## 4 Results of the Research
### Loaded Car Analysis
#### 4.2.1 Analysis on a Straight Track  
The periodic solutions that appeared on the ST at 30.8 m/s exist until the velocity \(v\) equals 49.9 m/s (Figure 7). Increasing \(v\) over this value by 0.1 m/s causes the bifurcation of periodic solutions to stationary ones. Stationary solutions persist up to a \(v\) value of 80.7 m/s. Increasing the velocity \(v\) by 0.1 m/s further causes bifurcation to periodic solutions again. The periodic solutions last until the end of the tested velocity range (200 m/s).  
Next, research was undertaken to eliminate the possibility of the occurrence of periodic solutions (self-exciting vibrations in real systems) in the modelled car's operating velocity range. Elements of the suspension system have a crucial impact on the dynamic properties of the vehicle. Changing the suspension parameter values is usually a relatively simple process. Accordingly, the influence of the longitudinal and lateral stiffness (\(k_{zx}\), \(k_{zy}\)) and damping (\(c_{2x}\), \(c_{zy}\)) on the primary suspension was examined by varying their values. The result of these tests is the determination of the lateral stiffness \(k_{zy}\) parameter for which a drop in value compared to the nominal value gives the most favourable effects, i.e., improvements in the analysed car's properties. The test results are shown in Figure 8.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline CC radius; \(R\) in m & 1200 & 2000 & 3000 & 4000 & 6000 & \(\infty\) \\ \hline Empty car; \(v_{n}\) in m/s & 45.2 & 44.4 & 44.4 & 42.6 & 40.1 & 37.2 \\ \hline Laden car; \(v_{n}\) in m/s & 55.8 & 43.7 & 33 & 45.6 & 45 & 30.8 \\ \hline \end{tabular}
\end{table}
Table 2: The nonlinear critical velocity values \(v_{n}\) of the modelled gondola car.  
Figure 8: Influence of the lateral stiffness \(k_{\rm zy}\) on laden car’s first wheelset. \(k_{\rm zy}=55.6*10^{5}\) N/m—nominal value (black line), \(k_{\rm zy}=50*10^{5}\) N/m——90% of nominal value (red line), \(k_{\rm zy}=44.5*10^{5}\) N/m——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 6: Stability map of laden gondola car: (**a**) maximum absolute value of the leading wheelset’s lateral displacement \(1y_{lu}\)_1 max_; (**b**) p-t-p value of \(y_{lu}\) as a function of vehicle velocity.  
A reduction in the lateral stiffness to 90% of its nominal value (90% \(\cdot k_{xy}\) = 0.9\(\cdot\)55.6\(\cdot\)10\({}^{5}\)\(\approx\) 50\(\cdot\)10\({}^{5}\) N/m) increases the \(v_{n}\) velocity up to 36.9 m/s. At this velocity, periodic solutions appear and exist up to 45.3 m/s. Then, bifurcation from periodic to stationary solutions in velocity \(v\) cause bifurcation from stationary to periodic solutions, with values close to those obtained for the nominal value of \(k_{zy}\). Periodic solutions persist up to velocities exceeding 200 m/s. A further decrease in the lateral stiffness value, down to 80% of the nominal value (80% \(\cdot k_{xy}\) = 0.8\(\cdot\)55.6\(\cdot\)10\({}^{5}\)\(\approx\) 44.5\(\cdot\)10\({}^{5}\) N/m), causes a jump in the critical velocity value. In this case, the range of periodic solutions which existed at velocities lower than 50 m/s disappears. Stationary solutions exist for a velocity of up to 136.1 m/s. Then, bifurcation to periodic solutions occurs. The values of the solutions are similar to those determined at the nominal value of the lateral stiffness. This type of solution lasts up to \(v\) = 200 m/s. So, a decrease in lateral stiffness \(k_{zy}\) by a dozen percent relative to the nominal \(k_{zy}\) value enables an improvement in the stability properties of the loaded car on an ST.