Definitions
\(a\): feed displacement (m)
\(h\): wear (m)
\(z\): compression (m)
Kinematic relation
\[ \frac{\mathrm{d}a(t)}{\mathrm{d}t} = \frac{\mathrm{d}h(t,r)}{\mathrm{d}t} + \frac{\mathrm{d}z(t,r)}{\mathrm{d}t} \]
Reye wear and Winkler contact
\[ \frac{\mathrm{d}h(t,r)}{\mathrm{d}t} = c\,f\,p(t,r)\,v(t,r) \]
\[ p(t,r) = k\,z(t,r) \]
Kinematic relation upgrade
\[ \frac{\mathrm{d}a(t)}{\mathrm{d}t} = c\,f\,k\,z(t,r)\,v(t,r) + \frac{\mathrm{d}z(t,r)}{\mathrm{d}t} \]
Variable change, from time to angle
With \( \mathrm{d}\alpha = \omega(t)\,\mathrm{d}t \) and \( v(t,r) = r\,\omega(t) \),
\[ \frac{\mathrm{d}a}{\mathrm{d}\alpha} = c\,f\,k\,z(\alpha,r)\,r + \frac{\mathrm{d}z(\alpha,r)}{\mathrm{d}\alpha} \]
Uniform feed rate
Assuming \( \mathrm{d}a/\mathrm{d}\alpha = a' \),
\[ a' = c\,f\,k\,z(\alpha,r)\,r + \frac{\mathrm{d}z(\alpha,r)}{\mathrm{d}\alpha} \]
Initial conditions
\[ z(0,r) = z_0,\qquad h(0,r) = 0 \]
Closed-form solution
\[ z(\alpha,r) = \left(z_0 - \frac{a'}{c f k r}\right)\mathrm{e}^{-c f k r\,\alpha} + \frac{a'}{c f k r} \]
\[ h(\alpha,r) = \left(z_0 - \frac{a'}{c f k r}\right)\bigl(1 - \mathrm{e}^{-c f k r\,\alpha}\bigr) + a'\,\alpha \]
\[ a(\alpha) = z_0 + a'\,\alpha \]