Structural Health Monitoring Using Smart Sensors - ideals ...

(a)
Figure 5.6. Effect of data loss and observation noise: coherence functions between (a)
nodes 11 and 19; and (b) nodes 19 and 25.
in Figure 5.6. The isolated downswings correspond to zeroes of the system; extremely
small responses of the structure at these frequencies at the zeros result in numerical error,
giving small coherence function.
The loss of 0.5 percent of data yields a much smaller coherence function, except at the
system’s poles (see Figure 5.6). Because of the random nature of data loss occurrence and
the excitation, the coherence function varies from simulation to simulation. Twenty time
average of the coherence functions are shown in this figure. The coherence function’s
discrepancy from unity depends on the two measurement nodes between which the
function is calculated. All of the investigated coherence function plots, however, support
that the loss of 0.5 percent of data affects the coherence function in a similar way as 5 to
10 percent measurement noise addition.
Though the data loss clearly impacts the PSD estimation and coherence function
negatively, the consequence in subsequent modal analyses is unclear. The modal
properties are mainly represented by frequency components near the natural frequency.
The PSD and coherence function are not affected by data loss and noise as much near the
poles. The ERA modal analysis method is then applied to the impulse response functions
to estimate the modal properties. Impulse response functions are the inverse Fourier
transform of the transfer functions from the input force to measured outputs, which is
calculated from CSD and PSDs.
Figure 5.7 shows how the estimates of the first natural frequency and the damping
ratio vary as a function of the data loss level. For comparison, these modal parameters are
also estimated from the simulation model subjected to measurement noise (see Figure
5.8). Again, a loss of 0.5 percent of data is approximately equivalent to 5 to 10 percent
measurement noise.
The first four modes are identified and these modal properties are used to estimate the
flexibility matrix F
. The flexibility matrix estimate Fˆ
is compared with F
and the
estimation error is shown in Figure 5.9 in terms of the Frobenius norm. represents the
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(b)
F