7 In-Situ Source Characterization for NVH Analysis of the Engine-Transmission Unit 81 Fig. 7.2 The procedure of Virtual Point Transformation. The substructure is equipped with acceleration sensors (blue), which measure the displacement through excitation (red arrows). The FRFs are then transformed to the virtual point (green) Note that blocked force calculation requires direct access to the input DoFs at the interface, which is often infeasible to measure and source measurement errors. A geometrical transformation is therefore needed and will be explained in the following. 7.2.2 Virtual Point Transformation Figure 7.2 shows an application of the virtual point transformation (VPT). Helderweirt et al. present the so-called IDM filtering, on whose idea the VPT is based [5]. With the definition of six rigid interface displacement modes (IDM) per interface point and the projection of an admittance matrix with nine degrees of freedom onto this subspace, initially only the dynamics that leave this area rigid will be preserved. Originally, IDM was used for interface deformation modes, but since we assume only rigid displacements, the designation of [6] is chosen. If in the following a substructuring with this filtered admittance is carried out, only those nine degrees of freedom exhibit locally rigid behavior are required to be compatible and balanced. The residual flexibility remains uncoupled. The interface problem is weakened and measurement errors are averaged out due to the reduction. This guiding principle of IDM filtering is now used in the VPT methodology presented in [7, 8]. With the definition of the IDMs for a single interface point, the dynamics is limited to a six degrees of freedom per node kinematics instead of the nine or more degrees of freedom of a measured FRF matrix. If the IDMs are defined to describe three translations (X,Y,Z) and three rotations (φx,φy,φz) of a single node in a global coordinate system, this kinematic description is equivalent to that of an FE node. In this case, the IDMs can be used to transform translational displacements to the motion of a single point in all six directions. Similarly, this can be used to extract a six-axis excitation to a point from a series of measured excitations. In this minimal example, it can be seen, that in the experimental case, the interface can neither be directly excited nor response signals acquired. Since only the points around the interface are of interest in the following, these are designated u for illustrative purposes. This results in the descriptive equation for the dynamics in the usual admittance form: u=Yf u∈Rn (7.3) The idea now is to represent the n interface displacements u by m < n rigid interface modes IDMs q. The IDMs are contained in the columns of the n ×mmatrix R, which is a frequency-independent eigenform matrix. Since the number of IDMs is less than the number of interface displacements, a residual term called μis added to the displacements. This residual term contains all displacements that cannot be mapped with the subspace of the IDMs. These displacements usually correspond to the flexible deformations. Thus the motion can be expressed as follows: u=Rq +μ q ∈Rm (7.4)
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