Influence of the Relative Width of Gears on the Volume of Gear-Cutting Work

A. E. Belyaev

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

The tendency to reduce the increase in the nonuniformity of load distribution with increasing gear width limits the latter. Therefore, at present designers give preference to comparatively narrow gears, whose relative width \(\psi\) (the ratio of the width \(B\) to the center distance \(A\)) in completed transmissions varies over wide limits \((0.2 \div 1.2)\). In this connection it is of interest to estimate the volume of gear-cutting work for gears with different values of \(\psi\).

An estimate of the volume of gear-cutting work \(V_{\text{з.р}}\) can be determined by the relation [1]

\[ V_{\text{з.р}}=\Omega \cdot B \cdot d_{\text{д}}, \tag{1} \]

where \(\Omega\) is the coefficient of proportionality;
\(d_{\text{д}}\) is the pitch-circle diameter;
\(B\) is the width of the gear rim.

The quantity \(d_{\text{д}}\) is proportional \((\Omega')\) to the center distance \(A\) \((d_{\text{д}}=\Omega' \cdot A)\). Thus,

\[ V_{\text{з.р}}=\Omega \cdot B \cdot \Omega' \cdot A. \tag{2} \]

According to the procedure [3], the center distance can be expressed by the equation

\[ A=(i+1)\cdot \sqrt{\frac{M_{1}\cdot K_{k}}{2\cdot i \cdot [C_{k}]\cdot \vartheta_{k}\psi}}, \tag{3} \]

where \(M_{1}\) is the torque on the pinion;
\(i\) is the gear ratio;
\(K_{k}\) is a coefficient taking into account the nonuniformity of load distribution over the width of the gear rim and additional dynamic loads in calculation for contact strength;
\([C_{k}]\) is the allowable value of the contact-stress coefficient;
\(\vartheta_{k}\) is a coefficient characterizing the increased load-carrying capacity of helical transmissions;
\(c_{k}\) is a coefficient taking into account the increase in load-carrying capacity of corrected transmissions.

For any transmissions being compared with different values of \(\psi\) (\(\psi_1\) and \(\psi_2\)), formula 2 will be

\[ \frac{(V_{\text{з.р}})_{\psi_1}}{(V_{\text{з.р}})_{\psi_2}} = \frac{B_{\psi_1}\cdot A_{\psi_1}}{B_{\psi_2}\cdot A_{\psi_2}}. \tag{4} \]

Here \(B_{\psi_1}\), \(B_{\psi_2}\), \(A_{\psi_1}\), \(A_{\psi_2}\) are, respectively, the widths and center distances of the two transmissions being compared (with \(\psi=\psi_1\) and \(\psi=\psi_2\)).

Taking into account the nature of the comparison, the center distances may be written as

\[ A_{\psi_1}=\theta\cdot \sqrt[3]{\frac{1}{\psi_1}} \quad \text{and} \quad A_{\psi_2}=\theta\cdot \sqrt[3]{\frac{1}{\psi_2}}, \tag{5 and 5′} \]

where

\[ \theta=(i+1)\cdot \left( \frac{M_1\cdot K_k}{2\cdot i [C_k]\cdot \vartheta_k\cdot \varphi_k} \right)^3 . \tag{6} \]

Let us note that when comparing transmissions of one and the same stage of a reducer, the coefficients \(\theta\) of the transmissions being compared will be practically identical; when comparing high-speed and low-speed stages, with other conditions equal, in formulas (5 and 5′) there will appear the values \([C_k]\) corresponding to that stage of the reducer which has a specified value of \(\psi\). However, for two- and even three-stage reducers, at identical levels of the quantities \([C_k]\), which is one of the main tasks in splitting the gear ratio (see, for example, GOST 2185—55 [4]), the difference in speed (at \(V=3\div 15\) m/sec) does not exert any appreciable influence, and consequently \([C_k]\) may be taken as identical.

Substituting expressions (5 and 5′) into (4) and replacing \(B_{\psi_1}=A_{\psi_1}\cdot\psi_1\) and \(B_{\psi_2}=A_{\psi_2}\cdot\psi_2\), we obtain

\[ \frac{(V_{\text{з.р}})_{\psi_1}}{(V_{\text{з.р}})_{\psi_2}} = \frac{\psi_1\cdot A_{\psi_1}\cdot \sqrt[3]{\dfrac{1}{\psi_1}}} {\psi_2\cdot A_{\psi_2}\cdot \sqrt[3]{\dfrac{1}{\psi_2}}} = \sqrt[3]{\frac{\psi_1}{\psi_2}}. \]

In enclosed transmissions of general machine building, \(\psi=0.2\div 0.4\) is adopted. If, for example, one passes from \(\psi=0.4\) to \(\psi=0.3\), then

\[ (V_{\text{з.р}})_{\psi_1}\simeq 0.9(V_{\text{з.р}})_{\psi_2}, \]

i.e., the volume of gear-cutting work decreases by approximately \(10\%\).

REFERENCES

1. A. E. Belyaev. On the volume of gear-cutting work in the transition from herringbone transmissions to helical ones. Tekhnologiya sudostroeniya, No. 6, Sudpromgiz, L., 1965.

2. A. E. Belyaev. Influence of speed on the contact strength of gear transmissions. Izvestiya TPI, No. 147, published by TSU, Tomsk, 1966.

3. V. N. Kudryavtsev. Simplified Calculations of Gear Transmissions. Mashgiz, Moscow, 1960.

4. S. S. Milovidov. Machine Parts. Vysshaya shkola, Moscow, 1961.