4 Validation of Blocked-Force Transfer Path Analysis with Compensation for Test Bench Dynamics 39 Active component A f1 uA 2 u B 2 Passive component B YA 21 gA 2 gB 2 YB 32 uB 3 Fig. 4.1 Two substructures: the active component A with the unknown excitation at point 1 and passive component B with receiver at point 3 set uB 3. On the interfaces, both substructures are subjected to interface forces g2. The system of equations of this coupled assembly can be written in block diagonal form as: 2 66 4 uA 1 uA 2 uB 2 uB 3 3 77 5 D 2 66 4 YA 11 YA 12 0 YA 21 YA 22 0 YB 22 YB 23 YB 32 YB 33 3 77 5 2 66 4 f1 gA 2 gB 2 0 3 77 5 (4.1) Here the upper part contains all Frequency Response Functions4 (FRFs) of the active component, whereas the lower part contains all FRFs of the passive component. In order to couple both substructures, the interface is subjected to a compatibility condition and an equilibrium condition, written respectively as: Compatibility W BuD 0 I I 0 2 66 4 uA 1 uA 2 uB 2 uB 3 3 77 5 D0 (4.2a) EquilibriumW g D BT (4.2b) HereBrepresents a signed boolean matrix.5 To determine the coupled response of uB 3 due to excitationf1, first the equilibrium equation (4.2b) is substituted into (4.1): 2 66 4 uA 1 uA 2 uB 2 uB 3 3 77 5 D 2 66 4 YA 11 YA 12 0 YA 21 YA 22 0 YB 22 YB 23 YB 32 YB 33 3 77 5 2 66 4 f1 0 3 77 5 (4.3) Next the interface forces are determined by substituting (4.3) into (4.2a): D .YA 22 CYB 22/ 1 YA 21 f1 (4.4) Finally, the response of interest is found by substituting (4.4) into the last line of (4.3)6: uB 3 D YB 32 DYB 32 .YA 22 CYB 22/ 1 YA 21 f1 (4.5) Note that (4.5) can also be expressed in the assembled system’s receptance, using YAB 31 DYB 32 .YA 22 CYB 22/ 1 YA 21 (4.6) Where superscript AB, denotes the receptance pertains to coupled structure. This approach is consistent with LM-FBS method.7 In summary this section derived the response of the passive component caused by the excitation of the active component, e.g. uB 3 DYAB 31 f1, based on the FRFs of the subsystems. 4The FRFs in YA and YB are measured on the separate components. 5For more information on the expression of the compatibility condition using signed boolean matrices, the reader is referred to [4]. 6Note that additional excitations on the passive side can formally be included as well by defining f 3 and including it in the force vector in (4.3). 7It can be verified that (4.6) can also be obtained by development of the LM-FBS notationYAB DY YBT BYBT 1 BY.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==