A Coupled Torsional-Transition Nonlinear Vibration and Dynamic Model of a Two-Stage Helical Gearbox Reducer for Electric Vehicles

A coupled torsional-transition nonlinear dynamic model of a two-stage helical gear (TSHG) reduction system for electric vehicles (EVs) is presented in this paper. The model consists of 16 degrees of freedom (DOF), which includes factors such as the nonlinearity of backlash, time-varying mesh stiffness (TVMS), mesh damping, supporting bearings, static transmission error (STE), and the torsional damping and stiffness of the intermediate shaft, in which the fourth-order Runge–Kutta numerical integration method was applied to solve the differential equations. With the help of bifurcation diagrams, time-domain histories diagrams, amplitude-frequency spectrums, phase plane diagrams, Poincaré maps, root-mean-square (RMS) curves, peak-peak values (PPVs), and Lyapunov exponents, the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG system are investigated. The stability properties of steady-state responses are investigated using Lyapunov exponents. The results reveal various types of dynamic evolution mechanisms and nonlinear phenomena such as periodic-one responses, quasiperiodic responses, jumps phenomena, and chaotic responses. The research presents useful results and information to vibration control and dynamic design of the TSHG transmission system used in EVs.

Restricted Internal Rotation in 1-Arylhydantoins, 3-Arylhydantoins, and 3-Aryl-2-Thiohydantoins: Reversal of the Effective Sizes of Methyl and Chlorine

In 1-arylhydantoins an o-methyl group is effectively larger than an o-chloro substituent in restricting rotation about the aryl-C—N bond, whereas in 3-arylhydantoins and 3-aryl-2-thiohydantoins the reverse order of sizes is observed. This difference is attributed to repulsion between the chlorine atom and a carbonyl oxygen atom in the torsional transition state of the 3-aryl compounds.