## 4 Results and Discussion
This can be done based on the diagrams in Figure 6, which show the root mean square of acceleration in the reference points of the rigid carbody (diagram a) and the root mean square of acceleration in the reference points of the flexible carbody for \(f_{vb}=6\) Hz (diagram b). It is observed that the bending vibration does not change the position of the carbody critical point. For both models, the carbody critical point is located above one of the two bogies, and its position changes as a function of velocity. At velocities smaller than 92 km/h, the carbody critical points alternate either above the rear bogie or the front bogie. A large velocity interval follows afterwards, which extends up to 190 km/h, where the carbody critical point is the reference point situated above the front bogie. Figure 7 highlights the influence of the carbody vertical bending upon the root mean square of acceleration at a velocity of 200 km/h, in correlation with the damping ratio of the secondary suspension and defined according to the equation  
\[\zeta_{c}\ =\ \frac{4c_{zc}}{2\sqrt{4k_{zc}m_{c}}}. \tag{35}\]  
Figure 5: Increase of root mean square of acceleration at the carbody centre versus the rigid carbody. Figure 6: The position of the carbody critical point: (**a**) ‘rigid carbody’ model; (**b**) ‘flexible carbody’ model (\(f_{vb}=6\) Hz). In accordance with the reference parameters of the vehicle model (see Table 1), the reference value of the damping ratio of the secondary suspension is \(\zeta_{c}=0.12\). For both models, it is noticed that the root mean square of acceleration continuously decreases until it reaches a minimum value, while the damping ratio of the secondary suspension increases to a certain value. The minimum value of the root mean square of acceleration depends on the carbody reference point and then on the model type--the 'rigid carbody' model or the 'flexible carbody' model. In the case of the 'rigid carbody' model, the damping ratio that minimises the root mean square of acceleration at the carbody centre is 0.2, while the root mean square of acceleration is minimum for \(\zeta_{c}=0.32\) and \(\zeta_{c}=0.43\), respectively (see Figure 8), above the bogies. The value of the damping ratio minimizing the root mean square of acceleration at the carbody centre declines for the flexible carbody (Figure 8, diagram a); the more flexible the carbody, the smaller this value (for instance, \(\zeta_{c}=0.10\) for \(f_{vb}=6\) Hz; \(\zeta_{c}=0.18\) for \(f_{vb}=12\) Hz). In the points located above the two bogies, the value of the damping ratio that minimises the root mean square of acceleration coincides with the value identified for the rigid carbody, \(\zeta_{c}=0.32\)--above the front bogie, \(\zeta_{c}=0.43\)--above the rear bogie, for \(f_{vb}=8\) Hz, \(f_{vb}=10\) Hz and \(f_{vb}=12\) Hz (Figure 8, diagrams b and c). For a very flexible carbody (\(f_{vb}=6\) Hz), the damping ratio that minimises the root mean square of acceleration is smaller, \(\zeta_{c}=0.30\) and \(\zeta_{c}=0.40\), respectively. Figure 8: The damping ratio of secondary suspension minimizing the root mean square of acceleration: (**a**) at the carbody centre; (**b**) against the front bogie; (**c**) against the rear bogie. Figure 7: Influence of the carbody vertical bending upon the root mean square of acceleration in correlation with the damping ratio of secondary suspension: (**a**) at the carbody centre; (**b**) against the front bogie; (**c**) against the rear bogie. The Figure 9 is used to analyse the influence of the carbody vertical bending upon the root mean square of acceleration at a velocity of 200 km/h, in correlation with the damping ratio of primary suspension. The damping ratio of primary suspension is calculated with the relation:  
\[\zeta_{b}\ =\ \frac{4c_{zb}}{2\sqrt{4k_{zb}m_{b}}}. \tag{36}\]  
For the reference parameters of the vehicle model (see Table 1), the reference value of the damping ratio for the primary suspension is \(\zeta_{b}=0.22\).