Rotor-stator contact in multiple degree of freedom systems

Transcription

1 Rotor-stator contact in multiple degree of freedom systems A. D. Shaw 1, A. R. Champneys 2, M. I. Friswell 1 1 Swansea University, College of Engineering, UK 2 University of Bristol, Department of Engineering Mathematics, UK ABSTRACT Nonlinear effects due to impact between a rotor and the stator can lead to the sudden onset of violent chattering or whirling motions that can cause significant damage. Previous work has shown that a synchronisation condition between the forward and backward whirl modes makes a rotor system susceptible to bouncing limit cycles at certain drive speeds. This work considers a two disc system, with contact occurring at a single stator. Using a brute-force bifurcation approach, we demonstrate that there are drivespeeds where multiple steady state bouncing orbits can occur. Very rich dynamics are seen including chaotic motion. 1 INTRODUCTION The vibrations of rotating machinery are an important issue in many engineered systems, from aircraft engines to drilling platforms (1). Understanding of rotating machinery in linear operating regimes is well established, with classical matrix methods having been applied to deduce the whirl speeds of rotational systems with any number of degrees of freedom. However, there are many outstanding problems concerning nonlinearity in rotating systems, and these are leading to new approaches in a variety of industrial applications. Patel and Darpe noted that cracks in rotors induce nonlinear effects and explored these effects with bifurcation analyses (2). Ehrich has compiled a review of numerous nonlinear phenomena witnessed in tests on turbomachinery, including bifurcation routes to chaos, subharmonic resonance and other surprising effects such as bearing phenomenon that lead to a rotor slowly switching between two amplitudes of vibration (3). Contacts are a major source of nonlinearity in rotor systems; Jacquet-Richardet et al. published a comprehensive review on rotor-stator contact in turbomachinery (4). Many studies are inspired by the complex needs of the drilling industry; a modern deep hole drilling assembly is a highly flexible rotor structure (relative to its length), subject to strong nonlinearity through contact forces. The complexity of this system is such that few studies describe it entirely; the majority of papers consist of far simpler studies on small aspects of the mechanics present. Richard et al. modelled the drill bit behaviour, demonstrating a stick-slip motion caused by the contact friction in combination with both axial and torsional modes of the shaft (5). Germay et al. extended this work to consider the drill as a continuous structure, instead of as an equivalent discrete model (6). Many of these phenomena are reported in on-site measurements reported by Leine et al. (7). Liu et al. considered a model of the drill bit that allows non-smooth effects due to both friction and loss of contact when axial vibrations cause the drill bit to separate from the rock surface (8).

2 In other work, Karpenko et al. considered piecewise smooth models for rotors experiencing frictionless impacts with a snubber ring (9). Karpenko et al. went on to compare these predictions with experiment in (10). Edwards et al. considered a simple Jeffcott model with contact, and used time simulation to show that the bifurcation pattern changes significantly if torsional motion occurs. The majority of studies in this field concern systems with relatively few degrees of freedom typically they concern single disc systems, perhaps with the inclusion of torsional and flapping motions. There seems to be little literature aiming to handle nonlinearity in rotor systems with many more degrees of freedom. One exception to this is the experimental study by Chu and Lu on a 2 disc system (11). They showed some highly complex multi-period and quasiperiodic behaviour, although the authors comment that they could not directly trace the route into chaos due to experimental control issues. However, an alternative possible explanation for this is raised in a paper by Zilli et al. (12). This showed how a synchronisation between the forward and backward whirl modes and the forcing of a single snubbed rotor system could cause a sudden onset of complex bouncing behaviour at certain drive speeds that are not near the linear critical speeds. In (13), the authors generalised the model in (12) to work with any two modes of a multi disk system, with any combination of forward and backward whirling. However, it was shown that the predictions for bouncing orbits based on linear whirling speeds only gave very approximate predictions for when bouncing was likely to occur, because the nonlinearity caused the whirling speeds to increase. Furthermore, it predicted individual frequencies that would experience bouncing, whereas in truth a similar bouncing oscillation may be sustained at various drive speeds. In the present work, a brute force numerical bifurcation approach is used to demonstrate that multiple forms of the bouncing motion can occur at a single drive speed. However, in nearly all cases these can be related to underlying whirling modes of the system by the synchronisation condition. Firstly the explanation of how synchronisation is required to allow bouncing oscillation to be sustained is recalled. Then the methods used for simulation and brute-force bifurcation are detailed and results are presented, for both a system with low contact stiffness and a purely elastic stator which permits very rich responses, and then for a system with the more common situation of a highly stiff contact stiffness and damped contacts. 2 SYNCHRONISATION AS THE CAUSE OF BOUNCING ORBITS The vibrations of rotating systems can be decomposed into their underlying whirling modes; these result from solving the eigenproblem resulting from the linear parts of the equations of motion. These whirling modes are circular orbits of the shaft, where the amplitude is a function of the axial location on the shaft given by the relevant eigenvector. The whirl speed or frequency comes from the relevant eigenvalue, and varies with shaft drive speed due to gyroscopic effects (1). For rotating systems with multiple whirling modes, a necessary condition for a periodic motion with stator contacts is that at repeating contacts there is an identical state of the rotor and the forcing, relative to one another. This state can be considered as a superposition of phasors, representing the whirling modes of the shaft and the forcing. If it is assumed (without loss of generality) that all active phasors are in alignment at the start of a cycle, then it follows that they must be in alignment at the end of the cycle. In the following paragraphs this condition is developed mathematically for just two arbitrarily chosen whirling modes with

3 frequencies ω n1 and ω n2 that are functions of driving speed Ω and it emerges that this synchronisation cannot occur when more than two modes are active. The whirling speeds may be positive or negative, indicating forward or backward whirling respectively, whilst the forcing speed is always positive. A convention that ω n2 ω n1 is adopted. The two whirling phasors will align periodically, with period given by: τ = 2π ω n1 ω n2 (1) Note that if the two speeds are opposite, this period will be shorter than either of the whirling periods. If the two speeds have the same sign, then this period is always greater than that of mode 2, and can be very large if the two speeds are similar. During this period, the first mode phasor will advance by an angle given by: γ = ω n1 τ (2) At the same time, the forcing phasor will also advance by an angle α = Ωτ (3) A periodic alignment of all three phasors will occur if α = γ + 2πm (4) where m is an integer that will be positive if α > γ, and negative for α < γ. Substitution of Eqs. (1) to (3) into Eq. (4) gives a relationship for driving speeds that allow periodic contact motion: Ω = ω n1 + m ω n2 ω n1 (5) Equation (5) gives a simple frequency relationship that is necessary for sustained bouncing oscillation. Note that for further whirling modes to become synchronised, they would need to have similar integer based relationships with Ω, which is practically impossible, hence all motions can be explained in terms of just two modes. In (13) it was seen that predictions made with (5) are not very accurate because the system shows amplitude dependence, hence the natural frequencies can vary. This also means synchronisation can occur over a range of different drive speeds instead of just at one. However, the insight that exactly two modes are active in any bouncing motion is invaluable. 3 BRUTE FORCE SIMULATION ANALYSIS Bifurcation analysis is an invaluable and ubiquitous framework for understanding nonlinear systems, by identifying and classifying points at which their responses undergo qualitative changes (15). However, in cases such as this one with relatively little analytical knowledge of the underlying system and its responses, and with the further complication of a discontinuous nonlinearity, the only means of obtaining a bifurcation diagram can be through large numbers of numerical timesimulations. This approach, known as brute force bifurcation, is performed here.

4 At a given driving speed, a randomised initial condition is created. A time simulation is then run for sufficient time to achieve a state of dynamic stability. A simple projection is then used to calculate the final steady state motion - in this case the motion is simply characterised by the maximum and minimum radius of the stator node during oscillation. For these tests, the minimum and maximum are taken over 250 drive rotations once the data has settled. The above procedure is repeated 25 times at each driving frequency; this effectively forms a Monte-Carlo analysis of the stable steady state responses of the system at a given driving speed. The driving speed is then iterated at speeds from 10rpm to 500rpm at unit intervals to build up the bifurcation diagram, hence the resulting bifurcation diagrams represent the results of separate simulations. 3.1 Test system The system under test is the same simple two disc system as used in (13), shown in Fig. 1. The rotor is mounted on pinned bearings at each end of the shaft, with a stator with clearance at the centre of δ =1mm. The shaft is a steel tube with 5mm outer diameter and thickness 1mm. The discs are also made of steel with radii 0.15m and 0.2m and thicknesses 0.01m and 0.02m respectively. The shaft is 0.60m long, with the smaller disc located 0.15m from the left hand end, and the larger disc located 0.20m from the right hand end. The stator is modelled as an additional linear stiffness k s and linear damper c s always acting in the radial direction towards the centre. Forcing is provided by a small eccentricity ε = on the smaller disc. Damping is provided through a damping matrix that is equal to 1% of the shaft stiffness matrix. Figure 1: Test system 3.2 Details of time simulation The system is mostly modelled using the free Matlab (14) scripts that accompany (1). This was used to implement a finite element model of the structure, including matrices that capture the gyroscopic effects due to the rotating shaft. This system with 48 degrees of freedom was reduced to 8 degrees of freedom using the first 8 mode shapes of the non rotating system. This included all the major modes of the system; modes 9 and above were at very much higher modal frequencies. The system was then transformed to state space form for simulation, hence the number

5 of degrees of freedom became sixteen to include the time derivatives of the original degrees of freedom. The simulation itself which was carried out in ODE45 with event detection used to locate changes between contacting and non-contacting motion. The event detection function halted the simulation every time the radial displacement of the contact node crossed the clearance threshold, and then the simulation would be restarted with optimal properties for the new state of contact or noncontact. The nonlinear force was evaluated by locating the lateral displacements of the shaft within the stator {u c,v c} T in the spatial domain, then evaluating the force vector: [k s (δ r) c s r ] { u c v } /r if r δ F c = { c { 0 (6) 0 } otherwise where r = u c 2 + v c 2. This force was then transformed back to the reduced modal form. The random initial condition consisted of the rotor at zero displacement, but at a random velocity in terms of the first eight static mode shapes of the rotor. This was scaled so that all cases had identical kinetic energy within their transients. 4 RESULTS 4.1 A low-stiffness snubber ring In this subsection, we present results where k s is at the relatively low value of N/m and no damping is applied in the stator i.e. c s = 0. This is typical of a snubber ring type stator. The brute force analysis took 5 hours and 11 minutes to run on a PC with a 64 bit, 2.67Ghz, 12 core Intel Xeon CPU, with 24Gb of RAM installed. Figure 2 shows the results of the brute force bifurcation process as described in Section 3. As can be seen, below 280rpm all markers coincide; since in all cases the maximum and minimum of the orbit are equal, it can easily be concluded that these cases always settle to constant whirling motion. This motion is either linear forward whirling, or in the case of the higher amplitude oscillations at rpm, constant contact whirl. In the region of approximately rpm, both solutions are stable, and the initial condition determines which orbit will occur in the steady state. By following the line from this region, it can easily be confirmed that the linear forward whirl remains a stable solution all the way up to 500rpm. Figure 2 shows the results of the brute force bifurcation process as described in Section 3. As can be seen, below 280rpm all markers coincide; since in all cases the maximum and minimum of the orbit are equal, it can easily be concluded that these cases always settle to constant whirling motion. This motion is either linear forward whirling, or in the case of the higher amplitude oscillations at rpm, constant contact whirl. In the region of approximately rpm, both solutions are stable, and the initial condition determines which orbit will occur in the steady state. By following the line from this region, it can easily be confirmed that the linear forward whirl remains a stable solution all the way up to 500rpm.

6 At 280rpm, a bouncing motion begins to occur for many cases. This motion can be shown to be a synchronisation of the first backward and first whirling mode, with m=2. Details on how this is done is given in (13). By following the line of maxima and minima of the oscillating solution, it may be seen that this family of responses exists and is stable up to approximately 397rpm. However it may be seen that there are discontinuities in the families of points. It emerges that these correspond to change-of-period bifurcations, as seen in Fig. 3, which shows how the kink in the line of maxima at 303rpm corresponds to a period-doubling. This shows how at 302rpm all contacts are identical, whereas at 304rpm there is a gentle oscillation in the maximum r that is reached. By 312rpm this has grown into alternate contacts being very different to each other. Figure 2: Results of brute force bifurcation for k s = N/m. Figure 3: Excerpt from the time series of the radial distance from centre r of the stator node, for the bouncing orbits at 302rpm, 304rpm and 312rpm, respectively from left to right. Crosses indicate a transition from contacting to noncontacting motion, or vice versa.

7 4.2 A rigid and damped stator In many cases, the stator will be highly rigid compared to the shaft. Furthermore, the stator will have an additional damping effect, and this will act to suppress many of the oscillations that make large incursions into the contact region as seen in Fig. 2. In order to see these effect, a second brute force diagram is presented with k s = N/m and also giving the contact model a damper of c s = Ns/m. This analysis took 14 hours and 27 minutes to run on, on the same computer used in the previous case. The results of this are shown in Fig. 4. It is clear that the stiffer but less elastic contacts have had the effect of stabilising the constant contact response, at the expense of causing the majority of stable bouncing orbit solutions to disappear. However, the same response family that begins at 280rpm still exists, although it now begins at 308rpm. Furthermore, the way that the response evolves as the driving speed is different. Figure 5 shows a series of time signals of r, at increasing rotor speed. The period-doubling seen earlier does not occur, however the orbits start to contain multiple impacts in each cycle (this may be seen by observing the crosses that signify crossings of the clearance threshold). However, despite the increasing complexity of the motion, the two underlying whirling speeds (and the forcing frequency) remain clearly visible in the spectrum of displacement as shown in Fig. 6. Figure. 4: Results of brute force bifurcation for k s = N/m with contact damper c s = Ns/m.

8 Figure 5: Time series of bouncing oscillations for k s = N/m with contact damper c s = Ns/m. Crosses indicate a transition from contacting to noncontacting motion, or vice versa. The final region of particular interest is the response at a drive speed of 357rpm, where Fig. 4 shows a large of slightly different minimum values for r. This signifies the onset of chaotic motion, as the period of oscillation is now so long that even the long time period used for the sample is insufficient to establish a consistent minimum. An example of the chaotic orbit is shown in 7. In this case, the orbit can no longer be represented with just two whirl modes. However it appears that the synchronisation effect at least provides the pathway to chaos. Figure 6: Frequency spectrum of x response for rotor with k s = N/m with contact damper c s = Ns/m, driven at 335rpm and exhibiting bouncing response.

9 Figure 7: Chaotic orbits for k s = N/m with contact damper c s = Ns/m, at 357rpm. Crosses indicate a transition from contacting to noncontacting motion, or vice versa. 5 CONCLUSIONS AND FUTURE WORK A phasor analysis of when contacting oscillations of a snubbed rotor shaft may occur has been extended to shafts with higher numbers of degrees of freedom. Whilst there is a clear link between the underlying whirl modes and the onset of these oscillations, predictions are somewhat approximate, and the effect of nonlinearity on the whirling frequencies needs to be understood to improve accuracy. It has been seen that while these bouncing orbits are not in themselves chaotic, they can lead to chaotic behaviour. The demonstration that bouncing orbits can be largely represented in terms of just two underlying modes is an invaluable insight that should lead to the development of more accurate predictive models. Furthermore, the authors are currently developing an experimental means of verifying these findings. ACKNOWLEDGEMENTS The research leading to these results has received funding from the EPSRC grant Engineering Nonlinearity EP/G036772/1. REFERENCE LIST (1) Friswell, M. I., Penny, J. E. T., Garvey, S. D. and Lees, A. W., Dynamics of Rotating Machines, Cambridge, (2) Patel, T. H. and Darpe, A. K., Influence of crack breathing model on nonlinear dynamics of a cracked rotor, Journal of Sound and Vibration, Vol. 311, No. 3, pp , (3) Ehrich, F. F., Observations of nonlinear phenomena in rotordynamics, Journal of system design and dynamics, Vol. 2, No. 3, pp , 2008.

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