with different dimensionless time steps in the case of typical TMD system The following graphs show the transfer functions h(r) of the substructure solution for a typical TMD system (case 2 in Table 7-3) with ηm=5%, ζ =5%, ζs =10%, ηf =ηfo (at the optimum frequency), with different numbers of sub steps ksub=2, ksub=5 or
=10
ksub (Figure 8.1, Figure 8.2 and Figure 8.3). Each graph shows the exact transfer function and the transfer functions of the substructure solutions with different values of the dimensionless time step (ηT= 0.05, 0.10, 0.15 and 0.20).
Figure 8.1: Transfer function of the substructure solution in series of ηT with ksub=2, typical TMD system (ηm =5%, ζ =5%, ζs =10%), case 2 in Table 7-3
Figure 8.2: Transfer function of the substructure solution in series of ηT with ksub=5, typical TMD system (ηm =5%, ζ =5%, ζs =10%), case 2 in Table 7-3
0 1 2 3 4 5 6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dim ensionless frequency r
Transfer function h
Figure 8.3: Transfer function of the substructure solution in series of ηT with ksub=10, typical TMD system (ηm =5%, ζ =5%, ζs =10%), case 2 in Table 7-3
There are some comments regarding these graphs. Firstly, the peaks of the transfer functions at two frequencies are different from the respective peaks of the exact transfer function. These differences vary with the dimensionless time step ηT. When the dimensionless time step ηT increases from 0.05 to 0.2, the amplitudes of the peaks of the transfer function may increase or decrease while the frequencies decrease significantly. The amplitude and frequency errors can be evaluated directly on the data of the transfer functions. Secondly, the transfer function of the substructure solution may be strongly different from the exact once in certain cases and the graph of the transfer function may have only one peak. For example, the graph of the case of
2 . 0
T =
η and ksub=10 in Figure 8.3 shows clearly only the first frequency.
8.1.2 Variation of the transfer function
with different numbers of sub steps in the case of typical TMD system
The following graphs show the transfer functions of the substructure solution of a typical TMD system (the case 2 in Table 7-3) with ηm=5%, ζ =5%, ζs =10%,
fo
f η
η = , with ηT =0.1 (Figure 8.4) or ηT =0.2 (Figure 8.5).
Within a graph, the variation of the transfer function can be seen in different values of the number of sub steps ksub. On the other hand, by comparing the graphs in Figure 8.4 and those in Figure 8.5, the effects of ksub on the transfer functions in the different cases of ηT are compared.
0 1 2 3 4 5 6
0.50 0.75 1.00 1.25 1.50
Dim ensionless frequency r
Transfer function h
0 1 2 3 4 5 6
0.50 0.75 1.00 1.25 1.50
Dim ensionless frequency r
Transfer function h
Figure 8.4: Transfer function of the substructure solution in series of ksub, with ηT =0.1, typical TMD system (ηm =5%, ζ =5%, ζs =10%), case 2 in Table 7-3
Figure 8.5: Transfer function of the substructure solution in series of ksub, with ηT =0.2, typical TMD system (ηm =5%, ζ =5%, ζs =10%), case 2 in Table 7-3
Theoretically, when the number of sub steps approaches infinity, the substructure solution obtained from the sub-step control will approach the one of the Newmark integration. The transfer function of the numerical solution, with the use of the Newmark integration for the numerical part and the use of the exact integration for the experiment part, is shown with the pink color.
It is known that the frequency of the numerical part decreases numerically due to the period elongation of the Newmark integration while there is no frequency distortion in the exact simulation of the experimental substructure. Therefore, the ratio between the frequencies of the TMD and the numerical structure will increase and it becomes larger than the optimum value. This is the reason why the amplitude of the transfer function in the Newmark case at the first frequency is slightly larger than the one at the second frequency in the case of the dimensionless time step ηT =0.1 (Figure 8.4).
When the dimensionless time step increases to ηT =0.2 (in Figure 8.5), the effect can be seen more strongly.
There are some recognizable notes on the effect of the number of sub steps ksub on the transfer function. Firstly, when the number of sub steps ksubvaries from 2 to 5, 10 and 20, the substructure solution moves closer and closer to the Newmark one. This is true in both of the cases ηT =0.1 and ηT =0.2. Especially, when the dimension less time step is small (ηT =0.1), the substructure solution using large number of sub steps such
=10
ksub or ksub=20 are very close to the Newmark one. Secondly, although the substructure solution in the case of large number of sub steps is close to the Newmark one, they may differ from the exact solution. For this reason, a number of sub steps ksub in the range between 2 and 5 can be selected appropriately in order to obtain better results (see Figure 8.4 and Figure 8.5).
Based on the knowledge of variation of the transfer function, large number of sub steps is not recommended in the case of large time step and an appropriate value for a parameter can chosen in certain conditions of the other parameters. The errors including period elongations (EP1 and EP2) and amplitude increments (EA1 and EA2) of the transfer functions are shown in next section for the detailed evaluation on the accuracy of the substructure solution.
8.1.3 Effect of the number of sub steps on the errors in the case of typical TMD system
The graphs in Figure 8.6 and Figure 8.7 show the errors of the substructure solution for a typical case of TMD system (ηm =5%, ζ =5%, ζs =10%) in series of different numbers of sub steps ksub in range from 1 to 20.
When ksub increases from 1 to 20, the period elongation of the substructure solution decreases at the first frequency but it does not change clearly at the second frequency (see Figure 8.6). The amplitude increment at the first frequency changes from negative value to positive value while the amplitude increment at the second frequency varies in the opposite side when ksub increases from 1 to 20 (see Figure 8.7).
Am plitude Error at First Frequency
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
ksub=1 ksub=2 ksub=5 ksub=10 ksub=20
ηT
EA1
Am plitude Error at Second Frequency
-0.40 -0.20 0.00 0.20 0.40 0.60 0.80
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
ksub=1 ksub=2 ksub=5 ksub=10 ksub=20
ηT
EA2
Dimensionless time step ηT Dimensionless time step ηT Period Error at First Frequency
0.00 0.02 0.04 0.06
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
ksub=1 ksub=2 ksub=5 ksub=10 ksub=20
EP1
Period Error at Second Frequency
-0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
ksub=1 ksub=2 ksub=5 ksub=10 ksub=20
EP2
Dimensionless time step ηT Dimensionless time step ηT
Figure 8.6: Period errors of the substructure solution at two frequencies, in series of ksub, typical TMD system (ηm =5%, ζ =5%, ζs =10%, ηf =ηfo), case 2 in Table 7-3
Figure 8.7: Amplitude errors of the substructure solution at two frequencies, in series of ksub, typical TMD system (ηm =5%, ζ =5%, ζs =10%), case 2 in Table 7-3
In addition, with a certain number of sub steps ksub (from 2 to 5), an appropriate dimensionless time step can be chosen in order to obtain very small amplitude error at the first or at the second frequency. For example, in the case ksub=5 in Figure 8.7, a choosing ηT to be about 0.09 results in that EA1 is nearly zero while EA2 is very small.
Oppositely, if ηT is about 0.06, the error EA1 will be small while the EA2 is nearly zero.
A range between 0.06 and 0.09 for ηT may be referred for this case instead of choosing a very small or large value such as η <0.06 or η >0.09.
Period Error at Second Frequency
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Period Error at First Frequency
0.00 0.02 0.04 0.06 0.08
0.00 0.05 0.10 0.15 0.20
Non-dim ension step tim e
Period Elongation EP2
EP1
Dimensionless time step ηT Dimensionless time step ηT
Am plitude Error at Second Frequency
-0.40 -0.30 -0.20 -0.10 0.00 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
Am plitude Error at First Frequency
-0.10 0.00 0.10 0.20 0.30 0.40
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
ηT
ηT
EA2
EA1
Dimensionless time step ηT Dimensionless time step ηT
8.1.4 Effect of the mass ratio on the errors in the case of typical TMD system The graphs in Figure 8.8 and Figure 8.9 show the errors of the substructure solution for a typical TMD system (ζ =5%, ζs =10%), with different values of the mass ratio ηm (2%, 5% and 10%) using a typical number of sub steps ksub=5.
Figure 8.8: Period errors of the substructure solution in series of mass ratio ηm, typical TMD systems (ζ =5%, ζs =10%), cases 1, 2 and 3 in Table 7-3
Figure 8.9: Amplitude errors of the substructure solution, in series of mass ratio ηm, typical TMD systems (ζ =5%, ζs =10%), cases 1, 2 and 3 in Table 7-3
When the mass ratio ηm increases from 2% to 10%, the period elongation decreases at the first frequency while it does not change at the second frequency. The amplitude errors at both frequencies vary significantly with the mass ratio and these errors may have negative or positive values. When the mass ratio ηm increases from 2% to 10%, the amplitude increment EA1 decreases while the amplitude increment EA2 increases.
Period Error at Second Frequency
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Period Error at First Frequency
0.00 0.01 0.02 0.03 0.04 0.05
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation EP2
EP1
Dimensionless time step ηT Dimensionless time step ηT
Am plitude Error at Second Frequency
-0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
Am plitude Error at First Frequency
-0.10 0.00 0.10 0.20 0.30 0.40
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
ηT
ηT
EA2
EA1
Dimensionless time step ηT Dimensionless time step ηT
8.1.5 Effect of the damping ratio of the numerical substructure on the errors in the case of typical TMD system
The graphs in Figure 8.10 and Figure 8.11 show the errors of the substructure solution for a typical TMD system (ηm =5%, ζs =10%) with different values of the damping ratio ζ (1%, 2% and 5%) using a typical number of sub steps ksub=5.
Figure 8.10: Period errors of the substructure solution in series of damping ratio ζζζζ, typical TMD system (ηm =5%, ζs =10%), cases 4, 5 and 2 in Table 7-3
Figure 8.11: Amplitude errors of the substructure solution in series of damping ratio ζζζζ, typical TMD system (ηm=5%, ζs =10%), cases 4, 5 and 2 in Table 7-3
The period errors at both frequencies do not vary much with the damping ratio ζ (Figure 8.10). The amplitude errors at both frequencies in Figure 8.11 vary significantly with the damping ratio ζ and these errors may have negative or positive values. When the damping ratio ζ increases from 1% to 5%, the amplitude increment
1
EA decreases while the amplitude increment EA2 increases. However, these changes cannot be seen clearly if the dimensionless time step ηT is smaller than 0.075.
Period Error at Second Frequency
0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Period Error at First Frequency
0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
ηT
ηT
EP2
EP1
Dimensionless time step ηT Dimensionless time step ηT
Am plitude Error at Second Frequency
-0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
Am plitude Error at First Frequency
-0.20 -0.10 0.00 0.10 0.20
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment EA2
EA1
Dimensionless time step ηT Dimensionless time step ηT
8.1.6 Effect of the damping ratio of the experimental substructure on the errors in the case of typical TMD system
The graphs in Figure 8.12 and Figure 8.13 show the errors of the substructure solution for TMD systems (ηm =5%, ζ =5%) with different values of the damping ratio ζs (2%, 5% and 10%) using a typical number of sub steps ksub=5.
Figure 8.12: Period errors of the substructure solution, in series of damping ratio ζs in typical TMD systems (ηm =5%, ζ =5%), cases 6, 7 and 2 in Table 7-3
Figure 8.13: Amplitude errors of the substructure solution, in series of the damping ratio ζs in typical TMD systems (ηm=5%, ζ =5%), cases 6, 7 and 2 in Table 7-3
The period errors in Figure 8.12 at both frequencies vary slightly with the damping ratio ζs. When the damping ratio ζs increases from 2% to 10%, the period elongation
1
EP increases slightly while the period elongation EP2 does not change clearly. The amplitude errors in Figure 8.13 vary significantly with the damping ratio ζs. When the
Period Error at First Frequency
0.00 0.01 0.02 0.03 0.04 0.05
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Period Error at Second Frequency
0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Dimensionless time step ηT
EP2
EP1
Dimensionless time step ηT
Am plitude Error at Second Frequency
-0.40 -0.30 -0.20 -0.10 0.00
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
Am plitude Error at First Frequency
-0.05 0.00 0.05 0.10 0.15 0.20
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment EA2
EA1
Dimensionless time step ηT Dimensionless time step ηT
damping ratio ζs increases from 2% to 10%, the amplitude increment EA1 increases while the amplitude increment EA2 decreases.
8.1.7 Effect of the mistuned-frequency phenomenon on the errors in the case of typical TMD system
The graphs in Figure 8.14 and Figure 8.15 show the errors of the substructure solution for a typical TMD system (ηm =5%, ζ =5%, ζs =10%) with different frequency ratios ηf (0.95ηfo, ηfo and 1.05ηfo, where ηfo is the optimum frequency ratio of the TMD system) using a typical number of sub steps ksub=5.
Figure 8.14: Period errors of the substructure solution in series of frequency ratio ηf, typical TMD system (ηm =5%, ζ =5%, ζs =10%), cases 8, 2 and 9 in Table 7-3
Figure 8.15: Amplitude errors of the substructure solution in series of frequency ratio ηf, typical TMD system (ηm =5%, ζ =5%, ζs =10%), cases 8, 2 and 9 in Table 7-3
Period Error at Second Frequency
0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Period Error at First Frequency
0.00 0.01 0.02 0.03 0.04 0.05
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation EP2
EP1
Dimensionless time step ηT Dimensionless time step ηT
Am plitude Error at Second Frequency
-0.40 -0.30 -0.20 -0.10 0.00
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
Am plitude Error at First Frequency
-0.50 -0.40 -0.30 -0.20 -0.10 0.00
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment EA2
EA1
Dimensionless time step ηT Dimensionless time step ηT
An increase of the frequency ratio ηf causes a small increase of the period error at the first frequency but it does not result in any significant change of the period error at the second frequency (see Figure 8.14). When ηf increases from 0.95 to 1.05 times of the optimum frequency ηfo, the amplitude increment EA1 does not change much while the amplitude increment EA2 decreases slightly (see Figure 8.15).
8.1.8 Effect of the mass ratio on the errors
in the case of coupled system with large mass ratio
The graphs in Figure 8.16 and Figure 8.17 show the errors of the substructure solution for a coupled substructure (ζ =5%, ζs =5%, ηf =1) with large values of the mass ratio ηm (20%, 50%, 100%) using a typical number of sub steps ksub=5.
Figure 8.16: Period errors of the substructure solution in series of mass ratio ηm, coupled system (ζ =5%,ζs =5%, ηf =1) with large mass ratio, cases 10, 11 and 12 in Table 7-3
Figure 8.17: Amplitude errors of the substructure solution in series of mass ratios ηm, coupled
Period Error at Second Frequency
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation
Period Error at First Frequency
0.00 0.01 0.02 0.03 0.04 0.05
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Period Elongation EP2
EP1
Dimensionless time step ηT Dimensionless time step ηT
In Figure 8.16, when the mass ratio ηm increases from 20% to 100%, the period elongation at the first frequency decreases while the period elongation at the second frequency does not change clearly. In Figure 8.17, when the mass ratio ηm increases from 20% to 100% the amplitude increment EA1 decreases while the amplitude increment EA2 increases.
8.1.9 Effect of the frequency ratio on the errors in the case of coupled system with large mass ratio
The graphs in Figure 8.18 and Figure 8.19 show the errors of the substructure solution for coupled substructures (ζ =5%,ζs =5%,ηm=50%) with different values of the frequency ratio ηf (0.5, 0.75, 1.0, 1.25, 1.5) using a typical number of sub steps
=5 ksub .
Figure 8.18: Period errors of the substructure solution, in series of frequency ratio ηf , coupled system ηm =50%, ζ =5%, ζs =5%), cases 13, 14, 10, 15, 16 in Table 7-3
In Figure 8.18, the period error at the second frequency does not change much while the one at the first frequency varies dramatically with the frequency ratio ηf . When the frequency ratio ηf is small as 0.5, the period error at the first frequency is very small (about 0.0012 or less). However, the period error at the first frequency increases dramatically when the frequency ratio ηf increases from 0.5 to 1.5.
The graph in Figure 8.19 shows that there are two different variations of the amplitude errors at the two frequencies.
At the first frequency, when the frequency ratio increases from 0.5 to 1.25, the amplitude increment decreases significantly. However, it does not change much when
Am plitude Error at Second Frequency
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment
Am plitude Error at First Frequency
-0.50 -0.40 -0.30 -0.20 -0.10 0.00
0.00 0.05 0.10 0.15 0.20
Dim ensionless step tim e
Amplitude Increment EA2
EA1
Dimensionless time step ηT Dimensionless time step ηT
Figure 8.19: Amplitude errors of the substructure solution, in series of frequency ratio ηf , coupled system (ηm=50%, ζ =5%, ζs =10%), cases 13, 14, 10, 15, 16 in Table 7-3
At the second frequency, the amplitude increment increases when the frequency ratio increases from 0.5 to 1.0. If the frequency ratio increases from 1.0 to 1.25, the graph of amplitude increment changes with a special trend in which the error will decrease in the range of ηT between 0 to 0.15 and will increase in the rest of the range of ηT (ηT >0.15). The phenomenon happens again and more strongly when the frequency ratio ηf is 1.5.