3. Operands

3.1. Generalities

The MAC matrix (M odal A insurance C riterion) indicates the correlation between two lists of vectors. It is often used to check the orthogonality of two modal bases (experimental or numerical).

The i-th row, j-th column of the MAC matrix is defined by the following relationship:

\({\mathit{MAC}}_{\mathit{ij}}=\frac{{\left({\Phi }_{i}^{H}W{\Phi }_{j}\right)}^{2}}{\left({\Phi }_{i}^{H}W{\Phi }_{i}\right)\left({\Phi }_{j}^{H}W{\Phi }_{j}\right)}\)

Where: \({\Phi }_{i}\) is the i-th vector of the first base,

\({\Phi }_{j}\) is the j-th vector of the second base,

\(W\) is a weighting matrix,

The exponent \({}^{H}\) indicates the conjugate transpose.

This MAC matrix is diagonal if the two bases are formed from modal vectors from the same structure and if the weighting matrix is equal to the mass or stiffness matrix of the structure in question. We then speak of the orthogonality of the natural modes in relation to the mass or stiffness matrix. Quantity \({\Phi }_{i}^{H}W{\Phi }_{j}\) corresponds to the term of the generalized stiffness (or mass) matrix if the weighting matrix is equal to the stiffness (or mass) matrix.

Criterion IERI (I E energy indicator of R, R regularity of I interface) is an energy indicator for calculating the difference between two vectors. It is defined by the following relationship:

\({\mathrm{IERI}}_{\mathrm{ij}}=\frac{{({({\Phi }_{i}-{\Phi }_{j})}^{H}W({\Phi }_{i}-{\Phi }_{j}))}^{2}}{{({\Phi }_{i}^{H}W{\Phi }_{i})}^{2}+{({\Phi }_{j}^{H}W{\Phi }_{j})}^{2}}\)

The weighting matrix is either the mass matrix or the stiffness matrix of the structure in question.

This criterion IERI tends to 0 if the two vectors are very close.

3.2. Operand BASE_1/BASE_2

♦ BASE_2 = base_2

Name of the modal concepts (mode_meca, mode_meca_c, mode_flamb) to compare. In general, we compare a numerical modal base resulting from a calculation (by CALC_MODES) and an experimental modal base, imported by LIRE_RESU. Both bases must be defined on the same model and have the same numbering.

3.3. Operand IERI

This operand is used to specify that we want to calculate the criterion IERI. In this case, the MATR_ASSE operand must be filled in. A mass matrix or a stiffness matrix must be associated with it.

3.4. Operand MATR_ASSE

◊ MATR_ASSE = matr

Operand designating an assembled matrix used as weighting.

matr is the weighting matrix that is applied to the base vectors. It must have the same numbering as base1 and base2. The mass matrix or the stiffness matrix is usually chosen as the weighting matrix.

If this operand is not entered, the weighting matrix chosen is equal to the identity matrix.

This weighting matrix is mandatory for the calculation of criterion IERI.

3.5. Operand TITRE

Optional name to give to the table.

3.6. Operand INFO

◊ INFO = 1 or 2

The MAC or IERI matrix and the generalized matrix are displayed as a table in file MESSAGE if INFO =2.

The name of the output table parameter associated with the generalized matrix is Y1_W_Y2. This generalized matrix is calculated if the base vectors are of the real type.