Modeling of Dynamics of Planetary Gears with Elements of the Increased Flexibility
The increased elements flexibility of planetary gear promotes load alignment in wheel meshing, but creates danger of undesirable vibrations emergence. The research problem of dynamics of planetary gears with elements of the increased flexibility is relevant. The planetary gear is presented by a set of solid bodies (gearwheels) connected by the springs modeling elastic constraints in gear pair. The flexible epicycle is presented in the form of gear rim parts and separate teeth with elastic constraint between them which are at present in meshing. The mathematical model of dynamics of the planetary mechanism is made on the basis of Lagrange's equations of the second kind. Nine generalized coordinates are considered. The generalized forces are: moments of the engine, forces of friction and useful resistance; the moment caused by torsion of an elastic axle of a sun gearwheel; elastic forces in wheels meshing, in satellites axles, between separate teeth of an epicycle. As a first approximation it is considered that planetary carrier's angular speed is constant. It is also accepted that the sum of the moments of the engine, forces of friction and useful resistance is equal to zero; and the mass center of the satellite with respect to the planetary carrier doesn't move. The system of nine differential equations is reduced to two equations of free vibrations of the satellite and a sun gearwheel. The meshing rigidity is expressed via the Young modulus and face width of gearwheels. The moments of inertia of wheels are accepted as for continuous disks; masses of wheels are determined with account of the filling coefficient. Analytical dependences for natural frequencies of free vibrations of the satellite and a sun gearwheel owing to teeth flexibility are received. Influence of a sun gearwheel diameter and gear ratio of the planetary mechanism on values of natural frequencies is investigated. The conclusion is made for the influence of the increased compliance of planetary gear elements on undesirable vibrations of the satellite and the sun gearwheel.