Determination of basic parameters of the wave gearings with intermediaterolling bodies

In article is described the constructions wave gearings with intermediate rolling bodies. The main relationships between the size of details transmission are proved, the forces exert on intermediate rolling bodyare determined, and general force laws are given. It is shown that rotational torque from cam to separator or centerwheel, can be transfer by intermediate rolling body only insofar as angles γ and ψ are greater 0° and less 90° together.


Introduction
Nowadays the wave gearings with intermediate rolling bodies are gain grounded in oil and gas industry, as well as in other industrial settings (aerospace, lift and carryetc.) [1][2][3][4]. Introduction of them into constructions make it possible to design machines with improved capability, because they have a wide range of transmission rate, action flexibility and multiple tooth contact gearing [5][6][7]. However, insufficient information obstructs their production and wide spread occurrence.

Theoretical background
Constructions wave gearings with intermediate rolling bodies are right amount, in the paper, a power transmission on the ground of mechanism present on the fig. 1 was considered.
Mechanism works following manner: during the propulsion to eccentric disk of cam 1, the intermediate rolling bodies 4 are received radial motion in mortices of separator 3 and rolled wavelike on toothing of center wheel 2. In the wake of such movement, the intermediate rolling elements 4 can revolve or separator 3, or wheel 2, depending of what detail is fixed in gear box.
As far as the geometric axes of the intermediate rolling bodies change positions in space and output element can be center wheel or separator, as such power transmission fits into differential-planet gears, which has two degrees of freedom.

Basic geometrical relationships in power transmission
The majority elements of details of power transmission can be procure base geometrical practice, a different matter stand with crown end of center wheel. We determine the parametric equations of curve, which governs tooth space of crown end of center wheel.
As far as intermediate rolling body undergoes different motion, as expand him on radial and tangential components. (sin tgψcos ) The peripheral component of conveyance speed come out of angular motion of separator Where z 2 is the number of teeth of center wheel Velocity vector of moving of center of rolling element for center wheel It follows by the gearing theory that line of action of relative velocity vector is perpendicular line of base tangent, as carried out to the joint surfaces in the point of contact.Let us call the normal vector to the joint surfaces n (fig. 2)

Transmission ratio
Let us derive the formula for determination of transmission ratio such mechanism, deprive of him single degree of freedom using latching control of center wheel. Because the mechanism fall into epicyclic gear train, as for determination the formula, which bind the corner frequencies ω 1 and ω 3 , let us use a Willis's method [8][9][10][11][12][13], which based on a concept planet carrier shutdown. We call the cam as planet carrier and draw up a table of the corner frequencies of crew of unreversed and reversed mechanism.

Table 1.The corner frequencies of crew of unreversed and reversed mechanism Parts
Keep of mechanism 1 2 3 Unreversed Let us define the transmission ratio from third part to second part for reversed mechanism where the transmission ratio from separator 3 to cam 1 for unreversed mechanism is plugging the (2) in (1), we get Transmission ratio from cam 1 to separator 3 is ( ) 13 1 where ( Found expression shows, that in case of a positive difference between lot intermediate rolling bodies and number of teeth of center wheel, sense of rotation of cam and separator will be coincide and conversely, in negative difference, how on fig. 1, sense of rotation will be the other way round.
Different sense of rotation of separator will action on part of epicycloid, which circumscribethe tooth space of toothing of center wheel, will be contained in force interaction with intermediate rolling body. Proceeding from this formula, during the known geometry and rotational power Т 1 on the output link, we can determine the forces from following formula: where ψ is the angle between line of action Р and axis Оу (Fig. 3

Conclusion
Following from the gotten formulas, rotational torque from cam to separator or centerwheel, can be transfer by intermediate rolling body only insofar as angles γ and ψ are greater 0° and less 90° together. The gotten formulas also illustrate, that during other factors being equal, the most favorable construction is been when output link is separator and center wheel is fixed in case of power transmission, by transmit drive's eyesight.