# Generation of helical gear profile curve and helix

To establish a curve in CAD, we first need to establish the parameter equation of the curve. The involute equation of helical gear profile is a polar coordinate equation with the rotation center of helical gear as the pole. However, it is difficult to construct the tooth profile curve by using the polar coordinate equation in UG, and the result is inaccurate, so the polar coordinate equation will not be discussed. In order to establish accurate tooth profile curve in CAD, it is necessary to deduce the rectangular coordinate parameter equation of involute tooth profile according to the generation principle of involute. The tooth profile of the standard helical gear is an involute, as shown in Figure 1. When the straight line BK makes pure rolling motion along the circumference with radius BR, the track AK of any point K on the straight line is the involute of the circle. This circle is the base circle of involute, BR is the radius of the base circle; Straight line BK is the generating line of involute; horn θ k. That is ∠ AOK, which is called the spread angle of point K on the involute. In Figure 2, l = L ‘, l = RB* ϕ ， In the coordinate system shown in the figure, the rectangular coordinate equation of involute is:

The calculation equation of (XK, YK) coordinate value is:

According to the formula, the involute equation of tooth profile in rectangular coordinate system is:

The formation of the helix is shown in Figure 3. The oblique line KK on the occurrence surface is no longer parallel to the base cylinder bus, but deviates by an angle relative to the base cylinder bus β ， When the surface makes pure rolling motion around the base cylinder, the intersection line between any section of the vertical gear axis and the tooth profile surface is involute, and the collection of these involutes is involute helicoid. The part of the involute helical surface within the tooth top cylinder is the tooth profile surface of the helical gear. At the same time, the intersection of cylindrical surfaces with different radii and tooth profile surface is a helix, β It is called the helix angle on the helical gear base cylinder β The larger, the more skewed the tooth direction of the gear teeth, when β = 0, it becomes a spur gear, so the spur gear can be regarded as a special case of helical gear.

According to the formation principle of involute helicoid above, the involute parameter equation in UG can be obtained as follows:

Of which:

In the above formulas,

U – Involute expansion angle (angle system);

S – Involute expansion angle (radian system);

T – UG system variable, t = 0.

As for the helix, in UG, you can directly use the “helix” tool in the toolbar to generate it. The number of revolutions of the helix is 0.05. As long as the length of the helix is greater than the tooth width. Since the helix angle of helical gear refers to the angle between the tangent line of helix on the indexing circle and the axis, the radius is taken as the radius of the indexing circle: R D = / 2, and the pitch is p d = π β Tan, and finally input the rotation direction according to the helical gear parameters.

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