104 D. Ocepek et al. force set is calculated that causes the same responses on the passive side when the source is turned off. Special attention is given to the in-situ approach [3] even eliminating the need to dismount any part of the assembly to obtain equivalent force set. An open-source implementation of the TPA techniques is pyFBS [4], a python package for frequency based substructuring and transfer path analysis. Additionally, in recent years, pyFBS has developed into a compact and versatile package that expanded beyond FBS and TPA techniques only. Most recent additions to the package include various expansion techniques [5] and multi-reference modal identification tool [6]. This paper however, the use of transfer path analysis within the pyFBS package is introduced. First, short theoretical background onin-situTPA is given, followed by a commonly adopted interface model using virtual point transformation [7]. Usage of the package is then demonstrated using numerical case study. Theoretical background and notation Consider a linear and time-invariant assembly of substructures A and B, coupled at the interface (Fig. 1a). Substructure A is an active component with the operational excitation f1. Meanwhile, no excitation force is acting on the passive substructure B. The responses on B are observed in three different sets of DoFs: at the interface DoFs (u2), in the proximity of the interface at the indicator DoFs (u4), and away from the interface at the target DoFs (u3). A B u1 f1 u4 u4 u2 Y42 AB u3 (a) A B u1 f1 eq -f 2 Bu 0= (b) A B u1 eq f 2 u3 u2 Y32 AB (c) Fig. 1 In-situ TPA: a) assembly of substructures A and B, b) feq 2 blocking the motion at the interface, c) replicating operational responses withfeq 2 . Source excitations f1 are often not measurable in practice; therefore, in-situTPA adopts a different approach for describing the operational excitations. A set of equivalent forces f eq 2 is introduced, applied at the interface DoFs. If the source is deactivated, f eq 2 yields the same responses on the passive side as f1 1: 0=YAB 41 f1 | {z } u4 +YAB 42 −f eq 2 . (1) Expressing the equivalent forces f eq 2 from the indicator responses u4 yields: f eq 2 = YAB 42 + u4. (2) Receiver operational responses for the arbitrary assembly (with the same source) can be replicated by the set of equivalent forces (Fig. 1c) as follows: u3 =YAB 32 f eq 2 . (3) Interface modeling is critical to prevent redundancy/bad conditioning or neglecting important transfer paths through the interface. This means the experimentalist must account for all significant DoFs at the interface. Recently, virtual point transformation (VPT) has been proposed to model the interface where interface DoFs (rigid or flexible) are selected manually by the experimentalist to avoid redundancy yet retain the full controllability of the interface. The loads mat the virtual point (VP) are obtained for a given vector of forces f in the proximity of the VP (Fig. 2). The contribution from all the input forces can be combined and expressed as follows: m=RT f f, (4) where the IDM matrix RT f ∈ Rm×nf contains the positions and orientations for all the excitation locations with respect to the VP (Fig. 2). The inverse relationship of Eq. (4) is derived with a constrained minimization for forces: f =Rf RT f Rf −1 m=TT f m ⇒ TT f =Rf RT f Rf −1 . (5) 1An explicit dependency on the frequency is omitted to improve the readability of the notation, as will be the case for the remainder of the paper.
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