54 K. L. Napolitano {a(ω)}=[H (ω)]{f (ω)}= ai (ωk) ab (ωk) = Hif (ωk) Hib (ωk) Hbf (ωk) Hbb (ωk) ff (ωk) fb (ωk) . (6.1) Hereafter, the frequency, (ω), will be removed to simplify the equations. The responses or outputs, {a}, are on the left hand side. The references or inputs, {f }, are on the right hand side, and the FRF matrix, [H], describes their relationship at a given frequency. A partial inversion of the FRF matrix can be performed to make the boundary accelerations references and the boundary forces responses. This is done by rearranging the second row of Eq. (6.1) such that fb =−Hbb−1 Hbfff +Hbb−1 ab (6.2) Plugging Eq. (6.2) into the top row of Eq. (6.1) yields the following equation ai fb = Hif −HibHbb−1 Hbf HibHbb−1 −Hbb−1 Hbf Hbb−1 ff ab = Hif Hib Hbf Hbb ff ab (6.3) Many derivations yield the same results for Hif by making the unnecessary assumption that ab =0. Doing so obscures quite a bit of useful information. For any linear relationship in the form of {x}=[C]{y}, that the matrix element CJK is equal to the value of xJ due to a unit input at yK, holding all other elements in{y} equal to zero. Thus, there is no reason to enforce ab =0 in the derivation and all four elements of H can be used. Each element in has physical meaning. Hif is the FRF matrix of outputs ai due to unit input forces inff , holding all acceleration DOF in ab fixed. Hbf is the FRF matrix of reaction forces fb due to unit inputs forces at ff , holding all acceleration DOF inab fixed. Hib is the FRF matrix of outputs ai due to unit inputs in ab and not applying forces inff . Hbb is the FRF matrix of reaction forces fb due to unit inputs at ab and not applying forces inff . Note that this partial matrix inversion is equivalent to techniques described in other frequency based substructuring techniques. However, it should be noted that the calculation of the matrix H can also be calculated directly by using both forces ff andab as references when calculating FRF from time domain data. 6.3 Test Setup The test setup is shown in Fig. 6.1. A 73.1 in. cylindrical aluminum beam with a diameter of 2 in. was suspended in a free-free boundary condition with two soft rubber bands. A total of 26 accelerometers were bonded to the beam at 13 equally spaced locations. The accelerometers were mounted to measure both out of plane axes of the beam. The beam was impacted at five equally spaced locations along the beam in two axes; 1X+, 1Y+, 4X+, 4Y+, 7X+, 7Y+, 10X+, 10Y+, 13X+and 13Y+. 6.4 Signal Processing The test data was appended into a single time history file which was then processed to calculate FRF in one of two ways. The first method used all the force channels as references to calculate baseline FRF matrix [H]. Then the partial matrix inversion method was used to fix whichever sets of accelerometers were defined for each boundary condition case. The second method involved using the time histories of different sets of accelerometers directly when calculating FRFs. Processing the time history data into frequency response functions was complicated by the fact that the aluminum beam had very low levels of damping. These light levels of damping magnified small errors in the test setup and test conduct to produce drive point FRF that are physically unrealizable. The phase of all drive-point FRF should stay within a 180◦ band. Zeros in all FRF should be associated with a positive phase shift, and poles should be associated with a negative phase shift. Figure 6.2 presents two example FRF with two different exponential windows applied. The red FRF has a conventional 10% exponential window, but it has a negative phase shift associated with the zero near 300 Hz. The blue curve has a very drastic exponential window used in this paper that inputs enough damping into the FRF so that the phase shift is positive. However, there is still a frequency range from 360 to 510 Hz where the phase is slightly outside the 180◦ band.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==