Stability analysis of a real-time shake table hybrid simulation for linear and non-linear SDOF systems
Gidewon G. Tekeste1,2, António A. Correia1, Aníbal G. Costa2
1 Laboratório Nacional de Engenharia Civil
Av. do Brasil 101, 1700-066 Lisboa
gtekeste@lnec.pt
aacorreia@lnec.pt
2 University of Aveiro
Campus Universitário de Santiago, 3810-193 Aveiro
gidewon.tekeste@ua.pt
agc@ua.pt
Abstract. Real-Time Hybrid Simulation (RTHS) is an experimental testing technique capable of representing the dynamic behaviour of complex structures. The testing methodology is composed of inner and outer loops; typically executed at different rates. The inner loop is characterized by the application of target displacements, defined by a numerical model, to the physical element using servo-hydraulic actuators, while the measured responses are feedback to the numerical model through the outer loop. The former is of major importance as its failure, or an error (delay and amplitude) in measured forces, can significantly alter the fidelity of RTHS. In RTHS using shake tables, the dynamics of the shake table is mainly responsible for the errors in the measured forces. Thus, a proper assessment of the shake table dynamics is indispensable so as to reduce the experimental errors through delay compensation techniques. A parametric system identification of the uniaxial shake table of LNEC (ST1D) was carried out in support of the development of a shake table RTHS testing platform. A mechanical system identification which includes the estimation of the effective horizontal mass and the dissipative forces of the moving components of ST1D was carried out during a testing campaign which utilizes periodicity of measured responses under a bare shake table condition. A series of sinusoidal and triangular input displacements of varying frequency and amplitude were used for this purpose. Using the above estimates, the complete dynamics of the shake table – comprising the dynamics of the controller, servo-valve, actuator and platen – was identified through a constrained non-linear least square optimization in MATLAB/Simulink.
The stability of a linear SDOF structure in a RTHS was simulated by assuming that it is partly modelled as an experimental substructure and partly as a numerical substructure, while the identified shake table transfer function simulates the experimental conditions. Routh’s stability criterion was used at varying controller gains to determine the percentages of mass, stiffness and damping of the SDOF that can be physically sub-structured without introducing instability issues. The instability of the linear SDOF at various frequencies and viscous damping ratios was also addressed under worst case scenario of component sub-structuring.
In order to extend the analysis to the non-linear range, the shake table transfer function and a non-linear SDOF, using the Bouc-Wen hysteretic model, was numerically modelled in Simulink to mimic a RTHS test. The stability margins identified for the linear and non-linear cases were then compared. Finally, the results found will be used to set up shake table RTHS testing conditions using ST1D.
Keywords: Shake table real-time hybrid simulation, Periodicity, Parametric system identification, Constrained non-linear least square optimization, Routh stability criterion.
