A typical small sized \( (< 10 \mathrm{~m} \) or \( < 33 \mathrm{ft} \) in length) sailboat propellor shaft is machined from stainless steel (see Table A-5) with a diameter of \( 25 \mathrm{~mm} \) and a length of 914.4 \( \mathrm{mm} \) (36 in.). Determine the intrinsic critical speed for this shaft.
Answer
6. Calculate the intrinsic critical speed \(\omega_c\) using the formula: \(\omega_c = \sqrt{\frac{GI}{L^2\mu}} = \sqrt{\frac{80\times10^9 \mathrm{~Pa} \cdot \frac{\pi (0.0125 \mathrm{~m})^4}{4}}{(0.9144 \mathrm{~m})^2(8000 \mathrm{~kg/m^3} \cdot \pi (0.0125 \mathrm{~m})^2)}}\)
Step 1 ：1. Calculate the radius \(r\) of the shaft: \(r = \frac{D}{2} = \frac{25 \mathrm{~mm}}{2} = 12.5 \mathrm{~mm}\)
Step 2 ：2. Calculate the second moment of area \(I\) of the shaft: \(I = \frac{\pi r^4}{4} = \frac{\pi (12.5 \mathrm{~mm})^4}{4}\)
Step 3 ：3. Find the corresponding value for stainless steel density \(\rho\) and modulus of rigidity \(G\) from Table A-5: \(\rho = 8000 \mathrm{~kg/m^3}\) and \(G = 80\times10^9 \mathrm{~Pa}\)
Step 4 ：4. Convert the length \(L\) of the shaft to meters: \(L = 914.4 \mathrm{~mm} = 0.9144 \mathrm{~m}\)
Step 5 ：5. Calculate the linear mass density of the shaft \(\mu\): \(\mu = \rho \pi r^2 = 8000 \mathrm{~kg/m^3} \cdot \pi (0.0125 \mathrm{~m})^2\)
Step 6 ：6. Calculate the intrinsic critical speed \(\omega_c\) using the formula: \(\omega_c = \sqrt{\frac{GI}{L^2\mu}} = \sqrt{\frac{80\times10^9 \mathrm{~Pa} \cdot \frac{\pi (0.0125 \mathrm{~m})^4}{4}}{(0.9144 \mathrm{~m})^2(8000 \mathrm{~kg/m^3} \cdot \pi (0.0125 \mathrm{~m})^2)}}\)
