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Shaft with four keyways

Shaft_with_four_keyways
Shaft with four keyways

Values for calculation

$T$ $\mathrm{Nm}$
$r$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$L$ $\mathrm{mm}$
$G$ $\mathrm{MPa}$

Calculation

Coefficient $ C_1 $

$$C_1=0.7854$$

Coefficient $ C_2 $

$$C_2=-0.1496+0.2773\cdot\cfrac{a}{b}-0.211\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ C_3 $

$$C_3=-2.9138-8.2354\cdot\cfrac{a}{b}+2.5782\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ C_4 $

$$C_4=2.2991+12.097\cdot\cfrac{a}{b}-2.2838\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ C $

$$C=C_1+C_2\cdot\cfrac{b}{r}+C_3\cdot\left(\cfrac{b}{r}\right)^2+C_4\cdot\left(\cfrac{b}{r}\right)^3$$

Coefficient $ B_1 $

$$B_1=1.0434+1.0449\cdot\cfrac{a}{b}-0.2977\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ B_2 $

$$B_2=0.0958-9.8401\cdot\cfrac{a}{b}+1.6847\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ B_3 $

$$B_3=15.749-6.965\cdot\cfrac{a}{b}+14.222\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ B_4 $

$$B_4=-35.878+88.696\cdot\cfrac{a}{b}-47.545\cdot\left(\cfrac{a}{b}\right)^2$$

Coefficient $ B $

$$B=B_1+B_2\cdot\cfrac{b}{r}+B_3\cdot\left(\cfrac{b}{r}\right)^2+B_4\cdot\left(\cfrac{b}{r}\right)^3$$

Polar moment of inertia

$$K=2\cdot C\cdot r^4$$

Angle of twist

$$θ=\cfrac{T\cdot 10^3\cdot L}{K\cdot G}$$

Torsion stress at $ M $

$$τ_{max-at-M}=\cfrac{10^3\cdot T\cdot B}{r^3}$$

Requirements

$$0.2 \le\cfrac{b}{r} \le 0.4$$$$0.5 \le\cfrac{a}{b} \le 1.2$$