2.4 Control methods for implicit schemes
2.4.3 The sub-step control method
The first continuous PsD test using this control method was applied in a test of ductile specimen with time scale 1:10 in 1987 (Roik and Dorka 1989). Then, the method has been developed further and applied in an aerospace application (Dorka et al. 1998, Bayer et al. 2000a, 2000b, 2002, 2005) and currently applied in substructure tests of Tuned Mass Dampers (TMD) in Saclay, France (Dorka et al. 2007).
The method uses implicit three-point integration in general formulation (Eq. 2.4) and digital feedback technique to control a linear control equation in order to solve for the implicit solution at the next step. The general solution of the three-point integration equation (Eq. 2.4) can be written as a linear control equation as Eq. (2.23) for only experimental coupling force or Eq. (2.24) for both experimental and numerical nonlinear forces.
)
( 1 1
1
1 + + +
+ = oi + ⋅ ci + ni
i u G f f
u (2.24)
where u0i+1 is the explicit displacement, see Eq. (2.25), and fni+1 is a vector of nonlinear forces comming from numerical nonlinear substructures.
[ ]
( ) ( )
[ ]
( ) ( )
[ ]
( )
( )
∆
− + +
∆ +
− +
∆ +
∆
− + +
∆
−
−
−
∆ +
−
−
∆
−
−
⋅
∆ +
∆ +
=
− +
− − +
1
* 2 2
1
* 2 2
1 1 2
1 2 2
1
2 2
1
2 1 1
0 2
. 1
. 2
2 1 2
i
i i
l
i i
i
f t
f t f
t
u K t tC
M
u K t tC
M K
t tC M
u
γ β
γ β β
γ β γ
γ β γ
β
γ (2.25)
With sub-step control, Eq. (2.24) is satisfied via ksub sub steps from the current step to the next step, see Figure 2.5.
In the flow chart in Figure 2.5, the coupling force at the last sub step is fed back in order to calculate the displacement at each sub step. When the coupling force is updated again at the end of the next step, the equilibrium equation may have a small unbalanced amount. This unbalanced amount is called error force (Dorka et al. 1998)
Figure 2.5: Substructure algorithm with digital feed back and error force compensation (Roik and Dorka 1989, Dorka et al. 1998, Dorka 2002)
or unbalanced force (Nguyen and Dorka 2007) and measured as Eq. (2.26). An error force compensation should be used (Figure 2.5(b)) for minimizing the error force. A detailed discussion on error force compensation will be presented in section 4.3.
) (
)
( 1 1 1 1 1 1
1 + + + + + +
+ = li + ci + ni − i + i + i
i
e f f f Mu Cu Ku
f
. ..
(2.26) In other formulation forms, Dorka et al. (1998) presented the formulations of this algorithm in displacement, velocity and acceleration control modes. Here the acceleration control mode is presented (Eqs. (2.27) - (2.29)) (Dorka et al. 1998) for the implementation of this algorimthm in the simulation in Chapter 6. The linear control equation in acceleration control mode is shown as Eq. (2.27).
G(fn+fc)
u0i-1
u0i+1
i-1 i i+1 step
displacement
j=1 2 ksub
ui-1 ui ui+1
(a) – linear control mechanism
j = ksub ?
calculate explicit displacement j = 1
calculate the time derivatives
.i+1
u and
..i+1
u
calculate the error force
1 + i
fe
apply displacement at each sub step
( n c)
sub i sub
i G f f
k u j k u j
u= 0(1− )+ 0+1( )+ +
j = j+1 error force
compensation
measure fc
no yes
(b) – flow chart of substructure algorithm i = i+1
u0i
u0i+1
simulate fn
( 1 1)
1 0
1 + + +
+ = i + a ci + ni
i u G f f
u
..
.. (2.27)
[ ]
∆
− +
∆ +
−
∆
− +
−
∆ +
∆ +
=
+ + −
i i
i
i i i
l i
u t u
t u K
u t u
C f K t tC M
u . ..
..
. ..
2 1
2 1 0 1
2 ) (1
) 1 (
β γ β
γ (2.28)
[ + ∆ + ∆ 2 ]−1
= M tC t K
Ga γ β (2.29)
At the end of each step, the velocity and displacement vectors are calculated as Eqs.
(2.30) and (2.31).
1
1 (1 ) +
+ = i+ − ∆ i+ ∆ i
i u tu tu
u
..
..
.
. γ γ (2.30)
2 1 1 2
2 )
(1 +
+ = i+∆ i+ − ∆ i+ ∆ i
i u tu t u t u
u
..
..
. β β (2.31)
The substructure algorithm with sub-step control has some important advantages.
Since the method does not assume any physical term on the coupling force, the algorithm can be applied on any kind of coupling force. Other methods based on predictions of physical quantities do not have this powerful feature. Different applications such as test of a stiff and ductile specimen (Roik and Dorka 1989), test of a vibrating model in aerospace application with high frequencies up to 50 Hz (Dorka et al. 1998) and distributed tests of TMDs (Dorka et al. 2007) are good examples that show the wide range of applications of this method.
The substructure algorithm with sub-step control has also certain disadvantages. The sub-step control with error force compensation is more complicated than a simple explicit integration such the CDM. Theoretically, if the number of sub steps reached to infinity, the solution of the digital control feed back would converge to the exact one.
However, the number of sub steps in a substructure test cannot be infinite due to certain reasons, eg. limitations of speed and/or resolution of the digital hardware. The solution may have small errors and this may cause instability in the substructure tests with very low damping or non-damping structures.
Up to now, the method has not been investigated deeply in terms of accuracy and stability in different cases of numbers of sub steps, without or with time lag in the hydraulic system. Once there is an appropriate compensation mechanism, the high accuracy and unconditional stability of the Newmark implicit method can be used without restrictions. On the other hand, the concept of force compensation can be developed further by using various techniques. Therefore, there should be further
investigating the accuracy and stability analyses as well as error compensation for the substructure algorithm with sub-step control.
3 ERRORS AND THEIR EFFECTS IN SUBSTRUCTURE TESTING Since the purpose of a substructure test is to investigate the dynamic response of whole structures by means of testing parts of the structures, differences between response of the substructure test and that of the full system should be identified. A number of factors can cause errors in the solution of a substructure test. Errors of substructure solution due to integration methods have been discussed in Chapter 2. The following sections discuss various errors resulting from modeling and loading assumption, control hardware, measurement and data conversion.