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249 IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 2
L. P. PERLIN | R. C. A. PINTO
is less than the number of unknowns, and thus there is not a
unique solution (Figure 4-a);
n determined: the number of independent linearly equations is
equal to the number of unknowns, and thus there is a unique
solution (Figure 4-b);
n overdetermined: the number of independent linearly equations
is greater than the number of unknowns. If the readings are ac-
curate, the system will be considered consistent with a unique
solution (Figure 4-c). If, on the other hand, the readings are not
accurate, the system will be inconsistent without a proper solu-
tion to all readings (Figure 4-d).
The tomographic system can be classified as underdetermined,
determined or overdetermined depending on the concrete element
to be study and its number of available sides to perform ultrasound
readings. For beams and slabs, there are usually only two sides
available, according to Figure 5-a. For columns, there are usu-
ally four sides that can be used, which allow the readings to be as
shown in Figure 5-b. It may be necessary to use different modes of
readings depending on how the objects are on the field Perlin [11].
T: a column vector with m lines that records the total travel-time of
each reading i.
It can be observed that the elements of matrix D are determined
since the grid and the position of the transducers are known for
each different reading. The elements of vector T are also known
since they are the results of each reading. Therefore, only the ele-
ments of vector P are to be determined.
3.2 Resolution
Equation 6 represents a system of simultaneous linear equations,
which could apparently be solved by simplified methods. However,
the matrix D is usually rectangular, with a number of equations that
differs from the number of the unknowns wave slowness. More-
over, there are some equations that are linearly dependent, which
may lead to a singular problem, without a unique solution.
Therefore, depending on the number of readings and on which
readings are performed, the problem may be:
n undertermined: the number of linearly independent equations
Figure 4 – Classification of the simultaneous linear equations system – (a) underdetermined
(b) determined – (c) overdetermined consistent – (d) overdetermined inconsistent
A
C
B
D