This dissertation aims at investigating the dynamic response of double-helical planetary gear sets theoretically. A three-dimensional discrete dynamic model of a double-helical planetary gear set is proposed, including all gear mesh, bearing and support structure compliances. The model is presented in three levels of complexity: (i) a linear time-invariant (LTI) model, (ii) a LTI model with gyroscopic effects included, and (iii) a nonlinear time-varying (NTV) model with parametrically time-varying gear mesh stiffnesses and nonlinear tooth separation effects included.
As the first step, a generic linear (no tooth separations), time-invariant (constant gear mesh stiffnesses) dynamic model is formulated to analyze any N-planet double-helical planetary gear system. The model includes any planet phasing conditions dictated by the number of planets, number of gear teeth and planet position angles as well as any phase shifts due to the designed stagger between the right and left sides of the gear set. The forced response due to gear mesh transmission errors excitations is computed by using the modal summation technique with the natural modes found from the corresponding Eigen value problem for the undamped system. The strain energies of the mode shapes are computed to identify the modes excitable by these excitations. Parametric studies are presented to demonstrate sizable influences of planet phasing, stagger conditions, gear and carrier support conditions as well as the number of planets on the steady-state forced response.
In the second modeling step, the LTI model is modified to include a class of gyroscopic effects due to vibratory skew of spinning gears for the case of a stationary carrier. The complex Eigen solutions are examined to quantify the influence of rotational speed of the gear set through gyroscopic effects on the natural modes. A complex modal summation formulation is used to compute the forced response with gyroscopic effects. Results indicate that the influence of gyroscopic moments on natural frequencies is modest within typical speed ranges, with only a sub-set of modes exhibiting dominant tilting motions impacted by the gyroscopic effects. Effect of gyroscopic moments on forced response curves is found to be limited to slight changes in amplitudes and frequencies of certain resonance peaks.
As the final step, mesh stiffness fluctuations due to change in number of tooth pairs are introduced as internal parametric excitation along with the transmission error excitations at the same phasing relations. Tooth separation functions are also applied to obtain a set of NTV equations of motion, which are solved by using direct numerical integration. Differences observed between the forced response curves for time-varying and time-invariant systems are characterized by additional resonance peaks and overall increases in response amplitudes while no signs of nonlinear behavior are noted.