160 F. Wenneker and P. Tiso Note that the rigid body modes are simply the free VMs associated to zero eigenfrequency. Residual attachment modes represent the deformation of the substructure when a unit force is applied to one boundary DOF. The reduction proposed in Eq. (14.9) therefore leads to a dual system of equations, i.e. displacement and force DOFs combined in q. In contrast to the CB method, the component modes act on the full set of DOFs rather than on the internal DOFs only. In our framework, we require a primal system (displacement DOFs only), where the boundary DOFs are not reduced. Therefore, the expression in Eq. (14.9) needs to be manipulated. Let us first partition the system of equations as: ub ui ‰r;b ˆr;b Nˆb ‰r;i ˆr;i Nˆi 2 4 gb r N 3 5 (14.10) Solving the first row in this expression for the boundary connection forces gb gives: gb ‰ 1 r;b ub ˆr;b r Nˆb N (14.11) From this expression a second coordinate transformation can be defined as: 2 4 gb r N 3 5 2 4 ‰ 1 r;b ‰ 1 r;bˆr;b ‰ 1 r;b Nˆb 0 I 0 0 0 I 3 5 2 4 ub r N 3 5D R2q2 (14.12) The total coordinate transformation that is done to obtain an approximation of the displacement field u, while retaining the boundary DOFs is now found by multiplying Eqs. (14.9) and (14.12): ub ui RR2q2 D RRq 2 D I 0 0 ‰r;i ‰ 1 r;b ˆr;i ‰r;i ‰ 1 r;bˆr;b Nˆi ‰r;i ‰ 1 r;b Nˆb 2 4 ub r N 3 5 (14.13) 14.3 Modal Derivatives Internal and free VMs in the CB and Rubin reduction bases describe linear dynamic behaviour only. The equations of motion in Eq. (14.2) are linearised around position ueq. In the following we will consider the dynamic contribution to the response, i.e. ui;dyn in Eq. (14.5). If we assume that the motion u around this position is small, i.e. u Dueq C u, the linearised reduced dynamic response can be written as: u m Xi D1 i .ueq/ i (14.14) where the free or internal VMs i are computed by solving the eigenvalue problem associated with the linearised equations of motion; free VMs if the free floating structure is considered and internal VMs if the structure is fixed on its boundary nodes. In geometrically nonlinear systems, the VMs are configuration dependent. Therefore, the basis needs to be updated during the response analysis to account for the nonlinearities. If this needs to be done too frequently, the advantage of the reduction step might be surpassed by the computational effort necessary to extract the new modal basis. To overcome this difficulty, the modal derivatives (MDs) of the VMs can be included in the basis [11]. When the displacements can no longer be considered small, the dynamic response of Eq. (14.14) should be expressed as: u m Xi D1 i .ueq C u/ i (14.15) where it is highlighted that the VMs are displacement dependent. Equation (14.15) can be expanded in Taylor series aroundueq: u m Xi D1 i .u/ i m Xi D1 @u @ i ˇ ˇ ˇ ˇ uD0 i C m Xi D1 m Xj D1 1 2 @2u @ 2 i jˇ ˇ ˇ ˇ uD0 i j (14.16)
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