First order (i.e. `once per revolution') forced bending vibration of high speed flexible shafts is caused by the small defects of initial bend and lack of mass balance that are inevitably present in any rotor. It can be reduced to an acceptable level by modal balancing. Large modern alternator rotors are particularly sensitive to vibration and it has been found that, while accurate balancing is of cardinal importance, it is not sufficient to remove all vibration. There remains, in particular, second order (or `twice per revolution') forced vibration which arises from the dual flexural rigidity that is virtually inescapable in a two-pole machine; the motion is excited by the weight of the rotor. This has now emerged as the source of considerable difficulty, largely because it can be cured only at the design stage and cannot be `balanced'. (Certain `trimming' modifications can be made, of course, but these present formidable problems of their own.) A theoretical treatment of the problem is given which is much less restrictive than that previously available. An analytical basis is provided for further work of a more specific nature, should it be required. The motion is examined mode by mode and various properties of second order vibration are exposed. In particular it is shown that the polar representation that has been successfully used in the analysis of first order vibration is also of value with second order vibration. This is illustrated and confirmed with results taken from a 350 MW rotor.