## 4 Results of the Research
### Empty Car Analysis at Reduced \(k_{zy}\) Values  
A moderate effect of reducing the lateral stiffness of the primary suspension can be observed on motion along an ST (Figure 11). The critical velocity \(v_{n}\) decreases from 37.2 m/s at a nominal \(k_{zy}\) value (38.9\(\cdot 10^{5}\) N/m) to 36.5 m/s for 90% \(\cdot k_{zy}\approx 35\cdot 10^{5}\) N/m  
Figure 10: Influence of lateral stiffness \(k_{zy}\) on laden car’s first wheelset. \(k_{zy}=55.6\cdot 10^{5}\) N/m—nominal value (black line), \(k_{zy}=50\cdot 10^{5}\) N/m—90% of nominal value (red line), \(k_{zy}=44.5\cdot 10^{5}\) N/m—80% of nominal value (blue line). Motion on a CC of \(R=3000\) m: (**a**) maximum absolute value of lateral; (**b**) peak-to-peak value of wheelset lateral displacement.  
and to 35.8 m/s for 80% \(\cdot\)\(k_{zy}\)\(\approx\) 31\(\cdot\)10\({}^{5}\) N/m. So, \(v_{n}\) is higher than the required minimum \(v_{n}\) = 33.3 m/s. The nature of the model solutions is analogous for each \(k_{zy}\) value. So, the solutions are stationary only for velocities \(v\) < \(v_{n}\), while for \(v\) > \(v_{n}\), periodic solutions appear, which exist up to \(v\) > 200 m/s. The solutions' values are very similar for each value of \(k_{zy}\).  
The influence of \(k_{zy}\) on the investigated model parameters is also moderate in the case of motion on curved tracks. The critical velocity is 40.1 m/s at the nominal \(k_{zy}\) value and with a large \(R\) = 6000 m (Figure 12). Reducing the lateral stiffness causes a slight decrease in the critical velocity to the values \(v_{n}\) = 39.6 m/s for 90% \(k_{zy}\) and \(v_{n}\) = 39.2 m/s for 80% \(k_{zy}\), respectively. Stable periodic solutions persist for an increased velocity up to about 151 m/s, followed by bifurcation to stationary solutions. Stable stationary solutions exist up to a velocity of about 176 m/s. The model solution values for the particular \(k_{zy}\) values applied are similar in the entire velocity range.  
The critical velocity is 44.4 m/s on a CC with a smaller \(R\) = 3000 m for the nominal value of \(k_{zy}\) = 38.9\(\cdot\)10\({}^{5}\) N/m (Figure 13). Reducing the lateral stiffness \(k_{zy}\) to 90% decreased  
Figure 11: Influence of lateral stiffness \(k_{zy}\) on the empty car’s first wheelset \(k_{zy}\) = 38.9 \(\ast\) 10\({}^{5}\) N/m—nominal value (black line), \(k_{zy}\) = 35 \(\ast\) 10\({}^{5}\) N/m \(\cong\) 90% of nominal value (red line), \(k_{zy}\) = 31 \(\ast\) 10\({}^{5}\) N/m \(\cong\) 80% of nominal value (blue line). Motion on the ST: (**a**) maximum absolute value of lateral displacement; (**b**) peak-to-peak value of wheelset’s lateral displacement.  
Figure 12: Influence of lateral stiffness \(k_{zy}\) on the empty car’s first wheelset \(k_{zy}\) = 38.9 \(\ast\) 10\({}^{5}\) N/m—nominal value (black line), \(k_{zy}\) = 35 \(\ast\) 10\({}^{5}\) N/m \(\cong\) 90% of nominal value (red line), \(k_{zy}\) = 31 \(\ast\) 10\({}^{5}\) N/m \(\cong\) 80% of nominal value (blue line). Motion on a CC with \(R\) = 6000 m: (**a**) maximum absolute value of lateral displacement; (**b**) peak-to-peak value of wheelset’s lateral displacement.  
the critical velocity to 44.2 m/s. A further reduction in the lateral stiffness \(k_{zy}\) to 80% reduced the critical velocity to 44 m/s only. So, a moderate influence of decreasing \(k_{zy}\) on the critical velocity on this route is also observed. The values of the model solutions (\(\left|\,y_{lu}\,\middle|\,max\) and p-t-p \(y_{lu}\right)\) for individual \(k_{zy}\) values are similar to the values on the previously studied routes. Thus, it can be observed that reducing \(k_{zy}\) to 80% of its nominal value reduces the model's critical velocity when the car is empty but not more than by 1 m/s.