MATHEMATECAL MGDELS FOR THE ‘FIME DOMAIN
ANALYSIS OF HYDRAULIC SYSTEMS

Thesis for the Degree of Ph. D.

MICHIGAN SKATE UNWERSITY

Hit-Erich Robert Martens
1962

 

This is to certify that the
thesis entitled

MATHEMATICAL MODELS FOR THE TD’IE-DQ’IAIN
ANALYSIS OF HYDRAULIC SYSTEMS

presented by

Hinrich Robert Martens

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in M.E.

MA};

Major professor

Date August 2, 1962

0-169

  
   
   

  

LIB RA R Y
Michigan State
University

 

MATHEMATICAL MODELS FOR THE TIMErDOMAIN

ANALYSIS OF HYDRAULIC SYSTEMS

By

Hinrich Robert Martens

AN ABSTRACT OF A THESIS

Submitted to the
School of Advanced Graduate Studies of
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1962

ABSTRACT

MATHEMATICAL MODELS FOR THE TIME-DOMAIN
ANALYSIS OF HYDRAULIC SYSTSNS

by Hinrich Robert Martens

Systematic procedures have recently been developed to
formulate the mathematical models of systems of discrete physical
components regardless of whether or not the characteristics of
the components are linear or nonlinear. These formulation
procedures lead to a time-domain mathematical model suitable
for computer processing. When applied to hydraulic systems,
this time-domain model makes it possible to include a multitude
of continuous and discontinuous nonlinearities as they are
encountered in modeling the characteristics of hydraulic com-
ponents. When this facility is combined with the ability of
the digital computer to obtain solutions the accuracy obtainable
in the study of nonlinear systems is limited only by the accuracy
with which the individual components can be modeled.

This thesis presents a general methodology for implementing
these procedures in the analysis and design of hydraulic
systems. Nonlinear mathematical models are established for a
number of typical hydraulic components. Conditions and
procedures are developed for the generation of time—domain models
suitable for computer processing of typical subassemblies and

functional hydraulic systems.

Hinrich Robert Martens
It is shown that the desired form of the mathematical

models may always be established when the effect of hydraulic
capacitance is included in the component models. The inclusion
of leakage effects in the models of hydraulic components is
readily accomplished although results indicate that they may be
neglected in most cases.

Comparisons made at various stages and levels of system
complexity establish a high level of correlation between
solutions based on these models and the performance of the actual

system.

MATHEMATICAL MODELS FOR THE TIME~DOMAIN

ANALYSIS OF HYDRAULIC SYSTEMS

By

Hinrich Robert Martens

A THESIS

Submitted to the
School of Advanced Graduate Studies of
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

1962

ACKNOWLEDGMENT

The author is very grateful to Dr. H. E. Koenig for his
guidance and encouragement during the preparation of this
thesis and during the period of research. The author also
wishes to gratefully acknowledge the support of the National

Science Foundation.

1.!

TABLE OF CONTENTS

SECTION Page
I.INTRODUCTION.................. 1
II, MATHEMATICAL MODELS OF CONTROL VALVES . . . . . 4
III. MATHEMATICAL MODELS OF HYDRAULIC MACHINES. . . . 29
IV. MATHEMATICAL MODELS OF HYDRAULIC LINES . . . . . #8

V. FORMULATION AND SOLUTION OF TYPICAL
SUBASSEMBLIES. . . . . . . . . . . . . . r . . 56
VI. FORMULATION AND SOLUTION OF TYPICAL SYSTEMS . . 90

VII 0 CONCLUSION O O O O O O O O O O O O O O O O O O O 112

LIST OF FIGURES

Figure Page
2.1.1 Single Orifice . . . . . . . . . . . . . . . . . . A
2.1.2 Pressure-Flow Curve for Single Orifice . . . . . . 5

2.1.3 Flow Reaction Force as a Function of Valve

Opening and Pressure . . . . . . . . . . . . . . 6
2.1.# Flow Reaction Force for a Compensated Valve. . . . 6
2.2.1 Terminal Graph of Basic Valve Configuration. . . . 7
2.2.2 System Graph of the General Case . . . . . . . . . 8
2.2.3 Steady State Flow Reaction Force . . . . . . . . . 9
2.2.h Three-Way Valve . . . . . . . . . . . . . . . . . 10
2.2.5 Four-Way Valve . . . . . . . . . . . . . . . . . . 11
2.2.6 System Graph of Flow Source-Valve Combination. . . 1h
2.3.1 Typical Four-Way Valve Electromagnetic

Transducer Assembly . . . . . . . . . . . . . . 15
2.3.2 Typical Electrohydraulic Servo Valve . . . . . . . 16
2.3.3 Jet-Pipe Valve . . . . . . . . . . . . . . . . . . l7
2.3.h Push-Pull Ram Actuator . . . . . . . . . . . . . . 19
2.3.5 Booster Stage . . . . . . . . . . . . . . . . . . 20
2.3.6 PEGASUS Electrohydraulic Servo Valve . . . . . . . 21
2.3.7 Double Flapper-Nozzle Valve. . . . . . . . . . . . 22
2.3.8 Booster Stage . . . . . . . . . . . . . . . . . . 23
2.A.1 RAD A10 Servo Valve. System Graph . . . . . . . . 2A
2.5.1 Typical Family of Pressure-Flow Curves . . . . . . 27
3.1.1 Basic Hydraulic Machines . . . . . . . . . . . . . 29

iv

LIST OF FIGURES (con't)

Figure Page
3.3.1 Assembly Drawing of Axial Piston Machine . . . . . 3h
3.3.2 Terminal Graph . . . . . . . . . . . . . . . . . . 3h
3.3.3 Terminal Graph of Subassembly . . . . . . . . . . 35
3.3.# Plate Valve Position . . . . . . . . . . . . . . . 36
3.3.5 Plate Valve Position . . . . . . . . . . . . . . . 37
3.3.6 System Graph of Axial Piston Pump . . . . . . . . 38
3.3.7 Terminal Graph . . . . . . . . . . . . . . . . . . 40
3.3.8 Pump Friction for Variable Displacement Pump . . . #2
3.3.9 Effect of Coulomb Friction . . . . . . . . . . . . #3

3.3.10 Pump Displacement . . . . . . . . . . . . . . . . 4%

3.3.11 Tilt-plate Torque vs Velocity . . . . . . . . . . #4
3.4.1 Fixed Displacement Machine Friction

Characteristics . . . . . . . . . . . . . . . . #7
4.1.1 Hydraulic Line and Terminal Graph . . . . . . . . 49
#.l.2 Lumped Model with n Sections; Closed Ends . . . . 50
4.2.1 System Graph for Terminal Representation . . . . . 52
h.2.2 Pressure Response for 16 m flexible line . . . . . 55
5.1.1 Graph of Subassembly l . . . . . . . . . . . . . . 57
5.2.1 Graph of a Two-Element Line in CO

Representation . . . . . . . . . . . . . . . . . 60
5.2.2 Graph of Subassembly 2 . . . . . . . . . . . . . . 60
5.2.3 Effect of variation of capacitance on

Velocity Response of Subassembly 2 . . . . . . . 6A
5.2.# Velocity Response Curves of Subassembly 2 . . . . 65

V

Figure
5.2.5
5.2.6
5.3.1
5.3.2
5.3.3
5.#.l
5.5.1
5.6.1
5.6.2
5.6.3
5.7.1
5.7.2
5.7.3
5.8.1
6.1.1
6.1.2

6.2.1

6.2.2

6.2.3

6.3.1
6.#.1

Velocity
Velocity
Graph of
Velocity
Velocity
Graph of
Graph of
Graph of
Velocity
Velocity
Graph of
Velocity
Velocity

Graph of

LIST OF FIGURES (can't)

Response Curves of
Response Curves of
Subassembly 3 . .
Response Curves of
Response Curves of
Subassembly 4 . .
Subassembly 5 . .
Subassembly 6 . .
Response Curves of
Response Curves of
Subassembly 7 . .
Response Curves of
Response Curves of

Subassembly 8 . .

Subassembly
Subassembly
Subassembly

Subassembly

Subassembly
Subassembly
Subassembly

Subassembly

Valve Controlled Positioning System . .
Valve Controlled Speed Control System .

Position Response Curves of Positioning
System . . . . . . . . . . . . . . .

Pressure Response Curves of Positioning
SYBteneoeeeeeeeeeeeee

Valve Stem Position as a Function of
Position Load Factor . . . . . . . .

Response Curves of Speed Control System
Pump Controlled Speed Control System .

vi

Page
66
67
69
72
73
74
76
78
81
82
8O
85
86
88
9O
91
94
95

96
105

106

LIST OF FIGURES (can't)

Figure Page
6.4.2 Speed Response Curves of Speed Control System . . . 109

6.4.3 Pressure Response Curves of Speed Control
System..................... 110

vii

LIST OF APPENDICES

APPENDIX

A. Numerical Values of Coefficients .

viii

I. INTRODUCTION

A complete analysis of systems of discrete parameter
components describable by linear algebraic and differential
equations is generally obtainable. Transform techniques which
have been highly developed are effectively utilized. However,
when systems contain nonlinearities transform procedures no
longer apply. Hydraulic systems are a noted example.

The first attempts in the analysis of hydraulic systems
have been based on linear approximations. In fact, this is still
widespread practice. Despite the awareness of the inherent
nonlinearities, an investigation based upon linear approximations
offered the only alternative for lack of other approaches. With
the deve10pment of techniques in handling nonlinear systems, a
great deal of effort, for example, was directed at the study of
hydraulic systems by means of describing function techniques
(2, 3, 9). However, all these techniques lean heavily toward
linearization. Thus either misrepresentation or erroneous
prediction of the physical system is the consequence. As a
result of the history of nonlinear studies of hydraulic systems,
it has become of prominent importance that a more effective
procedure is needed for a complete and reliable investigation.

Through the advance of automatic computing equipment, it
is now practical to simulate the mathematical models of systems

of discrete parameter components regardless of degree of com-

2

plexity and regardless of the nature of the nonlinearities
appearing in the model. Systematic procedures have recently
been developed to formulate the mathematical models of systems
of discrete parameter components regardless of whether or not
the characteristics of the components are linear (4). These
formulation procedures lead to a time-domain mathematical model

referred to as "normal form", i.e., a model of the form

dJ-dt)
at

where ;(t) and git) are vectors of system variables. When applied

 

= F5“), §(t))

to hydraulic systems, this time-domain model makes it possible

to include a multitude of continuous and discontinuous non-
linearities as they are encountered in the description of
hydraulic components. When this facility is combined with the
ability of the digital computer to obtain solutions, the accuracy
obtainable in the study of nonlinear systems is limited only by
the accuracy with which the individual components can be modeled.

One example based upon a time domain study was introduced
by Wang (5). He presented the mathematical model of an electro-
hydraulic servomechanism and included a discussion of how this
model was derived. However, a general and systematic procedure
for mathematically modeling hydraulic systems in the time-domain
has not been reported.

This thesis presents a systematic method for the analysis
of nonlinear hydraulic systems which involves l) establishing
nonlinear models of hydraulic components, 2) establishing a
nonlinear time-domain model of the hydraulic system in a form

(normal form) suitable for computer processing and 3) actual

solution of the system model.

The type of nonlinearities include both continuous and
discontinuous functions. Comparisons are made at various stages
and levels of system complexity between computer solutions and
actual performance characteristics.

Normal form formulation procedures require the following
information: 1) a mathematical model of the components of the
system and 2) a mathematical statement of the interconnection of
the system components. The first requirement involves the modeling
of the components in a way such that when the components are
assembled to form a system a normal form system model can be
generated.

The results of extensive research on the characteristics of
hydraulic components and typical subassemblies available in the
literature along with experimental investigations form the general
basis for the development of the component models presented in

this thesis.

II. MATHEMATICAL MODELS OF CONTROL VALVES

2.1 The Orifice

The basic component of a control valve is the orifice. As
shown in Fig. 2.1.1, the orifice may be represented as a four-
terminal component; two terminals for the measurement of force
and displacement of the valve stem, and two terminals for the

measurement of flow and pressure of the fluid.

 

 

 

 

 

 

 

 

V .
gm h2
mrvx—
gm h

(a) (b)

Fig. 2.1.1.--Sing1e orifice. Schematic and terminal graph
The terminal equations associated with the terminal graph of

Fig. 2.1.1(b) for a single orifice can be shown to have the form

:1 = 31(51) + M1 51 + be( 81);:1 (2.1.2)

where Kv = orifice flow coefficient

F13 = a switching function defined by

F (x)

13 X X ;> 0

(2.1.3)
:0 x<o

be( £1) = Flow reaction force coefficient

4

5
M1 = Combined mass of stem and spool

B1(5:1) = Compound friction function including the effects
of viscous, dry, and static friction.
*
\/ = a sign sensitive square root defined by
V? = 7%- Jlxl (2.1.10
[1 = Amount of Underlap or Overlap defined by
A = An > 0 for Underlap
£1 ==Z§O <: O for Overlap

Equation (2.1.1) is sometimes called the orifice equation.
This equation is based on the usual assumptions that (1) all
measurements are instantaneous, (2) the flow coefficient is a
constant, and (3) the orifice area varies linearly with the stem
displacement (8). A typical experimental pressure-flow curve is
shown in Fig. 2.1.2.

0
a

8,1

 

 

Fig. 2.1.2.--Pressure-flow curve for single orifice ( 8v = constant)
Because of uneven pressure distribution in the valve
spools during flow, a force term must be included in (2.1.2).
This flow reaction force is normally a function of both valve
opening and pressure drop across the valve. A typical set of

curves is shown in Fig. 2.1.3.

pl=lOOO psi

p1=2OO psi

 

 

X

Fig. 2.1.3.--Flow reaction force as a function of valve
opening (X) and pressure.

Since the force-displacement gradient is pdsitive, the valve
tends to close under flow condition. This, of course, is a
desirable stability feature used to a great advantage in more
complicated valve configurations. Through suitable geometric
design, the flow reaction force may be controlled to a minimum,
yet exhibit a desirable degree of stability (8). Figure 2.1.4
shows the flow reaction curves for an orifice which has been
compensated. Note that the curve now is essentially independent
of the pressure drop.

F

 

 

 

X
Fig. 2.1.4.--Flow reaction force for a compensated valve

2.2 Basic Valve Configurations.

Although a single orifice may be used as a control element

7

in a hydraulic system, a functional combination of several
orifices is usually more effective. Several basic valve con-
figurations are considered in this section and convenient terminal
representations developed. All terminal representations are

given in terms of the terminal graph of Fig. 2.2.1. Of interest
is information regarding the force and displacement variables of
the valve stem shaft,and pressure and flow variables of the
various ports to which system connections are made. These are

normally the two load ports, h and h2’ a supply port, h}, and a

1

return port, gh. Terminal gh is conveniently

 

 

hl
v
P sé
h h h
v 3 1 9
fv’ 5v n F, .
th 10h
3 3
gm

 

Fig. 2.2.l.--Termina1 graph of basic valve configuration
taken as the atmospheric reference. It will be seen later that
only when the terminal gh is included in a system graph can
systems whose component characteristics depend on the atmospheric
reference be properly formulated.

Consider first the mathematical model of a general valve
configuration. From this the characteristics of other configur-
ations are obtained as special cases or modifications. A valve
configuration consisting of four orifices is considered as the
general case. A larger number of orifices may be used to form a
control valve. Such an example was presented by Dushkes (10).

Blackburn has shown that a four-orifice valve is most

effectively analyzed when the valve is compared with a four-

8
arm bridge (7). Accordingly, in the system graph in Fig. 2.2.3,

elements 1, 2, 3, 4 constitute the four orifices. In general, the
orifices may be fixed or variable, and the supply may be either a

constant pressure or constant flow source.

 

 

 

 

 

 

 

Fig. 2.2.3.--System graph of the general case
If the elements in the graph representing external measure-

ments (measure graph) constitute a tree, all orifice elements
can be made chord elements. For the given nonlinear forms this
is a necessary condition to arrive at mathematical models
explicit in the terminal variables. Since the respective
orifice-across variables appear implicitly they must be related,
through the circuit equations, to those tree-across variables
that either are specified or otherwise known. Combination of
the relatively simple graph equations with the orifice terminal

equations yield the desired terminal representation.

9

 

 

éhl = xv<r13<a + 8") . ph1 - p113 . rum - 5‘) q)
éhz = KV(F13(A - 5v) v 1oha - 1a,,3 + F13” + 5,) V—p—hz)
9.} = Kvam + 5.) (1p113 - phl . rum - 5,) W)
t' = Lv(d/dt) 6' + K"(8v)

(2.2.1)

where Lv(d/dt) n' d/dt + Bv(év)d/dt

K

fv positive gradient coefficient of the flow
reaction force.

In the development of this result all orifices are considered
identical although other conditions may also be considered.

As long as valve configurations exhibit symmetric design
features, the steady-state flow reaction forces may be very
effectively shaped by design to give a characteristic, as indi-
cated for example, in Fig. 2.2.4.

f
v

 

 

Fig. 2.2.4.--Steady-state flow reaction force
Several special cases are now considered. These special
cases give the terminal equations of typical commercially produced

control valves.

10
The Three-way Valve with Constant Pressure Supply

 

The schematic of a typical three-way valve is shown in
Fig. 2.2.5. It consists of two variable orifices, both of which
are underlapped. The terminal representation of the three-way
valve may be obtained from (2.2.1) by making the following

simplifications:

Set ph = P

Orifices l, 2 variable with A = A u 2 'gvl

Orifices 3, 4 nonexistent.

h3 gh
A_inl_J -
Ll

 

 

 

 

 

 

 

 

 

S
'VTSfir

m

Fig. 2.2.5.--Three-way valve

This equation then reduces to

ehl Kvuau + 5v) uphl - P. + (Au- 8v) {r—phl)

 

ph = P8/2

H
I

2 2 '
MV (1 /dt 5v + Bv(5v) + Kfv( 5v)
(2.2.2)
A typical system connection with a three-way valve shows the

load inserted between h1 and h2’ so that ph is controlled by the
l

11
valve and ph is biased at half the system pressure.
2
The Four-way Valve with Constagt Pressure Supply
The schematic of a typical four-way valve is shown in

Fig. 2.2.6. It consists of 4 variable orifices all of which are

of equal dimensions.

D
L]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b"
:3‘

Fig. 2.2.6.--Four-way valve
The orifices may be either overlapped or underlapped.
Consequently, ZS is included as arbitrary parameter. By simply

setting ph = p8 in (2.2.1), there results
3

 

 

ghl = xv [rum + 5,) ephl - p8 . F13(A - 5v) .r—phlj
Shz = Kv [F13(A - 5v) {p112 - 138 + F13(A + 5v) {I phZJ

(202.3)
IV = Lv(d/dt)5v + Kh(5v)

l
2 2
Mvd /dt v + Bv(5v)d/dt

This result is valid for all values of valve lapping ZS .
When [5 = O, the valve is of zero-lap design, and (1.2.3)

reduces to

 

m.
D“
II

Kv ISVI Jpn - F12(5.)P. (2.2.4)
2 Kv ISVI \/th ' F:|_2('5v)Ps

 

09-
33'
ll

12

where F12 is a switching function defined by

F (X) = +1 X 2 0
12 (2.2.5)
= 0 X <: 0

Equations (2.2.3) and (2.2.4) are associated with the terminal
graph of Fig. 2.2.1. As mentioned before, the hydraulic reference
is required in many system studies. In cases where the hydraulic
reference may be omitted and A = 0, (2.2.4) may be simplified to

give the following terminal representation:

 

 

 

e -5!- [<91 0? r (5 no
h yg' v h 14 v 3
(2.2.6)
fv = Lv(d/dt) 5v + be( 5V)
hl v
éh’ph V \/ rv, 5v
h l
2 8m
where F14 is a switching function defined by
+1 X :> O
F1#(x) = o for x == 0 (2.2.7)
-1 X <: 0

The Four-way Valve with Constant FIgw Suppiy

The design of a four-way valve operating from a constant
flow supply is basically identical to the schematic of Fig.
2.2.5. However, one important restriction must be imposed. The
orifices must be underlapped so as to provide a shunt at all

times, i.e.,

13
Au >A1 a |6v| (2.2.8)

The formulation of an explicit mathematical model is com-
plicated by the condition that as is a specified flow driver
whose graph element must be included in the chord set. The
difficulty is evident when one lets 3h} = as and ph3 = p8 in
(2.2.1); this results in a model non-explicit in Ps' 0n the
other hand, the selection of a different tree in the graph of
Fig. 2.2.3, such that element hi is a chord element, as it
should be for a constant flow driver, interferes with the
requirement of putting all orifice elements in the chord set.

An explicit terminal representation may be obtained when
the connection line from the flow source to terminal h3 of the
valve is assumed to possess capacity. The capacity of the flow

source to line connection may be represented by a single element

with the terminal characteristics

ha

V d/dt pe .-. cl 5. (2.2.9)
3

 

whereCe is the capacitance of
line.
The system graph showing the interconnection of the four-
way valve with the flow source is shown in Fig. 2.2.7.
An explicit terminal representation may now be obtained by
combining the graph equations and terminals equations (2.2.1)

and (2e2e9).

14

 

 

Fig. 2.2.7.--System graph of flow source-valve combination

 

éhl = Kv((Au + 5 v) tphl ' ps + (Au - 5v) lphl)
€112 = xvuau - 5v) uphz - ps + (An + 5,) fphz)

l .
d/dt p8 =0: (Kvusu +5.) F—phB - pa + (Au -5v) ¢/——_ph2 - pa) + 8.)

iv = Mv dZ/dtz 5v + Bv(5v) + Kfv(5v)
(2.2.10)

Terminal representations of any other valve configuration
may be obtained on the basis of the techniques applied in the
derivation of the above three cases. However, if an explicit
mathematical model is to be realized it is necessary that all
orifice elements be made part of the chord-set in the system
graph. In pressure Operated valves this requirement is easily
satisfied; in flow operated valves special considerations must

be taken by including the effect of capacitance of lines and oil

volumes. Section 5.2 deals in more detail with this problem.

15
2.2. Valve Actuator;

To effectively incorporate a control valve in a hydraulic
system, provision must be made for controlling the stem position.
An electromagnetic transducer may be used to directly manipulate

the stem, as shown in Fig. 2.3.1.

 

 

 

 

 

 

 

 

 

 

D
3

 

 

 

{Jog—[3L

51 az b

 

 

 

 

P—
O'—
N

h1 h2

Fig. 2.3.l.--Typical four-way valve electromagnetic transducer
assembly.

Quite frequently, however, the output impedance of such a
transducer is not sufficiently low, unless the device is very
large, to adequately handle the load requirements imposed by the
valve-stem forces. A booster amplifier deriving its power from
the hydraulic power supply is frequently used to amplify the
action of the electromagnetic transducer. A typical design of
this type is shown in Fig. 2.3.2. Such an assembly is referred
to as an electrohydraulic servo valve.

In this section, the terminal characteristics of several
typical actuating mechanisms are developed.

Electromagnetic Actuator
The terminal characteristics of an electromagnetic trans-

ducer are modeled by equations of the form

16

R +L d/dt 0 K d/dt i
aa an ac a
= o Raa+Laad/dt "Kacd/dt 1b
2
-Kac +Kac ch+Bccd/dt+Mccd /dt 5c
(2.3.1)
a1 b1 m
'a vb re
1a 1b 5c
a2 b2 8m

where Raa = coil resistance

ch’ Bcc’ Mcc

L = coil inductance
aa

Kac' Kca = electromechanical coupling constants

= mechanical constants.

H

 

 

 

 

 

 

pressure

 

Fig. 2.3.2.--Typical four-way valve with electromagnetic

transducer and booster stage.‘

 

*Raymond Atchley Inc., Los Angeles, California, Model 410.

17

This linearized model of the electromagnetic transducer is
usually quite justifiable for small signal operation of the
component (8). The mechanical variables represent either rota-
tion or translation whichever is more convenient.

Booster Amplifiers

Several effective designs for a booster amplifier are
employed in the many commercially available servo valves. In
order to define the terminal characteristics of a servo valve
assembly it is convenient to consider the booster amplifier as a
subassembly of the servo valve and derive its terminal character-
istics separately. Two typical booster amplifiers are discussed
here.

Booster Amplifier of RAD Model 410

 

The booster stage may be considered as a subassembly ofthree
components, namely a jet-pipe valve, a push-pull hydraulic piston,
and a position feedback spring. The terminal characteristics of
the three components are considered first.

A. Jet-pipe valve

 

 

 

Ps c' d e
\/ c' d e
8m 8c:
““" c" «<s—4»

Fig. 2.3.3.--Jet pipe valve (a) schematic (b) terminal graph

For the terminal graph of Fig. 2.3.3(b), one has

 

 

 

 

 

 

 

 

”f A ' o o o - '5 v- ' o 1
C C
Pe J'kd 0 Rd .89 . Ps/2J

 

Equations (2.3.2) may be derived from (2.2.3) of section
2.2. If in Fig. 2.2.6, orifices l and 2 are considered variable
and orifices 3 and 4 are identical and fixed, and appropriate
modifications are made in (2.2.10), the characteristics of the

jet-pipe valve are obtained.

 

éhl = Kv(A ” 8v) #1511 ' Ps “ K1 {IF—111
éhz = Kv(A - 8v) 61:51:: + K1 zip—ha
IV = Lv(d/dt) 8v

Kfv E- O , A > 6v

For small signal operation about the operating point

 

 

 

 

FPhl- FPS/21
th = ps/2
-5 vJ - 0 J

the equations are linearized to a form identical to 2.3.2., i.e.,

 

 

 

 

 

 

- w - . - r -
p F p O FEB/2 2 p

h1 Kv A -Kl ZS-K1 hl s/2

p / p
s 2 s/2 .

p = O s + p

ha KVTA 4;) 1-31 112 s/2
r o o L (d/dt) 8' 0
_. v J - m J _. v J _ J

 

 

It is assumed that the static pressure at the two receiving
pipe openings varies linearly wth the position of the jet about
a point equal to one half the supply pressure p8. Since the

mechanical characteristics of the jet-pipe have been already

19
included in the description of the electromagnetic transducer,

and since for small displacements about the outer position no
net jet reaction force exists, the top row of (2.3.2) contains
only zero entries.

B. Push-Pull Ram Actuator

For the terminal graph

 

 

 

 

 

 

 

 

 

d' e f
3h
|____,
\/f
r
an an
W
Fig. 2.3.4.--Push-pull ram actuator
of Fig. 2.2.3, the terminal equations are
I. '- I o o d/dt A vp '-
d d
gov = o o —d/dt A pd' (2.3.3)
If j L.A A Lf(dm/dt)JLSf -

 

 

 

 

 

 

where A is the area of the spool ram, and Lf(d/dt) is a second
order differential operator. (See also section 3)
Any compressibility effects are negligible in (2.3.3)

since the actual volume in question is very small.

20

C. Feedback Spring

6"

f = 6
b Kb b (2.3.4)

 

f.
Components A, B, and C are now assembled for the booster
stage subassembly according to the system graph of Fig. 2.3.4(a)

and a terminal representation derived, retaining the three

terminals as shown in Fig. 2.3.4(b).

 

   

(b)

(a)

Fig. 2.3.5.--Booster stage: (a) system graph; (b) terminal graph

 

 

 

 

 

 

This results in
. F _ F
£11) Kb "Kb 5h
= (2.3.5)
2 2 2

Ark. _"Kb'2AK2 Mdd /dt + (Bd + 21 R)d/dt + xb‘ .ka

or when written in normal form

f11 = Kbsh ' Kb 5k ‘

d/dt 5 k = .ml—d((13d + 2A2R) 6k + xbsk 4k + (Kb + 2Akd) 8h)(2.3.6)

.1

 

d/dtSk = 5k

21
Booster Amplifier of Pegasus Model 120-F*

Terminal characteristics of essentially the same form as
(2.3.5) may be derived for the booster amplifier of the Pegasus
Model 120-F servo valve. As shown in Fig. 2.3.6 a dual flapper
nozzle valve in combination with a push-pull ram actuator is

utilized. As for the RAD design, the booster amplifier may be

     
   

\\\\\\\\\\\\\\\\ ‘

_n ‘\ Vk“
lifil'll' Y§§§§§§§§§F §§§Dirfii

     
      

 
    
   

  
  

 

     
 

    

       

 

       
  
  
 

 
 
 
      

'7 WWI/WWW
f
‘9‘ - /"// fl 7 7/, 7/ , gm, / 55;" ‘~:' ‘
51/ g'l/Illll III W '5 ' ‘

"I" ,”’,//? ,/’;/
' I’I/I/I/I” III, [I]: ‘.
a3Rdagflzzagzzzafifizza‘flzzaazSN“

.K'

  
  
 
 

 

azaezzeeazzasazzceg

“1‘“ I .\\\\\\\\l\\\- '2‘"
. a II \
5 g,

’/
M.

 
 
  

m

       

Ex C2 Ps CI

Fig. 2.3.6.--PEGASUS electrohydraulic servo valve
conveniently modeled as a three-terminal subassembly made up of
two components. The representations of the two components are
given first.

Dual Flapper - Nozzle Valve Configuration
In Fig. 2.3.7 are shown a schematic of the flapper-nozzle
configuration and a desired terminal graph.

The terminal equations for a linear model of the valve are

 

‘Pegasus, Inc., Berkeley, Michigan.

 

 

 

.1

'Lc(d/dt)
.Kjf
ij
_'K3p
c.

l\)
R)

 

 

 

 

 

 

 

I \ \ \\ \\\\\\J

 

 

 

where Ld(d/dt)

 

 

 

 

 

Lf(d/dt)

Kjf

Kip

R

M

(a)

The linear

operation of the

pull action.

and the graph of Fig.

It

- "‘ " ' "' F '1
K o o gc o
Lf(d/dt) o 0 5f' 0
+ (2.3.7)
'ij R O éd ps/2
KJP O R - _ge d _p5/2j
F" c” r9 d e
P .
s s
m c' f' d o
1
L‘ 8m gh

(b)

Fig. 2.3.7.--Double Flapper-Nozzle Valve

(a) Schematic (b) Terminal Graph

flapper characteristics

nozzle characteristics

jet force reaction coefficient

jet pressure reaction coefficient

hydraulic resistance of restriction

model is justified on the basis of small signal
variables and the cancelling effect of the push-

may also be derived as a special case of (2.2.3).

The push-pull hydraulic position is represented by (2.2.3)

The overall terminal represen-

2.3.4(b).

tation of the booster subassembly is obtained upon combining

(2.3.7) with (2.3.3), according to the system graph of Fig.

2 0308(51).

23

 

r L (d/dt) -K 5
h = h :3 h (2e308)
Ik -Kfj-2A Kpj Lk(d/dt) + 2A2R d/dt + 2AK 8k
.. - pi
e f c f
f?
kl
h' d h k
8,, .
”m

(a) (b)

Fig. 2.3.8.--Booster stage (a) system graph (b) terminal graph
A comparison of (2.3.8) with (2.3.5) reveals complete

agreement in terms of the form of the equations except for one
term in the lower right hand corner of the coefficient matrix.
In fact, it is this term which enters into the expression for
the steady state output impedance of the booster stage. As the
output impedance is, to a significant extent, a measure of the
effectiveness of the booster amplifier, a comparison reveals that
the dual flapper-nozzle assembly appears to be a more effective

design.

RAD 410 Pegasus l20-F

51:16 1 1

r = — 2A1:
k 11:0 k-b Pd

Steady State Output Impedances of Two Typical Booster Amplifiers
2.4 Terminal Representation of a typical electrohydraulic servo-

valve

When equations (202010), (20301) and (20305) are combined

24
according to the system graph of Fig. 2.4.1, the overall terminal

representation of the electrohydraulic servovalve of Fig. 2.3.2 is

obtained as

 

 

 

h1 h1
al b1 cl f a
Ph ’éh
\ h k 8 vab 1 l
\/ / c s h \/i gh
ab
P sé‘
a b b ha ha
2 2 3m
h
h2 2

(a) (b)

Fig. 2.4.l.--Model RAD 410 Servo Valve (a) System Graph
(b) Terminal Graph

 

 

 

 

 

 

d/dt 12b— 21/ Laamaaiab " vab * Kac 5c -
5c -l/Mcc(Bcc (Sc + (ch + Kb)5c - Kcauab)
8c = 5. . .
és -l/M8(A2R + Bs( 55))55 + Kbés + (Kb + AKd) Sc bes(55)
.55. L 5. _

(2.4.1)
éhl = Kv(F13(A - 8") {thl - pa + F13(A - 8v) {1%)
ghz = KV(F13(A - 8v) dpha - pa + F13(A + 5v) {Ira—ha)

2.5 Experimental Evaluation of Terminal Equations

Although the coefficients in (2.4.1) are shown as explicit
functions of the component parameters, these relations cannot be
used directly to determine the magnitudes of the coefficients.
Further, since not all variables appearing in the model are
accessible for external measurement, many of the coefficients

cannot be determined directly from measurement.

25
A fairly good approximation to the terminal representation

of the servo valve may be obtained by considering the valve to
be a second order system. In this approximation, the coefficients
L

aa’ Mcc' and Bcc are neglected. If in addition, the flow

reaction force, bes( 38), on the valve stem is neglected the
form of the model reduces to two first order differential and

three algebraic equations.

2 2
d 6v -2zwn -0 n 5v (on kavab (
a? = + 205e1)
5' 1 0 5v 0
1ab = Raavab
éhl = KV(F13(A + 8v) {I phl - p8 + F13(A - 6v) d phz)
éhz . K.(F13(A - 5.) d—phz - p8 . rum. 5.) J—; p112)

(2.5.2)

To complete the mathematical model of the servo vase it
is necessary to determine the numerical values of 2:, can, A , ka,
Kv and Raa' All of these coefficients are usually obtainable
from the manufacturer's specifications. In general, these
coefficients may also be determined experimentally by selection
of suitable test conditions. These test conditions are simple
for a servo valve with a zero-lap design (A = O). The test
procedures outlined below refer specifically to a zero-lap
servo valve.
a) No-load Gain Characteristics

To determine the gain characteristics of a zero-lapped
servo valve it is convenient to eliminate the hydraulic reference

from the terminal graph. Then the two algebraic flow equations

26

are reduced to a single equation.
K

.1
x/E'

 

8h = '6 vl V{[ph - F1I+ (SV)PS (20503)

By short-circuiting h and h2’ the valve load terminals,

1

ph is set to zero. For the steady state, 8’v may be expressed

in terms of v Thus, a convenient relation involving only

ab'
terminal variables results which may be employed to experimentally
determine the steady-state flow vs. input gain characteristics.
Kv

k
v5“

Although (2.5.4) was developed for a steady-state measure-

éh = J5: vab (20504)

ment, it will also serve as a basis for a frequency response test.
For
vab(t) = Vabsin w t

(2.5.4) in combination with the second-order valve model
of (2.5.2) yields

gh(t) = G sin (u)t + C)

It is therefore possible from the resulting frequency
response to determine the constants C and w n in (2.5.1).
Due to the fact that 6v’ the valve stem position, is not a
terminal variable, one cannot find the individual factors of the
product Kvka; only their combined value is experimentally
available.
b) The Pressure-Flow Characteristics

On the basis of one of the flow equations of (2.5.2), one
may specify convenient test conditions to obtain a pressure-

flow characteristic for a zero-lapped valve. For the second

equation, let

27

so that

éh2 = K167 :phz

.Again for the steady state, ¢§',is expressed in terms of vab so
that
en "' Kvka'ab ° 911 (2‘5'5)

2 2
Equation (1.5.5) is the expression for a family of square-

root curves with va as a parameter, as shown in Fig. 2.5.1.

b

 

 

 

 

Fig. 2.5.1.--Typica1 family of pressure flow curves:
(a) experimental (solid) (b) Calculated (dotted).

Equation (2.5.5) and the curves of Fig. 1.5.1 not only
provide a second method for determining the gain characteristic
of a servo valve, but they also offer a very effective check on
the assumption of the form of the orifice flow equations.

Usually the electro-hydraulic servo valve is operated in
connection with a servo amplifier. The mathematical model for

such a combination is obtained very readily by realizing that the

28

amplifier has essentially a zero-output impedance and negligible
time-dependent characteristics. For such a combination (2.5.1)
and (2.5.2) may be used as the mathematical model, with the equa-

tion containing ia omitted.

b

III. MATHEMATICAL MODELS OF HYDRAULIC MACHINES

3.1 Gyrators as Mathematical flpdels for Machines

Three basic types of positive displacement machines are of
interest in the study of hydraulic systems. They are shown in
Fig. 3elele

l. Hydraulic Cylinder

I“

 

 

 

 

 

 

 

 

 

l 2
s
r
8n
W
2. Hydraulic Motor
m
__ s
m
m2 8r
'—<nrvxnr

3. Variable Displacement Hydraulic Pump

 

8n /. 0
m / 2
/"at

Fig. 3.1.l.--Basic hydraulic machines

29

30

All three types are classified as multi-terminal components,

whose terminal representationsinvolve measurement of hydraulic

and mechanical variables.

When treated as "ideal" components,

i.e., no internal "losses", the mathematical models for all three

of these components are of a form referred to as "gyrators".

Such

models include only two hydraulic terminals as shown below.

For instance, the terminal representation of an ideal

cylinder is

 

where A
P

 

 

Fpl

stz

 

 

1

 

‘

crossectional area.

1‘1

 

For an ideal fixed motor one has ml

All

3’

 

I 0

-V

 

 

- 2J

L.

V

0.J_d2

' r
p1

 

 

1

 

\

 

m2

where V = volumetric displacement/rad.

An ideal pump is given by

F

t1

0

+V

 

 

 

é2

. .J

where V

tgt

b

tgt

-vt¢t

 

0 J

(5,1

 

 

Paj

b

01

\

zi¢l\

 

O2

volumetric displacement/rad.

and fit = volumetric control variable.

It is well recognized, however, that

in hydraulic machines are not in the least negligible.

8

r

/

 

/

/ f2’332

/’z2’¢2

 

/

(3.1.2)

(3.1.3)

(same (3.1.4)

 

the internal losses

The

losses which have to be considered are due to the inertial and

31
frictional properties of the moving parts of the machines. This

includes all mechanical and hydraulic effects. Since the fric-
tional properties in part depend on the pressures develOped
inside the machines during operation, the terminal graphs must
be constructed to include the hydraulic reference as one termi-
nal. Thus in the terminal representation for an ideal machine
both the form of the equations and terminal graph must be changed
to apprOpriately include loss characteristics.

Some basic assumptions are made to hold in general in the
developments for terminal characteristics of the machines.

1. All mechanical and hydraulic friction effects appear
lumped.

2. A11 mechanical and hydraulic inertia effects appear
lumped.

3. Leakage flows are assumed zero, unless otherwise
stated.

4. Oil compressibility is neglected,unless stated
otherwise.
3.2 Hydraulic Cylinder

The representation is given in two forms. The first form
neglects compressibility; the second form includes it.

Without compressibility, the terminal characteristics are

 

 

 

 

 

 

. - r - .

A, 0 0 A 9,1 A

81.2 = 0 0 -A pr

3r _ :A A Mrd/dt+Fr(cSri_5rd
r1 r2 3

1‘

(3.2.1)

where M
r
M
P

I

M
P

32

+ A2I

P
total mechanical mass

total hydraulic inertance

P

Fr( 81‘) 3 Fr + F14(8r)(fcr + Fcr(Pr + P

with St = o
I I
for fr < for + Fcr(Pr1
for =

4— pr ) and whenever 8 r

))

I 1'2

changes sign
2

constant Coulomb friction.

The primed quantities are the respective stiction terms.

The Coulomb friction characteristics and the stiction are

attributable to the seal pressure which is a function of the

operating pressures.

When oil compressibility effects are included the terminal

representation is modified:

 

 

 

13:1. - -/6/V( 5r)d/dt o
1
é’a = o IB/V(-5r)d/dt
f -A A
_ r .. _ r I‘

 

 

 

.1 ..
A, W

-Ar pr
Mrd/dt+Fr( 5r). .8? 4
(3.2.2)

where V( 8r) = total volume enclosed in one side of cylinder.

Equations (3.2.2) may be written in normal form:

d/dt pr1
pr2
d/dt St =
5 =
r

i 0
fr < for + Fcr(pr + p )

v(8r)
0

(Fr

.1.
M
r

0 whenever

I

5
r

O

v(-5r)

—AV( 5!.)
AV(-5r)

gr
ér

 

_$

+ Flt” 5r><fcr + Fcr(pr

5
r

1‘2

1

2

r

l

—

 

d

+ pr2))

changes sign and as long as

 

(3.2.3)

33
3.3 Variable Displacement Hydraulic Pump‘

 

The type of variable displacement pump most commonly
encountered is the axial piston machine. This unit consists of
a set of small pistons and cylinders uniformly arranged in a
cylinder block as shown in Fig. 3.3.1.

.The piston rods engage through a ball and socket Joint in
the drive shaft flange which in turn is rigidly connected to the
drive shaft. The cylinder block's motion is slaved to the drive
shaft by a universal joint. A plate valve containing two oval
shaped slots directs the flow of oil. In Operation, each piston
draws oil into the cylinder block bores via the inlet port
during one half of each revolution and forces it out through
the outlet port during the other half. Twice each revolution
the plate valve seals off any oil flow between the two adjacent
cylinder bores and the ports. The tilt-angle of the yoke
supporting the cylinder block determines the length of the stroke
of the pistons, and thus the oil flow in pump operation.

The terminal variables used in modeling the component are

identified by the terminal graph of Fig. 3.3.2.

 

‘This development is based upon an earlier development by
Kbenig.& Blackwell, Electromechanical System Theory, McGraw Hill,
1961, page 233.

31+

DRIVE SH APT

     
  
   

DRIVE SHAFT BEARINGS

DRIVE SHAFT FL ANGE

PISTON ROD

VALVE PLATE SLOT

OUTLET
PORT

Fig. 3.3.l.--Assemb1y drawing of axial piston machine‘

5d at O1 02
5'; §
,5 ‘3 ,L, ,3 01 02
d’ d t’ t pO pO
1 2
an em sh

Fig. 3.3.2.--Termina1 graph

 

‘Vickers, Inc., Detroit, Michigan.

35

The form of the equations relating these variables can be derived
from the characteristics of the various elementary components
which go to make up the unit.

As a first step one may consider the characteristics of
the tilt plate with the piston rods removed from their sockets.
The machine is assumed to have 7 pistons. The equations

relating the variables

 

 

 

 

 

 

sd st h i n
/ r I V . . V
d \ t\ IN 2’ 7
8m 8m 8m

Fig. 3.3.3.--Terminal graph of subassembly

of the terminal graph of Fig. 3.3.3 are

 

 

Th = Bd + Jdd/dt _ O ¢d
T; O Jtd/dt ¢d
- 7- fl
sin 95d sin (fid-OU ... sin (fid- 600
f
- kt sin fit :2
-cos ”d -cos (fld- «) ... -cos (fid- 64 f
- “.7

 

 

(3.3.1)

36

 

 

 

. " ‘1
51.1 008 {ad
52 cos (¢d-0<)
I = kt sin gt 2 (3.3.2)
'57. boos (fad - 6o< )_

 

where the subscript d refers to all axially rotating parts.
Consider now a relative motion between the rotating

cylinder block and the plate valve through an angle of ééflw In

particular let the motion be according to Fig. 3.3.h and 3.3.5,

first from - 27C< ”a < 0 then from o< ¢d< + It is

.21.
7
assumed that when ¢d = O the port of cylinder 1 is completely

closed by the plate valve. However, for any small angle ¢d = :8 ,

the piston port is connected to either the 01 or 02 region.

   

- 2C 2;
7 < ¢d< O O < ¢d< 7
F180 30304 F180 30305

As one of the cylinders (1) passes from one pressure region
to the other, the equations relating terminal variables of the

cylinder block are shown below.

 

 

 

 

 

 

'1 — d w ' 1
fi “3?) o o o o o o 0 mp 5'1
{2' o Mfg) o o o o o 0 -AP g2
f5 0 . . 0 -AP .
f4 . . . 0 -AP .
: = . ' . -AP 0 .
. . O 0 “AP 0
L} 0 o Mfg) -AP 0 5'7
gol O O O 0 AP AP Ap G011 6012 p01
_é02d -+Ap AP AP AP 0 o o G021 6022- -p02J
(3.3.3)
where L(d/dt) = Bp + MP d/dt
Bp = total frictional characteristics of a piston
Mp = total inertial characteristics of a piston
AP = crossectional area of piston
Go'j = leakage coefficient between regions i and J
l
h' i' n' 01 02
V1' V2' ° . ' 7W 01
8m

 

 

 

 

 

For the range 0 <: ¢O‘<: %F the system of equations is
identical to (3.3.3) except that the 8th and 9th term in the first
row are interchanged.

When the pistons are assembled with the cylinder block the

terminal graphs of the two subassemblies are assembled as shown

by the system graph of Fig. 3.3.6.

38

 

 

 

 

 

 

 

sd st h i n
\/ \/ \/ \l \/

d t 1 1 2 2' 7 '
8m 8m 8m

Fig. 3.3.6.--System graph of axial piston pump
Now through use of the simple cutset and circuit equations
of the system graph, (3.3.3) are first substituted into (3.3.7)

and the result is then combined with (3.3.2) to yield

 

 

 

 

 

 

 

 

 

 

- q - .. .. _ . -
15 Bd + Ja-%? O ll¢d F'sin2¢t (3d
= . + % ktZBP
.715. b O Bt + Jt ijgty b-sin gtj m
sin2¢t é;— .
+ g kt2 Mp (3.3.5)
:sin ¢t

 

 

sin(¢d-4d) + ... + sin(¢d-6d) sin ¢d + ... + sin(¢d'3x) Pol

-k A ¢
t p t -cos(¢d-Qd) - ... - cos(¢d-6dD-cos ¢d - ... - cos(¢d-3d) po
2
and
éol Bdgt sin(¢d- ’+°<) + ... + sin(¢d- 6x)
= Ktip . (3-3-5)
Léoz Lfldfld sin ¢d + ... + sin(¢d- 3cK)

 

 

 

 

39

-;3 cos (fl -‘+o<)+...+cos(¢ -6o<) G G p
.t d d + 011 012 01

-¢ cos {2' + ... + cos (95 - 3a) G G p
t d d 021 022 02

Similarly, a set of equations corresponding to (3.3.5) and
(3.3.6) may be obtained for the range 0 < Q'd < 27: . It is
evident that the case ¢d = O is the common borderline between

the ranges - .73 < {5d < O and O < 53d < and requires no

7
special consideration.

Note that the expressions relating the torques to the
pressures and the flows to the displacements are functions of
the shaft position fid. Due to the periodic pattern of the
switching action of the plate valve the above equations, as set
up for the range - 17: < ¢d< % , hold for other ranges as

well. Since the shaft torque 2’ and tilt-plate torque ft

d
contain a ripple at a frequency seven times the frequency of
rotation.

These pulses are noticable only at very low speeds.
Similar pulses of course, appear in the flow variables.

When the torques and flows are averaged (3.3.5) and (3.3.6)

over the range - 2I. < 55d < O and O < ¢s < 21 the terminal

7 7
equations reduce to
H W - d q (Fe 1
rd 3.0”.) + Jd<¢t)d—t- Bdt<¢t> -vt¢t W’s 95d
2. AL n O .
3% ’ thgt gd Btgt + Jt dt * th t O 0 gt
a

é V fl 0 G G p

01 t t 011 012 01
g -V ¢ 0 G G p
_°2_ _ t t 012 Ozzy L C’2‘

 

 

 

 

 

 

(3.3.7)

40

 

 

 

Fig. 3.3.7.--Terminal graph
where the coefficients are related specifically to the detailed
characteristics of each component in the assembly by:
_ Z 2
Bd(¢t) - Bd + 2 Kt BP g:

2
Bdt(¢t) = 7 Kt Mp ¢t

Jd(¢t) = Jd + g. i MP ¢i

B; = Bt - glxi Bp

J4=J.-%Kinp¢i

J3 ='§ Ki Mp ¢i

th(¢t) = E'Ki Mp g:

vt=.’J-TZ-K12:Ap

The term ch ¢t can be identified as the volumetric dis-

placement per radian. This can be seen by calculating VP’ the
volume displaced per piston per revolution. It is equal to the

length of stroke times the area of the piston.

VP = 2 sin fit Kt Ap = 2 ¢t Kt AP
Hence, the total volume of the pump per radian is
_ .JL.
Vm ’ 23f vp

or
_ JL

vm ' fr gt Kt Ap

Equations (3.3.7) and (3.3.8) associated with the terminal

graph of Fig. 3.3.7 represent the terminal characteristics of the

#1

variable displacement axial piston pump. An inspection of
equations (3.3.8) shows that a determination of the coefficients
of equation (3.3.?) is highly impractical. Despite the fact that
the coefficients are expressed explicitly in terms of the
characteristics of the individual parts of the pump, many of

the characteristics take on meaning only with the parts installed
in the pump. Thus, experimental means must be employed to find
the coefficients. Through suitable test conditions, one can
single out the coefficients of equation (3.3.7). Furthermore,
using the form of equations (3.3.7) as a guide, meaningful
relations can be established.

Pump friction test.

The following test conditions are specified: p01 = p02 = 0,
it = O, and fit taken as a parameter. For steady state conditions,
then, the curves of Fig. 3.3.8 can be obtained.

If the test conditions are altered, such as to introduce
the effect of pressure on the Coulomb friction, one obtains the
curves of Fig. 3.3.9, for which pc = pd = P, ét = 0 and ¢t = 0.

On the basis of these curves and equations (3.3.7), one

can establish the following steady state torque equation for the

pump

1+2

 

 

 

 

Fig. 3.3.8.--Pump friction for variable displacement pump

 

 

,5 IP=1+OO psi
d
200 psi
0 psi
¢d
Fig. 3.3.9.9-Effect of Coulomb friction
drive shaft:
2 C O O
2h = (Ba * Btzt) ¢d + ¢d/ l¢d| (the + Bed(Pol + P02»
with stiction component (3.3.9)

_ ,.._ , .
2.. ' T. + Be(Po1 * 902)
The coefficient Vt is easily calculated. Nevertheless, a

laboratory measurement will also quickly lead to a result.

Pol 31302 = 0, ¢d = constant, ¢t = O
A go vs ¢t curve is obtained as shown in Fig. 3.3.10.
’1
A great amount of simplification in (3.3.8) is pre-

cipitated by the experimental result that all terms containing
MP are significant.

The only term yet to be determined is the diagonal entry

4a

.-

+25° ¢t

 

Fig. 3.3.lO.—-Pump displacement
relating t; with ¢t. Experimental results indicate friction

characteristics of the nature

ii = gt/ |¢t|(tct + Bct(p01 + p02))
with a stiction torque

.. C
2t ‘ $2. + Bct(p01 + P02)
as indicated by the curves of Fig. 3.3.11.

2%

p0 =p02=200 ps1

0 psi

 

 

 

 

 

Fig. 3.3.ll.--Tilt-plate torque vs velocity

No significant viscous friction term is present.

45

In correspondence with the measured characteristics, a

meaningful terminal representation is now given in normal form

d/dt ¢d = ‘ l/Jd((Bd * Bdtgi) gs ' vt ¢t(Pol '
+ Flh(¢d)(zbd + Bcd<p01 + P02)))
+ F14(¢t)(tct + Bct(p01 + p02)))
_ '
¢t - O for tt-< ‘Zét + Bct (po + pO )
1 2
Pé‘- F l- —G G - p q
C . 011 012 ( c
= Vt¢d ¢t +
é -1 G G p
L 3, _ _ L 012 02a_ _ 3_

 

 

 

 

 

 

 

 

p )-7:
02 d

(3.3.19)

Since normally ¢d # 0, no stiction characteristics need be con-

sidered for the drive shaft.

3.4,Fixed Displacement Motor.

 

The terminal representation of the fixed displacement motor

evolves quite automatically as a special case of the previous

development. By setting
¢t=0
¢t = constant

one has immediately:

d/dt én

...

with fin O

for Z <: ‘2'
m cm

m1

 

 

 

 

 

o
+ ch(pm1

g 1 s

g -l e

- 1/Jm(Bm¢m - v

F14(¢m)(tcm + B

+ P

(p

cm(pm

1

l

 

 

2

+ pm )))

2

 

- pm ) -

(3.4.1)

#6

 

8

 

m

The frictional characteristics must again be determined
through a laboratory measurement. Since a motor may be operated
as a pump also, the frictional characteristics may be determined
through two independent procedures. First, in motor Operation,
and under no-load steady state conditions (1; = 0, Ci = constant)
the frictional characteristics may be determined through a
measurement of differential port pressure and flow. Secondly,
in pump Operation, and also under no-load steady state conditions

(p = pm = 0, gm = -ém = constant) a measurement of torque

l 2 1 2
and velocity will reveal the frictional characteristics. The

two results may be checked against one another very effectively
by plotting on the same figure the hydraulic measurements from

the motor operation on a (p - p ) . V vs g scale and the
m1 m2 m m1
mechanical measurements from the pump Operation on a

Z; vs Vm ' ¢m scale. The results from a typical machine are shown

in Fig. 3.4.1.

 

 

 

Fig. 3.#.l.--Fixed displacement machine friction character-
istics (a) pump friction (b) motor friction.

IV. MATHEMATICAL MODELS OF HYDRAULIC LINES

In this section several models of hydraulic lines are
developed. They are based upon lumped parameter approximations,
so that they may be effectively used to establish a system
model in normal form.

In the modeling process emphasis is placed more on the
presentation of models for convenient system formulation than
a study of time dependent fluid-flow. From this point of view
the derived models will be representative of terminal measure-
ments rather than an analytical account of fluid particle
behavior. Included in the modeling process are the effects of
resistance, inertance, fluid compressibility and line elasticity.

A hydraulic line, or a section of a hydraulic line, is
viewed as a three-terminal component - the two ends of the line
and a hydraulic reference which may be viewed as the atmospheric
region surrounding the line. That this latter point is required
in models that include fluid compressibility and line elasticity
is evident from the manner in which the hydraulic capacity is
measured. This effect is measured by closing one end of the
line to the atmosphere and supplying the other end from a
hydraulic power supply. The pressure and flow measurements
under these conditions are identified by element 1 in the terminal
graph of Fig. ’+.l.l.

The hydraulic capacitance is to include the effect of both

48

49

 

\/N

 

Fig. #.l.l.--Hydraulic line and terminal graph

line elasticity and fluid compressibility. It is defined by

31 = C(p1)p1 (l..1.1)

V
where C = (Ke + Fr)

and Ko coefficient of elastic expansion of line

V

l5

The elastic expansion coefficient should be treated as a

total volume included

Bulk Modulus.

function of the pressure as the elastic characteristics of lines
are generally nonlinear; this is true in particular for
flexible hoses.

The resistance and inertance effects are conveniently
modeled in terms of pressure and flow measurements between the
two ends Of the line. These measurements are indicated by
element 2 in the terminal graph of Fig. #.l.1.

' The inertance coefficient I is easily calculated from the
specific density, 9 , of the fluid, the hose length, L, and

area A, by

50
I = f 111 (4.1.2)
On the basis of typical measurements the terminal equation for
element 2 is established as
p2 = R ‘(é2)K + I AL E2 (4.1.3)
where R is the resistance

K a number normally of the range lg K g 2
«

*(x3‘=-l%— ( IXI )

By forming an assembly of line sections such as modeled
by the terminal graph of Fig. 4.1.1 lumped parameter models of
hydraulic transmission lines are obtained. The number of R-I
elements and C elements depends on the length of line and the
accuracy desired. An example is shown in Fig. 4.1.2, by the
graph of a lumped model with n sections.

For the convenience of later applications of lumped models
the following terminology is introduced. The graph of Fig.
#.1.2 is called a closed-closed (CC) graph, as both the first

and last elements of the graph are C-elements. In contrast,

 

 

3h

Fig. h.l.2.--Lumped model with n-sections, closed ends

51

an Open-open graph (00) begins and ends with an R-I element.
Thus, an n-section (CC) lumped parameter model has (n+1)
C~elements and n R-I elements, while an (00) model with n
sections contains n-l C elements and n . R-I elements.

The values of the individual elements of the graph are
computed on the basis of their relation to the measurements
indicated by the graph of Fig. #.l.l.

Under steady-state flow conditions only resistive effects
enter into a pressure-flow measurement. One has then the

relationship
11
R = 2 R1 (ital-0"")
1:].

When one end is blocked off, the line becomes simply a
fluid storage unit. Thus, under steady-state conditions, only

capacitive effects are measured so that

c = Z 01 (4.1.5)

The line inertance is simply related to the section iner-
tances by
n
I = 2 I1 (4.1.6)
1:].

2.2 Models of Lines

 

The terminal representation for the lumped parameter models
of hydraulic lines may be given in (CC), (00), (0C), or (CO)
form. For a particular application, the choice will depend upon
how the representation "fits" together with the rest of the

system in order to achieve normal form representation.

52

The terminal representation for a line with three sections

is now being developed in (CC) forms

1 \5 a2 \6 a3 \7 a,+ a b
/' /’

     
   

8h
Fig. 4.2.2.--System graph for terminal representation
The terminal equations for the elements in the system of

Fig. 4.2.2 are

 

 

 

 

 

 

 

 

 

 

- - -1 I r. _
d/dt pl 5 o o o gl
1 .
p2 _ O 5' O O 82
- l .
p3 O O C O g3
1 .
p“ o o o E g,M
(4.2.1)
'. ‘ " 1 R ‘ ' “
d/dt gS +f’ O -I' O O p5
e 1 R
86 ' II 0 “I 0 P6
0 l R
-g7J - 0 +1 0 O -I _ p7
é
5
is
é

 

 

The terminal equations of the capacitive elements appear

explicit in the derivative of the across variable; hence elements
1, 2, 3, and 4 are contained in the tree. Since these elements
complete the tree, the measure graph elements a and b must go

into the chordset. Upon expressing the right hand column

53

vector Of equation (#.2.1) in terms of the branch across and chord

through variables we have

 

 

 

 

 

d/dt -pl- '0 o .% o o) .131. 1% 0- ea.
pa 0 %;- 7% 0 pa 0 éb.
PB O O 3 ’3 P3 0 O
ph = o . . . o o o g;- ph + o 7% (#.2.2)
g5 % ’?1[' o o -213 o o g5 o o
36 o % -% o o -3;- o :6 o o
37‘ L0 O I'% O 0 "II 97. .0 OJ

 

 

 

The terminal representation is completed by introducing the

remaining terminal variables

 

 

 

 

 

 

 

 

 

 

 

 

p17 Tea.)
.P‘t .pb'l
and by changing from the measure graph to the terminal graph:
r - - '1 ~
d/dt pa) 0 o 73 o 0) pa) 5- o éa
1 1 .
Pb ' O O "6 Ph 0 c 8h
l l
I’2 ° 6 "('2' 0 P2 0 O
l 1
p3 = 0 o'- O O (3- -C jp3 + .
l l R
l l R
E6 0 O I -I O -I O g6 .
l 1 R
£7- L0 'I O 'I' O O "I- .é7. .0 0.1
(40203)

Equations (#.2.3) show that there are exactly two terminal
equations and five auxiliary equations to describe,in CC form, a
three-section hydraulic line. The total number of equations for
a line thus corresponds to the total number of elements chosen

to describe the line.

5#

The development of the terminal representation for an 00
model is handled similarly. Again, a three section model is

considered and the result is

d/dt éa L; o -% 0 ga I 0 pa
E1:: 0 ”I O “I O é1:: O I Pb
P1 = %’ 0 0 -% p1 + O 0
Pa O 3 O 3' p2 O
_éz._ _0 0 i “i 3}: fled _0 0_

 

 

 

 

 

 

 

 

(4.2.5)

It is to be noted that a CC representation has both ter-
minal variables explicit in the derivatives of the across
variables. Exactly the opposite is true for a 00 representation.
In a CO or DC representation, the terminal equations are
explicit in the derivatives of exactly one across and one through
terminal variable.

It is significant to note that a normal form represen-
tation, as obtained for these two examples directly, is an
immediate consequence of the topological development of the
graphs of lumped parameter models of lines.

Fig. 4.2.3 shows a comparison between the measured and
computed pressure response of a closed line. The computed

result is based upon a #-section CC model.

55

 

p
ne./m' x 10

2.0 " o ...-O°—°‘-°‘—'

1.0 - /

.5. /.,

 

 

 

w
0
4:-
5')
m
o

.1 .2 .

Fig. 4.2.3.--Pressure response for 16 m flexible line.

V. FORMULATION AND SOLUTION OF TYPICAL SUBASSEMBLIES

In the foregoing section, detailed mathematical models of
typical hydraulic system components were developed. Some of the
typical nonlinear phenomena inherent in the characteristics of
hydraulic components were included in the models. Two classes
of nonlinearities may be identified in the models. One class
consists of nonlinear functional forms which are generally
continuous functions of one or several variables. Examples
include the square root relation of the orifice equation, or the
quadratic form of the pump flow equation. The other class
consists of nonlinear functional forms which are discontinuous
or have discontinuous derivatives. The switching functions

F and F1#, or the Coulomb friction are examples of this

12’ F13
class. Also included are the limits on the range of values the
variables may assume, which are imposed during the solution
process. These limits depend on the geometric design of the
components and other physical restrictions.'

In the formulation of a mathematical model of a system
consisting of typical hydraulic components, two factors act as
a guide. First, the desired mathematical model is to be in
a form amenable to existing solution techniques, either analy-
tical or computational. It will be seen that a normal form

model offers the only acceptable mathematical form in the

presence of the nonlinearities. Secondly, for simplicity, only

56

57

as many of the detailed characteristics of the system components
as are necessary for a meaningful analysis are to be included

in the mathematical model. The second factor clearly requires a
knowledge of the solution prior to passing judgement whether or
not a particular characteristic has any significant effect on
the solution, i.e., the only way to determine whether certain
simplifying approximations are acceptable is to compare com-
puted results with actual performance. In keeping with this
Objective several typical subassemblies are analyzed and their
predicted performance is compared with measured performance.

5.1 Valve-Motor. (Subassembly l)

 

This subassembly consists of a fixed displacement motor
_cascaded with a four-way valve operating from a constant pressure
supply. The terminal equations of the two components are given by
equations (2.2.3) and (3.4.1), respectively. It is assumed that
no appreciable capacitive effect is present in the connection
lines between the components. The graph of this subassembly

is shown in Fig. 5.1.1.

 

v s
m

r
\/v \/.m
8 gm

 

 

Fig. 5.1.1.’-Graph Of subassembly 1

By using the simple graph equations Of Fig. 5.1.1, a

58

mathematical model in the form of three simultaneous differen-
tial and algebraic equations is obtained.

0 Vs”. + Kv(F13( A + 5 v) dph1'_. pa + F13(A - 5') d phl)

(5.1.1)
o = ”.5”. + KV(F13( A - 5v) dp‘h2—. pa + F13( A + 8 v> el'p'ha)

O 1 O
d/dt 9’. = - 3;- (3.”. - vmuohl - ph2> - rm

' + F14(¢m)(thc + Bmc(Ph + ph )))
l 2
To investigate the performance of this subassembly it is

necessary to determine a solution vector (ph (t), Ph (t),

. 1 .2

¢ (t))when the initial conditions (p (O), p (0), ¢ (0)) and
m h1 h2 m

the input function 8v(t) are specified. An analytical solution
process is unobtainable. On the other hand, the direct genera-
tion of a solution by a computer is not feasible either, due to
the implicit form of the algebraic equations. Automatic compu-
tation procedures based upon sumessive approximations such as
the Newton-Raphson method, or the Seidel iteration may be
solved for ph and ph for any ¢ calculated on a point for

1 2 m

point basis from a numerical integration of the differential
equation. Actual solution attempts on this basis, however,

have not been successful for the following three factors.

(1) Presence of the switching function.

.59
(2) Rapidly changing pressures.

(3) Smallness of the leakage coefficients.
For the slightly less general case of a zero-lapped valve
([1 = O) and no motor leakage, equations (5.1.1) may be reduced

to a single equation.

 

The result is .
. . V
d/dt¢=--1-(B¢-VF (5 )( -2(m¢m)2)
m Jm m m m 14 v ps Kv5'

- rm + Fl#(ém)(’l’mc + s 118)) (5.1.2)

mc
Equation (4.1.2) is a mathematical model of subassembly 1
in the form of a single differential equation. In order to
generate a solution, the following condition must be satisfied.
vée

m m
p8 - 2(K )

 

2: 0 (5.1.3)
v5v

This conditien implies that the pressures are limited to
the following range.

13" s (phl, phz) g p8, where Pv > 0 (5,1,5)

The lower limit is set by the vapor pressure of the fluid.
The upper limit, however, is a restriction which is not neces-
sarily valid under certain transient conditions. It is quite
likely that for sudden changes in S-v(t) the system pressures
can increase beyond the supply pressure. This is particularly
true when 5v(t) is set to zero from a non zero value so that

the motor is forced to come to an instantaneous halt.

5.2 Valve-Short-Line-Motor. (Subassembly 2)

 

The following system configuration is essentially identical
to the previous one. However, while in the previous case compres-

sibility is neglected, the effect of line capacity of the short

60

lines in this subassembly is taken into consideration.
For a short line, an adequate representation may be obtained
through a two-element lumped-parameter model as indicated by

the graph of Fig. 5.2.1 and (5.2.1).

 

a1 >> b
R-I element 2

C element

3h
Fig. 5.2.l.--Graph of a two-element line in (CO) representation
_ ,1 _ -. _
d/dt p - O 0 g
a1 C al
1
. O I _B .
8b I I 8h
L 1. L . L 1-.

 

 

 

 

 

 

Identical equations are used for the second line.
The mathematical model of this subassembly is formulated

according to the graph of Fig. 5.2.2.
h

 

Vm

 

 

 

 

Fig. 5.2.2.--Graph of subassembly 2
The formulation tree is selected by observing the rules of
normal form formulation (4): the terminal equation of the

capacitive elements of the lines, al and a2, have the across

61
variable explicit in the first derivative and the terminal

equations of elements b and b have the through variable ex-

1 2
plicit in the first derivative. Thus, a unique tree is deter-
mined.

First, combining the terminal equations of the capacitive

elements of the line with the motor flow equations as dictated

by the outset equations Of the system graph, gives

1 .
- - (é + v + G p - e p )
1 Cl hl Fgml m1 m1 m2 m2

d/dt pa

(5.2.2)

1

- -—- - V + Gm pm
2- C1(mg.16”12m1m1m2

Secondly, combining2 the remaining line equations with the

d/dt pa

circuit equations and the motor flow equations gives

p = p - V (R ¢ + I ld/dt ¢m )
'11 a1 “ 1." (5.2.3)
pie = pa2 + Vm(Rl¢m + I 1d/dtflm )

Equations (5.2.3) are subtracted and added and the
respective results are substituted into the remaining motor
equations (3.4.4). In addition, the valve equations (2.2.3) are
substituted into (5.2.2). With the assumption that for a short

line R1 and I1 are small, i.e.,

 

 

 

 

the final result is obtained as

62

1 O
d/dt phl e "6;”. .JV‘Fu‘A” 5.) r—phl-ps + F13(A- 5v) d—phl)
+ G p - G p )
“11 h1 m12 h2
1 O
d/dt p112 = -C—]:(Vm¢m+Kv(F13(A- 5") V phZ-ps + F13(A+5v) 6’ pha)
- G p + G P ) (SOZOLI’)
n‘12 hl m22 h2
d/dt 91‘ = - 3]: (Bn'. 91m - Vm(ph1 - ph?) - em

+ F14(¢m)(thc + Bmc(phl + ph2)))

where J' a J + 2V21
m m

B' = B + 2V2R
m m

Equations (5.2.4) represent three simultaneous first order
nonlinear differential equations. They are in normal form and
hence a computer solution is directly applicable.

Of interest is a comparison of (5.2.4) and (5.1.3). One
may consider the limit as the parameters of the lines, C1, R1,
and Il’ vanish in (5.2.4). As I and R vanish, the third
equation becomes identical to the corresponding equation of
(5.1.3). As C1 vanishes, the two top equations of (5.2.4)
become algebraic and identical to the algebraic equations Of
(5.1.5). oh the basis of this relationship, (5.2.4) may be
effectively employed to find an approximate solution to (5.1.3).

Although the derivatives of the pressure may become
rather large for small values of C1 and thus cause numerical

problems in the solution, accuracy may be maintained through

time scaling of analog computer solutions or by reducing the

63

increment of integration in a digital computer solution. The
curves of Fig. 5.2.3 show a sequence of solutions for various
values of C1 to demonstrate the implications of the capacity
assumption. The effects of stiction and leakage are neglected.
To avoid any possible difficulties introduced by the discontinu-
ities when 5 '(t) is taken as a step function, (Sv(t) is

taken as the solution of a first order differential equation.
Curves A, B, C are the solution of the differential equations
for three different values of capacitance, which correspond to
the following physical lines.

A - 30 cm of %" I.D. flexible hose

B - 100 cm of %" I.D. copper tubing

C - 10 cm of %" I.D. copper tubing

Curve D is the solution to the single dflferential equation
(5.1.6). It is quite clearly demonstrated that cases C and D
are not distinguishable. Already solution B is a very good
approximation to the equations describing the incompressible
case. Curve E represents the measured characteristic for con-
ditions corresponding to the solution indicated by curve A.

To enhance the significance of this comparison, it is
important to keep Jm as small as possible, so that the time
constant of the motor differential equation is in magnitude
close to the time constants of the pressure equations. For
instance, for a large inertial load, the effect of a change in
C1 is almost negligible.

The curves of Fig. 5.2.4 and 5.2.5 show velocity responses
of the subassembly to various magnitudes of inputs. Computed

and experimental results are shown for two types of lines,

7O

6O

50

4O

3O

20

10

64

 

Rad./sec.

 

 

    

 

 

 

o o O
_ O
\‘ . e o
A - %.= 1014, computed
.. \ 1
D - C :00 , computed
C - % = 1016, computed
.- O
\\\\\ E - % = 1014, measured
4g '\\\\\ ‘1 _ 15
o B - C _ lO , computed
1A
0
l I l I l
.02 .04 .06 .08 .1 sec

Fig. 5.2.3.--Effect of variation of capacitance on velocity

response (Subassembly 2).

 

v
/'/.
v .
//
Rad./sec. °
./
v /
300 — /
/
/
v.
/
/
/ //o——A—u——A~—n—JL—u——A__
-A
200 _ V/ ,X

100 - 7/

 

Valve Opening 100%

no:

/

 

 

YIA 0/0’O’N—O— —°"' _°_ 10%
I °/'
1,; ./
I./
17
IU
o l I 1 l J
O]. .2 39¢

Fig. 5.2.4.--Response curves for subassembly 2;lines:

50

cm copper

 

9&-
/

siy

.)

 

o O ’9’...—-—0'——O—"—0—‘—-r
O ./" —"
o ./
O 0/
,/
Rad./sec.
° / Valve Opening 100%

300 _ 0//

40,2;

AFOAF—CHFO-AFO—‘PO—o—O—

 

 

I °...O._O—. —o—o
. °/‘
7’ ”‘
200 ,. °
7/.
//
//
J-°
,‘l
l/
100 .. I!
U“ .-.,
[/ .’/. '\\ we
.1! ./° ° w... 0/0,.-:.--\.2,__.__.+._._._._._
II /'
1,
by /
o/ .
0 L11 1 I 1 1 1
.1 .2 sec

 

Fig. 5.2.5.--Response curves for subassembly 2;lines l m flexible

67

 

 

0—0" O'“'0—‘ —'
300 . Rad./sec. 9,0

200 . o.

100 p '

 

 

-2OO '

-3OO ?
b

 

 

 

l 2 3 4 5 sec.

Fig. 5.2.6.--Square-wave response curves for subassembly 2
with load;lines: l m flexible.

68

representing a capacity ratio of approximately 10. A comparison
of the curves reveals significant differences between the two
sets of curves only for small valve Openings. Two important
conclusions may be drawn from this observation: (1) The

damping effect of the control valve on the dynamic response of
the subassembly increases with an increase in valve opening.

(2) The effect of a change in capacitance of the amount indicated
in the lines is relatively insignificant for valve openings

above 10% of full opening.

As will be seen later in section 5, the variation of the
damping effect of the control valve as a function of the valve
opening has an extreme influence on the stability behavior of a
closed lOOp system containing this subassembly. The other
conclusion establishes the validity of the technique of
including a capacity in the system in order to facilitate the
computer solution.

An interesting observation is obtained when the effect
of the presence of leakage in the mathematical model is investi-
gated. It is immaterial whether or not the leakage terms are
included in the model, since for the numerical magnitudes in-
volved no detectable influence was discerned in the solutions
presented by the previous figures.

Fig. 5.2.6 shows the computed and experimental results
of a square wave response of this subassembly with a large
inertia load connected.

5.3 Valve Long-Line Motor. (Subassembly 3)
In order to obtain a mathematical model in normal form

for this subassembly it is necessary that the lines be modeled in

69
CC form. This condition is evident from an inspection of the

system graph and the terminal equations of the components.

 

v sIn
V/‘V \/:n
8m 8m

 

 

Fig. 5.3.l.--Graph of subassembly 3
To avoid nonlinear algebraic equations in the final mathematical
model elements a1 and a2 must be branch elements and as such
their across variables must be explicit in the first derivatives.
This condition is achieved by a closed representation for the
lines at those ends to which the valve is connected.

The motor equations require that p and pn2 be solved
explicitly. These pressures may be solved by the differential
equations of the line when a closed representation is chosen
for the other end also.

Let it be assumed that the characteristics of the lines
are adequately given by a 5-element representation. A three-

terminal representation for a 5-element line is

 

 

 

 

 

 

 

 

 

 

- .- _ - - - 7 _
d/dt pu O O C -% o pa % Q éa'l
“l 1 1 l 1
pb O O O O -5 pb O 6 gb
1 1 1 1 1
P = O O - - p + O O ‘ ‘
cl C C cl
. 1 1 R . (5.3.1)
sac I O ‘I “I 0 8as O O 7
. l o 1 1 o n l '
8 - ~' ~- g 0 O
h Cbl, L I I [_j - Cbl_ _
al bl ‘
p .Q, / p .é.
all 01 bl bl ‘
\.‘
gh

The development leading to (5.3.1) runs parallel to the
development of ecuatinns (4.2.3). Again, it is quite permissible
to let the C and R coefficients of (5.3.1) be functions of the
pressures and flows, respectively.

The model of the subassembly is obtained by combining
(5.3.1) with the other component equati ns as dictated by the
graph equations of Fig. 5.3.1.

Fig. 5.3.2 shows a set of velocity response curves for
this subassembly. The OOMputed results are b1“ed on the model
given by (5.3.3). The measured characteristics clearly show
the transportation leg characteristic of long lines. The
computed results also give correct indication of this lag,
but include a certain amount of oscillation not present in the
measured result. This is, of course, attributable to the
particular mathematical model chosen to describe the lines.

If the lines are represented by a lumped parameter model con-

taining more elements, thus forming a better approximation, the

71

- - - 1 _
d/dt Ph], 'C-(éacl + K417‘13(A + 6v) C/ phl - pa + F13(A- 5v) ‘V 131))

1 d"““ ' tf_-'

p112 “6(éac2 + K‘F13(A "' 5V) Pha T PB + F13(A+6v) P2»
1

Pc1 C(éac1 + gobl)
1

pea C(éac2 + écbz)

éac 'I‘Réac + Pcl ' ph )

1 1 1 2

éac2 = 'Iméaca + pc‘2 ' th)
l

g --(Ré + p . p ) (50303)

cb1 i cb1 c1 m1

écb2 Imécb2 + pc - pm2)

pm1 -C(écb1 + vmgm)
1 C

pm2 -C(écb; ' vmgm)

° 1

¢m '3F(Bm¢m vm<pm1 - sz) 'ltm

. F14(ém)(thc + B..<Pm + pm )>>
L a _ l 2

 

 

 

predicted response is found to agree more closely with the
measured response.

Fig. 5.3.3 shows another velocity response of this
subassembly with an inertia load connected. The measured result
coincides very closely with the mathematical solution in this
case. Since the inertia load is the primary time lag effect,
effects due to any differences between the physical and mathe-
matical model Of the lines are concealed.

5.4 Valve Short-Line cylinder. (Subassembly 4)

A combination of components occurring frequently as
subassembly in a hydraulic control system is a valve-line-cylinder
combination. The formulation for this subassembly is identical
in structure to the subassemblies l, 2, and 3. One can obtain
the corresponding mathematical models for the valve-line-
cylinder subassembly directly from equations (5.1.1), (5.2.4),

and (5.3.3) by simply replacing motor parameters by corresponding

 

300

200

100

72

 

 

Rad./sec.

Valve Opening

 

 

100/0
.\\. ’0’.’ o O
40%
\\. \\ \
o “*
o o
10%
0 fl.“
.Lxr“’”"u
3”
/
n
.2 sec

Fig. 5.3.2.--Response curves for subassembly 3.

Lines:

16 m flexible

73

 

 

 

 

Rad./sOC.
.6“.
/. *2.
/ to
I0 \00
I \. Oo /-"'6‘°° o O o ,_..._..
lo \. 0° )0/0 .\.___ /"° 0
I \J'
50 .. I '
I
I
F
I
I
I
o I
I
I
Io
I
I
I
-50 ’I
J.
l I I I
l 2 3 sec 4

Fig. 5.3.3.--Response curves for subassembly 3 with load.
Lines: 16 m flexible.

 

74

piston parameters. In all three cases it is, however, assumed
that the capacity of the cylinder chambers is negligible.
Special consideration must be given to applications where the
chamber capacity is not negligible. This particular case is
discussed here.

For this subassembly the following components are used:
(1) a four-way valve, (2) a short line modeled in terms of two
elements and (3) a cylinder with appreciable chamber capacity.

The system graph of the subassembly is shown in Fig. 5.4.1.

 

 

 

V Sr
v v V r
8m 8m

 

 

Fig. 5.4.1.--Graph of subassembly 4

The equations for the subassembly are found to be

d/dt

 

 

 

--(R e - p + p
1 b2 ( h2
- g. + A
Cp(lzr) b p
'c (-5 ) (3b ' A
l(003 r A2(
-'1' r-5r p prl
5 r

1
(IA-6') (P's—2'1“. + F13(A.5 v> e! p112)». 31.2

75

1 d d
'51(Kv (F13(A+5v) ph -pa + F13(A'5v) Phl)+ éb )

_1
C(K'(F13

1
(5.4.1)

F

5 + G p - G p )

O’ 11 1 ’12 ’2

6 P+Gp)
P ’ ’12 ’1 ’11 ’2
' Pr ) - 1’r + I:Il4(5r)(frc+ Brc<pr1+ pr2m

l

 

From (5.4.1), it is seen that 8 differential equations

are required when the capacitive effects of the line and the

cylinder are taken into account separately.

When the two capa-

citive effects are lumped together, as may well be done for a

short line, equations (5.4.1) reduce to a set of equations similar

to (5.2.4).

d/dt

where C(6r)

agree closely with experimental results.

by Wang (5).

P

P
P

 

’1

1

 

 

_

1
-ir(F

3.

1‘

6

1‘

cylinder.

--Y-—-7(Kv (F1M(Z§+¢S )

--(3_(:1-5-:).V(K (F: 3(A- 5 v)’p
-A:(pr-

1

r2

 

"‘ pr - Ps + F13(A- 5V) "‘ pr +Apér)
Prea- P813(A+6v) * pra-Ap6r)
)- r + F14(16r)(r +13 (1::r +p2)))
rc rc rl
(5.4.2)

combined capacitance of one line and one side of

Solutions as predicted by the mathematical model (5.4.2)

This was shown earlier

Due to lack of suitable experimental facilities,

the effect of a change in capacitance as a function of piston

rod position could not effectively be demonstrated.

76
5.5 Pump Motor. (Subassembly 5)

 

The combination of a variable displacement pump and a fixed
displacement motor is also a basic subassembly in hydraulic
systems. It is frequently employed to serve as an infinitely
variable speed transmission. The terminal representations of the
two components are given by equations (5.5.14) and (3.4.1). If
the subassembly is connected according to the graph of Fig.

5.5.1, the mathematical model (5.5.1) and (5.5.2) is obtained.

 

 

 

6
8d 81: m
v d v t v m
8m gm 8m

 

Fig. 5.5.l.--Graph of subassembly 5

The resulting mathematical model clearly is not in normal
form since a combination of nonlinear algebraic and differential
equations is involved. The difficulty involved in realizing an
exact solution is increased by the fact that a lower limit on
the pressures must be observed. This lower limit is set by the
pressure at which the make-up fluid is supplied to the system.
Normally, proper limiting of variables is handled utilizing
information on their derivatives. Since in (5.5.1) and (5.5.2),
the pressure derivatives do not appear, it is extremely diffi-
cult to properly determine when the pressure limits are in

effect. As an auxiliary measure, one might use information on

77

P- '1 r- -

 

 

 

 

C 1 C O
“t ”t
C 1 O
d/d" ¢d " '3;(Bd¢d ' t3 ' Vt(p01 ' Pozmt
+ medXz'dc + Bdc(p01 + p02)»
0 1 0
gm '3;(Bm¢m "tm - vm<PO1 - p02)
+ F14<¢.’(za. + Bdm(p01 + P02))) J
(5.5.1)
r- - - a *- - W
V” . . Vt (G01 + 6’1) -(G012 + G312) FP°1
O = gm + gd fit +
-v ..v -(G + G ) (G + G ) p
. m1 . t- - O12 m12 01 m1 J _ 02)

 

 

 

 

 

 

 

 

(5.5.2)
the velocities éd and 6m and position fit. By monitoring the
change in these three variables, sufficient information can be
made available to decide at what points the pressures should be
limited. This technique is discussed in detail in the next
section.

For the less general case in which the drive shaft speed

fl is held constant, the system equations reduce to
d

d/dt ét 7,1- (Btét - ”at + F1#(§5t)(7:tc + Btc(p01 + p02)»
= .t
¢t ”t
. Vt . Po
gm = — ‘7; ¢d¢t - [(GOl + Gml) - (G012 + Gm12)] P01 (5.5.2)
2
0 = 33%" '51. ' "1:5901 " 1°02) ” Fl4(¢m)(rmc + Bmc (For 1’62»

Equations (5.5.2) indicated that ¢m is algebraically related to

¢t’ i.e., the output speed changes instantaneously with the

. 78
tiltplate angle, regardless of load effects.

The development of the mathematical models of the valve-
motor and pump-motor subassemblies, namely, equations 5.1.1 and
5.5.1, were based on the assumption that the effect of oil
compressibility can be negligible. In both cases, the mathema-
tical models appeared as combinations of nonlinear algebraic
and differential equations, i.e., gg£_in normal form. When,
however, oil compressibility is included, a normal form model
will result. This was already demonstrated in subassembly 2.
Subassembly 6 demonstrates this for the pump motor combination.
5.6 Pump Short-Line Motor (Subassembly 6)

When the effects of a short line are added to the confi-
guration of subassembly 5, a normal form model is made possible.
The effects of a short line are accounted for in precisely the
same manner as for subassembly 2. The system graph for sub-

assembly 6 is shown in Fig. 5.6.1.

 

 

d t m
Vd Vt \/m
8m 8m 8m

 

 

 

 

Fig. 5.6.l.--Graph of subassembly 6
Following the formulation procedure of subassembly 2, the

normal form representation for subassembly 6 is obtained.

79

 

 

 

F. 1 F l . . .
¢t '3:(Bt¢t ' zt * Fl4(¢t)(ftc + Bt<:(1"o1 * P02”)
gt gt
(.1 --L(B5;5-’t:-V(p -p )1?)
d Jd d d d . t 01 02 t
d/dt a + F14(¢d)(zhc + Bdc(p01 + p02)))
1 C O
p --(V ¢ + V ¢ ¢ + (G + G ) p - (G + G )p )
01 C m m t d t 01 m1 O1 012 ’12 02
1 . .
p +-(V + V ¢ ¢ - (G + G )p + (G + G )p )
02 C m¢m t d t 012 mlz 01 01 m1 02
O 1 O
¢m -3;(Bm¢m ' in ..vm(p0l ' p03)
_ j . + Fl4(¢m)(¢mc + Bmc(p0l + P02))) j
(5.6.1)

Again, as previously, if the parameters of the short line
are permitted to approach zero in equation (5.6.1), equations
(5.5.1) and (5.5.2) result. In the solution of (5.6.1) the limits
for p01 and p02 must be observed. Since the pressure time
derivatives are directly available as calculated quantities,
information for proper limiting is attainable. Although the
pressure equations give the appearance that the pressures are
of equal magnitude, but opposite in sign, this is not at all
true because of the limit requirements. The limit requirements
may be interpreted as imposing a rectifying action on the
solution: only pressures above a positive make-up pressure,
pm, are permitted. Without leakage both pressures are main-
tained above pm in the steady state, with a pressure difference
as determined by the load requirements. But, due to leakage

to the atmosphere, one pressure will always return to the make-

up pressure after a transient, with the other following at a

 

8O

constant difference. Since the Coulomb friction characteristics
of the pump and motor are adversely affected by high system
pressure levels, it is clear that leakage to the atmosphere in
this application is of advantage. In fact, the pump-line-motor
configuration is one of the very few cases where it is important
to consider leakage effects in the mathematical model.

Fig. 5.6.2 and 5.6.3 show velocity response curves of this
subassembly fartwo different connection lines. The tiltplate was
stroked by a position servo whose response lag is indicated.
These curves indicate very sharply that a pump-motor subassembly
is much less danped than a corresponding valve—motor subassembly.
Therefore, the effect of a variation in the line capacity is much
more noticeable for this subassembly than for subassembly 2.

5,7 pump Long-Line Motor. (Subassembly 7)

 

Similar to the formulation procedure used in subassembly 3,
the mathematical model for subassembly 7 is developed. According
to the system graph of Fig. 5.7.1, one has with a 5-element line

approximation the mathematical model given in (5.7.1).

01 m1
8d at 5m
V d V t v m
8211 gm 8m

 

 

 

 

Fig. 5.7.1.--Graph of subassembly 7

81

 

 

 

//’°\\
Rad./sec. / ° .
/ ° .\\ / \-
/ \. / .
o 6 \o '/ °
15 ' /-/- \ ./'
/ I \ /
/ / u o /
/ \ /
/ \.
Tilt-plate lag/ 7/ \‘ ’/
/ i
100 . /
/ /
/ /
/ I‘ o
o/ /"\O
/ // . fl \\ / \
/ - / \\ ./ o
50 ' lf" / °\ / °
/ // o / \ 7
/ /, / \°\ ~’°/
// ‘; //
/ / /.
/
// 0/. //
// / //
0 [ .fi-K‘L’; l l I I
.025 .05 see

Fig. 5.6.2.--Response curves for subassembly 6.
Lines: 50 cm copper.

 

 

 

 

 

f.- \\
Rad./sec. /’ \.
/ \
’ \
/ o, . 0/.
150 - r__ 10 \ /
/ / \ O /'
- /
/ / o\. o /
Tilt-plate / \. ./
lag / \\‘_’.//
/
/ /
/
100 - / / °
/ /
/ /
/ /
/ /
/ / . /-/.°_ >‘\.
/ / ./ °‘\. ,,.
5° ' / /__ / /. \, .,/°
/ I / \13//
/
l/ I/ .l /
/ / / o
// / . ./
// /.’ /
0' /7 //. '
O ./ /.n i I
.05 .1 sec

Fig. 5.6.3.--Response curves for subassembly 6.
Lines: 1 m flexible

d/dt

 

 

83

- q

 

 

1 ' '
gt
1 O
-3;(Bd¢d ‘ ta ‘ Vt(Po1 ' p02)¢,
+ F14(¢d)(tac + Bdc(p01 + p02)))
1 Q
._(g + v ¢ ¢ + G p - G p )
c acl t d t 01 01 012 c1
1 O
'5(éac2 ' vt¢d¢t ' Go12 po1 * Go1 P02)
1
aw“, + t»;
1
-(s 1 g ) (5.7.1)
C ac2 cbz
-l(R g + - )
I a2 Pol pO1
-.1.(R g + - )
I a2 pc2 p02
-g-(R g + - )
I cb1 Pol ’-
-luz g + p - )
I cbz c2 Pnz
1 O
--(3 + V - G P ' G P )
C °’1 "g“ ‘12 I1 I'12 l'12
1 0
--(g - V - G p + G p )
0 ch. .15. ‘12 n1 '1 '2
1 O
--—(B - V (p - p ) -‘3
JI mfim m m1 n2 m
+ F14<¢n><t.. + Bnc(pm1 + pn2))) J

8’+

Fig. 5.7.2 shows a set of response curves for changes in
the tilt plate stroke as indicated. It is very clearly demonstrated
by these curves that the high absorption and dissipation
characteristics of the long line have practically eliminated any
oscillatory response. If, however, an inertial load is connected

to the motor, oscillation again is observed as is shown in

Fig. 5.7.}.

85

4/’//

System Parameters:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

16 m flexible lines
$6 = O
Rad./sec.
150 ._ r—-— o
O
O /./‘
o ,/'/.
./
./
/
0 /°
/
//
Tilt-plate lag /;
100 /
/
/
/° °
/
/
/ o
/
/
/° °
50 _— r-‘I‘ / O O O
I . . /......—-»~
I / ° ,/
/ ° .z’
|’ /° -/
I o o /
I / /‘
I - /
I / ° /°'
I , .
I - o /
/ ./°
_“_i a/ 0| /n/.
.05 .l .15 .2 .25
Fig. 5.7.2.--Velocity response of subassembly 7.

 

 

 

 

 

 

 

 

 

 

 

 

 

58¢

120

110

100

90
80

7O

6O

50

40
30

20

10

-10

-20

-30

-40

-50

-60

86

 

 

 

 

 

I ~\
/
\
- éo<>o\
I 9
0 I
b I o .-
. \ .’ X
I .Y ./ o \
F I \ /°0 0"° .- .................
0 \ / ....... o
r- I .0 ,0 °:°,/¢;oooo°°o
I \ o /o 1
- I \, o d
I \ x/
- I -.
I
’ I
I
' I
O
_ I
I
I
I
_I
_Io
I
_I
I
I
TI
0
"I
I
be,
1 L l 1 l
1 2 3 k 5 sec.

Fig. 5.7.3.~-Velocity response for subassembly 7 with load

87
5.8 Pump Short-Line Valve Cylinder. (Subassembly 8)

In the subassemblies discussed so far, only one control
element was employed to actuate a motor shaft or cylinder rod.
This section presents the mathematical model of a subassembly
involving two control elements, both a variable displacement
pump and four-way control valve. The potentials of such a
combination have well been recognized. The highly effective
control action of the four-way control valve is combined with
the pump which serves as a power supply, self-adjusting to power
needs.

Since the vabe in this subassembly is operating from a
specified flow supply, the pump, it must be of totally underlapped
design. It was indicated in section 2.2 for (2.2.10) that an
explicit terminal representation for a valve operating from a
specified flow supply is obtained by including the capacity in
the connection between the flow source and the valve. For this
reason, a short line is analytically inserted between the pump
and the valve. The terminal representation of the valve operating
under the stated conditions is given by (2.2.16). In (2.2.16)
information regarding the pressures phl, th and ph3 is required.
The existence of capacitance in both the lines connecting the
pump to the motor and the cylinder chambers result in first
order differential equations explicit in these pressures. Thus,
the mathematical model of this subassembly may be developed in
normal form.

In a pump-vare combination, it is appropriate to restrict

the pump flow to one direction. One pump port may then be

permanently "grounded" to the reservoir. Thus, the system graph

88
of subassembly 8 is established as shown in Fig. 5.8.1.

 

 

 

5d 8t 8r
V d V t vr
3m 8m 8m

 

Fig. 5.8.l.--Graph of subassembly 8
The tree of this graph is selected on the basis of normal
form requirements (4). This forces the line element into the
tree despite the fact that the corresponding terminal equation is
explicit in the derivative of a through variable. But, there
is no need to solve for this through variable as it is equal to
the specified flow driver as the cutset equation shows. The

formulation process then yields the mathematical model

d/dt

 

89

n l .
Jd d d t 113 t

1 C .
'3:(Bt¢t ' 7t + Flhmtxztc * Btc Ph

5’.

 

1
-C-(Kv((Au+ 5v) {/ph -ph+ (A
e 3 1
+ Vt¢d((l + GORegt + GOI

l
-—-(-5—)-Cr r (Kv((Au+ 5') {Iphl- Ph

4.

 

1
_ (K ((A - 5 > (I p - p
Cr(- 61.7 v u v he h3

1 0
Mr r r p h1 2

0'5.

 

 

' Ta + Flhwdfltdc + Bdcph

))
3

))
3

 

n-5

)dp ~19)
v h3 h2

99%) + Go P113»

4- (Au' 5vt’ph1)

Apér)

+ (An +5?) {/phz)
AP 5,)
trc+ Brc(Ph1+ ph2))

fr)

(5.8.1)

 

VI. FORMULATION AND SOLUTION OF TYPICAL SYSTEMS

The subassemblies investigated in section 5 may be very
effectively utilized in the construction of complete hydraulic
control systems. Of interest are control systems that may be
distinguished as the following types: Systems which are controlled
primarily by (1) a valve, (2) a variable displacement pump, and
(3) a combination of both a valve and pump.

This section deals with the presentation of mathematical
models of systems of the first two types. A discussion of a
system of the last type is given by Dushkes (10).

6.1 _Control Systems with Valves as the Primary Control Element.

The systems under consideration are a position and a speed
control system. The physical schematics of these systems are
shown in Fig. 6.1.1 and 6.1.2.

The principal subassembly of these valve controlled systems

is a valve—line—motor (piston) combination.

 

 

 

 

 

 

 

 

 

 

 

 

 

{5.5T15‘H; ““"“7.£‘I

I f I

' * 5 IT

‘1... I .3: g E
‘ Motor |-- E
I - I . _ Input
@hgn_h____"_‘%_l ' _

4L

 

 

 

 

 

 

 

Fig. 6.1.1.--Valve controlled positioning system
. 90

91

 

 

 

Torque Feedback

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I—__hl__—_—;l-:
I a I I I-
9.1—3.3; .I + —E
: ’ L Motor m I '
. L__i _____ fl“

 

 

 

 

Velocity Feedback

 

Fig. 6.1.2.--Valve controlled speed control system with

load compensation.

The valves are taken as amplifier controlled four-way servovalves

with characteristics as developed in section 2.5. The amplifier

and feedback components are treated as ideal unilateral components

with linear algebraic terminal equations. On the basis of the

information provided earlier, the normal form mathematical model

of the two system configurations may be readily established.

d/dt

(a) Position Control System:

P. - - . -

5v -C1 5‘ - C2( 5v + KaKp(¢O - ¢1))

5 5,
P1 -%'(KVF13(A+ 8') {I p1 - pa + F13(A- 5') fiH-Vmfio)
p2 --1-(K F (A- 5') J“ p2- pa + F13(A+ 5') V: p—2)- v.80)

C v 13
go '8‘380 ' T. ' vm(Pl ' P2) * F14(¢o)(tk * Bc(P1+ P2)))

 

 

- u- L.

 

Q'o
(6.1.1)

Where Cl’ 02 are valve constats as specified by equations (2.5.1).

 

92

2
J = Jn + JL + 2 VhI
B=B +13 +2V2R
m L m

= position feedback constant

(b) Speed Control System:

 

 

 

 

P8: I-clév - 02(8' - Ka('i - Ktéo - Keto))
5. 6'.
a/a. p1 = -%<x'r~13( A+ 5 v) m . F13(A- 5') {/33 + véo)
p2 '%(Kr_FlS(A'8v) m+F13(A* 5') W‘VBO)
_éo‘ Jig-(13550- 2'0 - Vm(pl - p2) . F14(530)(’Zc . Bc(p1+ p2))) .
(6.1.2)
where K - velocity feedback constant

t _

Re a load feedback constant

Equations (6.1.1) and (6.1.2) are the equations of valve

controlled systems with rotational output. Similar equations may

be written for valve controlled systems with translational output.

A solution of (6.1.1) and (6.1.2) is subject to a number

of constraints. These are:

(a) p g (p1. p2) s p , in which (6.1.3)

vap max.

the lower limit is set by the vapor pressure of the fluid. An

upper limit (corresponding to a relief valve) may be arbitrarily

imposed on the operation of the system to protect it from over-

load during transients.

(b) I‘SVI < gmax (valve stem travel) (6.1.10
(c) 80 = 0, whenever 80 changes sign
and V(p1 - p2) < - to + 7:“ . Bca(p1 + p2) (6.1.5)

Equation (6.1.5) specifies the stiction characteristics of

the motor and load combined.

93
6.2 Solution of Position Control System.

The combination of the inherent system nonlinearities and
the constraints of (6.1.3) to (6.1.5) make an analytical solu-
tion process impossible. A solution is considered for the input
vector function (610:), to“) . A solution may be described as
the 6-dimensional trajectory with t as parameter of the state
vector or more practically, a 6-variab1e trace with t as the
independent variable. A digital computer is ideally suited to
r.
57(t)T
5.,(t)
T(t) == p1(t) (6.2.1)
p2(t)
£50m
_¢O(t)j

determine such a solution vector. The various nonlinearities,

 

 

logic functions and constraints are very conveniently incorporated
in the numerical integration process. A solution trace for
position and one of the pressures is shown in Fig. 6.2.1 and Fig.
6.2.2, with experimental traces for comparison.

Despite the insurmountable difficulty of obtaining a
complete analytical solution to equations (6.1.1), some analytical
investigation is possible for certain special performance
characteristics: (1) steady state solution, (2) effect of
variation in parameters on system performance and (3) stability
behavior.

Of considerable interest in the performance study of a
position control system is the steady-state solution due to the

input vector.

14

13

12

ll

10

94

 

 

System Parameters:

 

Ka = .25 x 10"3
pa = (+00 P51
Radians ’L‘ = O
O
__ A = O
- o o
o o
h o
9/.-.\.\ O /".3\! o 0..
I— I .\ O / \ o /.’ \
/ x f d\ . o
, \ 1 . ,/
P- / 8. ° / °\. ° /
p \\ f] \ ° ./
— I, o. 0/ ° \-9 .. J
I °\0-—0/. o
- p o
.’
I
" I
I
I0
” I
I
_ l
.6
I
” I
I
.’
F 1°
I
/
/
J
' /
.6
L_‘{__ I L l I I sec
.05 .1 .15 02 .25

Fig. 602010--Sy5tem 601:

Position vs time for step response

 

when the derivative vector of (6.1.1) vanishes.

95

 

 

 

 

A solution to (6.1.1) with the inputs (6.2.2) is obtained

state solution T(°°) is obtained from an investigation of the

condition that the right side of (6.1.1) vanishes.

#00

300

200

100

From the fifth equation, one has for a given torque load

Thus, the steady

.911“; tbi 1
= (6.2.2)
:50“). _TO 4

 

 

 

Fig. 6.2.2.--System 6.1: Pressure vs time for

step response

psi
0
o
0/"‘5\
./ °‘
. ‘\°
I I’ ,’“\.
I \O [.0 O '\ r”
.I ‘o i! (‘3. f/ o .\
0
II \. \ , o \.
o ' \ o \
\ I \ / \
\ O . O
I, o e I O .I O
‘I I ‘\ /
\ 0’ O)
‘o / ‘| I X
\ J \ ¢/ 0 o\
I
o / §_ . oh 0
\ \ o/ \g”
I, ’1 \° °,-’
\.° ° 0/,
\uuf
sec
l J J 1 4
.05 .1 .15 .2 .25

 

96
_ - 32. (6 2 3)
P1 P2‘ v 00

It is reasonable to limit the torque to
to = «p tux = 9(PVp8 (6.204)
where ofip, defined as the position load factor; may vary between
0 SE 41 SE 1.
P

If it is assumed that the line pressures (pl, p2) have a

solution symmetrically snacod about pc/B, then this solution is

g p /2 (6.2.5)

 

 

 

 

- -4 - P.
Using equation (6.2.5) and the requirement that ¢O vanishes,

one has from the third and fourth equations of (6.1.1)

= +“1-1-“2A=/8(0()A (6.2.6)
vdl+ocp+J1-9(P p

 

The dimensionless ratio Sv/Zk is plotted as a function of

«P in Fig. 6.1.h).

1.0

 

.51;
A

 

 

O A I l l O(
.2 .u .6 .8 1.0 p

 

Fig. 6.2.3.--Va1ve stem position as a function of position
load factor.

When (6.2.6) is substituted into the right hand side of

97

the top equation of (6.1.1), one has:

[5

Now a complete solution vector for [S f O is

T(OO) =

T(O°) =

Févwo)
5500)
p1(O°) =
P2(oO)
5106.6)

¢Ohx>)

 

 

 

 

ap

O

1:
Pulp)

l-‘X

 

 

 

(6.2.?)

(6.2.8)

(6.2.9)

98
When o(p = O and/orA = O, T(oo) reduces to

T(00) = pB/2 (6.2.9)

 

 

Several interesting conclusions may be drawn from the above
development:

(1) In order to maintain the load factor within the range
0 $ 0(1) S 1, one must satisfy, through suitable design, the
condition

25‘s; vps

(2) For a given 0(p, the steady state output position error
is directly preportional to the amount of underlap Asa. This
result may be utilized very effectively in experimentally
determining the lap of a valve.

Linear approximations are quite successfully employed in
predicting, with some degree of accuracy, the behavior of a
nonlinear system about a point of equilibrium, i.e., a steady-
state solution point.

Equations 6.1.1 and 6.1.2 may be written in the form

g
dt

where E and 5 are vector functions of the system variables and

__. 56", §) ‘ (6.2.10)

the inputs, respectively. ;’= E6 is caned a solution of (6.2.10)

if F(;b, 5) = O° x0, of course, is the steady state response to

the input §.

99

If :6 is a solution to (6.2.10), one may express the right-

hand side of (6.2.10) in the form
fl}, 5) = AG - i0) + FIG, 3') (6.2.11)
where A is a constant matrix. If Fi(§i;)satisfies the condition

lim §l(x,y)

x-a>x0 x - xo

then A is a unique matrix and is computable as the Jacobian matrix
of (6.1.1) or (6.1.2), i.e.,

1, 2, 000’ n
1’ 2’ 000’ n

I"
II II

 

3 F165) ] 3
13 = D
b 13' 2:10

The equation

= A(; - Eb) (6.2.12)

is called the linear approximation to (6.2.10). The Jacobian

2|? I

matrix is the predominant part of (6.2.10) and hence (6.2.12)
may be used to predict approximately the solution of (6.2.10)
at E near :60.

For (6.2.1), the steady-state solution is given by (6.2.8)
with appropriate nodifications when T0 = 0 and/or $1 = 0. With
the linear approximation of the form (6.2.12), one may make
valid predictions on the effect of parameter variations on the
performance of the system and the stability behavior at the
steady state solution.

A linear approximation can be obtained for the above
equations, since the nonlinear function F1(;,y) corresponding to
(6.1.1) or (6.1.2) satisfies the requirements for the existence
of the Jacobian matrix at the steady state solution.

The linearized model for the position control system is

100

 

 

 

av -01 -02 o o o
5' 1 o o o 0
P1 0 ‘EF1%° ’6
= K p /2
p: K A
p o -1 o - ' X
2 c 2 p /2 c
8
° v v B
d o o o o 1
L OJ _

 

0
(6.2.13)

 

From (6.2.13), a linear open-loop transfer function is

 

found to be
Ko
F (8) =
0 .(1 + .1.>(1 + t2.><1 . 23L- . .(—1—) 2.2)
43n 0’n
where t1, t2 are computable by equation (2.5.1) and

 

(Bet/2p8 + Jszs)2
hc~J2ps(2v3J2ps - BKle)J

 

59

 

2v2./2§; + Bszx

 

 

QDn =
J2}:s JC
?K K K V ps
K _ a p v
0 -
2 J21): V2

(6.2.14)

(6.2.15)

The condition [1 = 0 results in a substantial simplification

in the above expressions, namely

 

101

“ix/E

Mn = V453 for = O (6e2016)
K K K
a E V .

Ko = 2v VZPB

Equations (6.2.15) and (6.2.16) may be quite successfully
employed to predict the effect of variation in parameter on the
open-loop system behavior and serve as reliable guide lines for
design. Exactly how extensive a closed-loop performance predic-
tion may be made on the basis of (6.2.15) and (6.2.16) is
questionable. Approximate relations exist between open-loop
and closed-loop response for higher order system; but at best,
they are only qualitative relations. However, to remain within
any given degree of accuracy, only a complete solution by com-
putation is acceptable. Thus, if direct computation offers the
only reliable method of design, the computation, of course,
should be based on the nonlinear model of the system and not on
its linear approximation.

The lack of reliability in a design of hydraulic system
using linear approximation, is underscored by the fact that the
actual system may be operating in a stable manner, while the
linear model indicates instability. Computed results of an
actual example represented by (6.1.1) show that the linear model
is unstable at a value of loop gain of about one third the value
at which the nonlinear model becomes unstable. This phenomenon
is directly attributable to the fact that the third and fourth

diagonal coefficients of (6.2.13) have their minimum value at

102
the steady-state solution. If permitted to change with the

operating conditions they would monotonically increase as the
actual operating point deviates from the steady-state point.
6.3 Solution of Speed Control System.

The mathematical model for the speed control system of Fig.
6.1.2 is given by (6.1.2). A steady-state solution for the input

vector
- _. 7

’31”) 431
= (6.3.1)

 

 

 

 

tb(t)J To

_- b .1

is determined by finding the solution of the right hand side of
(6.2.2). The bottom equation of (6.1.2) yields

1», - p2 = $3510 + to + Flhéo(te + Benn, + p2» (6.3.2)
Similar to the development of section 6.2, a load factor

ds’ 0 s (18$ 1, may be defined such that

p1 - p2 = 0L8 p,3 . (6.3.3)

wherecx.B necessarily is a function of the output speed.

By using equation (6.2.5), the valve stem position is

determined from the third and fourth equation of (6.1.2) with
A=OO

 

VB
5' -.- ° (6.3.4)
K, sips/2 J1 - “a

 

It is immediately evident from (6.3.4) that «s cannot
be equal to unity because of the restriction (6.1.4). Sub-
stituting this result into the top equation of (6.1.2), an
expression for the load feedback constant required to main-

tain the output at the desired level is obtained as

103

 

 

vac
K. = (60305)
KaK' ‘1 p8/2 \f3- - 0‘ ago
When ’to is assumed to be a viscous friction load
7:0 = B0 (to ' (6.3.6)
then Re is
K = V (6.3.7)

 

 

e
K‘Kvfia/z JT - o( sBL

(Bn + BL)¢o + C; + Bcpa

 

where 0L s
VP

s
In order to maintain zero steady-state error, Ke must be
readjusted for any changes in the input function. For zero

steady-state error, the steady-state solution is then given by
0
V¢i
Kvxlia/Exfl - «a

 

 

T(OO) = (6.3.8)

 

 

A linear approximation of (6.1.2) about the steady state

operating point given in (6.3.8) is

10#

 

 

 

 

 

'5' ' '.c1 .0 o o -KaKt-[ ‘6;

5' l 0 0 0 0 ‘5‘
V

P1 = 0 $32 -a33 0 *5 P1 (6.309)
V

p2 0 -a32 0 -a33 5' p2

e v v B °

_¢._ 0 3 *3 '3 _ W2-

 

 

 

O
K I - o(
7 8
where 8.32 .. ? 2 p3

a = V@P1
33 C(l - “JP.

 

It is obvious that as both the load and the operating speed
increase, a32 decreases while a33 increases. This has a profound
influence on the stability of the system when operating about an
equilibrium point. This can be seen from the open-loop transfer

function Fo(s) of the linear model which reads

K

 

 

 

 

o
F (s) =
0 (1 + .1.)(1 + .2.)(1 + E. + J—2 .2)
“’n “’11 (6.3.10)
anaxt vc .32
where KO = 2
2V + BC a
2 33
2V + BC a
w - ——__22
n ' JC 2
:2 (132+ J a”) c
(2v + BC a33)AJ

For an increase in either operating load or speed, the linear
small-signal model becomes more stable. The change in linear
stability characteristics as a function of operating point fis very
effectively demonstrated by the response curves of Fig. 6.3.1.

The velocity and torque feedback constants are adjusted at a

105

 

 

 

 

 

 

 

 

 

 

. a Open loop transient
1.3 _ 96950
o o b Velocity feedback applied
all-
ffflo \f c Load applied
1.2 _ ,0
f ) d Torque feedback applied
/
/ | e Operating speed reduced
10]- _ I e .
.o ‘
. I i (.
1.o__’_09313tiis__\_°_o._.____ -_ ________
I . Speed ‘.,/°’" 54’ ° 01)
I \ ./ t
I \. 1° I
.9 - ,1, \. / |
l ‘\°\?.9-9—é E
’ I
e8 .. l ‘
l
aeT a >F+—-b ——aL——¢3 d i e
07 .. lo ‘
' 1
l I“-
.6 \
”I ‘ -/ ( . /
i ‘ / .\ o/'
.s \ , ‘ /
\O 'O 0‘ O
I I \ I
I \ I 1 /
"* ,' ‘ '° \ I
l \ / 0\ ° /'
.3 l \ / \
’(° .° I
\ I \
l \/ ° I
I: - seconds P \o
'9 J 1 1 n 1 1 \l I
' 1 2 3 4 5 6 7 8

Fig. 6.3.l.--Response curves for system 6.2.

106
chosen level to maintain the desired speed irregardless of load.

When the command speed is sufficiently lowered, the system becomes

unstable.

6.# Control System with Variable Displacement Pump as the
Primary Control Element.

The principal component of the speed control system to be
discussed in this section is the subassembly of section 5.6 and
consists of a pump, a short line, and a motor. The tilt-plate
control mechanism is a subassembly consisting of a valve, a
short line and a drive piston. The schematic of the entire

system is shown in Fig. 6.4.1.

 

 

 

 

 

 

 

 

 

 

 

 

. I '

._ ._2_ _____...__ ....

 

 

 

 

 

 

 

 

 

 

(”H—'7
l

a?

J?

 

 

 

 

{i}

Fig. 6.4.1.--Pump-controlled speed control system
As in the valve controlled system, all components employed

in the feedback networks are treated as ideal unilateral

10?
characterized by linear algebraic terminal equations.

The control valve of the tilt-plate stroke assembly is

assumed to be of zero-lap design. The piston rod is connected to

the tiltplate by a lever of length Rt' The mathematical model

of the system as obtained by combining the component equations

with the circuit and cut-set equations of the system graph is

 

 

 

 

 

 

 

 

F . '1 e 2 .
5v 'Zgwn Sv' wn (5v - Kahi ' Kpsr ' thmn
5' 5v
1 .
l’hl 'C:(Kv (8" {113,11 ' F12( 5v) pa + Ap 5r)
1 g 0
p112 -5:(x' )5" <rph2 - F12(- 5') p, - A196,.)
' 1 .
d/dt 6r = -h +an ((Br+ 12:13,) - Ap(ph1- phz)
r t t .
. r14(5r)(£n+ RiBt°(pn1+ 13‘2”)
é5r ‘8r
1 C .
Pml 'C-iwtgs 5r + V-flm + (601 + Gml) pml ' (602+ sz) pm2,>
1 O .
pma +5:(Vt¢8 5r + vmflm + (602 + Gm2) Pml ' (GO? Gml) 1)ma)
0 1 0
gm J + 2V2I ((Bm + 2v:lem - ti ' vm<Pm1 - pmz)
m m l .
.. _J L- + Fll+(¢m)(fmc + Bmc(pml + pm2))) —
(6.4.1)

The variables in (6.6.1) are subject to the following

bounds imposed by physical limitations and geometric design

considerations:

108

(a) The tilt-plate stroke is limited to a maximum (5r ,
max

i.e.

Iéij .: (Sr (6.h.2)
max
(b) -
pn s; (pml, pug) (6.#.3)
The line pressures in the pump-motor connection cannot

dr0p lower the make-up system pressure, which itself must be
above atmospheric pressure to be effective.

Additional bounds are set by equations (6.2.2) through
(6O3Ol‘) 0

Insight to the steady state performance of the system in
Fig. 5.h.l may be obtained by solving (5.4.1) when the derivative

vector is zero. The input vector for which a solution is of

interest is

th(t) Ta
= (6.6.6)
L '1. .yi.

 

 

 

 

For simplicity, let the make-up system pressure be pn = 0.

Then, the last equation of (6.4.1) gives
1

pm = V - B

2 0
1 . me (Tn + the + (Bm + 2vll R.) flg)

p =.0

Substituting (6.4.5) into the seventh equation of (6.#.l)
the required tilt-plate displacement is obtained as
+ G

5r = —-—-(vm¢m+ _ 01

2 0
v tg Bmc (Tm + tic + (Bm + 2Vh R2 05))
d

(6.4.6)

109

 

System parameters:

Rad./sec. P

150 .-

 

 

 

O KaKt = 1+
0 O .135 x 10' computed
O

0‘\

./O\. o l 0

° / \ 0 o I \

Input 0 j \ / \
A — -I—— —\— ——Io— —\

1°0— I \ / 3 /

 

.169 x 10'4measured

 

 

.’ '\ /' I ° .
I . \
/ O \ / \e l
. . /. \“I
I \. I . .
I \./ O
I
50 t i
I o
I
/ o
I
/ o
/
O dz/clog lpm:50psli 1
01 92 93 01+ BQCe

Fig. 6.#.2.--Response curves for speed control system.

500

400

300

200

100

110

 

 

 

 

 

System Parameters:
psi
K = O
P
_ .169 x lO-hmeasured
K K =
A t a .135 x 10-#computed
A A
- A
A
A A
P /e\ A A A
.l‘e /e
// \. / \ A . \A A A
I \ / \ I; \
A \ . A '\ ./ ’
I . I / \
' f \ /‘¥ \ \
\ .\ /
\ ‘ e e
I ::" ‘7 v "’
e / O
/ .l \ /.‘\ V /.\.
A/ / \ V ‘l V ./ V\
_ /. / v . / V \. l/v
. /e \ ./ V \ /
A/ ’ \. / \. ./
L———VLV._V.—._.—v x. V
v v
L J l l
.1 02 03 cl‘l’_

Fig 6.4.2.--Pressure

response curves for

sec

speed control system

111
It can be seen from (6.#.6) that the tilt plate displacement is

a function of the load torque only through the pump to atmosphere

leakage coefficients. Thus, for a system with no leakage, the

output speed is independent of the load torque. Equation (5.2.6)

offers a convenient method of determining the leakage coefficients.
On the basis of the steady-state analysis of system 5.1,

it can be said that

 

 

 

 

Pphi- (p3)
= >5 (601“?)
P P
- h21 - 81
g. = '1 - thn = KP 8r (6.1‘08)

Hence, the steady-state velocity error is directly proportional
to the tilt-plate angle when tilt-plate position feedback is
employed. It is zero when no tilt-plate position feedback is

used.

VII. CONCLUSION

In the presence of the inherent nonlinearities in the
mathematical models of hydraulic system components the analysis
and design of hydraulic systems is most effectively carried out
in the time-domain.

Time-domain solutions by computer methods require that
the derived mathematical models of the systems is given in normal
form. Normal form models for hydraulic systems may always be
established when the effect of hydraulic capacitance is included
in the models of the components.

Computer solutions are most conveniently generated on
the digital computer as the nonlinearities encountered are
more suitably handled by numerical integration procedures than
through analog calculations.

0n the basis of correlation obtained between computer
solutions and laboratory measurements for the functional
subassemblies and systems considered, it can be said that the
nonlinear mathematical models serve effectively to predict
performance characteristics, and thus may be reliably employed

in the design of hydraulic system.

112

APPENDIX A

NUMERICAL VALUES OF COEFFICIENTS

Note: All units are consistent with the MKS system.

1. Valve. RAD Model #10, Raymond Atchley

K' = 07 X 1.0-h
Ka = .5 x 10'5 variable to .25 x 10-3
_ -3
max. - .2 x 10
n = 250
= 1.0

2. Hydraulic Cylinder. %” Bore, 6" Stroke,

Cylinders & Valves, Inc.

AP = .253 x 10‘3
Fr = .665 x 10‘3
Mr 2 o 21"‘5
F = 0
er
I = 0
cr
F' = 0
or
' -
fcr — 1.25

3. Variable Displacement Pump
0 - .2 cu in / rev.
Modified from Vickers Hydraulic Transmission.

.161 x 10’3
2

C...
II

a:
ll

d .395 x 10'

113

5.

116

Hi Bdt = 32

Vt = .967 x lO-I+
tcd = .54 x 10-1
Bed = .587 x 10'7
RE Jt = 2.3

tct = 10

13ct = .35 x 10'5
tét = 15 I
Bét = .63 x 10"5

Fixed Displacement Motor
.08 cu. in. / rev

Vickers Aircraft Hydraulic Motor

Jm = .513 x 10"+

Bm = .77 x 10’3
Vm = 0219 x 10-6
-1

tom = .392 x 10
B = .213 x 10’7

cm
-1
tc'm = .5 x 10

_ -7

Bém - .613 x 10

Hydraulic Line
%" Io DO H‘C-l+

Weatherhead Hose

R = .85 x 109
C = .33 x 10-12 per meter
I = .269 x 108

1.

2.

3.

5.

LIST OF REFERENCES

Koenig, H. E. and Blackwell, W. A. Electromechanical System
Theory. McGraw-Hill Book Co., N. Y., 1961.

Reeves, E. I., "Contributions to Hydraulic Control-7,
Analysis of the Effects on Nonlinearity in a Valve-
Controlled Hydraulic Drive. Trans. ASME vol. 79 (1957),
pp. 627-33.

Zaborsky, J. and Harrington, H. J. A series of 6 papers
on Describing Functions for Electrohydraulic Valves.

AIEE Transactions, vol. 76, pt. I, May, 1957, pp. 183-98
and vol. 76, pt. II, January, 1958, pp. 396-608.

Wirth, J. L. Analysis in the time domain and existence of
solutions, Ph.D. Thesis, 1962, Michigan State University.
Wang, P. C, K. Mathematical Models for Time-Domain Design
of Electra-Hydraulic Servomechanisms. AIEE ITG on Auto-
matic Control, Volume 1, No. 1, Winter 61/62.

Blackburn, J. F. Contributions to Hydraulic Control-3,
Pressure-Flow Relationships for 6-Way Valves. Trans.
ASME, vol. 76, 1952, pp. 1163-70.

Blackburn, J. F. Contributions to Hydraulic Control-6,
Notes on the Hydraulic Wheatstone Bridge. Trans. ASME,
v.1. 75. (1953), pp. 1171-73.

Blackburn, J. F., Shearer, J. L. and Reethof, G. Fluid

Power Control. Wiley--Technology Press. 1960.

115

9.

10.

116
Thaler, G. J. and Pastel, E. S. Analysis and Design of

Nonlinear Feedback Control Systems. McGraw-Hill Book Co. N. Y.
1962.

Dushkes, S. Z. Bootstrapped Variable Displacement Pump

Servo. Control Engineering. McGraw-Hill Book Co., April,

1962, pp. 123-25.

Struble, G. Nonlinear Differential Equations. McGraw-Hill

Book Co. N. Y. 1962.

' . n n-
”‘E'a U6; 0’:
,1”; ”:23" ‘ “'5"-

I' I

l;