# Virtual Scene Construction for Seismic Damage of Building Ceilings and Furniture
## 3 Technical Method
### Simulation of the Movement of Damaged Nonstructural Components
#### 3.3.1 Suspended Ceilings
According to the rules for inelastic collisions, the vertical velocity of the tile after collision (i.e., \(v_{t2}\)) can be calculated using Equation (5)  
\[v_{t2}=\frac{C_{R}m_{f}\Big{(}v_{f1}-v_{t1}\Big{)}+m_{t}v_{t1}+m_{f}v_{f1}}{m_{ t}+m_{f}} \tag{5}\]  
where \(v_{t1}\) and \(v_{t2}\) are the vertical velocities of the tile before and after collision, respectively; \(v_{f1}\) is the velocity of the floor before collision, which is equal to zero (i.e., \(v_{f1}=0\)); \(m_{t}\) and \(m_{f}\) are the masses of the tile and floor, respectively; and \(C_{R}\) is the coefficient of restitution. Because \(m_{f}\) is much larger than \(m_{t}\), Equation (5) can be simplified into Equation (6) as  
\[v_{t2}=-C_{R}v_{t1} \tag{6}\]  
In the third stage, the tile will be bounced. Its path on the \(x\)- and \(y\)-direction can be calculated by Equation (1) and Equation (7), respectively. Figure 2:0  
In the last stage, the tile will slide along the floor until the velocity decreases to zero. The path in the \(x\)-direction can be calculated using Equation (8)  
Figure 2:1  
where \(\mu\) is the kinetic friction coefficient between the tile and floor. 3. Validation and coefficient determination  
For the above physical model of falling tiles, the four-stage movement needs to be validated and the key coefficients (i.e., \(C_{R}\), and \(\mu\)) also need to be determined. To this end, an experiment to investigate the falling process of a ceiling tile is performed. A typical aluminum ceiling tile with a size of \(300\times 300\times 0.8\) mm is used as the falling tile. In this experiment, the tile is projected horizontally at three heights (i.e., 165 cm, 180 cm, and 195 cm), and then falls to a concrete floor. The entire tile falling process is recorded by video cameras, and thus the movement path of the tile can be obtained by measuring the images in the video. In this experiment, the tile fell three times with different initial horizontal velocities for each height, and thus a total of nine movement paths of the tile were obtained. A typical movement process of the tile is shown in Figure 4, which is in agreement with the four stages in the proposed physical model, and therefore demonstrates that the proposed physical model is reasonable. Based on the recorded movement paths and Equation (4), the drag coefficient, \(c\), can be calculated; it is in the range of 0.92-1.05. In this study, the mean value of \(c\) (i.e., 0.97) is used. Through measuring the images in the video of the experiment, the velocities before and after collision are obtained, which allows the coefficient of restitution, \(C_{R}\), to be calculated. The range and mean value of \(C_{R}\) are 0.23-0.37 and 0.30, respectively. The kinetic friction coefficient, \(\mu\), between the tile and concrete floor is also measured, and the value is 0.37 \(\pm\) 0.01. These key coefficients will be used to simulate a falling tile in the physics engine. 4. Implementation using a physics engine  
Although the proposed physical model can describe the dynamic nature of the movements of the damaged ceiling tiles, an implementation approach based on the physical model is still needful for simulating such movements. As mentioned previously, PhysX is employed as the default physical engine in Unity. Furthermore, the proposed four-stage physical model has been embedded in PhysX. Hence, PhysX is a good simulation platform for the damaged ceiling tiles. Four steps need to be done for implementing the movement simulations of the damaged ceiling tiles, as follows:  
(1) Create rigid-body models  
Using PhysX, each ceiling tile is defined as a rigid body. In this step, the geometrical information of ceiling tiles in Unity is imported to PhysX to create the corresponding rigid bodies. (2) Determine key coefficients  
The key coefficients of rigid-body models have significant influences on the accuracy of the movement simulation of ceiling tiles.