An application of force compensation can be used to compensate the error force of the equilibrium equation at the next step. Error force compensation is an important technique in the substructure algorithm with sub-step control (Roik and Dorka 1989, Dorka et al. 1998, Bayer et al. 2005).
e(t) uctrl(t) um(t)
+ -
+ +
uc(t)
GFB(s) GFF(s)
Gxu(s) Hydraulic system Feedback controller
Feed forward controller
In order to obtain the solution of implicit scheme, the method accepts the error force due to the digital feedback mechanism and then uses an appropriate error force compensation to minimize the error. For a test without force compensation, error force can be known and limited in certain range by choosing appropriate testing parameters.
However, when a highly accurate integration without numerical damping is implemented, the substructure tests with low damping or non-damping structures may not be possible if small error forces cause instability problems. Therefore, error force compensation is necessary for the substructure algorithm with sub-step control.
An error force compensation based on PID minimization as Eq. (4.7) has been proposed and used successfully to compensate error force in an aerospace substructure test. This compensation is called PID error force compensation (Dorka et al. 1998).
( )
−
+∆
∆ +
= −
=
+ ∑ 1
1
1 i
e i e i
j i e i
e
i f f
t f D t I f P
∆f (4.7)
where P, I and D are the proportional, integral and differential gains respectively.
The PID error force compensation has been applied in a simple case with 0<P<1, I=0 and D=0 for testing an aerospace model representing the Ariane IV rocket with payload (Dorka et al. 1998). Usually, the value of P is recommended to be about 0.9 and it should not be larger than 1.0 to prevent the amplification of noise. The test results show that a simple PID compensation can significantly improve the stability and accuracy of the test.
The application of the PID error force compensation is still limited because of following reasons. There is not enough investigation to understand effects of I and D parameters in the dynamic response of the compensation. Then, this compensation may depend on the dynamic properties of the specimen (in similar meaning to the PID control) therefore requiring an appropriate set of values of P, I and D parameters. It is difficult to apply an automatic process to select appropriate values of the parameters, which are suitable for different tests, specially in case of specimen with time-varying and nonlinear properties.
5 SCOPE OF THESIS AND CONTRIBUTIONS
In the first four chapters, the fundamentals of substructure testing regarding methodology and control algorithms have been reviewed. The principle, the history and the state-of-the-art of substructure testing have been presented in Chapter 1, the control algorithms for substructure in Chapter 2, errors in substructure testing in Chapter 3, while the available methods for error compensation in substructure testing have been reviewed in Chapter 4. These discussions have shown that RTST is currently facing the following problems.
Firstly, although the substructure control algorithm has been studied during the last decades, accuracy and stability of real-time substructure tests are still needed to be investigated further. Secondly, because errors can cause unexpected effects in the substructure solution and may cause instability in the real-time control of substructure tests, researches on advanced error compensation methods are strictly required in this field.
The substructure algorithm with sub-step control developed by Dorka (Roik and Dorka 1989, Dorka et al. 1998) is chosen for the investigation and development in this thesis because of the following reasons. The substructure algorithm is based on the general integration, thus it can be applied with any implicit integration. Especially, the method uses the Newmark-β integration with no numerical damping and the smallest period elongation for general applications, thus it can provide high accuracy and unconditional stability in RTSTs. In addition, the method does not assume any physical term and there is no restriction on what types of substructures and/or applications are used. Also the method has been applied in a wide range of applications including a stiff-ductile structure (Roik and Dorka 1989), vibrating substructures such as a TMD in civil engineering (Dorka et al. 2006, 2007) or a two- DOF vibrating system in aerospace engineering (Dorka et al. 1998, Bayer et al. 2005).
However, the accuracy and stability of the substructure algorithm with certain numbers of sub steps should be investigated further. Once these issues have been clarified, a much better understanding of the substructure algorithm is achieved and it enables us have better results in RTSTs. In addition, the results can be improved further with the help of advanced compensation methods to deal with the error force and the phase lag.
This thesis focuses on accuracy and stability analyses of the substructure algorithm, the effects of phase lag phenomenon on the substructure solution, and develops
advanced error force and phase lag compensations for RTST. The four problems that have been investigated in depth are as follows:
The first problem: although the substructure algorithm with sub-step control uses the Newmark-β integration, the substructure solution may be different from that of the Newmark-β one because the numbers of sub steps in RTSTs are always limited. The error force depends strongly on the number of sub steps and it may destabilize the numerical solution if the number of sub steps is not properly selected. Thus, accuracy and stability of the algorithm with sub-step control should be investigated further. The methodologies for investigating both accuracy and stability are presented in section 6.1 while the investigated cases are listed in section 7.1. Both the numerical and experimental substructures in these analyses are vibrating SDOF substructures.
Consequently, the results of the accuracy analysis are presented in section 8.1 while the results of the stability analysis are presented in section 8.4.
The second problem: phase lag always exists in hydraulic actuators and it causes certain errors in RTSTs. Thus, the effect of phase lag on the substructure solution should be analyzed. The errors of the substructure solution of a linear SDOF substructure in different cases of phase lags are analyzed and presented in section 8.2.
The third problem: since the sub-step time interval cannot approach zero, the error force in the substructure algorithm does not annul. The error force should be compensated in order to avoid any destabilizing effect. Because the application of the PID force compensation has certain limitations, a new force compensation method that can adapt to the change of the test system during a substructure test is needed. The methodology for developing new error force compensation is presented in section 6.2.1 and the details of the development and implementation are presented and discussed in section 9.1.
The fourth problem: because the existing phase lag compensations have certain limitations, it is needed a more advanced compensation in order to compensate effectively the phase lags in hydraulic systems for different RTSTs. The theories of estimation based on black-box data model (section 6.2.4) and online system identification (section 6.2.5) are used to develop a new phase lag compensation method (sections 6.2.2 and 9.2). The new phase lag compensation is implemented in a control system (section 9.3) and the effectiveness of the phase lag compensation is validated on a realistic testing system using hydraulic cylinder (section 9.4).
The research of this thesis makes three contributions in the substructure testing field.
The first contribution is that both accuracy and stability of the substructure algorithm with different numbers of sub steps have been analyzed and discussed. Understanding the accuracy and stability of the substructure algorithm is helpful for selecting the appropriate time step and the number of sub steps in substructure testing with sub-step control. The second contribution is that a new error force compensation is proposed and the error force can be compensated effectively (sections 6.2.1, 9.1, 9.3 and 9.4).
The third one is a proposal of a new adaptive phase lag compensation for real-time substructure testing (sections 6.2.2, 9.2, 9.3). The phase lag of a realistic hydraulic system in RTST can be compensated effectively (section 9.4). The error compensations are presented in the context of a linear system; however, with the adaptive capability in the identification mechanism, the compensation methods could be developed further for different nonlinear systems.
6 METHODOLOGIES