# Interpolation mechanism of differential eccentric gear system based on unequal division model

Two identical standard involute circular gears are installed eccentrically at their respective axes O1 and O2 in the same phase, with an axis distance of a and an eccentricity of E = lo1c1 = lo2c2. In the initial position, the geometric centers C1 and C2 of the two gears are biased to the horizontal left of the axis, and the paired teeth of the two wheels (marked 0 and 9 in the figure) mesh at the intersection K1 of their working tooth profile and the inner common tangent N1N2, as shown in Fig. 1A. Taking the rotation center O1 of driving gear 1 as the coordinate origin, a fixed coordinate system xo1 y is established, as shown in Fig. 1b. When driving gear 1 turns φ 1 angle, driven gear 2 turns φ 2 horns, φ 1 start with the negative direction of x-axis and turn clockwise to o1c1 measurement (negative value), φ 2 take O2 A2 as the starting edge and turn counterclockwise to O2 C2 measurement (positive value). A2 point is the intersection of the tooth shape coordinate starting point of the mating tooth and the connecting line of the center of the driven axle on the base circle during the initial meshing of the driven gear 2. According to the bisection model, the working tooth profile of paired teeth will cross or separate. Order φ The initial value of 2 is the solution of the bisection model φ~ 2 is:

Where ε——— Gear eccentricity

If driving gear 1 turns φ 1 is fixed in angular increments Δφ Rotate the driven gear 2 slightly to continuously obtain the changing tangent coordinates, so that the intersection K1 of the internal common tangent N1N2 and the tooth profile working tooth profile of the paired teeth of the two wheels coincides with K’1 or K2 and K’2, or K1 coincides with K’1 and K2 coincides with K’2 at the same time, so as to obtain the actual rotation angle of the driven gear φ 2。 In this way, the driven gear angle φ 2 can be expressed as:

The formula is an unequal model of eccentric involute gear transmission, which needs to be solved iteratively by numerical method. Δφ The “±” before 12 indicates the relative initial solution of driven gear 2 φ~ 2 turns along wheel 2 (forward rotation) and against wheel 2 (reverse rotation). Take “+” for forward rotation and “-” for reverse rotation.

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