An involute curve is defined only outside the basic circle.
In an involute tooth profile, the locus of the contact point is the area where the standard tangent line \(EF\) of the basic circle is sandwiched by the tooth tip circles of both gears. If the tooth tip circle of Gear 1 greatly exceeds the F point of Gear 2, Gear 1 will come into contact with the non-involute portion of the tooth root of Gear 2, and regular rotation cannot be transmitted. This phenomenon is called interference. This phenomenon is called interference. This occurs when the tooth tip circle of the other gear exceeds the contact point \(F\) or \(E\) of the base circle, so \(E\) and \(F\) are called the interference points.
If a pinion cutter (gear-shaped cutter) is used instead of gear 1, the interfering part of the tooth root of gear 2 is cut away, and the involute part near the tooth bottom is also removed. This phenomenon is called undercut.
The larger the number of teeth in Gear 1, the easier it is to exceed the interference point \(F\), even though the size of the teeth is the same. Therefore, interference is most likely to occur when a rack is used. When creating a gear with a rack cutter, the teeth are created gradually from the tooth tips, and when the cutter reaches the interference point \(F\), the involute is entirely created up to the base circle. However, if the cutting edge of the cutter exceeds \(F\), the trajectory drawn by the angle of the cutting edge concerning the gear is a kind of trochoid, which gouges the tooth base, cutting away the involute that has been created, resulting in the phenomenon of devaluation. The actual cutting edge of a rack cutter has rounded edges, as shown in the dashed line, and devaluation occurs when the rounded edge of the straight cutting edge exceeds the interference point.
The smaller the number of teeth of the generated gear is, the more likely it is that devaluation will occur, and the critical number of teeth is calculated by the following equation.
$$ z_u=\frac{2h_k}{m\sin^2\alpha} $$
where \(h_k\): tooth end length, \(m\): module, \(\alpha\): pressure angle
Since \(h_k=m\) for parallel teeth,
$$ z_u=\frac{2}{\sin^2\alpha} $$
\(h_k=m\) for the parallel teeth, we have If \(\alpha=20^{\circ}\), then \(z_u=17.1\), so the minimum number of teeth is \(18\).
shift
To avoid devaluation, the cutter should be positioned farther from the gear center so that the edge of the straight portion of the cutter’s cutting edge does not exceed the interference point. This shifting of the cutter position is called shifting. Moving the cutter position away from the base circle from the standard position is called forward dislocation, and moving it closer to the base circle is called negative dislocation.
The gear created by this process is called profile-shifted gear. The amount of profile shifting is called the amount of profile shifting, and the value obtained by dividing the amount of profile shifting by the module is called the shifting coefficient of the profile.
Dislocation changes the position of the reference, tip, and root circles. The tooth thickness must be maintained at \(\displaystyle \frac{m}{2}\pi \). Positive inversion results in the use of a position where the curvature of the involute curve is greater, resulting in a tooth profile that is thicker at the root and thinner at the tip, which increases the bending strength of the tooth.
Width of tooth face
The width of the tooth tip varies with the amount of dislocation. As the amount of negative dislocation increases, the width of the tooth tip decreases, and at \(0\), the apex point is generated, after which the position of the apex point drops, and the tooth tip length cannot be maintained.
Taking the tooth-cutting pressure angle as \(\alpha_0\), the module as \(m\), the number of teeth as \(z\), and the dislocation coefficient as \(x\), we obtain the following equation.
$$ d_b=mz\cos\alpha_0, d=m(z+2x), d_a=d+2m=m(z+2+2x) $$
Considering the involute function,
$$ \theta=\cos^{-1}\frac{d_b/2}{d/2}=\cos^{-1}\frac{mz\cos\alpha_0}{m(z+2x)}=\cos^{-1}\frac{z\cos\alpha_0}{(z+2x)} $$
$$ \nu=\tan\theta-\theta $$
$$ \phi=\cos^{-1}\frac{d_b/2}{d_a/2}=\cos^{-1}\frac{mz\cos\alpha_0}{m(z+2+2x)}=\cos^{-1}\frac{z\cos\alpha_0}{(z+2+2x)} $$
$$ \mu=\tan\phi-\phi $$
Also, \(\gamma\) is for taking the tooth thickness equal to the tooth groove width,
$$ \gamma=\frac{2\pi}{4z}=\frac{\pi}{2z} $$
The angle \(2\sigma\) of the width of the tooth end is given by
$$ 2\sigma=\gamma+\nu-\mu $$
module
If the pitch is \(t\), the diameter of the pitch circle is \(d\), and the number of teeth is \(z\)
$$ t=\frac{\pi d}{z} $$
and the value of \(t\) becomes an irrational number, and even if it is stopped at the appropriate place after the decimal point, it will be a half number. For this reason, when \(d\) is expressed in units (mm),
$$ m=\frac{d}{z} $$
\(m\) is called the module and is used as the standard value for determining the dimensions of each part of the gear. For example, for a tooth with standard dimensions, i.e., a full-depth tooth, the height at the end of the tooth \(h_k\) equals \(m\). The value of the crest should be at least \(0.25m\), and so on. The value of \(m\) that should be used for actual gears is defined. The pitch is denoted by \(t=\pi m\). For a pair of gears that mesh with each other, the pitch must be equal, so \(m\) must also be equal.
If the diameters of the pitch circles of the two meshing gears are \(d_1\) and \(d_2\) and the number of teeth are \(z_1\) and \(z_2\), then.
$$ d_1=mz_1, \quad d_2=mz_2 $$
Therefore, if the center distance of the gear is \(a\), the following equation can be expressed.
$$ a = \frac{d_1}{2}+\frac{d_2}{2}=\frac{m}{2}(z_1+z_2)$$
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