The Cost Function section determines how the cost (error) is calculated for each accelerometer set. The drop-down box selects the cost function and the Normalize Shapes? checkbox determines whether the shapes are normalized to the candidate TAM mass matrix before performing the cost function calculation. The cost is calculated for each accelerometer set at the end of every generation and the sets are ranked according to their cost. Accelerometer sets with the lowest cost are reproduced (if non-zero Percent Reproduce is specified) and are most likely to be selected for the mutation and crossover process to populate the new generation. At the conclusion of the GA solution the only accelerometer sets available to review results are the original Seed TAMs and the GA DOF with the lowest cost.
The four cost available cost functions are:
1. Pseudo-Orthogonality (RSS)
2. Pseudo-Orthogonality (Max)
3. Self-MAC (RSS)
4. Self-MAC (Max)
All four cost functions are based on either the Pseudo-Orthogonality (PO) or the Self Modal Assurance Criterion (Self-MAC) which are two industry standards for measuring linear independence of test-measured modeshapes. The PO is calculated as:
where:
is the
GA TAM modeshapes, expanded to the candidate DOF set size
is the candidate DOF set mass matrix
For perfect linear independence the PO matrix would have 1.0 along the diagonal and 0.0 as all off-diagonal terms. The PO error is calculated as
where:
diag takes the diagonal of the resulting matrix
I is an identity matrix
W is a vector of the target mode weighting factors
is the term-by-term multiplication
The FEM and TAM MAC are calculated as follows
where:
is the
candidate TAM target modeshapes, partitioned to the GA TAM DOF
is the
GA TAM modeshapes
The MAC error for each target modeshape is calculated as:
Error vectors are computed for each FEM configuration using the appropriate equation (PO error for PO cost functions, MAC error for MAC cost functions). These two error measures result in a vector showing the error of each target mode. The error vectors are reduced to a single value—the cost—using either the root-sum-square (RSS) of the error vector or the maximum value, as determined by the selected cost function. This value gives a quantitative measure of the cost associated with each accelerometer set for each FEM configuration. If multiple FEM configurations are simultaneously processed, the costs of each individual FEM configuration is either RSS’ed or the maximum is taken, according to the selected cost function. This overall cost gives a measure of the quality of a specific accelerometer set to render all target modes of all FEM configurations linearly independent. The RSS error measures how well all target modes are represented, while the maximum value error is a measure of the worst target mode representation. The GA uses the selected cost function to rank each accelerometer set.
The results plot on the right side of the main GA form shows the minimum cost for each generation in blue on the left axis and the average cost of all genes in each generation is plotted in red on the right axis.
If a Pseudo-Orthogonality cost function is used (cost functions 1 or 2) then the candidate TAM/GA TAM mode matching and frequency comparisons are based on the Cross-Orthogonality. If a Self-MAC cost function is used (cost functions 3 or 4) then the candidate TAM/GA TAM mode matching and frequency comparison is based on the Cross-MAC.
All comparisons are performed at the candidate-set-size level by back expanding the TAM modeshapes to the candidate-set size using a Guyan back expansion matrix.