Limit cycles at oversteer vehicle

Handling and stability properties of automobiles are most often studied from a practical point of view by applying a reduced set of equations, where the forward velocity is kept constant. At studying the full set of equations of a basic nonlinear two-wheel vehicle model, a supercritical Hopf bifurcation is found for an oversteer vehicle. All state variables of the vehicle are involved at small amplitude limit cycles in the vicinity of the Hopf bifurcation point with the steering angle (drive torque) as bifurcation parameter. At the transition to large amplitude relaxation cycles, the cyclic motion of the vehicle may be separated into ‘slow’ longitudinal velocity-related segments, and ‘fast’ vehicle yaw and side slip-related segments, indicating a singular perturbed system. Moreover, Canard phenomenon is observed for both steering angle and drive torque bifurcation parameters

On Hopf bifurcation in the problem of motion of a heavy particle on a rotating sphere: the viscous friction case

We investigate the Hopf bifurcation of a mass on a rotating sphere under the influence of gravity and viscous friction. After determining the equilibria, we study their stability and calculate the first Lyapunov coefficient to determine the post-critical behavior. It is found that the bifurcating periodic branches are initially stable. For several inclination angles of the sphere’s rotation axis, the periodic solutions are calculated numerically, which shows that for large inclination angles turning points occur, at which the periodic solutions become unstable. We also investigate the limiting case of small friction coefficients, when the mass moves close to the equator of the rotating sphere.404940601

Travelling interface waves in a brake-like system under unilateral contact and Coulomb friction

International audienceThis article considers the frictional interface waves generated by the flutter instability of the sliding steady state for an elastic tube in frictional contact with a rigid and rotating shaft. According to the values of the contact pressure, the rotation velocity and the friction coefficient, several periodic dynamical responses can be found under the form of travelling surface waves. Examples of stick-slip, stick-slip-separation and stick-slip-separation-reverse-slip waves are reported here. Some discussions on the stability of these waves are also given

On the stability of the track of the space elevator

A string moving with geostationary angular velocity in its radial relative equilibrium configuration around the Earth, reaching from the surface of the Earth far beyond the geostationary height, could be used as track for an Earth to space elevator. This is an old dream of mankind, originating about 100 years ago in Russia. Besides the question of feasibility from a technological point of view also the question concerning the stability of such a configuration has not yet been completely solved. Under the assumption that a proper material (carbon nanotubes) is available, making the connection possible technologically, we address the question of existence and stability of the radial relative equilibrium of a tapered string on a circular geosynchronous trajectory around the Earth, reaching from the surface of the Earth far beyond the geostationary heigh

Analyzing the non-smooth dynamics induced by a split-path nonlinear integral controller

In this paper, we introduce a novel non-smooth integral controller, which aims at achieving a better transient response in terms of overshoot of a feedback controlled dynamical system. The resulting closed-loop system can be represented as a non-smooth system with different continuous dynamics being active in dedicated regions of the state-space. The dynamical behavior of the hybrid system will be studied through simulations