**1. Introduction**

Substructure shake table testing (SSTT) is one of the most advanced experimental techniques in structural and earthquake engineering [1]. It combines the advantages of the real-time dynamic loading of a shake table and the substructure technique from hybrid simulation (HS) and real-time hybrid simulation (RTHS). The best use of SSTT is for investigating the vibration control effects of dampers, such as the classical TMDs and TLDs, and emerging dampers, such as particle dampers (PDs) [2] and particle-tuned mass dampers (PTMDs) [3]. These dampers can be regarded as secondary structures with respect to the primary structure. To conduct SSTT, the damper and the primary structure are taken as the experimental and analytical substructures, which are experimentally tested on the shake table and numerically simulated on a computer, respectively.

The shake table tests are generally used for obtaining the dynamic responses and dynamic characteristics of structures [4,5]. The conventional experimental method of investigating the control effects of dampers is to carry out shake table tests for complete structure–damper systems. For instance, Kang et al. [6] conducted a series of 1:30 scaled model shake table tests and numerical simulations for a coal-fired power plant equipped with a large mass ratio multiple-tuned mass damper (LMTMD), and found that the LMTMD is effective and robust in reducing structural dynamic responses. Wang et al. [7] evaluated

**Citation:** Fu, B.; Jiang, H.; Chen, J. Substructure Shake Table Testing of Frame Structure–Damper System Using Model-Based Integration Algorithms and Finite Element Method: Numerical Study. *Symmetry* **2021**, *13*, 1739. https://doi.org/ 10.3390/sym13091739

Academic Editor: Yang Yang

Received: 4 August 2021 Accepted: 13 September 2021 Published: 18 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the performance of a pendulum pounding-tuned mass damper (PPTMD) by carrying out a series of shake table tests. They reported that the inherent damping of the primary structure decreases the control efficiency of the PPTMD. Zhao et al. [8] conducted shake table tests on a 1:8 scaled transmission tower equipped with and without TMDs, and found that the TMD's control performance is related to earthquake type and excitation intensity. Vafaei et al. [9] proposed a modified-tuned liquid damper (MTLD) to attenuate multiple mode vibration. The effectiveness of the MTLD was demonstrated by several shake table tests on a scaled three-story structure with and without an MTLD. Lu et al. [3] explored the damping performance of a PTMD by comparing the shake table results of a scaled five-story steel frame with and without a PTMD. Based on the shake table tests, they conducted comprehensive parametric analyses on the reduction effects of PTMD, including the auxiliary mass ratio, gap clearance, and the mass ratio of particles to the total auxiliary mass. Shen et al. [10] investigated the influence of a double-layer-tuned particle damper (DTPD) on the seismic performance of super high-rise structures by conducting a series of 1:20 scaled model shake table tests on high-rise structures with and without DTPDs. They concluded that the effectiveness of the DTPD is closely related to the ground motion.

Compared with the conventional shake table tests for a complete structural system, one of the notable advantages of SSTTs is that any size effect of the specimen can be reduced. SSTT has also been applied to structure–damper systems. For instance, numerous researchers [11–14] investigated the performance of TLDs in controlling the seismic response of structures using a SSTT. Fu et al. [15,16] recently conducted the first SSTT of a single degree-of-freedom (SDOF)-PD system and compared the vibration control effects of TLDs and PDs based on the experimental results of a series of SSTTs. It should be noted, however, that the analytical substructures in most existing studies were optimized to be SDOF structures or shear-type structures with few DOFs. This is because these structures do not require considerable computational time to solve their equations of motion (EOMs). However, oversimplification of the analytical substructure may hinder the application of SSTT in real engineering practice. Therefore, it is essential that more refined models of the analytical substructure are applied and examined for SSTT.

It is well known that the finite element (FE) method is an accurate and reliable approach for simulating structures [17–22]. However, it requires significantly longer computational time compared with the simplified SDOF or shear-type model. Therefore, if the FE method is applied to simulate the analytical substructure in SSTT, an integration algorithm with high computational efficiency must be used. In the past two decades, a new class of integration algorithms, called model-based integration algorithms [23], have been developed for the application of HS and RTHS. Model-based integration algorithms are computationally competitive because they combine the advantages of explicit displacement formulation and unconditional stability, which is not possible for traditional integration algorithms such as Newmark algorithms. Model-based integration algorithms have been successfully applied in actuator-type RTHS with the FE model of the analytical substructure [24–26]. However, there are few studies on SSTT with the FE method using model-based integration algorithms. In this study, we numerically investigate the seismic response reduction effects of four types of dampers, i.e., TMD, TLD, PD, and PTMD, on a four-story steel frame by conducting a series of virtual SSTTs. The steel frame is simulated by stiffness-based beam-column elements with fiber sections and bar elements. The EOM of the frame structure is solved using a family of model-based integration algorithms.

The formulation and basic features of model-based integration algorithms are summarized in Section 2. Section 3 explains the FE modeling of the four-story steel frame. The analytical models of the four types of dampers are given in Section 4. Section 5 provides the procedure of SSTT and numerical results of a series of SSTTs. The influences of the auxiliary mass ratio, integration parameters, time step, and time delay on the SSTT are also investigated in Section 5. Some important conclusions are drawn in Section 6.
