3 Controls Based Hybrid Sub-Structuring Approach to Transfer Path Analysis 23 Fig. 3.9 Six degree of freedom Shore Western shake table For the TPHS method to work, the physical loading of the full system needs to be quantified so that it is the force due to the physical loading at the interaction point between the physical and numerical substructures. In this test case the physical loading is the displacement excitation of the Quanser Shake Table. This was measured by testing the physical substructure with the Quanser Shake Table excitation with the output being tri-axial force sensors below the physical substructure’s isolators at the base, which is the interaction point between the numerical and physical substructures. Lastly, the physical substructure was experimentally tested. In the TPHS arrangement, this is a reactant force due to a base displacement excitation. To experimentally measure this relationship, a six degree of freedom (6DOF) Shore Western Shake Table, shown in Fig 3.9, was used. The physical substructure was excited with a band-limited white noise (BLWN) displacement from 0 to 20 Hz in all six Cartesian directions. Similar to when the physical loading was recorded, the physical substructure had tri-axial force sensors below its isolators at the base to record the reactant force. This arrangement allowed for the direct measurement of the desired physical substructure transfer function. To experimentally verify the TPHS calculation using a PSD matrix as the physical loading, the numerical substructure model and the physical substructure experimental test were coupled together using Eq. (3.4) and then compared to the experimental full system which was constructed and tested. Figure 3.10 shows the comparison of the TPHS method vs. the experimental measurement of the mechanical systems base force PSDs. Being able to calculate the PSDs of the coupled systems response is typically more advantageous because it is measure of the actual levels of response instead of transfer function calculations which are normalized measurements of the mechanical systems response. This comparison shows that TPHS is a viable substructuring method that can use PSDs of mechanical system excitation to calculate the coupled dynamics of the mechanical system. This is the major advantage of this method. 3.6 Conclusion This paper demonstrated a new frequency based substructuring method referred to as Transfer Path Hybrid Substructuring. It was demonstrated that this method is mathematically equivalent to traditional dynamic substructuring. In addition, it was shown that this method can be used to accurately couple physical loading with unknown vibration excitation, with the dynamics of a numerically modeled support structure. This is the main advantage of this method over other substructuring methods since typically it is very difficult to quantify the exact source of the system excitation. This method does have required conditions. The physical loading should be measured with the physical substructure having a perfectly rigid interface to the test base in the frequency range of interest. This method also requires that the physical substructure transfer function (reactant force due to an applied base motion excitation) can be measured using linear signal processing techniques. This obviously assumes that this substructure is a stationary, ergodic system. In the case study
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