A discrete, non-linear, time-varying, torsional dynamic model of a multi-stage planetary train that is formed by any number of simple planetary stages is proposed in this study. Each planetary stage has a distinct fundamental mesh frequency and any number of planets spaced in any angular positions. The model allows the analysis of the gear train in all possible power flow configurations suitable for various gear drive ratios. It includes periodic variation of gear mesh stiffnesses as well as clearance (backlash) non-linearities that allow tooth separations. Equations of motion for the general case are formulated and solved semi-analytically using a hybrid harmonic balance method (HBM) in conjugate with inverse Fourier transform. Relative mesh displacements along lines of action of individual gear pairs were used as the continuation parameters to pass singular points and ill-conditioned equations in their proximity. At the end, a case study of a two-stage planetary train is used to demonstrate the effectiveness of the model and solution methods. The HBM solutions are compared to those obtained by a direct numerical integration method to assess their accuracy.
The FEM is employed to study the effect of notch depth on a new strain-concentration factor (SNCF) for rectangular bars with a single-edge notch under pure bending. The new SNCF K new e is defined under the triaxial stress state at the net section. The elastic SNCF increases as the net-to-gross thickness ratio h 0 /H 0 increases and reaches a maximum at h 0 /H 0 = 0.8. Beyond this value of h 0 /H 0 it rapidly decreases to the unity with h 0 /H 0 . Three notch depths were selected to discuss the effect of notch depth on the elastic-plastic SNCF; they are the extremely deep notch (h 0 /H 0 = 0.20), the deep notch (h 0 /H 0 = 0.60) and the shallow notch (h 0 /H 0 = 0.95). The new SNCF increases from its elastic value to the maximum as plastic deformation develops from the notch root. The maximum K new e of the shallow notch is considerably greater than that of the deep notch. The elastic K new e of the shallow notch is however less than that of the deep notch. Plastic deformation therefore has a strong effect on the increase in K new e of the shallow notch. The variation in K new e with M/M Y , the ratio of bending moment to that at yielding at the notch root, is slightly dependent up to the maximum K new e for the shallow notch. This dependence is remarkable beyond the maximum K new e . On the other hand, the variation in K new e with M/M Y is independent of the stress-strain curve for the deep and extremely deep notches.
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