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A discrete non-linear torsional vibration model of a single-stage planetary set is proposed in the current study. The model includes all possible power flow configurations, any number of planets in any spacing arrangement and any planet mesh phasing configurations. It also includes time variation of gear mesh stiffnesses as well as clearance (backlash) nonlinearities. The non-linear equations of motion are solved semi-analytically using multi-term harmonic balance method (HBM) in conjunction with inverse discrete Fourier transform and Newton-Rapson method. The HBM solutions are compared with numerical simulation results to demonstrate the accuracy of the HBM formulation. Another comparison with predictions of a deformable-body dynamic model is also provided to assess the accuracy of the discrete model. Limited parametric studies are presented at the end, to show the influence of key gear design parameters on dynamic response.

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.

A heat transfer model was developed to predict a transient thermal behavior of asphalt concrete during service life at different weather conditions. The developed model has the capability to predict the distribution of temperature field with respect to time within the Hot-Mix asphalt body based on surrounding environmental conditions. This will greatly help pavement engineers to select the suitable asphalt grade to achieve the best pavement performance and avoid pavement distresses might be caused due to extreme pavement temperatures. These distresses include fatigue cracks, rutting, and thermal cracking. The resulted model required data on asphalt mixture, incident radiation, surface, and ambient temperatures in addition to thermal properties of Hot-Mix asphalt including absorptivity, heat transfer coefficient, and the emissivity. A sensitivity analyses wasperformed to study the impact of a number of thermalenvironmental and pavement geometric parameters on predicted temperature responses. The results of analysis indicated that the incident radiation, absorptivity, and the heat transfer coefficient have the most significant effect on Hot-Mix asphalt temperature. Also, the emissivity has insignificant effect on surface temperature Hot-Mix asphalt.