The weights of two gears rigidly attached to a shaft at two different locations, denoted as 1 and 2, are W1 = 200 N and W2 = 350 N, respectively. The shaft is rotating counterclockwise with a constant operating speed ω = 100 rad/s.

From a deflection analysis of the shaft, the known influence coefficients are a11 = 35 ×

10−6 mm/N, a12 = 50 × 10−6 mm/N, and a22 = 90 × 10−6 mm/N. Neglecting gravity and the mass of the shaft, determine the first and second critical speeds of the shaft using the exact equation. Is the operating speed of the shaft acceptable? Determine the first critical speed of the shaft using:

(a) the Rayleigh-Ritz method; and

(b) the Dunkerley approximation.