![Preface
Science is for those who learn;
poetry for those who know.
—Joseph Roux
This book is a continuation of my previous book, Dynamics and Control of
Structures [44]. The expanded book includes three additional chapters and an
additional appendix: Chapter 3, “Special Models”; Chapter 8, “Modal Actuators and
Sensors”; and Chapter 9, “System Identification.” Other chapters have been
significantly revised and supplemented with new topics, including discrete-time
models of structures, limited-time and -frequency grammians and reduction, almost-
balanced modal models, simultaneous placement of sensors and actuators, and
structural damage detection. The appendices have also been updated and expanded.
Appendix A consists of thirteen new Matlab programs. Appendix B is a new
addition and includes eleven Matlab programs that solve examples from each
chapter. In Appendix C model data are given.
Several books on structural dynamics and control have been published.
Meirovitch’s textbook [108] covers methods of structural dynamics (virtual work,
d’Alambert’s principle, Hamilton’s principle, Lagrange’s and Hamilton’s equations,
and modal analysis of structures) and control (pole placement methods, LQG design,
and modal control). Ewins’s book [33] presents methods of modal testing of
structures. Natke’s book [111] on structural identification also contains excellent
material on structural dynamics. Fuller, Elliot, and Nelson [40] cover problems of
structural active control and structural acoustic control. Inman’s book [79]
introduces the basic concepts of vibration control, while Preumont in [120] presents
modern approaches to structural control, including LQG controllers, sensors, and
actuator placement, and piezoelectric materials with numerous applications in
aerospace and civil engineering. The Junkins and Kim book [87] is a graduate-level
textbook, while the Porter and Crossley book [119] is one of the first books on
modal control. Skelton’s work [125] (although on control of general linear systems)
introduces methods designed specifically for the control of flexible structures. For
example, the component cost approach to model or controller reduction is a tool
frequently used in this field. The monograph by Joshi [83] presents developments on](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-14-320.jpg)
![x Preface
dissipative and LQG controllers supported by numerous applications. Genta’s book
[65] includes rotor dynamics; the book by Kwon and Bang [92] is dedicated mainly
to structural finite-element models, but a part of it is dedicated to structural
dynamics and control. The work by Hatch [70] explains vibrations and dynamics
problems in practical ways, is illustrated with numerous examples, and supplies
Matlab programs to solve vibration problems. The Maia and Silva book [107] is a
study on modal analysis and testing, while the Heylen, Lammens, and Sas book [71]
is an up-to-date and attractive presentation of modal analysis. The De Silva book
[26] is a comprehensive source on vibration analysis and testing. Clark, Saunders,
and Gibbs [17] present recent developments in dynamics and control of structures;
and Elliott [31] applies structural dynamics and control problems to acoustics. My
book [47] deals with structural dynamics and control problems in balanced
coordinates. The recent advances in structural dynamics and control can be found in
[121].
This book describes comparatively new areas of structural dynamics and control
that emerged from recent developments. Thus:
x State-space models and modal methods are used in structural dynamics as well
as in control analysis. Typically, structural dynamics problems are solved using
second-order differential equations.
x Control system methods (such as the state-space approach, controllability and
observability, system norms, Markov parameters, and grammians) are applied to
solve structural dynamics problems (such as sensor and actuator placement,
identification, or damage detection).
x Structural methods (such as modal models and modal independence) are used to
solve control problems (e.g., the design of LQG and Hf controllers), providing
new insight into well-known control laws.
x The methods described are based on practical applications. They originated from
developing, testing, and applying techniques of structural dynamics,
identification, and control to antennas and radiotelescopes. More on the
dynamics and control problems of the NASA Deep Space Network antennas can
be found at http://tmo.jpl.nasa.gov/tmo/progress_report/.
x This book uses approximate analysis, which is helpful in two ways. First, it
simplifies analysis of large structural models (e.g., obtaining Hankel singular
values for a structure with thousands of degrees of freedom). Second,
approximate values (as opposed to exact ones) are given in closed form, giving
an opportunity to conduct a parametric study of structural properties.
This book requires introductory knowledge of structural dynamics and of linear
control; thus it is addressed to the more advanced student. It can be used in graduate
courses on vibration and structural dynamics, and in control system courses with
application to structural control. It is also useful for engineers who deal with
structural dynamics and control.
Readers who would like to contact me with comments and questions are invited to
do so. My e-mail address is Wodek.K.Gawronski@jpl.nasa.gov. Electronic versions](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-15-320.jpg)
![xxii List of Symbols
i
] ith modal damping
= matrix of modal damping coefficients
o
B nodal input matrix
oq
C nodal displacement output matrix
ov
C nodal velocity output matrix
m
B modal input matrix
mq
C modal displacement output matrix
mv
C modal velocity output matrix
m
C modal output matrix, 1
m mq m
C C C
: v
mi
b input matrix of the ith mode, ith row of m
B
mi
c output matrix of the ith mode, ith column of m
C
2
m
B modal input gain
2
m
C modal output gain,
2
2 2
1
2 2
2
m mq mv
C C C
:
2
mi
b input gain of the ith mode
2
mi
c output gain of the ith mode
i
Z
' ith half-power frequency, 2
i i i
Z ] Z
'
2ij
V H2 placement index for the ith actuator (sensor)
and the kth mode
ij
Vf Hf placement index for the ith actuator (sensor)
and the kth mode
2
6 H2 placement matrix
f
6 Hf placement matrix
( )
I k membership index of the kth sensor
i
E pole shift factor
d
n number of degrees of freedom
n number of modes
N number of states
s number of inputs
r number of outputs
S number of candidate actuator locations
R number of candidate sensor locations](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-27-320.jpg)
![4 Chapter 1
tachometers. For more details about the antenna and its control systems, see [59] and
[42], or visit the web page http://ipnpr.jpl.nasa.gov/. The finite-element model of the
antenna structure consists of about 5000 degrees of freedom, with some nonlinear
properties (dry friction, backlash, and limits imposed on its rates, and accelerations).
However, the model of the structure and the drives used in this book are linear, and
are obtained from the field test data using system identification procedures.
Figure 1.5. The Deep Space Network antenna at Goldstone, California (courtesy of
NASA/JPL/Caltech, Pasadena, California). It can rotate with respect to azimuth (vertical)
axis, and the dish with respect to elevation (horizontal axis).
In the following we briefly describe the field test. We tested the antenna using a
white noise input signal of sampling frequency 30.6 Hz, as shown in Fig. 1.7(a). The
antenna elevation encoder output record is shown in Fig. 1.7(b). From these records
we determined the transfer function, from the antenna rate input to the encoder
output, see Fig. 1.8(a),(b), dashed line. Next, we used the Eigensystem Realization
Algorithm (ERA) identification algorithm (see [84], and Chapter 9 of this book) to
determine the antenna state-space representation. For this representation we obtained
the plot of the transfer function plot as shown in Fig. 1.8(a),(b), solid line. The plot
displays good coincidence between the measured and identified transfer function.](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-31-320.jpg)
![2
Standard Models
ª how to describe typical structures
Equation Chapter 2 Section 1
The best model of a cat is another, or
preferably the same, cat.
—Arturo Rosenblueth with Norbert Wiener
In this and the following chapter we explain structural models that describe standard
—or more common—structures. The standard models include structures that are
stable, linear, continuous-time, and with proportional damping.
We derive the structural analytical models either from physical laws, such as
Newton’s motion laws, Lagrange’s equations of motion, or D’Alembert’s principle
[108], [111]; or from finite-element models; or from test data using system
identification methods. The models are represented either in time domain
(differential equations), or in frequency domain (transfer functions).
We use linear differential equations to represent linear structural models in time
domain, either in the form of second-order differential equations or in the form of
first-order differential equations (as a state-space representation). In the first case,
we use the degrees of freedom of a structure to describe structural dynamics. In the
second case we use the system states to describe the dynamics. Structural engineers
prefer degrees of freedom and the second-order differential equations of structural
dynamics; this is not a surprise, since they have a series of useful mathematical and
physical properties. This representation has a long tradition and using it many
important results have been derived. The state-space model, on the other hand, is a
standard model used by control engineers. Most linear control system analyses and
design methods are given in the state-space form. The state-space standardization of
structural models allows for the extension of known control system properties into
structural dynamics. In this chapter we use both second-order and state-space
models, and show their interrelations.](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-40-320.jpg)
![Standard Models 17
0.01
i
d i
k , i = 1, 2, 3, 4. There is a single input force at mass 3, and three outputs:
displacement and velocity of mass 1 and velocity of mass 3.
For this system the mass matrix is 1 2 3
( , , ),
M diag m m m thus 3
M I . The
stiffness and damping matrices are
1 2 2
2 2 3 3
3 3 4
0
0
k k k
K k k k k
k k k
ª º
« »
« »
« »
¬ ¼
,
1 2 2
2 2 3 3
3 3 4
0
0
d d d
D d d d d
d d d
ª º
« »
« »
« »
¬ ¼
,
therefore,
6 3 0
3 6 3
0 3 3
K
ª º
« »
« »
« »
¬ ¼
, and
0.06 0.03 0.00
0.03 0.06 0.03
0.00 0.03 0.03
D
ª º
« »
« »
« »
¬ ¼
.
The input and output matrices are
,
0
0
1
o
B
ª º
« »
« »
« »
¬ ¼
1 0 0
0 0 0
0 0 0
oq
C
ª º
« »
« »
« »
¬ ¼
, and
0 0 0
1 0 0
0 0 1
ov
C
ª º
« »
« »
« »
¬ ¼
.
On details of the derivation of this type of equation, see [70], [120].
2.2.2 Modal Models
The second-order models are defined in modal coordinates. These coordinates are
often used in the dynamics analysis of complex structures modeled by the finite
elements to reduce the order of a system. It is also used in the system identification
procedures, where modal representation is a natural outcome of the test.
Modal models of structures are the models expressed in modal coordinates. Since
these coordinates are independent, it leads to a series of useful properties that
simplify the analysis (as will be shown later in this book). The modal coordinate
representation can be obtained by the transformation of the nodal models. This
transformation is derived using a modal matrix, which is determined as follows.
Consider free vibrations of a structure without damping, i.e., a structure without
external excitation (u { 0) and with the damping matrix D = 0. The equation of
motion (2.7) in this case turns into the following equation:
(2.8)
0.
Mq+ Kq =](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-44-320.jpg)
![Standard Models 19
1
2
i
i
i
in
I
I
I
I
½
° °
° °
® ¾
° °
° °
¯ ¿
#
. (2.13)
The modal matrix ) has an interesting property: it diagonalizes mass and stiffness
matrices M and K,
,
T
m
M M
) ) (2.14)
.
T
m
K K
) ) (2.15)
The obtained diagonal matrices are called modal mass matrix and modal
stiffness matrix ( The same transformation, applied to the damping matrix
( )
m
M
).
m
K
,
T
m
D D
) ) (2.16)
gives the modal damping matrix , which is not always obtained as a diagonal
matrix. However, in some cases, it is possible to obtain diagonal. In these cases
the damping matrix is called a matrix of proportional damping. The proportionality
of damping is commonly assumed for analytical convenience. This approach is
justified by the fact that the nature of damping is not known exactly, that its values
are rather roughly approximated, and that the off-diagonal terms in most cases—as
will be shown later—have negligible impact on the structural dynamics. The
damping proportionality is often achieved by assuming the damping matrix as a
linear combination of the stiffness and mass matrices; see [18], [70],
m
D
m
D
1 2
D K M
D D
, (2.17)
where 1
D and 2
D are nonnegative scalars.
Modal models of structures are the models expressed in modal coordinates. In
order to do so we use a modal matrix to introduce a new variable, , called modal
displacement. This is a variable that satisfies the following equation:
m
q
.
m
q q
) (2.18)
In order to obtain the equations of motion for this new variable, we introduce (2.18)
to (2.7) and additionally left-multiply (2.7) by ,
T
) obtaining
,
.
T T T T
m m m
oq m ov m
o
M q D q K q B
y C q C q
) ) ) ) ) ) )
) )
u](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-46-320.jpg)
![20 Chapter 2
Assuming a proportional damping, and using (2.14), (2.15), and (2.16) we obtain the
above equation in the following form:
,
.
T
m m m m m m o
oq m ov m
M q D q K q B u
y C q C q
)
) )
Next, we multiply (from the left) the latter equation by 1
m
M
, which gives
1 1 1
,
.
T
m m m m m m m m o
oq m ov m
q M D q M K q M B u
y C q C q
)
) )
The obtained equations look quite messy, but the introduction of appropriate
notations simplifies them,
(2.19)
2
2 ,
.
m m m
mq m mv m
q q q B
y C q C q
=: :
mu
In (2.19) : is a diagonal matrix of natural frequencies, as defined before. Note,
however, that this is obtained from the modal mass and stiffness matrices as follows:
2 1
.
m m
M K
: (2.20)
In (2.19) is the modal damping matrix. It is a diagonal matrix of modal damping,
=
1
2
0 0
0 0
0 0 n
]
]
]
ª º
« »
« »
=
« »
« »
« »
¬ ¼
, (2.21)
where i
] is the damping of the ith mode. We obtain this matrix using the following
relationship , thus,
1
2
m m
M D
=:
1 1
2 2
1 1
0.5 0.5
m m m m m
M D M K
= : D . (2.22)
Next, we introduce the modal input matrix m
B in (2.19),
1
.
T
m m o
B M B
) (2.23)
Finally, in (2.19) we use the following notations for the modal displacement and rate
matrices:](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-47-320.jpg)
![Standard Models 21
,
mq oq
C C ) (2.24)
.
mv ov
C C ) (2.25)
Note that (2.19) (a modal representation of a structure) is a set of uncoupled
equations. Indeed, due to the diagonality of : and = , this set of equations can be
written, equivalently, as
(2.26)
2
1
2
, 1, ,
,
mi i i mi i mi mi
i mqi mi mvi mi
n
i
i
q q q b u
y c q c q i n
y y
] Z Z
¦
! ,
where is the ith row of
mi
b m
B and are the ith columns of and
, respectively. The coefficient
,
mqi
c mvi
c mq
C
mv
C i
] is called a modal damping of the ith mode. In
the above equations is the system output due to the ith mode dynamics, and the
quadruple
i
y
( , , , )
i i mi mi
b c
Z ] represents the properties of the ith natural mode. Note
that the structural response y is a sum of modal responses yi, which is a key property
used to derive structural properties in modal coordinates.
This completes the modal model description. In the following we introduce the
transfer function obtained from the modal equations. The generic transfer function is
obtained from the state-space representation using (2.6). For structures in modal
coordinates it has a specific form.
Transfer Function of a Structure. The transfer function of a structure is derived
from (2.19),
(2.27)
2 2 1
( ) ( )( 2 ) .
mq mv n m
G C j C I j Z B
Z Z Z Z
: :
However, this can be presented in a more useful form, since the matrices : and =
are diagonal, allowing for representation of each single mode.
Transfer Function of a Mode. The transfer function of the ith mode is obtained
from (2.26),
2 2
( )
( ) .
2
mqi mvi mi
mi
i i
c j c b
G
j
Z
Z
i
Z Z ] Z Z
(2.28)
The structural and modal transfer functions are related as follows:](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-48-320.jpg)
![22 Chapter 2
Property 2.1. Transfer Function in Modal Coordinates. The structural
transfer function is a sum of modal transfer functions
(a)
1
( ) ( )
n
mi
i
G G
Z Z
¦ (2.29)
or, in other words,
2 2
1
( )
( ) ,
2
n
mqi mvi mi
i i
i
c j c b
G
j
Z
Z
i
Z Z ] Z Z
¦ (2.30)
and the structural transfer function at the ith resonant frequency is approximately
equal to the ith modal transfer function at this frequency
(b) 2
( )
( ) ( )
2
mqi i mvi mi
i mi i
i i
jc c b
G G
Z
Z Z
] Z
# , 1, , .
i n
! (2.31)
Proof. By inspection of (2.27) and (2.28). ‹
Structural Poles. The poles of a structure are the zeros of the characteristic
equations (2.26). The equation 2
2 i i i
s s
] Z Z2
0
is the characteristic equation of
the ith mode. For small damping the poles are complex conjugate, and in the
following form:
2
1
2
2
1 ,
1 .
i i i i
i i i i
s j
s j
] Z Z ]
] Z Z ]
(2.32)
The plot of the poles is shown in Fig. 2.1, which shows how the location of a pole
relates to the natural frequency and modal damping.
Example 2.2. Determine the modal model of a simple structure from Example 2.1.
The natural frequency matrix is
3.1210 0 0
0 2.1598 0
0 0 0.7708
ª º
« »
: « »
« »
¬ ¼
,
and the modal matrix is](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-49-320.jpg)
![Standard Models 23
(a)
0.5910 0.7370 0.3280
0.7370 0.3280 0.5910
0.3280 0.5910 0.7370
ª º
« »
)
« »
« »
¬ ¼
.
The modes are shown in Fig. 2.2.
i i
] Z
arcsin( )
i i
D ]
i
Z
Im
0 Re
s2
s1
2
1
i i
Z ]
2
1
i i
Z ]
i
Z
Figure 2.1. Pole location of the ith mode of a lightly damped structure: It is a complex pair
with the real part proportional to the ith modal damping; the imaginary part approximately
equal to the ith natural frequency; and the radius is the exact natural frequency.
The modal mass is 3
m
M I , the modal stiffness is 2
m
K : , and the modal
damping, from (2.22), is
0.0156 0 0
0 0.0108 0
0 0 0.0039
ª º
« »
= « »
« »
¬ ¼
.
We obtain the modal input and output matrices from (2.23), (2.24), and (2.25):
0.3280
0.5910 ,
0.7370
m
B
ª º
« »
« »
« »
¬ ¼](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-50-320.jpg)
![Standard Models 27
harmonics are shown on the impulse response plot, but are more explicit at the
impulse response spectrum plot, Fig. 2.7, as the spectrum peaks at these frequencies.
Impulse response is the time-domain associate of the transfer function (through
the Parseval theorem); therefore, Property 2.1 can be written in time domain as
1
( ) ( )
n
i
i
h t h t
¦
where h(t) is the impulse response of a structure and is the impulse response of
the ith mode. Thus, the structural impulse response is a sum of modal responses.
This is illustrated in Fig. 2.8, where impulse responses of modes 1, 2, and 3 are
plotted. Clearly the total response as in Fig. 2.7 is a sum of the individual responses.
Note that each response is a sinusoid of frequency equal to the natural frequency,
and of exponentially decayed amplitude, proportional to the modal damping
( )
i
h t
i
] .
Note also that the higher-frequency responses decay faster.
10
–1
10
0
10
1
10
–2
10
–1
10
0
10
1
10
2
2
Z
frequency, rad/s
10
–1
10
0
–400
–200
0
200
2
Z
1
Z
1
Z (b)
(a)
3
Z
3
Z
10
1
magnitude
phase,
deg
frequency, rad/s
Figure 2.5. Transfer function of a simple system: (a) Magnitude shows three resonance
peaks; and (b) phase shows three shifts of 180 degrees; 1 2
, ,
Z Z and 3
Z denote the natural
frequencies.](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-54-320.jpg)
![with God's help, to justify our guardianship if need be against all
who would silence them.'
This kind of language, so different from that usually used by Eastern
politicians in their conversations with Europeans, impressed me very
deeply, and I made a strong mental contrast between Arabi and that
other champion of liberty whom I had met and talked with at
Damascus, Midhat Pasha, altogether in Arabi's favour. Here was no
nonsense about railroads and canals and tramways as nostrums that
could redeem the East, but words that went to the root of things
and fixed the responsibility of good government on the shoulders
which alone could bear it. I felt that even in the incredulous and
trifling atmosphere of the House of Commons words like these would
be listened to—if only they could be heard there!
With regard to the Sultan and the connection of Egypt with Turkey,
Arabi was equally explicit. He had no love, he told me, for the Turks
who had mis-governed Egypt for centuries, and he would not hear of
interference from Constantinople in the internal affairs of the
country. But he made a distinction between the Ottoman
Government and the religious authority of the Sultan, whom, as Emir
el Mumenin, he was bound, as long as he ruled justly, to obey and
honour. Also the example of Tunis, which the French had first
detached from the Empire, and then taken possession of, showed
how necessary it was to preserve the connection of Egypt with the
Head of the Moslem world. We are all, he said, children of the
Sultan, and live together like a family in one house. But, just as in
families, we have, each of us provinces of the Empire, our separate
room which is our own to arrange as we will and where even the
Sovereign must not wantonly intrude. Egypt has gained this
independent position through the Firmans granted, and we will take
care that she preserves it. To ask for more than this would be to run
a foolish risk, and perhaps lose our liberty altogether.[7] I asked him
rather bluntly whether he had been, as was then currently asserted,
in personal communication with Constantinople, and I noticed that
he was reserved in answering and did so evasively. Doubtless the](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-56-320.jpg)
![missed his mark entirely. The timid Khedive was only frightened, and
the Nationalists, instead of being frightened, were enraged. The
Egyptians for the first time found themselves quite united. Sheykh
Mohammed Abdu and the cautious Azhar reformers from that point
threw in their lot wholly with the advanced party. All, even the
Circassians, resented the threat of foreign intervention, and on the
other hand the most anti-Turkish of the Nationalists, such as my
friend Hajrasi, saw that Arabi had been right in secretly leaning upon
the Sultan. Arabi thus gained immensely in popularity and respect,
and for many days after this I hardly heard anything from my
Egyptian friends but the language of Pan-Islamism. It was a
Roustan[8] policy over again, they said.
I did my best to smooth down matters with them till the explanation
should arrive which Malet had promised us; but I found my efforts
useless. It was an alarming three weeks for us all, from the delivery
of the Note till Gambetta's fall. News came that a French force was
being assembled for embarkation at Toulon, and that was the form
of intervention generally expected. Indeed, I think it is not too much
to say that Gambetta's resignation on 31st January alone saved
Egypt from the misfortune, even greater perhaps than what
afterwards befell her, of a French invasion avowedly anti-
Mohammedan and in purely European interests.](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-71-320.jpg)
![FOOTNOTES:
[7] Sir William Gregory, who saw Arabi about the same date as I
did, has recorded in the Times very similar language as used by
him.
[8] Roustan was the French diplomatist at Tunis who had
engineered the French designs on the Regency.](https://image.slidesharecdn.com/1021619-250521192058-c7b0d19d/85/Advanced-Structural-Dynamics-And-Active-Control-Of-Structures-Gawronski-Wk-72-320.jpg)
Advanced Structural Dynamics And Active Control Of Structures Gawronski Wk
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- 6. Mechanical Engineering Series Frederick F. Ling Series Editor
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- 8. Wodek K. Gawronski Advanced Structural Dynamics and Active Control of Structures With 157 Figures
- 9. Wodek K. Gawronski Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109, USA wodek.k.gawronski@jpl.nasa.gov Series Editor Frederick F. Ling Ernest F. Gloyna Regents Chair in Engineering, Emeritus Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712-1063, USA and William Howard Hart Professor Emeritus Department of Mechanical Engineering, Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy, NY 12180-3590, USA Library of Congress Cataloging-in-Publication Data Gawronski, Wodek, 1944– Advanced structural dynamics and active control of structures/Wodek Gawronski. p. cm. — (Mechanical engineering series) ISBN 0-387-40649-2 (alk. paper) 1. Structural dynamics. 2. Structural control (Engineering) I. Title. II. Mechanical engineering series (Berlin, Germany) TA654.G36 2004 624.1′71—dc22 2003058443 Based on Dynamics and Control of Structures: A Modal Approach, by Wodek K. Gawronski, 1998 Springer-Verlag New York, Inc. ISBN 0-387-40649-2 Printed on acid-free paper. 2004 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 10943243 www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH
- 10. To my friends— Jan Kruszewski and Hans Günther Natke —scholars of dedication and imagination Although this may seem a paradox, all exact science is dominated by the idea of approximation. —Bertrand Russell
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- 12. Mechanical Engineering Series Frederick F. Ling Series Editor The Mechanical Engineering Series features graduate texts and research mono- graphs to address the need for information in contemporary mechanical engineer- ing, including areas of concentration of applied mechanics, biomechanics, compu- tational mechanics, dynamical systems and control, energetics, mechanics of materials, processing, production systems, thermal science, and tribology. Advisory Board Applied Mechanics F.A. Leckie University of California, Santa Barbara Biomechanics V.C. Mow Columbia University Computational Mechanics H.T. Yang University of California, Santa Barbara Dynamical Systems and Control K.M. Marshek University of Texas, Austin Energetics J.R. Welty University of Oregon, Eugene Mechanics of Materials I. Finnie University of California, Berkeley Processing K.K. Wang Cornell University Production Systems G.-A. Klutke Texas AM University Thermal Science A.E. Bergles Rensselaer Polytechnic Institute Tribology W.O. Winer Georgia Institute of Technology
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- 14. Preface Science is for those who learn; poetry for those who know. —Joseph Roux This book is a continuation of my previous book, Dynamics and Control of Structures [44]. The expanded book includes three additional chapters and an additional appendix: Chapter 3, “Special Models”; Chapter 8, “Modal Actuators and Sensors”; and Chapter 9, “System Identification.” Other chapters have been significantly revised and supplemented with new topics, including discrete-time models of structures, limited-time and -frequency grammians and reduction, almost- balanced modal models, simultaneous placement of sensors and actuators, and structural damage detection. The appendices have also been updated and expanded. Appendix A consists of thirteen new Matlab programs. Appendix B is a new addition and includes eleven Matlab programs that solve examples from each chapter. In Appendix C model data are given. Several books on structural dynamics and control have been published. Meirovitch’s textbook [108] covers methods of structural dynamics (virtual work, d’Alambert’s principle, Hamilton’s principle, Lagrange’s and Hamilton’s equations, and modal analysis of structures) and control (pole placement methods, LQG design, and modal control). Ewins’s book [33] presents methods of modal testing of structures. Natke’s book [111] on structural identification also contains excellent material on structural dynamics. Fuller, Elliot, and Nelson [40] cover problems of structural active control and structural acoustic control. Inman’s book [79] introduces the basic concepts of vibration control, while Preumont in [120] presents modern approaches to structural control, including LQG controllers, sensors, and actuator placement, and piezoelectric materials with numerous applications in aerospace and civil engineering. The Junkins and Kim book [87] is a graduate-level textbook, while the Porter and Crossley book [119] is one of the first books on modal control. Skelton’s work [125] (although on control of general linear systems) introduces methods designed specifically for the control of flexible structures. For example, the component cost approach to model or controller reduction is a tool frequently used in this field. The monograph by Joshi [83] presents developments on
- 15. x Preface dissipative and LQG controllers supported by numerous applications. Genta’s book [65] includes rotor dynamics; the book by Kwon and Bang [92] is dedicated mainly to structural finite-element models, but a part of it is dedicated to structural dynamics and control. The work by Hatch [70] explains vibrations and dynamics problems in practical ways, is illustrated with numerous examples, and supplies Matlab programs to solve vibration problems. The Maia and Silva book [107] is a study on modal analysis and testing, while the Heylen, Lammens, and Sas book [71] is an up-to-date and attractive presentation of modal analysis. The De Silva book [26] is a comprehensive source on vibration analysis and testing. Clark, Saunders, and Gibbs [17] present recent developments in dynamics and control of structures; and Elliott [31] applies structural dynamics and control problems to acoustics. My book [47] deals with structural dynamics and control problems in balanced coordinates. The recent advances in structural dynamics and control can be found in [121]. This book describes comparatively new areas of structural dynamics and control that emerged from recent developments. Thus: x State-space models and modal methods are used in structural dynamics as well as in control analysis. Typically, structural dynamics problems are solved using second-order differential equations. x Control system methods (such as the state-space approach, controllability and observability, system norms, Markov parameters, and grammians) are applied to solve structural dynamics problems (such as sensor and actuator placement, identification, or damage detection). x Structural methods (such as modal models and modal independence) are used to solve control problems (e.g., the design of LQG and Hf controllers), providing new insight into well-known control laws. x The methods described are based on practical applications. They originated from developing, testing, and applying techniques of structural dynamics, identification, and control to antennas and radiotelescopes. More on the dynamics and control problems of the NASA Deep Space Network antennas can be found at http://tmo.jpl.nasa.gov/tmo/progress_report/. x This book uses approximate analysis, which is helpful in two ways. First, it simplifies analysis of large structural models (e.g., obtaining Hankel singular values for a structure with thousands of degrees of freedom). Second, approximate values (as opposed to exact ones) are given in closed form, giving an opportunity to conduct a parametric study of structural properties. This book requires introductory knowledge of structural dynamics and of linear control; thus it is addressed to the more advanced student. It can be used in graduate courses on vibration and structural dynamics, and in control system courses with application to structural control. It is also useful for engineers who deal with structural dynamics and control. Readers who would like to contact me with comments and questions are invited to do so. My e-mail address is Wodek.K.Gawronski@jpl.nasa.gov. Electronic versions
- 16. Preface xi of Matlab programs from Appendix A, examples from Appendix B, and data from Appendix C can also be obtained from this address. I would like to acknowledge the contributions of my colleagues who have had an influence on this work: Kyong Lim, NASA Langley Research Center (sensor/actuator placement, filter design, discrete-time grammians, and Hf controller analysis); Hagop Panossian, Boeing North American, Inc., Rocketdyne (sensor/actuator placement of the International Space Station structure); Jer-Nan Juang, NASA Langley Research Center (model identification of the Deep Space Network antenna); Lucas Horta, NASA Langley Research Center (frequency- dependent grammians for discrete-time systems); Jerzy Sawicki, Cleveland State University (modal error estimation of nonproportional damping); Abner Bernardo, Jet Propulsion Laboratory, California Institute of Technology (antenna data collection); and Angel Martin, the antenna control system supervisor at the NASA Madrid Deep Space Communication Complex (Spain) for his interest and encouragement. I thank Mark Gatti, Scott Morgan, Daniel Rascoe, and Christopher Yung, managers at the Communications Ground Systems Section, Jet Propulsion Laboratory, for their support of the Deep Space Network antenna study, some of which is included in this book. A portion of the research described in this book was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Wodek K. Gawronski Pasadena, California January 2004
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- 18. Contents Series Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1 Introduction to Structures . . . . . . . . . . . . . . . . . . . . 1 1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 A Simple Structure . . . . . . . . . . . . . . . . . . . 1 1.1.2 A 2D Truss . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 A 3D Truss . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.4 A Beam . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.5 The Deep Space Network Antenna . . . . . . . . . . . . 3 1.1.6 The International Space Station Structure . . . . . . . . . 6 1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Standard Models . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Models of a Linear System . . . . . . . . . . . . . . . . . 14 2.1.1 State-Space Representation . . . . . . . . . . . . . . . 14 2.1.2 Transfer Function . . . . . . . . . . . . . . . . . . . 15 2.2 Second-Order Structural Models . . . . . . . . . . . . . . . 16 2.2.1 Nodal Models . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Modal Models . . . . . . . . . . . . . . . . . . . . 17 2.3 State-Space Structural Models . . . . . . . . . . . . . . . . 29 2.3.1 Nodal Models . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Models in Modal Coordinates . . . . . . . . . . . . . . 31 2.3.3 Modal Models . . . . . . . . . . . . . . . . . . . . 35
- 19. xiv Contents 3 Special Models . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Models with Rigid-Body Modes . . . . . . . . . . . . . . . . 41 3.2 Models with Accelerometers . . . . . . . . . . . . . . . . . 45 3.2.1 State-Space Representation . . . . . . . . . . . . . . . . 45 3.2.2 Second-Order Representation . . . . . . . . . . . . . . . 48 3.2.3 Transfer Function . . . . . . . . . . . . . . . . . . . . 49 3.3 Models with Actuators . . . . . . . . . . . . . . . . . . . 50 3.3.1 Model with Proof-Mass Actuators . . . . . . . . . . . . . 50 3.3.2 Model with Inertial Actuators . . . . . . . . . . . . . . . 53 3.4 Models with Small Nonproportional Damping . . . . . . . . . . 54 3.5 Generalized Model . . . . . . . . . . . . . . . . . . . . . 58 3.5.1 State-Space Representation . . . . . . . . . . . . . . . . 59 3.5.2 Transfer Function . . . . . . . . . . . . . . . . . . . 59 3.6 Discrete-Time Models . . . . . . . . . . . . . . . . . . . . 60 3.6.1 State-Space Representation . . . . . . . . . . . . . . . . 61 3.6.2 Transfer Function . . . . . . . . . . . . . . . . . . . 63 4 Controllability and Observability . . . . . . . . . . . . . . . . . 65 4.1 Definition and Properties . . . . . . . . . . . . . . . . . . . 65 4.1.1 Continuous-Time Systems . . . . . . . . . . . . . . . . 66 4.1.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . 68 4.1.3 Relationship Between Continuous- and Discrete-Time Grammians . . . . . . . . . . . . . . . . . 69 4.2 Balanced Representation . . . . . . . . . . . . . . . . . . . 71 4.3 Balanced Structures with Rigid-Body Modes . . . . . . . . . . 73 4.4 Input and Output Gains . . . . . . . . . . . . . . . . . . . . 74 4.5 Controllability and Observability of a Structural Modal Model . . . 76 4.5.1 Diagonally Dominant Grammians . . . . . . . . . . . . . 76 4.5.2 Closed-Form Grammians . . . . . . . . . . . . . . . . . 79 4.5.3 Approximately Balanced Structure in Modal Coordinates . . . 80 4.6 Controllability and Observability of a Second-Order Modal Model . . 85 4.6.1 Grammians . . . . . . . . . . . . . . . . . . . . . . 85 4.6.2 Approximately Balanced Structure in Modal Coordinates . . . 87 4.7 Three Ways to Compute Hankel Singular Values . . . . . . . . 91 4.8 Controllability and Observability of the Discrete-Time Structural Model . . . . . . . . . . . . . . . . . . . . . . 91 4.9 Time-Limited Grammians . . . . . . . . . . . . . . . . . . 94 4.10 Frequency-Limited Grammians . . . . . . . . . . . . . . . . 99 4.11 Time- and Frequency-Limited Grammians . . . . . . . . . . . 103 4.12 Discrete-Time Grammians in Limited-Time and -Frequency Range . 107 5 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.1 Norms of the Continuous-Time Systems . . . . . . . . . . . 109 5.1.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 109 5.1.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . 111 5.1.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . 112
- 20. Contents xv 5.2 Norms of the Discrete-Time Systems . . . . . . . . . . . . . 113 5.2.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 113 5.2.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . 114 5.2.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . . 114 5.3 Norms of a Single Mode . . . . . . . . . . . . . . . . . . . 115 5.3.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 115 5.3.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . . 117 5.3.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . 118 5.3.4 Norm Comparison . . . . . . . . . . . . . . . . . . . 119 5.4 Norms of a Structure . . . . . . . . . . . . . . . . . . . . 120 5.4.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 121 5.4.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . 121 5.4.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . 123 5.5 Norms of a Structure with a Filter . . . . . . . . . . . . . . 124 5.5.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 124 5.5.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . . 126 5.5.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . 127 5.6 Norms of a Structure with Actuators and Sensors . . . . . . . . 127 5.6.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 128 5.6.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . . 130 5.6.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . 132 5.7 Norms of a Generalized Structure . . . . . . . . . . . . . . 135 5.8 Norms of the Discrete-Time Structures . . . . . . . . . . . . 137 5.8.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 138 5.8.2 The Hf Norm . . . . . . . . . . . . . . . . . . . . . 139 5.8.3 The Hankel Norm . . . . . . . . . . . . . . . . . . . 140 5.8.4 Norm Comparison . . . . . . . . . . . . . . . . . . . 140 6 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . 143 6.1 Reduction Through Truncation . . . . . . . . . . . . . . . 143 6.2 Reduction Errors . . . . . . . . . . . . . . . . . . . . . . 145 6.2.1 H2 Model Reduction . . . . . . . . . . . . . . . . . . 145 6.2.2 Hf and Hankel Model Reduction . . . . . . . . . . . . 146 6.3 Reduction in the Finite-Time and -Frequency Intervals . . . . . 147 6.3.1 Reduction in the Finite-Time Interval . . . . . . . . . . . 148 6.3.2 Reduction in the Finite-Frequency Interval . . . . . . . . 150 6.3.3 Reduction in the Finite-Time and -Frequency Intervals . . . 151 6.4 Structures with Rigid-Body Modes . . . . . . . . . . . . . 155 6.5 Structures with Actuators and Sensors . . . . . . . . . . . . 159 6.5.1 Actuators and Sensors in a Cascade Connection . . . . . . 159 6.5.2 Structure with Accelerometers . . . . . . . . . . . . . 161 6.5.3 Structure with Proof-Mass Actuators . . . . . . . . . . . 162 6.5.4 Structure with Inertial Actuators . . . . . . . . . . . . 165
- 21. xvi Contents 7 Actuator and Sensor Placement . . . . . . . . . . . . . . . . . 167 7.1 Problem Statement . . . . . . . . . . . . . . . . . . . . 168 7.2 Additive Property of Modal Norms . . . . . . . . . . . . . . 168 7.2.1 The H2 Norm . . . . . . . . . . . . . . . . . . . . . 169 7.2.2 The Hf and Hankel Norms . . . . . . . . . . . . . . . 169 7.3 Placement Indices and Matrices . . . . . . . . . . . . . . . 170 7.3.1 H2 Placement Indices and Matrices . . . . . . . . . . . 170 7.3.2 Hf and Hankel Placement Indices and Matrices . . . . . . 172 7.3.3 Actuator/Sensor Indices and Modal Indices . . . . . . . . 173 7.4 Placement for Large Structures . . . . . . . . . . . . . . . 180 7.4.1 Actuator Placement Strategy . . . . . . . . . . . . . . 182 7.4.2 Sensor Placement Strategy . . . . . . . . . . . . . . . 182 7.5 Placement for a Generalized Structure . . . . . . . . . . . . 187 7.5.1 Structural Testing and Control . . . . . . . . . . . . . 187 7.5.2 Sensor and Actuator Properties . . . . . . . . . . . . . 189 7.5.3 Placement Indices and Matrices . . . . . . . . . . . . . 192 7.5.4 Placement of a Large Number of Sensors . . . . . . . . . 193 7.6 Simultaneous Placement of Actuators and Sensors . . . . . . . 197 8 Modal Actuators and Sensors . . . . . . . . . . . . . . . . . . 203 8.1 Modal Actuators and Sensors Through Modal Transformations . . 204 8.1.1 Modal Actuators . . . . . . . . . . . . . . . . . . . 204 8.1.2 Modal Sensors . . . . . . . . . . . . . . . . . . . . 208 8.2 Modal Actuators and Sensors Through Grammian Adjustment . . 213 9 System Identification . . . . . . . . . . . . . . . . . . . . . . 219 9.1 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . 220 9.2 Markov Parameters . . . . . . . . . . . . . . . . . . . . 221 9.3 Identification Algorithm . . . . . . . . . . . . . . . . . . 221 9.4 Determining Markov Parameters . . . . . . . . . . . . . . 224 9.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 226 9.5.1 A Simple Structure . . . . . . . . . . . . . . . . . . 226 9.5.2 The 2D Truss . . . . . . . . . . . . . . . . . . . . 230 9.5.3 The Deep Space Network Antenna . . . . . . . . . . . . 232 10 Collocated Controllers . . . . . . . . . . . . . . . . . . . . . 235 10.1 A Low-Authority Controller . . . . . . . . . . . . . . . . 236 10.2 Dissipative Controller . . . . . . . . . . . . . . . . . . . 237 10.3 Properties of Collocated Controllers . . . . . . . . . . . . . 239 10.4 Root-Locus of Collocated Controllers . . . . . . . . . . . . 241 10.5 Collocated Controller Design Examples . . . . . . . . . . . 245 10.5.1 A Simple Structure . . . . . . . . . . . . . . . . . . 245 10.5.2 The 2D Truss . . . . . . . . . . . . . . . . . . . . 246 11 LQG Controllers . . . . . . . . . . . . . . . . . . . . . . . 249 11.1 Definition and Gains . . . . . . . . . . . . . . . . . . . . 250 11.2 The Closed-Loop System . . . . . . . . . . . . . . . . . . 253
- 22. Contents xvii 11.3 The Balanced LQG Controller . . . . . . . . . . . . . . . 254 11.4 The Low-Authority LQG Controller . . . . . . . . . . . . . 255 11.5 Approximate Solutions of CARE and FARE . . . . . . . . . 257 11.6 Root-Locus . . . . . . . . . . . . . . . . . . . . . . . 260 11.7 Almost LQG-Balanced Modal Representation . . . . . . . . . 262 11.8 Three Ways to Compute LQG Singular Values . . . . . . . . 264 11.9 The Tracking LQG Controller . . . . . . . . . . . . . . . 264 11.10 Frequency Weighting . . . . . . . . . . . . . . . . . . . 266 11.11 The Reduced-Order LQG Controller . . . . . . . . . . . . . 269 11.11.1 The Reduction Index . . . . . . . . . . . . . . . . . 269 11.11.2 The Reduction Technique . . . . . . . . . . . . . . 271 11.11.3 Stability of the Reduced-Order Controller . . . . . . . . 272 11.11.4 Performance of the Reduced-Order Controller . . . . . . 274 11.11.5 Weights of Special Interest . . . . . . . . . . . . . . 275 11.12 Controller Design Procedure . . . . . . . . . . . . . . . . 276 11.13 Controller Design Examples . . . . . . . . . . . . . . . . 277 11.13.1 A Simple Structure . . . . . . . . . . . . . . . . . . 277 11.13.2 The 3D Truss . . . . . . . . . . . . . . . . . . . . 279 11.13.3 The 3D Truss with Input Filter . . . . . . . . . . . . . 281 11.13.4 The Deep Space Network Antenna . . . . . . . . . . 283 12 Hf and H2 Controllers . . . . . . . . . . . . . . . . . . . . . 287 12.1 Definition and Gains . . . . . . . . . . . . . . . . . . . 288 12.2 The Closed-Loop System . . . . . . . . . . . . . . . . . 291 12.3 The Balanced Hf Controller . . . . . . . . . . . . . . . . 292 12.4 The H2 Controller . . . . . . . . . . . . . . . . . . . . 294 12.4.1 Gains . . . . . . . . . . . . . . . . . . . . . . . . 294 12.4.2 The Balanced H2 Controller . . . . . . . . . . . . . 296 12.5 The Low-Authority Hf Controller . . . . . . . . . . . . . 296 12.6 Approximate Solutions of HCARE and HFARE . . . . . . . 298 12.7 Almost Hf-Balanced Modal Representation . . . . . . . . . 300 12.8 Three Ways to Compute Hf Singular Values . . . . . . . . . 301 12.9 The Tracking Hf Controller . . . . . . . . . . . . . . . . 301 12.10 Frequency Weighting . . . . . . . . . . . . . . . . . . . 301 12.11 The Reduced-Order Hf Controller . . . . . . . . . . . . . 304 12.11.1 The Reduction Index . . . . . . . . . . . . . . . . . 304 12.11.2 Closed-Loop Poles . . . . . . . . . . . . . . . . . 304 12.11.3 Controller Performance . . . . . . . . . . . . . . . 306 12.12 Controller Design Procedure . . . . . . . . . . . . . . . . 307 12.13 Controller Design Examples . . . . . . . . . . . . . . . . 308 12.13.1 A Simple Structure . . . . . . . . . . . . . . . . . . 308 12.13.2 The 2D Truss . . . . . . . . . . . . . . . . . . . . 310 12.13.3 Filter Implementation Example . . . . . . . . . . . . 312 12.13.4 The Deep Space Network Antenna with Wind Disturbance Rejection Properties . . . . . . . . . 313
- 23. xviii Contents Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 A Matlab Functions . . . . . . . . . . . . . . . . . . . . . . . 319 A.1 Transformation from an Arbitrary State-Space Representation to the Modal 1 State-Space Representation . . . . . . . . . . . 320 A.2 Transformation from an Arbitrary State-Space Representation to the Modal 2 State-Space Representation . . . . . . . . . . . 322 A.3 Transformation from Modal Parameters to the Modal 1 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . 324 A.4 Transformation from Modal Parameters to the Modal 2 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . . 325 A.5 Transformation from Nodal Parameters to the Modal 1 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . 326 A.6 Transformation from Nodal Parameters to the Modal 2 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . 328 A.7 Determination of the Modal 1 State-Space Representation and the Time- and Frequency-Limited Grammians . . . . . . . . . . 329 A.8 Open-Loop Balanced Representation . . . . . . . . . . . . . 331 A.9 H2 Norm of a Mode . . . . . . . . . . . . . . . . . . . . 332 A.10 Hf Norm of a Mode . . . . . . . . . . . . . . . . . . . . 333 A.11 Hankel Norm of a Mode . . . . . . . . . . . . . . . . . . 333 A.12 LQG-Balanced Representation . . . . . . . . . . . . . . . 334 A.13 Hf-Balanced Representation . . . . . . . . . . . . . . . . 335 B Matlab Examples . . . . . . . . . . . . . . . . . . . . . . . 337 B.1 Example 2.5 . . . . . . . . . . . . . . . . . . . . . . . 337 B.2 Example 3.3 . . . . . . . . . . . . . . . . . . . . . . . 341 B.3 Example 4.11 . . . . . . . . . . . . . . . . . . . . . . . 342 B.4 Example 5.3 . . . . . . . . . . . . . . . . . . . . . . . 344 B.5 Example 6.7 . . . . . . . . . . . . . . . . . . . . . . . 347 B.6 Example 7.2 . . . . . . . . . . . . . . . . . . . . . . . 348 B.7 Example 8.1 . . . . . . . . . . . . . . . . . . . . . . . 353 B.8 Example 9.1 . . . . . . . . . . . . . . . . . . . . . . . 356 B.9 Example 10.4.2 . . . . . . . . . . . . . . . . . . . . . . 359 B.10 Example 11.13.1 . . . . . . . . . . . . . . . . . . . . . 361 B.11 Example 12.13.2 . . . . . . . . . . . . . . . . . . . . . 365 C Structural Parameters . . . . . . . . . . . . . . . . . . . . . 371 C.1 Mass and Stiffness Matrices of the 2D Truss . . . . . . . . . 371 C.2 Mass and Stiffness Matrices of the Clamped Beam Divided into 15 Finite Elements . . . . . . . . . . . . . . . . . . . . 373 C.3 State-Space Representation of the Deep Space Network Antenna . 376 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
- 24. List of Symbols Each equation in the book ... would halve the sales. —Stephen Hawking General T A transpose of matrix A * A complex-conjugate transpose of matrix A 1 A inverse of square nonsingular matrix A tr(A) trace of a matrix A, tr( ) ii i A a ¦ 2 A Euclidean (Frobenius) norm of a real-valued matrix A: 2 2 , tr( ) T ij i j A a A ¦ A diag( ) i a diagonal matrix with elements along the diagonal i a eig(A) eigenvalue of a square matrix A ( ) i A O ith eigenvalue of a square matrix A max ( ) A O maximal eigenvalue of a square matrix A ( ) i A V ith singular value of a matrix A max ( ) A V maximal singular value of a matrix A n I identity matrix, n u n 0n m u zero matrix, n u m
- 25. xx List of Symbols Linear Systems (A,B,C,D) quadruple of the system state-space representation (A,B,C) triple of the system state-space representation ( , , ) d d d A B C discrete-time state-space representation ( , , lqg lqg lqg A B C ) ) LQG controller state-space representation ( , , A B C f f f Hf controller state-space representation ( , , ) o o o A B C closed-loop state-space representation G transfer function d G discrete-time transfer function 1 H Hankel matrix 2 H shifted Hankel matrix k h kth Markov parameter U input measurement matrix Y output measurement matrix x system state e x system estimated state u system (control) input y system (measured) output z performance output w disturbance input 1 B matrix of disturbance inputs 2 B matrix of control inputs 1 C matrix of performance outputs 2 C matrix of measured outputs 2 G continuous-time system H2 norm G f continuous-time system Hf norm h G continuous-time system Hankel norm 2 d G discrete-time system H2 norm d G f discrete-time system Hf norm d h G discrete-time system Hankel norm controllability matrix ' observability matrix c W controllability grammian o W observability grammian i J ith Hankel singular value max J the largest Hankel singular value of a system * matrix of Hankel singular values CARE controller algebraic Riccati equation FARE filter (or estimator) algebraic Riccati equation HCARE Hf controller algebraic Riccati equation
- 26. List of Symbols xxi HFARE Hf filter (or estimator) algebraic Riccati equation c S solution of CARE e S solution of FARE c Sf solution of HCARE e Sf solution of HFARE i P ith LQG singular value i Pf ith Hf singular value 0 matrix of the LQG singular values, diag( ) i P 0 f 0 matrix of the Hf singular values, diag( ) i P f f 0 U parameter of the Hf controller c K controller gain e K estimator gain H tracking error t time sequence t ' sampling time N number of states s number of inputs r number of outputs Structures D damping matrix K stiffness matrix M mass matrix m D modal damping matrix m K modal stiffness matrix m M modal mass matrix q structural displacement (nodal) m q structural displacement (modal) ab q structural displacement (almost-balanced) i q displacement of the ith degree of freedom mi q displacement of the ith mode abi q displacement of the ith almost-balanced mode i I ith structural mode abi I almost-balanced ith structural mode ) modal matrix ab ) almost-balanced modal matrix i Z ith natural frequency : matrix of natural frequencies
- 27. xxii List of Symbols i ] ith modal damping = matrix of modal damping coefficients o B nodal input matrix oq C nodal displacement output matrix ov C nodal velocity output matrix m B modal input matrix mq C modal displacement output matrix mv C modal velocity output matrix m C modal output matrix, 1 m mq m C C C : v mi b input matrix of the ith mode, ith row of m B mi c output matrix of the ith mode, ith column of m C 2 m B modal input gain 2 m C modal output gain, 2 2 2 1 2 2 2 m mq mv C C C : 2 mi b input gain of the ith mode 2 mi c output gain of the ith mode i Z ' ith half-power frequency, 2 i i i Z ] Z ' 2ij V H2 placement index for the ith actuator (sensor) and the kth mode ij Vf Hf placement index for the ith actuator (sensor) and the kth mode 2 6 H2 placement matrix f 6 Hf placement matrix ( ) I k membership index of the kth sensor i E pole shift factor d n number of degrees of freedom n number of modes N number of states s number of inputs r number of outputs S number of candidate actuator locations R number of candidate sensor locations
- 28. 1 Introduction to Structures ª examples, definition, and properties A vibration is a motion that can't make up its mind which way it wants to go. —From Science Exam Flexible structures in motion have specific features that are not a secret to a structural engineer. One of them is resonance—strong amplification of the motion at a specific frequency, called natural frequency. There are several frequencies that structures resonate at. A structure movement at these frequencies is harmonic, or sinusoidal, and remains at the same pattern of deformation. This pattern is called a mode shape, or mode. The modes are not coupled, and being independent they can be excited separately. More interesting, the total structural response is a sum of responses of individual modes. Another feature—structural poles—are complex conjugate. Their real parts (representing modal damping) are typically small, and their distance from the origin is the natural frequency of a structure. 1.1 Examples In this book we investigate several examples of flexible structures. This includes a simple structure, composed of three lumped masses, a two-dimensional (2D) truss and a three-dimensional (3D) truss, a beam, the Deep Space Network antenna, and the International Space Station structure. They represent different levels of complexity. 1.1.1 A Simple Structure A three-mass system—a simple structure—is used mainly for illustration purposes, and to make examples easy to follow. Its simplicity allows for easy analysis, and for
- 29. 2 Chapter 1 straightforward interpretation. Also, solution properties and numerical data can be displayed in a compact form. The system is shown in Fig. 1.1. In this figure m1, m2, and m3 represent system masses, k1, k2, k3, and k4, are stiffness coefficients, while d1, d2, d3, and d4, are damping coefficients. This structure has six states, or three degrees of freedom. 1 m 2 m 3 m 1 d 4 k 3 k 2 k 2 d 3 d 4 d 1 1 , f q 2 2 , f q 3 3 , f q 1 k Figure 1.1. A simple structure. 1.1.2 A 2D Truss The truss structure in Fig. 1.2 is a more complex example of a structure, which can still easily be simulated by the reader, if necessary. For this structure, l1=15 cm, l2=20 cm are dimensions of truss components. Each truss has a cross-sectional area of 1 cm2 , elastic modulus of 2.0u107 N/cm2 , and mass density of 0.00786 kg/cm3 . This structure has 32 states (or 16 degrees of freedom). Its stiffness and mass matrices are given in Appendix C.1. Figure 1.2. A 2D truss structure. 2 l 10 9 1 6 7 2 8 4 3 1 l 5 1.1.3 A 3D Truss A 3D truss is shown in Fig. 1.3. For this truss, the length is 60 cm, the width 8 cm, the height 10 cm, the elastic modulus is 2.1u107 N/cm2 , and the mass density is 0.00392 kg/cm3 . The truss has 72 degrees of freedom (or 144 states).
- 30. Introduction to Structures 3 14 1 26 27 28 25 24 23 22 21 20 19 18 17 16 15 13 8 11 12 10 9 4 5 6 7 3 2 Figure 1.3. A 3D truss structure. 1.1.4 A Beam A clamped beam is shown in Fig. 1.4. It is divided into n elements, with n–1 nodes, and two fixed nodes. In some cases later in this book we use n=15 elements for simple illustration, and sometimes n=60 or n=100 elements for more sophisticated examples of beam dynamics. Each node has three degrees of freedom: horizontal displacement, x, vertical displacement, y, and in plane rotation, T. In total it has 3(n–1) degrees of freedom. The beam is 150 cm long, with a cross-section of 1 cm2 . The external (filled) nodes are clamped. The beam mass and stiffness matrices for n=15 are given in Appendix C.2. 1 2 0 3 n2 n1 n Figure 1.4. A beam divided into n finite elements. 1.1.5 The Deep Space Network Antenna The NASA Deep Space Network antenna structure illustrates a real-world flexible structure. The Deep Space Network antennas, operated by the Jet Propulsion Laboratory, consist of several antenna types and are located at Goldstone (California), Madrid (Spain), and Canberra (Australia). The Deep Space Network serves as a communication tool for space exploration. A new generation of Deep Space Network antenna with a 34-m dish is shown in Fig. 1.5. This antenna is an articulated large flexible structure, which can rotate around azimuth (vertical) and elevation (horizontal) axes. The rotation is controlled by azimuth and elevation servos, as shown in Fig. 1.6. The combination of the antenna structure and its azimuth and elevation drives is the open-loop model of the antenna. The open-loop plant has two inputs (azimuth and elevation rates) and two outputs (azimuth and elevation position), and the position loop is closed between the encoder outputs and the rate inputs. The drives consist of gearboxes, electric motors, amplifiers, and
- 31. 4 Chapter 1 tachometers. For more details about the antenna and its control systems, see [59] and [42], or visit the web page http://ipnpr.jpl.nasa.gov/. The finite-element model of the antenna structure consists of about 5000 degrees of freedom, with some nonlinear properties (dry friction, backlash, and limits imposed on its rates, and accelerations). However, the model of the structure and the drives used in this book are linear, and are obtained from the field test data using system identification procedures. Figure 1.5. The Deep Space Network antenna at Goldstone, California (courtesy of NASA/JPL/Caltech, Pasadena, California). It can rotate with respect to azimuth (vertical) axis, and the dish with respect to elevation (horizontal axis). In the following we briefly describe the field test. We tested the antenna using a white noise input signal of sampling frequency 30.6 Hz, as shown in Fig. 1.7(a). The antenna elevation encoder output record is shown in Fig. 1.7(b). From these records we determined the transfer function, from the antenna rate input to the encoder output, see Fig. 1.8(a),(b), dashed line. Next, we used the Eigensystem Realization Algorithm (ERA) identification algorithm (see [84], and Chapter 9 of this book) to determine the antenna state-space representation. For this representation we obtained the plot of the transfer function plot as shown in Fig. 1.8(a),(b), solid line. The plot displays good coincidence between the measured and identified transfer function.
- 32. Introduction to Structures 5 The flexible properties are clearly visible in the identified model. The identified state-space representation of the antenna model is given in Appendix C.3. EL torque ELEVATION DRIVE AZIMUTH DRIVE ANTENNA STRUCTURE XEL error EL error AZ encoder EL encoder AZ torque EL rate AZ rate wind AZ pinion rate EL pinion rate Figure 1.6. The open-loop model of the Deep Space Network antenna (AZ = azimuth, EL = elevation, XEL = cross-elevation): The AZ and EL positions are measured with encoders, EL and XEL errors are RF beam pointing errors. 0.4 (a) 0.2 rate input, V 0 –0.2 –0.4 0 100 200 300 time, s 400 500 600 0.04 (b) 0.03 encoder, deg 0.02 0.01 0 0 100 200 time, s 300 400 500 600 Figure 1.7. Signals in the identification of the antenna model: (a) Input white noise (voltage); and (b) output–antenna position measured by the encoder.
- 33. 6 Chapter 1 10 –1 10 frequency, Hz 0 10 –3 10 –2 10 –1 (a) 0 10 magnitude frequency, Hz 10 –1 10 0 (b) –100 phase, deg –200 –300 Figure 1.8. The antenna transfer functions obtained from the data (dashed line), and obtained from the identified model (solid line): (a) Magnitude; and (b) phase. 1.1.6 The International Space Station Structure The Z1 module of the International Space Station structure is a large structure of a cubical shape with a basic truss frame, and with numerous appendages and attachments such as control moment gyros and a cable tray. Its finite-element model is shown in Fig. 1.9. The total mass of the structure is around 14,000 kg. The finite- element model of the structure consists of 11,804 degrees of freedom with 56 modes, of natural frequencies below 70 Hz. This structure was analyzed for the preparation of the modal tests. The determination of the optimal locations of shakers and accelerometers is presented in Chapter 7. 1.2 Definition The term flexible structure or, briefly, structure has different interpretations and definitions, depending on source and on application. For the purposes of this book we define a structure as a linear system, which is x finite-dimensional; x controllable and observable; x its poles are complex with small real parts; and x its poles are nonclustered.
- 34. Introduction to Structures 7 Beam Control moment gyros Cable tray Antenna boom Figure 1.9. The finite-element model of the International Space Station structure. Based on this definition, we derive many interesting properties of structures and their controllers later in this book. The above conditions are somehow restrictive, and introduced to justify the mathematical approach used in this book. However, our experience shows that even if these conditions are violated or extended the derived properties still hold. For example, for structures with heavy damping (with larger real parts of complex poles), or with some of the poles close to each other, the analysis results in many cases still apply. 1.3 Properties In this section we briefly describe the properties of flexible structures. The properties of a typical structure are illustrated in Fig. 1.10. x Motion of a flexible structure can be described in independent coordinates, called modes. One can excite a single mode without excitation of the remaining ones. Displacement of each point of structure is sinusoidal of fixed frequency. The shape of modal deformation is called a modal shape, or mode. The frequency of modal motion is called natural frequency. x Poles of a flexible structure are complex conjugate, with small real parts; their locations are shown in Fig. 1.10(a).
- 35. 8 Chapter 1 Figure 1.10. Properties of a typical flexible structure: (a) Poles are complex with small real parts; (b) magnitude of a transfer function shows resonant peaks; (c) impulse response is composed of harmonic components; and (d) phase of a transfer function displays 180 deg shifts at resonant frequencies. Figure 1.11. Structure response depends strongly on damping: (a) Poles of a structure with small (x) and larger (u) damping – damping impacts the real parts; (b) impulse response for small (solid line) and larger (dashed line) damping – damping impacts the transient time; (c) magnitude of the transfer function for small (solid line) and larger (dashed line) damping – damping impacts the resonance peaks; and (d) response to the white noise input for small (solid line) and larger damping (dashed line) – damping impacts the rms of the response. –0.02 –15 (a) –0.01 real 0 –10 –5 0 5 10 15 10 –1 10 0 10 frequency, rad/s 1 (b) 10 2 10 2 magnitude imaginary 10 0 10–2 –1 (c) 10 –100 –1 10 0 10 frequency, rad/s 1 (d) 100 1 impulse response 50 0.5 phase, deg 0 0 –0.5 –50 0 5 10 time, s 15 20 10 2 –0.4 –15 (a) 15 –1 (b) 1 10 impulse response 0.5 5 imaginary 0 0 –5 –0.5 –10 –0.3 –0.2 real –0.1 0 0 5 10 time, s 15 20 10 2 10 0 –0.5 0 0.5 1 (c) (d) response –1 magnitude 10 0 10–2 0 10 20 30 time, s frequency, rad/s
- 36. Introduction to Structures 9 x The magnitude of a flexible structure transfer function is characterized by the presence of resonance peaks; see Fig. 1.10(b). x The impulse response of a flexible structure consists of harmonic components, related to complex poles, or to resonance peaks; this is shown in Fig. 1.10(c). x The phase of a transfer function of a flexible structure shows 180 degree shifts at natural frequencies, see Fig. 1.10(d). Poles of a flexible structure are complex conjugate. Each complex conjugate pair represents a structural mode. The real part of a pole represents damping of the mode. The absolute value of the pole represents the natural frequency of the mode. Consider two different structures, as in Fig. 1.11(a). The first one has poles denoted with black circles (x), the second one with crosses (u). The locations of the poles indicate that they have the same natural frequencies, but different damping. The structure with poles marked with black circles has larger damping than the one with poles marked with the crosses. The figure illustrates that structural response depends greatly on the structural damping. For small damping the impulse response of a structure decays slower than the response for larger damping, see Fig. 1.11(b). Also, the magnitude of the response is visible in the plots of the magnitude of the transfer function in Fig. 1.11(c). For small damping the resonance peak is larger than that for larger damping. Finally, the damping impacts the root-mean-square (rms) of the response to white noise. For example, Fig. 1.11(d) shows that for small damping the rms response of a structure is larger than the response for larger damping. When a structure is excited by a harmonic force, its response shows maximal amplitude at natural frequencies. This is a resonance phenomenon – a strong amplification of the motion at natural frequency. There are several frequencies that structures resonate at. A structure movement at these frequencies is harmonic, or sinusoidal, and remains at the same pattern of deformation. This pattern is called a mode shape, or mode. The resonance phenomenon leads to an additional property – the independence of each mode. Each mode is excited almost independently, and the total structural response is the sum of modal responses. For example, let a structure be excited by a white noise. Its response is shown in Fig. 1.12(a). Also, let each mode be excited by the same noise. Their responses are shown in Fig. 1.12(b),(c),(d). The spectrum of the structural response is shown in Fig. 1.13(a), and the spectra of responses of each individual mode are shown in Fig. 1.13(b),(c),(d). Comparing Fig1.13a with Fig.1.13b,c,d we see that the resonance peak for each natural frequency is the same, either it was total structure excited, or individual mode excited. This shows that the impact of each mode on each other is negligible. The independence of the modes also manifests itself in a possibility of exciting each individual mode. One can find a special input configuration that excites a selected mode. For example, for the simple structure presented above we found an excitation that the impulse response has only one harmonic, see Fig. 1.14(a), and the magnitude of the transfer function of the structure shows a single resonance peak, see Fig. 1.14(b). However, there is no such input configuration that is able to excite a single node (or selected point) of a structure. Thus structural modes are independent, while structural nodes are not.
- 37. 10 Chapter 1 Figure 1.12. Response to the white noise input: (a) Total structure response is composed of three modal responses; (b) mode 1 response of the first natural frequency; (c) mode 2 response of the second natural frequency; and (d) mode 3 response of the third natural frequency. Figure 1.13. Spectra of the response to the white noise input: (a) Total structure spectrum consists of three modal spectra; (b) mode 1 spectrum of the first natural frequency; (c) mode 2 spectrum of the second natural frequency; and (d) mode 3 spectrum of the third natural frequency. 0 5 10 time, s 15 –0.5 (a) 20 –1 1 0 5 10 time, s 15 –0.5 (b) –1 20 1 mode 1 response structure response 0.5 0.5 0 0 (c) 0 5 10 time, s 15 –0.5 20 –1 1 0 0.5 mode 2 response (d) 0 5 10 time, s 15 –0.5 –1 20 1 mode 3 response 0.5 0 0 0 (a) 15 0 0 (b) 15 structure spectrum mode 1 spectrum 10 10 5 5 5 10 frequency, rad/s 15 5 10 frequency, rad/s 15 (c) 0 0 15 (d) 0 0 15 mode 2 spectrum mode 3 spectrum 10 10 5 5 5 10 frequency, rad/s 15 5 10 frequency, rad/s 15
- 38. Introduction to Structures 11 0.4 (a) 0.2 impulse response 0 –0.2 –0.4 0 2 4 6 8 10 12 14 16 18 20 time, s (b) 1 magnitude 0.01 1 10 frequency, rad/s Figure 1.14. An input configuration that excites a single mode: (a) Impulse response; and (b) magnitude of the transfer function.
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- 40. 2 Standard Models ª how to describe typical structures Equation Chapter 2 Section 1 The best model of a cat is another, or preferably the same, cat. —Arturo Rosenblueth with Norbert Wiener In this and the following chapter we explain structural models that describe standard —or more common—structures. The standard models include structures that are stable, linear, continuous-time, and with proportional damping. We derive the structural analytical models either from physical laws, such as Newton’s motion laws, Lagrange’s equations of motion, or D’Alembert’s principle [108], [111]; or from finite-element models; or from test data using system identification methods. The models are represented either in time domain (differential equations), or in frequency domain (transfer functions). We use linear differential equations to represent linear structural models in time domain, either in the form of second-order differential equations or in the form of first-order differential equations (as a state-space representation). In the first case, we use the degrees of freedom of a structure to describe structural dynamics. In the second case we use the system states to describe the dynamics. Structural engineers prefer degrees of freedom and the second-order differential equations of structural dynamics; this is not a surprise, since they have a series of useful mathematical and physical properties. This representation has a long tradition and using it many important results have been derived. The state-space model, on the other hand, is a standard model used by control engineers. Most linear control system analyses and design methods are given in the state-space form. The state-space standardization of structural models allows for the extension of known control system properties into structural dynamics. In this chapter we use both second-order and state-space models, and show their interrelations.
- 41. 14 Chapter 2 Besides the choice of form of equations, we represent the analytical model in different coordinates. The choice of coordinates in which the system model is represented is rather arbitrary. However, two coordinate systems, nodal and modal, are commonly used. Nodal coordinates are defined through displacements and velocities of selected structural locations, called nodes; and modal coordinates are defined through the displacements and velocities of structural (or natural) modes. In this book we use both coordinate systems; however, we put more weight on the modal coordinate system. At the beginning of this chapter we present a generic state-space system model and its transfer function; next, structural state-space models follow the second-order models of flexible structures. 2.1 Models of a Linear System Models of a linear system are described by linear differential equations. The equations can be organized in a standard form called state-space representation. This form is a set of first-order differential equations with unit coefficient at the first derivative. The models can also be represented in the form of a transfer function, after applying the Laplace or Fourier transformation. The state-space representation carries information about the internal structure (represented by states) of the model, while the transfer function describes the model in terms of its input–output properties (although its internal state can be somehow recovered). Also, the state- space models are more convenient and create less numerical difficulties than transfer functions when one deals with high-order models. 2.1.1 State-Space Representation A linear time-invariant system of finite dimensions is described by the following linear constant coefficient differential equations: , , x Ax Bu y Cx (2.1) with the initial state (0) o x x . In the above equations the N-dimensional vector x is called the state vector, o x is the initial condition of the state, the s-dimensional vector u is the system input, and the r-dimensional vector y is the system output. The A, B, and C matrices are real constant matrices of appropriate dimensions (A is NuN, B is Nus, and C is ruN). We call the triple the system state-space representation. ( , , ) A B C Every linear system, or system of linear-time invariant differential equations can be presented in the above form (with some exceptions discussed in Chapter 3). It is important to have a unique form as a standard form in order to develop
- 42. Standard Models 15 interchangeable software and compatible methods of analysis. However, for the same system presented by the state equations (2.1) the matrices A, B, C and the state vector are not unique: different representations can give an identical input– output relationship. Indeed, one can introduce a new state variable, ( , , ) A B C , n x such that , n x Rx (2.2) where R is a nonsingular transformation matrix. Introducing x from (2.2) to (2.1) we obtain the new state equations , , n n n n n n x A x B u y C x (2.3) where . (2.4) 1 1 , , n n n A R AR B R B C CR Note that u and y are identical in (2.1) and (2.3); i.e., the input–output relationship is identical in the new representation ( and in the original representation (A,B,C). This might suggest that there is no difference as to what coordinates we use for a system analysis. But this is not necessarily true. Although input–output relations remain invariant, it makes a difference for system analysis or controller design what state or representation is chosen. For example, some representations have useful physical interpretations; others are more convenient for analysis and design. , , ) n n n A B C 2.1.2 Transfer Function Besides the state-space representation a linear system can be alternatively represented by its transfer function. The transfer function G(s) is defined as a complex gain between y(s) and u(s), ( ) ( ) ( ), y s G s u s (2.5) where y(s) and u(s) are the Laplace transforms of the output y(t) and input u(t), respectively. Using the Laplace transformation of (2.1) for the zero initial condition, x(0) = 0, we express the transfer function in terms of the state parameters (A,B,C), (2.6) 1 ( ) ( ) . G s C sI A B The transfer function is invariant under the coordinate transformation (i.e., , which can be checked by introducing (2.4) into the above equation. 1 ( ) ( ) n n C sI A B C sI A B 1 n
- 43. 16 Chapter 2 2.2 Second-Order Structural Models In this and the following sections we will discuss the structural models. One of them is the second-order structural model. It is represented by the second-order linear differential equations, and is commonly used in the analysis of structural dynamics. Similarly to the state-space models the second-order models also depend on the choice of coordinates. Typically, the second-order models are represented either in the nodal coordinates, and are called nodal models, or in the modal coordinates, and are called modal models. 2.2.1 Nodal Models The nodal models are derived in nodal coordinates, in terms of nodal displacements, velocities, and accelerations. The model is characterized by the mass, stiffness, and damping matrices, and by the sensors and actuators locations. These models are typically obtained from the finite-element codes or from other Computer-Aided- Design-type software. As a convention, we denote a dot as a first derivative with respect to time (i.e., / x dx dt ), and a double dot as a second derivative with respect to time (i.e., 2 / 2 x d x dt ). Let be a number of degrees of freedom of the system (linearly independent coordinates describing the finite-dimensional structure), let r be a number of outputs, and let s be a number of inputs. A flexible structure in nodal coordinates is represented by the following second-order matrix differential equation: d n , . o oq ov Mq+ Dq+ Kq= B u y = C q+C q (2.7) In this equation q is the 1 d n u nodal displacement vector; is the nodal velocity vector; is the q 1 d n u q 1 d n u nodal acceleration vector; u is the 1 s u input vector; y is the output vector, ; M is the mass matrix, 1 r u d d n n u ; D is the damping matrix, ; and K is the stiffness matrix, d d n n u d d n n u . The input matrix o B is , the output displacement matrix is d n s u oq C d r n u , and the output velocity matrix is . The mass matrix is positive definite (all its eigenvalues are positive), and the stiffness and damping matrices are positive semidefinite (all their eigenvalues are nonnegative). ov C d r n u Example 2.1. Determine the nodal model for a simple system from Fig. 1.1. For this system we selected masses 1 2 3 1, m m m stiffness 1 2 3 3, k k k and a damping matrix proportional to the stiffness matrix, D = 0.01K, or 4 0, k
- 44. Standard Models 17 0.01 i d i k , i = 1, 2, 3, 4. There is a single input force at mass 3, and three outputs: displacement and velocity of mass 1 and velocity of mass 3. For this system the mass matrix is 1 2 3 ( , , ), M diag m m m thus 3 M I . The stiffness and damping matrices are 1 2 2 2 2 3 3 3 3 4 0 0 k k k K k k k k k k k ª º « » « » « » ¬ ¼ , 1 2 2 2 2 3 3 3 3 4 0 0 d d d D d d d d d d d ª º « » « » « » ¬ ¼ , therefore, 6 3 0 3 6 3 0 3 3 K ª º « » « » « » ¬ ¼ , and 0.06 0.03 0.00 0.03 0.06 0.03 0.00 0.03 0.03 D ª º « » « » « » ¬ ¼ . The input and output matrices are , 0 0 1 o B ª º « » « » « » ¬ ¼ 1 0 0 0 0 0 0 0 0 oq C ª º « » « » « » ¬ ¼ , and 0 0 0 1 0 0 0 0 1 ov C ª º « » « » « » ¬ ¼ . On details of the derivation of this type of equation, see [70], [120]. 2.2.2 Modal Models The second-order models are defined in modal coordinates. These coordinates are often used in the dynamics analysis of complex structures modeled by the finite elements to reduce the order of a system. It is also used in the system identification procedures, where modal representation is a natural outcome of the test. Modal models of structures are the models expressed in modal coordinates. Since these coordinates are independent, it leads to a series of useful properties that simplify the analysis (as will be shown later in this book). The modal coordinate representation can be obtained by the transformation of the nodal models. This transformation is derived using a modal matrix, which is determined as follows. Consider free vibrations of a structure without damping, i.e., a structure without external excitation (u { 0) and with the damping matrix D = 0. The equation of motion (2.7) in this case turns into the following equation: (2.8) 0. Mq+ Kq =
- 45. 18 Chapter 2 The solution of the above equation is j t q e Z I . Hence, the second derivative of the solution is 2 j t q e Z Z I . Introducing the latter q and into (2.8) gives q 2 ( ) j t K M e Z Z I 0. (2.9) This is a set of homogeneous equations, for which a nontrivial solution exists if the determinant of is zero, 2 K Z M 2 det( ) 0. K M Z (2.10) The above determinant equation is satisfied for a set of n values of frequency Z. These frequencies are denoted 1 2 , ,..., n Z Z Z , and their number n does not exceed the number of degrees of freedom, i.e., d n n d . The frequency i Z is called the ith natural frequency. Substituting i Z into (2.9) yields the corresponding set of vectors ^ ` 1 2 , ,..., n I I I that satisfy this equation. The ith vector i I corresponding to the ith natural frequency is called the ith natural mode, or ith mode shape. The natural modes are not unique, since they can be arbitrarily scaled. Indeed, if i I satisfies (2.9), so does i DI , where D is an arbitrary scalar. For a notational convenience define the matrix of natural frequencies 1 2 0 0 0 0 0 0 n Z Z Z ª º « » « » : « » « » « » ¬ ¼ (2.11) and the matrix of mode shapes, or modal matrix ) , of dimensions which consists of n natural modes of a structure , d n n u @ 11 21 1 12 22 2 1 2 1 2 ... d d d n n n n n nn I I I I I I I I I I I I ª º « » « » ) « » « » « » ¬ ¼ ! ! ! , (2.12) where ij I is the jth displacement of the ith mode, that is,
- 46. Standard Models 19 1 2 i i i in I I I I ½ ° ° ° ° ® ¾ ° ° ° ° ¯ ¿ # . (2.13) The modal matrix ) has an interesting property: it diagonalizes mass and stiffness matrices M and K, , T m M M ) ) (2.14) . T m K K ) ) (2.15) The obtained diagonal matrices are called modal mass matrix and modal stiffness matrix ( The same transformation, applied to the damping matrix ( ) m M ). m K , T m D D ) ) (2.16) gives the modal damping matrix , which is not always obtained as a diagonal matrix. However, in some cases, it is possible to obtain diagonal. In these cases the damping matrix is called a matrix of proportional damping. The proportionality of damping is commonly assumed for analytical convenience. This approach is justified by the fact that the nature of damping is not known exactly, that its values are rather roughly approximated, and that the off-diagonal terms in most cases—as will be shown later—have negligible impact on the structural dynamics. The damping proportionality is often achieved by assuming the damping matrix as a linear combination of the stiffness and mass matrices; see [18], [70], m D m D 1 2 D K M D D , (2.17) where 1 D and 2 D are nonnegative scalars. Modal models of structures are the models expressed in modal coordinates. In order to do so we use a modal matrix to introduce a new variable, , called modal displacement. This is a variable that satisfies the following equation: m q . m q q ) (2.18) In order to obtain the equations of motion for this new variable, we introduce (2.18) to (2.7) and additionally left-multiply (2.7) by , T ) obtaining , . T T T T m m m oq m ov m o M q D q K q B y C q C q ) ) ) ) ) ) ) ) ) u
- 47. 20 Chapter 2 Assuming a proportional damping, and using (2.14), (2.15), and (2.16) we obtain the above equation in the following form: , . T m m m m m m o oq m ov m M q D q K q B u y C q C q ) ) ) Next, we multiply (from the left) the latter equation by 1 m M , which gives 1 1 1 , . T m m m m m m m m o oq m ov m q M D q M K q M B u y C q C q ) ) ) The obtained equations look quite messy, but the introduction of appropriate notations simplifies them, (2.19) 2 2 , . m m m mq m mv m q q q B y C q C q =: : mu In (2.19) : is a diagonal matrix of natural frequencies, as defined before. Note, however, that this is obtained from the modal mass and stiffness matrices as follows: 2 1 . m m M K : (2.20) In (2.19) is the modal damping matrix. It is a diagonal matrix of modal damping, = 1 2 0 0 0 0 0 0 n ] ] ] ª º « » « » = « » « » « » ¬ ¼ , (2.21) where i ] is the damping of the ith mode. We obtain this matrix using the following relationship , thus, 1 2 m m M D =: 1 1 2 2 1 1 0.5 0.5 m m m m m M D M K = : D . (2.22) Next, we introduce the modal input matrix m B in (2.19), 1 . T m m o B M B ) (2.23) Finally, in (2.19) we use the following notations for the modal displacement and rate matrices:
- 48. Standard Models 21 , mq oq C C ) (2.24) . mv ov C C ) (2.25) Note that (2.19) (a modal representation of a structure) is a set of uncoupled equations. Indeed, due to the diagonality of : and = , this set of equations can be written, equivalently, as (2.26) 2 1 2 , 1, , , mi i i mi i mi mi i mqi mi mvi mi n i i q q q b u y c q c q i n y y ] Z Z ¦ ! , where is the ith row of mi b m B and are the ith columns of and , respectively. The coefficient , mqi c mvi c mq C mv C i ] is called a modal damping of the ith mode. In the above equations is the system output due to the ith mode dynamics, and the quadruple i y ( , , , ) i i mi mi b c Z ] represents the properties of the ith natural mode. Note that the structural response y is a sum of modal responses yi, which is a key property used to derive structural properties in modal coordinates. This completes the modal model description. In the following we introduce the transfer function obtained from the modal equations. The generic transfer function is obtained from the state-space representation using (2.6). For structures in modal coordinates it has a specific form. Transfer Function of a Structure. The transfer function of a structure is derived from (2.19), (2.27) 2 2 1 ( ) ( )( 2 ) . mq mv n m G C j C I j Z B Z Z Z Z : : However, this can be presented in a more useful form, since the matrices : and = are diagonal, allowing for representation of each single mode. Transfer Function of a Mode. The transfer function of the ith mode is obtained from (2.26), 2 2 ( ) ( ) . 2 mqi mvi mi mi i i c j c b G j Z Z i Z Z ] Z Z (2.28) The structural and modal transfer functions are related as follows:
- 49. 22 Chapter 2 Property 2.1. Transfer Function in Modal Coordinates. The structural transfer function is a sum of modal transfer functions (a) 1 ( ) ( ) n mi i G G Z Z ¦ (2.29) or, in other words, 2 2 1 ( ) ( ) , 2 n mqi mvi mi i i i c j c b G j Z Z i Z Z ] Z Z ¦ (2.30) and the structural transfer function at the ith resonant frequency is approximately equal to the ith modal transfer function at this frequency (b) 2 ( ) ( ) ( ) 2 mqi i mvi mi i mi i i i jc c b G G Z Z Z ] Z # , 1, , . i n ! (2.31) Proof. By inspection of (2.27) and (2.28). ‹ Structural Poles. The poles of a structure are the zeros of the characteristic equations (2.26). The equation 2 2 i i i s s ] Z Z2 0 is the characteristic equation of the ith mode. For small damping the poles are complex conjugate, and in the following form: 2 1 2 2 1 , 1 . i i i i i i i i s j s j ] Z Z ] ] Z Z ] (2.32) The plot of the poles is shown in Fig. 2.1, which shows how the location of a pole relates to the natural frequency and modal damping. Example 2.2. Determine the modal model of a simple structure from Example 2.1. The natural frequency matrix is 3.1210 0 0 0 2.1598 0 0 0 0.7708 ª º « » : « » « » ¬ ¼ , and the modal matrix is
- 50. Standard Models 23 (a) 0.5910 0.7370 0.3280 0.7370 0.3280 0.5910 0.3280 0.5910 0.7370 ª º « » ) « » « » ¬ ¼ . The modes are shown in Fig. 2.2. i i ] Z arcsin( ) i i D ] i Z Im 0 Re s2 s1 2 1 i i Z ] 2 1 i i Z ] i Z Figure 2.1. Pole location of the ith mode of a lightly damped structure: It is a complex pair with the real part proportional to the ith modal damping; the imaginary part approximately equal to the ith natural frequency; and the radius is the exact natural frequency. The modal mass is 3 m M I , the modal stiffness is 2 m K : , and the modal damping, from (2.22), is 0.0156 0 0 0 0.0108 0 0 0 0.0039 ª º « » = « » « » ¬ ¼ . We obtain the modal input and output matrices from (2.23), (2.24), and (2.25): 0.3280 0.5910 , 0.7370 m B ª º « » « » « » ¬ ¼
- 51. 24 Chapter 2 0.5910 0.7370 0.3280 0 0 0 0 0 0 mq C ª º « » « » « » ¬ ¼ , and 0 0 0 0.5910 0.7370 0.3280 0.3280 0.5910 0.7370 mv C ª º « » « » « » ¬ ¼ . 0.591 1 m 2 m 3 m 0.328 0.591 0.737 1 m 2 m 3 m 0.737 0.328 –0.591 1 m 2 m 3 m 2 m 3 m 1 m –0.737 0.328 equilibrium 1 I —mode 1 2 I —mode 2 3 I —mode 3 Figure 2.2. Modes of a simple system: For each mode the mass displacements are sinusoidal and have the same frequency, and the displacements are shown at their extreme values (see the equation (a)). Example 2.3. Determine the first four natural modes and frequencies of the beam presented in Fig. 1.5. Using the finite-element model we find the modes, which are shown in Fig. 2.3. For the first mode the natural frequency is 1 72.6 Z rad/s, for the second mode the
- 52. Standard Models 25 natural frequency is 2 198.8 Z rad/s, for the third mode the natural frequency is 3 386.0 Z rad/s, and for the fourth mode the natural frequency is 4 629.7 Z rad/s. –1 –0.5 0 0.5 1 mode 3 mode 1 mode 2 mode 4 displacement, y-dir. 0 2 4 6 8 10 12 14 node number Figure 2.3. Beam modes: For each mode the beam displacements are sinusoidal and have the same frequency, and the displacements are shown at their extreme values. Example 2.4. Determine the first four natural modes and frequencies of the antenna presented in Fig. 1.6. We used the finite-element model of the antenna to solve this problem. The modes are shown in Fig. 2.4. For the first mode the natural frequency is 1 13.2 Z rad/s, for the second mode the natural frequency is 2 18.1 Z rad/s, for the third mode the natural frequency is 3 18.8 Z rad/s, and for the fourth mode the natural frequency is 4 24.3 Z rad/s. Example 2.5. The Matlab code for this example is in Appendix B. For the simple system from Fig. 1.1 determine the natural frequencies and modes, the system transfer function, and transfer functions of each mode. Also determine the system impulse response and the impulse responses of each mode. Assume the system masses stiffnesses 1 2 3 1, m m m 1 2 3 3 k k k , 4 0 k , and the damping matrix proportional to the stiffness matrix, D = 0.01K or 0.01 i d i k , i = 1, 2, 3, 4. There is a single input force at mass 3 and a single output: velocity of mass 1. We determine the transfer function from (2.27), using data from Example 2.2. The magnitude and phase of the transfer function are plotted in Fig. 2.5. The magnitude plot shows resonance peaks at natural frequencies 1 0.7708 Z rad/s, 2 2.1598 Z rad/s, and 3 3.1210 Z rad/s. The phase plot shows a 180-degree phase change at each resonant frequency.
- 53. 26 Chapter 2 (b) (c) (d) (a) Figure 2.4. Antenna modes: (a) First mode (of natural frequency 2.10 Hz); (b) second mode (of natural frequency 2.87 Hz); (c) third mode (of natural frequency 2.99 Hz); and (d) fourth mode (of natural frequency 3.87 Hz). For each mode the nodal displacements are sinusoidal, have the same frequency, and the displacements are shown at their extreme values. Gray color denotes undeformed state. We determine the transfer functions of modes 1, 2, and 3 from (2.28), and their magnitudes and phases are shown in Fig. 2.6. According to Property 2.1, the transfer function of the entire structure is a sum of the modal transfer functions, and this is shown in Fig. 2.6, where the transfer function of the structure was constructed as a sum of transfer functions of individual modes. The impulse response of the structure is shown in Fig. 2.7; it was obtained from (2.19). It consists of three harmonics (or responses of three modes) of natural frequencies 1 0.7708 Z rad/s, 2 2.1598 Z rad/s, and 3 3.1210 Z rad/s. The
- 54. Standard Models 27 harmonics are shown on the impulse response plot, but are more explicit at the impulse response spectrum plot, Fig. 2.7, as the spectrum peaks at these frequencies. Impulse response is the time-domain associate of the transfer function (through the Parseval theorem); therefore, Property 2.1 can be written in time domain as 1 ( ) ( ) n i i h t h t ¦ where h(t) is the impulse response of a structure and is the impulse response of the ith mode. Thus, the structural impulse response is a sum of modal responses. This is illustrated in Fig. 2.8, where impulse responses of modes 1, 2, and 3 are plotted. Clearly the total response as in Fig. 2.7 is a sum of the individual responses. Note that each response is a sinusoid of frequency equal to the natural frequency, and of exponentially decayed amplitude, proportional to the modal damping ( ) i h t i ] . Note also that the higher-frequency responses decay faster. 10 –1 10 0 10 1 10 –2 10 –1 10 0 10 1 10 2 2 Z frequency, rad/s 10 –1 10 0 –400 –200 0 200 2 Z 1 Z 1 Z (b) (a) 3 Z 3 Z 10 1 magnitude phase, deg frequency, rad/s Figure 2.5. Transfer function of a simple system: (a) Magnitude shows three resonance peaks; and (b) phase shows three shifts of 180 degrees; 1 2 , , Z Z and 3 Z denote the natural frequencies.
- 55. Other documents randomly have different content
- 56. with God's help, to justify our guardianship if need be against all who would silence them.' This kind of language, so different from that usually used by Eastern politicians in their conversations with Europeans, impressed me very deeply, and I made a strong mental contrast between Arabi and that other champion of liberty whom I had met and talked with at Damascus, Midhat Pasha, altogether in Arabi's favour. Here was no nonsense about railroads and canals and tramways as nostrums that could redeem the East, but words that went to the root of things and fixed the responsibility of good government on the shoulders which alone could bear it. I felt that even in the incredulous and trifling atmosphere of the House of Commons words like these would be listened to—if only they could be heard there! With regard to the Sultan and the connection of Egypt with Turkey, Arabi was equally explicit. He had no love, he told me, for the Turks who had mis-governed Egypt for centuries, and he would not hear of interference from Constantinople in the internal affairs of the country. But he made a distinction between the Ottoman Government and the religious authority of the Sultan, whom, as Emir el Mumenin, he was bound, as long as he ruled justly, to obey and honour. Also the example of Tunis, which the French had first detached from the Empire, and then taken possession of, showed how necessary it was to preserve the connection of Egypt with the Head of the Moslem world. We are all, he said, children of the Sultan, and live together like a family in one house. But, just as in families, we have, each of us provinces of the Empire, our separate room which is our own to arrange as we will and where even the Sovereign must not wantonly intrude. Egypt has gained this independent position through the Firmans granted, and we will take care that she preserves it. To ask for more than this would be to run a foolish risk, and perhaps lose our liberty altogether.[7] I asked him rather bluntly whether he had been, as was then currently asserted, in personal communication with Constantinople, and I noticed that he was reserved in answering and did so evasively. Doubtless the
- 57. recollection of his conversation with Ahmed Ratib, of which I then knew nothing, crossed his mind and caused his hesitation, but he did not allude to it. Finally we talked of the relations of Egypt with the Dual Government of France and England. As to this he admitted the good that had been done by freeing the country of Ismaïl and regularizing the finances, but they must not, he said, stand in the way of the National regeneration by supporting the Khedive's absolute rule or the old Circassian Pashas against them. He looked to England rather than to France for sympathy in their struggle for freedom, and especially to Mr. Gladstone, who had shown himself the friend of liberty everywhere—this in response to what I had explained to him of Gladstone's views—but like everybody else just then at Cairo he distrusted Malet. I did what I could to ease his mind on this point, and so we parted. This first interview gave me so favourable an opinion of the fellah Colonel that I went immediately to my friend, Sheykh Mohammed Abdu, to tell him how he had impressed me, and suggested that a program, in the sense of what Arabi had told me, ought to be drawn up which I might send to Mr. Gladstone, as I felt certain that if he knew the truth as to the National aspirations, in an authoritative way, he could not fail to be impressed by it in a sense favourable to them. I spoke, too, to Malet on the same subject, and he agreed that it might do good, and I consequently, in conjunction with Sheykh Mohammed Abdu and others of the civilian leaders, drew up, Sabunji being our scribe, a manifesto embodying succinctly the views of the National party. This Mohammed Abdu took to Mahmud Pasha Sami, who was once again Minister of War, and gained his adhesion to it, and it was also shown to and approved by Arabi. This done I forwarded it, with Malet's knowledge and approval, to Gladstone, explaining to him the whole situation and inviting his sympathy for a movement so very much in accordance with his avowed principles. I cannot understand, I said, in concluding my letter to Gladstone, that these are sentiments to be deplored or actions to be crushed by an English Liberal Government. Both may be easily guided. And I think the lovers of Western
- 58. progress should rather congratulate themselves on this strange and unlooked for sign of political life in a land which has hitherto been reproached by them as the least thinking portion of the stagnant East. You, sir, I think, once expressed to me your belief that the nations of the East could only regenerate themselves by a spontaneous resumption of their lost national Will, and behold in Egypt that Will has arisen and is now struggling to find words which may persuade Europe of its existence. While sending this Program of the National Party to Gladstone, I also at the same time, by Sir William Gregory's advice, sent it to the Times. Of this course Malet disapproved as he thought it might complicate matters at Constantinople, an idea strongly fixed in his cautious diplomatic mind. But Gregory insisted that it ought to be published, as otherwise it might be pigeon-holed at Downing Street and overlooked; and I think he was right. Gregory was a personal friend of the then excellent editor of the Times, Chenery, whose services to the National cause in Egypt at this date were very great. Chenery was a man of a large mind on Eastern affairs, being a considerable Arabic scholar, and had published a most admirable English translation of the Assemblies of Hariri; and he was able thus to take a wider view of the Egyptian question than the common journalistic one that it was a question primarily concerning the London Stock Exchange—this although he was himself an Egyptian Bondholder. He consequently gave every prominence to the letters Gregory and I wrote to him during the next few months in support of the National movement, and to the last, even when the war came, continued that favour. In the present instance, indeed, Chenery somewhat overdid his welcome to our program, stating that it had been received from Arabi himself, an inaccuracy which enabled Malet, who knew the facts, to disown it through Reuter's Agency as an authentic document. It will perhaps be as well to explain here the way in which the London Press and especially Reuter's News Agency was at this time manipulated officially at Cairo and made subservient to the intrigues
- 59. of diplomacy. Very few London newspapers had any regular correspondent in Egypt, the Times and the Pall Mall Gazette being, as far as I know, the only two that were thus provided. Both, as far as politics were concerned, were practically in the hands of Sir Auckland Colvin, the English Financial Controller, an astute Indian official, with the traditions of Indian diplomacy strongly developed in his political practice. He had some experience of journalism, having been connected with the Pioneer in India, an Anglo-Indian journal of pronounced imperialistic type with which he was still in correspondence. He was also Morley's regular correspondent in the Pall Mall Gazette, and had through him the ear of the Government. The importance of this unavowed connection will be seen later when he made it his business to bring about English intervention. Lastly, on all important diplomatic matters he inspired the Times, whose regular correspondent, Scott, depended on him for his information. With regard to Reuter and Havas, the Telegraphic Agencies, both were heavily subventioned by the Anglo-French Financial Control, receiving £1,000 a year each, charged on the thin resources of the Egyptian Budget. Reuter especially was the servant and mouthpiece of the English Agency, and the telegrams despatched to London were under Malet's censorship. This sort of manipulation of the organs of public news in the interests of our diplomacy exists in nearly all the capitals where our agents reside, and is a potent instrument for misleading the home public. The influence is not as a rule exercised by any direct payment, but by favour given in regard to secret and valuable information, and also largely by social amenities. In Egypt it has always within my knowledge been supreme, except at moments of extreme crisis when the body of special Press correspondents at Cairo or Alexandria has been too numerous to be kept under official control. In ordinary times our officials have had complete authority both as to what news should be sent to London, and what news, received from London, should be published in Egypt. It is very necessary that this, the true condition of things, should be steadily borne in mind by historians when they consult the newspaper files of these years in search of information.
- 60. Down, however, to near the end of the year 1881, except for this small difference of opinion, my relations with Malet remained perfectly and intimately friendly. He made me the confidant of his doubts and troubles, his anxiety to follow out the exact wishes of the Foreign Office, and his fears lest in so difficult a situation he should do anything which should not gain an official approval. He professed himself, and I think he was, in full sympathy with my view of the National case, and he leaned on me as on one able, at any rate, to act as buffer between him and any new violent trouble while waiting a decision in Downing Street as to clear policy. Thus I find a note that on the 19th December I was asked by him and Sir Auckland Colvin, whose acquaintance I had now made and who affected views hardly less favourable than Malet's to the Nationalists, to help them in a difficulty they were in about the Army Estimates. It was the time of year when the new Budget was being drafted, and the Nationalist Minister of War, Mahmud Sami, had demanded £600,000 as the amount of the year's estimates for his department. It was an increase of I forget how many thousand pounds over the estimate of 1881, and was necessitated, Mahmud Sami said, by the Khedive's promise of raising the army to the full number of men allowed by the Firman, 18,000. The Minister had explained his insistence on the plea that a refusal would or might cause a new military demonstration, the bug-bear of those days; and I was asked to find out what sum the army would really be satisfied with for their estimates. Colvin authorized me to go as far as £522,000, and to tell Arabi and the officers that it was financially impossible to give more. He had no objection, he said, to the army's being increased so long as the estimates were not exceeded. He thought, however, the sum proposed would suffice for an increase up to 15,000 men. I consequently went to Arabi and argued the matter with him and others of the officers; and persuaded them, on my assurance that Colvin's word could be trusted, to withdraw all further objection. They said they would accept the increased sum of £522,000 as sufficient, and make it go as far in the increase of soldiers as it could. They meant to economize, they said, in other ways, and
- 61. hoped to get their full complement of men out of the balance. They promised me, too, on this occasion to have patience and make no further armed demonstrations, a promise which to the end they faithfully fulfilled. Arabi's last words to me on this occasion were men sabber dhaffer, he who has patience conquers. I sent a note the same day to Colvin informing him of the result, and I was also thanked by Malet for having helped them both out of a considerable difficulty. Nevertheless Malet, about a week later, surprised me one afternoon, 28th December, when I had been playing lawn tennis with him, as I often did at the Agency, by showing me the draft of a despatch he had just sent to the Foreign Office mentioning my visit to Egypt and the encouragement I had given to the Nationalists, and without mentioning what I had done to help him, complaining only of my having sent the Program against his wishes to the Times. As we had up to that moment been acting in perfect cordiality together, and nothing whatever had occurred beyond the publication of the manifesto, I took him pretty roundly to task for his ill faith in concealing my other services rendered to his diplomacy, and insisted that he should cancel this misleading despatch, and with such energy that he wrote in my presence a cancelling telegram, and also a second despatch repairing in some measure the injustice he had done me. I have never quite understood what Malet's motive was in this curious manœuvre. I took it at the time to be a passing fit of jealousy, a dislike to the idea that it should be known at the Foreign Office that he owed anything to me in the comparatively good relations he had succeeded in establishing with the Nationalists; but on reflection I have come to the conclusion, as one more in accordance with his cautious character, that he was merely guarding himself officially against public responsibility of any kind being fixed on him for my Nationalist views, should these be condemned in Downing Street. It is the more likely explanation because his private conscience evidently pricked him about it to the extent of avowing to me what he had officially done. The insincerity, however, though repented of, was a warning to me which I did not forget, and while I
- 62. continued for some weeks more to go to the Agency it was always with a feeling of possible betrayal at Malet's hands. I was ready, nevertheless, to help him, and it was not long before he was again obliged, by the extreme circumstances of his political isolation at Cairo, to resort to my good offices, and, finding himself in flood water altogether beyond his depth, to send me once more as his messenger of peace to Arabi and the other Nationalist leaders. All had gone well so far, as far as any of us knew, in the political situation at Cairo down to the end of the year, and during the first week of the new year, 1882. There was a good understanding now between all parties in Egypt, the army was quiescent, the Press was moderate under Mohammed Abdu's popular censorship, and the Nationalist Ministers, undisturbed by menace from any quarter, were preparing the draft of the Organic Law which was to give the country its civil liberties. On the 26th of December, the Chamber of Delegates summoned to discuss the articles of the promised Constitution had met at Cairo, and had been opened formally with a reassuring speech by the Khedive in person, whose attitude was so changed for the better towards the popular movement that Malet was able, on the 2nd of January, to write home to Lord Granville: I found His Highness, for the first time since my return in September, cheerful in mood and taking a hopeful view of the situation. The change was very noticeable. His Highness appears to have frankly accepted the situation. Arabi had ceased to busy himself personally with the redress of grievances, and it had been arranged with the approval of the French and English agents that Arabi should, as they expressed it, regularize his position and accept the responsibility of his acknowledged political influence by taking office as Under- Secretary at the War Office. This it had been thought would be putting the dangerous free lance in uniform and securing him to the cause of order. The only doubtful point was now the attitude of the Deputies in regard to the details of the Constitution they had been assembled to discuss; and the majority of them, as were my reforming friends at
- 63. the Azhar, seemed disposed to moderation. We have waited, said Sheykh Mohammed Abdu, so many hundred years for our freedom that we can well afford now to wait some months. Certainly at that date Malet and Colvin, and I think also Sinkiewicz, were favourably disposed to the claim of the Nationalists to have a true Parliament. They had begun to see that it was the universal national desire, and would act as a safety-valve for ideas more dangerous. A frank public declaration of goodwill at that moment on the part of the English and French Governments towards the popular hopes would have secured a workable arrangement between the Nationalist Government and the Dual Control, which would have safeguarded the bondholders' interests no less than it would have secured to Egypt its liberty. Nor did we think that this would be long delayed. On the first day of the New Year the National Program I had sent to Mr. Gladstone was published in the Times, with a leading article and approving comments, and in spite of Malet's prognostication of evil had been well received in Europe, and even at Constantinople where it had drawn down no kind of thunderbolt. Its tone was so studiously moderate, and its reasoning so frank and logical that it seemed impossible the position in Egypt should any longer be misunderstood. Especially in England, with an immense Liberal majority in the House of Commons, and Mr. Gladstone at the head of affairs, it was almost inconceivable that it should not be met in a friendly spirit—quite inconceivable to us who were waiting anxiously for Gladstone's answer at Cairo, that at that very moment the English Foreign Office should be proceeding to acts of menace and the language of armed intervention. Unfortunately, however, though none of us, not even Malet, at the time knew it, the decision, adverse to the Egyptian hopes, had already been half taken. The program reached Mr. Gladstone, as nearly as I can calculate it, a fortnight too late. We were all expecting a message of peace, when, like thunder in a clear sky, the ill-omened Joint Note of January 6th, 1882, was launched upon us. It upset all our hopes and calculations and threw back Egypt once more into a sea of troubles.
- 64. It is right that the genesis of this most mischievous document, to which is directly due the whole of the misfortunes during the year, with the loss to Egypt of her liberty, to Mr. Gladstone of his honour, and to France of her secular position of influence on the Nile, should be truly told. Something regarding it may be learned from the published documents, both French and English, but only indirectly, and not all; and I am perhaps the only person not officially concerned in its drafting who am in a position to put all the dots with any precision on the i's. In Egypt it has not unnaturally been supposed that, because in the event it turned to the advantage of English aggression, it was therefore an instrument forged for its own purposes at our Foreign Office, but in reality the reverse is true and the note was drafted not in Downing Street but at the Quai d'Orsay, and in the interests, so far as these were political—for they were also financial—of French ambition. I have told already how I travelled with Sir Charles Dilke from London to Paris, and of our conversation on the way and of the impression left on me by it that he would sell Egypt for his Commercial Treaty; and this is precisely what in fact had happened. The dates as far as I can fix them were these: On the 15th of November St. Hilaire had gone out of office, and had been succeeded by Gambetta, who found himself faced with a general Mohammedan revolt against the French Government in Tunis and Algeria. He was alarmed at the Pan-Islamic character it was taking, and attributed it largely to the Sultan Abdul Hamid's propaganda, and he thought he saw the same influence at work in the National movement in Egypt, as well as the intrigues of Ismaïl, Halim, and others. France had been traditionally hostile to the sovereign claims of the Porte in North Africa, and Gambetta came into office determined to thwart and deal with them by vigorous measures. He was besides, through his Jewish origin, closely connected with the haute finance of the Paris Bourse, and was intimate with the Rothschilds and other capitalists, who had their millions invested in Egyptian Bonds. Nubar Pasha and Rivers Wilson were then both living at Paris, and his close friends and advisers in regard to
- 65. Egyptian matters, and it was from them that he took his view of the situation. He had, therefore, not been more than a few days in office before he entered into communication with our Foreign Office, with the object of getting England to join him in vigorous action against the National movement, as a crusade of civilization and a support to the established order at Cairo of Financial things. In London at the same time there was a strong desire to get the Commercial Treaty, which was about to expire, renewed with France as speedily as possible, and advantage was taken at the Foreign Office of Sir Charles Dilke's personal intimacy with the new French Premier to get the negotiation for it finished. A commission for this purpose, of which Dilke and Wilson were the two English members, had been sitting at Paris since the month of May, and so far without result. Dilke's visit to Paris was in connection with both matters, and was resolved on within a week of Gambetta's accession to power. Reference to newspapers of that date, November 1881, will show that the negotiations between the two Governments about the Commercial Treaty were just then in a highly critical state, and it was even reported that they had been broken off. Dilke's presence, however, gave them new life, or at least prevented their demise. Between the 22nd of November and the 15th of December he passed to and fro between the two capitals; and at the latter date we find Gambetta (Blue Book Egypt 5, 1882, page 21) approaching Lord Lyons, our Ambassador at Paris, with a proposal to take common action in Egypt. He considers it to be extremely important to strengthen the authority of Tewfik Pasha; every endeavour should be made to inspire him with confidence in the support of France and England, and to infuse into him firmness and energy. The adherents of Ismaïl and Halim and the Egyptians generally should be made to understand that France and England would not acquiesce in his being deposed.... It would be advisable to cut short the intrigues of Constantinople, etc. This language is communicated by Lord Lyons to the Foreign Office, and on the 19th Lord Granville agrees in thinking that the time has come when the two Governments should
- 66. consider what course had better be adopted, etc. Thus encouraged, Gambetta on the 24th proposes to take occasion of the meeting of the Egyptian Notables to make a distinct manifestation of union between France and England so as to strengthen the position of Tewfik Pasha and discourage the promoters of disorder. The Egyptian Chamber meets on the 26th, and on the 28th Dilke, who has returned the day before to Paris, has a long conversation with Gambetta about the Treaty of Commerce (Times), while on precisely the same day Lord Granville agrees to give assurance to Tewfik Pasha of the sympathy and support of France and England, and to encourage His Highness to maintain and assert his proper authority. This identity of date alone suffices to fix the connection between the two negotiations, and shows the precise moment at which the fatal agreement was come to, and that my communication of the National Program to Gladstone, which was posted on the 20th, must have been just too late to prevent the disaster. Letters then took a week to reach London, and Gladstone was away for the Christmas holidays, and cannot have had time, however much he may have been inclined to do so, to forward it on to the Foreign Office. Our Government thus committed to Gambetta's policy, Gambetta on the 31st (Blue Book Egypt 5, 1882) presents to Lyons the draft, drawn up with his own hand, of the Joint Note to be despatched to Cairo in the sense of his previous communication of the 24th—and, be it noted, on the same day negotiations for a renewal of the Commercial Treaty are announced as formally renewed. On the 1st of January the Paris correspondent of the Times sends a précis of the Joint Note to London, explaining that he only now forwards it, having been instructed by M. Gambetta only to divulge it at the proper moment. This is understood to mean the final success of Dilke's commercial mission, and the following day, 2nd January, he returns to London. I trace, nevertheless, the influence of my appeal to Gladstone in the delay of five days, still made by Granville before he unwillingly signs the Note, and the reservation he stipulates for on the part of Her Majesty's Government that Her Majesty's
- 67. Government must not be considered as committing themselves thereby to any particular mode of action, a postscript typical of Granville's character, and, as I think too, of a conflict in ideas, afterwards very noticeable, between the Foreign Office, pushed on by Dilke, and Gladstone as Prime Minister. Such is the evidence which, intelligently read, can be gathered from the published documents of the day. I have, however, a letter from Sir Rivers Wilson dated a few days later, 13th January, in answer to one of mine, which explains in a few words the whole situation. I am above all pleased, he writes, at the interest you are taking in Egyptian politics. You confirm what I believe to be the case in two particulars at least, viz., that the soldiers express the feeling of the population, and that Tewfik has been working with the Sultan. As regards the latter circumstance I must say there is nothing surprising in it. Six weeks ago Gambetta said to me, 'Le Khedive est aux genoux du Sultan.' But the reason is plain. Tewfik is weak and cowardly. His army is against him. The Harems hate him. He found no support there where he naturally might have looked for it, viz., at the hands of the English and French Governments, and so he turned to the only quarter where sympathy and perhaps material assistance were forthcoming. It was to remedy this state of things that the idea of the Joint Declaration was conceived, whatever gloss or subsequent explanation may be now put forward, and I shall be disappointed if it does not produce the desired effect and cause the officers, Ulemas, and Notables to understand that renewed disturbance means armed intervention in Europe. Our Government may not like it, but they are bound now by formal engagement to France and cannot withdraw. This letter, coming from Wilson at Paris, holding the official position there he did, and being, as he was, on intimate terms both with Dilke and Gambetta, is a document of the highest historical importance, and fixes beyond the possibility of doubt on the French Government the initiative in the designed intervention, though the Yellow Books also are not altogether silent. These, though most
- 68. defective in their information, do not hide Gambetta's initial responsibility. I heard at the time, and I believe that the form of joint intervention he designed for Egypt was that England should demonstrate with a fleet at Alexandria while France should land troops. Had that come to pass we cannot doubt that French influence would now be supreme in Egypt. It was only frustrated that winter by the accident of Gambetta's unlooked-for fall from power by an adverse vote on some domestic matter in the Chamber at the end of the month, for Gladstone at that time was far too averse from violent measures to have sent an English fleet with a French army, and the landing of troops would have been certainly needed. There is more than one moral to be drawn from this historic episode, and the most instructive is, perhaps, the fact that neither of the two Ministers, with all their cleverness and in spite of their apparent success each in his own scheme, really effected his purpose. Gambetta and Granville in the first weeks of January doubtless plumed themselves on having gained an important object and strengthened the friendly link between their two Governments by a common agreement. Gambetta had got his note, Granville his treaty. But neither rogue was really able to bring home his booty. Gambetta, though he exerted all his influence with the Chamber to get the Commercial Treaty with England renewed, failed to obtain a majority and the treaty lapsed, and with it the Liberal argument that Free Trade was not isolating England. On the other hand, though he had got Granville unwillingly to sign the Note, which he intended to use for the glory of France, Gambetta found that he had forged a weapon which he could not himself wield and which within six months passed into his rival's hand, while the friendly arrangement proved almost as soon as it was come to, to be the destruction of all cordial feeling between the two nations for close on a generation. Personally, in the disappointment of the two intriguers and the rival interest of the two nations, I am able to hold a detached attitude. What seems to me tragic in the matter is that for the sake of their paltry ambitions and paltrier greeds a great national hope was
- 69. wrecked, and the cause of reform for a great religion postponed for many years. The opportunity of good thrown away by the two statesmen between them can hardly recur again in another half century. The effect of Gambetta's menace to the National Party was disastrous at Cairo to the cause of peace. I was with Malet soon after the note arrived, and he gave it me to read and asked me what I thought of it. I said: They will take it as a declaration of war. He answered: It is not meant in a hostile sense, and explained to me how it might be interpreted in a way favourable to the National hopes. He asked me to go to the Kasr el Nil and persuade Arabi, who had just been made Under-Secretary of War, to accept it thus, authorizing me to say, that the meaning of the Note as understood by the British Government was that the English Government would not permit any interference of the Sultan with Egypt, and would also not allow the Khedive to go back from his promises or molest the Parliament. He also told me, though he did not authorize me to repeat this on his authority, that he hoped to get leave to add to the Note a written explanation in the sense just given. I know that he telegraphed repeatedly for some such permission, and that he wrote strongly condemning the note as impolitic and dangerous. Not a word, however, of these important protests and requests is to be found in the Blue Books, though the Blue Books show that Lord Granville must have paid attention to them to the extent of expressing himself willing to give some such explanation of the Note but being prevented from doing so by Gambetta. Sinkiewicz seems also to have asked his Government to be allowed to explain the Note, but was forbidden. Sir Auckland Colvin, too, condemned the Note in conversation with me quite as strongly as Malet had done. I went accordingly to the Kasr el Nil about noon on the 9th (the text of the Note had reached us on the 8th) and found Arabi alone in his official room. For the first and only time I have seen him so, he was angry. His face was like a thundercloud, and there was a peculiar gleam in his eye. He had seen the text of the Note though it had not
- 70. been published—indeed, it had only as yet been telegraphed—and I asked him how he understood it. Tell me, rather, he said, how you understand it. I then delivered my message. He said: Sir Edward Malet must really think us children who do not know the meaning of words. In the first place, he said, it is the language of menace. There is no clerk in this office who would use such words with such a meaning. He alluded to the reference to the Notables made in the first paragraph of the Note. That, he said, is a menace to our liberties. Next, the declaration that French and English policy were one meant that, as France had invaded Tunis, so England would invade Egypt. Let them come, he said, every man and child in Egypt will fight them. It is contrary to our principles to strike the first blow, but we shall know how to return it. Lastly, as to the guarantee of Tewfik Pasha's throne. The throne, he said, if there is one, is the Sultan's. The Khedive needs no foreign guarantees. You may tell me what you will, but I know the meaning of words better than Mr. Malet does. In truth, Malet's explanation was nonsense, and I felt a fool before Arabi and ashamed of having made myself the bearer of such rubbish. But I assured him I had delivered the message as Sir Edward had given it me. He asks you to believe it, I said, and I ask you to believe him. At leaving he softened, took me by the arm to lead me down and invited me still to come as before to his house. I said: I shall only come back when I have better news for you, by which I intended to hint at a possible explanation of the Note such as Malet had telegraphed for permission to give. None however came. Nor did I see Arabi again till more than three weeks later, when a letter from Mr. Gladstone reached me which I interpreted in a more hopeful sense and which caused us great rejoicing. On returning to the Residency, Malet asked me how I had fared. They are irreconcilable now, I answered. The Note has thrown them into the arms of the Sultan. Such indeed was the effect, and not with the soldiers alone, but as soon as the Note was published with all sections of the National Party, even with the Khedive. Gambetta, if he had expected to strengthen Tewfik's hands, had
- 71. missed his mark entirely. The timid Khedive was only frightened, and the Nationalists, instead of being frightened, were enraged. The Egyptians for the first time found themselves quite united. Sheykh Mohammed Abdu and the cautious Azhar reformers from that point threw in their lot wholly with the advanced party. All, even the Circassians, resented the threat of foreign intervention, and on the other hand the most anti-Turkish of the Nationalists, such as my friend Hajrasi, saw that Arabi had been right in secretly leaning upon the Sultan. Arabi thus gained immensely in popularity and respect, and for many days after this I hardly heard anything from my Egyptian friends but the language of Pan-Islamism. It was a Roustan[8] policy over again, they said. I did my best to smooth down matters with them till the explanation should arrive which Malet had promised us; but I found my efforts useless. It was an alarming three weeks for us all, from the delivery of the Note till Gambetta's fall. News came that a French force was being assembled for embarkation at Toulon, and that was the form of intervention generally expected. Indeed, I think it is not too much to say that Gambetta's resignation on 31st January alone saved Egypt from the misfortune, even greater perhaps than what afterwards befell her, of a French invasion avowedly anti- Mohammedan and in purely European interests.
- 72. FOOTNOTES: [7] Sir William Gregory, who saw Arabi about the same date as I did, has recorded in the Times very similar language as used by him. [8] Roustan was the French diplomatist at Tunis who had engineered the French designs on the Regency.
- 73. CHAPTER IX FALL OF SHERIF PASHA The political crisis at Cairo, by the middle of January, was evidently approaching fast. Indeed it had become inevitable. The publication of the Joint Note happened to coincide with the drafting of the new Leyha or Organic Law, which was to define the power of the Representative Chamber in the promised Parliament. In regard to this, the Financial Controllers had been insisting with the Ministry that the power they had been exercising for the last two years of drawing up the yearly Budget, according to their own view of the economic requirements of the country, should remain intact, that is to say, that it should not be subject to discussion or a vote in the Chamber; and to this Sherif Pasha had agreed, and had already drafted his project of law without assigning to the Chamber any right in money matters. The majority of the delegates, however, were not unnaturally dissatisfied at this, arguing that the Foreign Financial Control, having its sole status in the country as guardian of the foreign obligations, and as the interest on the debt amounted only to one-half of the revenue, the remaining half ought to be at the disposal of the nation. Nevertheless, there is no reason to suppose that the point would not have been conceded by them, especially as Sultan Pasha, who had been named their President, was with Sherif in considering it prudent to yield, had things remained during the month as they were at the beginning. It has been seen how readily the War Office had come to terms with the Controllers in the matter of the Army Estimates. Now, however, under the menace of the Note, the Notables were no longer in a mood of conciliation, and met Sherif's draft with a counterdraft of their own, adding a number of new articles to the Leyha, largely extending the Parliamentary powers,
- 74. and subjecting the half of the Budget not affected to the interest of the debt to vote by the Chamber. This brought the Controllers into active conflict with them, M. de Blignières taking the lead in it and bringing Colvin into line with him. The Controllers declared it absolutely necessary that the Budget should remain whole and undivided in their hands, and denounced the counter-draft as being a project, not of a Parliament, but of a Convention. The phrase, founded on memories of the French Revolution, was doubtless de Blignières', but it was adopted by Colvin, and pressed by him on Malet. The dispute was a serious one, and might lead to just such mischief as Malet feared, and give excuse to the French Government for the intervention it was seeking. Sherif having already committed himself to the Controllers' view, was being persuaded by them to stand firm, and the Khedive's attitude was doubtful. A quarrel between the Khedive and his Parliament on a financial question involving European bondholding interests was just such a case as the French Government—for Gambetta was still in office—might be expected to take advantage of for harm. In this emergency Malet—and Colvin, who though he wished to get his way as Financial Controller had no mind for French intervention— joined in asking me yet once again to help them, and to make a last effort to induce the extreme party among the Notables to yield something of their pretensions, and after consultation with Sheykh Mohammed Abdu, who as usual was for prudence and conciliation, it was arranged that I should have a private conference at his house with a deputation from them, and argue the case with them, and show them the probable consequences of their resistance—namely, armed intervention. Accordingly, I got up the case of the Controllers with Colvin, and drew up with Malet the different points of the argument I was to use. These I have by me in a paper headed, Notes of what I have to say to the Members of the Egyptian Parliament, 17th January, 1882. According to this my instructions were to represent to the Members of the Deputation that the existing procedure respecting the Budget
- 75. was an international affair, which neither Sherif nor the Parliament had any right to touch without gaining the consent of the two controlling Governments. I was to recite the history of the Control's establishment, and show them a private Note which had been appended by Malet and Monge (the French Consul-General), 15th November, 1879, to the Decree instituting it. I was to invite the members to consider whether an alteration in the form of determining the Budget was not an international matter, and, as such, outside the sphere of their action. They had admitted that international matters must be left untouched by them. The control of the Budget was an international matter. Therefore it should be left untouched by them. I was, however, authorized by Colvin to say that personally he had no objection to a slight modification of the present arrangement, such as should give the Parliament a consultative voice which might later become a right of voting. Should they accept such a compromise, Malet would represent the matter favourably to his Government, though he had no authority to promise its acceptance by France or England. All other differences with Sherif they must settle with him themselves, etc., etc. On this basis, with Sabunji's help and Mohammed Abdu's, I argued the case thoroughly with them, and convinced myself that there was no possibility of their yielding. They agreed, indeed, to modify three or four of the articles which the Controllers had principally objected to as giving the Chamber powers of a Convention, and the amendments I proposed in these were in fact incorporated later in the published Leyha. But on the Article of the Budget they were quite obdurate, notwithstanding the support Sheykh Mohammed Abdu gave me. They would not yield a line of it, and I returned crestfallen to report my failure, nor did I again undertake any mission of mediation between Malet and the Nationalists. I had done my best to help him to a peaceful solution of his difficulties, but our points of view from this time forth became too divergent for me any longer to be able to work with him. Although I had done my very best to persuade the Notables to give way—for I was then firmly convinced that they were menaced with intervention—I could not
- 76. help in my inner mind agreeing with them in their claim of controlling the free half of the Budget as a sound one, if Parliamentary Government was to be a reality for them, not a sham. Malet's despatches of the time show that they were all of one mind on this point, and even Sultan Pasha, who was a timid man and easily frightened, declared roundly that Sherif's draft was like a drum; it made a great sound but was hollow inside. As between Sherif and the Notables in the quarrel which followed, my anti- Turkish sympathies put me on their side rather than on his. At Malet's suggestion I had a little before called on Sherif and had discussed the matter with him, and had been unfavourably impressed. Sherif was a Europeanized Turk of good breeding and excellent manners, but with all that arrogant contempt of the fellahin which distinguished his class in Egypt. Malet had a high opinion of him because he was a good French scholar and so was easy to deal with in the ordinary diplomatic way, but to me he showed himself for this very reason in disagreeable contrast with the sincere and high- minded men who were the real backbone of the National movement, and for whom he expressed nothing but the superior scorn of a fine French gentleman. He was cheerfully convinced of his own fitness to govern them and of their incapacity. The Egyptians, he told me, are children and must be treated like children. I have offered them a Constitution which is good enough for them, and if they are not content with it they must do without one. It was I who created the National Party, and they will find that they cannot get on without me. These peasants want guidance. When, therefore, a fortnight later the quarrel became an open one between him and them I had no difficulty in deciding which way my sympathies lay. I was no longer at Cairo when the news of Sherif's resignation on the 2nd of February reached me. The failure of my negotiation, just described, with the Notables, had depressed my spirits. I felt that by undertaking it I had risked much of my popularity with my European friends, and that they perhaps distrusted me for the pains I had
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