First find the expression for T = f(d/D)

Let d/D = x

For wear theory $\left(\mathrm{P_{. r}=C}\right)$

$\mathrm{P}_{\mathrm{m} \cdot} \frac{\mathrm{d}}{2}=\mathrm{C}$

$\mathrm{T}=\frac{\pi \mu \mathrm{C}}{4}\left(\mathrm{D^{2}-d}^{2}\right)$

$\mathrm{T}=\frac{\pi \mu}{4} \mathrm{P}_{\mathrm{m}} \cdot \frac{\mathrm{d}}{2}\left(\mathrm{D^{2}-d}^{2}\right)$

$\mathrm{T}=\frac{\mathrm{P_{m}} \cdot \pi \mu}{8} \cdot \mathrm{D}^{2}\left(\mathrm{d}-\frac{\mathrm{d}^{2}}{\mathrm{D}^{2}} \mathrm{d}\right)$

$\mathrm{T}=\frac{\mathrm{P_{m}} \pi \mu}{8} \cdot \mathrm{D}^{3}\left(\frac{\mathrm{d}}{\mathrm{D}}-\frac{\mathrm{d}^{3}}{\mathrm{D}^{3}}\right)$

$\mathrm{T}=\frac{\mathrm{P_{m}} \pi \mu}{8} \cdot \mathrm{D}^{3}\left(\mathrm{x-x}^{3}\right)$

D = constant

$\mathrm{T}=\frac{\mathrm{P_{m}} \pi \mu \mathrm{D}^{3}}{8} \cdot\left(\mathrm{x-x}^{3}\right)$

For maximum torque or minimum torque:

$\frac{\mathrm{d T}}{\mathrm{d x}}=0$

$\frac{\mathrm{d T}}{\mathrm{d x}}=\frac{\mathrm{P_{m}} \cdot \pi \mu \mathrm{D}^{3}}{8} \cdot\left(1-3 \mathrm{x}^{2}\right)=0$ [where, $\frac{\mathrm{P_{m}} \cdot \pi \mu \mathrm{D}^{3}}{8}$ is constant]

$\frac{\mathrm{d T}}{\mathrm{d x}}=0=\left(1-3 \mathrm{x}^{2}\right)$

$\Rightarrow 1=3 \mathrm{x}^{2}$

$\Rightarrow \mathrm{x}=\frac{1}{\sqrt{3}}$

$\Rightarrow\mathrm{x}=0.577=\mathrm{d} / \mathrm{D}$

$\frac{\mathrm{d^{2} T}}{\mathrm{d x}^{2}}=\frac{\mathrm{d}}{\mathrm{d x}}\left(1-3 \mathrm{x}^{2}\right)=-6 \mathrm{x}$

$\frac{\mathrm{d^{2} T}}{\mathrm{d x}^{2}}=-6 \mathrm{x}=-6 \times 0.577=\mathrm{-v e}$

So, at $\mathrm{x}=0.577,\mathrm{ T=T_{\max }}$

$\mathrm{x=d / D}=0.577$

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Static And Fluctuating Load

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