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Shaft with two splines

Shaft with two splines M r a b
Shaft with two splines

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 $

$$C_1=0.7854$$
$C_1=0.7854$
Equations in LaTeX code
C_1=0.7854

Coefficient $ C_2 $

$$C_2=0.0204-0.1307\cdot\cfrac{a}{b}+0.1157\cdot\left(\cfrac{a}{b}\right)^2$$
$C_2=0.0204-0.1307\cdot\cfrac{a}{b}+0.1157\cdot\left(\cfrac{a}{b}\right)^2$
Equations in LaTeX code
C_2=0.0204-0.1307\cdot\cfrac{a}{b}+0.1157\cdot\left(\cfrac{a}{b}\right)^2

Coefficient $ C_3 $

$$C_3=-0.2075+1.1544\cdot\cfrac{a}{b}-0.5937\cdot\left(\cfrac{a}{b}\right)^2$$
$C_3=-0.2075+1.1544\cdot\cfrac{a}{b}-0.5937\cdot\left(\cfrac{a}{b}\right)^2$
Equations in LaTeX code
C_3=-0.2075+1.1544\cdot\cfrac{a}{b}-0.5937\cdot\left(\cfrac{a}{b}\right)^2

Coefficient $ C_4 $

$$C_4=0.3608-2.2582\cdot\cfrac{a}{b}+3.7336\cdot\left(\cfrac{a}{b}\right)^2$$
$C_4=0.3608-2.2582\cdot\cfrac{a}{b}+3.7336\cdot\left(\cfrac{a}{b}\right)^2$
Equations in LaTeX code
C_4=0.3608-2.2582\cdot\cfrac{a}{b}+3.7336\cdot\left(\cfrac{a}{b}\right)^2

Coefficient $ C $

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

$$B_1=0.6366$$
$B_1=0.6366$
Equations in LaTeX code
B_1=0.6366

Coefficient $ B_2 $

$$B_2=0.0069-0.0229\cdot\cfrac{a}{b}+0.0637\cdot\left(\cfrac{a}{b}\right)^2$$
$B_2=0.0069-0.0229\cdot\cfrac{a}{b}+0.0637\cdot\left(\cfrac{a}{b}\right)^2$
Equations in LaTeX code
B_2=0.0069-0.0229\cdot\cfrac{a}{b}+0.0637\cdot\left(\cfrac{a}{b}\right)^2

Coefficient $ B_3 $

$$B_3=-0.0675+0.3996\cdot\cfrac{a}{b}-1.0514\cdot\left(\cfrac{a}{b}\right)^2$$
$B_3=-0.0675+0.3996\cdot\cfrac{a}{b}-1.0514\cdot\left(\cfrac{a}{b}\right)^2$
Equations in LaTeX code
B_3=-0.0675+0.3996\cdot\cfrac{a}{b}-1.0514\cdot\left(\cfrac{a}{b}\right)^2

Coefficient $ B_4 $

$$B_4=0.3582-1.8324\cdot\cfrac{a}{b}+1.5393\cdot\left(\cfrac{a}{b}\right)^2$$
$B_4=0.3582-1.8324\cdot\cfrac{a}{b}+1.5393\cdot\left(\cfrac{a}{b}\right)^2$
Equations in LaTeX code
B_4=0.3582-1.8324\cdot\cfrac{a}{b}+1.5393\cdot\left(\cfrac{a}{b}\right)^2

Coefficient $ B $

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

$$K=2\cdot C\cdot r^4$$
$K=2\cdot C\cdot r^4$
Equations in LaTeX code
K=2\cdot C\cdot r^4

Angle of twist

$$θ=\cfrac{T\cdot 10^3\cdot L}{K\cdot G}$$
$θ=\cfrac{T\cdot 10^3\cdot L}{K\cdot G}$
Equations in LaTeX code
θ=\cfrac{T\cdot 10^3\cdot L}{K\cdot G}

Torsion stress at $ M $

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

Requirements

$$0 \le\cfrac{b}{r} \le 0.5$$
$0 \le\cfrac{b}{r} \le 0.5$
Equations in LaTeX code
0 \le\cfrac{b}{r} \le 0.5
$$0.2 \le\cfrac{a}{b} \le 1.4$$
$0.2 \le\cfrac{a}{b} \le 1.4$
Equations in LaTeX code
0.2 \le\cfrac{a}{b} \le 1.4
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