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design of the bridge. A more detailed description of the finite element model of the bridge
can be found in Dyke et al, (2002).
PROBLEM FORMULATION
The objective of this research is to investigate the capabilities of the NExT/ERA technique
for identification of the modal parameters of this structure (Caicedo et al., 2002). Herein,
the excitation is assumed to be ambient, and is considered to be unknown for the identification
procedure. Ambient inputs are simulated by applying uncorrelated, stationary
broadband random forces at all nodes of the structure. The modal parameters are determined
through acquisition of free response data from ambient responses and then identification
of modal parameters. The methodology is described in the following sections.
Acquiring Free Response Data
Here it is assumed that the excitation to the system consists of ambient vibrations, which
are not measurable. James et al. (1993) showed that the matrix of cross-correlation functions
between the responses of the system and a response selected to be the reference response
are a solution to the homogenous equation of motion. We refer to this step as
application of NExT. Consider the equation of motion for a N degree of freedom classically
damped linear system
Mjc(f) + Cjc(/) + Kx(0 = f(r)
where x is the N x 1 vector of displacements, M , C , and K are the N x N mass, damping,
and stiffness matrices, respectively, and f(t) is the vector offerees acting on the system.
Assuming the excitation and responses are each stationary random processes, Eq. (1)
is written
MX(t) + CX(r) -r KX(r) = F(t) (2)
where X(t) is a displacement stochastic vector process and F(t) is the stochastic excitation
vector process. Assuming that the structural parameter matrices are deterministic,
postmultiplying Eq. (2) by a reference scalar response process X t (s) , and taking the expected
value of each side yields
, s) + CR^t, s) + KJR^f, s) = R FX (t, s) (3)
where R(-) denotes a vector of correlation functions. Assuming that X, X and X are
weakly stationary processes, and the excitation is uncorrelated with the responses, Eq. (3)
can be written as (Bendat and Pierson, 2000)
(I)
+ Ktf^Cc) = 0 (4)
Thus, the vector of displacement process correlation functions, R^r(t) , satisfies the homogeneous
differential equation of motion.
To implement this method, one of the available responses is selected as the reference signal,
jc r (r) , the cross spectral density functions between the reference signal and each of the
response signals are obtained, and an inverse fast Fourier transform is performed to determine
the cross correlation functions. The reference channel should be selected such that
all of the modes are observed in the responses at that location. If the reference channel lo-