For the power transmission system (as shown in Figure 2.10), power flows from input gear A to the output gear D. If the pitch diameters of the gears B and C are in the ratio 1:2, what is the proper ratio of the shaft diameters d1:d2 if both the shafts have same maximum stress.

Question

For the power transmission system (as shown in Figure 2.10), power flows from input gear A to the output gear D. If the pitch diameters of the gears B and C are in the ratio 1:2, what is the proper ratio of the shaft diameters [latex] d_1: d_2 [/latex] if both the shafts have same maximum stress.

Accepted Answer

Let P be the power which is being transmitted through the system and ω be the angular speed. Then torque acting on the input shaft is

[latex] T_{ ext {in }}=\frac{P}{\omega_{ ext {in }}} [/latex]

Now, let us draw the free-body diagrams of the shafts

If we consider the free-body diagram shown in Figures 2.11(a) and (b), we can write

[latex] F_{ t } imes \frac{d_{ b }}{2}=T_{ ext {in }} \quad ext { and } \quad T_{ ext {out }}=F_{ t } imes \frac{d_{ c }}{2} [/latex]

Therefore,

[latex] \frac{T_{ out }}{T_{ in }}=\frac{d_{ c }}{d_{ b }} [/latex] (1)

This can be verified considering no power loss in the transmission system and writing

[latex] \frac{T_{ ext {out }}}{T_{ ext {in }}}=\frac{\omega_{ ext {in }}}{\omega_{ ext {out }}}=\frac{\omega_{ B }}{\omega_{ C }}=\frac{d_{ c }}{d_{ b }} [/latex]

This equality [latex] \left(\omega_{ B } / \omega_{ C } ight)=\left(d_{ c } / d_{ b } ight) [/latex] holds goods as the tangential pitch line velocity is same for both B and C. If we assume shear stress developed in the shafts to be same, we can write

[latex] \frac{T_{ ext {out }}}{T_{ ext {in }}}=\frac{\frac{\pi}{16} d_2^3}{\frac{\pi}{16} d_1^3}=\frac{d_2^3}{d_1^3} [/latex] (2)

From Eqs. (1) and (2),

[latex] \frac{d_2^3}{d_1^3}=\frac{d_{ c }}{d_{ b }} [/latex]

or [latex] \left\lgroup \frac{d_2}{d_1} ight group =\sqrt[3]{\frac{d_{ c }}{d_{ b }}}=\sqrt[3]{2} \quad\left\lgroup ext { as } \frac{d_{ c }}{d_{ b }}=\frac{2}{1} ight group [/latex]

Therefore,

[latex] \frac{d_1}{d_2}=\frac{1}{\sqrt[3]{2}}=0.7936 [/latex]

Thus, the ratio of shaft diameters is 0.7936 : 1.

Question 2.9: For the power transmission system (as shown in Figure 2.10),...

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