## 3 Frequency Domain Methods of Calculation  
### Simple Vehicle and Multi-Beam-on-Soil Track  
The dynamic stiffness \(K_{V}\) of the vehicle can be approximated by the mass \(m_{W}\) of the wheelset and the circular frequency \(\omega\) as:  
\[K_{V}(\omega)=-m_{W}\omega^{2} \tag{1}\]  
The minor effects of more detailed vehicle models have been discussed in [31].  
Many track systems have been calculated by the combined finite-element boundary-element method [32] or by a faster multi-beam-on-soil track model in the frequency-wavenumber domain (Figure 2) as follows. The track model consists of \(n\) beams that represent the rails and track slabs, and which are described by the bending stiffnesses \(EI_{j}\) and the masses per length \(m_{j}\) in the diagonal matrices \(\mathbf{EI}\) and \(\mathbf{M}\), and by a stiffness and damping matrix \(\mathbf{K}\) and \(\mathbf{C}\) representing rail pads, sleepers, the ballast and any isolation element. The multi-beam system fulfils the set of differential equations for the displacements \(\mathbf{u}\) and the load (per length) \(\mathbf{F^{\prime}}\mathbf{\tau}\):  
\[\mathbf{EI}\frac{\partial^{4}\mathbf{u}}{\partial y^{4}}+\mathbf{M}\frac{ \partial^{2}\mathbf{u}}{\partial t^{2}}+\mathbf{C}\frac{\partial\mathbf{u}}{ \partial t}+\mathbf{K}\mathbf{u}=\mathbf{F^{\prime}}_{\mathrm{T}} \tag{2}\]This equation is transformed to the frequency-wavenumber domain with wavenumber \(\xi\):  
\[(\xi^{4}\mathbf{EI}-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}) \mathbf{u}(\omega,\xi)=(\xi^{4}\mathbf{EI}+\mathbf{K}_{\mathrm{TS}}(\omega,\xi)) \mathbf{u}(\omega,\xi)=\mathbf{F}_{\mathrm{T}}^{{}^{\prime}}(\omega,\xi) \tag{3}\]  
where \(\mathbf{K}_{\mathrm{TS}}(\omega,\xi)=\mathbf{K}_{\mathrm{T}}(\omega)+\mathbf{K }_{\mathrm{S}}(\omega,\xi)\) is the dynamic track-soil stiffness matrix. The track stiffness \(\mathbf{K}_{\mathrm{T}}(\omega)\) is calculated by transfer matrices for each support element [33], and the soil stiffness \(\mathbf{K}_{\mathrm{S}}(\omega,\xi)=k_{\mathrm{S}}(\omega,\xi)\mathbf{e}_{ \mathrm{n}}^{{}^{\prime}}\mathbf{e}_{\mathrm{n}}\) is added to the last diagonal element. The soil stiffness \(k_{\mathrm{S}}(\omega,\xi)\) is calculated as a wavenumber integral and can be approximated by a Winkler approach \(k_{\mathrm{S}}\mathrm{*}(\omega)=k_{W}+\mathrm{i}\omega c_{W}\) (see Appendix A).  
The dynamic track stiffness \(K_{\mathrm{T}}(\omega)\) between the axle load \(F_{T}\) and the corresponding rail displacement \(u_{R}\) can be calculated from the wavenumber integral:  
\[\frac{1}{K_{T}(\omega)}=\frac{u_{\mathrm{R}}}{F_{T}}(\omega)=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathbf{e}_{1}^{{}^{\mathrm{T}}}\Big{(}\mathbf{EI}^{ \mathrm{s}4}+\mathbf{K}_{\mathrm{TS}}(\omega,\xi)\Big{)}^{-1}\mathbf{e}_{ \mathrm{1}}\mathrm{d}\xi \tag{4}\]  
The force \(F_{\mathrm{S}}\) on the soil follows from the displacement \(u_{\mathrm{S}}\) of the soil and the soil stiffness \(k_{\mathrm{S}}\) as:  
\[F_{\mathrm{S}}(\omega)=k_{\mathrm{S}}u_{\mathrm{S}}(\omega,\xi=0)=k_{\mathrm{ S}}(\omega,\xi=0)\mathbf{e}_{\mathrm{n}}^{{}^{\mathrm{T}}}\mathbf{K}_{\mathrm{TS}}( \omega,\xi=0)^{-1}\mathbf{e}_{1}F_{\mathrm{T}}=H_{\mathrm{T}}(\omega)F_{ \mathrm{T}}(\omega) \tag{5}\]  
The base vector \(\mathbf{e}_{\mathrm{n}}\) stands for the top of the track, the base vector \(\mathbf{e}_{\mathrm{n}}\) stands for the bottom of the track (the soil).