In section 8.1, both amplitude and frequency errors at two frequencies of the substructure solution for the SDOF substructure system with typical dimensionless parameters have been presented. The parameter variations are:
- damping of the numerical structure: ζ = 0.01, 0.02 and 0.05;
- damping of the experiment: ζs = 0.02, 0.05 and 0.1;
- mass ratio: ηm = 0.02, 0.05, 0.1, 0.2, 0.5 and 1.0;
- frequency ratio: ηf = 0.5, 0.75, 0.95ηf0, ηf0, 1.05ηf0, 1.25 and 1.5, where
0
ηf is the optimum frequency ratio of the TMD system;
- the number of sub steps ksub=1, 2, 5, 10 and 20;
- and the dimensionless time step ηT in range from 0 to 0.2.
In section 8.2, phase lag effect is considered and the errors of the substructure solution for a typical case with a TMD (ηm =5%, ζ =5%, ζs =10%) are analyzed with the following variations of parameters:
- dimensionless phase lag ηp= 0, 0.5, 1.0, 1.5 and 2.0 - the number of sub steps ksub=2 and ksub=10.
In section 8.3, the substructure response of a TMD system in free vibration has been compared with the exact one to determine the errors at two frequencies. The errors in this case have been compared with the respective errors of the accuracy analysis in section 8.1. It has been shown that these errors fit well together.
For more details, the errors of the substructure solution in the different variations of the dimensionless parameters can be summarized in following.
• When the number of sub steps ksub increases from 1 to 20: The period elongation decreases slightly at the first frequency and does not change significantly at the second frequency (Figure 8.6). In difference with this, the amplitude increment increases at the first frequency and decreases at the second frequency (Figure 8.7). Moreover, when the number of sub steps, ksub, is in range between 2 and 5, certain values of the dimensionless time step ηT can be chosen in order to achieve very small amplitude errors.
• When the mass ratio ηm increases from 0.02 to 0.1 or from 0.2 to 1.0, the period elongation decreases slightly at the first frequency but it does not change at the second frequency (Figure 8.8 and Figure 8.16). In addition, the amplitude increment decreases at the first frequency and increases at the second frequency (Figure 8.9 and Figure 8.17).
• When the damping ratio ζ of the numerical substructure increases from 0.01 to 0.05, the period elongations at both frequencies do not change significantly (Figure 8.10) while the amplitude increment decreases at the first frequency and increases slightly at the second frequency (Figure 8.11).
• When the damping ratio ζs of the experimental substructure increases from 0.02 to 0.10, the period elongation increases slightly at the first frequency and does not change at the second frequency (Figure 8.12), the amplitude increment increases at the first frequency and decreases at the second frequency (Figure 8.13).
• When the frequency ratio ηf changes from 0.95 to 1.05 times of the optimum value of a TMD system, the period elongation increases at the first frequency and but it does not change at the second frequency (Figure 8.14), the amplitude increment does not change at the first frequency and decreases at the second frequency (Figure 8.15).
• In other coupled systems with large mass ratio, when the frequency ratio ηf changes from 0.5 to 1.5, the period elongation increases dramatically at the first frequency but it does not change at the second frequency. On the other hand, the amplitude increment at the first frequency decreases significantly while the the amplitude increment at the second frequency varies in a complex trend (see Figure 8.18, right).
• It was shown that the phase lag phenomenon causes larger amplitude and period errors at the peaks of transfer function (Figure 8.20). When the phase lag ratio ηp increases from 0 to 2, the amplitude increment at the first frequency decreases dramatically at the first frequency and increases extremely at the second frequency increases (Figure 8.22 and Figure 8.24). In difference with this, period elongation at the first frequency increases dramatically while period elongation at the second frequency decreases with the increase of ηp from 0 to 0.05 and increases with the increase of ηp from 0.05 to 2 (Figure 8.21 and Figure8.23).
8.6.2 Summary of stability analysis
In section 8.4, the stability of the substructure solution for the SDOF substructure system with typical dimensionless parameters has been presented. The parameter variations are:
- damping ratio of the numerical substructure: ζ ranges from 0 to 0.1, - damping ratio of the experimental substructure: ζs ranges from 0 to 0.1, - mass ratio: ηm ranges from 0 to 1.0,
- frequency ratio: ηf ranges from 0 to 4,
- dimensionless frequency: Ω ranges from 0 to 2π, - the number of sub steps: ksub is 1, 2, 3, 4 or 5.
Depending on the variation of the parameters, the different effects of the parameters on the stability can be summarized as follows.
• When the damping ratio of the numerical or the experimental substructures (ζ or ζs) increases, the stability of the substructure solution is clearly improved (Figure 8.28 to Figure 8.37). When a typical TMD system (ηm=5%, ζ =5%) is considered, the stability is concerned in the case of ksub=1 if the damping ratio ζs is about 5% or less. However, the other cases, with ζs> 5%, have not shown the stability problem. When a typical TMD system (ηm=5%, ζs =10%) is considered, the stability is only concerned when the damping ratio ζ is less than 1.1%. This should not be of concern in realistic applications because the realistic damping ratio is usually larger than 1.1%.
• When the mass ratio ηm increases from 0 to 1, the maximum spectral radius increases and the unstable region becomes larger (Figure 8.41 to Figure 8.45).
In most of cases of ηm , when ksub increases from 1 to 5, the stable region is lager and the limited frequency increases. In addition, when the mass ratio ηm is less than 17%, there is no concern of stability for any number of sub steps in range from 2 to 5.
• When the frequency ratio ranges between 0 and 4, the stability in the case of SDOF substructure (ηm =5%, ζ =5%, ζs =10%) has a certain unstable region when ksub takes a value of 1 (Figure 8.38). However, it has only a stable region when ksub ranges from 2 to 5 (Figure 8.39 to Figure 8.44).
In addition, the stability analysis has been validated in section 8.5 by checking the substructure solution in the time domain. The validation has shown that the stability states of the selected cases in the time domain are exact in the same as those of the stability analysis.
Moreover, at the end of this chapter, a simple error force compensation with P=1, D=0 and I=0 is used in the substructure algorithm to perform a VST of the substructure system at a checked point. The substructure solution without the force compensation was unstable but it became stable when the PID force compensation was applied.
9 DEVELOPMENT OF COMPENSATION METHODS
Based on the error force compensation in sections 2.4.2 and 4.3 and the methodology in section 6.2.1, new error force compensation is developed and presented in section 9.1. In addition, new phase lag compensation is proposed in section 9.2. Both two compensations use the same estimation and online system identification methods. The implementation of the new error compensations is presented in section 9.3. In section 9.4, some VSTs are performed on a SDOF substructure under earthquake load by a using hydraulic system to validate the compensations.