# Vibro-Acoustic Numerical Simulation for Analyzing Floor Noise of a Multi-Unit Residential Structure
## 2 Description of the Methodology  
### Vibro-Acoustic Analysis  
Typically, floor noise, in multi-unit residential structures, refers to noise transferred to adjacent households. When floor impact sources such as children's running, adults' walking, etc., are applied on the upper floor, they generate a vibration at the upper floor slabs that is transferred to the lower floors through the walls and the ceiling slabs. When slabs or walls vibrate the air particles adjacent to them are also vibrated, and the vibration of the air particles causes sound. Accordingly, in order to analyze noise caused by floor impact sound, the speed of the air particle vibrations occurring due to vibration transfer is the boundary condition of the noise analysis. The precise vibration velocity of structures such as slabs or walls can be computed through the analysis of structural vibration, and the precise analysis of noise can be conducted based on the computed vibration velocity [3; 13; 14]. Generally, Equation (1) is applied to calculate the structural vibration of the test structure, which is caused by an external impact source (\(F\)). \[[m]\ddot{[x]}+[c]\dot{[x]}+[k][x]=\{F\} \tag{1}\]  
where, \(x\), \(\dot{x}\), \(\ddot{x}\) are the displacement, velocity, and acceleration, respectively. \(m\), \(c\), \(k\) are the mass, damping value, and stiffness of the structure, respectively. \(F\) is the external load applied to the structure. The structural vibration of a wall or slab in a multi-unit residential building is caused by an external impact force, i.e., an impact source in this study. A bang machine is adopted as the standard impact source, specified by KS F 2810 [19], ISO 10140 [20], ISO 16283 [21], and JIS A 1418 [22], for acoustic vibration evaluation. An acoustic wave equation can be derived by utilizing the equation of the state, which indicates the relationship between density and pressure. The equation of motion showsthe relationship between pressure and displacement, and the equation of continuity indicates the displacement-density relation. Equation (2) below is the acoustic wave equation [13; 23]. \[\
abla^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0 \tag{2}\]  
\[\mathrm{c}^{2}=\frac{\kappa}{\rho} \tag{3}\]  
where, \(p\) is the sound pressure, \(c\) is the velocity of air, \(\rho\) is the fluid density, \(\kappa\) is the fluid bulk modulus, and \(t\) is the time. Figure 1 shows graphically the vibro-acoustic analysis process mentioned above. The structural vibration for Equation (1) was analyzed using OptiStruct of Altair Engineering, Inc. [24], which is a commercial finite element analysis program using the numerical model of the test structure. Then, modal analysis was conducted to determine the natural mode shapes and frequencies of the structure, and to calculate the vibrational responses of the structures under the impact force source, i.e., the bang machine in this study. During the analysis, proper boundary conditions (e.g., fixed conditions for the bottom regions of the test structure) have been applied. Then, the acoustic analysis of Equation (2) was performed using Virtual LAB [25] of SIEMENS. To do this, an acoustic model for a test floor in the structure has been constructed. It should be clarified that the meaning of the test floor is defined as the floor on which the impact force source is applied. A 2D plane grid mesh for a microphone array has been employed to measure the sound pressure level at the target area. A 3D tetra mesh has been used to create the acoustic mesh model. A detailed explanation of the structural analysis and acoustic analysis for interested readers can be found in [3; 14; 26; 27; 28; 29; 30]. It should be noted that a wavelength is important to solve the acoustic wave equation and determine the mesh size of the acoustic regions. The wavelength equation to calculate the proper wavelength for the acoustic mesh of the test structure [31] can be found as Equation (4):  
\[\lambda=\frac{c}{f} \tag{4}\]  
Figure 1: Schematic overview of vibro-acoustic analysis procedure.