min υ1,...,υ‘X k j¼1 yj X ‘ i¼1 yj, υi D E W υi 2 ð15:20aÞ so that υi, υj W ¼ δij for i, j ¼1, . . . , ‘ ð15:20bÞ where a, b h iW ¼ a, Wb h i¼ Wa, b h i¼a TWb ð15:21Þ denotes the weighted inner product of two vectors with a positive definite weighting matrix W. A basis V¼ ½v1, . . . , vℓ , which is a solution to the above minimization problem, can be obtained by solving the symmetric eigenvalue problem YTWY υ i ¼λiυi for i ¼1, . . . , ‘ ð15:22aÞ where Y¼ ½y1, . . . , yk . The final POD basis vectors are obtained by evaluating vi ¼ 1 ffiffifffii λi p Y υi: for i ¼1, . . . , ‘ ð15:22bÞ This procedure is sometimes called the method of snapshots [7]. 15.3.3 Test Load Based Joint Interface Modes In [11] it was suggested to define test loads within the contact interface and utilize the POD on the resulting displacement fields for computation of the joint interface modes. The respective steps are briefly recalled in this subsection. First step is the decomposition of the discretized contact interface bΓjc into n subareas as suggested in [9]. Each of these subareas is loaded by a unit pressure distribution, according to Newton’s 3rd law on both sides of the contact interface. The resulting equivalent nodal force vectors are collected in the matrix of test loadcases Fð ∗Þ jc ¼ f ð∗Þ jc,1, . . . , f ð∗Þ jc,n h i: ð15:23Þ These test load cases are utilized within the static equilibrium KbUð∗Þ ¼Fspc þFð ∗Þ jc ð15:24Þ where the body is fixed at bΓd andbΓt as well. The constraints at bΓd are already considered within the principle of virtual work, the constraints at bΓt are enforced by the respective column vectors of the matrix Fspc. The resulting displacement fields are collected in the matrix bUð∗Þ ¼ buð ∗Þ 1 , . . . , buð∗Þ n h i. By settingY¼ bUð∗Þ andW¼Ithe application of the snapshot method Eq. (15.22) for computation of the joint interface modes ΦAjim ¼Vis pretty straight forward. Furthermore [11] suggests to use the stiffness matrix for weighting, W¼K, which lead to promising results during own investigations of the authors. This requires a positive definite stiffness matrix, which is ensured in our case as Krepresents the constrained body. This method will be denoted as “method A” for the remainder of this contribution. 172 M. Breitfuss and H.J. Holl
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