Modelling, analysis and mitigation of self-excited vibrations of a magnetic track brake

Magnetic track brakes work independently of the wheel–rail contact as an additional braking system in railway vehicles. In the past, magnetic track brakes were usually deactivated at velocities below 25 km/h in mainline applications to avoid stopping jerks. However, current demands on braking performance require activation until full stop. During field tests on this subject, severe self-excited vibrations were measured at velocities below 25 km/h. This study analyses the oscillations observed by focusing on the stability behaviour of a simplified linear coupled electro-magneto-mechanical model of the track brake. Key parameters are identified by applying established criteria of linear stability theory. To account for essential nonlinear effects, a detailed multibody dynamics model with flexible bodies including a more enhanced electro-magnetic model is introduced and parameterised by using measurement data. Simulation results reveal two separate (or combined) mechanisms that may lead to self-excitation. On the one hand, friction-induced vibrations between the magnets and the rail; on the other hand, coupling effects between the electro-magnetic and the mechanical subsystems of the track brake may be the cause of the self-excited vibrations observed. Stability behaviour is influenced by various design parameters and in particular by the contact conditions between track brake and rail. Finally, a few passive methods are briefly discussed that may help to mitigate or reduce self-excited vibrations at low velocities in future designs of track brakes.

Asymptotic Stability of the Pexider–Cauchy Functional Equation in Non-Archimedean Spaces

In this paper, we investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We also showed that, under some conditions, if ∥f(x+y)−g(x)−h(y)∥⩽ε, then f,g and h can be approximated by additive mapping in non-Archimedean normed spaces. Finally, we deal with a functional inequality and its asymptotic behaviour.

Practical Application of the Whipple and Carvallo Stability Model on Modern Bicycles with Pedal Assistance

Increasingly more people cycle with electrically-powered pedal assistance. The reduced pedalling effort attracts physically challenged people and seniors, who have a higher risk of falling. Since electric bicycles are heavier and the centre of masses are located higher, accidents happen easily. This study analyses the influence of the addition of a battery and motor unit on the stability behaviour of common bicycles for women. Based on market research, seven typical bicycle configurations are determined. Geometrics, mass values, and cycling postures are measured, and the theoretical stability behaviour is determined analytically based on the stability model of Whipple and Carvallo. The research shows that bicycles without pedal assistance have a smaller self-stable and semi-stable range than most electric bicycles. The electric bicycle with a motor implemented in the front wheel perform best, as the extra weight of the motor enhances the gyroscopic self-stabilization of the front wheel. Furthermore, a battery in the lower mid-tube is preferred over one in the luggage rack as it lowers the center of mass of the rear frame assembly. Knowledge about the optimal configuration to maximize the stability will enhance the cycling comfort and minimize the chance of accidents.