ABSTRACT
A technique for identifying the modal properties of an elastic
structure in a testing laboratory is presented. The technique is
based upon the use of digital processing and the fast Fourier
transform (FFT) to obtain transfer function data, and then the use
of a least squared error estimator to identify modal properties from
the transfer function data. Both analytical and experimental results
are presented.
INTRODUCTION
In recent years the implementation of the fast Fourier transform
(FFT) in low cost minicomputer systems has provided the
environmental testing laboratory with a faster and more powerful
tool for acquisition and analysis of vibration data. from mechanical
structures. The results are used by analysts and designers alike as
an aid to better understanding and improving mechanical designs.
In this paper an analytical technique is presented which as been
implemented in a Fourier Analyzer to provide modal data on site in a
testing laboratory. The technique is based upon the application of a
least squares estimator to measured transfer function data. During
the process the natural frequencies, damping factors, and mode
shapes of all the predominant modes of vibration of a structure are
identified.
A brief review of the modal theory and a derivation of the
analytical form used in the estimation process are given in the
following section. Following that is a discussion of how parameters
are obtained from a single transfer function, and some experimental
results are given. Lastly the global nature of a mode is discussed
and verified with experimental results.
MODAL THEORY
Assume that an elastic system has ndegrees of freedom and
that its motion can be adequately described by nlinear
differential equations with constant coefficients. written as
M d2x(t)/dt2 + C dx/dt(t) + Kx(t) = ft)
(1)
where x(t) and f(t) are displacement and force nvectors
respectively, and M, C, and K are real symmetric matrices. M is
called the mass matrix, C the damping matrix, and K the stiffness
matrix.
Taking the Laplace transform of equation (1) gives
where X(s) <> x(t) and F(s) <>
f(t) are vector Laplace transform pairs. B is defined as the (n x n)
system matrix. Eq. (2) is often referred to as an expression of the
dynamic flexibility of the structure.
Eigenvectors (yk) and eigenvalues () of the matrix can be defined in the
usual way, i.e. to satisfy the equation
where yk is an nvector and is a constant. The system Matrix B
has (n) eigenvalues and (n) eigenvectors; each eigenvaluevector
pair is defined by equation (3).
It is straightforward to show that the yk eigenvectors
are orthogonal, provided all values are different, as follows:
For two different eiqenvalues (k) and (j)
Note that equation (5) can be rewritten as
since B is symmetric. (The superscript t denotes the transpose
operation.) Premultiplying equation (4) by
ytj. and post multiplying equation (6) by
yk
so
Thus for not equal , ytj
yk = 0 which defines orthogonality between two
eigenvectors.
As usual, the eigenvalues can
be expressed as roots of the determinant
where I is an (n x n) identity matrix, and
each value can be found by
solving the polynomial equation defined by (9).
We define a transformation matrix as
where the n columns of are
the eigenvectors yk. We also define a diagonal matrix
of eigenvalues as
Then, the above definition of eigenvalues and eigenvectors can be
expressed in matrix form as
By defining the generalized inverse of as
equation (12) can be rewritten as
If the eigenvectors are normalized to unit magnitude, so that
= I, then
=. In any case, is
an (n x n) diagonal form of B. The general form Dt BD is
called a congruence transformation, and if the columns of D are
orthogonal (so that -DtD is diagonal), it is called an
orthogonal transformation. Thus,
B =
represents an orthogonal diagonalization of B. If the eigenvalues of
B are unique, then the eigenvectors are also unique except for an
arbitrary normalization constant, so this orthogonal diagonalization
of B must be unique to the same extent.
The transfer function matrix H of this linear system (1) is
defined as
assuming that the indicated matrix inverses exist.
We can also write
Thus, is
the orthogonal diagonalization of H. which is unique except for
normalization constants. Note that both B and H are diagonalized by
the same orthogonal transformation.
Recall that the elements of B are quadratic functions of s. Both
the eigenvalues and the
eigenvector components yk are generally rather
complicated (usually irrational) functions of s. This means that the
eigenvector components in the time domain are each changing in some
complicated way with respect to one another, and that each
corresponding eigenfunction (time domain representation of )
is a complicated time waveform. The only real advantage to this
formulation is that each eigenvector distribution is orthogonal with
respect to all other eigenvectors.
It is preferable to decompose B or H into a set of time invariant
vectors (independent of s), and put all s dependence into some
diagonal representation of the system or transfer matrix. Practical
experience indicates that this possibility exists, i.e. physical
structures exhibit "standing wave" vibration patterns at certain
frequencies, in which a "global" vibration mode shape is associated
with each "resonant" frequency. We are further encouraged by the
fact that the solution of the homogeneous wave equation can be
expressed as the product of a time function and a space function.
Finally the driving function can be decomposed into a linear
combination of these homogeneous solutions, and the complete
solution obtained in terms of linear combinations of these
homogeneous "eigenfunctions". It should be apparent that the key to
this desired decomposition of B lies in the solution to the
homogeneous equation Bu = 0.
It is now shown how this homogeneous equation can be solved in
terms of the previously defined eigenvalues and eigenvectors of B.
We begin by recognizing that each element of H = B-1 is a
rational fraction in s, with denominator given by detail.
Thus, the roots of this denominator, called the poles of H. are the
values of s = sk for which det|B| = 0. These values of s
also satisfy the above homogeneous equation Bu = 0.
Each element of H can be expanded into a partial fraction
expansion about each pole so that H can be written in the following
form:
where the ak's are matrices independent of s. Recall
the representation of H in terms of eigenvalues and orthogonal
eigenvectors.
Each ak matrix can be found by multiplying H times
ssk, and then setting s = sk, provided
all Sk values are different. Thus,
Recognize that there are 2n poles because each element of B is of
quadratic form. Further, the poles generally appear in complex
conjugate pairs because the elements of M, C, and K are real
numbers, and hence each quadratic element of B has real
coefficients. If poles do not appear in conjuqate pairs, then they
must be real.
Now, is a diagonal matrix
whose elements are functions of s. Furthermore,
because is similar to B ( =
B), and hence has the same
eigenvalues. Thus, any value of s = sk which satisfies
det [B] = 0, will also force one of the 's, say, to zero. Rewriting the eigenvector
definition
Thus, = 0 for either s =
sk or s = s*k , and the homogeneous
solution at sk= sk is the original eigenvector
evaluated at s = sk. Also, the solution corresponding to
the conjunate poles
Note that uk utk is an (n x n)
symmetric (complex) matrix, while utk
uk is a complex scalar.
Therefore, the ak matrix is determined by a mode shape
vector uk, which is simply the solution to the
homogeneous system equation B uk = 0 for s =
sk . Furthermore, each column of ak is this
same bode shape vector (within a constant multiplier), and each row
is the transpose of the vector. From a measurement standpoint this
implies that the same mode shape is obtained regardless of which
spatial point is excited or monitored. This pervasiveness of the
mode shapes throughout the transfer matrix is verified with
experimental results later in the paper.
Returning to the partial fraction expansion of H.
We can represent this summation of partial fraction terms in
matrix form by defining the following matrices:
is called the modal
transformation matrix, and t is
the transfer matrix in modal coordinates. Note that the columns of
are not orthogonal (even
though the parent eignevectors yk are orthogonal) because
each uk is evaluated at a different value of s. However,
the elements of are not
functions of s. All of the s dependence is contained in. Each column of represents a normalized mode shape
vector for the corresponding pole of H. It should be apparent that
this normalization is arbitrary, and could be absorbed into the matrix if desired.
As discussed previously, the poles of H usually occur in
conjugate pairs, and for this case the mode shape vectors associated
with the negative poles (lower half of splane) are simply the
conjugates of the vectors associated with the positive poles. Thus,
if 1*, is
defined as that (n x n) part of
associated with positive poles, then of will correspond to the
negative poles. Similarly, can be
broken into two parts, 1 comprising the positive
poles, and A2 comprising the negative poles. H can then be
represented
or in partitioned form as
Each of these submatrices is (n x n) and only 1 and 2 are functions of s.
Define
we define the modal mass as the coefficient of s2 ien
the denominator of each element of H. However, we recognize that
this coefficient is arbitrary, depending on the numerator
normalization. Notice that Ak has the dimensions of
(s.mass)-1, so the numerator should be normalized by
dividing by something of the form AkSk. We can
use the rather arbitrary normalization factor
Notice that each element of the H matrix has a different zero in
the s-plane, depending upon the angle of Ak and
uk at each point, but the poles of each element of H are
common, and occur at s = sk and s =
sk*.
For the special case of zero damping (ck = 0),
called the normal mode case, we find that sk =
sk* is purely imaginary. Thus, the B matrix becomes real
symmetric, and it's eigenvalues and eigenvector components become
real. This means that uk = uk* ,
and we can show that Ak becomes purely imaginary, so
Ak = Ak*. In this case, the numerator
zero in each element of H goes to infinity, and H becomes
Thus it has been shown that two transfer function forms of
interest in modal analysis, namely the complex
eigenvalueeigenvector case (eq. 34) and the normal mode case
(eq. 35) can be obtained from a more general
eigenvalueeigenvector diagonalization of the system or transfer
matrix. In the next section the identification of modal parameters
from measured transfer function data using eq. 34 is discussed.
IDENTIFICATION OF MODAL PARAMETERS
The technique used to obtain the results presented here involves
the curvefitting of analytical expression (34) to a set of
measured transfer function data. The curve fitting is performed in a
manner which minimizes the squared difference between the complex
data and the complex valued analytical function form, i.e. a least
squared error estimate of the data is determined.
Recall that according to the modal theory, only one row or one
column of the transfer matrix need be measured since all other rows
and columns contain redundant information. During the process of
determining the least squared estimator for the transfer matrix,
complex values of Sk and the residues of one column or
one row of the transfer matrix H for all predomenant modes of
vibration are determined.
For example, the qth column of H would
have the residues
After measuring these n residues of H, we form the sum of the
squares of these numbers giving
Taking the square root, and normalizing the measured residues by
this quantity gives
,
which are the elements of the normalized mode shape vector. The
Ak coefficients are readily found from any residue of
Hpq by dividing by the product of the pth and
qth components of the normalized mode shape vector. The
modal system parameters (mass, stiffness, damping) are obtained from
Ak and sk, and the mode shape is given by the
uk vectors (generally normalized by
.
The pole location of mode (k) in the splane, also called the
complex frequency, can be written in terms of the coordinates
is called the damping
factor and wk the natural frequency of mode
(k). Other related and commonly used terms are the damping ratio
and resonant frequencv.
These terms are shown in the splane in figure (1)
The experimental data was taken from the metal T-plate mounted on
a foam rubber base shown in Figure 2.
A hammer was used to provide the broad band excitation force,
with a load cell attached to it to measure the force. An
accelerometer mounted on the plate was used to measure
responses.
The transfer function data was obtained using a
HewlettPackard 5451B Fourier Analyzer, and the modal
parameter identification ::as performed using the
HewlettPackard Modal Analysis Package.
Transfer functions were measured between 22 different points
evenly spaced along the outer periphery of the Tplate. Figure 3
shows a typical transfer function in rectangular or coquad
form.
Figure 4 shows the least squares estimate of this transfer
function and Table 1 contains its corresponding modal parameters.
These results were generated on the Fourier Analyzer using the Modal
Package in about 30 seconds.
A MODE AS A GLOBAL PROPERTY
By far the most fundamental assumption of modal testing is that a
mode of vibration can be excited from anywhere on an elastic
structure, except of course along its node lines where it can't be
excited al all. This is another way of stating the result derived
earlier, i.e. that the same mode shape vector (scaled by a different
component of itself) is contained in every row and column of the
transfer matrix. In addition, modal frequency and damping are
constants which can be identified in any element of the transfer
matrix, i.e. any transfer function taken from the structure.
It is important to recognize that this global mode shape concept
implies some sort of spatial boundaries, beyond which vibrations
will not readily propagate. Any attempt to extent B or H beyond
these boundaries will result in singular matrices, and a breakdown
of the modal concept. This behavior implies that B and H must be
partitioned into submatrices, some of which will be
nonsingular, and will possess well defined vibration modes. If two
linear systems are completely isolated, then a single composite mode
including both systems is not meaningful.
Conversely, it is important to include enough spatial points to
describe all of the vibration modes of interest. If some region of a
bounded system is not monitored or excited, or if points are not
chosen sufficiently close together, then some modes cannot be
adequately represented.
Following are the results of two separate modal tests that were
performed on the Tplate. In test #1 the accelerometer was
mounted on the bottom plate as shown in figure 2. and the plate was
impacted with the hammer at the 22 peripheral locations. Using the
Fourier Analyzer a transfer function was measured between each of
the 22 impact points and the single response point (accelerometer
location). Since the transfer function is the same between two
points regardless of which one is the excitation or response point
this test is equivalent to impacting the plate in one spot and
moving the accelerometer to all 22 positions. This reciprocity or
symmetry assumption is also fundamental to modal analysis and is
reflected in the symmetry of the system and transfer matrices.
Test #2 was the same as test #1 except that the accelerometer was
mounted at position #2.
Table 2. contains the least squared estimates of the natural
frequency and damping factor of a single mode from the 22 transfer
function measurements. These are remarkably good when one considers
that the resolution between spectral lines is 10 Hz. Working in a
narrower bandwidth or using more data points to describe each
transfer function should give better results.
Table 3 contains the corresponding normalized mode shape
vectors from the two tests.
CONCLUSIONS
The results indicate that by applying an analytical transfer
function expression through least squares estimation to measured
data from linearly behaving (small displacements) structures, modal
parameters consistent with the theory can be obtained. The
vibrations specialist must be continually aware however of the
important assumptions necessary for obtaining valid modal results
from test specimens.
REFERENCES
1. Foss K.A "CoOrdinates which Uncouple the Equations of
Motion of Damped Linear Dynamic Systems" J. of Applied Mechanics
ASME paper 57A86, 1958
2. Flannelly, W.G. McGarvey, J.H., Berman, A., "A Theory of
Identification of the Parameters in the Equations of Motion of a
Structure Through Dynamic Testing" paper No. C1 Symposium on
Structural Dynamics, U. of Technology, Loughborough, England, March
1970
3. Thoren, A.R. "Derivation of Mass and Stiffness Matrices form
Dynamic Test Data", AIAA paper 72346, 13th Structures,
Structural Dynamics, and Materials Conf., San Antonio, Texas April
12, 1972
4. Hurty, W.C. and Rubinstein, M.F. "Dynamics of Structures"
PrinticeHall Inc., Englewood Cliffs, N.J., 1965
5. Caughey, T.K., O'Kelly, M.E.J. "General Theory of Vibration of
Damped Linear Dynamic Systems" Cal. Inst. of Tech., Dynamics Lab.,
Pasadena, California, June 1963
6. Klosterman, A.L., "On the Experimental Determination and Use
of Modal Representations of Dynamic Characteristics", Ph.D.
Dissertation, University of Cincinnati, 1971