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Study of the Omnidirectional Driving Gear Corresponding to Free-Form Surfaces: —Basic Operating Characteristics of a Drive Unit with Two Curvatures—
Abstract
Omnidirectional driving gear is a proprietary technology designed and possessed by Tadakuma laboratory in Yamagata University. It is a next generation gear mechanism having two degrees of freedom on one gear surface. We have newly developed a free form omnidirectional driving gear by combining flat and convex arc surfaces. A v-grove rail guide mechanism is adopted to enable stable horizontal motion along the planar gear surface and rotational motion round the convex gear surface. With this setup, we were able to design and operate an omnidirectional driving gear corresponding to a free-form surface. Although this mechanism adopt rail and cantilever structure, it was confirmed by experiments that its performance is comparable to that of the conventional flat plate omnidirectional driving gear. In this design, the motion range is small and is limited to the shape of the curvature. We plan to come up with a 2nd prototype design which will be more flexible and that will be able to achieve sophisticated motion path trajectories.
The torque variation of gears is caused by mesh error and tooth friction. As the torque variation due to mesh error is given by differentiation of mesh error, the torque variation due to tooth friction can be obtained experimentally by subtracting the derivative of mesh error from total torque variation. The wave forms of torque variation due to tooth friction give the reliable and detailed informations of tooth contact, such as the ratio of share of tooth load, the alternative contact of two pairs of meshing teeth, the coefficient of friction of tooth surface and others.The main conclusions in this study are as follows:1) The mean peak values of torque variations retain about a constant value independently of any contact ratios in this experiment.2) Practically, only one pair of tooth is in contact independently of any contact ratios.3) The mean peak values of total torque variations are not affected by the torque variations due to mesh errors.
It is difficult to obtain a characteristic, periodical signal of gears, that is, the transmission error, as many irregular signals due to lubrication, frictional losses and so on are relatively larger and involved. This report tries to obtain the transmission error by processing method with average, using a new developed machine which can directly and rapidly show an error spectrum and phases of the static transmission error.(1) The processing method with average is very useful to obtain the transmission error of gears during rotation.(2) The limit of rotational speed to measure the static transmission error of the above mentioned machine is about 300 rpm, and agrees with the theoretical limit. Therefore the limit can be estimated theoretically in general case.(3) The dynamic motion of gears, which have a known static transmission error, obtained by the experiment agrees with the theoretical motion. Therefore the measured values by the above mentioned machine are very useful to estimate any motion of gears.
In the present paper the authors studied the position of the weakest section of gear tooth and the stress concentration factor at the tensile-side fillet of tooth. Then the authors calculated the stress concentration factors for some components of the force on the tooth profile, and found the relation among these factors. Consequently the following formula has been introduced for the bending stress of gear tooth : σt=(1+0.08 S/ρ){0.66σNb+0.40√(σNb2+36τN2)+1.15σNc} By this formula the authors calculated the bending stress of gear tooth of some shapes, and compared the result with that of photoelastic experiments.
Gear vibration is induced by the excitation due to manufacturing/alignment errors and periodical chasnge of meshing tooth stiffness with progress of gear rotation. As an index for evaluating the degree of gear vibration and noise as functions of gear dimensions, accuracy and driving conditions, transmission error of gears and total vibrational excitation are widely accepted. This report clarifies that the influence of transmission error and that of total vibrational excitation on gear vibration are equal in their first order frequency component, and the difference between them is due to the higher order frequency components of the periodical change of meshing tooth stiffness. The difference in the degree of influence of these two indices on the relative rotational vibration of gears is discussed using numerical examples. Propriety between both indices is also discussed to predict the actual gear noise of automotive power transmission of personal cars in production.
