*4.4. Particle-Tuned Mass Damper (PTMD)*

The particle-tuned mass damper (PTMD) is the combination of TMD and PD. The simplified analytical model proposed by Lu et al. [3] is also adopted. The PTMD can be idealized as a 2DOF system, which includes an SDOF TMD connected to the primary structure and an SDOF PD attached to the TMD. For the PTMD, the mass of the container cannot be neglected because it constitutes the TMD. Therefore, the total auxiliary mass of the PTMD is divided into two parts: PD and TMD. Lu et al. [3] investigated the influence of the ratio of the particle mass (*MPD*) to the total auxiliary mass (*MPTMD*) on the structural control effects of PTMD and found that the vibration attenuation of the PTMD can be improved to a certain extent by increasing the mass proportion of the PD. Therefore, an 80% mass ratio of *MPD* to *MPTMD* is used. Four PTMDs with varying auxiliary mass ratios from 1% to 4% are selected. The parameters of the TMD and PD are determined following the procedures presented in Sections 4.1 and 4.3, respectively. The corresponding parameters of PTMDs can be found in Table 3.

### **5. Modeling of Substructure Shake Table Testing of Frame Structure–Damper System**

### *5.1. Procedure of Substructure Shake Table Testing of Frame Structure–Damper System*

To carry out SSTT of the frame structure–damper system, the primary structure (frame) and the secondary structure (damper) are assigned as the analytical and experimental substructures, respectively. The analytical substructure is numerically simulated, and the damper is mounted on and excited by the shake table [15,16]. A schematic diagram of SSTT of the frame structure–damper system is illustrated in Figure 8.

**Figure 8.** Schematic diagram of shake table testing of the frame structure–damper system.

> Figure 9 shows a flowchart of SSTT of the frame structure–damper system using GCR algorithms and the finite element method.

**Figure 9.** Flowchart of substructure shake table testing of the frame structure–damper system using GCR algorithms and the finite element method.

In Figures 8 and 9, it should be noted that the excitation signal of the shake table is the total displacement at the interface instead of the relative displacement. Therefore, two additional calculation steps are required:

$$\mathbf{x}\_{i+1}^{l} = \mathbf{T}\_1 \mathbf{X}\_{i+1} \tag{31}$$

$$
\mu\_{i+1}^{I} = \mathbf{x}\_{i+1}^{I} + \mathbf{x}\_{\mathbf{\tilde{g}},i+1} \tag{32}
$$

where *<sup>x</sup>Ii*+<sup>1</sup> and *<sup>u</sup>Ii*+<sup>1</sup> are the relative and total displacements at the interface, respectively; *xg*,*i*+<sup>1</sup> is the ground displacement; *T*1 is the matrix transforming the DOFs of the structure to the interface DOF.

In addition, the EOM of the primary structure should be modified due to introduction of the damper force (interface force):

$$\mathbf{M}\ddot{\mathbf{X}}\_{i+1} + \mathbf{C}\dot{\mathbf{X}}\_{i+1} + \mathbf{R}\_{i+1} = \mathbf{F}\_{i+1} + T\_2 f\_{D, i+1} \tag{33}$$

where *fD*,*i*+<sup>1</sup> is the damper force; *T*2 is the matrix converting the interface DOF to the DOFs of the structure. This study aimed to conduct a series of virtual SSTTs, so the damper forces of the four types of dampers are numerically simulated instead of experimentally measured in a real SSTT. All numerical simulations were performed using MATLAB software and the Simulink toolbox.
