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

/e 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. /e 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.


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][2][3][4][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][5][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.
e 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.developed a mathematic model including the shaking table base-table interaction and the table-payload interaction and compared the numerical analysis with experimental results.e 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.e natural frequency of the payload becomes the second resonance frequency of the shaking table system except the oil column resonant frequency.
e 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.e 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.e 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.e frequency effect is biggest of all, the damping ratio effect is secondary, and the mass effect is least.e 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 amplitudefrequency response in a frequency band close to the natural frequency of the specimen.e 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.
e 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 outerloop 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.
e 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.e 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.e 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.
e 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.
e 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.e 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.

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 2 Shock and Vibration 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 M T is the mass of the table, x is the piston position, A p is the cross-sectional area of the actuator piston, p L is the pressure di erence between actuator chambers, Q L is the total oil ow, V is the compressed oil volume in the actuator cylinder, β is the oil bulk modulus, C c is the leakage coe cient, k q is the ow gain, K c is the ow-pressure coe cient, s is the Laplace transform variable, x v is the valve spool displacement, and F L is the force in the actuator.e 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 exible connection specimen were analyzed as a numerical model.Figure 1 shows the mechanical model of the dual shaking tables with exible connecting specimen.
e dynamic equilibrium equations of the tables and specimen can be written in terms of Laplace transform as where M p is the mass of the exible connecting specimen, k is the connection sti ness coe cient between each table and specimen, c is the damping coe cient between each table and specimen, x p is the relative displacement of the specimen, x 1 is the displacement of table 1, x 2 is the displacement of table 2, F 1 is the force in the actuator of the shaking table 1, and F 2 is the force in the actuator of the shaking table 2. Equations ( 4)-( 6) can be simpli ed to Combining the dynamic equilibrium equations of the tables and specimen with the basic equations of the electrohydraulic servo-shaking table system, the basic equations of the dual electrohydraulic servo-shaking table system with specimen can be written in terms of Laplace transform as where In order to build the transfer function of dual shaking tables with specimen, three-variable feedback was introduced into the open-loop transfer function of the dual electrohydraulic servo-shaking tables, and second-order characteristics of the servo-valve and the sensor were considered.
e transfer function block diagram of the system is shown in Figure 2.
e transfer function matrix of the dual shaking tables system with specimen is where e system performance e ects on the dual shaking tables caused by the mass, frequency, and damping ratio of the exible connecting specimen were obtained as shown in Figures 3-8.Figures 3, 5, and 7 show that the interaction e ects on the transfer function H 11 are characterized by a peak-notch distortion in the amplitude-frequency response at the natural frequency of the specimen and in a close frequency band.Figures 4, 6, and 8 show that the interaction e ects on the transfer function H 12 are characterized by a peak distortion in the amplitude-frequency response at the natural frequency of the specimen and in a close frequency band.Figures 3 and 4 present the e ects of the specimen's mass on frequency response of transfer functions H 11 and H 12 when the frequency of the specimen is 5 Hz and the damping ratio of the specimen is 0.035.Comparison of the frequency response characteristic curves in Figure 3 indicates that the dual shaking tables- 4

Shock and Vibration
specimen interaction e ect on transfer function H 11 increases with the mass of the specimen.Comparison of the frequency response characteristic curves in Figure 4 indicates that the dual shaking tables-specimen interaction e ect on the transfer function H 12 also increases with the mass of the specimen.Figures 5 and 6 present the e ect of specimen's frequency on frequency response of the transfer functions H 11 and H 12 when the mass of the specimen is 12000 kg and the damping ratio of the specimen is 0.035.Comparison of the frequency response characteristic curves in Figure 5 indicates that the dual shaking tables-specimen interaction e ect on the transfer function H 11 increases with the frequency of the specimen.
Comparison of the frequency response characteristic curves in Figure 6 indicates that the dual shaking tables-specimen interaction e ect on the transfer function H 12 also increases with the frequency of the specimen.Figures 7 and 8 present the e ects of the specimen's damping ratio on frequency response of transfer functions H 11 and H 12 when the mass of the specimen is 12000 kg and the frequency of the specimen is 5 Hz.Comparison of the Bode diagrams in Figure 7 indicates that the dual shaking tables-specimen interaction e ect on the transfer function H 11 decreases with the damping ratio of the specimen.Comparison of the Bode diagrams in Figure 8 indicates that the dual shaking tables-specimen interaction e ect on the transfer function H 12 also decreases with the damping ratio of the specimen.Comparing with the interaction e ects generated by the single shaking table-specimen [7], the interaction e ects generated by the dual shaking tables-specimen are more complicated and obvious.

Force Feedback Compensation for the Interaction Effects
Performance analysis of the dual shaking tables with specimen showed that the interaction e ect on the transfer function H 11 is characterized by a peak-notch distortion in the amplitude-frequency response at the natural frequency of the specimen and in a close frequency band, and the interaction e ect on the transfer function H 12 is characterized by a peak distortion in the amplitude-frequency response at the natural frequency of the specimen and in a close frequency band.e interaction generated by the specimen loaded on the dual shaking tables a ected the output of the tables.In order to eliminate the interaction e ects on system performance of the dual shaking tables, force feedback compensation was introduced into the existing three-variable control in this paper.e compensation principle is that the interaction can be calculated by

Shock and Vibration
Equation (12), in which the force in the actuator F L is measured by the force sensor, or the pressure di erence between actuator chambers p L is measured by the di erential pressure sensor, and the acceleration of the table s 2 x T is measured by the acceleration sensor.en, the compensation signal is formed by multiplying the interaction between the shaking table and specimen by the inverse transfer function from the valve spool displacement x v to the force in the actuator F L .e new equivalent drive signal is synthesized by the compensation signal negative feedback to the command signal.Figure 9 shows the block diagram of the dual shaking tables system in which force feedback compensation was applied to the block diagram in Figure 2. e interaction between the dual shaking tables and specimen is e inverse transfer function G −1 Fu (s) from the valve spool displacement x v to the force in the actuator F L obtained by Equations ( 1)-( 3) is e compensation signal Δu can be expressed as

Simulation Analysis of the Control System of Dual Shaking Tables
In order to verify the control e ect of the proposed method for the dual shaking tables loading with specimen, the dual electrohydraulic servo-shaking tables connected by the exible specimen were selected for simulation.Table 1 shows the characteristic parameters of the dual electrohydraulic servoshaking tables.e time history of El-Centro NS reduced 10 times scale was employed as the input signal.e simulation analysis of the dual electrohydraulic servo-shaking tables was conducted by using MATLAB/Simulink.Figure 10 shows the input signal.
A exible specimen was selected for numerical simulation.
e mass of the specimen is 12000 kg, the nature frequency of the specimen is 9 Hz, and the damping ratio of the specimen is 0.03. Figure 11 shows the Bode diagram of the transfer function H 11 of the dual shaking tables system with specimen controlled by three-variable control and force feedback compensation.Comparison of three-variable control and force feedback compensation indicates that the Bode diagram of the transfer function H 11 under threevariable control has a peak-notch distortion in the amplitude-frequency response at the natural frequency of the specimen and in a close frequency band.When force feedback compensation is introduced into three-variable control, the interaction e ect on the transfer function H 11 can be completely compensated.Figure 12 shows the Bode diagram of the transfer function H 12 of the dual shaking tables system with specimen for the cases controlled by three-variable control and force feedback compensation.e Bode diagram of the transfer function H 12 under threevariable control has a peak distortion in the amplitudefrequency response at the natural frequency of the specimen and in a close frequency band.As the acceleration of the shaking table 1   Frequency (Hz) 6 Shock and Vibration force feedback compensation.Figure 13 shows the acceleration time histories of the dual shaking tables without/with specimen under three-variable control.Figure 14 shows the acceleration time histories of the dual shaking tables with specimen under force feedback compensation and without specimen under three-variable control.Comparison of Figures 13 and 14 indicates that the acceleration time history of the dual shaking tables with specimen under force feedback compensation control coincides exactly with the acceleration time history of the dual shaking tables without specimen under three-variable control.Figure 15 shows the acceleration Fourier amplitude spectrum of the dual shaking tables without/with specimen under three-variable control.
e interaction e ect on the acceleration Fourier amplitude spectrum is manifested by a peak-notch distortion at the natural frequency of the specimen and in a close frequency band.Figure 16 shows the acceleration Fourier amplitude spectrum of the dual shaking tables with specimen under force feedback compensation and without specimen under   indicates that the acceleration Fourier amplitude spectrum of the dual shaking tables with specimen under force feedback compensation control coincides exactly with the acceleration Fourier amplitude spectrum of the dual shaking tables without specimen under three-variable control.e peak-notch distortion in the acceleration Fourier amplitude spectrum caused by the specimen is well compensated by force feedback compensation introduced into three-variable control.

Simulation Analysis of Error Influence
Force feedback compensation proposed in this paper was based on the assumptions that the force in the actuator F L and the acceleration of the table s 2 x T were measured accurately.Based on the above ideal state, the control e ect of force feedback compensation for the interaction between the shaking table and specimen were analyzed.Actually, some errors in real time and accuracy of information acquisition and feedback might exist in the shaking table testing, such as the measurement of force in the actuator and the acceleration of the shaking table, and so on.ose errors have an e ect on the control e ect of force feedback compensation.e e ects of force sensor error (or the di erential pressure sensor error) and the acceleration sensor error on force feedback compensation were simulated for the interaction between the dual shaking tables and specimen.
Considering the errors of the force sensor or the differential pressure sensor and acceleration sensor, Equation  Shock and Vibration (12) of the interaction between the shaking table and specimen can be expressed as where α is the error coe cient of acceleration sensor and c is the error coe cient of force sensor or di erential pressure sensor.
Figures 17 and 18 present the e ect of varying acceleration sensor error on the frequency characteristic of transfer functions H 11 and H 12 when the dual shaking tables with specimen are controlled by introducing force feedback compensation into three-variable control.e frequency characteristic curves, which are obtained by the error coe cient of the acceleration sensor varied from 0 percent to 15 percent, indicate that the e ect caused by some error of the acceleration sensor is characterized by a peak distortion in the high frequency of the amplitude-frequency response of the transfer functions H 11 and H 12 .However, the peak value in the amplitude-frequency response of the transfer function H 11 caused by the acceleration sensor error is less than 3 dB, which is within the range of acceptable error.e peak in the amplitude-frequency response of the transfer function H 12 caused by the acceleration sensor error is less than −50 dB.So the error e ect caused by the acceleration sensor can be neglected.erefore, the error of acceleration sensor does not signi cantly a ect the interaction between the dual shaking tables and specimen compensated by introducing force feedback into three-variable control.
Figures 19 and 20 present the e ect of varying force sensor error or di erential pressure sensor error on the frequency characteristic of transfer functions H 11 and H 12 when the dual shaking tables with specimen are controlled by introducing force feedback into three-variable control.
e frequency characteristic curves in Figure 19, which are obtained by the error coe cient of the force sensor or the di erential pressure sensor varied from 0 percent to 15 percent, indicate that the e ect caused by some error of force sensor or the di erential pressure sensor is still characterized by a peak-notch distortion in the amplitudefrequency response of the transfer function H 11 at the natural frequency of the specimen and in a close frequency band.However, the absolute values of the peak and notch are less than 3 dB.e e ect of the force sensor error or the di erential press sensor error on the compensation for the interaction e ect can be neglected.At the same time, the high-frequency amplitude in the amplitude-frequency response of the transfer function H 11 is slightly reduced by the error of the force sensor or the di erential pressure sensor.A frequency band of 3 Hz can be reduced by per 5 percent error.Comparison of the frequency characteristic Shock and Vibration curves in Figure 20 indicates that the peak distortion in the amplitude-frequency response of the transfer function H 12 can be still suppressed by introducing force feedback into three-variable control.e amplitude of the peak distortion is down from 7 dB to −12 dB when the error of the force sensor or the di erential pressure sensor is 15 percent.e control e ect is slightly decreased by the error of the force sensor or the di erential pressure sensor.

Conclusion
In the dual shaking tables test, the interaction force, which is generated by the specimen and nonsynchronous output of shaking tables, a ects the system characteristics of the dual shaking tables; therefore, the precision of the output of the dual shaking tables is decreased.In order to improve the precision of the output of the dual shaking tables, the interaction e ect and compensation on the system characteristics of the dual shaking tables with specimen were studied in this paper.
(1) e interaction e ect on the transfer function from u x1 to s 2 x 1 (H 11 ) was characterized by a peak-notch distortion in the amplitude-frequency response at the natural frequency of the specimen and in a close frequency band, and the interaction e ect on the transfer function from u x1 to s 2 x 2 (H 12 ) was characterized by a peak distortion in the amplitude-frequency response at the natural frequency of the specimen.10 Shock and Vibration (2) In order to improve the output precision of the dual shaking tables, force feedback compensation was introduced into three-variable control to compensate the interaction between the dual shaking tables and specimen.e simulation analysis of the dual shaking tables with specimen shows that the interaction effect on the transfer function from u x1 to s 2 x 1 (H 11 ) can be completely compensated, and the interaction effect on the Bode diagram of the transfer function from u x1 to s 2 x 2 (H 12 ) can be suppressed by introducing force feedback compensation into threevariable control.(3) e information acquisition and feedback errors of force in the actuator and the acceleration of the shaking table exist in the dual shaking tables testing.But the error influence on the system performance of the dual shaking tables was not obvious if the information acquisition and feedback errors were less than 15 percent.e error influence on the control effect of force feedback compensation can be neglected.Force feedback compensation can be applied to compensate the interaction between the dual shaking tables and specimen.

Figure 2 :Figure 3 :
Figure 2: Transfer function block diagram of dual shaking tables with specimen.

Figure 4 :Figure 5 :Figure 6 :
Figure 4: E ect of specimen's mass on frequency response of H 12 .

Figure 8 :
Figure 8: E ect of specimen's damping ratio on frequency response of H 12 .

Figure 7 :
Figure 7: E ect of specimen's damping ratio on frequency response of H 11 .

Figure 9 :
Figure 9: Block diagram of force feedback compensation of the dual shaking tables with specimen.

Figure 15 :Figure 16 :Figure 17 :
Figure 15: Acceleration Fourier amplitude spectrum of the dual shaking tables without/with specimen under TVC.

Figure 19 :Figure 20 :Figure 18 :
Figure 19: E ect of di erential pressure sensor error on the Bode diagram of H 11 .

Table 2
Figure 1: Mechanical model of dual shaking tables with specimen.H 11 is the transfer function from the command signal of the shaking table 1 u x1 to the acceleration of the shaking table 1 s 2 x 1 , H 12 is the transfer function from the command signal of the shaking table 1 u x1 to the acceleration of the shaking table 2 s 2 x 2 , H 21 is the transfer function from the command signal of the shaking table 2 u x2 to the acceleration of the shaking table 1 s 2 x 1 , and H 22 is the transfer function from the command signal of the shaking table 2 u x2 to the acceleration of the shaking table 2 s 2 x 2 .
v1 is the valve spool displacement of the shaking table 1, x v2 is the valve spool displacement of the shaking table 2, s 2 x 1 is the acceleration of the shaking table 1, s 2 x 2 is the acceleration of the shaking table 2, u x1 is the command signal of the shaking table 1, u x2 is the command signal of the shaking table 2, G 3 is the transfer function of the threevariable generator, G 4 is the transfer function of the threevariable feedback loop, G sv is the transfer function of the electrohydraulic servo-valve when the ow gain is k q 1, s 2 x 1 H 11 u x1 + H 12 u x2 , the acceleration of the shaking table 1 s 2 x 1 is obviously in uenced by the motion of the shaking table 2 in the natural frequency of the specimen and in a close frequency band.When force feedback compensation is introduced into three-variable control, the amplitude of the Bode diagram of the transfer function H 12 is down from 7 dB to −40 dB at the natural frequency of the specimen.e interaction e ect on the Bode diagram of the transfer function H 12 is suppressed by