# Effect of Control Measures on Wheel/Rail Noise When the Vehicle Curves
## 2 The Time Domain Model for Predicting Wheel/Rail Noise under Lateral Excitation
### The Transient Finite and Boundary Element Model for Wheel/Rail Vibration and Sound Radiation  
Three-dimensional (3D) finite element (FE) models of a standard metro wheel and a conventional slab track were developed to obtain the vibrations (Figure 6). In the model, the normal forces and lateral creep forces obtained by the vehicle/track interaction dynamic model in Section 2.1 are applied to the tread of wheelset and rail head. Figure 6 also shows the FE models of resilient wheel and embedded track to analyze their effect on wheel/rail noise attenuation under lateral excitation. The FE model was developed in commercial software ANSYS. Once the vibrations of the wheel and track are determined using by 3D FE model, sound radiations from the wheel and track can be evaluated using an appropriate vibro-acoustic prediction method. Figure 5: Dynamic model of resilient wheel. Figure 6: Dynamic model of resilient wheel and embedded rail. There are three radiators: the wheels, the rails and the slabs. Equation (20) shows the 3D acoustic integral equation [36] for a sound field of frequency \(\omega\). \[\overline{p}(x,y,z)=-\int\limits_{S}\biggl{(}\overline{p}(x^{\prime},y^{\prime},z^{\prime})\frac{\partial G(R)}{\partial n}+i\rho\omega G(R)v_{n}(x^{\prime}, y^{\prime},z^{\prime})\biggr{)}dS \tag{20}\]  
where \(\overline{p}(x,y,z)\) denotes sound pressure amplitude in the acoustic domain, \(v_{n}(x^{\prime},y^{\prime},z^{\prime})\) denotes air particle velocity amplitude in the normal direction of the domain boundary (the unit normal vector of the boundary, denoted by \(n\), is pointing out of the domain, and finally,  
\[G(R)=\frac{e^{-ikR}}{4\pi R},R=\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{ \prime}\right)^{2}+\left(z-z^{\prime}\right)^{2}} \tag{21}\]  
is the Green's function with the source at \((x,y,z)\) and observer at \((x^{\prime},y^{\prime},z^{\prime})\), and \(k\) is the wavenumber. Equation (20) is solved by 3D boundary element method (BEM). Discrete boundary surface into several sub boundary elements \(\Omega_{\text{ae}}\). In this way, the distribution of sound pressure \(p\) and normal velocity \(v_{n}\) on the boundary surface is approximated to the sound pressure \(\hat{p}\) and normal velocity \(\hat{v}_{n}\) in a series of sub boundary elements. Then, the sound pressure \(p\) and normal velocity \(v_{n}\) can be written as:  
\[p(x^{\prime},y^{\prime},z^{\prime})\approx\hat{p}(x^{\prime},y^{\prime},z^{ \prime})=\sum\limits_{i=1}^{n_{\text{e}}}N_{i}^{\text{e}}(x^{\prime},y^{\prime },z^{\prime})\cdot a_{pi},\quad(x^{\prime},y^{\prime},z^{\prime})\in\Omega_{ \text{ae}} \tag{22}\]  
\[v_{n}(x^{\prime},y^{\prime},z^{\prime})\approx\hat{v}_{n}(x^{\prime},y^{\prime },z^{\prime})=\sum\limits_{i=1}^{n_{\text{e}}}N_{i}^{\text{e}}(x^{\prime},y^{ \prime},z^{\prime})\cdot a_{pi},\quad(x^{\prime},y^{\prime},z^{\prime})\in \Omega_{\text{ae}} \tag{23}\]  
The \(N_{i}^{\text{e}}\) is the shape function of node \(i\). Defined that \(N_{i}^{\text{e}}\) is one at the node \(i\), and zero at other nodes. \(a_{pi}\) and \(a_{pi}\) are the sound pressure and normal velocity of the node \(i\). Defined that [\(N\)] is the (\(1\times n_{\text{a}}\)) matrix of shape function \(N_{i}^{\text{e}}\). \(\{\hat{p}_{i}\}\) is the (\(n_{\text{a}}\times 1\)) matrix of sound pressure \(a_{pi}\). \(\{\hat{v}_{ini}\}\) is the (\(n_{\text{a}}\times 1\)) matrix of normal velocity \(a_{pi}\). Here \(n_{\text{a}}\) is the total number of nodes. \(\hat{p}(x^{\prime},y^{\prime},z^{\prime})\) and \(\hat{v}_{\text{n}}(x^{\prime},y^{\prime},z^{\prime})\) can be further written as:  
\[\hat{p}(x^{\prime},y^{\prime},z^{\prime})=[N]\cdot\{\hat{p}_{i}\},\quad(x^{ \prime},y^{\prime},z^{\prime})\in\Omega_{\text{a}} \tag{24}\]  
\[\hat{v}_{n}(x^{\prime},y^{\prime},z^{\prime})=[N]\cdot\{\hat{v}_{ini}\},\quad(x ^{\prime},y^{\prime},z^{\prime})\in\Omega_{\text{a}} \tag{25}\]  
To use (24) and (25) solving Equation (20), the relationship of \(\hat{p}_{i}\) and \(\hat{v}_{ni}\) needs to be determined.