# The Two-Parameter Bifurcation and Evolution of Hunting Motion for a Bogie System
## 5 Conclusions  
A lateral dynamic model of the 7-DOFs bogie system is constructed, in which the nonlinear, non-smooth wheel-rail contact relations and the piecewise linear flange force make it a typical non-smooth dynamic system. Based on Poincare mapping theory and two-parameter simulation, patterns and distributions of hunting motions are accurately identified and depicted in the (\(v\), \(\mu\))-plane, and 3D cloud maps of the maximum lateral wheel-rail impact velocity are obtained. In the (\(v\), \(\mu\))-plane, for low \(v\), the pattern types of periodic hunting motions are relatively simple and appear as vertical band distributions. As \(\mu\) increases, the critical speed of the bogie system increases accordingly, and the regions of 1-0-0 and 1-1-1 motions are significantly narrowed. Among them, wheel-rail impactless motion 1-0-0 is vital in detecting hunting instability. For high \(v\), the change in \(\mu\) initiates significant changes in the type of periodic motion patterns, accompanied by the emergence of subharmonic and chaotic motions. The maximum lateral wheel-rail impact velocities of wheelset-1 are generally higher than those of wheelset-2, and the overall trend of wheel-rail impact velocities increases with the increasing \(v\). The correlations between the lateral wheel-rail impact velocities and the patterns of hunting motions were found. The 1-1-1 motion was associated with the low maximum lateral wheel-rail impact velocities. In contrast, unidentified gray areas (dominated by chaotic motions and part of subharmonic motions) commonly corresponded to the peak regions. The bifurcations and the evolution of periodic hunting motions are further revealed by selecting \(\mu\) = 0.12 and \(\mu\) = 0.18 within typical parameter intervals, respectively. The Hopf bifurcation initiated the hunting instability of the bogie system, and the grazing bifurcation initially caused the wheel-rail impact. Based on the combined bifurcation diagram, it is observed that the subharmonic motions in specific periodic motion windows exhibit symmetry, so there is no coexistence of periodic motions. Conversely, within some subharmonic motion windows, the coexistence behavior of periodic motion is caused by pitchfork bifurcation. The coexisting periodic solutions with different initial values belong to two symmetric branches of pitchfork bifurcation. At the same time, the pitchfork bifurcation caused asymmetric impact velocities between the wheelset and the two sides of the rail. At higher speed intervals, jump bifurcation causes wheel-rail impact velocities' amplitude jumps and produces a coexistence behavior of periodic motions. Based on the numerical simulation, the bogie system exhibits abundant periodic hunting motion patterns when operating above the critical speed and correlation between the pattern types and the wheel-rail impact characteristics. The coexisting behavior of periodic hunting motions in some speed intervals could initiate abnormal vibrations and  
Figure 18: Phase portraits of coexisting motions in Figure 15 (the magenta and black phase trajectories correspond to the same-colored branches in the bifurcation diagram, respectively). (**a1**) \(v\) =113 m/s, 1-2-2 motion (in black) and 1-2-2 motion (in magenta) of wheelset-1; (**a2**) \(v\) =113 m/s, 1-2-2 motion (in black) and 1-2-2 motion (in magenta) of wheelset-2; (**b1**) \(v\) =122 m/s, 1-2-2 motion (in black) and 1-2-2 motion (in magenta) of wheelset-1; (**b2**) \(v\) =122 m/s, 1-1-1 motion (in black) and 1-2-2 motion (in magenta) of wheelset-2. increased wear. The two-parameter co-simulation research method used in this paper can provide some theoretical guidance for the prediction of hunting motion patterns and the parameter design and selection of a bogie system. As wheel-rail impacts occur at speeds higher than critical speeds, relevant experimental validation is seldom found to be carried out. Therefore, the validation of the effectiveness of numerical simulations still needs to be realized in subsequent studies. Conceptualization, S.W.