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Edited and reviewed by: Abid Ali Shah, University of Science and Technology Bannu, Pakistan

Reviewed by: Lihua Zhu, Xi'an University of Architecture and Technology, China; Yanbin Shen, Zhejiang University, China

This article was submitted to Structural Materials, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The viscoelastic material is one of most popular shock absorbing materials for mitigating vibrations of building structures due to earthquake. Its dynamic performance is affected by the temperature, the excitation frequency and the excitation amplitude. Therefore, in order to study the non-linear dynamic performance of the viscoelastic materials, a hybrid test system using the electric exciter is proposed, in which the electric exciter is the actuator, MATLAB is used for the simulation of the numerical substructure and the communication between the upper computer and the lower machine, and STM32 single-chip microcomputer is used to control the work state of the electric exciter. Based on the equivalent force control method and the incremental PID control algorithm, a controller is designed to make sure the electric exciter produces accurate control forces to the non-linear test component in the physical substructure. It can be shown from the test results that the developed whole hybrid test system is feasible and effective.

In the process of the research of the structural dynamic performance induced by different earthquake, sometimes the core materials or components of the whole structure have very strong non-linearities and unknown properties, which accurate mathematical model cannot be obtained, at the same time, the structure is some kind of large-scale structures, so how to analysis the effect or functions of the core components and the whole structural dynamic performance? Hybrid Test, also called as on-line experiment (Hakuno et al.,

Since the twenty-first century, with the advent of large-scale and complex structures, the hybrid test is developing in two directions: In the spatial domain, in order to integrate the experimental resources from different regions, the hybrid test is developed from local partial tests to network collaborative tests (Mosqueda et al.,

The viscoelastic material is one of most popular shock absorbing materials for mitigating vibrations of building structures due to earthquake (Xu et al.,

The hybrid test system using the electric exciter mainly includes two parts: the upper computer and the lower machine (that is the physical substructure), as shown in

The hybrid test system.

The detail design scheme diagram of the hybrid test system.

In this paper, the electric exciter is as the actuator, which works on the non-linear test component according to the control signal coming from the upper computer. Here the viscoelastic damper is chosen as the non-linear test component. In order to make the electric exciter to produce accurate control forces, the equivalent force control method and the incremental PID control algorithm are used at the same time, as shown in

Equivalent force control schematic diagram.

For the electric exciter, its control current is linear with the force it produces. While the test component is non-linear, the equivalent force control method (Wu et al.,

Where, _{N} is the reaction force vector of the numerical substructure; _{E} is the reaction force vector of the non-linear test component; _{i}_{i}_{i} are the displacement, the velocity and the acceleration, respectively; Δ

According to Equations (2, 3), the speed and the acceleration of step

Substitute Equations (4, 5) into Equation (1):

_{PD} is a pseudo-stiffness matrix. _{EQ, i+1} is the equivalent force for each load cycle, which consists of two parts, the external excitation force _{i+1} in the current loading cycle and the pseudo-dynamic effect calculated according to the displacement response in this period. Equation (6) is a non-linear equation about the displacement variable _{i+1}, which can also be regarded as an equilibrium equation about the equivalent force _{EQ, i+1}. Meanwhile it can be seen from Equation (6) that the left side of the equation is added by the damping force _{N}_{i+1} of the numerical substructure, the pseudo-dynamic _{PD}_{i+1} and the experimental reaction force _{EQ}_{i+1} of the non-linear test component, and the right side of the equation can be regarded as the equivalent external force _{EQ, i+1}. So the solution of the equation is the displacement _{i+1} of the effect system under the influence of equivalent external force.

The equivalent force control method uses a closed-loop control system, that is, a feedback control method, the control method makes the feedback force [left side of Equation (6)] equal to the equivalent force [right side of Equation (6)]steadily and asymptotically, as shown in _{F}, the equivalent force difference _{EQ, i+1}(_{EQ, i+1}(_{EQ, i+1}(_{i+1}(_{F} is the force distribution coefficient, its effect is equivalent to Newton iterative method in the Jacobian matrix, the value of the force distribution coefficient _{F} is as follows:

Where, _{N} and _{E} are the initial stiffness matrices of the numerical substructure and the non-linear test component respectively.

The function of the controller is enabling the equivalent force feedback value track the equivalent force command accurately, the controller is divided into the inner-loop force controller and the outer-loop equivalent force controller. The outer-loop controller is equal-effect control to calculate the force loading command of the electric exciter by the force distribution coefficient. The inner-loop controller is the force control of the electric exciter, so that the electric exciter can accurately reach the force command. The PID controllers are adopted as these two controllers.

In the control system involving computer technology, most of the traditional analog PID cannot be used successfully, because the computer cannot perform integral or differential operation directly, computers can only be simulated infinitely to approach this mathematical calculation in other ways. On the other hand, in the single-chip microcomputer technology, the signal acquisition is also discrete, only can collect signal for feedback periodically through the signal acquisition module. Digital PID algorithm is divided into two kinds, the incremental PID algorithm and the position PID algorithm. The position PID algorithm directly produces the final output of the system in each control period, which is a relatively direct control method. But in this approach, each of the previous errors are accumulated, and the entire system is closely linked before and after each adjustment cycle. It can easily lead to excessive adjustment of the amplitude, and even cause serious control accidents. The incremental PID algorithm converts the analog signal into a digital signal, which is convenient for computer calculation and the single-chip acquisition. On the other hand, compared with the position PID algorithm, the incremental PID algorithm has a smaller calculation, which can guarantee the reliability and real-time performance of the system. Therefore, the incremental PID controller is chosen, and the discrete incremental PID algorithm is as follows:

Where, _{P} is the proportional coefficient of the control system; _{i} is the integral time constant of the control system; _{d} is the differential time constant of the control system; and

The each adjustment cycle output of the incremental PID controller is the increment of adjustment on the basis of the first 2 times adjustment cycles, so by subtracting Equation (11) from Equation (10), the output equation of the incremental PID controller can be obtained:

Where,

According to the above principle of equivalent force control, the Matlab/Simulink simulation model of the equivalent force control system of the hybrid test system is shown in

Simulink simulation model of the equivalent control system.

In the simulation model the viscoelastic damper is chosen as the non-linear test component, and its transfer function is:

Where _{e} is the equivalent stiffness of the viscoelastic damper. According to the basic theory of the viscoelastic damper, its parameters can be obtained. The shear modulus is:

The loss factor is:

The loss modulus is:

The equivalent stiffness is:

The equivalent damping is:

Where _{0} is the maximum displacement of the viscoelastic damper in horizontal direction; _{0} is the maximum damping force of the viscoelastic damper; _{1} is the damping force at the maximum displacement of the viscoelastic damper; _{2} is the damping force at zero displacement of the viscoelastic damper; ω is the loading circular frequency.

The test device of the hybrid test system designed in this paper is shown in

The test device of the hybrid test system.

The viscoelastic damper.

Before testing the performance of the whole hybrid test system, the dynamic performance of the physical substructure is tested firstly. Three sets of sinusoidal signals with the same frequency and different amplitudes are used as the control signals of the electric exciter, which will make the electric exciter to produce the frequencies of the forces all are 5 Hz, and the amplitudes of the forces are 5, 100, and 150 N, respectively. Under these three sets of the forces the force-displacement relationship of the viscoelastic damper are detected by using the force sensor and the displacement sensor in the hybrid test system, and then the results are shown in

Force-displacement hysteretic curve of the viscoelastic damper.

El-Centro earthquake wave and Tianjin earthquake wave are used as the excitation signals of the whole hybrid test system to test its performance. Under different earthquake wave excitation, the output forces of the electric exciter collected in real-time are compared with expected output forces, as shown

Comparison between experimental value and expected value under different earthquake wave.

In this paper, a hybrid test system is proposed, in which MATLAB is used for the simulation of the numerical substructure and the communication between the upper computer and the physical substructure; the electric exciter is the actuator; STM32 single-chip microcomputer is used to control the work state of the electric exciter; the viscoelastic damper adopted as the non-linear test component; the force sensor and the displacement sensor are used to measure the force and the displacement of the non-linear test component. In order to make the electric exciter to produce accurate control forces to the non-linear test component in the physical substructure, the controller is designed based on the equivalent force control method and the incremental PID control algorithm. The dynamic performance of the physical substructure and the performance of the whole hybrid test system are tested. The test results show that the physical substructure can work normally; the viscoelastic damper has strong non-linearity and good energy dissipation characteristics; and the whole system is feasible and effective.

All datasets generated for this study are included in the manuscript and/or the supplementary files.

Y-QG proposed the idea of this paper. Under the guidance of Y-QG, YaL, XC, XJ, and YiL finish end the numerical analysis. YaL, T-TY, and XC finished the experiment of the hybrid test system. Y-QG, YaL, T-TY, and XJ jointly completed the writing of the article. YiL helped in proof reading of overall presentation and experimental data.

XC was employed by company Nanjing Dongrui Damping Control Technology Co., Ltd, Nanjing, China. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The work was supported by The National Key R&D Programs of China with Grant Numbers (2016YFE0119700 and 2016YFE0200500), Jiangsu International Science and Technology Cooperation Program with Grant Number (BZ2018058), the Program of Chang Jiang Scholars of Ministry of Education, Ten Thousand Talent Program of Leading Scientists and the Program of Jiangsu Province Distinguished Professor.