9.4 Verification of the compensations by using virtual substructure test
9.4.5 Comparison and discussion on the test results
Figure 9.22 and Figure 9.23 show the comparisons between the responses of VSTs without any compensation (test No. 4 in Table 7.9) and the reference solution.
Figure 9.22: Comparison between the displacements of the VST without compensation and the reference solution
Figure 9.23: Comparison between the coupling forces of the VST without compensation and the reference solution
-40 0 40
0 20 Tim e (s) 40
Displacement (mm)
Substructure Reference
0.0 1.6
0.0 2.5 Frequency (Hz)5.0
Spectrum of displacement (mm)
Substructure Reference
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Coupling force (N)
Substructure Reference
0 100
0.0 2.5 Frequency (Hz)5.0
Spectrum of coupling force (N)
Substructure Reference
The comparisons are displayed in the time domain (left graphs) and the frequency domain (right graphs). The VST is performed on a hydraulic system with a typical phase lag (time lag approximately 15 ms) while the reference solution is obtained by using the mode superposition method and the exact direct integration method (Appendix A1).
From these comparisons, there is a large error in both the time domain and the frequency domain. In frequency domain, the amplitudes at the second frequency are gained while the amplitudes at the first frequency are decreased. The reason of the large error in this VST is that the effects of the phase lag and the error force in the substructure algorithm have not been compensated. This confirms the effect of phase lag on the substructure solution as presented in section 8.2.1.
Figure 9.24: Comparison between the displacements of the VST with the force compensation and the reference solution
Figure 9.25: Comparison between the coupling forces of the VST with the force compensation and the reference solution
Figure 9.24 and Figure 9.25 show the comparisons between the results of VST with error force compensation (Test No. 5 in Table 7.9) and the reference solution. The
-40 0 40
0 20 Tim e (s) 40
Displacement (mm)
Substructure Reference
0.0 1.6
0.0 2.5 Frequency (Hz)5.0
Spectrum of displacement (mm)
Substructure Reference
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Coupling force (N)
Substructure Reference
0 40
0.0 2.5 Frequency (Hz)5.0
Spectrum of coupling force (N)
Substructure Reference
results of the VST with force compensation have large differences in comparison with the reference solution in both the time and frequency domains.
Figure 9.26 and Figure 9.27 show the comparisons between the results of VST with phase lag compensation (Test No. 6 in Table 7.9) and the reference solution. By comparing them with the reference solution, the results of the VST have very small differences in both time and frequency domains. In the time domain, the response of the VST fits well with the reference solution. However, small errors at the second frequency in both displacement and force spectrums can be easily recognizable.
Figure 9.26: Comparison between the displacements of the VST with the phase lag compensation and the reference solution
Figure 9.27: Comparison between the coupling forces of the VST with the phase lag compensation and the reference solution
Figure 9.28 and Figure 9.29 show the comparisons between the responses of a VST with both error compensations (Test No. 7 in Table 7-9) and the reference solution. By comparing them with the reference solution, the results of the VST have very small differences in both the time and frequency domains. In the frequency domain, very
-800 0 800
0 20 Tim e (s) 40
Coupling force (N)
Substructure Reference
0 40
0.0 2.5 Frequency (Hz)5.0
Spectrum of displacement (mm)
Substructure Reference
small errors at the second frequency on both displacement and coupling force can be observed.
Figure 9.28: Comparison between the displacements of the VST with both force and phase lag compensations and the reference solution
Figure 9.29: Comparison between the coupling forces of the VST with both force and phase lag compensations and the reference solution
From the results above, a comparison between these VSTs with different configurations of two compensations could be made in order to see the effectiveness of two compensation methods on the substructure solutions. A comparison on the coupling forces is shown in Figure 9.30.
The graphs in Figure 9.30 show that when there is no compensation, the difference between the response of the VST and the reference solution is quite large at the second frequency. When the force compensation or the phase lag compensation is applied (blue graph or pink graph), the test results are significantly improved. Especially, when both compensations are used (green graph), the test result fits very well with the reference solution.
0.0 1.6
0.0 2.5 Frequency (Hz)5.0
Spectrum of displacement (mm)
Substructure Reference
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0 20 Tim e (s) 40
Displacement (mm)
Substructure Reference
0 120
0 1 2 Frequency (Hz) 3
Spectrum of coupling force (N)
VST no compensation VST w ith force compensation VST w ith phase lag compensation VST w ith both compnsations Reference
Figure 9.30: Force comparison in VSTs with different configurations of the error force and phase lag compensations and the reference solution
In addition, the phase lag compensation in this case is much more effective than the error force compensation. This is likely due to the fact that phase lag compensation has dealt with a delay time that is approximately equivalent to six subs steps (about 15 ms) while the force compensation has dealt with a smaller error due to a shorter time interval of the sub step (2.5 ms).