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On post-resonance backward whirl in an overhung rotor with snubbing contact

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Abstract

Rotordynamic systems are central to many aerospace and heavy-duty industrial applications. The vibrational response of such systems is usually associated with forward whirl (FW) and backward whirl (BW) precessions. It is well known in the literature that the BW precession generally precedes the passage through the critical FW resonance precession. Therefore, it can be named as a pre-resonance BW frequency (Pr-BW). However, another kind of BW has been recently observed to be immediately excited after the passage through the critical FW resonance frequency in cracked rotors with anisotropic supports during run-up and coast-down operations. Consequently, this kind of BW can be named as a post-resonance backward whirl (Po-BW) precession. The Pr-BW and Po-BW phenomena are investigated here with an overhung rotor system that exhibits snubbing contact and stiffness anisotropy in the supports. Incorporating the snubbing moment couple into the equations of motion of the considered overhung rotor model yields a piecewise and strongly nonlinear system. Full-spectrum analysis is employed to capture the BW zones of rotational speeds in the whirl response. Wavelet transform spectrum analysis is also employed to determine the frequency content in the Pr-BW and the Po-BW zones. Three cases are considered in this numerical study to explore the effect of the support stiffness isotropy and anisotropy with active and inactive snubbing contact on the Po-BW excitation. For all cases, the Po-BW zones of rotational speeds are found. Moreover, the broadness and recurrence of the Po-BW zones of rotational speeds are more prominent for the cases of active snubbing contact. Even though the Pr-BW and Po-BW zones are excited at different shaft rotational speeds, they are found to possess nearly similar BW frequencies which are less than the FW resonance frequency of the considered system.

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  • 28 August 2020

    On the title page, author Oleg Shiryavev should be spelled as Oleg Shiryayev.

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Correspondence to Mohammad A. AL-Shudeifat.

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Appendix

Appendix

The kinetic energy equation of the considered system can be written according to [33, 34] as

$$ T = \frac{1}{2}I_{0} \left( {\dot{\theta }^{2} + \dot{\psi }^{2} } \right) + \frac{1}{2}I_{p} \varOmega \left( {\dot{\theta }\psi - \dot{\psi }\theta } \right) $$
(A.1)

where the potential energy is written as

$$ V = \frac{1}{2}k_{\theta } \theta^{2} + \frac{1}{2}k_{\psi } \psi^{2} $$
(A.2)

Considering for an accelerated rotor, \( \varOmega \left( t \right) = \alpha t \) where the application of the Euler–Lagrange equation yields

$$ \begin{aligned} I_{0} \ddot{\theta } + I_{p} \alpha t\dot{\psi } + k_{\theta } \theta + \frac{1}{2}\alpha I_{p} \psi & = \sum {M_{\theta } } \\ I_{0} \ddot{\psi } - I_{p} \alpha t\dot{\theta } + k_{\psi } \psi - \frac{1}{2}\alpha I_{p} \theta & = \sum {M_{\psi } } \\ \end{aligned} $$
(A.3)

For the considered physical parameters of \( k_{\theta } = k_{\psi } = 10^{6} \,{\text{N/rad}} \) and \( I_{p} = 0.3105\,{\text{kg}}\,{\text{m}}^{2} \), then \( 0.5\alpha I_{p} = 0.1553\alpha \). Therefore, the effects of the off-diagonal added terms given by \( 0.5\alpha I_{p} \) to the diagonal stiffness matrix were found to be negligible for the considered acceleration rates in this study. Considering the damping, gravity, mass unbalance, snubbing and friction moments, the equations of motion can be rewritten as in Eq. (2) in the paper

$$ \begin{aligned} I_{0} \ddot{\theta } + I_{p} \alpha t\dot{\psi } + k_{\theta } \theta + C_{\theta } \dot{\theta } = &mga - mea\left( {\left( {\alpha t} \right)^{2} \sin \left( {0.5\alpha t^{2} \, } \right) - \alpha \cos \left( {0.5\alpha t^{2} \, } \right)} \right) + M_{\theta } \\ I_{0} \ddot{\psi } - I_{p} \alpha t\dot{\theta } + k_{\psi } \psi + C_{\psi } \dot{\psi } = &mea\left( {\left( {\alpha t} \right)^{2} \cos \left( {0.5\alpha t^{2} \, } \right) + \alpha \sin \left( {0.5\alpha t^{2} \, } \right)} \right) + M_{\psi } \\ \end{aligned} $$
(A.4)

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AL-Shudeifat, M.A., Friswell, M., Shiryayev, O. et al. On post-resonance backward whirl in an overhung rotor with snubbing contact. Nonlinear Dyn 101, 741–754 (2020). https://doi.org/10.1007/s11071-020-05784-3

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