## 4 Results of the Research
### Loaded Car Analysis
#### 4.2.2 Analysis on Curved Tracks  
The following step in the research examined whether the beneficial effect on an ST (regarding the improvement in the system properties) due to reducing the lateral stiffness \(k_{xy}\) would also appear as a favourable improvement on a CC with \(R\) = 6000 m. For the nominal value \(k_{xy}\) = 55.6\(\cdot\)10\({}^{5}\) N/m, the corresponding critical velocity was \(v_{n}\) = 45 m/s (Figure 9). This was determined to be to some extent arbitrary because at a lower \(v\) value (about 20 m/s), \"periodic\" solutions of the minimal p-t-p value of \(y_{lm}\) appeared. A velocity \(v_{n}\) = 45 m/s was adopted, judging that the p-t-p value of \(y_{lm}\) was large enough (about 0.001 m) at this velocity to talk about periodic solutions in the practical sense. However, a considerable increase in the p-t-p value of \(y_{lm}\) to a dozen millimetres is observed at a velocity of 57.1 m/s. Stable periodic solutions exist up to \(v\) = 61.6 m/s only, thereafter being replaced by bifurcation to stationary solutions. In further increasing the velocity, stationary solutions exist up to 92.3 m/s. Then, bifurcation to stable periodic solutions is observed. Periodic solutions exist until \(v\) = 131 m/s. Next, bifurcation to stable stationary solutions appears. Such solutions exist for a high velocity range from 131 to 148 m/s. At even greater velocities, no stable solutions are observed. In the following research step, the lateral stiffness decreased to 90% of its nominal value, \(k_{zy}\) = 0.9\(\cdot\)55.6\(\cdot\)10\({}^{5}\)\(\approx\) 50\(\cdot\)10\({}^{5}\) N/m. Stable stationary solutions persist up to \(v\) = 102 m/s. Similarly to the model with a nominal value for lateral stiffness, some \"periodic\" lateral displacements of the wheelset appear on the curve, too. Nevertheless, they have minimal  
Figure 9: Influence of lateral stiffness \(k_{xy}\) on laden car’s first wheelset. \(k_{xy}\) = 55.6 \(\cdot\) 10\({}^{5}\) N/m—nominal value (black line), \(k_{xy}\) = 50 \(\cdot\) 10\({}^{5}\) N/m—90% of nominal value (red line), \(k_{xy}\) = 44.5 \(\cdot\) 10\({}^{5}\) N/m—80% of nominal value (blue line). Motion on CC of \(R\) = 6000 m: (**a**) maximum absolute value of lateral displacement; (**b**) peak-to-peak value of wheelset’s lateral displacement. p-t-p values. Increasing the vehicle velocity to \(v=102.1\) m/s induces bifurcation to periodic solutions. This type of solution persists when increasing \(v\) up to 131 m/s. The next step was reducing the lateral stiffness to 80% of its nominal value (80% \(\cdot k_{zy}=0.8\cdot 55.6\cdot 10^{5}\approx 44.5\cdot 10^{5}\) N/m). Stable stationary solutions occur up to a velocity \(v=107.1\) m/s. Bifurcation from stable stationary solutions to stable periodic one occurs at this velocity. Periodic solutions last up to approximately \(v=130\) m/s. The values of \(\mid y_{lu}\mid max\) and p-t-p \(y_{lu}\) for all the \(k_{zy}\) values tested are similar. To conclude, a reduction in the lateral stiffness \(k_{zy}\) by a dozen percent compared to its nominal value results in a considerable rise in the critical velocity on a CC of \(R=6000\) m.  
In the case of motion on a CC of \(R=3000\) m and the nominal value of \(k_{zy}\), the critical velocity \(v_{n}=33\) m/s (Figure 10). This is slightly lower than the maximum operating (admissible) velocity \(v_{ad}\) of the studied gondola (\(v_{ad}=120\) km/h \(\approx 33.3\) m/s). Stable periodic solutions occur up to a velocity \(v=103.6\) m/s in this case. Then, bifurcation to other periodic solutions appears but with lower values of \(\mid y_{lu}\mid max\) and p-t-p \(y_{lu}\) (approx. p-t-p \(y_{lu}=0.004\) m). The maximum velocity for which a stable solution exists is \(v=109\) m/s. A decrease in the stiffness \(k_{zy}\) to 90% of its nominal value (\(k_{zy}=50\cdot 10^{5}\) N/m) results in a critical velocity increase up to \(v_{n}=34.6\) m/s.