Structural Health Monitoring Using Smart Sensors - ideals ...

Table 7.1. SVD Accuracy Check on Matrices of Various Size
|USV-A| 2 /|A| 2
matrix size
Numerical Recipes CLAPACK
3x3 1.01 x e-15 1.01 x e-15
30x30 6.39 x e-16 4.96 x e-15
50x50 1.09 x e-15 4.0337 x e-16
100x100 9.72 x e -16 4.1957 x e-15
and the diagonal matrix of singular values require n + n B of memory. Even when
the entire 256 kB of RAM on the Imote2 is used for storing these matrices, the maximum
size of the matrix to be analyzed is .
7.1.3 Eigensolver
ERA requires an eigensolver to estimate modal parameters. The eigensolver is
employed to the estimated system matrix, A. Modal frequencies and damping ratios are
estimated from its eigenvalues, while mode shape estimation requires the eigenvectors.
Therefore, both eigenvalues and eigenvectors need to be calculated. Moreover, the A
matrix is not necessarily symmetric. A complex eigensolver capable of calculating both
eigenvalues and eigenvectors is needed.
CLAPACK contains a double precision complex eigensolver. The algorithm reduces
a matrix to its upper Hessenberg form. The Hessenberg matrix is then reduced to its Schur
form. The eigenvalues are obtained as the diagonal elements of the Schur form matrix.
The eigenvectors of this matrix is then calculated and transformed back considering the
Hessenberg and Schur transforms. The source code is modified for use on the Imote2.
The accuracy of the implementation is examined. A matrix is constructed in the same
way as in the SVD function accuracy test. The eigensolver is applied to this matrix on both
the Imote2 and the PC. Figure 7.4 shows eigenvalues estimated on the two platforms and
the normalized difference between the two eigenvalues. Eigenvalues on the Imote2 and on
the PC are considered to be numerically equal. As another accuracy indicator, the Modal
Assurance Criterion (MAC) for eigenvectors calculated on the Imote2 and on the PC is
investigated. The MAC between two vectors and are defined as follows:
MAC =
(7.1)
where * denotes complex conjugate. Identical vectors give a MAC value of 1.0, even if
one of them is multiplied by a constant. Uncorrelated vectors give a MAC value of zero.
The deviation of the MAC values from one is plotted on Figure 7.5. These values are as
small as – when the corresponding eigenvalues are sufficiently large. Although the
eigenvectors corresponding to eigenvalues smaller than the precision of the double data
type are different from those on the PC, the eigensolver implementation on Imote2 is
considered to be accurate with the precision of the data type.
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