# Tooth surface construction of pure rolling internal meshing line gear based on cylindrical helix

On the basis of space curve meshing theory, the tooth surface construction of parallel shaft pure rolling internal meshing line gear has been described in detail. In this part, the cylindrical helix is used as the contact line of pinion tooth surface, and the condition of pure rolling meshing of parallel shaft internal meshing line gear is discussed.

In the coordinate system S1, the parameter equation of the contact line R1 on the tooth surface of the pinion is as follows:

Where: m is the helix radius of cylinder helix; N is the pitch coefficient, t is the variable parameter of cylinder helix. The helix angle of cylinder helix R1 is.

According to the formula, the principal normal vector of R1 at the contact point can be obtained.

According to the formula, the meshing equation of the contact point can be obtained.

T and t can be obtained by simplifying the formula θ1

According to the joint formula, the contact line parameter equation of the ring gear is as follows:

If M2 = m + e, t = t / i12, the formula can be transformed into:

Where: N2 = i12n, by comparing the formula, it is not difficult to find that the parameter equations of contact line R1 and contact line R2 are the same, they are cylindrical contact lines, and both are right-handed helix, as shown in Figure 1. The helix rising angle of cylinder helix R 2 is as follows:

In this case, the conjugate curve can keep tangent contact all the time; In order to realize the pure rolling engagement, it must meet the following requirements υ( 1) 12 = 0, the relevant parameters are brought into the formula as follows:

By simplifying the formula, we can get the following results

Combined with the formula, the helix angle of contact wire R2 can be expressed as:

According to the formula, when the center distance and transmission ratio of pure rolling parallel shaft internal meshing line gear are determined, the spiral radius of cylindrical helix is also determined. When the center distance and helix radius meet the formula, the helix angle of the driven helix and the driven helix is equal.

Through the above discussion, the pure rolling Meshing Conditions for selecting cylindrical helix as contact line are summarized

(1) The helix angle of contact wire R1 and contact wire R2 are equal and the rotation direction is the same;

(2) The helix radius m and center distance E satisfy the relation formula.

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