Features of the dynamics of a rotating shaft with nonlinear models of internal damping and elasticity

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Abstract

The paper analyzes the influence of nonlinear (cubic) internal damping (in the Kelvin–Feucht model) and cubic nonlinearity of elastic forces on the dynamics of a rotating flexible shaft with a distributed mass. The shaft is modeled by a Bernoulli–Euler rod using the Green function, the discretization and reduction of the problem of rotating shaft dynamics to an integral equation are performed. It is revealed that in such a system there is always a branch of limited periodic movements (self-oscillations) at a supercritical rotation speed. In addition, with low internal damping, the periodic branch continues into the subcritical region: when the critical velocity is reached, the subcritical Poincare–Andronov–Hopf bifurcation is realized and there is an unstable branch of periodic movements, below the branch of stable periodic self-oscillations (the occurrence of hysteresis with a change in rotation speed). With an increase in the internal friction coefficient, the hysteresis phenomenon disappears and at a critical rotation speed, a soft excitation of self-oscillations of the rotating shaft occurs through the supercritical Poincare–Andronov–Hopf bifurcation.

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About the authors

А. А. Azarov

Bauman Moscow State Technical University

Author for correspondence.
Email: 13azarov.ru@gmail.com
Russian Federation, Moscow

A. М. Gouskov

Bauman Moscow State Technical University; Mechanical Engineering Research Institute of RAN

Email: gouskov_am@mail.ru
Russian Federation, Moscow; Moscow

G. Y. Panovko

Mechanical Engineering Research Institute of RAN

Email: gpanovko@yandex.ru
Russian Federation, Moscow

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Supplementary files

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2. Fig. 1. Calculation scheme of a rotating shaft: 1 – precession trajectory, 2 – direction of rotation.

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3. Fig. 2. Argand diagram in the range of rotation speeds.

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4. Fig. 3. Displacements of the middle node at the supercritical speed at , , : (a) , ; (b) , ; (c) , .

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5. Fig. 4. Bifurcation diagram along the upper periodic branch, A – limit point, B – subcritical bifurcation ( , , , , ).

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6. Fig. 5. Precession of the cross-section of the rod corresponding to the middle node.

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7. Fig. 6. Distribution of the average value of the precession coefficient along the upper stable branch ( , , , , ).

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8. Fig. 7. Trajectories of collocation nodes during steady motion (black lines) and the shape of the curved axis (red lines) ( , , , , , ).

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9. Fig. 8. Dependence of and on the coefficient (a); distribution of the amplitudes of the displacements of the middle node during subcritical (b) and supercritical (c) bifurcations ( , , , ).

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