INSTABILITY OF A FLEXIBLE ROTOR PARTIALLY FILLED WITH FLUID

Transcription

1 INSTABILITY OF A FLEXIBLE ROTOR PARTIALLY FILLED WITH FLUID Zhu Changsheng College of Electrical Engineering Zhejiang University, Hangzhou, , Zhejiang, P. R. of China cszhu@hotmail.com Tel: Fax: ABSTRACT The instability of an over-hung flexible rotor partially filled with fluid is experimentally studied. More attentions were paid on the developing process of the unstable motion, the dynamical behavior of the rotor system during the instability occurs, and the whirling characteristics of the rotor system within the unstable speed region. The effect of the fluid-filled ratio on the whirling frequency, the unstable region and the rotor vibration amplitude are also dealt with. It is shown that the rotor system in the unstable region of rotational speeds does not whirl at either a constant speed or the critical speed of the empty rotor system. The whirling frequency of the rotor system in the unstable region depends on not only the rotational speed, but also the volume of the fluid filled in the chamber. The unstable motion characterized by the sub-synchronous forward motion occurs when the rotational speed is just over the critical speed of the rotor system fully filled with the fluid. 1. INTRODUCTION There are some cavity components partially filled fluid in part rotating machinery. The interaction between the motion of the rotor and the motion of the rotating fluid in its cavity may lead to self-excited vibration and even instability. The resulting instability causes sharply increase of the rotor vibration amplitude and introduces a dangerous asynchronous whirling motion. These phenomena are of technical importance with fluid-filled centrifuges, fluid-cooled gas turbines, spin-stabilized satellites and rockets containing liquid fuels. Many researchers [1-8] have theoretically explained the instability of a rotor partially filled with fluid observed experimentally by Kollmann in 1960s[1], but some of conclusions obtained theoretically are contradictive. The experimental results[9-13] re not enough to verify these theoretical models. It is necessarily to carry out thoroughly the experimental research on the instability of a rotor partially filled with fluid. The objective of this paper is experimentally to investigate the instability of an over-hung flexible rotor partially filled with fluid. More attentions are paid on the developing process of the unstable motion and the dynamical behavior of the rotor system during the instability occurs, and the whirling characteristics of the rotor system within the unstable speed region. The effect of the fluid-filled ratio on the whirling frequency, the unstable region and the rotor vibration amplitude are also dealt with. 2. EXPERIMENTAL APPARATUS An over-hung flexible rotor with a symmetrical cylindrical chamber is shown in Fig. 1. The rotor was driven via a flexible coupling by a speed-controlled motor that could drive the rotor to a maximum rotational speed of rpm at a given acceleration rate or maintain a given steady rotational speed. A clear acrylic plate covered the top of the cylindrical chamber with a hole in the center for adding or removing fluid. A displacement limitation or safely bearing in the previous rigid rotor rig [13], was not used in order to clearly show the dynamic behavior of the rotor partially filled with fluid within the unstable region and to avoid the impact/rub problem cased by the limitation of safely bearing. The bearing used was carefully chosen in order to reduce the effect of the bearing on the rotor vibration signals near the unstable region of the rotational speeds when the rotor vibration is relative small. After comparing the ball bearing and the journal bearing, two journal bearings with a external flexible element were chosen to support the flexible shaft at the upper and the lower ends.

2 The inner diameter of cylindrical chamber in the overhung disk was 120 mm and its height was 40 mm. The total volume of the chamber was 427 ml. The total mass of the over-hung disk was 2.40 kg. The shaft was with diameter of 10 mm and length of 600 mm. The span between two bearings was 325 mm. The first critical speed of the empty rotor system was at 770 rpm, and the second one is much over the speed of 3500 rpm. The external damping ratio of the empty rotor system is about In order to study the effect of the fluid free surface on the rotor instability, a special flange structure was designed in bottom of the chamber. When the volume of the fluid is over a certain volume, the flange will be steeped in the fluid and only the part of the chamber forms the fluid free surface and the fluid free surface will be decreased. The rotor system was balanced at a steady speed of 700 rpm, in order to reduce the vibration amplitude in the resonant region and to avoid exciting other nonlinear effects in the rotor system Figure 1. A cross-section of the experimental rig Vibrations of the rotor were measured by pairs of eddycurrent proximity probes located below the over-hung disk. In order to reduce the effect of operation conditions on measured results as possible, the rotor rig was not dismantled during the tests and only the fluid amount was changed in experiments. For the increasing rotational speed operation, a known amount of water was poured into the chamber while the rotor was at rest, then the rotational speed was slowly increased with a constant acceleration rate of 8 rpm/s until the unstable region was crossed, and reached the maximum rotational speed. For the decreasing rotational speed operation, if the rotor system partially filled fluid can pass through the unstable region, after the rotor system stabilized, the rotational speed slowly decreased with the same acceleration rate as used in the increasing speed operation to rest. If the rotor system partially filled fluid cannot pass through the unstable region, the rotor, without fluid, was first run up to a desired high rotational speed well over the unstable region, after the rotor system stabilized, a known amount of water was added into the chamber, and then the rotational speed slowly decreased to rest While the rotor was accelerated or decelerated, the vibrations of the rotor were continually sampled by a data acquisition system. The rotor vibration was obtained by averaging the measured vibration amplitudes within 0.2 second. It means that the vibration amplitude value is averaged at lest one revolution in the lowest rotational speed, and more revolutions in the high rotational speed. In order to measure the characteristics of rotor motion in some steady state speeds, the rotational speed gradually increased step by step, the vibrations of the rotor system, after the rotor system stabilized, were measured and recorded. After analyzing the vibration data, the motion characteristics of rotor system, such as rotor orbit, direction of the rotor whirling motion, whirling frequency and spectrum, can be obtained. Water was used as working fluid in experiment. The fluid-filled ratio H = 1 a/ b, which is defined as the ratio of unperturbed thickness of rotational fluid layer ba to the inner radius of the chamber b, where a is the inner radius of unperturbed rotational fluid free surface, varied from EXPERIMENTAL RESULTS AND DISCUSSION Dynamics of the Rotor System with no Fluid in the Chamber When the rotational speed increases or decreases at the given acceleration rate, the imbalance response curves of the rotor system with no fluid in the chamber show that there was no obvious difference in the imbalance response curves between the increasing and decreasing rotational speed operations except in a very narrow speed region near resonant speed. It is of importance to note that the rotor vibration was very small in the super-critical rotational speed region, in which rotor instability could be generated when water was poured into the chamber. The rotor system always whirled in a synchronous forward motion and no nonsynchronous motion was observed. There were other frequency components in the spectrum of vibration signals, in addition to the synchronous rotational speed frequency, but their vibration intensities were very small and did not changed. In order to check for any non-linearity in the rotor system when the rotor vibration becomes large, which also might result in rotor instability and nonsynchronous motion, and to determine the effect of such non-linear factors on the rotor stability, an additional imbalance mass was added to the disk. The steady state imbalance response test shows that the rotor system still whirled in a synchronous forward motion. The instability of the journal bearing, characterized by the oil whirl or oil whip motion, did not observed. Therefore, any non-synchronous motion and instability occurring when the rotor is partially filled with fluid must therefore be produced by interaction between the rotational fluid in the chamber and the rotor.

3 a. Imbalance response b. Waterfall diagram of vibration c. Orbit and corresponding spectrum Figure 2. Characteristics of rotor motion with 280 ml water in increasing speed operation Developing Process of the Rotor Instability Figure 2 shows motion characteristics of rotor system filled with 280 ml water (i.e., the fluid -filled ratio was 0.383) in increasing speed operation with the given acceleration rate. The waterfall spectrum of the vibration signals, the rotor orbits and the vibration spectrum at some special steady rotational speeds were obtained, when the rotor was stabilized. The numbers in Figure 2(c) correspond to ones in Figures 2(a). The frequency axis in the power spectrum plot is normalized using the rotor rotational frequency. When the rotational speed slowly increased from low rotational speed, the rotor system whirled in a synchronous forward motion about its geometric center in the sub-critical rotational speed region and the rotor vibration amplitude increased. The resonance occurred below the critical speed of the empty rotor system. When the rotor passed through the resonant speed, the rotor vibration amplitude decreased and the rotor system also whirled in a synchronous forward motion as shown in case (1) in Fig.2 (c). If the rotational speed was above a certain value, a sub-synchronous frequency, lower than the rotor rotational frequency, appeared, but the rotor system still whirled in a synchronous forward orbit with a small vibration amplitude as the vibration intensity of the sub-synchronous frequency component was weaker than that of the synchronous one, as shown in case (2). Further increase in rotational speed leaded to an increase in the vibration intensity of the subsynchronous frequency and the rotor motion changed to a forward sub-synchronous orbit from the synchronous one. Even if the vibration intensity of the sub-synchronous frequency component was higher than that of the synchronous rotation frequency component, the sub-synchronous whirling orbit was stable and bounded as shown in case (3). With subsequent increase of the rotational speed, the rotor vibration amplitude grew rapidly and the vibration intensity of the sub-synchronous frequency component was much greater than that of the synchronous one, and the rotor whirled at the sub-synchronous frequency, as shown in case (4). In fact, the rotor system has run in an unstable region of the rotational speeds. Within the unstable region, the whirling orbit may be a forward circular, but the rotor motion was sub-synchronous. The rotor vibration amplitude in the unstable region increased with the increase of the rotational speed. In some rotational speeds, since the rotor vibration amplitudes were very large, therefore the minimum gap between the eddy-current proximity and the shaft surface was so small that the probes was in saturation state in this region, the orbits with a straight side as shown in cases (5) and (6). Only when the rotational speed was much higher, did the rotor vibration suddenly jumped to a subsynchronous state with low vibration, since then the

4 rotor returned to a synchronous orbit again as show in case (7). However, in some cases, the rotor vibration did not decrease even if the rotor ran to very high rotational speed, it being very difficult to drive the rotor through the unstable region of rotational speeds. When the rotational speed was slowly decreased from a state with a small vibration at a very high rotational speed, the motion phenomena of the rotor system are very similar to that observed in the increasing speed operation, and the rotor imbalance response curve in the resonant region is the same as that in the increasing speed operation, but the rotational speeds at which the sub-synchronous frequency first occurred and the vibration sharply growing were not at the same speeds as in the increasing speed operation. Generally, the lower and upper bounds of the unstable region determined in the increasing speed operation are higher than that determined in the decreasing speed operation. When the rotor acceleration or deceleration rate is small enough, the lower bounds measured in the increasing and decreasing speed operations will be agreement, but there is still a great gap at the upper bound of the unstable region between in the increasing and decreasing speed operations. That is to say, there exists a wide hysteresis range of rotational speeds at the upper bound, in which the rotor motions are different for different operating directions and the hysteresis range did not narrow even if the rotational speed was changed very slowly. Therefore, the acceleration or deceleration rate should keep as small as possible in order to accurately determine the unstable bounds of the rotor system partially filled with fluid. Within the unstable region, the a-periodic break-down phenomenon of the fluid free-surface, i.e., the fluid free-surface transition from uniform to separation and break-down, and then reformation of a uniform fluid free-surface, was also observed in the flexible rotor system with medium fluid fill-ratios. The a-periodic break-down process of fluid free-surface would be modulated by the rotor motion, but the frequency of the break-down motion was much lower than the rotor whirling frequency or the rotor rotational frequency. Whirl Frequency of the Rotor System within the Unstable Region From waterfall spectrum diagram in Figure 2, it is found that the rotor whirling frequency within the unstable region was not constant, but increased with the rotational speed. The variation of the non-dimensional whirling frequency with the rotational speed ratio in increasing and decreasing speed operations is shown in Figure 3. Where the rotational speed ratio is the ratio of the rotor rotational speed to the first critical speed of the empty rotor system. The non-dimensional whirling frequency in the upper set is defined by the ratio of the rotor whirling frequency to the first critical speed of the empty rotor system. The non-dimensional whirling frequency in the lower set is defined by the ratio of the rotor whirling frequency to the rotational speed, which is often called as the whirling frequency ratio. It is shown that the non-dimensional whirling frequency measured in the increasing speed and decreasing speed operations are basically same. As the rotational speed increases within the unstable region, the whirling frequency of the rotor systen increases, but the whirling frequency ratio decreases. The rotor whirling frequency within the unstable region was not constant, neither did it equal the first critical speeds of the empty rotor system nor the reduced critical speed of the rotor completely filled with fluid or a fixed fraction of the first critical speed of the empty rotor. Figure 3. The variation of non-dimensional whirling frequency ratio with the rotational speed for fluid-filled ratio Since the whirling frequency ratio in the unstable region of rotational speeds is varied with the rotational speed, so when the rotor enters the unstable region in the increasing and decreasing speed operations, the whirling frequency ratios are different, which can be used to explain why the whirling frequency ratio in the increasing rotational speed operation entering the unstable region was higher than that in the decreasing rotational speed oobserved in the rigid rotor partially filled with fluid[13]. The variation of the whirling frequency ratio with the fluid-filled ratio at the different rotational speeds is shown in Figure 4. For a given rotational speed, the whirling frequency ratio first decreases with an increase in the fluid-fill ratio in the region of lower fluidfilled ratios, and slightly increases in the region of higher fluid-filled ratios. The rotor whirling frequency was essentially proportional to the rotational speed with a fixed fraction depending on the fluid -fill ratio, and decreased with an increase in the fluid -filled ratio, which is very quite different from classical rotor instability caused by the journal bearing, inter friction and others, where the rotor whirling frequency is usually locked on to the lowest critical speed of the rotor system and depends only weakly on the rotational speed and other rotor system parameters.

5 Figure 4. The variation of the whirling frequency ratio with fluid-filled ratio at the different rotational speeds Figure 5. The variation of the unstable region with the fluid-filled ratio Effect of Fluid-Fill Ratio on the Unstable Region and Rotor Vibration From the developing process of the unstable motion, it is clear that the unstable region of rotational speeds determined by first occurrence of a sub-synchronous whirling frequency is the widest. In this paper, the occurrence of the rapidly-growing sub-synchronous vibration amplitude over a certain value is used as a criterion to obtain the instability bounds. The variation of rotor instability boundaries with the fluid-fill ratio defined by occurrence of the rapidly-growing subsynchronous vibration amplitude over 0.2 mm is shown in Fig.5. The lower bound of the rotor unstable region in increasing speed operation is very close to that in decreasing speed operation. The derivation between them is caused by the transient motion of the rotor system. The upper bound of the rotor unstable region determined in the increasing speed operation is much higher than that in decreasing speed operation due to the hysteresis region. As the fluid-filled ratio increases, the lower unstable boundary speed always increases, but the upper unstable boundary speed first increases for the fluid -filled ratios of up to , and then decreases. The unstable region of rotational speeds for increasing speed operation is wider than that for decreasing speed operation, due to the hysteresis range at the upper bound of the unstable region. As the fluid-filled ratio increases, the unstable region of rotational speeds first widens, and then narrows at larger fluid -filled ratios. Generally, the instability of the rotor with low fluid-filled ratio starts just above the first resonant speed of the rotor system partially filled fluid. Since the resonant speed of the rotor filled with the fluid is always lower than the first critical speed of the empty rotor, it is possible to observe the instability in the speed less than the critical speed of the empty rotor system when the fluid-filled ratio is small. Figure 6. The variation of rotor imbalance response for different fluid-filled ratios The rotor imbalance responses for different fluid-filled ratios in increasing and decreasing speed operations are shown in Figure 6. It is shown that the resonance of the rotor filled with different volumes of water occurred almost at the same speed, i.e., about rpm, which is lower that the first critical speed of the empty rotor. It is shown that the resonant speed of the rotor partially filled fluid did not change with the volume of fluid, which coincides with the results in [1,2] and others. The rotor vibration amplitude in the unstable region is different at the different volume of the water and decreases with an increase in the fluidfill ratio. The more the volume of the fluid filled in the chamber, the smaller the rotor vibration amplitude will be. The rotor system filled with more flu id is more easily gone through the unstable region with small vibration amplitude. From Figure 6, a very important phenomenon can be found in case with 410 ml water. When the 410 ml water was filled on the chamber, the flange was steeped in the fluid and only the part of the chamber formed the fluid free surface and the high of the fluid free surface only 20 mm. In this case, the instability did not observed at all and the rotor imbalance response curve measured in the increasing speed operation is the same as that in the decreasing speed operation. It

6 is shown that the fluid free surface has a great effect on the instability occurred in the rotor partially filled with the fluid and the instability is caused by the motion of the fluid free surface. If the area of the fluid free surface can be reduced, the instability will be reduced or suppressed in some cases. 4. CONCLUSIONS From our detailed experimental study on the instability of a flexible rotor partially filled with fluid, the following conclusions may be drawn: a. The instability of rotor partially filled with fluid occurs near the first critical speed of the empty rotor system. b. The instability of rotor partially filled with fluid is characterized by a sub-synchronous forward whirl. The rotor whirling frequency within the unstable region did not equal either the first critical speeds of the empty rotor system or the reduced critical speed of the rotor completely filled with fluid or a fixed fraction of the first critical speed of the empty rotor. The rotor whirling frequency increases with an increase in the rotational speed. The whirling frequency ratio is essentially proportional to the rotational speed with a fixed fraction depending on fluid-fill ratio. c. There exists a wide hysteresis range of rotational speeds at the upper bound of the unstable region, which did not caused by the rotor transient motion. 6. Holm-Christensen, O., and Träger, K., A Note of Instability Caused by Liquid Motions, J. of Applied Mechanics, 58(3), pp: , Lichtenberg, G., Vibrations of an Elastically Mounted Spinning Rotor Partially Filled with Liquid, J. of Mechanical Design, 104(2), pp , Hendricks, S.L., Stability of a Clamped Free- Rotor Partially Filled with Liquid, J. of Applied Mechanics, 53(2), pp , Kaneko, S., and Hayama, S., Self-excited Oscillation of a Hollow Rotational Shaft Partially Filled with a Liquid, Buletin of JSME, 28, pp: , Ota, H.,et al., Experiments on Vibrations of a Hollow Rotor Partially Filled with Fluid, Buletin of JSME, 29, pp: , Cheng, C.A., Berman, A.S., and Lundgren, T.S., Asynchronous Instability of a Rotational Centrifuge Partially Filled with Fluid, J. of Applied Mechanics, 52(3), pp: , Colding-Jorgensen,J., Rotor Whirl Measurements on Long a Rotating Cylinder Partially Filled With Liquid, J. of Vibration and Acoustics, 115(2), pp , Zhu, C.S., Experimental Investigation on the Instability of an Over-hung Rigid Centrifuge Rotor Partially Filled with Fluid, Journal of Vibration and Acoustics, d. The instability of the rotor system partially filled with fluid is caused by the fluid free surface motion. Reducing the fluid free surface can reduce or suppress the rotor instability. ACKNOWLEDGEMENTS The supports of the National Natural Science Foundation, China, and Alexander von Humboldt Foundation, Germany, are gratefully acknowledged. REFERENCES 1. Kollmann, F.G., Experimentelle und theoretische Untersuchungen uber die Kritischen Drehzahlen flussigkeitsgefulter Hohlkorper, Forschund auf dem Gebiete des Ingenieurweasns, Ausgabe, B, 2, pp , and pp , Ehrich, F.F.; The Influence of Trapped Fluids on High Speed Rotor Vibration, J. of Engineering for Industry, 89(4), pp , Wolf, J.A., Jr., Whirl Dynamics of a Rotor Partially Filled with Liquid, J. of. Applied Mechanics, 35(3), pp , Hendricks, S.L., and Morton, J.B.; Stability of a Rotor Partially Filled with a Viscous Incompressible Fluid, J. of Applied Mechanics, 46(4), pp , Saito, S., and Someya, T., Self-Excited Vibration of a Rotational Hollow Shaft Partially Filled with Liquid, J. of Mechanical Design, 102(2), pp: , 1980