Seismic energy response analysis of equipment-structure system via real-time dynamic substructuring shaking table testing

Abstract

Dynamic equations are presented that have been deduced for a real-time dynamic substructuring shaking table test of an equipment-structure system, based on the branch mode substructure method. The equipment is adopted as the experimental substructure, which is loaded by the shaking table, while the structure is adopted as the numerical substructure. Real-time data communication occurs between the two substructures during the test. A real-time seismic energy calculation method was proposed for the calculation of energy responses, both in the experimental substructure and the numerical substructure. Taking a representative four-story steel frame/equipment model, real-time dynamic substructuring shaking table tests and overall model tests were executed. The proposed real-time dynamic substructuring shaking table testing method was verified by comparing the test results with shaking table test results for the overall model. The energy responses of each component in the equipment-structure system, using different connection types, also were studied. Changes in the connection types can lead to changes in the energy responses of the equipment-structure system, especially with respect to the equipment. The choice of the connection for the equipment-structure coupled system should take into account the operational performance objective of the equipment.

Keywords

branch mode substructure equipment-structure system real-time substructure test seismic energy response shaking table

Introduction

Owing to the increasing number of non-structural components that are present in modern buildings, the seismic performance of equipment-structure (ES) systems has been widely investigated (Filiatrault and Sullivan, 2014; Zhu et al., 2014). Current experimental studies of ES systems can be considered either as overall model tests (Chen et al., 2015; Furukawa et al., 2013; Li et al., 2003) or as individual equipment tests (Mosqueda et al., 2009). Conventional overall model tests are limited by complicated procedures and high costs, while individual equipment tests require complicated and specialist test devices. The real-time dynamic substructuring (RTDS) testing method can provide a novel way to investigate system interactions (Nakashima et al., 1992). Using this approach, the overall model is divided into experimental substructures and numerical substructures. Experimental substructures are loaded using the shaking table, while the numerical substructures are simulated using computing software. Data exchange between the experimental substructures and the numerical substructures can reduce the required scope of test procedures without affecting test accuracy (Günay et al., 2015; Mosalam et al., 2016). The real-time dynamic substructuring shaking table (RTDSST) test method was applied to investigate aseismic performance of the structure equipped with control devices (Lee et al., 2007; Lin and Christenson, 2011; Mercan and Ricles, 2009). Tu et al. (2018) utilized the RTDSST test method on the composite structure, they studied the behavior of a traditional light-frame wood shear walls with post-tensioned rocking cross-laminated timber panels. Considering the advantages of RTDSST test method on reducing the test scale, the method was also applied to the soil-structure interaction system (Wang et al., 2011; Yan et al., 2014); the soil was adopted as the numerical substructure thus simplifying the boundaries treatment of the soil and was convenient for changing the soil type. For the application of real-time substructure tests on ES systems, a key point is the calculation of the interaction effects between the equipment and the structure. Jiang and Yan (1994) proposed motion equations for soil-structure interaction systems using the branch mode substructure method. Therein, the effects of soil-structure interaction were reflected by loads on the term for soil-structure coupling. Connections between the equipment and the structure include rigid connections (e.g. screw bolts, soldered joints) (Heredia-Zavoni et al., 2010) and flexible connections (e.g. rubber supports) (Fan et al., 2009; Lu and Lin, 2008; Reggio and De Angelis, 2014). Previous studies have indicated that changes in the connection types can lead to changes in the dynamic characteristics of the ES system, thus affecting the seismic responses of the equipment and the structure (Alhan and Gavin, 2005; Han and Qin, 2004). It is desirable, therefore, if the effects of different connection types on the seismic response of equipment and structure are well understood.

In order to reflect the overall performance of ES systems under seismic excitations, the concept of energy was introduced into the interaction study (Chen and Soong, 1994). During the process of earthquake, the phenomena of energy input, transmission, and dissipation exist in the structure. Compared with acceleration, displacement, and other indicators of general seismic responses of the system, the energy component can give an improved reflection of seismic mechanisms. Besides, the energy component has the potential to address the effect of earthquake duration and cumulative damage of the system directly (Bojórquez et al., 2011; Kalkan and Kunnath, 2008; Wang et al., 2015). Under seismic excitations, dynamic energy is exchanged between the equipment and the structure, and this is more prevalent when frequency tuning occurs between the two parts (Li and Guo, 2006). Analyzing the energy flow in the overall ES system leads to a better understanding of the mechanism of interactions. Previous studies revealed that energy consumption by the secondary structure can change the energy distribution of the primary structure (Ahn et al., 2013; Chen et al., 2016; Tan et al., 2014; Wong and Chee, 2004). Current studies of the energy response of ES systems focus mainly on the energy responses of the structure, while the energy response of each component in the coupled system has not been properly understood and current calculation methods are relatively complicated.

Based on the branch mode substructure method, motion equations for the ES system were developed during the present investigation, which reduces the degree of freedom (DOF) of the equipment and the structure. A scale model of the ES system is used as the research object, where the equipment is modeled as a 2-DOF communicator model and the structure is modeled as a four-story steel frame. An RTDSST testing method for the ES system has been proposed. Combined with the RTDSST test, a real-time seismic energy calculation method for ES system has been proposed for the calculation of energy levels both in the numerical substructure and the experimental substructure. RTDSST tests and overall model shaking table tests for the ES system model, under rigid and flexible connections, were conducted. The proposed RTDSST testing method, based on branch mode substructure method, was verified by comparing the test results with those of tests for the overall model. Finally, the energy responses from each component in the ES system, under different connection types, were investigated, and the effects of the connection type on energy distributions in the equipment and the structure were studied.

ES system motion equations and energy balance equations

First, the motion equation of a coupled ES system was derived using the traditional branch mode substructure method. The number of DOFs of the structure and equipment model is reduced. Based on that, a four-story steel frame/equipment system is taken as an example. The completed computational characteristics matrices for the equipment and the structure are reserved to consider the effects of nonlinear factors. After transformation, the calculation formula for the RTDSST test of the ES system is given. Combined with the RTDSST test, the seismic energy response calculation equations for both the structure and the equipment during tests can be derived.

ES system motion equation, based on the branch mode substructure method

The general system model for an ES system is shown in Figure 1. The system consists of both the structure and the equipment. The equipment and its connection supports are referred to collectively as “the equipment.” According to the branch mode substructure method, the ES system (see Figure 1(a)) can be divided into branch s (rigid equipment on the elastic structure, see Figure 1(b)) and branch d (deformed equipment on the rigid structure, see Figure 1(c)).

Figure 1.

Schematic diagram of the motion equation for the general ES system: (a) ES system, (b) rigid equipment on the elastic structure, and (c) deformed equipment on the rigid structure.

The characteristic equations for branch s are

\begin{matrix} [k_{s}] {ϕ_{s}} = λ_{s} [m_{s}] {ϕ_{s}} \\ [Φ_{s}] = [{ϕ}_{s}^{1} \dots {ϕ}_{s}^{i} \dots {ϕ}_{s}^{m}] \end{matrix}

(1)

where k_s and m_s denote the stiffness matrix and mass matrix in branch s, respectively, and Φ_s is the modal transformation matrix consisting of m order modes of branch s.

The computational matrices of branch s after modal transformation are

\begin{matrix} {u_{s}} = [Φ_{s}] {q_{s}} \\ [{\tilde{k}}_{s}] = {[Φ_{s}]}^{T} [k_{s}] [Φ_{s}] \\ [{\tilde{m}}_{s}] = {[Φ_{s}]}^{T} [m_{s}] [Φ_{s}] \\ [{\tilde{c}}_{s}] = {[Φ_{s}]}^{T} [c_{s}] [Φ_{s}] \\ [{\tilde{f}}_{s}] = {[Φ_{s}]}^{T} [f_{s}] \end{matrix}

(2)

where u_s and q_s denote the displacement coordinates and modal coordinates of the structure in branch s; c_s denotes the damping matrix of the structure in branch s, which can be obtained using general damping theory; and f_s denotes the load matrix of structure in branch s.

Similarly, the characteristic equations of branch d are

\begin{matrix} [k_{d}] {ϕ_{d}} = λ_{d} [m_{d}] {ϕ_{d}} \\ [Φ_{d}] = [{ϕ}_{d}^{1} \dots {ϕ}_{d}^{i} \dots {ϕ}_{d}^{n}] \end{matrix}

(3)

where k_d and m_d denote the stiffness matrix and mass matrix in branch d, respectively, and Φ _d is the modal transformation matrix, composed of n order modes of branch d. The computational matrices of branch d after modal transformation are

\begin{matrix} {u_{d}} = [Φ_{d}] {q_{d}} \\ [{\tilde{k}}_{d}] = {[Φ_{d}]}^{T} [k_{d}] [Φ_{d}] \\ [{\tilde{m}}_{d}] = {[Φ_{d}]}^{T} [m_{d}] [Φ_{d}] \\ [{\tilde{c}}_{d}] = {[Φ_{d}]}^{T} [c_{d}] [Φ_{d}] \\ [{\tilde{f}}_{d}] = {[Φ_{d}]}^{T} [f_{d}] \end{matrix}

(4)

where u_d and q_d denote the displacement coordinates and modal coordinates of the equipment in branch d, respectively, and c_d and f_d denote the damping matrix and load matrix of the equipment in branch d, respectively.

The displacement of the equipment consists of rigid body displacement caused by deformation of the structure (see Figure 1(b)) and self-displacement (see Figure 1(c)). This can be expressed as

{u_{d}} = [R] [Φ_{s}] {q_{s}} + [Φ_{d}] {q_{d}}

(5)

where R represents the displacement transformation matrix of the equipment, which can be obtained based on the coordinate relationship of the deformation in branch s. Displacement of the equipment and the structure then can be expressed as

{\begin{matrix} u_{s} \\ u_{d} \end{matrix}} = [\begin{matrix} Φ_{s} & 0 \\ R Φ_{s} & Φ_{d} \end{matrix}] {\begin{matrix} q_{s} \\ q_{d} \end{matrix}}

(6)

According to the principles of the branch mode substructure method (Jiang and Yan, 1994), the motion equation of the ES system can be obtained by

\begin{matrix} [\begin{matrix} {\tilde{m}}_{s} & Φ_{s}^{T} R^{T} m_{d} Φ_{d} \\ Φ_{d}^{T} m_{d} R Φ_{s} & {\tilde{m}}_{d} \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{q}}_{s} \\ {\overset{\cdot\cdot}{q}}_{d} \end{matrix}} \\ + [\begin{matrix} {\tilde{c}}_{s} & 0 \\ 0 & {\tilde{c}}_{d} \end{matrix}] {\begin{matrix} {\overset{\cdot}{q}}_{s} \\ {\overset{\cdot}{q}}_{d} \end{matrix}} + [\begin{matrix} {\tilde{k}}_{s} & 0 \\ 0 & {\tilde{k}}_{d} \end{matrix}] {\begin{matrix} q_{s} \\ q_{d} \end{matrix}} = {\begin{matrix} {\tilde{f}}_{s} \\ {\tilde{f}}_{d} \end{matrix}} \end{matrix}

(7)

The traditional branch mode method can reduce greatly the DOFs of an ES system while ensuring accuracy. However, this method is only applicable to linear analysis, whereas the structure and equipment may be in a nonlinear state when subjected to seismic excitations. The traditional branch mode substructure method was improved by Wang and Jiang (2010), and thus can consider the effects of nonlinear factors. Based on that approach, motion equations for the ES system that are applicable to both linear and nonlinear analyses have been proposed in this study.

The RTDSST testing method and its associated energy balance equations

A four-story steel frame/equipment model was used as the research object. Due to the low DOFs of the model, and system nonlinearity, complete matrices of computational characteristics were used without modal transformations, based on the concept of the branch mode substructure method (Figure 2).

Figure 2.

Schematic diagram of the motion equations of an ES system model: (a) ES system, (b) rigid equipment on the elastic structure, and (c) deformed equipment on the rigid structure.

Displacement of the structure and the equipment (equation (6)) changes to

{\begin{matrix} u_{s} \\ u_{d} \end{matrix}} = [\begin{matrix} I & 0 \\ R & I \end{matrix}] {\begin{matrix} q_{s} \\ q_{d} \end{matrix}}

(8)

and the ES system equation of motion (equation (7)) changes to

\begin{matrix} [\begin{matrix} m_{s} & R^{T} m_{d} \\ m_{d} R & m_{d} \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{q}}_{s} \\ {\overset{\cdot\cdot}{q}}_{d} \end{matrix}} + [\begin{matrix} c_{s} & 0 \\ 0 & c_{d} \end{matrix}] {\begin{matrix} {\overset{\cdot}{q}}_{s} \\ {\overset{\cdot}{q}}_{d} \end{matrix}} \\ + [\begin{matrix} k_{s} & 0 \\ 0 & k_{d} \end{matrix}] {\begin{matrix} q_{s} \\ q_{d} \end{matrix}} = {\begin{matrix} f_{s} \\ f_{d} \end{matrix}} \end{matrix}

(9)

In order to realize an RTDSST test of the ES system, the items relating to the coupling terms R^Tm_d and m_dR in equation (9) are shifted to the right side of the equation and appear as the load. Therefore, the motion equation for the structure and equipment with interaction effects taken into consideration can be described by

m_{s} {\overset{\cdot\cdot}{q}}_{s} + c_{s} {\overset{\cdot}{q}}_{s} + k_{s} q_{s} = f_{s} - R^{T} m_{d} {\overset{\cdot\cdot}{q}}_{d}

(10)

m_{d} {\overset{\cdot\cdot}{q}}_{d} + c_{d} {\overset{\cdot}{q}}_{d} + k_{d} q_{d} = f_{d} - m_{d} R {\overset{\cdot\cdot}{q}}_{s}

(11)

According to equations (10) and (11), the coupled term load due to interaction effects appears on the right side of both the structure and equipment equations, and an RTDSST based on the branch mode substructure method can be implemented, due to the presence of the coupling terms. In the RTDSST test, the data exchange between the substructures is realized by transmission of the coupling term load.

The procedure for an RTDSST test for an ES system is as follows:

Assume that the total load on the right-hand side of the structure equation of motion (equation (10)) is known, and hence the acceleration responses of the structure at this moment can be calculated in the software.

Substitute the acceleration responses of the structure into equation (11), and the excitation of the equipment then can be calculated and applied to the equipment via the shaking table.

Based on the acceleration responses collected from the equipment, the overall load experienced by the structure at the next moment can be calculated using equation (10), and the corresponding acceleration responses of the structure can be calculated again.

In this way, the earthquake responses of the structure and the equipment can be obtained. From the point of view of energy, equations (10) and (11) are integrated separately in the time range of 0-t₀ with respect to displacements d_qs and d_qd, and the relative energy equations applicable to the calculation of the individual energy responses of the structure and equipment are

\begin{matrix} \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{s}^{T} m_{s} {\overset{\cdot\cdot}{q}}_{s} d (t) + \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{s}^{T} c_{s} {\overset{\cdot}{q}}_{s} d (t) + \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{s}^{T} k_{s} q_{s} d (t) \\ = \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{s}^{T} (f_{s} - R^{T} m_{d} {\overset{\cdot\cdot}{q}}_{d}) d (t) \end{matrix}

(12)

\begin{matrix} \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{d}^{T} m_{d} {\overset{\cdot\cdot}{q}}_{d} d (t) + \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{d}^{T} c_{d} {\overset{\cdot}{q}}_{d} d (t) + \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{d}^{T} k_{d} q_{d} d (t) \\ = \int_{0}^{t_{0}} {\overset{\cdot}{q}}_{d}^{T} (f_{d} - m_{d} R {\overset{\cdot\cdot}{q}}_{s}) d (t) \end{matrix}

(13)

Terms in equations (12) and (13), from left to right, are the kinetic energy, the damping energy, the strain energy, and the total input energy of the structure and the equipment (i.e. the energy transfers from the structure to the equipment), respectively.

The kinetic energy, damping energy, strain energy, and the total input energy of the structure in equation (12) are expressed as E_KS, E_CS, E_SS, and E_IS, respectively. The kinetic energy, damping energy, strain energy, and the total input energy of the equipment in equation (13) are expressed as E_KE, E_CE, E_SD, and E_ID, respectively.

The elastic strain energy E_SS in equation (12) can be redefined as equation (14), according to the principle of equivalence. Then the plastic strain energy of the structure can be calculated using equation (15), where k is the stiffness matrix of the structure when under an elastic state, and f(x) is the resilience of the structure. The elastic and plastic strain energy calculation principles for the equipment are same as those for the structure

E_{SS} = \frac{0.5 f^{2} (x)}{k}

(14)

E_{PS} = \int_{0}^{t_{0}} \frac{f (x) d (x) - 0.5 f^{2} (x)}{k}

(15)

Once the structure is in the plastic state, the strain energy of the structure includes the elastic strain energy (E_ES) and the plastic strain energy (E_PS)

E_{SS} = E_{ES} + E_{PS}

(16)

The ratios of kinetic energy, elastic strain energy, damping energy, and plastic strain energy to the total input energy of the structure can be defined as R_KS, R_SS, R_CS, and R_PS, respectively

R_{KS} = \frac{E_{KS}}{E_{IS}}, R_{SS} = \frac{E_{SS}}{E_{IS}}, R_{CS} = \frac{E_{CS}}{E_{IS}}, R_{PS} = \frac{E_{PS}}{E_{IS}}

(17)

Once the equipment is in the plastic state, the strain energy of the equipment includes elastic strain energy (E_SE), plastic strain energy (E_PE), and energy dissipated by connection in the equipment (E_HR)

E_{SD} = E_{SE} + E_{PE} + E_{HR}

(18)

The ratios of kinetic energy, elastic strain energy, damping energy, plastic strain energy, and energy dissipated by connection to the total input energy of the equipment can be defined as R_KE, R_SE, R_CE, R_PE, and R_HR, respectively

\begin{matrix} R_{KE} = \frac{E_{KE}}{E_{ID}}, R_{SE} = \frac{E_{SE}}{E_{ID}}, R_{CE} = \frac{E_{CE}}{E_{ID}}, R_{PE} = \frac{E_{PE}}{E_{ID}}, \\ R_{HR} = \frac{E_{HR}}{E_{ID}} \end{matrix}

(19)

ES system—shaking table test

To verify the reliability of the proposed RTDSST testing method, RTDSST tests and overall model tests were undertaken for the ES system under rigid and flexible connections.

Tests were carried out on the 3 m × 3 m shaking table at Beijing University of Technology. The tests used one-way loading and three earthquakes that were consistent with Site Class III conditions were selected (National Standard of the People’s Republic of China, 2010): the El Centro, PerSon, and Tianjin earthquakes. The acceleration time history curves of these quakes are shown in Figure 3 (solid black line). The time scale factor of the three earthquakes was 0.45. The amplitudes of earthquake acceleration were adjusted according to the requirements of Level VIII seismic intensity, and amplitudes of accelerations during minor earthquakes and moderate earthquakes were modulated to 0.7 m/s² and 2.0 m/s², respectively. As equipment may collapse and the structure may be greatly damaged during major earthquakes, considering that the frame structure will be used in the following related researches, this study did not consider major earthquake conditions.

Figure 3.

Comparative results of measured acceleration and reference acceleration of the shaking table: (a) El Centro,(b) PerSon, and (c) TianJin.

RTDSST tests

In the RTDSST tests, the equipment and the structure were regarded as the “experimental substructure” and the “numerical substructure,” respectively. By establishing data exchange routes and using appropriate control methods, real-time substructure test results can be obtained that meet the required predictive accuracy.

Test model

A 1/5 scale model of ES system that was used in the RTDSST tests is shown in Figure 4. The equipment model had a layer height of 0.25 m and a layer mass of 90 kg. Made of circular steel pipes, the equipment scale model had an elastic modulus and yield strength of 192.0 GPa and 421.4 MPa, respectively. The stiffness and damping of each layer of the equipment scale model were 744368 N/m and 174.5 N/(m/s) in accordance with design parameters. The structure model had a total of four layers with longitudinal and horizontal spans of 1.6 m. The height of the first layer was 0.68 m and the heights of other layers were 0.63 m. The mass of first, second, and third layers of the structure model were 1700 kg, and the mass of the fourth layer was 1540 kg. The structure used H-section steel beams and stanchions (100 mm × 45 mm × 6 mm × 8 mm). The elastic modulus and yield strength of the structure were 202.0 GPa and 339.6 MPa, respectively. The numerical model of the structure can be simplified into a 4-DOF shear model by referring to the deformation characteristics of the structure in the subsequent test results. The material of the steel frame was tested and the material properties of the beam and column were obtained. Sine sweep test was conducted for the frame structure fixed on the shaking table, to investigate its vibration characteristics and then system identification were executed to determine the parameters of the numerical substructure model both in the linear and nonlinear state. The systemic identification of the frame structure was conducted using the command “fmincon” in MATLAB. Thus ensuring the numerical substructure model can capture the actual responses of the experimental structure. The parameters of the numerical substructure are shown in Figure 4.

Figure 4.

Design of ES system test model.

To investigate the effects of connection types on the responses of equipment and structure, both rigid and flexible connections were involved (see Figure 4). The rigid connection support was made of Q345 steel, while the flexible connections mainly were made of rubber material. In the real-time substructure tests, the connecting support and the equipment were assembled first, using bolts into an equipment subsystem, and the subsystem then was fixed on the shaking table. Sensors were arranged on the connection support, the top of the equipment and the shaking table. The results of a sine sweep for the structure indicated that the fundamental frequency of the structure was 2.38 Hz, while the fundamental frequencies of the equipment with connections F, R1, and R2 were 4.58, 4.42, and 3.58 Hz, respectively, which indicates that the decrease in the stiffness of the connection results in decrease in the frequency of the equipment subsystem.

Test system and control method

The experimental control system consisted of a specialized engineering computer, compiler, and input/output (I/O) devices. SIMULINK simulation software was installed on the computer for calculating the response of the numerical substructure. The I/O devices linked the experimental substructure and input the test data to the numerical substructure, then output computational data to the shaking table. The dynamic characteristics of the shaking table have a crucial impact on the accuracy of the test results. In order to track accurately the performance of the shaking table, an approach that combined real-time numerical feedback control and physical feedback control was used for the controller design (see Figure 5(a)). The inverse dynamic compensation strategy, as reported elsewhere (Tagawa et al., 2010; Tang et al., 2016) was used for modeling certain dynamics of the shaking table in a real-time numerical feedback control loop.

Figure 5.

Principle and accuracy of the control method: (a) principle of control method and (b) comparison of the experimental and approximated transfer functions of shaking table.

The transfer function which was used in the numerical feedback control loop was identified using MATLAB based on the measured acceleration of the shaking table and the reference signal within the control computer (see Figure 3)

G = \frac{2.611 \times 10^{8}}{s^{4} + 913.6 s^{3} + 9.525 \times 10^{4} s^{2} + 1.245 \times 10^{7} s + 2.613 \times 10^{8}}

(20)

Figure 5(b) shows the comparative results of amplitudes and phases from the shaking table testing and the systematic identification. As can be observed, the results in these two cases coincide in terms of their overall trends, demonstrating that the transfer function fitted by the systematic identification is capable of describing the actual vibrational motion of the shaking table. Then the physical part is driven by the shaking table controller, which helps to provide more accurate tracking and control of the shaking table trajectory.

Test calculation module

The real-time seismic response outputs of the numerical substructure and the experimental substructure were encompassed in the design of the control module. The numerical substructure calculation module is shown in Figure 6(a). The nonlinear dynamic time history calculation of the structure used the combined Newmark-β incremental algorithm and the Newton-Raphson iterative solution method, while the convergent solution at each step was obtained according to the force convergence criterion. A nonlinear structure calculation program was inserted into an embedded MATLAB function module in SIMULINK and this could be compiled in C programming code to run on the computer.

Figure 6.

Module design of numerical substructure: (a) calculation module and (b) kinetic energy calculation module.

In Figure 6(a), ü_g denotes the seismic input; f_cp denotes the coupling term load of the structure; acc, vel, and dsp denote the acceleration, velocity, and displacement response of each floor of the structure, respectively; ü_out denotes the acceleration responses of the equipment; and ü₄ denotes the acceleration response at the top of the structure, which was selected and summed with the seismic excitation ü_g to obtain the excitation of the equipment (i.e. acceleration associated terms on the right side of equation (10)), then the excitation was input to the equipment via the shaking table. The acceleration responses collected by the sensors on the equipment were used to calculate f_cp for the structure, which then was used together with the external load f_s for the calculation of the force on the structure at the next moment.

During the experiment, the seismic energy responses of both the structure and the equipment can be output in real time. Both the damping values of the equipment and the structure were approximately assigned as the their initial damping values. Taking the kinetic energy calculation module of the structure as an example, Figure 6(b) shows a schematic of the kinetic energy calculation module. According to the calculated acceleration, velocity, and displacement of the structure, the kinetic energy of the structure can be obtained using the Matrix Multiplier and the Discrete-Time Integrator module, according to equation (12). Other types of energy calculations for the structure are similar.

Unlike the structure energy calculation, the acceleration, velocity and displacement responses for the equipment energy calculation were measured by the sensors. The equipment energy calculation model uses a 2-DOF model. ${\overset{\cdot\cdot}{q}}_{d}, {\overset{\cdot}{q}}_{d}, and q_{d}$ in equation (13) are the acceleration, velocity, and displacement responses of each layer of the equipment, relative to the top of the structure, respectively. The acceleration responses of the equipment were measured by 941B sensors and the collected data were filtered before being used. Integration of these data leads to the corresponding velocity and displacement responses. The energy terms can be obtained by accumulations of the energy responses at each time interval of the seismic input. The energy consumption of flexible connection is calculated using the area enclosed by the restoring force-displacement curve at the connection support (Reggio and De Angelis, 2013).

Overall model—shaking table test

To verify the reliability of the RTDSST tests, the overall model test of the ES system was executed. Accordingly, the four-layer frame structure was processed into a test model (Figure 4). Acceleration sensors were arranged at the center of each floor. A_i1 and A_i2 denote the sensor locations parallel and perpendicular to the loading direction of the ith floor, respectively. During the test, the subsystem composed of the equipment and the connecting support was connected to the top of the structure using bolts.

Results

The reliability of the RTDSST testing method was verified by collation with the acceleration response of the equipment top, using two testing methods as the index. As the connection type affects the internal energy responses of an ES system, the effects of the connection type between structure and equipment on their energy responses were studied.

Experimental verification

The acceleration response of the equipment top under rigid connection at an input peak ground acceleration (PGA) of 0.7 m/s² was used as an indicator. Figure 7 shows the comparison of the real-time substructure test results and the overall model test results. As can be observed, the equipment top acceleration time history curves in these two cases coincide in terms of their overall trends. Compared with that during the overall model test, the peak acceleration of the equipment top in the substructure tests was 8.6%, 9.2%, and 6.2% smaller under the ground motion of the El Centro, PerSon, and TianJin quakes, respectively. The underestimates in the real-time substructure tests were attributable mainly to differences between the numerically modeled structure model and the structure test model and the accuracy of the test control method. Overall, the proposed RTDSST testing method based on the branch mode substructure approach met the accuracy requirements and was reliable and effective.

Figure 7.

Comparisons of the acceleration response of the equipment top for the different testing methods.

Analysis of test results

As the accuracy of the RTDSST tests was confirmed by comparing the test results with those from overall model shaking table tests, the data used for the following analysis were those collected in the RTDSST tests. Then the effects of the connection types on the seismic energy responses of the equipment and the structure were analyzed from the perspectives of their energy responses. Energy response values at the end of each seismic excitation were used as the values taken for the calculation.

Structure and equipment energy responses for different connection types

Tables 1 to 6 show the energy responses of the structure and the equipment using different connections, where Acc denotes the peak acceleration of the original seismic input of the shaking table. EL, PS, and TJ denote the ground motion of the El Centro, PerSon, and TianJin earthquakes, respectively. According to the energy response values of the structure and the equipment, the final value for the total input energy was the sum of the other energy response final values, which confirmed the validity of the energy response formula. Errors in the calculated data are inevitable due to the presence of “noise” even after data filtration.

Table 1.

Energy responses of the structure using rigid connection F.

Type	Acc (m/s²)	E_IS (J)	E_KS (J)	E_SS (J)	E_CS (J)	E_PS (J)
EL	0.7	40.263	0.001	0.598	39.086	0.000
EL	2.0	329.601	0.324	1.059	329.483	0.000
PS	0.7	8.319	0.084	0.070	8.174	0.000
PS	2.0	98.396	0.227	0.069	97.776	0.000
TJ	0.7	108.030	23.382	0.569	85.153	0.000
TJ	2.0	915.950	22.979	1.553	617.493	286.370

Table 2.

Energy responses of structure using flexible connection R1.

Type	Acc (m/s²)	E_IS (J)	E_KS (J)	E_SS (J)	E_CS (J)	E_PS (J)
EL	0.7	37.002	0.027	0.278	35.910	0.000
EL	2.0	316.562	0.131	0.904	309.787	0.000
PS	0.7	8.047	0.045	0.074	7.873	0.000
PS	2.0	96.279	0.115	0.104	95.403	0.000
TJ	0.7	104.060	22.214	0.004	80.897	0.000
TJ	2.0	892.870	21.762	1.689	596.080	278.013

Table 3.

Energy responses of structure using flexible connection R2.

Type	Acc (m/s²)	E_IS (J)	E_KS (J)	E_SS (J)	E_CS (J)	E_PS (J)
EL	0.7	27.486	0.302	0.979	25.633	0.000
EL	2.0	254.267	0.005	8.060	241.808	0.000
PS	0.7	8.293	0.001	0.422	7.714	0.000
PS	2.0	96.667	0.043	1.199	93.825	0.000
TJ	0.7	86.424	16.801	2.904	65.414	0.000
TJ	2.0	742.858	29.418	20.592	482.857	194.629

Table 4.

Energy responses of equipment using rigid connection F.

Type	Acc (m/s²)	E_ID (J)	E_KE (J)	E_SE (J)	E_CE (J)	E_PE (J)
EL	0.7	0.891	0.001	0.005	0.829	0.000
EL	2.0	6.014	0.008	0.004	5.160	0.789
PS	0.7	1.143	0.004	0.024	1.111	0.000
PS	2.0	8.203	0.001	0.113	8.151	0.204
TJ	0.7	0.870	0.007	0.039	0.836	0.000
TJ	2.0	6.043	0.080	0.229	3.926	2.000

Table 5.

Energy responses of equipment using flexible connection R1.

Type	Acc (m/s²)	E_ID (J)	E_KE (J)	E_SE (J)	E_CE (J)	E_PE (J)	E_HR (J)
EL	0.7	1.232	0.066	0.003	0.605	0.000	0.556
EL	2.0	7.831	0.837	0.008	3.686	0.090	3.248
PS	0.7	0.941	0.068	0.008	0.476	0.000	0.371
PS	2.0	7.183	0.931	0.022	3.418	0.000	2.672
TJ	0.7	1.867	0.376	0.002	0.936	0.000	0.551
TJ	2.0	12.377	0.781	0.037	3.663	5.950	2.125

Table 6.

Energy responses of equipment using flexible connection R2.

Type	Acc (m/s²)	E_ID (J)	E_KE (J)	E_SE (J)	E_CE (J)	E_PE (J)	E_HR (J)
EL	0.7	9.455	1.260	0.009	0.502	0.000	7.682
EL	2.0	58.528	8.097	0.010	2.894	0.000	46.682
PS	0.7	6.567	1.199	0.006	0.361	0.000	4.963
PS	2.0	13.211	2.448	0.002	1.086	0.000	9.422
TJ	0.7	17.011	2.659	0.298	1.058	0.000	12.814
TJ	2.0	55.101	1.891	0.523	2.513	32.868	15.880

The energy distribution ratios of the structure were calculated based on its energy responses. As can be observed, the total input energy of the structure using different connections mainly was dissipated by its damping energy. When the structure is in an elastic state, the proportion of damping energy R_CS increases with the amplitude of the seismic input. Also, as can be demonstrated from the energy responses of the structure, the deformation state has a significant influence on the energy distribution within the structure; an increase in the plastic energy consumption leads to a decrease in damping energy consumption of the structure. At an input PGA of 2.0 m/s² under the Tianjin ground motion, the presence of plastic energy leads to a decrease in damping energy, compared with those values under the input PGA of 0.7 m/s². The damping energy consumption to total input energy R_CE decreased by 11.41%, 10.98%, and 10.69%, respectively, under the connections of F, R1, and R2, demonstrating the energy balance within the structure. By comparing the energy responses of the structure under different connections, it can be concluded that changes in the connection types can lead to changes in the structure energy responses, this phenomenon is significant when using connection R2. For instance, at an input PGA of 2.0 m/s² under the El Centro and Tianjin ground motions, the input energy of the structure E_IS decreased by 31.73% and 20.00%, respectively, compared with that of the connection F. While the change of connection has a relatively small effect on the energy distribution within the structure. For instance, the ratios of the plastic energy consumption of the structure R_SE decreased by 0.13% and 5.06%, respectively, when using the connection of R1 and R2, compared with the connection F. These phenomena demonstrate that as the stiffness of the connection reduces, the fundamental frequency of the equipment subsystem draw closer to that of the structure, which results in more energy transferring from the structure to the equipment, thus reducing the energy responses of the structure.

Tables 4 to 6 show the energy responses of the equipment using different connections.

It can be concluded that the energy responses of the equipment using different connections also follow the law of energy balance. The connection type has a key effect on the energy responses of the equipment. When the stiffness of the connection reduces, the changing of connection leads to more significant effects on the energy responses of the equipment. For instance, at an input PGA of 0.7 m/s² under the ground motions of the El Centro quake, the changes in the input energy of the equipment was not so significant when using R1, compared with that for the connection F. While the differences in the input energy when using connection R2 were significant, the input energy increased by 98.99% and 96.98%. Under the ground motion of the TianJin quake, the changes in the input energy and plastic energy dissipation were also significant. For instance, the ratios of the plastic energy consumption of the equipment R_PE increased by 14.98% and 26.55%, respectively, when using the connection of the R1 and R2, compared with that for the connection F. These phenomena indicate that when the fundamental frequency of the equipment subsystem is close to that of the structure, the energy transmission within the ES system is more prevalent; more energy transfers from the structure to the equipment leads to adverse effects on the responses of the equipment.

The above analysis demonstrates that changes in the connection types can lead to significant changes in the energy responses of the coupled ES system, especially with respect to the equipment. When the first-order frequency of the equipment is close to that of the structure, the energy transmission within the ES system is prevalent, which leads to adverse effects on the responses of the equipment. An effective connection design should take into account the operational performance objective of the equipment. For the equipment with operational functions (structural, architectural, electrical, mechanical, etc.), the use of connection should keep the frequency of the equipment subsystem away from that of the structure, while for the equipment designed for shock absorption of the structure, a frequency tuned or nearly tuned to that of the structure can ensure its effectiveness.

Conclusion

In this study, the equation of motion of an ES system was deduced, based on the branch mode substructure method, and was applied to RTDSST tests. The RTDSST tests and overall model tests of an ES system were executed to verify the reliability of the proposed RTDSST testing method. Considering the effects of different connection types on the interaction, the relationship of energy distribution in the structure and the equipment with different types of connection was analyzed and the following conclusions were drawn:

The equation of motion for the ES system, which was derived using the branch mode substructure method, can reduce the scale of system calculations. A formula for substructure tests that is suitable for the ES system has been proposed. The reliability of the method was verified using a four-story steel frame/equipment model.

An energy calculation method for the energy responses of each part in the overall ES system has been proposed, and the reliability of the method was verified by the test results. Combined with the RTDSST tests, a method of real-time energy calculation also has been proposed to evaluate the energy variations of the structure and equipment during the process of testing, thus facilitating a study of the effects of different connection types on the energy responses of the equipment and structure.

Changes in the connection types can lead to changes in the energy responses of the ES system, especially with respect to the equipment. When the first-order frequency of the equipment subsystem is close to that of the structure, the energy transmission within the ES system is prevalent, which leads to adverse effects on the responses of the equipment. The choice of the connection for the ES coupled system should take into account the operational performance objective of the equipment.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Science Foundation Project, China (research project no. 51478312).

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