Proper orthogonal modes of rank g minimize a cost function J in the form of o ¦ ¦ " 2 g m T 1 g j j i i j 1 i 1 J( , , ) min u u y y u u , (9) so that ® ¯ z T i j 1 i j 0 i j u u , (10) where the Euclidian norm is defined as T y y y. (11) The POM 1u to gu can be found by the solution of an eigenvalue problem in the form of O T i i i Y Yv v for i = 1, … ,g (12) and a subsequent transformation 1 O i i i u Yv for i = 1, … ,g. (13) The maximum error in a Euclidean sense can be given as error of the function J in the form of O ¦ ¦ ¦ " 2 g m m T 1 g j j i i i j 1 i 1 i g 1 J( , , ) u u y y u u . (14) The slightly different POD method with a weighted inner product minimizes a cost function in the form of D o ¦ ¦ " 2 g m T 1 g j j j i i j 1 i 1 A J( , , ) min u u y y Au u , (15) where the matrix A hast to be symmetric, of the dimension (m x m) and positive semi definite, D j are nonnegative weights and the induced norm is of the form T W W y y y . (16) Now the eigenvalue problem takes on the form O T i i i Y AYv v for i = 1, … ,g (17) and the final POM can be computed along equation (13). From a mechanical point of view, the characteristics of the matrix A are the same as the ones of the stiffness matrix of a linear FE model. In that context, the POM can be interpreted as kind of ‘energy’ modes. For more information on the physical interpretation of POD refer to [15]. The singular values Oi can be used as an estimation of how many number of POM are required because they hold a ratio of the modeled to the total energy contained in the system Y, which is expressed by O O ¦ ¦ p i i=1 g i i=1 w(g) (18) 5. POD based Joint Interface Modes The application of the POD method to the problem under consideration is quite straight forward. The space spanned by Y is replaced by X from equation (3) and, in case of POD weighted inner product, the matrix A of equation (17) is replaced by the stiffness matrix K from the equation of motion (1). 22
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