96 J. Korbar et al. Primal Formulation of Modal Substructuring Dynamic substructuring in the modal domain allows to combine individual component models to estimate assembly dynamics or remove parts of the structure based on the individual substructures’ modal parameters. The governing linear equation of motion for a substructure s in the physical domain can be written as: M(s)¨u(s) +C(s) ˙u(s) +K(s)u(s) =f(s) +g(s), (1) where M(s), C(s), and K(s) are the mass, damping, and stiffness matrices, respectively, u(s) is the displacement vector, f(s) is the external force vector, and g(s) is the connecting interface force vector. The velocity and acceleration vectors are denoted as the first and second time derivatives ˙u(s) and ¨u(s) of displacements u(s). The equations of motion for N individual substructures can be combined into a single uncoupled equation of motion: M¨u+C˙u+Ku=f +g, (2) whereM=diag M(1), . . . , M(N) , C=diag C(1), . . . , C(N) , K=diag K(1), . . . , K(N) , u=nu(1)⊤, . . . , u(N)⊤o ⊤ , f =nf(1)⊤, . . . , f(N)⊤o ⊤ , and g =ng(1)⊤, . . . , g(N)⊤o ⊤ . Coupling the individual dynamic models requires satisfying the constraints regarding the compatibility of interface displacements and equilibrium of interface forces: Bu=0, (3) L⊤g =0. (4) The physical DoFs are reduced to a smaller set of generalized DoFs ηusing a block-diagonal reduction matrixRm: u=Rmη, (5) where Rm = diag R(1) m , . . . , R(N) m , η =nη(1)⊤, . . . , η(N)⊤o ⊤ , and R (s) m denotes the reduction matrix. In this study, a set of mass-normalized mode shapes Φ(s) is considered as a reduction matrix. Pre-multiplying Eq. (2) with R⊤ m and considering the reduction in Eq. (5) yields the following equation of motion and corresponding compatibility and equilibrium constraints: Mm¨η+Cm˙η+Kmη =fm+gm, (6) Bmη =0, (7) L⊤mgm=0, (8) where Mm = R⊤mMRm, Cm = R⊤mCRm, Km = R⊤mKRm, Bm = BRm, Lm = null (Bm), fm = R⊤mf, and gm = R⊤mg. Eq. (7) enforces exact/strong compatibility of physical DoFs in the reduced DoF space. If the number of constraints exceeds the combined number of the individual substructures’ modes, the set compatibility equations is overdetermined and generally cannot be satisfied exactly. In addition, the number of modes which can be calculated for the coupled structure decreases with the increasing number of constraints and decreasing number of modes of the individual substructures. Therefore, it can be reasonable to reduce the number of constraints by weakening the compatibility conditions, i. e. approximately satisfying Eq. (7) by pre-multiplication with a weakening matrixW: eBmη =0, (9) eLmgm=0, (10) where eBm =WBm and eLm = null (WBm)⊤. Substructuring in the primal formulation is performed by expressing the dual set of generalized coordinates ηusing a unique set of generalized coordinates ξ: η =eLmξ, (11) therefore, the coupled equation of motion is written as: fMm ¨ξ+eCm ˙ξ+ eKmξ =efm, (12) where fMm=eL⊤mMmeLm, eCm=eL⊤mCmeLm, eKm=eL⊤mKmeLm, and efm=eL⊤mfm.
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