The dedendum tooth profiles of the internal teeth and external teeth of a strain wave gearing are prescribed by a first homothetic curve BC and a second homothetic curve AC obtained from a curve segment from a point A, at which the angle formed by the tangent to a movement locus Mc when meshing is approximated by rack meshing and the major axis is A, to a low point B. The dedendum tooth profile of the internal teeth is prescribed by a curve formed on the internal teeth in the course of the addendum tooth profile of the external teeth moving from an apex of the movement locus to point A. The dedendum tooth profile of the external teeth is prescribed by a curve formed on the external teeth when the addendum tooth profile of the internal teeth moves from the apex to arrive at point A.

A plastic helical gear has a three-dimensional tooth surface modification section on a tooth surface of each of involute-shaped teeth. The three-dimensional tooth surface modification section is a combined surface of a tooth top modification surface and an arc crowning surface. The tooth top modification surface is reduced in tooth thickness from a position between a tooth top and a tooth root toward the tooth top. The arc crowning surface is reduced in tooth thickness from a position between one end in the tooth width direction and the other end in the tooth width direction toward both ends in the tooth width direction. A line extending from an intersection point P0 between the starting position of the tooth top modification surface and the apex position of the arc crowning surface to the tooth root is aligned with a line on the tooth surface of the involute-shaped tooth.

A planetary gear device configured by combining a plurality of planetary gear mechanisms includes first and second planetary gear mechanisms sharing a carrier, wherein each planetary gear mechanism is composed of an internal gear I.sub.k (k is an integer equal to or larger than 2) and a planetary gear P.sub.k which is engaged with the internal gear I.sub.k and revolves in a circumferential direction of the internal gear, the planetary gear P.sub.k of each planetary gear mechanism is composed of a spur gear in the form of an external gear, the planetary gears P.sub.k of the planetary gear mechanisms share a central axis or have central axes integrally connected to integrally rotate on a common rotation central axis line or are integrated with each other to integrally rotate on the common rotation central axis line in order to configure the entire planetary gear device as a two-stage gear mechanism, the planetary gear device is configured such that the number of teeth z.sub.p1 of a first planetary gear constituting the first planetary gear mechanism and the number of teeth z.sub.p2 of a second planetary gear constituting the second planetary gear mechanism are different from each other, the number of teeth on the internal gear I.sub.1 is z.sub.i1, and the number of teeth on the internal gear I.sub.2 is z.sub.i2, an addendum modification coefficient of the first planetary gear is x.sub.p1, an addendum modification coefficient of an internal gear which is engaged with the first planetary gear and constitutes the first planetary gear mechanism is x.sub.i1, an addendum modification coefficient of the second planetary gear is x.sub.p2, an addendum modification coefficient of an internal gear which is engaged with the second planetary gear and constitutes the second planetary gear mechanism is x.sub.i2, a power transmission efficiency of the planetary gear device having the addendum modification coefficients x.sub.p1, x.sub.i1, x.sub.p2, and x.sub.i2 is , an addendum modification coefficient of the internal gear I.sub.1 is x.sub.i1, and an addendum modification coefficient of the internal gear I.sub.2 is x.sub.i2, and the addendum modification coefficients have relationships in which values selected from combinations of the addendum modification coefficients which maximize or submaximize the power transmission efficiency within an allowable range of design specifications given in advance are combined.

A drive arrangement for a bicycle may be provided with various interactive components configured to reliably and repeatably engage and disengage with one another both when new and after a period of wear. The drive arrangement may include a drive sprocket assembly connected to a driven sprocket assembly with a chain movable by a gear changer.

A flexible gear coupling includes two external gears and two internal gears meshing with the respective two external gears. A tooth root crowning radius of each external gear is smaller than a tooth tip crowning radius of the external gear. The external gear is formed such that a reference tooth height that is a tooth height at a tooth width direction middle position is smaller than an end tooth height that is a tooth height at a tooth width direction end position. A ratio of the end tooth height to the reference tooth height is set to 1.21 or more, and/or a ratio Rc/Rb of the tooth root crowning radius to the tooth tip crowning radius is set to 0.37 or less.

A resin helical gear is formed by setting a first machining reference line obliquely coupling a tooth tip side of a first tooth to a tooth root side of a second tooth on another end side in the tooth width direction along a tooth surface, and a second machining reference line obliquely coupling a tooth tip side of the second tooth to a tooth root side of the first tooth along the tooth surface. Then, the tooth surface is cut out from the first machining reference line to the tooth root of the first tooth while the tooth surface is cut out from the second machining reference line to the tooth root of the second tooth. Then, an involute tooth profile form is left on a tooth tip side of the tooth with respect to the first machining reference line and the second machining reference line.

A method for generating a tooth space profile between two teeth of a wheel engaging with a roller pinion in a reference plane perpendicular to a central axis (X1) whose intersection with the reference plane defines a centre (C). The method comprises defining a nominal tooth space with symmetrical first and second profiles so that the space is suitable for receiving a roller which comes into contact, without clearance, with the facing active portions of the first and second profiles at the level of a primitive diameter. The method further comprises the steps of deforming the bottom portion of the first profile so that the bottom point is brought nearer to the centre by an amount equal to a given radial clearance, and moving the first profile angularly through a given angular half-clearance around the centre to shift it away from the radial axis.

Each of an external tooth-bottom surface and an external bottom-side contact surface has a cross-sectional shape formed by a hypocycloid curved line. An external front-side contact surface has a cross-sectional shape formed by an epicycloid curved line. An external tooth-front surface is formed at a radial-inside position of a reference epicycloid curved line, that is, a position closer to a first pitch circle of an external gear. A possible contact between a tooth-front portion of the external gear and a tooth-front portion of an internal gear can be avoided. A connecting point between the external tooth-front surface and the external front-side contact surface is located at such a position that a contact surface length of an external tooth-contact surface is larger than twenty percent of a total tooth surface of the external gear.

An object of the present invention is to provide a sprocket that mitigates the impact of tension fluctuations resulting from load torque changes, suppresses noise and vibration, ensures stable chain behavior, prevents sprocket durability loss and the generation of harmonics during rotation, along with a chain drive system. Tooth roots between adjacent teeth of the sprocket include those with a radial deviation from a root circle of a standard tooth profile, set within a predetermined numerical range corresponding to a tooth pitch. A phase variation pattern is created by changing a root radius in accordance with an angular position. The phase variation pattern has a waveform with an amplitude varied within a range of ( 1/7)Amax, where Amax represents a maximum amplitude. The chain drive system includes a plurality of sprockets and a chain passed over the sprockets, at least one of them being the above-described sprocket.

In a bevel gear pair, a first gear (G1) and a second gear (G2) are applied with tooth top modification, and a ratio of a distance (L1b) from a pitch circle (P1) to a starting position (R1) of the tooth top modification to a distance (L1a) from the pitch circle (P1) to a tooth top (T1) in the first gear (G1) is larger than a ratio of a distance (L2b) from a pitch circle (P2) to a starting position (R2) of the tooth top modification to a distance (L2a) from a pitch circle (P2) to a tooth top (T2) in the second gear (G2). Thus, the bevel gear pair that can achieve smooth meshing is provided.