Study on the load-sharing characteristics of face-gear four-branching split-torque transmission system

In order to solve the load-sharing characteristics of face-gear four-branching split-torque transmission system (FGFBSTTS), the static load-sharing mechanical analysis model was established. In the model, the deformation coordination conditions of torsional angle and torque balance condition were considered. By using Loaded Tooth Contact Analysis (LTCA) technology of face gear and herringbone gear, the time-varying meshing stiffness was calculated. The influences of manufacturing errors, installation errors, I-stage pinion floating, II-stage pinion spline clearance floating, and radial limit ring clearance floating on the load-sharing characteristics are analyzed. The results show that the LTCA technology is more accurate to reflect the load-sharing characteristics of each meshing position. When the I-stage pinion and the II-stage pinion floated at the same time, the best load-sharing characteristics can be obtained. The load-sharing characteristics affected by manufacturing errors showed obvious periodic change. The radial limit ring plays a better auxiliary role in load-sharing characteristics. The theoretical results were compared with the experiments to verify the correctness of the theoretical analysis. The research results can provide a theoretical basis for the optimal design of the load-sharing structure, error control, and assembly of the face gear four branch transmission system.

Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

Ramanujan's Radial Limits and Mixed Mock Modular Bilateralq-Hypergeometric Series

Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsomet al.for mock theta functions. Here we see that Mortenson's previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateralq-series with mixed mock modular behaviour is well suited for such radial limits. We present five more radial limit results, which follow from mixed mock modular bilateralq-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function of Gordon and McIntosh. The later bilateral series can be used to compute radial limits for many classical second-, sixth-, eighth- and tenth-order mock theta functions.

Spatial distributions of electromagnetic field variations and injection regions during the 20 November 2007 sawtooth event

We report multi-spacecraft and ground-based observations of a "sawtooth" event on 20 November 2007. For this event, data from three THEMIS, two GOES, and four LANL spacecraft are available as well as those from extensively distributed ground magnetometers and all-sky imagers. In the present paper we focus on the spatial extents of the electromagnetic and particle signatures of the first "tooth". In this event, auroral images and ground magnetic bays showed two activations: a pseudo onset and a major onset (we use the term pseudo onset since the former auroral brightening did not significantly expand poleward). Ground magnetic bay observations indicate that the substorm current wedge (SCW) developed after the major onset in an azimuthally wide region of ~14–3 h MLT. Similarly, broad magnetic bay distribution was observed also for the pseudo onset prior to the major onset. Furthermore, around the pseudo onset, magnetic dipolarisations were observed from 0.5 to 5 h MLT. These observations illustrate that, during sawtooth events, activities following not only the major onset but also the pseudo onset can extend more widely than those during usual substorms. Remarkable electromagnetic field fluctuations embedded in the dipolarisation trend were seen at 0.5 and 2.5 h MLT. In particular, comprehensive plasma and field data from THEMIS showed the presence of a long-excited weak magnetosonic wave and an impulsive large-amplitude Alfvén wave with an earthward Poynting flux at around the eastward edge of the SCW; the latter was sufficiently strong for powering aurora (140 mW/m2 when mapped to the ionosphere). These two activations of the electromagnetic wave were identified, corresponding to the pseudo onset and the major onset. On the other hand, the dipolarisation at geosynchronous 0 h MLT was observed only after the major onset, despite its closer location to the centre of the auroral activity in terms of the MLT; this indicates that the inner radial limit of the dipolarisation region at the pseudo onset was tailward of geosynchronous altitude at 0 h MLT. The outer radial limit of the electron injection region was also found at ~10 RE by conjunction measurements with THEMIS satellites. These radial distributions are not significantly different to those expected for usual substorms.