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• W2003226394 abstract "Floor vibration researchers frequently use experimental modal analysis techniques to determine accelerance frequency response functions (FRFs), natural modes, and damping of laboratory specimens and building floors. Electrodynamic shakers (Fig. 1) are often used to provide excitation forces during these tests because they provide higher quality FRFs than can be estimated from tests performed using other forcing methods.1-5 Chirp excitations (Fig. 2) are often applied to the structure because they provide excellent signal-to-noise ratios.2-4, 6 To estimate FRFs, the floor acceleration response and input force must be measured, either directly or indirectly. Acceleration responses are directly measured using accelerometers placed on the slab. Applied forces may be directly measured using a force transducer (force plate) or indirectly measured using an armature accelerometer. Using the latter method, the armature acceleration is multiplied by the armature mass to estimate the force applied by the shaker. The goal of this paper is to investigate force measurement methods as they relate to floor vibration modal testing. The basics of shaker–structure interaction and consequences of this interaction are presented. An experimental program and its results are described, followed by conclusions that can be made from its results. Floor shaker, force plate, driving point accelerometer, and armature accelerometer Force waveform showing “force drop-off” at resonance It is well known that shakers interact with structures under test (SUT) and produce force drop-offs when the SUT is excited at or near resonance (Fig. 2). In the literature,7, 8 the effect is sometimes referred to as a force spectrum “glitch” which is composed of a peak and a notch as shown in Fig. 3. Because an armature's natural frequency is lower than the floor's natural frequency, a glitch begins with a sharp increase in the force spectrum followed by a sharp decrease.9 As noted by several researchers, amplifier mode (voltage vs. current), damping of the SUT, and SUT-to-armature mass ratio are the parameters that most significantly affect force notch depth and bandwidth. Notches are smaller for shakers run in current mode and for SUT with higher damping. For grounded SUT, increased SUT-to-armature mass ratio results in smaller notches.6-13 Although floor vibration researchers have the ability to use amplifiers in current mode, they have no control over SUT damping and have limited control over SUT-to-armature mass ratio. Note that floors are continuous structures, so the “modal” or “effective” mass is the most appropriate measure of the SUT mass. Force spectrum showing force “glitch” The force autospectrum is used during estimation of the FRF. Using the most popular FRF estimator (H1), the accelerance FRF, a complex quantity at each spectral line, is expressed as: The magnitude of the accelerance FRF, a real number at each spectral line, can be expressed as the ratio of the acceleration and frequency autospectra, given by Eq. 2. The force autospectrum appears in the denominator of Eqs. 1 and 2, and has a direct influence on the FRF estimate. where Gxx(ω) = Autospectrum of the acceleration response at frequency ω When using the force plate to directly measure applied forces, force glitches are detected but are still significant due to three related issues: SUT nonlinearity, low signal-to-noise ratio at the notch, and the lack of repeatability of FRF estimates at frequencies between the peak and the notch.12 All structures are nonlinear to some degree and drastically different force levels will be input at adjacent spectral lines if a large glitch exists. This concern can be reduced for floor vibration testing, however, by limiting the force input to levels, which will cause very small accelerations, which are of interest for floor vibration serviceability. Insufficient signal-to-noise ratio remains a possible problem near the notch, but observation of the coherence function indirectly indicates whether sufficient force was applied at these spectral lines. It is the authors' experience that large force spectrum glitches decrease the repeatability of FRF estimates for lightly damped specimens even when a force plate is used. For example, Fig. 4 shows sample driving point acceleration and force plate autospectra. The acceleration autospectrum varies rapidly near the natural frequency, and the force spectrum contains a very large discontinuity between the glitch peak and notch, which are also very near the natural frequency. Therefore, very small differences from test to test often result in FRF estimates that are significantly different. (a) Driving point acceleration and (b) force plate autospectrum Using the armature accelerometer method, the consequences are more severe because the computed force will be incorrect at most spectral lines near SUT natural frequencies. The situation is not improved by the use of the H1 FRF estimator which only minimizes uncorrelated response content.6, 10, 11 Of the parameters influencing force spectrum glitch severity, the mass ratio is of primary interest for this study because the researcher has some control over this ratio. As previously mentioned, for grounded structures, higher structure-to-armature mass ratios result in decreased glitch severity. The mass of the structure is not the most appropriate numerator because the glitch severity can vary for different locations on the same structure, as shown in Fig. 5. The force spectra shown in Fig. 5 were measured on the same single-span laboratory specimen using the same input current and armature mass. The location varied, however. Figure 5a and b shows the force spectra that were measured for tests conducted with the shaker at midspan and at a quarter point, respectively. Force autospectra. (a) Shaker at midspan; and (b) shaker at quarter point The force glitch is more severe for the test conducted at midspan because the structure's modal mass is smaller at that location than at the quarter point. Because there is no single definition of modal mass, it is defined in Eq. 3 for use in this study. Rearranging the steady-state amplitude for a lightly damped system gives the modal mass: |Ẍ| = Acceleration magnitude |F| = Force magnitude |A| = Accelerance magnitude Using this definition, it is possible to compute the modal mass at a given FRF peak. The key parameter is the ratio of modal mass-to-armature mass. The size of a force glitch is shown later in this paper to decrease with larger modal masses or equivalently, lower accelerance magnitudes. To illustrate the consequences of using the two force measurement methods, FRFs were measured for four structures. Widely varying mass ratios were achieved by selecting structures with different masses, varying the shaker location, and varying the armature mass. Structure A was a very light three-span footbridge laboratory specimen constructed using a composite slab supported by light steel trusses (joists). Each bay was 2.1 m (7 ft) wide by 9.1 m (30 ft) long and weighed approximately 58,000 N (13,000 lbf). Structure B was a lightweight building floor under construction, built using a noncomposite slab supported by light steel trusses (joists). Bay sizes varied slightly, but the average weight was approximately 125,000 N (28,000 lbf). Structure C was a long-span composite slab laboratory specimen, spanning 9.1 m (30 ft) between perimeter girders and weighing 280,000 N (63,000 lbf). Structure D was a heavier building floor under construction, built using a composite slab supported by conventional steel framing. The average bay weighed approximately 218,000 N (49,000 lbf). All four structures were bare slabs with very low critical damping ratios. For Structure A, the damping ratios were 0.2, 0.3, and 0.34% of critical for the three FRF peaks considered. For Structure B, the damping ratio was approximately 1%. For Structures C and D, the damping ratios were approximately 0.5% of critical. Driving point FRFs were measured at bay centers for all four structures. Because the modal mass varies with shaker location, additional measurements were taken at a one-eighth span point on Structure A. Additional measurements were also taken on Structure C at the quarter point between midspan, and one of the girders and at the edge of slab, directly over a girder. Burst chirp forces were applied using an APS Electro-Seis Model 400 electrodynamic shaker (APS Dynamics Inc., Carlsbad, CA). Chirp excitation levels were selected to result in midspan acceleration in the range of interest for floor vibration serviceability, that is, between approximately 0.5 and 2%g. During the majority of the tests, the entire shaker armature mass was used, 30.5 kg (67.4 lbf). Two of the four armature mass plates were removed during several of the Structure C tests, resulting in an armature mass of 17 kg (37.4 lbf). The APS Dual-Mode Model 144 Amplifier was operated in current mode for all the tests. Driving point floor accelerations were measured using a PCB Model 393C seismic accelerometer. A custom-built force plate was used to directly measure the applied forces. A PCB Model 393B04 accelerometer (PCB Piezotronics, Inc., Depew, NY) was used to measure armature accelerations. A Model 20-42 SigLab unit (Spectral Dynamics, Inc., San Jose, CA) was used to send signals to the shaker amplifier and to record force plate and accelerometer signals. Of the four specimens, Structures A and D illustrate the most severe and least severe consequences of force spectrum glitches. Therefore, detailed results for these two structures are presented. In all cases, both force measurement methods resulted in approximately equal natural frequencies. However, significant differences were observed in the accelerance peak magnitudes in several cases. Figure 6a and b shows the applied force waveforms from direct measurement and from the armature accelerometer method, respectively, for Structure A, the light footbridge. Figure 6a shows a large force drop-off which is not seen in Fig. 6b. Figure 6c shows the frequency content of the measured and computed force waveforms, indicating large force notches which are not detected using the armature accelerometer method. Figure 6d shows the accelerance FRFs, indicating very different results, and Table 1 summarizes the accelerance peaks estimated using the two references. It also includes the mass ratio determined using Eq. 3, and the accelerance peak magnitude estimated using the force plate as the reference. A1, A2, and A3 are peak magnitudes for the first three modes. Results for Structure A. (a) Measured force; (b) computed force; (c) force frequency content; and (d) accelerance FRFs Figure 6 and Table 1 indicate that the accelerance peak magnitudes estimated using the two references are different by as much as 60% for Structure A which has a very low modal mass-to-armature mass ratio. Estimates obtained using the force plate are presumed to be more accurate than those obtained using the armature accelerometer because the armature accelerometer does not detect the force drop-off and force glitch. Figure 7a and b shows applied force waveforms for Structure D, the heavy building floor, determined using the force plate and armature accelerometer, respectively. Figure 7a shows a small force drop-off, at approximately 17 s, which is not seen in Fig. 7b. Figure 7c shows the frequency content of the measured and computed force waveforms, indicating a small force glitch which is not detected using the armature accelerometer method. Figure 7d shows the accelerance FRFs, indicating almost identical results, and Table 2 lists the accelerance peak magnitudes and mass ratios for the first three modes. Results for Structure D (building floor). (a) Measured force; (b) computed force; (c) force frequency content; and (d) accelerance FRFs Figure 7 and Table 2 indicate that the accelerance peak magnitudes estimated using the two references are in reasonable agreement for Structure D which has a much more favorable modal mass-to-armature mass ratio than Structure A. Each accelerance peak obtained for the four specimens was analyzed to determine the ratio of accelerance peak magnitudes obtained using the two force measurement methods. Figure 8 shows the ratio of accelerance peak magnitudes plotted versus the modal mass-to-armature mass ratio. As expected, the results tend to be in better agreement for larger mass ratios. Accelerance ratios for tests with mass ratios exceeding 1000 were different by a maximum of 13.2% and an average of 2.9%. Peak accelerance magnitude ratio versus mass ratio for all FRF peaks Figure 9 shows the force autospectrum for the test with the highest mass ratio, approximately 7100. It was obtained during the Structure C test with the smaller armature mass and the shaker placed over a girder. The resulting accelerance peaks were 5% different for this test. Figure 10 shows a closer look at the force autospectrum, indicating that a small force glitch exists even at this high mass ratio, explaining why the accelerance ratio does not converge towards unity for the mass ratios considered. Force autospectrum, test with maximum mass ratio Force autospectrum, test with maximum mass ratio (closer look at force glitch) This paper presented an investigation of two common shaker force measurement methods, direct measurement using a force plate and indirect measurement using an armature accelerometer, and their effects on estimated accelerance FRFs. The investigation centers around shaker–structure interaction, specifically force drop-off (and accompanying “glitches” in the force spectrum) near resonance, and its effects. Although they are a concern regardless of the force measurement method, glitches are detected when using a force plate, but not when an armature accelerometer is used to indirectly measure the applied force. Therefore, results obtained using a force plate are considered to be more accurate than the ones obtained using an armature accelerometer. Four specimens with very different masses were modal tested to obtain driving point accelerance FRFs to evaluate the effect of the modal mass-to-armature mass ratio on glitch severity. Shaker location and armature mass were varied to obtain additional measurements with different modal mass-to-armature mass ratios. In practically all cases, the two methods indicated approximately equal natural frequencies. However, from a floor vibration serviceability perspective, natural frequency estimates are of secondary importance to acceleration response. The lightweight footbridge (Structure A) represented the worst-case scenario: small modal mass-to-armature mass ratio. Accelerance FRFs were very different depending on the force measurement method due to the presence of large force drop-offs and large glitches in the force spectrum, which go undetected using the armature accelerometer method. The heavier building floor under construction (Structure D) represented a much less severe scenario due to its higher modal mass-to-armature mass ratio. Accelerance FRFs were in good agreement for this specimen. The ratio of accelerance peak magnitudes obtained using both methods was determined and plotted versus the modal mass-to-armature mass ratio for all accelerance peaks included in the study. The plot indicates poor agreement for smaller mass ratios and much better agreement for larger ones, as expected. Agreement to within approximately 10–15% was indicated for tests with mass ratios exceeding 1000. Force glitches are observed during tests with large mass ratios, even as high as 7100, explaining the discrepancies between the two methods even at the highest mass ratios considered. Therefore, it is concluded that a force plate should be used during modal tests to increase the probability of estimating high quality accelerance FRFs. If lower quality modal estimates are acceptable for the intended research objective, then an armature accelerometer may be used in lieu of a force plate if the modal mass-to-armature mass ratio is large, say exceeding 1000. Very significant estimation errors are likely to result for mass ratios lower than approximately 1000, which would be very common for light laboratory footbridge and floor specimens and some steel framed floors. Even when using a force plate, it is the authors' experience that higher quality FRFs are obtained when the mass ratio is maximized, therefore minimizing the severity of force glitches. Therefore, manipulation of shaker armature mass and shaker location should be used when necessary to increase the quality of FRF estimates." @default.
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• W2003226394 date "2009-07-28" @default.
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• W2003226394 title "USE OF A FORCE PLATE VERSUS ARMATURE ACCELEROMETER FOR MEASURING FREQUENCY RESPONSE FUNCTIONS" @default.
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