Experiments in Fluids 33 (2002) 369–373 DOI 10.1007/s00348-002-0418-7
Flow instability and elastic vibration of shrouded corotating disk systems R. Fukaya, S. Obi, S. Masuda, M. Tokuyama
369 reducing the disk-to-shroud spacing, indirectly suggesting the importance of internal flow. It has been shown that the flow between SCD becomes unstable over a certain range of Reynolds numbers (Lennemann 1974; Kaneko et al. 1977; Abrahamson et al. 1989; Iglesias and Humphrey 1998; Herrero and Giralt 1999). This instability is manifest by the transition from a steady two-dimensional (axisymmetric) flow to unsteady three-dimensional (nonaxisymmetric) flows. The observed circumferential wave numbers reported so far are compiled in Fig. 1, which shows Reynolds number Re=qWR22 versus nondimensional disk spacing h=H/R2, where W is the angular velocity of rotation of the disks, l is the fluid viscosity, q is the fluid density, R2 is the outer radius of the disk and H is the spacing between a pair of disks. As indicated in the 1 figure, the wave number tends to decrease with increasing Introduction Reynolds number, while no systematic dependence on the With increasing track density and access speed of hard disk drives (HDD), disk vibration is becoming a serious spacing has been found. Two dashed lines denoted by a–a and b–b are the boundaries of the flow regimes subjecproblem even for small size disks. The ratio of the disk tively defined by Abrahamson et al. (1989). They observed vibration amplitude to the flying height of the head is expected to increase with increasing rotational speed and a stable vortical structure above line a–a, which becomes decreasing flying height. One promising way of minimiz- alternating or indeterminate as H/R2 decreases. Another ing the disk vibration is to reduce the unsteadiness of classification based on the direct numerical simulation by internal flow, since it is now believed to excite and Herrero and Giralt (1999) is denoted by I, II and III, which maintain the disk vibration. indicate a steady–axisymmetric flow (I), a flow with shiftMcAllister (1997) reported on the disk vibration of and-reflect symmetry with respect to the interdisk midshrouded corotating multiple disks (SCD) and concluded plane (II) and an asymmetric flow with it (III). Since these that the amplitude and frequency of disk vibration unsteady three-dimensional flows may exert time-dependepends on the density of the surrounding fluid. Imai et al. dent pressure on the surface of the disk, the vibration of (1999) reported the reduction of vibration amplitude by the multiple disk system may be excited. The coupling of flow instability between both sides of the disk through the shroud gap may become additional problems. In the present study, the interaction between flow inReceived: 18 June 2000 / Accepted: 28 January 2002 stability and elastic vibration of an axisymmetric model of Published online: 24 July 2002 a 3.5-in HDD is investigated. Simultaneous measurements Springer-Verlag 2002 of velocity fluctuation and disk vibration are performed in a wide range of Reynolds numbers. The cross-correlation R. Fukaya of velocity and disk vibration is examined in order to Toyota Motor Corp., 1 Toyota-cho, Toyota 471-8572, Japan obtain an insight into the effect of flow instability on disk S. Obi, S. Masuda (&) vibration. The two-point correlation of the velocity flucDept. of Mechanical Engineering, Keio University, tuations at two axially separated spaces is also examined to 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail:
[email protected] reveal the axial extent of the unstable flow structure. Abstract The interaction between flow instability and elastic vibration of shrouded corotating disks is investigated. Simultaneous measurements of velocity fluctuation and disk vibration are performed in a 3.5-in hard disk drive, which was modified to ensure geometrical axisymmetry. The dependence on Reynolds number of velocity fluctuation and disk vibration is examined. Spectral analysis and cross-correlation measurements employing hotwire anemometry reveal the wave number, the speed of rotation, the coupling of instabilities on both sides of a disk as well as the axial extent of the unstable flow structure. The role of flow instability on disk vibration is emphasized.
M. Tokuyama Mechanical Engineering Laboratory, Hitachi Ltd., 502 Kandatsu, Hitachi 300-0013, Japan
2 Experimental facility and methods The authors are indebted to Mr. S. Imai of Hitachi Ltd. for providing The experimental facility consists of a test section, a trathe experimental facilities and the FEM results. His stimulating versing device, DC power supply and a rigid base, as discussions are also gratefully acknowledged. The authors would like shown in Fig. 2. The test section is an axisymmetric model to thank Mr. K. Gomi and Mr. S. Uchida for conducting a part of of a ten-decker 3.5-in HDD, modified to avoid geometrical the experiment.
the periphery. The data were low-pass filtered at 2.5 kHz and 6·105 samples were obtained at the sampling frequency of 5 kHz. In addition to the Reynolds number defined above, the stiffness parameter =Eb2/{12(1–m2)qdW2R24}m, the density ratio L=qR22/(qdb2), the nondimensional disk spacing h=H/R2, the nondimensional disk thickness b/R2, the nondimensional shroud clearance a/R2 and the inner– outer radius ratio R1/R2 are defined, where qd, E and m are density, Young’s modulus and Poisson’s ratio of the disk material, respectively. The Reynolds number is varied from 4.57·103 to 1.10 ·105. As indicated in Fig. 1, this range of Reynolds number covers the transition from the steady two-dimensional flow to unsteady three-dimensional flow as well as the operating range of the current 3.5-in commercial HDD. In the present study, Reynolds number variation is implemented by reducing the ambient pressure in the evacuated chamber, thus reducing the density of the air, without varying the rotational speed. During this experiment, the stiffness parameter is kept constant at 0.547, while the density ratio L varies from 0.065 to 1.55. This method is more convenient than variable speed experiments, since the Reynolds number, which is the dominant parameter for flow instability, can be varied independently of the centrifugal stiffening of the disk.
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3 Experimental results Fig. 1. Existing data and the range of present investigation (numbers indicate circumferential wave number)
3.1 Velocity fluctuation and disk vibration The rms value of velocity fluctuation mh¢ is given in Fig. 3, together with the corresponding rms value of the out-ofplane displacement of the disk w¢. Velocity data were obtained at r/R2=0.96 in the interdisk midplane of the third spacing from the top, while vibration data were obtained at the outer rim of the third disk from the top. The rms qffiffiffiffiffi values are defined as v0h ¼ v2h Vh (Vh is mean circum-
Fig. 2. Experimental apparatus
asymmetry by employing a cylindrical shroud and by removing magnetic heads and arms. The outer and inner radius of the disk are R2=47.5 mm and R1=16 mm, respectively, the thickness b is 0.80 mm, the interdisk spacing H is 1.84 mm, and the angular velocity is kept constant at W=754 rad/s (7200 rpm). The disk-to-shroud clearance a is 1.2 mm. The circumferential component of flow velocity vh is measured using a constant temperature anemometer (CTA) equipped with a self-made single-sensor hot-wire probe. The sensor consists of 2.5-lm Wollaston wire (silver-coated 90%Pt–10%Rh) and the length-to-diameter ratio is 200. Displacement of the disk surface is measured using a capacitance-type displacement meter with the end face of a sensing head inclined 45 with respect to the Fig. 3. Reynolds number dependence of amplitude of velocity probe axis, so that the probe can be installed from outside fluctuation and disk vibration
pffiffiffiffiffiffi ferential velocity) and w0 ¼ w2 , where both are normalized by the corresponding values under atmospheric conditions, mh0¢and w0¢. Within the range of 104 £ Re £ 6·104, velocity fluctuation decreases gradually with decreasing Reynolds number. The laser Doppler velocimetry measurements conducted by Schuler et al. (1990) revealed that the transition from a steady axisymmetric laminar flow to an unsteady nonaxisymmetric laminar flow with a circumferential periodicity was observed at Re4580, which is denoted by line A–A in Fig. 3. Another line denoted by B–B is the direct numerical simulation result of Herrero and Giralt (1999), below which the flow is steady and axisymmetric in their simulation. The decrease in the present rms velocity is confined within these two lines, suggesting the weakening of the flow instability caused by the increasing viscous effect. It is worth noting that this change occurs gradually in the Reynolds number range of over nearly one order of magnitude, contrary to the finding of Herrero and Giralt (1999), who suggested an abrupt change at B–B called the Hopf bifurcation. The corresponding rms value of disk vibration exhibits a similar tendency to rms velocity. This result supports the idea that aerodynamic forcing caused by the flow instability excites disk vibration. According to Renshaw et al. (1994), the aerodynamic forcing term in the equation governing the transverse vibration of the disk has the form of LDp, where Dp is the normalized pressure difference on both sides of the disk, and L is the density ratio defined above. In addition to the Reynolds number, the decrease in ambient pressure also gives rise to the decrease in L as indicated on the horizontal axis in Fig. 3. The decreasing pressure may have a duplicate effect on the vibration by: a) decreasing the Fig. 4. Power spectral density of velocity fluctuation (a) and disk 5 pressure difference Dp because of viscous stabilization and vibration (b) for Re=1.10·10 b) decreasing its effect because of the decrease in L. n-nodal circles and m-nodal diameters, and B and F designate backward- and forward-traveling waves, respec3.2 tively. The important thing is that these vibration Frequency characteristics Power spectral densities (PSD) of velocity fluctuation and frequencies do not appear in the velocity spectra except for disk vibration at Re=1.10·105 (atmospheric condition) are 500 Hz (peak f and peak 4). Also interesting is that the given in Fig. 4. The measurement positions are the same as same velocity peaks were observed in the experiment with those in Fig. 3. The PSD of the velocity fluctuation given in the thicker disk at one order of magnitude higher eigenFig. 4a exhibits eight peaks (marked 1 to 8) below 1 kHz frequency (Ichitsubo 1997). These findings suggest two except for the electric noise (marked by A) and the rota- important things: first, the disk is essentially vibrating not tion frequency (marked by B). Assuming that these eight with the flow instability mode but with the eigenmodes of peaks are caused by the passing of the nonaxisymmetric the disk itself, and second, the instability of the surflow structure, the wave number and the speed of rotation rounding flow is not governed by the disk eigenmodes. In addition to these major peaks, there exist additional can be estimated. The results indicate that the nonaxisymmetric flow structure in this case is composed of eight broad peaks (marked a to e) with amplitudes one order of discrete components with wave numbers ranging from 2 to magnitude less than those of the major peaks. The fre9, and these components rotate on average at 84% of the quencies of these minor peaks are lower than those of the eigenmodes. It is interesting to note that the low-frequency disk rotational speed. Besides the run-out components marked by the open peaks marked c, d, e and f coincide well with the velocity squares, the PSD of disk vibration shown in Fig. 4b shows peaks 1, 2, 3 and 4, respectively. It is considered that the several broad peaks. The five major peaks (marked f to j) low-frequency weak vibration from the flow instability is coincide well with the eigenfrequency of a disk in vacuum superposed on the dominant eigenmodes. Although this estimated by FEM analysis. Among them, peak f and peak j result suggests the possible collaboration of the flow inare exactly equal to the eigenfrequencies of the (0–2)B and stability and elastic vibration, the very small correlation coefficient between velocity and vibration signals confirms (0–2)F modes, respectively, while peak h and peak i are the independence of disk vibration and velocity fluctuanearly equal to those of the (0–1)F and (0–3)B modes, respectively. Here, (m–n) designates the eigenmode with tion at each instant. Because of the extremely small
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amplitude of the disk vibration, it is likely that the effect of unsteady boundary condition imposed by the oscillating disk does not penetrate deep into the fluid, leaving negligible traces on the velocity fluctuation in the central region.
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3.3 Circumferential two-point velocity correlation In order to give an insight into the flow instability, the two-point correlation coefficient of velocity fluctuations RAB at circumferentially separated points A and B is shown in Fig. 5. It is defined by RAB ðDtÞ ¼
vhA ðtÞvhB ðt þ DtÞ ; v0hA v0hB
where the overbar denotes the time average. The double sensor hot-wire probe is placed in the interdisk midplane of the third spacing from the top. The radial position is r/R2=0.96 and the angle of separation of the two sensors is Dh=0.110 rad (5 mm). The time interval s0 between two major peaks (denoted by A and B) corresponds to the time required for the nonaxisymmetric flow structure to rotate once. The angular velocity estimated from s0 is about 85% of the rotational speed of the disk. This result is in good agreement with the estimation based on PSD discussed in Sect. 3.2. The circumferential wave number can also be estimated from Fig. 5 by counting the number of minor peaks between A and B, since they may correspond to the passage of similar flow patterns. In the present study, the dominant wave number is estimated as 6.
3.4 Coupling of flow instability on both sides of a disk The importance of the pressure difference between the two sides of a disk has been postulated in Sect. 3.1 and Sect. 3.2. To further improve our understanding, the coupling of flow instability on both sides of a disk is examined. The correlation coefficient of velocity fluctuations in the upper and lower sides of the sixth disk from the top is given in Fig. 6 for Re=1.10·105. The probes are located at the midplanes and the radial positions are the same as
Fig. 5. Correlation coefficient of velocity fluctuations at two circumferentially separated points (Re=1.10·105, Dh=0.110 rad)
Fig. 6. Correlation coefficient of velocity fluctuations on both sides of a disk (Re=1.10·105)
before. The profile shows large amplitude periodicity with the interval exactly equal to the time required for rotation of the nonaxisymmetric flow structure s0, as mentioned in Sect. 3.3. In addition, the six minor peaks with regular intervals can be observed. These results indicate that similar structures with a finite number of components exist on both sides of the disk, and these structures rotate at the same rotational speed. The first peak of the correlation coefficient appears at Dt=0.3 ms, corresponding to the angular displacement of the structures of 11. This angular displacement can effectively generate the wall pressure difference, resulting in the aerodynamic forcing of the disk systems.
3.5 Flow structure extending over the entire disk assembly Finally, the axial extent of the nonaxisymmetric flow structure is examined by placing one hot-wire probe in the midplane of the fifth space from the top, while the other probe is located in the Nth space. The radial positions are the same as before. The angular displacement between the structures in two axially separated spaces is estimated from the time delay Dt for the local maximum of the correlation coefficient multiplied by the angular velocity of the structure x. As shown in Fig. 7, it increases linearly with increasing axial distance and the displacement between the first and the ninth spaces is approximately p/2. This result indicates that the flow instabilities in different spaces are statistically dependent and the resulting nonaxisymmetric flow structures pile up with a gradual angular displacement and tend to form a single large structure extending over the entire disk assembly. 4 Concluding remarks The flow instability and the elastic vibration of shrouded corotating disks have been investigated. The parameter range corresponds to that of a current 3.5-in hard disk drive. The flow between disks undergoes transition from a steady axisymmetric flow to an unsteady nonaxisymmetric one. This transition does not occur abruptly at a specific
Fig. 8. Conceptual view of interaction between flow instability and elastic vibration
4. The resulting time-dependent wall boundary condition generates oscillatory flow, but it is confined only within Fig. 7. Angular displacement of nonaxisymmetric flow structures in the thin layer close to the surface. axially separated spaces (relative to the fifth space from the top, 5. Flow instability is thus independent of disk vibration, Re=1.10·105) as if the disk were rigid. Reynolds number, but occurs gradually in the Reynolds number range of over nearly one order of magnitude. The nonaxisymmetric flow structure is composed of a finite number of discrete components whose wave numbers are 2 to 9, and it rotates at 85% of the rotational speed of the disks. Although the amplitude changes markedly with Reynolds number, the frequency characteristics of the unsteady flow are only weakly dependent on the Reynolds number. They are also independent of disk vibration. The flow instabilities in different spaces are statistically dependent and the resulting nonaxisymmetric flow structures pile up with a gradual angular displacement and tend to form a single large structure extending over the entire disk assembly. The small angular displacement of the nonaxisymmetric flow structure on both sides of a disk may cause the elastic vibration of the disk. The rms value of disk vibration decreases in accordance with the decrease in rms velocity. In spite of this, PSD of the disk vibration is coincident not with the velocity fluctuation but with the eigenfrequency estimated by FEM analysis, neglecting the surrounding fluid. Besides the eigenmodes, there exists a low-frequency vibration that coincides with velocity fluctuation. This suggests fluid–disk collaboration, but is so weak that its contribution to the vibration is negligible. The interaction between the flow instability and elastic vibration assumed from the results discussed thus far is summarized in the conceptual illustration given in Fig. 8. The expected chain of phenomena is as follows: 1. At a certain Reynolds number, the transition starts away from the disk surface. 2. Because of the small phase difference on both sides of the disk, a time-dependent pressure force is exerted on the disk. 3. The eigenmodes of elastic vibration are excited by it.
As illustrated in this figure, the chain of the phenomena is not completed as a closed loop. Thus it can be said that, for small HDDs in the current speed range, disk vibration is not ‘‘flutter’’ in the common engineering terminology but natural vibration excited by a random aerodynamic forcing caused by an unstable flow.
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