
negative overlap and annular clearance using (3, 4)
the relation (5) will be obtained. Similar relations can
be obtained also for zones YS ≥ YSN
, YS
≤ −YSN
,
− Y < Y < Y . The complex function (5) represent
0i
S 0i
the generalized equation of the adjustment characteristic
for the flow capacity QMA;QMB = f (∆pMAB,
YS)∆p=ct with the pressure ∆pMAB = f(YS) and for the
load QMA;QMB = f (∆pMAB, YS)∆p=0, in laminar and
transition flow.
In Fig. 2 are given the adjustment characteristics
QMA; QMB = f (∆pMAB, YS)∆p=ct and QMA;QMB =
= f (∆pMAB, YS)∆p=0 for the flow passing in directions
A and B. The relations (5) emphasize the work of the
ensemble spool valve in three distinct domains:
negative or positive overlap zone Y S < Y0i
, linear
zone Y 0i
< YS
< YSN
and saturation zone Y S > YSN
.
The flow that passes through the distributor is
affected by the overlap degree Y0i (Y0i = 1, 2, 3, 4)
Fig. 2a, 2b, 2c. The selection of the overlap degree
Y0i and the size of the annular clearance J are of
great importance for numerous working characteristics
of the system such as: consumption, precision,
stability and elasticity.
306
Fig .2.a.
Fig. 2.b.
Fig. 2.c
The relations (3, 4, 5) put into evidence the influence
different overlap degrees (Y0i ≠ 0) of the throttle orifices
(Y01 ≠ Y02 ≠ Y03 ≠ Y04) which appear often
in practice because technological it is impossible to
obtain a perfect symmetry of the throttle edges. Simultaneously
these relations allow analyzing the influence
upon the adjustment characteristics of variations
(Y0i) of the four throttle edges.
3.THE DYNAMIC ECHILIBRIUM MODEL OF
THE SPOOL VALVE
During the work of distribution and adjustment
elements, upon the spool valve operates a series of
forces, their nature and magnitude determining the
running performances. The resultant of the acting
forces can be expressed as an algebraic sum:
Fpa + Ffrl
+ Ffrv
+ Fear
+ Fg
+ Fh
± Fin
= 0 (6)
which establishes the spool valve dynamics and finally
the EHSV dynamics. These forces are:
2
d Y
- F [ ] S
isv = mS
+ Kms(
ma1
+ ma2
) – inertia (7)
2
dt
force;
Fear = Kear(
YS
+ Yoa
) − pressur forces ; (8)
D
dY
F π.
ρ.
S . n . L . C . sign(
Y ) S
frv = −
&
m m fr S
(9)
J
dt
– viscous forces

In [1] there are given similar relations also for the
friction force, generated by non-balanced lateral forces.
The weight Fg is negligible in comparison with other
forces. The moving law of the spool valve is given by
the dynamic equilibrium of the acting forces. For the
distributor with non-null overlap and annular clearance
the moving law of the spool valve is:
{ K + K [ p − p − p − p ] }
2
d YS
MS
=
2 frv HDY 03 MB MA 04
dt
. sign(
Y&
). Y&
S S + [ KHS(
p03
− ∆pMAVB
− p04)
+ Kear
] . YS
−
2
πDS
− ∆pcab.
(10)
4
The relation (10) was used for modeling mathematically
the EHSV. Using together the equilibrium
equation, [1; 5] between the stage nozzle-flap and the
spool valve (Fig. 3).
VC
d(
∆pcab)
K ∆X − KQp.
∆pcab
= Sp.
Y&
S + . (11)
QX
2E
dt
and the simplified dynamic equilibrium equation of
the spool valve:
S . ∆p
= M . Y&
& + K . Y&
+ K . Y (12)
p
cab
S
S
it was obtained the III order transfer function [1;5] for
the ensemble spool valve- distributor body, in complete
form:
KQX
S
H
P
SV3(
S)
=
VC.
M
⎡
S 3 KQP.
MS
VC.
K ⎤
QP
S + ⎢ + ⎥S
2
+
2E.
SP
⎢ S
2
2
P
2E.
S ⎥
⎣
P ⎦
⎡ Kf
. KQP
VC.
K
⎤
S
KS.
K
⎢
QP
1 + + ⎥S
+
⎢ S
2
2 2
P
2.
E.
S ⎥
⎣
p ⎦
S
P
(13)
Simplifying these equation it was obtained:
YS
( S)
K
H
5
SV3(
S)
= =
=
∆X(
S)
3 3 3
K1.
S + K2.
S + K3.
S + K4
(14)
1
= KYS
3 2
S + Q2.
S + Q1.
S + Q0
which can be written in normalized form, taking into
account the experimental conditions:
Y ( S)
1
H ( S)
Sn
SV3n
= =
(15)
A . X ( S)
3 2
0 ∆ n A1.
S + A2.
S + A3.
S + 1
The equation (15) allows determining the theoretical
frequency characteristics in a plotting system comparable
with the experimental results.
f
S
u
S
S
3.1. THE EHSV ANALYSIS THROUGH
FREQUENCY
The frequency analysis is characterized by plotting
the transfer function in the imaginary plan and by the
functions that can be obtained at frequency variations
(ω = 0 ⇒ ∞ ) for an input signal iC(t) = i0sinωt.
Taking into account the transfer function equation
for the ensemble directional control valve in complete
and normalized form (13,14,15) the sinusoidal response
is determined by substituting the complex operator
S = jω:
H
SV3
K
( Jω
) =
5
3 3 2 2
(16)
K . j ω + K . j ω + K . jω
+ K
1
2
Developing and separating the terms in (16) it result:
2
K ( K K )
H Re( j ) JIm(
j )
5 4 − 3ω
SV3(
jω)
= ω + ω =
−
2 2
3 2
( K4
−K2ω
) + ( K3ω−K1
ω )
3 2
K ( K K )
J 5 3ω−
1ω
−
( 17)
2 2
3 2
( K4
−K2ω
) + ( K3ω
−K1ω
)
with the terms :
H ( j ) 20.
lg Re
2
( j ) Im
2
sv3
ω = −
ω + ( jω)
dB
0
Im( jω)
Φ
SV
( jω)
= −arctg
3
Re( jω)
3
4
(18)
The theoretical model was verified with the geometric
parameters of EHSV 2T-7.5 for the computation
utilizing the program SIST-SERV.
Fig. 3
In fig. 4a, b there are represented the frequency
characteristics and the transfer plan, putting into evidence
the influence of the overlap degree Y01, the input
pressure p03 = 5.5…10 MPa (Fig. 4a) and the influence
307

308
a.1.
a.2.
Fig.4.a
b.1.
b.2
Fig. 4.b.
of control current ∆iC = 5…15 mA for input pressures
p03 = 5.5…10 MPa (Fig. 4b).
For all analyzed cases there were obtained similar
frequency characteristics. On the whole, the studied
cases attest the presence of a dominant proper frequency
in the domain
3dB
( 10...
30 ) Hz
∈ ω −
,
which correspond to a no periodic oscillation model
and to a proper pulsation with great frequency
ω r ∈(
200...
400 ) Hz ,
which correspond to a damped oscillation model.
Taking into account the inertia of the dynamic
system, EHSV can work in a frequency band of
10… 30 Hz
3.2. THE TIME ANALYSIS OF THE EHSV
DYNAMIC BEHAVIOR
This study has as objective to determine the variation
in time of the system response YS(t) when it is excited
with an input value i(t) of the type unitary step, unitary
ramp or sinusoidal. This response is analyzed both for
the adaptation period (transition stage) and for stationary
regime. The output value is obtained as the solution
of the linear differential equation, which describes the
work of EHSV by III or V order transfer functions (14,
15). Applying the inverse Laplace transform to relation
(14) the indicial response is obtained:
−x
t
e 3
YS
( t)
=
+
2
β − 2.
ξgβ
+ 1
+
(19)
−ξg.
ωgt
β.
e
⎡
2 ⎤
. sin
⎢
ωgt
1−
ξg
− ψ
2 2
⎣
⎥
1−
ξ β − ξ β +
⎦
g 2 g 1
maintaining the notation given in [1]. The determination
of the response YS(t) gives the time Tr that
characterize the EHSV both in stationary and transient
regimes. In Fig. 5a,b,c is plotted this response
obtained for the EHSV 2T-7.5 with emphasize to
both the influence of control current ∆iC for various
input pressures (p03 = 5.5, 7.0, 10 MPa, Fig. 5a,b)
and the input pressure for a constant control current
∆iC = 10 mA (fig.5.c).
As become clear from Fig.5b the response time is
diminished with the increase of the input pressure
p03 for the constant control current ∆iC = 10 mA, that
means it is diminished with the opening of the spool
valve for the same input pressure (Fig. 5a, b). For all
studied cases the weight of the oscillatory component
is of little importance. In consequence it can be
stated that the response of EHSV 2T-7.5 at a unitary
step signal can be approximated with a transfer
function of II order, which is specific for a rapid
dynamic process.

4. CONCLUSIONS
Fig.5.a
Fig.5.b
Fig.5.c.
Electro hydraulic servo valve is one of the most
complex of the electro hydraulic driving systems,
both from the constructive and working point of
view. Establishing a mathematical model, which can
express satisfactory the static and dynamic properties
is prime order necessity for the analyses and synthesis
of servo valves. In the frame of the present mathematical
model there were obtained the following results:
• the adjustment characteristics for flow capacity,
pressure and load were defined in a unitary form
both for linear and nonlinear zones, the flow conditions
through the control directional valve being
laminar or transitional;
• with the view to put in evidence the influence of the
overlap degree on the stationary behavior of EHSV
the mathematical model was applied for eight values
of overlap between Y01 = ±(0….15)YSN;
• the equation of the dynamic equilibrium on the
spool valve was established;
• assuming as a basis the stability criteria enunciated
in the techniques of automatic stability systems
analysis it was effectuated the EHSV study both in
the frequency domain (faze amplitude-frequency
characteristics) and by the response characteristic
to a step signal;
Synthesizing the main elements of the dynamic
analysis it results:
• the dominant frequency ω-3dB = 10…40 Hz
corresponding to the time constant values TA =
= (0.0106… 0.0053 0 s, is significant to the
behavior of a damp oscillatory hydraulic system
(that means the system is stable in the analyzed
variation range of control currents, processes,
flow capacities and overlap degrees);
• in the frequency domain ω-3dB = 10…30 Hz the
EHSV behavior may be approximate with a linear
system of first degree or at most of second degree;
• the response time tr = 10…30 ms put into evidence
the feature of a rapid system with a high degree of
stability.
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