The expression looks to be one appropriate for a rotating shaft subject to presumably the first transverse flexural mode instability. Such that the shaft bows in the middle. Which is why engineers are careful to operate either well below resonant frequency or have the machinery driven up fast to well above it. Above/below critical speed is iirc how it's usually termed. Being close to critical speed would lead to small, 2nd-order axial displacements. But sorry to say such has no connection to the COM issue of #1.