The natural frequencies of a thin rotating cantilever with offset root

In this paper previous analyses of rotating cantilevers are extended to allow consideration of the case where the root offset from the spin axis is such that part of the beam is directed radially inwards towards the axis of rotation. In this situation the fundamental natural frequency is a function of the spin frequency. The form of the functional relationship depends on the value of r, the ratio of root offset to cantilever length. It is shown that, for certain values of r, it is possible to select a value of spin such that the fundamental natural frequency and the spin frequency coincide, the beam then being "tuned". It is also possible to choose values of r such that the fundamental natural frequency reduces to zero as the spin is increased, and buckling occurs. It is further shown that by suitable choice of beam length for a given root offset the tuning and buckling frequencies can be minimized. The behaviour of higher modes of vibration is also discussed.