Nonlinear dynamics of a spinning shaft with non-constant rotating speed
Nonlinear dynamics of a spinning shaft with non-constant rotating speed
Fotios Georgiades 0
0 F. Georgiades (
Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spindown operation. In this article, initially the equations of motion for a spinning shaft with non-constant speed are derived, then the system is discretised, and finally a nonlinear dynamic analysis is performed using multiple scales perturbation method. The system in firstorder approximation takes the form of two coupled sets of paired equations. The first pair describes the torsional and the rigid body rotation, whilst the second consists of the equations describing the two lateral bending motions. Notably, equations of the lateral bending motions of first-order approximation coincide with the system in case of constant rotating speed, and considering the amplitude modulation equations, as it is shown, there are detuning frequencies from the Campbell diagram. The nonlinear normal modes of the system have been determined analytically up to the second-order approximation. The comparison of the analytical solutions with direct numerical simulations shows good agreement up to the validity of the performed analysis. Finally, it is shown that the Campbell diagram in the case of spin-up or spin-down operation cannot describe the critical situations of the shaft. This work paves the way, for new safe operational 'modes' of rotating structures bypassing critical situations, and also it is essential to identify the validity of the tools for defining critical situations in rotating structures with non-constant rotating speeds, which can be applied not only in spinning shafts but in all rotating structures.
Spinning shaft; Non-constant rotating speed; Nonlinear normal modes; Critical speeds; Campbell diagram
1 Introduction
Starting about 93 years ago, with the pioneered
seminal work by Campbell [
1–3
], the main theory was
developed to examine critical situations in vibrations
of turbine wheels in constant rotating speeds. This
work is the basis of the current examination of
critical speeds of rotating structures in steady states using
the diagram that indicates how the natural frequencies
of the structure vary with the rotating speed (limited
to steady states) incorporating the excitation frequency
due to rotating speed, which forms the Campbell
diagram (CD). Since then, based on CD, plenty of research
articles have been reported about rotating structures and
spinning shafts but restricted mainly to steady states.
Extended literature review on critical speeds on steady
states is out of the scope of this work. Only a few
articles are related in examining their dynamics during
spin-up and spin-down operation, which corresponds
to non-constant rotating speed. Plaut and Wauer [
4
]
examined parametric, external and combination
resonances in coupled flexural and torsional oscillations of
an unbalanced rotating shaft with non-constant
rotating speed, but the rigid body equation of motion for
non-constant rotating speed was neglected. Suherman
and Plaut [
5
] used flexible internal support in order
to mitigate lateral bending vibrations. In [
5
], a model
was developed and dynamics for a spinning shaft with
non-constant rotating speed was examined including
a flexible internal support considering also the
equation of rigid body motion, but the torsional motion was
neglected in this treatment. Wauer [
6
] modelled and
formulated equations of motion for cracked beams
considering non-constant rotating speed, but without
considering the rigid body equation of motion. It should be
commented that in [
4–6
] the models are not considering
all the motions in order to perform nonlinear dynamic
analysis and the results are limited to these models. In
[
7
], the equations of motion of a spinning shaft with
dynamic boundary conditions (eccentric sleeves) were
derived, since the main work was about the dynamics
of the shaft due to the particular dynamic boundary
conditions; although non-constant rotating speed was
considered, it was not given any special attention.
Further work has been conducted in the so-called
non-ideal systems, which correspond to rotating
mechanical systems incorporating the electromechanical
coupling with the DC motor to examine Sommerfeld effect
but limited to discrete systems with the excitation of
natural frequencies by the external torque of the motor
[
8–10
]. The significance of considering non-ideal
systems is discussed in [
11
], whereas there is comparison
in dynamic results between ideal and non-ideal
systems. Although in this area of research it is considered
in some cases non-constant rotating speed, the work is
focused on the effect of external torque through the DC
motor in the nonlinear dynamics of these
electromechanical systems, and it is also restricted to discrete
models. In [
12
], nonlinear dynamics of rings rotating
w (...truncated)