Simon, let me take a "shot" at this. It's not the simplest theory to explain on a forum.

quote:

I am not able to understand why a flexible rotor flexes in the vicinity of 1st critical and then the flexure decreases,where as a rigid rotor does not flex at 1st critcal speed

First, by definition, a rigid rotor is just a rotor that operates below the 1st critical. If that rigid rotor is operated at sufficient speed, it will pass through a resonance. See response after next quote for second part of the question.

quote:

My be it will sound like a silly question,but please explain to me ,why it flexes becasues the stiffness is less and what makes the flexure to decrease after 1st critical

Don't think of it as the stiffness changes; it's related to the forces due to the square of the angular velocity and stiffness cancel each other at the resonance, leaving only the damping force to control the amplitude.

The square of angular velocity (acceleration) leads displacement by 180 degrees. Before the resonance the dominating force is due to the stiffness component (stiffness * displacment). Above the resonance the dominating force is due to the square of the angular velocity (angular velocity ^2 * mass). At resonance the 90 degree phase change is seen since the damping term (angular velocity * damping coefficient) is a function of angular velocity, and velocity leads displacement by 90 degrees and lags acceleration by 90 degrees. Below the resonance the rotor tends to rotate about its geometric center, while well above the resonance, the rotor tends to rotate about its mass center. Since the angular acceleration is the dominant factor, we see a 180 degree phase shift from the original response.