On the Estimation of Gearbox Weight for Various Values of the Relative Width of the Gear Rim

A. E. Belyaev

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

At present, a standard series of gearboxes of the RTsO, RTsD, and other types is produced in the USSR. One of the parameters adopted in forming these series is the relative width of the gear rim,

\[ \psi=\frac{B}{A}=0.4. \]

However, a number of experimental studies carried out in the gear-transmission laboratory (LZP) at the Leningrad Mechanical Institute (LMI), as well as experience in designing new gearboxes at the Special Design Bureau of LMI, have shown that, with a reduction in the value of \(\psi\), gearboxes can be designed with better structural and economic parameters. Unfortunately, up to the present time there has been no generally accepted estimate of the change in gearbox weight when the initial parameters are varied and, in particular, as a function of the value of \(\psi\).

Below an attempt is made to establish analytical relationships connecting the change in gearbox weight when passing from \(\psi=0.4\) to \(\psi=0.3\).

As the object of comparison, a two-stage production gearbox of the RTsD type is taken, with standard values of the gear ratio, pressure angle, helix angle, configuration of the gears, and gearbox housing.

1. Geometric and Force Relationships

According to [1], the center distance \(A\) of the gearbox can be determined by the following formula:

\[ A=(i+1)\cdot \sqrt[3]{\frac{M_1\cdot k}{2\cdot i\cdot \psi \cdot [C_k]\cdot \vartheta_k \cdot \xi_k}} . \tag{1} \]

In expression 1, all other conditions being equal (the comparison is made for \((M_1)_{0.4}=(M_1)_{0.3}\) and \([C_k]_{0.4}=[C_k]_{0.3}\), etc.), the value of \(A\) depends on \(\psi\), and consequently,

\[ \frac{A_{0.3}^{3}}{A_{0.4}^{3}}=\frac{\psi_{0.3}}{\psi_{0.4}} . \]

And finally,

\[ A_{0.3}=\sqrt[3]{\frac{\psi_{0.4}}{\psi_{0.3}}}\,A_{0.4} =\sqrt[3]{\frac{0.4}{0.3}}\,A_{0.4} =1.1\,A_{0.4}, \tag{2} \]

\(^1\) Here and below, the indices 0.3 and 0.4 will correspond to gearboxes with \(\psi=0.3\) and \(\psi=0.4\), respectively.

which is valid for any stage of the reducer.

Since the magnitude of the pitch-circle diameter \(d_o\) (of the pinion and wheel) is proportional to \(A\) \(\left(d_o=\dfrac{2A}{i+1}\simeq A\right)\), the ratio is

\[ \frac{(d_o)_{0.3}}{(d_o)_{0.4}}=\frac{A_{0.3}}{A_{0.4}}=1.1 . \tag{3} \]

The ratio of the widths \(B\) turns out to be

\[ \frac{B_{0.3}}{B_{0.4}}=\frac{\psi_{0.3}A_{0.3}}{\psi_{0.4}A_{0.4}}=0.825 . \]

The expressions given make it possible to assume that the forces arising in the mesh (with equality of the geometric parameters of engagement \(\alpha\), \(\beta\), etc.) are in a ratio inversely proportional to relation (3).

For identical torques but different shaft lengths, the support reactions (and shaft diameters) will be determined by the bending moments acting on them.

\[ M_{\Sigma\text{bend}}=R_{\Sigma}\cdot l_{\text{bend}}, \]

where \(l_{\text{bend}}\) is the arm of application of the force relative to the support of the shaft, considered as a beam on two supports.

Proceeding from the structural proportions adopted in practice for two-stage reducers, it may be assumed that \(l_{\text{bend}}\) can always be expressed in fractions of \(B\). Taking into account the fact that \(d_o\) is proportional to \(A\), and that the value \(d_{o1}\) (and \(d_{o2}\)) can be represented as

\[ d_{o1}=\frac{B}{0.5(i+1)\psi}, \]

then the value of the total reactions at the supports reduces to the form (for \(l_{\text{bend}}=xB\))

\[ R_{\Sigma}=P_{\text{circ}}\cdot B\left(\operatorname{tg}\beta_o-\frac{1}{(i+1)\cdot\psi}+x\frac{\operatorname{tg}\alpha_n}{\cos\beta_o}\right). \tag{4} \]

In the last expression \(x\) will be different for different stages (for example, for the low-speed stage \(x=1.7\)), but when comparing identical stages with different \(\psi\), this value may also be considered constant. When comparing \((R_{\Sigma})_{0.3}\) with \((R_{\Sigma})_{0.4}\), it turns out that

\[ \frac{(R_{\Sigma})_{0.3}}{(R_{\Sigma})_{0.4}} = \frac{(P_{\text{circ}})_{0.3}\cdot B_{0.3}}{(P_{\text{circ}})_{0.4}\cdot B_{0.4}} = 1.1\cdot 0.825\simeq 0.91 . \tag{5} \]

The analysis shows (Fig. 1) that, in addition to a decrease in the support reactions in absolute value, they are also redistributed more uniformly between the supports. This makes it possible to assume that, all other conditions being equal, the supports of a reducer with \(\psi=0.3\) will be less heavily loaded or (which is more important for the designer) a transition to a smaller size of rolling bearing may prove possible, on the basis of the adopted load-capacity coefficient; this will also lead to greater compactness of the bearing bosses and bearing covers.

Fig. 1. Load distribution on the supports of the high-speed shaft of a two-stage reducer in percent (hatched—for \(\psi=0.3\), unhatched—for \(\psi=0.4\)).
\(L^*\)—distance between shaft supports.

II. Determination of reducer weight

1. Determination of the weight of the reducer housing.

If the bosses are disregarded (assuming wall I to be smooth, Fig. 2a, b), then, in the general case, the weight of walls I (front and rear, \(2G_1\)) for reducers with the compared parameters will be

\[ 2(G_1)_{0.3}=2\cdot (G_1)_{0.4}\cdot \left(\frac{A_{0.3}}{A_{0.4}}\right)^2 . \tag{6} \]

Here \((G_1)_{0.3}\) is the weight of wall I of a gearbox with \(\psi=0.3\);
\((G_1)_{0.4}\) is the same for \(\psi=0.4\).

The weight of walls II \((G_2)\) can be determined as the product of the perimeter \(P\) of the indicated wall by the width \(Y\) (Fig. 2),

\[ G_2=P\cdot Y . \tag{7} \]

Fig. 2. For calculating the weight of the gearbox housing: a and c) simplified form of the gearbox outline; b) determination of the weight of the front (and rear) wall of the gearbox without taking the bosses into account, by areas \(S_1, S_2, S_3, S_4, S_5\); \(t\) is the boss thickness; \(l_1, l_2, l_3, l_4, l_5\) are the components of the gearbox-housing perimeter.

The quantity \(Y\) can be obtained by taking into account the widths of the gears of all stages and the clearances between the gears and the housing (the latter, for purposes of comparison, may be evaluated by a proportionality coefficient \(\Theta\), which for gearboxes with the compared values of \(\psi\) may be taken as identical)

\[ Y=\Theta(B_{\mathrm{T}}+B_{\mathrm{B}})=\Theta(A_{\mathrm{T}}\psi+A_{\mathrm{B}}\psi)=\Theta\psi A, \tag{8} \]

where

\(B_{\mathrm{T}}\) and \(B_{\mathrm{B}}\) are the gear widths of the low-speed and high-speed stages, respectively (for standard gearboxes \(\psi=\dfrac{B}{A}=\mathrm{const}\) for all stages);

\(A_{\mathrm{T}}\) and \(A_{\mathrm{B}}\) are the center distances of the low-speed and high-speed stages of the gearbox, respectively.

Consequently, taking (8) into account,

\[ \frac{y_{0.3}}{y_{0.4}} = \frac{\Theta\cdot\psi_{0.3}\cdot A_{0.3}} {\Theta\cdot\psi_{0.4}\cdot A_{0.4}} = \frac{0.3}{0.4}\cdot 1.1 = 0.825 . \tag{9} \]

Taking into account the proportionality of the quantities \(P\) and \(A^2\), the relationship between the weight of walls II in general form will be

\({}^{2)}\) This is evidently valid, since the length \(L\) and \(A\) are related by \(L=\gamma\cdot A\), and their ratio is

\[ \frac{L_{0.3}}{L_{0.4}} = \frac{\gamma\cdot A_{0.3}}{\gamma\cdot A_{0.4}} = 1.1 . \]

\[ (G_2)_{0.3}=(G_2)_{0.4}\cdot \frac{A_{0.3}}{A_{0.4}}\cdot \frac{y_{0.3}}{y_{0.4}} . \tag{10} \]

Thus, the total weight of the gearbox housing (see 6 and 7) (for example, for a gearbox with \(\psi=0.4\)) is

\[ (G)_{0.4}=2\cdot (G_1)_{0.4}+(G_2)_{0.4}. \tag{11} \]

Then the ratio of the housing weights (taking into account expressions 6, 9, 10, and 11) will be as follows

\[ \frac{(G)_{0.4}}{(G)_{0.3}} = \frac{2\cdot (G_1)_{0.4}+(G_2)_{0.4}} {2\cdot (G_1)_{0.4}\cdot \left(\frac{A_{0.3}}{A_{0.4}}\right)^2 +(G_2)_{0.4}\cdot \frac{A_{0.3}}{A_{0.4}}\cdot \frac{y_{0.3}}{y_{0.4}}} = \frac{2\cdot (G_1)_{0.4}+(G_2)_{0.4}} {2.42\,(G_1)_{0.4}+0.906\,(G_2)_{0.4}} . \tag{12} \]

However, the weight ratio cannot be obtained directly from expression (12), just presented, since the ratio between the weights of walls I and II is unknown.

Obviously, on a certain scale the weight of walls I can be expressed through \(A\) (for the time being disregarding the weight of the bosses), i.e., determined through the total center distance and the area of this wall, taking its thickness to be equal to unity. Then the area of wall I (Fig. 2, b) is

\[ S^{\mathrm I}=S_1+S_2+S_3+S_4+S_5, \tag{13} \]

where \(S_1, S_2, S_3, S_4, S_5\) are the components of the total area I.

On the basis of Fig. 2 we have

\[ S^{\mathrm I}=\frac{\pi R^2}{4}+A\left(x+R-\frac{R-r}{2}\right)+\frac{\pi r^2}{4}+x\cdot r+x\cdot R . \]

Expressing \(R=f(A)\), \(r=f(A)\), and \(x=f(A)\), we obtain (on the basis of the last dependence)

\[ 2\cdot S^{\mathrm I}_{0.4}=3.7\,A_{0.4}^2 \quad \text{and} \quad 2\cdot S^{\mathrm I}_{0.3}=3.54\,A_{0.3}^2 . \]

We carry out the same calculation in order to determine the weight of wall II. The perimeter \(\Pi\) consists of the following components (Fig. 2a)

\[ \Pi=l_1+l_2+l_3+l_4+l_5+L, \]

where

\[ l_1=l_5=x; \qquad l_2=\frac{\pi R}{2}; \qquad l_3=\sqrt{(R-r)^2+A^2}; \qquad l_4=\frac{\pi R}{2}; \qquad L=A+(R+r). \]

Expressing the quantities \(x, R, r\) through \(A\), we find that the weight of walls II in the gearboxes (according to 7) will be

\[ S^{\mathrm{II}}_{0.4}=\Pi_{0.4}\cdot y_{0.4}=2.64\,A_{0.4}^2 \quad \text{and} \quad S^{\mathrm{II}}_{0.3}=\Pi_{0.3}\cdot y_{0.3}=2\,A_{0.3}^2 . \]

Thus, if walls I are assumed to be smooth, then the weight of the gearbox housing will be proportional to

\[ S_{0.4}=2\cdot S^{\mathrm I}_{0.4}+S^{\mathrm{II}}_{0.4}=6.34\,A_{0.4}^2; \tag{14} \]

\[ S_{0.3}=2\cdot S^{\mathrm I}_{0.3}+S^{\mathrm{II}}_{0.3}=5.54\,A_{0.3}^2 . \tag{14'} \]

To simplify the accounting for the weight of the bosses (or, correspondingly, the area \(S^b\)), we take the quantities \(t\) for the gearboxes being compared to be identical (Fig. 2). In this case, the weight of the bosses may be considered proportional to the area of a ring with thickness equal to unity, \(t=\mu l\), where \(\mu\) is the number of these rings fitting into the quantity \(t\).

Expressing all the quantities \(r_1; R_1; r_2; R_2; r_3; R_3\) (the radii of the outer and inner circles of the bosses, Fig. 2, a) through the quantity \(A\),

and, taking into account the total number of bosses, we obtain, respectively, for gearboxes with \(\psi=0.4\) and \(\psi=0.3\):

\[ S^{\mathrm{B}}_{0.4}=1.69\cdot A_{0.4}^{2}, \tag{15} \]

\[ S^{\mathrm{B}}_{0.3}=1.19 A_{0.3}^{2}. \tag{15'} \]

On the basis of the above dependences \((14,\ 14',\ 15,\ 15')\), we find the total weight of the gearbox housing for

\[ \psi=0.4 \qquad (S_{\text{total}})_{0.4}=S_{0.4}+S^{\mathrm{B}}_{0.4}=8.03\,A_{0.4}^{2}, \]

\[ \psi=0.3 \qquad (S_{\text{total}})_{0.3}=S_{0.3}+S^{\mathrm{B}}_{0.3}=6.73\,A_{0.3}^{2}. \]

The ratio of the weights of the housings is

\[ \frac{(S_{\text{total}})_{0.3}}{(S_{\text{total}})_{0.4}} = \frac{6.73\cdot A_{0.3}^{2}}{8.03\cdot A_{0.4}^{2}} = \frac{6.73}{8.03}\cdot(1.1)^{2}=1. \tag{16} \]

Thus, the weight of the compared housings is practically the same for different \(\psi\).

2. Determination of the weight of the gear wheels.

The weight of a gear wheel proves to be proportional to

\[ G_k=\Omega\cdot d_0^{2}\cdot B, \tag{17} \]

where

\(d_0\) is the pitch-circle diameter,
\(\Omega\) is the coefficient of proportionality.

Taking the ratio of the weights of the wheels of gearboxes with \(\psi=0.3\) and \(\psi=0.4\) (on the basis of equations 3 and 17), we obtain

\[ \frac{(G_k)_{0.3}}{(G_k)_{0.4}} = \frac{\Omega\cdot(d_0)_{0.3}^{2}\cdot B_{0.3}} {\Omega\cdot(d_0)_{0.4}^{2}\cdot B_{0.4}} = \frac{A_{0.3}^{3}\cdot\psi_{0.3}}{A_{0.4}^{3}\cdot\psi_{0.4}} = (1.1)^3\,\frac{0.3}{0.4}=1. \tag{18} \]

Thus, for a sufficiently simple configuration of the wheels, their weight does not depend on the relative width.

3. Determination of the weight of the gearbox shafts.

The weight of the shafts in gearboxes can be determined as

\[ G_{\mathrm{v}}=\nu\cdot L^{*}\cdot d_{\mathrm{v}}^{2}, \tag{19} \]

where

\(L^{*}\) is the length of the shaft between supports;
\(d_{\mathrm{v}}\) is the mean shaft diameter;
\(\nu\) is the coefficient of proportionality.

If it is assumed that in the compared gearboxes the shaft diameters will be the same \((d_{\mathrm{v}0.4}=d_{\mathrm{v}0.3})\), then in this case as well the weight of the shaft in the gearbox with \(\psi=0.3\) will be less than that with \(\psi=0.4\);

\[ (G_{\mathrm{v}})_{0.3}<(G_{\mathrm{v}})_{0.4}, \quad \text{since } L^{*}_{0.3}<L^{*}_{0.4}, \]

namely (according to 19 and 8):

\[ \frac{(G_{\mathrm{v}})_{0.3}}{(G_{\mathrm{v}})_{0.4}} = \frac{\nu\cdot L^{*}_{0.3}\cdot(d_{\mathrm{v}})_{0.3}^{2}} {\nu\cdot L^{*}_{0.4}\cdot(d_{\mathrm{v}})_{0.4}^{2}} = \frac{L^{*}_{0.3}}{L^{*}_{0.4}} = \frac{\mu\,(B_{\mathrm{B}0.3}+B_{\mathrm{T}0.3})} {\mu\,(B_{\mathrm{B}0.4}+B_{\mathrm{T}0.4})} = \]

\[ = \frac{\psi_{0.3}\cdot A_{\mathrm{B}0.3}+\psi_{0.3}\cdot A_{\mathrm{T}0.3}} {\psi_{0.4}\cdot A_{\mathrm{B}0.4}+\psi_{0.4}\cdot A_{\mathrm{T}0.4}} = \frac{\psi_{0.3}\cdot A_{0.3}}{\psi_{0.4}\cdot A_{0.4}} = \frac{0.3}{0.4}\cdot 1.1=0.825. \]

Consequently, \((G_{\mathrm{v}})_{0.3}=0.825(G_{\mathrm{v}})_{0.4}\). Taking into account the fact that there are three shafts in the gearbox, the overall ratio of the weights of the shafts will be still higher.

Thus, the total weight of the gearbox with \(\psi=0.3\) will be somewhat less than with \(\psi=0.4\) only because of the reduction in the weight of the shafts, covers

and bearings. Knowing the weight of the gearbox (as a function of the center distance, the value of the weight of two-stage gearboxes is given, for

Fig. 3. Dependence of gearbox weight on the total center distance

example, in Fig. 3) and estimating the weight components of the new gearbox, it is possible to find the weight ratio of the gearboxes being compared.

REFERENCES

1. V. N. Kudryavtsev. Simplified Calculations of Gear Transmissions. Mashgiz, Moscow, 1969.