let us consider the eigensolutions of the statically condensed dual problem derived from(31): −μ2 k I 0 0 MI + 0 RTBT BR −FI xαk xλk =0 (32) The eigenvectors of this problem will form a basis to represent the interface dynamics. Note that those eigenvectors include not only the interface force modes xλk but also the associated rigid body displacements necessary to guarantee the selfequilibrium of the interface forces with the inertia forces in the substructures. Due to the symmetry of the statically-condensed dual interface problem, the eigensolutions satisfy the orthogonality properties XT α XT λ I 0 0 MI Xα Xλ =I XT α XT λ 0 RTBT BR −FI Xα Xλ =μ2 (33) where Xα, Xλ contain in their columns the eigenvectors of (32) and where μ2 is a diagonal matrix containing the associated eigenvalues. In these relations we have assumed that the eigenvectors have been normalized with respect to reduced mass matrix. Note that due to the weak compatibility imposed in this statically condensed problem, there will be as many positive eigenvalues as there are rigid body modes in R and as many negative eigenvalues as there are effective Lagrange multipliers. Non-effective Lagrange multipliers, namely those related to possibly redundant compatibility conditions (as encountered sometimes on corners) will generate zero eigenvalues. Since the compatibility is very weak in this statically condensed problem, one typically finds positive and negative eigenvalues with low absolute values. The strategy now consists in considering only a small number of interface modes to approximate the interface dynamics. For that we choose the nstat modes corresponding to the eigenvalues μk with the lowest absolute value. Storing those lowest eigensolutions in ˜Xα, ˜Xλ and ˜μ2, we propose to modify the approximations in the Dual Craig-Bampton by applying a second reduction step so that ⎡ ⎣ α η λ ⎤ ⎦ ⎡ ⎣ 0 ˜Xα I 0 0 ˜Xλ ⎤ ⎦ η ˜λ =TI η ˜λ (34) Applying now this reduction to the reduced system (25) leads to I −Ω−2ΘTBT ˜Xλ −˜XT λBΘΩ− 2 I ¨η ˜¨λ + Ω2 0 0 ˜μ2 η ˜λ =TT I ¯TT dualf (35) Once this reduced problem is solved, the physical dofs can be reconstructed by applying to back transformation Daniel J. Rixen 320
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