The First Half-Turn of an Inertial Vibrator

The paper concerns the starting of an inertial vibrator which, due to a high value of static unbalance, is unable to perform the first half-turn in the gravitational field of force. The problem plays an essential role in the selection of driving units for the wide class of over–resonance machines. On the basis of the Pontryagin’s Maximum Principle the problem was formulated in terms of a dynamic optimisation. Thus, the driving moment performs the function of control – which is being looked for – while equations of motion and start-end conditions related to the position of a vibrator, act as constrains equations and boundary conditions. The possible simplifications in description of the motion of the vibrator, as well as the influence of the variable asynchronous motor driving moment on the optimal solutions, are discussed in the paper. On the basis of the work-energy equivalence principle, the minimum values of the driving moment – which warrants the performance of the first half-turn of the vibrator at a given number of the moment switchovers – were determined. The problem of thermal loads of the motor during the first half-turn was also considered. The theoretical results were confirmed by the computer simulations.