Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Nonlinear Dynamics, Nov 2017

Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spin-down 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 first-order 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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2Fs11071-017-3888-0.pdf

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)


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs11071-017-3888-0.pdf

Fotios Georgiades. Nonlinear dynamics of a spinning shaft with non-constant rotating speed, Nonlinear Dynamics, 2017, pp. 1-30, DOI: 10.1007/s11071-017-3888-0