A 600-mm-long steel shaft is simply supported by two bearings at A and C. Flywheels 1 and 2 are attached to the shaft at locations B and D, respectively. Flywheel 1 at location B weighs 50 N, flywheel 2 at location D weighs 20 N, and the weight of the shaft can be neglected. The known stiffness coefficients are k11 = 25 000 N/m, k12 = 50 000 N/m, and k22 = 40 000 N/m. Determine

(a) the first and second critical speeds of the shaft using the exact solution, and

(b) the first critical speed using the Dunkerley, and

(c) Rayleigh-Ritz approximations.

(d) If flywheel 2 is then placed at location B and flywheel 1 is placed at location D, determine the first critical speed of the new system using the Dunkerley approximation.
Figure P13.24 Steel shaft simply supported by two rolling element bearings.