The 600-mm-long shaft is simply supported by the bearings at A and D. Flywheel 1 at location B weighs 30 N, and flywheel 2 at location C weighs 15 N. The weight of the shaft can be neglected.

The stiffness coefficients of the shaft are k11 =

1.0 × 106 N/m and k22 = 5.0 × 106 N/m. (a)

Using the Dunkerley approximation, determine the first critical speed of the shaft.

(b) If the flywheels are interchanged (that is, if flywheel 2 is at location B, and flywheel 1 is at location C), then use the Dunkerley approximation to determine the first critical speed of the new system.