A 30mmdiamter uniform stainless steel shaft is 1000 mm long

A 30-mm-diamter uniform stainless steel shaft is 1000 mm long between two bearings at two ends with the figure (a): Find the lowest critical speed of the shaft. Use the Rayleigh\'s method to calculate the lowest critical speed of the shaft by partitioning the shaft into two elements, as seen in figure (b). What do you find by comparing the results using the above two methods?

Solution

(a) In the first case, we can consider the shaft as a simply supported beam of uniformly distributed mass. Lets take uniformly distributed weight per unit length m and length is L.

E=Young\'s modulus, I= Area moment of inertia about an axis passing through the diameter of shaft

So, the maximum deflection will be D1 = 5mL⁴/384EI   

For this case, minimum speed will be W1 = sqrt (g/D1) = sqrt(384gEI/5mL⁴) [where g= gravitational acceleration]

(b) If we cut the shaft in two halves, we can consider each part as a cantelever beam of length L/2. Rest of the things are same as above.

Now the maximum deflection D2 = mL⁴/128EI

So the minimum speed W2 = sqrt (g/D2) = sqrt(128gEI/mL⁴)

(c) In the first case, shaft speed will be lower.