# Pedestrian Comfort Analysis and Parameterized Optimization Design for Raised Floors
## 3 Results  
### Model Analysis  
3.1.1. Theoretical Modal Analysis  
The finite element method, a prevalent numerical technique for solving differential equations of continuous fields, is a key design tool in architecture. Modal analysis, frequently employed using the finite element method, is instrumental in assessing the dynamic properties of building structures. The outcomes of modal analysis offer accurate predictions of dynamic characteristics and vibration response in structures, providing valuable theoretical insights for equipment operation.  
Based on the theoretical modal analysis, Equation (3) represents the characteristic equation of the natural frequency and mode shape for the undamped structure.  
\[\left(K-\omega^{2}M\right)\varphi=0 \tag{3}\]  
where \(\omega\) is the natural frequency of the system's free vibration; \(K\) is the stiffness matrix; \(M\) is the mass matrix; and \(\varphi\) is the eigenvector, which represents the normalized model of the structure.  
\(K-\omega^{2}M\) remains as a matrix, known as the characteristic matrix, denoted as \([A]\). The nontrivial solution of the equation occurs when the determinant of its coefficients is equal to zero, that is, \([A]\) = 0. Expanding \([A]\) gives rise to an algebraic equation of nth order in terms of, represented as Equation (4).  
\[\omega^{2n}+a_{1}\omega^{2(n-1)}+\cdots+a_{n-1}\omega^{2}+a_{n}=0 \tag{4}\]  
Solving for the eigenvalues of the system yields \(n\) natural frequencies, denoted as \(\omega_{1}^{2}\), where \(\omega_{1}\), \(\omega_{2}\), \(\ldots\), \(\omega_{n}\) represents the natural frequencies of the structural system. Consequently, a series of proportional vectors \(\varphi_{i}\) can be obtained, each corresponding to a specific natural frequency \(\omega_{i}\), describing the vibration modes at that frequency. These vectors are referred to as the mode shapes. These vectors are referred to as the mode shapes. Modal analysis focuses on investigating the natural vibration frequencies and mode shapes of structural systems.  
In this study, Abaqus 2020 was utilized to perform an in-depth analysis of four distinct types of raised floors, with a particular emphasis on their dynamic properties. Through a series of meticulous operations including mesh division, material assignment, boundary condition setting, and contact relationship establishment, the researchers were able to systematically examine the natural frequencies and mode shape variations of the raised floors. In order to simplify the calculations, the interaction relationships between components were set as bonded contacts, while the boundary conditions were established as hinged connections to mimic real-world scenarios where raised floors are assembled with wall systems. Utilizing the mapping meshing method in Abaqus, the mesh of the raised floor models was partitioned with a hexahedral mesh type. The results of the analysis, illustrated in Figure 4, reveal the specific calculation conditions and highlight the areas of maximum deformation within the structures. Notably, it was observed that the presence of beams at the bottom of the raised floor divides the structure into two vulnerable blocks, with the center of each block being the most prone to deformation. The magnitude of deformation is related to the material of the floor. The center of the raised floor supported by twelve legs is the most susceptible area to deformation.