ON THE DETERMINATION OF THE CONTACT RATIO OF GEARS

A. E. Belyaev

(Presented by the scientific seminar of the Department of Applied Mechanics)

In calculating gears for contact strength, all existing methods [1] [2] [4] use Hertz’s formula

\[ \sigma_k = 0.418 \sqrt{\frac{P_n}{L_k}\cdot \frac{E}{\rho}}, \tag{1} \]

where \(P_n\) is the normal force between the teeth,
\(L_k\) is the nominal length of the contact lines,
\(E\) is the reduced modulus of elasticity,
\(\rho\) is the reduced radius of curvature.

The quantity \(L_k\) is determined by the formula

\[ L_k=\frac{B\cdot \varepsilon_s}{\cos \beta_0}, \tag{2} \]

where \(B\) is the working face width of the gear wheels,
\(\varepsilon_s\) is the contact ratio,
\(\beta_0\) is the helix angle of the teeth on the base cylinder.

The contact ratio \(\varepsilon_s\) entering formula (2) is determined by the expression

\[ \varepsilon_s=\frac{b_1\cdot b_2}{t_{os}}=\frac{b_1p+b_2p}{t_{os}}, \tag{3} \]

where \(b_2p;\ b_1p\) are portions of the line of action,
\(t_{os}\) is the base pitch in the transverse section.

The latter quantity, in general form, can be determined from the formula

\[ t_{os}=\pi m_s\cdot \cos \alpha_{os}, \tag{4} \]

where \(m_s\) is the module of engagement in the transverse section,
\(\alpha_{os}\) is the profile angle of the generating rack in the transverse section.

According to the method [2], the quantities \(b_1p\) and \(b_2p\) can be expressed through the pressure angle in the transverse section \(\alpha_s\) and the diameters of the reference circles \(d\).

Thus,

\[ b_1p=0.5\left[ \sqrt{ \left(\frac{d_1}{h_1'}\cdot \sin \alpha_s\right) +4\left(\frac{d_1}{h_1'}+1\right) } -\frac{d_1}{h_1'}\cdot \sin \alpha_s \right]\cdot h_1' = \]

\[ = e_1\cdot h_1, \tag{5} \]

where by the quantity \(e_1\) is meant the expression in this formula that is enclosed in square brackets.

Similarly,

\[ b_2 p = e_2 h'_2 . \tag{6} \]

In formulas 5 and 6, \(h'_1\) and \(h'_2\) are the addendum heights of the pinion and the gear, respectively.

To simplify finding the contact ratio \(\varepsilon_s\) by formula 3, the method of [2] recommends a graph that makes it possible to find the values of \(e\) as a function of the ratio \(\dfrac{d}{h'}\) for various values of \(\alpha_s\). Determination of \(\varepsilon_s\) is reduced to determining the value of \(\dfrac{d}{h}\) and, from the found value of \(\alpha_s\), to determining (with the aid of the graph) the coefficient \(e\). After calculating the base pitch (formula 4), the coefficient \(\varepsilon_s\) is found.

Taking the method of [2] as a basis, one can simplify the determination of the contact ratio. The contact ratio \(\varepsilon_s\) [3] can be represented as the sum of the partial contact ratios of zones I and II (Fig. 1).

Fig. 1. For determining the contact ratio \(\varepsilon_s\)

\[ \varepsilon_s = \frac{b_1p+b_2p}{t_{os}} = \frac{b_1p}{t_{os}}+\frac{b_2p}{t_{os}} = \varepsilon_{\mathrm{II}}+\varepsilon_{\mathrm{I}}, \tag{7} \]

where \(\varepsilon_{\mathrm{I}}\) is the partial contact ratio of zone I, where engagement occurs between the dedendum flank of a tooth of the first gear and the addendum of a tooth of the second gear;

\(\varepsilon_{\mathrm{II}}\) is the partial contact ratio of zone II, where engagement occurs between the addendum of a tooth of the first gear and the dedendum flank of a tooth of the second gear.

Substituting the expressions for \(b_1p\) and \(b_2p\) (formulas 5 and 6) and \(t_{os}\) (formula 4) successively into the last equation, we obtain (after some transformations) the expression for the partial contact ratios \(\varepsilon_{\mathrm{I}}\) and \(\varepsilon_{\mathrm{II}}\):

\[ \varepsilon_{\mathrm{I}} = \frac{\cos \beta_0}{\pi \cdot \cos \alpha_{os}} \cdot e_2 \cdot \frac{h'_2}{m_n} = \Theta_{\varepsilon_{\mathrm{I}}}\cdot \frac{h'_2}{m_n}, \]

\[ \varepsilon_{\mathrm{II}} = \frac{\cos \beta_0}{\pi \cdot \cos \alpha_{os}} \cdot e_1 \cdot \frac{h'_1}{m_n} = \Theta_{\varepsilon_{\mathrm{II}}}\cdot \frac{h'_1}{m_n}, \]

where

\[ \Theta_{\varepsilon_{\mathrm{I}}} = \frac{\cos \beta_0}{\pi \cdot \cos \alpha_{os}}\cdot e_2, \qquad \Theta_{\varepsilon_{\mathrm{II}}} = \frac{\cos \beta_0}{\pi \cdot \cos \alpha_{os}}\cdot e_1. \]

\[ \Theta_\varepsilon=\frac{\cos \beta_0}{\pi\cdot \cos \alpha_{0s}}\cdot \varepsilon \]

Fig. 2. Graph for finding the value \(\Theta_\varepsilon\).

Thus,

\[ \varepsilon_s=\varepsilon_I+\varepsilon_{II} =\Theta_{\varepsilon I}\cdot\frac{h_2'}{m_n} +\Theta_{\varepsilon II}\cdot\frac{h_1'}{m_n}. \tag{8} \]

The coefficients \(\Theta_\varepsilon\) for both zones are determined with allowance for the coefficient \(e\). To simplify its determination, a graph has been constructed (Fig. 2). From the found value of the ratio \(\frac{h'}{m_n}\), for various angles of inclination of the tooth on the base cylinder \(\beta_o\), the coefficient \(\Theta_\varepsilon\) is found. In this case formula 8 is not transformed, since the found ratios \(\frac{h_2'}{m_n}\) and \(\frac{h_1'}{m_n}\) are also used for the graphical determination of the coefficient \(\Theta_\varepsilon\) and the coefficient \(\varepsilon_s\).

In the particular case when the gear train is uncorrected (i.e., the height of the tooth addenda is equal to the magnitude of the normal module of the mesh \((h'=m_n)\)), the coefficient \(\Theta_\varepsilon\) is found simply from the number of teeth \(Z\), and the expression for the contact ratio coefficient \(\varepsilon_s\) proves to be equal to

\[ \varepsilon_s=\Theta_{\varepsilon I}+\Theta_{\varepsilon II}. \]

REFERENCES

1. Ya. G. Kistyan. Method for Calculating Gear Transmissions for Strength, TsNIITMASH, book 107, Mashgiz, Moscow, 1963.
2. V. N. Kudryavtsev. Gear Transmissions, Mashgiz, Moscow, 1957.
3. V. N. Kudryavtsev. Simplified Calculations of Gear Transmissions, Moscow, Mashgiz, 1960.
4. A. I. Petrusevich. Gear transmissions, in the book Machine Parts, ed. by N. S. Acherkan, book 1, Mashgiz, Moscow, 1954.