The first critical speeds of a rotating shaft with two mass disks, obtained from three different mathematical techniques, are 110 rad/s, 112 rad/s, and 100 rad/s, respectively.

(a) Which values correspond to the first critical speed of the shaft from the exact solution, the Dunkerley approximation, and the Rayleigh-Ritz approximation?

(b) If the influence coefficients are a11 = a22 =

10−4 m/N and the masses of the two disks are the same, that is, m1 = m2 = m, then use the Dunkerley approximation to calculate the mass m.

(c) If the influence coefficients are a11 =

a22 = 10−4 m/N and the masses of the two disks are specified as m1 = m2 = m = 0.5 kg, use the Rayleigh-Ritz approximation to calculate the influence coefficient a12.