Structural Coupling Analyses of Experimental Models in a Virtual Shaker Testing Environment for Numerical Prediction of a Spacecraft
10.4 Experimental Results and Numerical Analysis for Coupled Shaker and Test Specimen Models
Fig. 10.8 Left—experimental and synthesised FRFs for data driven simulation model of HE and HE/beam structure coupled models in free-free boundary conditions.Right—acceleration spectra simulation results for preliminary sine control prediction using the experimental and synthesised FRFs and model from the left
coarse frequency resolution of 4 Hz of the measured FRF in combination with the interpolation and the frequency sweep rate. Consequently, the synthesised FRFs result in a smoother control since the frequency resolution of the FRF can be chosen arbitrarily and adapted to the sine control parameter, and no noise occurs. Despite this improvement, a deviation is also observed at the anti-resonance around 900 Hz. This is a result of the modal parameter estimation and the resulting models (dashed magenta and dotted black) which do not fully approximate the measured FRF (green) on the left of Fig.
10.8. Consequently, the resulting and predicted acceleration control spectra show this effect as a deviation in the control performance, too. The simulation results based on the dynamical models in z-domain transfer function and discrete state- space representation derived from the estimated s-domain PolyMAX model are equal to each other and show the same dynamic results as the data driven approach by using the synthesised LUT approach. All sine control predictions overlap and exhibit an equal characteristic. Consequently, it is summarised for this application and with the given settings, that both modelling approaches, either direct data driven LUT or the dynamic model approach are capable of generating a preliminary reliable sine control prediction. The deviations occurring at the anti-resonance are still being investigated but can be minimised by adapting the modal parameter estimation.
10.4 Experimental Results and Numerical Analysis for Coupled Shaker
Fig. 10.9 Left—impact testing FRFs of coupled Head Expander and beam structure in free-free and fixed-free (mounted on shaker table) boundary conditions.Right—open-loop random testing FRFs of coupled system.
• Additionally, first shaker table rotational (rocking) modes are excited during test in the fixed-free case and occur as additional resonance peaks in the HE sensor HeadExp:RB (red) and beam sensor Beam:B3 (magenta) FRFs at low frequencies at approx. 34 and 66 Hz, and a dominating mode at 100 Hz prior to the 1st beam bending mode as shown in Fig.10.9.
• The high frequency range is dominated by higher order modes at 660 and 960 Hz as clearly indicated by HeadExp:RB (red) and Beam:B3 (magenta) whereas for the free-free case only the clearly identified saddle mode at approx. 842 Hz is detected and shown by the dashed FRFs.
This shaker to test structure interaction will increase if heavier test specimens are tested and can lead to major challenges while performing a controlled closed-loop vibration test.
To asses such interactions and coupling effects low level open-loop random tests (0.3 Vrms and 0.5 Vrms over the entire frequency range <1000 Hz) are performed and presented in the right plot of Fig.10.9. The results show acceleration-over- current and voltage-over-current FRFs acquired during open-loop random testing. The modes which are identified by impact testing are clearly excited and well presented in the acquired FRFs. The FRFs of the two different excitation levels totally overlap and show now presence of structural non-linearities besides of the low frequency suspension mode (shaker table and coil move in phase w.r.t. the shaker body). This mode shows a softening effect which is decreased in frequency by increasing the excitation level (0.3Vrms: 14.6 Hz, 0.5Vrms: 12.8 Hz) and can be related to the guidance system of the shaker. In the future it is intended to investigate this occurrence more in detail.
The open-loop random tests are performed to generate a data set for system identification purposes as it is used during the self-check phase which is performed prior to a closed-loop sine control execution. A schematic sketch is given on the left of Fig.10.10. A test structure model is coupled to the electrodynamic shaker system and to the vibration controller C(s). The shaker facility and test specimen interacts with the vibration test controller by a closed-loop acceleration feedback control system to be capable of controlling the shaker table acceleration w.r.t. required test specifications by driving the electrodynamic shaker with an appropriate voltage signal. This voltage signal is calculated by the control algorithm based on the test specifications and internal vibration control parameter, e.g. sweep rate for sine sweep testing, to perform the controlled test execution.
Since, hereafter, an alternative method is presented to derive a coupled shaker and test specimen model to numerically predict sine vibration tests without the need to physically couple an electrodynamic shaker model (refer to Sect.10.2) with structural model representations but to use experimental test results in a data driven approach instead.
To validate the subsequent numerical sine control simulations, Fig.10.10on the right, shows experimental sine control test results of a logarithmic sweep of 4 oct/min up to 1000 Hz with a control reference spectrum of 1 g of the coupled HE/beam structure in blue and the HE without beam structure in red.
Subsequently, the open-loop random results are used to calculate acceleration-over-voltage transfer functions, estimate discrete model representations based on a PolyMAX modal parameter estimation approach and to use the derived z-domain models in a closed-loop simulation coupled to the sine control model. Consequently, this approach follows the equivalent steps as introduced in Sect.10.3.3but instead of using the modal models, the coupled electrodynamic shaker and test structure
Fig. 10.10 Left—sketch of coupled shaker and test specimen to vibration controller for system identification purposes.Right—experimental sine control test results (reference control spectrum: 1 g, sweep rate: 4 oct/min) of shaker system loaded with HE (red) and coupled HE/Beam (blue) system
Fig. 10.11 Left—experimental and synthesised acceleration-over-voltage FRFs calculated from open-loop random excitation tests of Fig.10.9 (right).Right—numerical closed-loop sine control predictions (reference control spectrum: 1 g, sweep rate: various) using the estimated z-domain model on the left and comparison to predicted results of Fig.10.8using the free-free model representation
dynamics is comprised in the discrete model representation of the acceleration-over-voltage transfer function (left sketch of Fig.10.10). The measured (black), synthesised (green) and z-domain polynomial model (red) of the transfer function between the beam sensor B3 and the electrical voltage are presented in the left plot of Fig.10.11.
The following numerical sine control predictions with a control reference spectrum of 1 g over the entire frequency range of 10 to 1000 Hz are presented on the right of Fig.10.11. The figure shows the acceleration spectra for different sine sweep rates: in blue a logarithmic sweep rate of 4 oct/min equivalent to the real experimental sine test results as shown in the right of Fig.10.10, in green and black linear sweep rates of 2 Hz/s and 1 Hz/s, and in magenta it reviews the results of Sect.10.3.3 considering the estimated modal models in free-free boundary conditions. It needs to be mentioned that different control sensors were selected.
Nevertheless, the comparison between the numerically simulated and predicted results (Fig. 10.11right) to the real physical test results (Fig.10.10right) can be used to draw some preliminary observations. By comparing the blue acceleration spectra of both figures similar control deviations in terms of amplitude and frequency occur at structural modes and coupled structural and shaker modes even if different sensors are used for control. The experimental results use the maximum signal of all HE acceleration sensors whereas the numerical model is based on the beam sensor B3. Consequently the simulations show the presence of the beam’s 1st bending mode clearly as control deviation at 157 Hz whereas the experimental results are less influenced. All other modes which have been detected while impact and open-loop random testing occur as control deviations in the acceleration spectra of the experimental test and numerical predictions in an equivalent manner.
Additionally, the numerical simulations can be used to simulate beating phenomena influenced by the sine control sweep rate as shown by the blue, green and black predictions with different sweep rates of 4 oct/min, 2 Hz/s and 1 Hz/s, respectively.
According to ECSS [2] and industrial publications of Thales Alenia Space [8,21] the beating effect occurs if the sweep rate is not sufficiently slow, then the dynamic and coupled system response is no longer stationary (steady-state) at each instant of time and spectral sine excitation. Consequently, it results in a modification of the shape and position of a resonance followed by a ringing effect as a function of the sweep rate and direction. These beating phenomena can be clearly observed after the beam’s first resonance at 157 Hz. It is clearly shown that with decreasing sweep rate (from blue to black) the beating phenomena and control deviations decrease at structural resonances. In practice, it is usually not possible to apply very low sweep rates to provide better control in this way, since the risk of damaging the structure under test increases as well as the testing time. Consequently, it is of major importance to include the virtual shaker testing approach in the S/C vibration testing procedure for being capable of including and predicting sweep rate effects and to avoid misinterpretations during tests.
If a preliminary sine control prediction is performed as explained in Sect.10.3.3and represented by the magenta control spectrum in Fig.10.11a significant deviation at shaker modes (34 Hz, 66 Hz and 100 Hz) or coupled HE structural modes (660 and 960 Hz) is detected compared to simulation approach considering the coupled shaker and structural dynamics. In contrary, the two approaches show similar results for the unchanged 1st beam resonance mode at 156 Hz and can be used as a fast methodology to assess the control performance in this frequency range. Considering the entire frequency range of testing and the increasing model and modal complexity the investigations clearly show that the virtual shaker simulation environment needs to comprise all three major contributors of the vibration test chain to predict the outcome of a S/C vibration test and to assess possible challenges which can occur during the test execution.