Reprinted from December
1987,
Vol.
109,
Journal
of
Applied Mechanics
S.
F.
Masri
R.
K.
Miller
A.
F.
Saud
Department
of
Civil
Engineering,
University
of
Southern California,
Los Angeles,
CA
90089-0242
T.
K.
Caughey
Division
of
Engineering and Applied Science,
California Institute
of
Technology,
Pasadena,
CA
91 106
1
Introduction
Identification of Nonlinear
Vibrating Structures:
Part I-Formulation
A
self-starting multistage, time-domain procedure is presented for the identification
of nonlinear, multi-degree-of-freedom systems undergoing free oscillations or sub-
jected to arbitrary direct force excitations
and/or nonuniform support motions.
Recursive least-squares parameter estimation methods combined with
non-
parametric identification techniques are used to represent, with sufficient accuracy,
the identified system in a form that allows the convenient prediction of its transient
response under excitations that differ from the test signals. The utility of this
pro-
ced~lre is demonstrated in a companion paper.
1.1
Background.
The identification and modeling of
nonlinear multidegree-of-freedom (MDOF) dynamic systems
through the use of experimental data is a problem of con-
siderable importance in the applied mechanics area. Since the
model structure in many practical dynamics problems is by no
means clear, an increasing amount of attention has recently
been devoted to nonparametric identification methods.
One rather general nonparametric nonlinear identification
approach is based on the expansion of the nonlinear restoring
force functions in a power series or generalized Fourier series
involving orthogonal polynomial functions. In applications, it
is generally assumed that such series are rapidly convergent so
that only a few terms need be retained for identification pur-
poses. In such an approach, the coefficients of the retained
terms from the series become parameters of the system which
may be identified by many well-known techniques, such as
least-squares fit in the time domain.
The origins of this basic approach are very classical and
diverse, with roots in the theory of analytic functions and in
the theory of Fourier series, and with applications in many
engineering disciplines as well as operations research,
economics, and the physical sciences. With regard to the
engineering literature, the basic approach is outlined in the
book by Graupe (1976). Applications of the method in the
mechanical sciences appear to have originated in the early
1950's in several NACA technical notes (Greenberg, 1951;
Shinbrot, 1951; Shinbrot, 1952; Briggs and Jones, 1953; and
Shinbrot, 1954) and in the papers by Klotter (1953) and
Shin-
brot (1957). In the following years, interest in similar time
series methods for nonlinear system identification of struc-
tures expanded, as attested to by the representative
publica-
Contributed
by the Appl~ed Mechanics Div~slon for publ~cat~on in the JOUR-
NAL
OF
APPLIED MECHANICS.
Discussion on this paper should be addressed to the
Ed~tor~al Department,
ASME, Un~ted Englneerlng Center, 345 East 47th Street, New York,
N
Y
10017, and will be accepted untll two months after final pubhcat~on of the paper
Itself In the JOURNAL
OF
APPLLED MECHANICS.
Manuscript
recelved
by
ASME
Appl~ed Mechanics Dlvis~on, January
13,
1987, final revlsion June
23,
1987
tions of Kohr (1963), Hoberock and Kohr (1967), Sprague and
Kohr
(1969), Sehitoglu and Klein (1975), Masri et al. (1982),
Natke (1982), Masri et al. (1984), Tomlinson (1985), and Hac
and Spanos (1987).
Most of the research in this area has been concerned with
SDOF systems with
nonlinearities of varying complexity. The
basic identification method becomes generally impractical for
complex MDOF systems due to excessive computation and
computer memory requirements caused by slow convergence
of the series expansions. However, Masri et al. (1982)
demonstrated by example that rapid series convergence (and
hence practical identification results) may be obtained in at
least some MDOF
stru~tural applications by basing the iden-
tification procedure on a set of generalized coordinates cor-
responding to the mode shapes of a comparison linear struc-
tural system.
In the paper by Masri et al.
(1982), certain restrictions were
made on the class of nonlinear structural systems to be iden-
tified. In particular, it was assumed that (1) the system mass
matrix
M
is diagonal and known; (2) the equivalent linear
system stiffness matrix
K
is symmetric and known; and (3) the
excitation to the system is furnished through forces directly
applied to the discrete mass locations. The requirement of
knowing the linearized system parameters pertaining to
M
and
K,
as well as the exclusion of the class of problems involving
support motion (such as in the case of earthquake ground mo-
tion), limited the utility of the approach in practical cases.
The present paper further extends the above-referenced
work by generalizing the approach to handle, approximately,
the case of arbitrary nonlinear MDOF dynamic systems with
multiple inputs and outputs under the action of force excita-
tions and/or nonuniform support motion. The method is
based on the use of time-domain estimation techniques to
identify the parameters of an equivalent linear model whose
eigenvectors are then used to estimate the "modes" of the
nonlinear system. Regression techniques involving the use of
two-dimensional orthogonal functions are then employed to
develop an approximate expression for the system generalized
9181Vol.
54,
DECEMBER
1987
Transactions of
the
ASME
Direct excitation
/
t,(t)
System
H
Response
x-,,(t)
Interface
motion
Fig.
1
Model of system
restoring forces in terms of the corresponding generalized
system state variables.
Section
2
of this paper extends the work of previous in-
vestigators by presenting a unified approach for handling the
time-domain identification of the system matrices associated
with a variety of classes of linear problems arising in the field
of structural dynamics. The formulation under discussion in-
cludes the cases of free vibrations as well as direct force
and/or independent support motion.
Section
3
incorporates the results of Section
2
in the iden-
tification of nonlinear vibrating structures. The "calibration"
of this approach is accomplished in the companion paper
(Masri et al.,
1987)
by applying the method under discussion
to a representative
multi-input/multi-output
nonlinear system
incorporating polynomial as well as hysteretic nonlinearities.
1.2
Formulation of Time Domain Identification
Pro-
cedure.
Consider a discrete multi-degree-of-freedom
(MDOF) system of the type shown in Fig.
1,
which is subjected
to directly applied excitation forces
f,
(t)
as well as prescribed
support motions
x, (t)
.
The motion of this multi-input/multi-
output nonlinear system is governed by the set of equations
fT(x,x,x)=f1 (t)
(1)
where:
fT
=
an
n,
column vector representing the total sum of
all the inertia and restoring forces acting on the
system,
f,
(t)
=
an
n,
column vector of directly applied forces,
x(t)
=
(x,(t), ~,(t))~
=
system displacement vector of
order
(n,
+
no),
x, (t)
=
active degree-of-freedom (DOE) displacement vec-
tor of order
n,,
x, (t)
=
prescribed support displacement vector of order
no.
Let
fT
(t)
be expressed as
where
Lfl
(t) =Mllxl
(f)
f
C1lXi (t) +Kllxl
(f),
(3)
Sf,
(t) =Mflxl (t) + Cf,x, (t) +Kflxl (t),
(4)
LfO(f)
=M10~0(f) +CIO~O(~) +K~oxo(~), (5)
'fo
(t) =Mfoxo(t)
+
Cfoio (t) +Kfoxo (t),
(6)
MI,, Cll
,
K,,
=
constant matrices that characterize the iner-
tia, damping, and stiffness forces
associated with the unconstrained
DOE of
the system, each of order
n,
x
n,
,
Mf,, Cf,, Kf,
=
response-dependent matrices that
characterize the inertia, damping, and stiff-
ness forces associated with the uncon-
strained
DOE of the system, each of order
n,
x
nl,
MI,, C,,,
K,,
=
constant matrices that characterize the iner-
tia, damping, and stiffness forces
associated with the support motions, each
of order
n,
x
no,
Mfo, Cfo, Kf0
=
response-dependent matrices that
characterize the inertia, damping, and stiff-
ness forces associated with the support mo-
tions, each of order
n,
x
no,
Lf,
(t)
=
an
n,
column vector of linear forces involv-
ing
XI
(t),
Sf1
(t)
=
an
n,
column vector of response-dependent
forces involving
x, (t)
,
Lfo
(t)
=
an
n,
column vector of linear forces involv-
ing
xo
(
t)
,
Sfo
(t)
=
an
n,
column vector of response-dependent
forces involving
x, (t)
as well as
xo
(t),
f,(t)
=
an
n,
column vector of nonlinear non-
conservative forces involving
x, (t)
as well
as
xo (t).
Making use of equation
(2),
the system equation of motion
(1)
can be expressed as
M:,x, (t) +CT,x, (t) +KT,x, (t)
+
M~oxo(t)+Cf~x~(t)+KToxo(t)+fN(t)
=fl
(t),
(7)
where:
MT1 =MI, +Mfl,
MTo
=
M1o
f
Mfo
,
CC;1
=
Cli
+
GI
9
CTo
=
Clo
+
C&,
(8)
KT, =K,l +Kfl, KTo=Kl0+K%,
This study is concerned with a time-domain method for the
identification of the system matrices appearing in equation
(7)
as well as the nonlinear forces acting on the system. The
representation of the identifed system will be in a form that
allows the prediction of its transient response under arbitrary
excitations, by using conventional numerical techniques for
initial-value problems in ordinary differential equations.
Note that equation
(7)
can be expressed as
xi (t)
=[~:,i-~(f,
(t)
-feet)
-f,ct)),
(9)
where:
fm
=bl(t)+bo(t), (10)
b1 (t) =C:I~I(~) +KT,x, (t),
(1 1)
bo
(1)
=MToxo (t)
+
CToxo (t) +KToxg(t).
(12)
Thus, by introducing the state vector
y
of order
2n,
where
Y2r-I
=XI!
(13)
y2!
=xlI
i=
1,
2,
.
.
.
,
n,
(14)
standard time-marching techniques can be used to solve
2
Time-Domain Identification of Linear System
Matrices
The use of least-squares methods to estimate unknown
parameters is a well known and developed approach which oc-
cupies significant portions of numerous books devoted to the
subject of parameter estimation, particularly in the field of
electrical engineering control and system theory
(Mendel,
1973;
Graupe,
1976;
Hsia,
1977;
Sorenson,
1980).
While this
approach has also been frequently applied in the field of struc-
tural dynamics
(Caravani and Thomson,
1974, 1977;
Ibrahim
and
Mikulcik,
1973, 1976, 1977;
Ibrahim and Pappa,
1982;
Ibrahim,
1977, 1978, 1983;
Junkins,
1978;
Beck and Jennings,
1980;
Yao,
1985;
Torkamani and Hart,
1975;
Shinozuka et
al.,
1982;
Rajaram and Junkins,
1985;
Hac and Spanos,
1987),
there is a paucity of studies that are concerned with the
problems encountered by this approach when applied to
realistic problems arising in the vibration field. Consequently,
the present section of this paper is devoted to presenting an in-
depth, unified, and efficient approach for using least-squares
Journal
of
Applied Mechanics
DECEMBER
1987,
Vol.
541919
parameter estimation methods to identify the needed system
matrices associated with a wide variety of realistic situations
commonly encountered when dealing with experimental
measurements of vibrating structures.
2.1
Formulation.
Consider a linearized version of the
system shown in Fig.
1,
and assume it is governed by
MIIXI (t)
+
CII~~ (t) +KIIx~ (t)
+
Mloio(t) +C~oxo(t) +K,oxo(t) =fl (t).
(16)
Let the response vector
r (t)
of order
3(nl
+
no)
be defined as
r(t)
=
(xF(t), xF(t), x:(t), x:(t), xT(t), ~;(t))~. (17)
For clarity of presentation, let the six matrices appearing in
equation
(16)
be denoted by
'A, ,A, . . .
,
6A,
respectively.
Let
(JA,)
=
ith row of a generic matrix
'A,
and introduce
the parameter vector
a,.
a,= (('A,), (2A,), (3A,), (4A,), (5AI), (6AI))T. (18)
Suppose that the excitation and the response of the system
governed by equation
(16)
is measured at times
t,
,
t,, . . .
,
t,.
Then at every
t,,
Introducing matrix
R
to the corresponding formulation in equation
(21),
the former
suffers from a significant (practical) limitation pertaining to
the number of system
DOFs simultaneously excited.
2.2.2
Recursive Solutions.
Suppose that a set of
m
equa-
tions
A
A
R,&= b'k' (26)
has been used to obtain a weighted least-squares estimate for
h,
denoted by
kck):
h'"
=
(R;
wkRk)
-lR;~,^b(~).
(27)
Using an additional set of relations
a new estimate of
&,
denoted by
&(k+'),
can be obtained
yitho~t reprocessing the whole set of equations involving
(b(,), b(,+l))
(Brogan,
1985).
2.3
Special Cases.
In the work of Masri et al.
(1987a),
special cases that influence the application of the method in
practical situations are discussed and steps are provided for
alleviating some of the problems appearing in realistic cases.
Among these topics are the uniqueness issues, partial
knowledge of system parameters, conditions under which the
approach fails to yield desired results, symmetry assumptions,
and recursive approaches to enhance computational
efficiency.
and using the notation above, the grouping of the
measurements can be expressed concisely as
R&=^b (21)
where
R
is
a
block-diagonal matrix whose diagonal elements
are equal to
R,
&
=
(a?, a:, .
.
.
,
and
b
is the cor-
responding vector of
excjtation measurements.
Keeping in mind that
R
is of order
rn
x n
where
m
=
Nn,,
and
n
=
3n, (n,
+
no),
then if a sufficient number of
measurements is taken, this will result in
m
>
n.
Under these
conditions, least-squares procedures can be used to solve for
all the system parameters that constitute the entries in
h:
&=$^b (22)
where
kt
is the pseudoinverse of
R
(Golub and Van Loan,
1983).
Using the weighted least-squares approximation to
minimize the cost function,
J,
results in the approximate solu-
tion
&=
(RTWR)
-1R~h,
.
(23)
where
W
is the error weighting matrix.
2.2
Computational Efficiency.
2.2.1
Decoupling.
One way to reduce the order of equa-
tion
(21)
to a manageable level is by making use of the
diagonal nature of partitioned matrix
R,
thus resulting in a set
of
n
,
decoupled matrix equations each of the form
Comparing the order of
R
in equation
(24)
with that of
k
in
equation
(21),
shows-that the order of
R
is smaller by a factor
of
n!
compared to
R.
Least-squares techniques can again be
used to obtain the components of the
n,
parameter vectors
a,:
hi=~+^bi; i=1,2,.
. .
,n,. (25)
Note that
R?
needs to be computed only
once.
While the formulation in equation
(24)
is obviously superior
3
Identification of Nonlinear Systems
Consider the nonlinear system governed by equation
(1)
and
assume that the identification procedure discussed in Section
2
has yielded the system matrices needed to determine the
equivalent linear internal force vector
f;(t)
appearing in
equation
(9)
and defined by equation
(10).
3.1
Restoring Force Estimation.
Solving equation
(7)
for
the nonlinear force vector
f,
(t)
results in
fN(t) =fl (t)
-
(M:~x~
(t) +fe(t)). (29)
Since all the terms appearing on the right-hand side of equa-
tion
(29)
are available from measurements or have been
previously identified, the time history of
f,
can be deter-
mined. Note from equation
(29)
that
f,(t)
can be interpreted
as the residual force vector corresponding to the difference
between the excitation vector
f,
(t)
and the equivalent linear
force vector composed of the inertia, damping, and the stiff-
ness terms.
An alternative form of equation
(29)
is
fR(t) 'fN(t)
+fz(f)
=fl(f) -Mrl~l(f),
(30)
where
fR
(t)
represents the difference between the excitation
and equivalent linear inertia forces associated with the active
degrees of freedom. The force
fR
can be thought of as the
"restoring force" of the system.
Let
h,(t)
represent either the ith component of the
nonlinear residual force vector
f,(t)
as defined by equation
(29)
or the restoring force vector
fR (t)
as defined by equation
(30).
In general, vector
h
depends simultaneously on all the
components of the system acceleration, velocity, and displace-
ment vectors associated with the
n,
active DOE as well as the
no
support components:
h (t)
=
h(x,
x,
x).
(3
1)
The central idea of the present method is that, in the case of
nonlinear dynamic systems commonly encountered in the ap-
plied mechanics field, a judicious assumption is that each
component of
h
can be expressed in terms of a series of the
form:
9201Vol.
54,
DECEMBER
1987
Transactions of
the
ASME
where the
v,'s
and
v2's
are suitable generalized coordinates
which, in turn, are linear combinations of the physical
displacements, velocities, and accelerations. The approxima-
tion indicated in equation
(32)
is that each component
h,
of
the
nonlinear force vector
h
can be adequately estimated by a
collection of terms
h:),
each one of which involves a pair of
generalized coordinates. The particular choice of combina-
tions and permutations of the generalized coordinates and the
number of terms
J,,,
needed for a given
h,
depends on the
nature and extent of the nonlinearity of the system and its ef-
fect on the specific DOF
i.
3.2
Eigenvector Expansion.
If
h,
(t)
is chosen as the ith
component of
f,(t),
then the procedure expressed by equa-
tion
(32)
will directly estimate the corresponding component
of the unknown nonlinear force. For certain structural con-
figurations
(e.g., localized nonlinearities) and/or relatively
low-order systems, the choice of suitable generalized coor-
dinates for the series in equation
(32)
is a relatively straightfor-
ward task. However, in many practical cases involving
distributed nonlinearities coupled with a relatively high-order
system, an improved rate of convergence of the series in equa-
tion
(32)
can be achieved by performing the least-squares fit of
the nonlinear forces in the "modal" domain as outlined
below.
Using the identification results for the
linear system dis-
cussed in Section
2,
the eigenvalue problem associated
with
MfilK,,
is solved resulting in the eigenvector or modal
matrix and the corresponding vector of generalized coor-
dinates
u:
h,(u,
u)=
aTf,(t)
with
With this formulation in mind, equation
(32)
can be viewed as
allowing for "modal" interaction between all generalized
coordinates, taken two at a time. Note that the formulation in
equation
(32)
allows for "modal" interaction between all
"modal" displacements, velocities, and accelerations.
3.3
Series Expansion.
The individual terms appearing in
the series expansion of equation
(32)
may be evaluated by us-
ing the least-squares approach to
de~ermine the optimum fit
for the time history of each
h,.
Thus
h!')
may be expressed as a
double series involving a suitable choice of generalized coor-
dinates:
where the
C,,'s are a set of undetermined constants and
Tk(.)
are suitable basis functions, such as orthogonal polynomials.
Let
h?),
the deviation (residual) error between
h,
and its first
estimate
hjl),
be given by
Equation
(32)
accounts for the contribution to the nonlinear
force
h,
of generalized coordinates
v\t)
and
viy
appearing in
the form
(v(:))~
(v$:))~.
Consequently, the residual error as
defined by equation
(32)
can be further reduced by fitting
hy)
by a similar double series involving variables
v(:)
and
v$):
h
j2)(x,
k,
i)
=
hj2)(v(:), v@)
(37)
where
By extending this procedure to account for all
DOFs that
have significant interaction with DOF
i,
equation
(32)
is ob-
tained with
Journal
of
Applied Mechanics
h:+ l)(x,
k,
2)
=
h:) (x,
x,
x)
-
h:)
(ve), ut));
j=
1,2,
.
-
.
,
Jmaxl (39)
where
h;l)(x,
x,
x)
=
hi (x,
x,
x),
(40)
and
Note that, in general, the range of the summation indices
k
and tappearing in equation
(41)
may vary with the series index
j
and DOF index
i.
Similarly,
JmaXl,
the total number of series
terms needed to achieve a given level of accuracy in fitting the
nonlinear force time history, depends on the DOF index
i.
3.4
Least Squares Fit for Nonlinear Forces.
Using two-
dimensional orthogonal polynomials
Tk(.)
to estimate each
h,
(x,
x.
x)
by a series of approximating functions
hl(J)
of the
form indicated in equation
(41),
then the numerical value of
the
C,,
coefficients can be determined
by
invoking the ap-
plicable orthogonality conditions for the chosen polynomials.
While there is a wide choice of suitable basis functions for
least-squares application, the orthogonal nature of the
Chebyshev polynomials and their "equal-ripple"
characteristics
make
them convenient to
use
in the present
work.
Let each generalized coordinate
v
appearing in equation
(32)
be normalized to lie in the range
-
1
to
1:
U'
=
tu-
(~max
+
~min)/2l/t(umax
-
vmin)/21 (42)
If the Chebyshev polynomials, given by
Tn(E)=cos(ncos-l~), -1<[<1
(43)
and satisfying the weighted orthogonality property
where
w
(x)
=
(1
-
x~)-"~
is the weighting function, are used,
then the
Ck,
coefficients would be given by
r
(2/3~)~S,,
k
and 1r0
[
(r2)Se,
K
and
1=
0
where
and
Note that in the special case when no cross-product terms
are involved in any of the series terms, functions
h
can be ex-
pressed as the sum of two one-dimensional orthogonal
polynomial series instead of a single two-dimensional series of
the type under discussion.
4
Summary and Conclusions
An approximate time-domain method is presented for the
identification of nonlinear multi-degree-of-freedom systems
subjected to arbitrary direct force excitations and/or
not-
necessarily-identical support motions. This self-starting
method uses recursive least-squares parameter estimation
methods, combined wtih nonparametric identification techni-
ques, to generate a reduced-order nonlinear mathematical
model suitable for use in subsequent studies to predict, with
DECEMBER
1987,
Vol.
54
1921
good fidelity, the response of the test article under arbitrary
dynamic excitations. The utility of this procedure is
demonstrated in a companion paper.
Acknowledgment
This study was supported in part by a grant from the Na-
tional Science Foundation and the National Institute of
Health Biomedical Simulations Resource at the University of
Southern California.
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