Non-linear dynamic interactions of a Jeffcott rotor with preloaded snubber ring
Introduction
Complex rotor systems have wide industrial applications, such as aero engines, steam and gas turbines, turbogenerators, etc. Forced vibrations in these systems are usually caused by centrifugal forces due to rotor mass imbalance, and various transmission forces. Even if the rotor is well balanced, the balance may deteriorate with use. To prevent the housing from deterioration caused by the excessive amplitude vibrations the preloading of the stator is often introduced. The level of the preloading together with imbalance, stiffness and damping crucially effect the dynamic system behaviour.
A full understanding of dynamics of real rotor systems (like gas turbines) is difficult and sometimes impossible without first considering the behaviour of simple models. To study the rotor–stator interactions, two- [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] or four-degrees-of-freedom [11], [12], [13] Jeffcott rotor models and simple structures like a shaft with one or more bearings [14], [15] are often used.
Unwanted contact between the rotating and stationary parts of a rotating machine, more commonly referred to as rub, is a serious problem that has been regularly identified as a primary mode of failure in rotating machinery [13]. Rub may typically be caused by mass imbalance, turbine or compressor blade failure, defective bearings and/or seals, or by rotor misalignment, either thermal or mechanical. Several different physical events may occur during a contact between the rotor and the stator: an impact, friction between the two contacting parts, and a significant increase in the stiffness of the rotating system whilst contact is maintained, to name just three. The behaviour of the system in this case is highly non-linear and responses may be chaotic.
Rotor–stator rub interactions in rotating assemblies have attracted great attention from researches [1], [2], [4], [9], [10], [11], [12], [13], [16], [17]. Choy and Padovan [1] studied the transient response associated with rubbing between a bladed rotor and its housing. Muszynska and Goldman [2] showed that during the impact of a rotating shaft against a stationary element, such as in the rotor-to-stator rub case, not only the radial (straight impact) effects must be taken into consideration, but, due to rotation, also the tangential effects. For the rub-impact Jeffcott rotor system supported on oil film bearings Chu and Zhang [11] found out four different scenarios to and out of chaos, namely, periodic, quasi-periodic, intermittent and period doubling bifurcation routes. In reference [9] Zhang et al. indicated that the non-linear rub-impact interactions caused by the rotor imbalance change the stable periodic motion of the system to quasi-periodic and chaotic motion and introduce period-doubling and grazing bifurcations. The reliability of the rotor system with rubbing was investigated theoretically by Chu and Zhang [10]. Dai et al. [12], [16] proved that it is useful to introduce stops to limit violent vibration of rotor–bearing systems and that the friction factor plays an important role in the performance of the stops. Edwards et al. [13] and Al-Bedoor [17] present the models for the torsional and lateral vibrations of imbalanced rotors that account for the rotor-to-stator rubbing.
Previous studies on rotor systems have also addressed many important scientific and technical issues regarding stability [3], [6], [18], [19]. A few papers concern control of the dynamic responses using smart materials and structures [8], [14]. Zeng and Wang [8] created the model of the electromagnetic balancing regulator allowing the non-contact electromagnetic force to drive the correction masses so as to generate a suitable correction weight. Vibration control of a rotor system by a disk type electrorheological damper was applied by Yao et al. [14] to a six-degrees-of-freedom shaft model. This study showed that a controllable damping can suppress the large amplitude vibrations and sudden unbalance responses. Spontaneous sidebanding was investigated by Ehrich [5].
Separate attention of the researchers was given to rotor–stator impacts, which were studied both in the presence of the friction between the contacting parts (see, for example, Ref. [20]), and without it [7], [21], [22], [23]. Three different models have been used so far [13]: (i) classical restitution coefficient approach, (ii) non-elastic impact with a zero restitution coefficient, where the impact is followed by a sliding stage, and (iii) piecewise approach, with extra stiffness and damping terms included during the contact stage. Using the third approach under assumption of absence of the friction force Neilson and Barr [21] and Gonsalves et al. [7] showed a reasonable correlation between the experimental and numerical results for four- and two-degrees-of-freedom Jeffcott rotor models. Gonsalves et al. [7] have also identified the existence of chaotic vibrations for a Jeffcott rotor purely due to impacts. The recent papers by Karpenko et al. [22], [23] studying the same type of rotor–stator impact confirmed these findings and have also showed an existence of multiple attractors and fractal basins of attractions.
Despite a significant number of theoretical and experimental investigations conducted so far, one of the important characteristics such as the effect of preloading of a snubber ring on the dynamics of a Jeffcott rotor model, which is an additional source of non-linearity in the rotor–stator system, has not been properly addressed yet. Therefore in this paper, a mathematical model of a Jeffcott rotor with a preloaded snubber ring is developed and the resultant system dynamics is studied.
Section snippets
Physical model
A two-degrees-of-freedom model of the rotor system with a preloaded snubber ring is shown in Fig. 1a. The excitation is provided by an out-of-balance rotating mass mρ. During operation the rotor of mass M makes intermittent contact with the snubber ring. It is assumed that contact is non-impulsive and that the friction between the snubber ring and the rotor is neglected. Since the mass ratio between the snubber ring and the mass of the rotor is small (for existing experimental rig it is equal
Force in the snubber ring
It is assumed that the rotor and the snubber ring are in contact and the rotor moves the snubber ring in the direction shown in Fig. 3. The forces F1, F2, F3 and F4 generated in the snubber ring as a result of the rotor and the snubber ring interactions can be described in vector form asIt is convenient to define the force in the snubber ring Fs
Contact regimes
Let one now consider the motion of the rotor in detail. As mentioned earlier the rotor can move either in or out of contact with the snubber ring. During the contact the force acting between the rotor and the snubber ring significantly depends on the “depth” of contact and four different regimes can occur as listed in Section 2. This can be clearly explained using (xr,yr) plane where each regime is mapped into an associated region as shown in Fig. 5. The boundaries between regimes I, II, III,
Numerical investigations
Numerical results presented in this section are to demonstrate the use of the developed analytical formulas and show the influence of the preloading on the dynamics of the rotor crossing different regions of operation.
Bifurcation diagrams shown in Fig. 8 were constructed for the displacement of the rotor under varying the frequency ratio η for the unpreloaded (Fig. 8a) and the preloaded (Fig. 8b) cases. The control parameter η was set to the leftmost value 2. Starting with zero initial
Conclusions
Two-degrees-of-freedom model of the Jeffcott rotor with the preloaded snubber ring subjected to out-of-balance excitation was developed, and dynamic interactions between the rotor and the preloaded snubber ring were studied. During operation the rotor can be in one of five different contact regimes, for which boundaries on (xr,yr) plane have been analytically determined. The current location of the snubber ring has been obtained using the principle of the minimum elastic energy in the snubber
Acknowledgements
The financial support provided by EPSRC, Rolls-Royce plc. and ORS award scheme is gratefully acknowledged.
References (23)
- et al.
Non-linear transient analysis of rotor-casing rub events
Journal of Sound and Vibration
(1987) - et al.
Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs
Chaos, Solitons and Fractals
(1995) Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances
Mechanism and Machine Theory
(1996)- et al.
The harmonic balance method with arc-length continuation in rotor/stator contact problems
Journal of Sound and Vibration
(2001) - et al.
The influence of the electromagnetic balancing regulator on the rotor system
Journal of Sound and Vibration
(1999) - et al.
Bifurcation and chaos in a rub-impact Jeffcott rotor system
Journal of Sound and Vibration
(1998) - et al.
Periodic quasi-periodic and chaotic vibrations of a rub-impact rotor system supported on oil film bearings
International Journal of Engineering Science
(1997) - et al.
The partial and full rubbing of a flywheel rotor–bearing–stop system
International Journal of Mechanical Sciences
(2001) - et al.
The influence of torsion on rotor/stator contact in rotating machinery
Journal of Sound and Vibration
(1999) - et al.
Vibration control of a rotor system by disk type electrorheological damper
Journal of Sound and Vibration
(1999)
Dynamic analysis and reduced order modelling of flexible rotor–bearing system
Computers and Structures
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