Single- and Dual-Plane Automatic Balancing of an Elastically-Mounted Cylindrical Rotor with Considerations of Coulomb Friction and Gravity
This work treats dual-plane automatic ball balancing of elastically-mounted cylindrical rotors. The application is primarily to systems with a vertically-oriented single-bearing support, but extension is also made to horizontally-oriented single-bearing support as typically found in a vehicle wheel. The rotor elastic mounting includes three translational degrees of freedom for the body geometric center and three rotational degrees of freedom. Damping is included for each of these six degrees of freedom. The model for the automatic ball balancer consists of up to two arbitrarily-located hollow circumferential races, each of which contains up to two sliding particles. The friction model for the particles includes both viscous and Coulomb friction forces. Of considerable complexity is the logic path for the individual particles being either in motion or stationary relative to the rotor. The exact equations of motion for the overall system are derived via a Newtonian approach. Numerical-integration results show that the balancer performance depends strongly on the friction levels as well as the operating speed of the body. Simulations conducted with a pure static imbalance show that ideal automatic balancing is possible only for vertical-axis rotors that have zero Coulomb friction levels between the balancing particles and the races. Simulations with a horizontal-axis statically-imbalanced rotor show that an automatic balancer can improve performance for certain operating speeds and non-zero Coulomb friction levels in the presence of gravitational forces. Simulations conducted with a pure dynamic imbalance show that there is no inherent mechanism to counteract rotational displacements of the rotor about its geometric center. As a result, the balancing particles exhibit several phenomena described in previous works such as synchronous motion and oscillatory behaviors within their respective races. Simulations for an arbitrarily located imbalance show that rotor performance can be improved using dual-plane balancing techniques for certain operating speeds and Coulomb friction levels. Due to the inherent complexity in eliminating an arbitrarily located mass imbalance, the system is generally unable to reach a perfectly balanced configuration, but performance can be improved for carefully-selected initial conditions.