A shaft rotating with a constant angular velocity is simply supported at A and B. Gears C and D are rigidly attached to the shaft at locations 1 and 2, respectively. The weight of gear C at location 1 is 13 N, and the weight of gear D at location 2 is 23 N. The known influence coefficients of the shaft are a11 = 6.8×10−6 m/N, a12 = 5.3×10−6 m/N, and a22 = 7.9 × 10−6 m/N. Determine the first critical speed of the shaft using:

(a) the Dunkerley approximation, and

(b) the Rayleigh-Ritz method.

If gear C is moved to location 2 and gear D is moved to location 1, then calculate the new first critical speed using:

(c) the Dunkerley approximation, and

(d) the Rayleigh-Ritz method.