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Torque and the Flexural Stability of a Cantilever: Extension of the Bernoulli‐Euler Theory of Bending to Cover the Whirling of Shafts Transmitting Power

Aircraft Engineering and Aerospace Technology

ISSN: 0002-2667

Article publication date: 1 December 1951


1. Introduction THE effect of torque is usually neglected in the Bernoulli‐Euler treatment of the flexure of thin rods. In practical cases the torque will rarely have any appreciable influence on static stability but from the dynamical aspect it may be the precursor of exponential instability in the absence of adequate damping forces. This phenomenon was suggested to me by Mr A. C. Hutchinson of Aliens, Bedford. Owing to torque there are cross deflexion coefficients which in particular cases are equal and opposite; and it is this negative reciprocity that gives rise to dynamic instability as we shall see. In the case, say, of a load on an overhung shaft it would appear that this instability is manifest at all speeds becoming more violent with the relative increase of torque. In practice it has frequently been found to be difficult if not impossible safely to run through particular ‘whirling’ speeds. In this connexion the problem of the stability of rotating shafts has reached a fresh peak of importance in consequence of the rapid development of the gas turbine as the prime mover in aircraft. The fundamental principles underlying the treatment of whirling phenomena are now well established and there is ample practical experience in support of the physical notion of shaft revolution propounded by Jeffcott in the Phil. Mag. for March 1919. Thus if a shaft is perfectly straight, completely balanced, and runs in bearings which are truly aligned; then, provided such masses as may be carried by the shaft have no appreciable moments of inertia, the normal ‘static’ flexural vibrations of the shaft will be independent of any imposed rotation, that is the path of any point on the neutral axis of the shaft will be uninfluenced by the rotation of the shaft section of which it is the centre. If the shaft is initially deflected by its weight, then the static vibrations will occur about the position of rest of the shaft and the imposed rotation will only apply to the cross‐sections of the shaft about their centres. If, however, the shaft is initially bent then each element may be considered as being ‘out of balance’ by the amount it is off the position it would occupy if the shaft were true and at rest. If we apply power to a bent shaft such as to maintain it in rotation with constant angular velocity then each element of shaft is regarded as being compelled to rotate with this constant angular velocity about the point on the effective elastic axis of the shaft, that is the axis of the deflected but corresponding unbent shaft. The unbalance will thus give rise to a forced vibration which will compel each element of the shaft to describe a circular path about its appropriate centre on the effective elastic axis; and this forced circular vibration will be independent of the static vibrations which are unforced and in consequence described as ‘free’. The radius of the path of any element in the forced circular vibration will depend inter alia on the speed of the imposed rotation, and the strain energy of the shaft so bent by flexure will be drawn or absorbed from the power which drives the shaft.


Morris, J. (1951), "Torque and the Flexural Stability of a Cantilever: Extension of the Bernoulli‐Euler Theory of Bending to Cover the Whirling of Shafts Transmitting Power", Aircraft Engineering and Aerospace Technology, Vol. 23 No. 12, pp. 375-376.




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