From the theory of accuracy analysis in section 6.1.2, the parameters of the substructure are the damping ratio ζ of numerical substructure, the damping ratio ζs of experimental substructure, the mass ratio ηm and the frequency ratio ηf ; while the parameters of the algorithm for substructure testing are the dimensionless time step ηT and the number of sub steps ksub. In addition, in order to investigate the accuracy of the substructure solution with phase lag, there should be one dimensionless parameter of phase lag.
From the discussion in section 3.2, a phase lag at a certain frequency can be represented by the time lag δt. In this research, the phase lag ratio ηp defined by the ratio between the time lag δt and the time step ∆t as Eq. (7.1) is used for discussion on the phase lag. The list of parameters for stability and accuracy analyses is shown in Table 7-1.
t t
p = ∆δ
η (7.1)
Table 7-1: List of parameters for stability and accuracy analyses Name
of parameters
Symbol Possible range
Practical ranges in civil engineering Damping ratio of the
numerical substructure
ζ 0 – 1 Usually between 1% and 5% ,
rarely smaller than 1% or larger than 5%
Damping ratio of experimental substructure
ζs 0 – 1 Usually from between 1% and 10% , rarely smaller than 1% or larger than 10%
Mass ratio
ηm ≥ 0 Usually between 0 and 1 Frequency ratio
ηf ≥ 0 Usually about 1 for TMD,
any value for other coupling systems Dimensionless time
step ηT > 0 Considered range between 0 and 0.2
Number of sub steps
ksub ≥ 1 Usually between 2 and 10 Value of 1 for explicit scheme Phase lag ratio
ηp ≥ 0 Usually between 0 and 2
There are some notes in the practical ranges of the parameters in Table 7-1. The damping ratio ζ is considered under damping and usually ranges between 1% and 5%.
For example, steel structures have a damping value less than 1% while concrete structure usually have a higher damping of approximately 5%.
The dimensionless time step ηT can be considered in a range between 0 and 0.2. That is due to the following reasons. The first reason belongs to the digital world.
According to the Nyquist theorem, digital data can represent properly a signal only if the sampling time interval is not larger than the half of period of the signal. Thus, the range ηT >0.5 is not used in the accuracy analyses in this thesis. In addition, when the sampling time interval is near to the half of period of the signal, the data will not represent well the signal. Another reason is that, when ηT is larger than 0.2, the errors in the integration scheme will be very large (see Figure 2.4 in section 2.2). From these, the considered range of ηT in this research is set to be between 0 and 0.2.
Concerning the number of sub steps ksub, Dorka et al. (1998) analyzed the effects of different values of ksub in a range between 2 and 9. Since the time step in most of applications in civil engineering is between 5 ms and 20 ms and the sub-step time interval is usually larger than 0.5 ms, the number of sub steps in this research is chosen within the range from 1 to 10. Exceptionally, in some analyses, ksub =20 is used to investigate the case with large number of sub steps.
The mass ratio is usually between 0 and 1 because the masses of the experimental substructures are usually smaller than those of the numerical substructures. Values of ηm about 2% and 5% are usually used for TMDs while any value in the range between 0 and 1 may be used for other coupled substructures.
The damping ratio ζs is considered in under damping and it usually ranges between 1% and 12%. The value of ζs about 10% is usually used for TMDs. Smaller values are used for normal coupled substructures.
The frequency ratio ηf is usually smaller and close to 1 for TMD substructures, while other values may be used for other coupled substructures. For TMD systems, the parameter ηf is used near to the optimum value ηf0 at which the two peaks of the transfer function have the same values. In addition, since the transfer function in a TMD system is sensitive to the frequency ratio ηf , the errors of the transfer function are also analyzed at 95% and 105% of the optimum one ηf0.
In civil engineering, hydraulic systems for RTST usually have time lag about 5 ms to 20 ms while most of the tests with earthquake load have a time step of 10 ms. Thus, the considered range of phase lag ratio is chosen between 0 and 2.
7.1.2 List of investigations on the amplitude and period errors
The purpose of the investigation is to analyze and discuss the amplitude increments and the period elongations in different variations of the dimensionless parameters (ηm, ηf , ζ, ζs, ksub and ηT) in the case of non-existance of phase lag (ηp= 0).
Table 7-2: Parameters for investigation on amplitude increments and period elongations of the substructure solution
Mass ratio
Damping ratio of the numerical
substructure
Damping ratio of the experimental
substructure
Frequency ratio
Number of sub steps
Dimensionless time step
ηm ζ ζs ηf ksub ηT
2%, 5%, 10%, 20%, 50%, 100%
1% , 2% , 5% 2%, 5%, 10% 0.5 - 1.5 1, 2, 5, 10, 20 0.0 – 0.2
Table 7-2 shows the list of parameters and their values for error analyses and the reasons for selecting the values are given as follows:
• The small values of ηm such 2%, 5% and 10% are usually used for the TMD system while larger values such as 20%, 50% and 100% are investigated for other coupled substructures.
• The values of the damping factor ζ of 1%, 2% and 5% are used for the typical structures in civil engineering. Higher damping values are not considered in this research because they are almost not a realistic value in structures.
• The values of the damping factor ζs of 2%, 5% and 10% are used for the typical TMD in civil engineering.
• The values of frequency ratio ηf in range of 0.9 to 1.0 are usually used for TMD systems while other values are used for other coupled systems. When ηf is smaller than 0.5 or larger than 1.5, two peaks of the transfer function are strongly different in terms of amplitude and frequency. In these cases, it is difficult to see both peaks on the graph of the transfer function. Moreover, when ηf > 1.5 the second frequency is large and it can reach the limitation of the
ηT = 0.2. In those cases, frequency analysis of the discrete response by using the FFT will have large errors. Those are the reasons why the parameter ηf is investigated in the range between 0.5 and 1.5.
• Values of the number of sub steps ksub ranging between 2 and 10 are usually used in RTST and the special case of ksub = 20 is implemented to show the errors for very large number of sub steps. In a special case, when ksub= 1, the algorithm becomes an explicit scheme because the substructure solution is based on the feedback force at the current time step. A comparison of different cases of ksub will also show the difference on the accuracy between the implicit integration (ksub>1) and the explicit integration (ksub=1).
• The dimensionless step time ηT is considered as a varying variable in the range between 0 and 2. Since the horizontal axis in all the error graphs is the dimensionless step time ηT, the errors are analyzed in series of values of ηT in the considered range (between 0 and 2).
Table 7-3 shows various combinations of the dimensionless parameters for different purposes of investigation as follows:
• Tests No. 1, 2 and 3: to investigate the effect of the mass ratio ηm (2%, 5% and 10%) on the errors of the substructure test of typical TMD system.
• Tests No. 4, 5 and 2: to investigate the effect of the damping ratio ζ (1%, 2%
and 5%) on the errors of substructure test of typical TMD system.
• Tests No. 6, 7 and 2: to investigate the effect of the damping ratio ζs (2% 5%
and 10%) on the errors of substructure test of typical TMD system.
• Tests No. 8, 2 and 9: to investigate the effect of mistuned frequency ηf (95%, 100% and 105% of the optimum frequencies) on the errors of substructure test of typical TMD system.
• Tests No. 10, 11 and 12: to investigate the effect of the large mass ratio ηm (20%, 50% and 100%) on the errors of substructure test of SDOF substructure.
• Tests No. 13, 14, 10, 15 and 16: to investigate the effect of the frequency ratio ηf (0.5, 0.75, 1.0, 1.25 and 1.5) on the errors of substructure test of a normal coupled substructure.
It is noted that the value ηfo in Table 7-3 is the optimum frequency ratio at which two peaks of the transfer function have the same values.
Table 7-3: Combination of parameters for accuracy analysis
No.
Mass ratio
Damping ratio of the
numerical substructure
Damping ratio of the experimental
substructure
Frequency ratio
Number of sub steps
Dimensionless time step
ηm ζ ζs ηf ksub ηT
1 2% 5% 10%
nfo= 0.957448 1, 2, 5, 10, 20
2 5% 5% 10%
nfo= 0.907664 1, 2, 5, 10, 20
3 10% 5% 10%
nfo= 0.836071 1, 2, 5, 10, 20
4 5% 1% 10%
nfo= 0.93300 1, 2, 5, 10, 20 0.0 – 0.2
5 5% 2% 10%
nfo= 0.92735 1, 2, 5, 10, 20
6 5% 5% 2%
nfo= 0.845107 1, 2, 5, 10, 20
7 5% 5% 5%
nfo= 0.893608 1, 2, 5, 10, 20
8 5% 5% 10% 95% of nfo
(nfo= 0.907664)
1, 2, 5, 10, 20
9 5% 5% 10% 105% of nfo
(nfo= 0.907664)
1, 2, 5, 10, 20
10 20% 5% 5% 1 1, 2, 5, 10, 20
11 50% 5% 5% 1 1, 2, 5, 10, 20
12 100% 5% 5% 1 1, 2, 5, 10, 20
13 20% 5% 5% 0.5 1, 2, 5, 10, 20
14 20% 5% 5% 0.75 1, 2, 5, 10, 20
15 20% 5% 5% 1.25 1, 2, 5, 10, 20
16 20% 5% 5% 1.50 1, 2, 5, 10, 20
7.1.3 List of investigations on the effect of phase lag on the amplitude and period errors
The investigations in section 7.1.2 take no consideration on the effect of the phase lag.
The following table lists the parameters for investigating the errors under the effect of phase lag. To see the effect of phase lag on the errors of the substructure solution, the errors in different cases of phase lag (ηp= 0.5, 1.0, 1.5 and 2.0) are compared with those in the case without phase lag (ηp = 0) (see No. 2 in Table 7-3).
Table 7-4: Combination of parameters for investigation on the effect of phase lag
No.
Mass ratio
Damping ratio
Damping ratio
Frequency ratio
Number of sub step
Phase lag ratio
Dimensionless time step
ηm ζ ζs ηf ksub
ηp ηT
1 5% 5% 10% ηfo 2 0.5, 1.0, 0-0.2
2 10 1.5, 2.0
7.1.4 List of investigations on stability of the substructure solution
The maximum spectral radius ρ(A) of the amplification matrix and the stability state depend on the dimensionless parameters (ζ, ζs, ηm, ηf , Ω). For various investigation purposes, different values of the parameters are listed in the following table.
Table 7-5: Combination of the parameters for stability analysis No.
Mass ratio Damping ratio
Damping ratio
Frequency ratio
Number of sub steps
Dimensionless frequency
ηm ζ ζs ηf ksub Ω
1 5% 5% 0%-10% 1 1, 2, 3, 4, 5
2 5% 0%-10% 10% 1 1, 2, 3, 4, 5 0-2π
3 10% 5% 10% 0-4 1, 2, 3, 4, 5
4 0%-100% 1% 10% 1 1, 2, 3, 4, 5
The cases No. 1 and 2 are used to investigate the variation of damping ratios ζs and ζ to the stability of the substructure solution. The case No. 3 is analyzed for investigation on the variation of frequency ratio ηf while the case No. 4 is focused on the effect of mass ratio ηm on the stability of the substructure solution. It should be mentioned that the number of sub steps ksub is investigated only in range from 1 to 5 because the time for processing the amplification matrix is extremely large when value of ksub is larger than 5. The results of stability analysis will be presented in section 8.4.