László E. Kollar

DYNAMICS OF PIECEWISE LINEAR DISCONTINUOUS MAPS

In this paper, the dynamics of maps representing classes of controlled sampled systems with backlash are examined. First, a bilinear one-dimensional map is considered, and the analysis shows that, depending on the value of the control parameter, all orbits originating in an attractive set are either periodic or dense on the attractor. Moreover, the dense orbits have sensitive dependence on initial data, but behave rather regularly, i.e. they have quasiperiodic subsequences and the Lyapunov exponent of every orbit is zero. The inclusion of a second parameter, the processing delay, in the model leads to a piecewise linear two-dimensional map. The dynamics of this map are studied using numerical simulations which indicate similar behavior as in the one-dimensional case.

The Effects of Wind Induced Conductor Motion on Accreted Atmospheric Ice

Galloping of transmission lines creates some cyclic stresses in the conductor and accreted atmospheric ice covering the conductor, which may result in ice failure leading to shedding. Attempts have been made in this research to estimate these cyclic stresses, and experimental tests have been conducted to study their effects on atmospheric ice. First, galloping of an ice-covered conductor was simulated by appropriate modification of existing models for bare conductors submitted to galloping. Then, the results of simulation were applied as input for a new model developed using ABAQUS. The results show that the layers of atmospheric ice at the top and bottom of the conductor endure maximum stress. The results of experimental tests with increasing cyclic stress show that ice does not break during galloping at wind velocities below 4.5 m/s. The tests under cyclic loads with a constant amplitude reveal that the ice does not fail under stresses corresponding to wind speeds of 3 and 4 m/s, and sometimes fails under stresses arising at a wind speed of 5 m/s.

Digital controlling of piecewise linear systems

A mechanical model of a digital balancing system is constructed and its stability analysis is presented. This model describes practical problems like backlash and sampling delay. Stationary and periodic solutions are determined numerically for the case of the system without sampling. The existence and stability of periodic solutions are checked analytically. Adding sampling delay to the system, the stability conditions change and above a critical value of the delay, the balancing is impossible. The stability chart is determined again, and stable motions are identified.