## 2 Wheel-Track Coupling Model
### Numerical Model
#### 2.2.5 Model Verification  
As it is known that the higher vibration mode may face a computational ill-conditioned problem due to the noisy calculation of eigenvalues, a series of models with different model reduction assumptions are analysed and compared all together with the results of the full FEM model. The numbers of eigenmodes of the rail submodel are chosen to be 100, 200, 400 and 800, and the harmonic analysis has been carried out to compare the track receptance results between the MBS and FEM models. The track receptance \(H(\omega)\) is usually described as the driving-point response of the track structure under frequency sweep sine waves and can be obtained as follows:  
\[H(\omega)=\frac{D\cdot e^{(ioct+\varphi)}}{F\cdot e^{ioct}} \tag{19}\]  
where \(D\) denotes the driving-point displacement response, \(F\) denotes the excitation force and \(\varphi\) denotes the phase angle difference between the excitation force and driving-point response. Using the receptance equations above, the track receptance at the mid-span between sleepers are compared with the exact solution in FEM in Figure 5. It can be seen that as the number of mode shapes increases, the results of the proposed model are becoming closer and are converged to the results of the fully-detailed FEM model. All models have shown the first two resonance frequency at around 51.7 Hz and 149.3 Hz. When only considering 100 and 200 modes, the results fail to simulate the third resonance frequency at around 1570.8 Hz. When the quantity of modes chosen is greater than 400, the  
Figure 4: Coupled wheel–track model.  
receptance of the proposed model are getting converged. So, in the following, the number of eigenmodes is chosen as 800.  
To ensure that the flexible model proposed here is reliable, the receptance and phase angle at the sleepers mid-span and above the sleepers are compared with a model based on Finite Element Method (FEM). The comparison results are shown in Figures 6 and 7.  
The receptance and phase angles of the rail at the sleepers mid-span and above the sleepers calculated with the proposed model and the FEM model show that the first resonance frequency is approximately 51.7 Hz and the second resonance frequency is approximately 149.3 Hz, which are the frequencies of the excitation force and track displacement with a phase angle of 90\({}^{\circ}\). The velocity of the excitation point of the track is in the same phase as the excitation force, and whenever the displacement becomes zero, the excitation force reaches the maximum and keeps amplifying the track displacement. Both methods indicate that the pinned-pinned resonance frequency is approximately 1570.8 Hz. When the track is excited at the mid-span, the excitation force amplifies the vibration of the track, and when the track is excited above the sleeper, the excitation force and excitation point move in anti-phase, i.e., with a phase angle of 180\({}^{\circ}\). The cut-off frequency is 3000 Hz for both methods because the modelling accuracy cannot be guaranteed under the beam assumption when the dynamic response is higher than 3000 Hz. It can be seen that the  
Figure 5: Acceptance and phase angle at mid-span between sleepers.  
Figure 6: Acceptance and phase angle at sleepers mid-span.  
receptance and phase angle are almost the same for both methods, which verifies the accuracy of the flexible track model established in this study.