depth of cut = diameter of rod − diameter of machined feature 2 {\displaystyle {\text{depth of cut}}={\frac {{\text{diameter of rod}}-{\text{diameter of machined feature}}}{2}}}
cutting velocity = cutting speed × π × diameter of rod 12 {\displaystyle {\text{cutting velocity}}={\frac {{\text{cutting speed}}\times \pi \times {\text{diameter of rod}}}{12}}}
cutting time = length of feature feed rate × cutting speed {\displaystyle {\text{cutting time}}={\frac {\text{length of feature}}{{\text{feed rate}}\times {\text{cutting speed}}}}}
material removal rate = cutting speed × feed rate × depth of cut × π × diameter of rod + diameter of feature 2 {\displaystyle {\text{material removal rate}}={\text{cutting speed}}\times {\text{feed rate}}\times {\text{depth of cut}}\times \pi \times {\frac {{\text{diameter of rod}}+{\text{diameter of feature}}}{2}}}
Line efficiency = Actual production Maximum possible production {\displaystyle {\text{Line efficiency}}={\frac {\text{Actual production}}{\text{Maximum possible production}}}}
Machine utilisation rate = Processing time Processing time + Idle time = Processing time Total available time {\displaystyle {\text{Machine utilisation rate}}={\frac {\text{Processing time}}{\text{Processing time + Idle time}}}={\frac {\text{Processing time}}{\text{Total available time}}}}
Slenderness ratio λ = L k = 2.5 1 2 × 25 × 10 − 3 = 200 {\displaystyle {\text{Slenderness ratio }}\lambda ={\frac {\text{L}}{\text{k}}}={\frac {\text{2.5}}{{\frac {1}{2}}\times 25\times 10^{-}3}}=200}
Euler failure:
σ e = π 2 E λ 2 = π 2 × 60 × 10 9 200 2 = 14.8 MPa {\displaystyle \sigma _{e}={\frac {\pi ^{2}E}{\lambda ^{2}}}={\frac {\pi ^{2}\times 60\times 10^{9}}{200^{2}}}=14.8{\text{ MPa}}}
Rankine-Gordon failure model:
σ c r = σ y 1 + σ y λ 2 π 2 E = 24 × 10 6 1 + 24 × 10 6 × 200 2 π 2 × 60 × 10 9 = 9.16 MPa {\displaystyle \sigma _{cr}={\frac {\sigma _{y}}{1+{\frac {\sigma _{y}\lambda ^{2}}{\pi ^{2}E}}}}={\frac {24\times 10^{6}}{1+{\frac {24\times 10^{6}\times 200^{2}}{\pi ^{2}\times 60\times 10^{9}}}}}=9.16{\text{ MPa}}}
Johnson's Parabola:
σ c r = σ y − ( σ y 2 4 π 2 E ) λ 2 = 24 × 10 6 − ( ( 24 × 10 6 ) 2 × 200 2 4 × π 2 × 60 × 10 9 ) = 14.3 MPa {\displaystyle \sigma _{cr}=\sigma _{y}-\left({\frac {\sigma _{y}^{2}}{4\pi ^{2}E}}\right)\lambda ^{2}=24\times 10^{6}-\left({\frac {\left(24\times 10^{6}\right)^{2}\times 200^{2}}{4\times \pi ^{2}\times 60\times 10^{9}}}\right)=14.3{\text{ MPa}}}
Perry-Robertson formula:
σ c r = σ 2 2 − σ 2 2 4 − σ y σ e {\displaystyle \sigma _{cr}={\frac {\sigma _{2}}{2}}-{\sqrt {{\frac {\sigma _{2}^{2}}{4}}-\sigma _{y}\sigma _{e}}}}
σ 2 = σ y + σ e ( 1 + 0.003 λ ) {\displaystyle \sigma _{2}=\sigma _{y}+\sigma _{e}\left(1+0.003\lambda \right)}
σ 2 = 24 × 10 6 + 14.8 × 10 6 ( 1 + ( 0.003 × 200 ) ) = 47.7 MPa {\displaystyle \sigma _{2}=24\times 10^{6}+14.8\times 10^{6}\left(1+\left(0.003\times 200\right)\right)=47.7{\text{ MPa}}}
σ c r = ( 47.7 × 10 6 2 ) − ( 47.7 × 10 6 ) 2 4 − ( 24 × 10 6 × 14.8 × 10 6 ) = 9.23 MPa {\displaystyle \sigma _{cr}=\left({\frac {47.7\times 10^{6}}{2}}\right)-{\sqrt {{\frac {\left(47.7\times 10^{6}\right)^{2}}{4}}-\left(24\times 10^{6}\times 14.8\times 10^{6}\right)}}=9.23{\text{ MPa}}}
