Cohen: Effect of the coupling between flexural and torsional vibrations on rotors driven by a flexible shafts

ABSTRACT: The present study treats the unstable regions of high speed rotors, caused by the coupling of torsional and flexural vibrations.

initially the transverse reactions, which appear at the ends of a weightless shaft, mounted on wide bearings at one of its ends, were calculated. These calculations were carried out as a function of the displacements, angular inclination and of the axial moment transmitted by the shaft. Integration of equilibrium equations, assuming that the axial moment and the displacements are small, revealed that here too - as in the case of a rigid shaft, driven by a constant velocity joint - there will appear at the ends of the shaft transverse moments equal to the product of the axial moment and one half of the inclination angle.

These results were used to define some of the forces and moments involved in deriving the equations of motion of an unbalanced rotor, driven by a weightless shaft with elastic response to torsional and flexural forces. The transformation of these equations to a rotary coordinate system yielded a set of nonlinear equations without periodic coefficients. Linearization of the equations near the quasi-stationary solution, constrained by the imbalance, made it possible to investigate stability by solving for the eigenvalues by numerical methods.

In addition an approximate analytical study of the instability was performed by expanding the characteristic polynomial by means of a Taylor's series around the eigen values of the free system. As a result eight focal points were defined around which instability zones may be generated. Of these, the most dangerous ones are the four combination resonance regions in which the frequency of the angular velocity of the rotor corresponds to the sum or to the difference between a flexural and torsional eigen value. The remaining two are produced if the system has multiple eigen values - are excited by constraint terms of the third order.

By using an approximate analysis method, we were able to study the effect of the load torque on the stability, disregarding the coupling between flexural and torsional vibrations. It was found, that to a first approximation, the diverging effect of the load torque is mainly due to the angular inclination of the rotor. Accordingly, deflection damping alone will not suffice to offset the diverging effect of the load torque, and the inclination, too, must be damped; this applies in particular to fast rates of rotation. However, only a small increment of displacement damping will make it possible to markedly reduce the inclination damping required to stabilize the system.

The nonlinear behaviour of the system in the combination resonance zones was investigated by an asymptotic method. This investigation yielded a number of interesting findings. Firstly, it is just when the system is unstable, that there is a tendency for the ratio between the amplitudes of transverse vibrations and torsional vibrations, and for the common phase angle to become stabilized. Secondly, the extension of instability regions, where only one kind of damping is found, is due to the fact that in such a case the undamped degree of freedom continues to be excited at a frequency which is very close to its eigen frequency, even if the rotating frequency is far away from the combination resonance frequency.

A study of the effect of the axial moment on the critical speeds of a continuous shaft (rotor) mounted on narrow bearing at both ends showed that in most previous studies on this subject, incorrect boundary conditions were chosen, and incorrect critical speeds were obtained as a result. This error increases with increasing load torque, and becomes particularly significant if the length-to-diameter ratio of the shaft is large.

ABSTRACT IN HEBREW

54 references:

Bousso, D. (1933-1971), "A stability criterion for rotating shafts," D.Sc. thesis, Technion, Haifa, Israel (1963) (In Hebrew) synopsis in English.
Bousso, D. (1933-1971), "A stability criterion for rotating shafts," Israel J. Technol., v. 10, pp. 409-423 (1972) Abstract and reviews in English.
Ben-Amots, N., "The motion of a high-speed rotor under the influence of a moment perpendicular to the axes of precession and nutation," D.Sc. thesis, Technion, Haifa, Israel (1975) (In Hebrew) Abstract and review in English.
Porat, I. (1934-2012), "Moments transmission by a universal joint," Mechanism and Machine theory, v. 15, No. 4, pp. 245-254 (1980).

and more 50 references.

● COMMENT (handwritten): To Netsivi, Thanks for the useful discussions, Rafi Cohen.

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