smart technologies for safety engineering

42 Smart Technologies for Safety Engineering
matrix is assembled as
Figure 3.1 Virtual distortion states for a beam element
MKL =
i
A
M i J
KL + M i
KL
where i denotes a part of the mass matrix in global coordinates, corresponding to the beam
element i. The increment of mass (cf. Equation (14)) is then expressed as
MKL = ˆMKL − MKL =
i
μ A
A
i − 1 M i KL + μ J J
i − 1 M i
KL
3.2.4 Problem Formulation and Optimization Issues
In most approaches to damage identification, the measured quantity is acceleration, because it
is relatively easy to obtain. However, the raw acceleration signal in time is never used directly –
it requires FFT processing to transfer the analysis into the frequency domain. In the proposed
approach a different quantity is measured. It is namely strain in time, measured by piezotransducers,
which is then directly used in the VDM-T approach (cf. Reference [29]). For
harmonic excitation (VDM-F) only frequency-dependent amplitudes of strains are examined.
Thus it is possible to speak about the VDM frequency-domain approach. It should be noted that
there is no need for an FFT processing of the time signal as performed in standard frequencydomain
methods.
As often happens in parameter estimation procedures, the identification task is posed as
a nonlinear least squares minimization problem. The objective function expressed in strains
collects time responses for the VDM-T approach:
F t (μ) =
εk − εM k
t
and amplitude responses from selected nω frequencies of operation for the VDM-F approach:
ε M k
F ω (μ) =
εk − εM k
nω
Both functions (16) and (17) collect responses from k sensors, placed in those elements where
nonzero strains of high signal-to-noise ratio are measured. The strain εk in an arbitrarily selected
location k is influenced by virtual distortions ε0 i , which may be generated in any element i
of the structure (cf. Equation (8) in Chapter 2). One should also note that the modification
coefficient μi, quantifying potential damage and used as a variable in optimization, depends
upon the virtual distortions ε0 i non linearly (cf. Equation (17) in Chapter 2). The VDM-T
ε M k
2
2
(14)
(15)
(16)
(17)