The steel shaft is simply supported by two bearings at A and D. Two gears are rigidly attached to the shaft at locations B and C. Gear 1 at location B weighs 150 N, gear 2 at location C weighs 90 N, and the weight of the shaft can be neglected. The stiffness coefficients are k11 = 2.5 × 104 N/m, k12 = 5.0 × 104 N/m, and k22 = 4.0 × 104 N/m. Determine the first critical speed of the shaft using:

(a) the Dunkerley approximation, and

(b) the Rayleigh-Ritz method.

Determine the first and second critical speeds of the shaft using the exact solution. If gear 2 is now placed at location B, and gear 1 is placed at location C, then determine the first critical speed of the new system using the Dunkerley approximation.