The weights of two masses m1 and m2 that are rigidly attached to the rotating shaft are 31.5 lb and 13.5 lb, respectively. The shaft is rotating counterclockwise with a constant angular velocity

ω = 100 rad/s. From a deflection analysis, the influence coefficients for the shaft are a11 =

3.56 × 10−6 in/lb, a22 = 21.36 × 10−6in/lb, and a12 = a21 = 7.12 × 10−6 in/lb. Neglecting 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 the Rayleigh-Ritz equation and the Dunkerley approximation. Determine the first critical speed of the shaft if mass m1 is moved to location 2 and mass m2 is moved to location 1.