Determining Gear Tooth and Gear Mesh Parameters.
Find the gear ratio, circular pitch, base pitch, pitch diameters, pitch radii, center distance, addendum, dedendum, whole depth, clearance, outside diameters, and contact ratio of a gearset with the given parameters. If the center distance is in‑ creased 2% what is the new pressure angle and increase in backlash?

Question

Given: A 6 [latex] p_{d} [/latex], 20° pressure angle, 19-tooth pinion is meshed with a 37-tooth gear.

Assume: The tooth forms are standard AGMA full-depth involute profiles.

Accepted Answer

1 The gear ratio is found from the tooth numbers on pinion and gear using equations 9.5a and 9.5c.

[latex] m_{V}=\pm \frac{d_{i n}}{d_{o u t}}=\pm \frac{N_{i n}}{N_{o u t}} [/latex] (9.5a)

[latex] m_{G}=\left|m_{V} ight| ext { or } m_{G}=\left|m_{T} ight|, ext { for } m_{G} \geq 1 [/latex] (9.5c)

[latex] m_{G}=\frac{N_{g}}{N_{p}}=\frac{37}{19}=1.947 [/latex] (a)

2 The circular pitch can be found either from equation 9.4a or 9.4d.

[latex] p_{c}=\frac{\pi d}{N} [/latex] (9.4a)

[latex] p_{d}=\frac{\pi}{p_{c}} [/latex] (9.4d)

[latex] p_{c}=\frac{\pi}{p_{d}}=\frac{\pi}{6}=0.524 ext { in } [/latex] (b)

3 The base pitch measured on the base circle is (from equation 9.4b):

[latex] p_{b}=p_{c} \cos \phi [/latex] (9.4b)

[latex] p_{b}=p_{c} \cos \phi=0.524 \cos \left(20^{\circ} ight)=0.492 ext { in } [/latex] (c)

4 The pitch diameters and pitch radii of pinion and gear are found from equation 9.4c.

[latex] d_{p}=\frac{N_{p}}{p_{d}}=\frac{19}{6}=3.167 ext { in, } \quad r_{p}=\frac{d_{p}}{2}=1.583 ext { in } [/latex] (d)

[latex] d_{g}=\frac{N_{g}}{p_{d}}=\frac{37}{6}=6.167 ext { in, } \quad r_{g}=\frac{d_{g}}{2}=3.083 ext { in } [/latex] (e)

5 The nominal center distance C is the sum of the pitch radii:

[latex] C=r_{p}+r_{g}=4.667 ext { in } [/latex] (f)

6 The addendum and dedendum are found from the equations in Table 9-1:

[latex] a=\frac{1.0}{p_{d}}=0.167 ext { in, } \quad b=\frac{1.25}{p_{d}}=0.208 ext { in } [/latex] (g)

7 The whole depth [latex] h_{t} [/latex] is the sum of the addendum and dedendum.

[latex] h_{t}=a+b=0.167+0.208=0.375 ext { in } [/latex] (h)

8 The clearance is the difference between dedendum and addendum.

[latex]c=b-a=0.208-0.167=0.042 ext { in }[/latex] (i)

9 The outside diameter of each gear is the pitch diameter plus two addenda:

[latex] D_{o_{p}}=d_{p}+2 a=3.500 ext { in, } \quad D_{o_{g}}=d_{g}+2 a=6.500 ext { in } [/latex] (j)

10 The contact ratio is found from equations 9.2 and 9.6a.

[latex]\begin{aligned}Z=& \sqrt{\left(r_{p}+a_{p} ight)^{2}-\left(r_{p} \cos \phi ight)^{2}}+\sqrt{\left(r_{g}+a_{g} ight)^{2}-\left(r_{g} \cos \phi ight)^{2}}-C \sin \phi \=& \sqrt{(1.583+0.167)^{2}-\left(1.583 \cos 20^{\circ} ight)^{2}} \&+\sqrt{(3.083+0.167)^{2}-\left(3.083 \cos 20^{\circ} ight)^{2}}-4.667 \sin 20^{\circ}=0.798 ext { in } \m_{p}=& \frac{Z}{p_{b}}=\frac{0.798}{0.492}=1.62\end{aligned}[/latex] (k)

11 If the center distance is increased from the nominal value due to assembly errors or other fac‑ tors, the effective pitch radii will change by the same percentage. The gears’ base radii will remain the same. The new pressure angle can be found from the changed geometry. For a 2% increase in center distance (1.02x):

[latex] \phi_{ ext {new }}=\cos ^{-1}\left(\frac{r_{ ext {base circle }_{p}}}{1.02 r_{p}} ight)=\cos ^{-1}\left(\frac{r_{p} \cos \phi}{1.02 r_{p}} ight)=\cos ^{-1}\left(\frac{\cos 20^{\circ}}{1.02} ight)=22.89^{\circ} [/latex] (l)

12 The change in backlash as measured at the pinion is found from equation 9.3.

[latex] heta_{B}=43200(\Delta C) \frac{ an \phi}{\pi d}=43200(0.02)(4.667) \frac{ an \left(22.89^{\circ} ight)}{\pi(3.167)}=171 ext { minutes of arc } [/latex] (m)

[latex]\begin{aligned}& ext { TABLE 9-1 AGMA Full-Depth Gear Tooth Specifications }\&\begin{array}{lll}\hline { ext { Parameter }} & ext { Coarse Pitch } \left( p_{ d }< 2 0 ight) & ext { Fine Pitch }\left(p_{ d } \geq 2 0 ight) \\hline ext { Pressure angle } \phi & 20^{\circ} ext { or } 25^{\circ} & 20^{\circ} \ ext { Addendum } a & 1.000 / p_{d} & 1.000 / p_{d} \ ext { Dedendum } b & 1.250 / p_{d} & 1.250 / p_{d} \ ext { Working depth } & 2.000 / p_{d} & 2.000 / p_{d} \ ext { Whole depth } & 2.250 / p_{d} & 2.200 / p_{d}+0.002 ext { in } \ ext { Circular tooth thickness } & 1.571 / p_{d} & 1.571 / p_{d} \ ext { Fillet radius-basic rack } & 0.300 / p_{d} & ext { Not standardized } \ ext { Minimum basic clearance } & 0.250 / p_{d} & 0.200 / p_{d}+0.002 ext { in } \ ext { Minimum width of top land } & 0.250 / p_{d} & ext { Not standardized } \ ext { Clearance } ext { shaved or ground teeth) } & 0.350 / p_{d} & 0.350 / p_{d}+0.002 ext { in } \\hline\end{array}\end{aligned}[/latex]

Question 9.1: Determining Gear Tooth and Gear Mesh Parameters. Find the ge...

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