
High-Order Time-Domain Algorithms. The algorithms that fall into the cate-
gory of high-order time-domain algorithms include the algorithms most commonly
used to determine modal parameters. The least squares complex exponential
(LSCE) algorithm is the first algorithm to utilize more than one frequency response
function, in the form of impulse-response functions, in the solution for a global esti-
mate of the modal frequency.The polyreference time-domain (PTD) algorithm is an
extension to the LSCE algorithm that allows multiple references to be included in a
meaningful way so that the ability to resolve close modal frequencies is enhanced.
Since both the LSCE and PTD algorithms have good numerical characteristics,
these algorithms are still the most commonly used today. The only limitations for
these algorithms are the cases involving high damping.As these are high-order algo-
rithms, more time-domain information is required than for low-order algorithms.
First-Order Time-Domain Algorithms. The first-order time-domain algorithms
include several well-known algorithms such as the Ibrahim time-domain (ITD) algo-
rithm and the eigensystem realization algorithm (ERA).These algorithms are essen-
tially a state-space formulation with respect to the second-order time-domain
algorithms.The original development of these algorithms is quite different from that
presented here, but the resulting solution of linear equations is the same regardless
of development.There is a great body of published work on both the ITD and ERA
algorithms, much of which discusses the various approaches for condensing the
overdetermined set of equations that results from the data (least squares, double
least squares, singular value decomposition).The low-order time-domain algorithms
require very few time points in order to generate a solution because of the increased
use of spatial information.
Second-Order Time-Domain Algorithms. The second-order time-domain algo-
rithm has not been reported in the literature previously but is simply modeled after
the second-order matrix differential equation with matrix dimension N
o
. Since an
impulse-response function can be thought to be a linear summation of a number of
complementary solutions to such a matrix differential equation, the general second-
order matrix form is a natural model that can be used to determine the modal
parameters. This method is developed by noting that it is the time-domain equiva-
lent to a frequency-domain algorithm known as the polyreference frequency-
domain (PFD) algorithm. The low-order time-domain algorithms require very few
time points in order to generate a solution because of the increased use of spatial
information.
High-Order Frequency-Domain Algorithms. The high-order frequency-domain
algorithms, in the form of scalar coefficients, are the oldest multiple degree-of-
freedom algorithms utilized to estimate modal parameters from discrete data. These
are algorithms like the rational fraction polynomial (RFP), power polynomial (PP),
and orthogonal polynomial (OP) algorithms. These algorithms work well for narrow
frequency bands and limited numbers of modes but have poor numerical character-
istics otherwise. While the use of multiple references reduces the numerical condi-
tioning problem, the problem is still significant and not easily handled. In order to
circumvent the poor numerical characteristics, many approaches have been used (fre-
quency normalization, orthogonal polynomials), but the use of low-order frequency-
domain models has proven more effective.
Orthogonal Polynomial Concepts. The fundamental problem with using a
rational fraction polynomial (power polynomial) method can be highlighted by
looking at the characteristics of the data matrices. These matrices involve power
EXPERIMENTAL MODAL ANALYSIS 21.63
8434_Harris_21_b.qxd 09/20/2001 12:09 PM Page 21.63