Identification of the dynamic characteristics of nonlinear structures

iv
ACKNOWLEDGMENTS
The author is most grateful to his supervisor, Prof. D. J. Ewins, for
his sustained encouragement, interest, stimulus and guidance
given over the period in which the research work presented
herein has been carried out.
Thanks are also due to the members of the Dynamics Section,
both past and present, especially to Dr. J. He and Mr. D. A. Robb
for their valuable help given in computing and conducting
experiments.
For the friendly cooperation and useful discussions throughout
the period of research, the author wishes to express his gratitude
to many past and present colleagues in the Dynamics Section, in
particular to Waiming To, Tom Pialucha and 
Nobari.
Finally, the author is indebted to the British Council and the
Chinese Government for providing the financial support.

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V
NOMENCLATURE
a
b
DC
E
constant (real)
real and imaginary parts of modal constant of 
mode (real)
real and imaginary parts of modal constant of 
mode which are
functions of vibration amplitude (real)
system 
mass matrix (real)
modal constant of 
mode of 
(complex)
constant (real)
coefficient vector (real)
system generalised stiffness matrix (real)
constant (real)
eigenvector scaling factor (complex)
viscous damping matrix (real)
hysteretic damping matrix
Jacobian matrix of vector field f(x) at 
(real)
Jacobian matrix of map P at (real)
capacity dimension of an attractor (real)
Young’s modulus (real)
a vector with its element unity and all the others zeros (real)
coefficient matrix (real)
force signal (real)
Fourier transform of force signal f(t) (complex)
restoring force (real)
general nonlinear function (real)
Wiener G-functional (real)

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vi
I
k
L
m
coefficient matrix (real)
frequency response function of linear systems (complex)
Volterra kernel (real)
Volterra kernel transform (Volterra transfer function) (complex)
Volterra kernel transform corresponding to excitation at coordinate
and response at coordinate (complex)
receptance-like Volterra kernel transform (complex)
mobility-like Volterra kernel transform (complex)
first-order frequency response function of
nonlinear systems (complex)
frequency response function defined as the
measured 
or 
(complex)
complex notation
current (real)
unit matrix (real)
constant (real or integer)
Wiener kernel (real)
derived Wiener kernel which is of 
(real)
expanded 
extensional and bending element
stiffness matrices (real)
Wiener kernel transform (complex)
system stiffness matrix (real)
analytical stiffness matrix (real)
stiffness error matrix (real)
stiffness error matrix due to nonlinearity (real)
number of independent design variables (integer)
number of measured or calculated modes (integer)
expanded 
element mass and stiffness matrices (real)

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vii
N
P
AR, AI
system mass matrix (real)
analytical mass matrix (real)
mass error matrix (real)
mass and stiffness matrices corresponding to
master (measured) 
(real)
mass and stiffness matrices corresponding to master
(measured) and slave (unmeasured) 
(real)
mass and stiffness matrices corresponding to slave 
(real)
dynamically or statically condensed mass and
stiffness matrices (real)
number of measured coordinates (integer)
number of degrees of freedom specified in the
analytical model (integer)
describing function coefficient (complex)
real and imaginary part of 
(real)
null matrices (real)
vector of displacement and acceleration of
principal coordinates (real)
mapping function (real)
probability function (real)
vector of design variable change (real)
polynomial function of (real)
real part and imaginary part of 
(real)
real and imaginary part of the residue term of 
mode
coefficient vector (real)
residual matrix representing the contribution of unmeasured
or uncalculated higher modes (complex)
first-order eigensensitivity coefficient (real)

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t
V
x(t)
 
auto-spectrum of force signal (real)
cross-spectrum of force and response signals (complex)
auto-spectrum of response signal (real)
eigensensitivity coefficient matrix (real)
time parameter (real)
Volterra functional (real)
coefficient vector (real)
left singular vector matrix (real or complex)
coefficient matrix (complex)
voltage (real)
coefficient vector (complex)
right singular vector matrix (real or complex)
Vandermonde matrix (real)
response signal (real)
equilibrium point (real)
fixed point of map P (real)
Fourier transform of response signal x(t) (complex)
vibration amplitude of certain coordinate (real)
vector of displacement and acceleration (real)
vector of displacement in state-space (real)
system impedance matrix (complex)
analytical impedance matrix (complex)
experimental impedance matrix (complex)
impedance matrix corresponding to master
(measured) 
(complex)
impedance matrix corresponding to master (measured)
and slave (unmeasured) 
(complex)
impedance matrix corresponding to slave 
(complex)

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impedance error matrix (complex)
receptance element and 
(complex)
{a,(o)} 
analytical receptance (complex)
experimental receptance (complex)
receptance error matrix defined as 
[a,(o)] (complex)
receptance matrix corresponding to the contribution of
calculated or measured modes (complex)
receptance matrix of updated analytical model (complex)
receptance corresponding to master (measured) 
(complex)
receptance corresponding to slave (unmeasured) 
(complex)
receptance matrix corresponding to master
(measured) 
(complex)
receptance matrix corresponding to master (measured)
and slave (unmeasured) 
(complex)
receptance matrix corresponding to slave 
(complex)
constant (real)
element of 
modeshape 
(real)
undamped 
modeshape and modeshape matrix (real)
flow which starts at x (real)
modeshape corresponding to master 
(complex)
modeshape corresponding to slave 
(complex)
analytical (real) and experimental (complex) modeshape man-ices
constant (real)
damping coefficient of 
mode (real)
respone-amplitude-dependent damping coefficient
of 
mode (real)
error on damping coefficient estimate (real)

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X
a
hysteretically damped 
modeshape and
modeshape matrix (complex)
eigenvalue matrix (complex)
analytical (real) and experimental (complex) eigenvalue matrices
eigenvalue and eigenvector of lower response level (complex)
eigenvalue and eigenvector of higher response level (complex)
eigenvalue and eigenvector derivatives (real)
coefficient matrix (complex)
density and probability density function (real)
singular value of a matrix (real)
constant coefficient vector (real)
singular value matrix (real)
time parameter (real)
circular frequency (real)
natural frequency of 
mode (real)
respone-amplitude-dependent natural frequency of 
mode (real)
error on the estimate of (real)
damping loss factor (real)
viscously damped 
modeshape and
modeshape matrix (complex)