10 Source Characterization for Automotive Applications Using Innovative Techniques 119 Fig. 10.2 Schematic showing the virtual point transformation [4] 10.1.2 Virtual Point Transformation Additionally, to obtain the FRFs in Eqs. 10.2 and 10.3 with references exactly at the interfaces, the virtual point (VP) transformation can be used. With respect to TPA, the virtual point transformation can be used to provide a full 6-DoF description of the forces and moments acting exactly at the interface. The VP transformation is shown schematically in Fig. 10.2 and further detailed in [4]. Using this technique, forces are applied in a typical manner using an impact hammer or shaker, and a geometry-based transformation matrix is used to transform the FRF to have inputs exactly at the desired interface node(s), including both translations and rotations. Essentially, this transformation can be written using interface displacement modes (IDMs) as u=Ruq ⇒ q=(Ru)+u (10.4) m=Rf Tf ⇒ f = Rf T +m (10.5) where the measured responses u are transformed to the q virtual point responses using the response IDM matrix Ru, and similarly the measured forces f are transformed to the mvirtual point forces and moments using the force IDM matrix Rf. The IDM matrices are constructed to accurately characterize the problem at hand, often using six degrees of freedom (three translations and three rotations). The sensors and excitations are typically placed close to the virtual point such that rigid IDMs are used; however flexible IDMs can also be included. The consistency functions detailed in [5] can be used to assess whether the chosen IDMs accurately represent the observed dynamics. The sensor consistency is essentially calculated by transforming the measured responses to the virtual point, projecting the virtual point responses back to the original coordinate system, and comparing the “filtered” responses with the original raw signals. This produces a frequency-dependent metric that is equal to 1 if all sensor responses are accurately described by the chosen IDMs. A similar calculation can be done for the impacts. The sensor and impact consistencies are typically assessed for the measurements made around each of the attachment points of an active source. 10.1.3 Techniques Presented Here In-situ blocked force TPA using the virtual point transformation is becoming increasingly common in the automotive world. Under typical testing conditions where impact testing performs well, the TPA results are generally quite good. However there are certain conditions where the results can be improved using the techniques described below. Rigidness Correction for Low Frequency TPA At low frequencies (less than 20 or 50 Hz), it is often challenging to get enough consistent energy into large structures using a regular impact hammer. Data quality issues at the low frequencies are generally not uncommon when performing impact testing; for an overview of practical test considerations see [6]. These issues can be accepted in many applications where higher frequencies are the source of noise and vibration problems and the low frequency data quality is not paramount to the success of the analysis. However, for certain applications such as ride comfort, the lower frequencies can be of high interest. For such cases, the consistency of the data at the low frequencies should be reviewed. At the very low frequencies, all sensors should move together, exhibiting rigid body motion. The sensor consistency can be used to assess the rigidness of
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