Shock and Vibration

Volume 2018, Article ID 6795763, 12 pages

https://doi.org/10.1155/2018/6795763

## Effects of Interaction between Dual Shaking Tables and Specimen and Force Feedback Compensation Control

^{1}Beijing Key Lab of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, Beijing 100124, China^{2}Institute of Geophysics, China Earthquake Administration, Beijing 100081, China

Correspondence should be addressed to Xiaojun Li; moc.anis.piv@ilreeb

Received 17 May 2018; Revised 18 July 2018; Accepted 31 July 2018; Published 16 September 2018

Academic Editor: Hamid Toopchi-Nezhad

Copyright © 2018 Fangfang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The shaking table array system is composed of multiple shaking tables for seismic response simulation tests of large-span spatial structures, bridge structures, slender structures such as pipeline and aqueduct, complex structures, and so on. In the process of testing with the multiple shaking tables, the interaction between the shaking tables and specimen affects the output accuracy of the shaking tables. The characteristics and rules of the dual shaking tables-specimen interaction effects on the system performance were analyzed in this paper. In order to improve the output accuracy of the dual shaking tables, force feedback compensation was introduced into three-variable control to reduce the interaction effects. However, the measurement errors of the force in the actuator and the acceleration of the shaking tables existed in the process of force feedback compensation. In order to verify the effectiveness of force feedback compensation for interaction between the dual shaking tables and specimen, the error influences on the system performance were simulated.

#### 1. Introduction

An electrohydraulic shaking table can be used to simulate the real-time process of different ground motions for seismic tests of engineering structures in the laboratory. Compared with the pseudostatic test and pseudodynamic test, the shaking table test can better reflect structural responses under earthquake excitation; therefore, the shaking table is very important in earthquake engineering research field [1–5]. When analyzing the electrohydraulic shaking table system, the specimen is generally assumed to be a rigid mass in the three continuous equations of the hydraulic system [4–6]. However, because the specimen is an elastomer with large mass, the interaction between the shaking table and specimen affects the frequency response characteristic and accuracy of reproduction of the command signal. In 1988, Blondet and Esparza [7] built the analytical model including the structural specimen represented by a single DOF viscoelastic oscillator and studied the shaking table-specimen interaction effects on the shaking table system based on displacement control. The results showed that the interaction effects can lead to degradation of the performance of the shaking table, and the interaction effects are characterized by a peak-notch distortion in the amplitude-frequency response and a violent phase lag in a frequency band close to the natural frequency of the specimen. Li and Tian [8] did the theory analysis of the shaking table loaded with the *N* degrees of freedom specimen. It indicated that the nonlinear property of the specimen decreases the reproductive accuracy, and increasing the mass ration between the table and the specimen is one way to improve the reproductive accuracy of the shaking table. Only when the specimen mass is far less than the table mass can the nonlinear specimen effects on the performance of the system be neglected. Dyke et al. [9] discussed the importance of considering control-structure interaction when modelling a control system and developed a model including control-structure interaction (CSI) in the field of structural control for mitigation of response due to environment loads. Maoult et al. [10] did the numerical analyses on the boundary conditions between the test structure and the platform of a shaking table and verified the influence of interaction between the shaking table and specimen. Conte and Trombetti [11–13] developed a mathematic model including the shaking table base-table interaction and the table-payload interaction and compared the numerical analysis with experimental results. The results showed that the interaction between the table and payload has more significant influence on the dynamic performance of the shaking table than the interaction between the shaking table base and table. The natural frequency of the payload becomes the second resonance frequency of the shaking table system except the oil column resonant frequency. The interaction between the shaking table and payload leads to the degradation of the performance of the shaking table. At the same time, the oil column resonant frequency decreases with the increase of the specimen mass. Li et al. [14] analyzed the influences of the specimen on stability of the shaking table system caused by SDOF and MDOF specimen models. The results showed that the stability of the shaking table designed for a nonrigid payload is greatly improved in the unload condition. Tang et al. [15] analyzed the effects of the mass, frequency, and damping ratio of the specimens on the control performance. The results showed that the mass, frequency, and damping ratio of the specimens make different degrees of influence on the interaction between the shaking table and specimen. The frequency effect is biggest of all, the damping ratio effect is secondary, and the mass effect is least. The reproduction accuracy of the command signal is poor when the frequency is close to the natural frequency of the specimen. In conclusion, the interaction effects of the shaking table and specimen on the electrohydraulic servo-shaking table system are characterized by a peak-notch distortion in the amplitude-frequency response in a frequency band close to the natural frequency of the specimen. The output of the table is affected by the shaking table-specimen interaction.

In order to improve the reproduction precision of the command signal of the shaking table, researchers did in-depth studies and developed many methods to compensate the interaction between the shaking table and specimen. Dozono et al. [16] developed adaptive filter compensation (AFC) to compensate the disturbance of the reaction force generated by a nonlinear specimen. Iwasaki et al. [17] adopted a disturbance observer-based control approach to compensate the reaction force generated by a nonlinear specimen on the shaking table. Seki et al. [18, 19] proposed an adaptive feedback compensator by applying an adaptive notch filter to identify the frequency in online manner to suppress the disturbance caused by the specimen loaded on the shaking table. Tang et al. [15] proposed a real-time compensation of reaction algorithm to improve the disadvantage caused by the interaction between the specimen and the shaking table. In the algorithm, the interaction between the shaking table and specimen was approximately calculated by the theoretical model of the specimen. The proposed control algorithm can be applied to completely compensate the disadvantage caused by the single degree of freedom specimen. However, the compensation reaction of the shaking table loaded with the multidegree of freedom specimen needs to be obtained by inertial force of the principal mode of the MDOF specimen. Phillips et al. [20] proposed a model-based multimetric control strategy applied to a small electric shaking table to improve tracking of the desired signal. In the approach, tracking over a broad frequency range was improved by using both displacement and acceleration measurements incorporated into the feedback control, the voltage command was modified through an outer-loop controller, the tracking error was reduced by applying LQR control to bring the deviation states to zero, and the transfer function iteration (TFI) was combined with the model-based feedback controller. However, velocity feedback was not included in the multimetric control strategy. Tian and Chen [21] proposed a control strategy in the shaking table test based on the elastic load. In the control strategy, the inner-loop control parameters were designed by analyzing physical models of the elastic load to realize that the transfer function of the shaking table keeps nearly unvaried with different loads. The outer-loop control was based on an adaptive control to compensate the peak and the notch generated by the elastic load. However, the apparent mass to set the parameters of the inner-loop was still obtained by the modal analysis of the elastic load.

Although the researchers already did a lot of research on the interaction between the shaking table and specimen and put forward a series of compensation control algorithms, various algorithms were for different research objects, and the scopes of application were also different. In particular, most of research studies concentrated on the interaction of a single shaking table loaded with specimen. The shaking table array system, which is composed of multiple shaking tables, can be applied for seismic response simulation tests of large-span spatial structures, bridge structures, slender structures such as pipeline and aqueduct, complex structures, and so on. The interaction forces between the multiple shaking tables and specimen are more complex than the interaction between the single shaking table and specimen, but the research on the interaction between the multiple shaking tables and specimen is lacking. The interaction effects on the system performance of the multiple shaking tables with specimen are more complicated than those of the single shaking table with specimen. Recently, the authors of this article, Li et al. [22], proposed a novel synchronous tracking strategy, differential movement synchronous tracking control (DMSTC) strategy, for double-shaking table system taking the interaction between the shaking tables and specimen into consideration. The DMSTC strategy could extend and improve the frequency bandwidth of the double-shaking table system and also improved the replication accuracy and tracking accuracy of each shaking table. In this paper, the further analysis of the interaction effects on system performance of the dual shaking tables with specimen was carried out, and force feedback compensation was introduced into three-variable control to reduce the interaction effects between the dual shaking tables and specimen. The influences of measurement errors of the force of the actuator and the acceleration of the shaking table on the system performance were simulated to verify the effectiveness of force feedback compensation.

#### 2. Performance Analysis of Dual Shaking Tables with Specimen

In the traditional modelling of the electrohydraulic shaking table system, the table and specimen are generally assumed to be a single DOF system, and the specimen is assumed to be a rigid mass. Equations (1)–(3) are the basic equations of the hydraulic system [4, 6], in which (1) is the equilibrium equation between the inertial force of the table and the force in the actuator.where is the mass of the table, is the piston position, is the cross-sectional area of the actuator piston, is the pressure difference between actuator chambers, is the total oil flow, is the compressed oil volume in the actuator cylinder, is the oil bulk modulus, is the leakage coefficient, is the flow gain, is the flow-pressure coefficient, is the Laplace transform variable, is the valve spool displacement, and is the force in the actuator.

The shaking table array system is composed of multiple shaking tables. In addition to the force in the actuator and the interaction between the shaking table and specimen, the force acting on the shaking table also includes additional force generated by the nonsynchronous output of shaking tables. In this paper, the dual shaking tables with flexible connection specimen were analyzed as a numerical model. Figure 1 shows the mechanical model of the dual shaking tables with flexible connecting specimen.