This is well-known theory in finite element analysis. There are two different effects that contribute to the effective stiffness of the structure in addition to the elastic stiffness $K_e$ given by Euler-Bernoulli-Timoshenko beam theory, and you need to include both of them to get the right answer.
(1) Rotating the beam creates axial tension in the beam, which will be zero at the tip and increases to a maximum value at the rotation axis. This tension creates an additional stiffness that opposes lateral bending of the beam, in the same way that the tension in a guitar string affects its vibration frequency and hence the pitch of the note produced. (Of course in a guitar string, $K_e$ is negligible, unlike your beam). 333
This stiffness term was traditionally called the "geometric stiffness" (for reasons which aren't worth explaining) but a more logical name is "stress stiffness" often denoted by $K_\sigma$, since it depends only on the internal stresses in the structure and not on the material properties.
Incidentally, $K_\sigma$ for a compressive stress is what causes <PERSON> buckling in columns. At the "critical load" for buckling, $K_\sigma$ "cancels out" the elastic stiffness and when $K_e + K_\sigma = 0$, the structure can't resist the applied loads. For tension stresses, $K_\sigma$ increases the stiffness of the structure.
(2) If the beam deflects from its original straight configuration, the direction and/or the point of application of the centrifugal loads changes.
If the beam rotates about a vertical axis and the beam deflects vertically, the CF load on each particle of the beam is still in a horizontal direction, and therefore the CF loads at the beam tip will create a bending moment that tends to straighten out the beam.
But if the beam deflects tangentially (for example while you are ramping up the rotation speed from zero) the direction of the CF load is radially away from the axis, and that generates a bending moment that tends to increase the bending of the beam, not reduce it.
The usual name for both these effects is "follower forces", because the direction of the applied loads "follows" the deflection of the structure. The corresponding stiffness terms are usually called the "load stiffness" and denoted by $K_L$.
The original equation for statics using only the elastic stiffness, $$K_e u = F$$ becomes $$[K_e + K_\sigma(\omega) + K_L(\omega)]u = F.$$ In principle you could get expressions for $K_\sigma$ and $K_L$ for a simple situation (e.g. a rectangular straight beam) but in practice it's much easier to use a Finite Element package that includes these effects. 333