WESPAC IX 2006
Proceedings of the 9th Western Pacific Acoustics Conference
Seoul, Korea, June 26-28, 2006

Fluid-Structure-Acoustic Analysis of a Polygon Motor
Osamu AKIYAMA, Naoki WATANABE

Chisachi KATO

Quality Assurance Operations
Konica Minolta Business Technologies, Inc., Japan
E-mail: osamu.akiyama@konicaminolta.jp

Institute of Industrial Science
The University of Tokyo, Japan
E-mail: ckato@iis.u-tokyo.ac.jp

Takuya YOSHIMURA

Masashi MIYAZAWA

Department of Mechanical Engineering
Tokyo Metropolitan University, JapanABSTRACT
A polygon motor consists of several mirror faces and it is one of the core components for alaser beam printer and a copier. Under high-speed rotation, aeroacoustic noise with a peakfrequency of rotational speed times the number of mirror faces and its harmonics is
prominently generated from a polygon motor covered with a casing. This paper describes amethod to predict such noise that is generated from the flow in between the rotating mirror andstationary casing, propagates through the casing and is radiated into the ambient air. The
unsteady flow is firstly computed by large-eddy simulation. The computed pressure
fluctuations on the inner wall of the casing are then fed to structural analysis based on adynamic explicit finite element method that computes the propagation of the elastic waves inthe casing. Resulting external surface velocities are finally used for the computation of theacoustical field. A good correlation between the computed and measured results has beenobserved both in terms of velocity spectra on the casing and spatial distribution of soundpressure level. The proposed method thus seems a promising engineering tool for predicting
and identifying the generation mechanism of such noise that is generated from unsteady flow,
propagates through solid walls and is radiated to ambient air, in general.
KEYWORDS: Structural Noise, Polygon Motor, Aeroacoustic, Large Eddy Simulation

INTRODUCTION
A rotating polygon mirror reflects laser beam and plots a latent image on a photo conductor drum. A polygon
motor generates peak noise at a frequency of nz (n : rotational frequency, z : number of mirror faces) and its
harmonics similarly to turbomachinery noise at the blade passing frequency. For high-speed operation of apolygon motor, reduction in the peak noise is demanded although the noise level is not necessarily high when
it is covered with a casing. This is because such a peak noise has, in general, a high frequency such as ten

Copyright © The Acoustical Society of Korea. All Rights Reserved. WESPAC IX 2006

KONICA MINOLTA TECHNOLOGY REPORT VOL.4（2007）

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Fig. 1 Polygon motor

Fig. 2 Computational mesh for the polygon motor

thousand hertzs and is very uncomfortable.
The mechanism of noise generation is roughly described as follows: 1) rotating polygon mirror faces
generates pressure fluctuations on the internal surface of the casing, 2) sound waves generated by the vibration
source propagates through the solid casing, and 3) vibration velocity on the external surface radiates soundwave into the ambient air. Regarding the flow-induced noise inside a structure like this case, a prediction
method based on an one-way coupling analysis of fluid flow, structural and acoustical computations has been
applied to a 5-stage centrifugal pump [1, 2, 3, 4].
In this paper is, pressure fluctuations are computed by large-eddy simulation (LES), frequency transfer
characteristics and operational deflection shape radiating noise are then computed by dynamic explicit
structural analysis, and finally radiated sound fields are predicted by acoustical analysis. Prediction accuracyof the present method is verified by comparisons with measurements in terms of the vibration velocity as wellas the radiated sound pressure level (SPL). In addition, sound generating mechanism and propagation
characteristic are clarified by the computed results.

INNER FLOW-FIELD COMPUTATION USING LES
Numerical Method and Computational Conditions Pressure fluctuations in the casing are computed by
Front Flow - Blue [5, 6], which is an LES code based on a streamline upwind finite element formulation of
second order both in time and space and numerically computes the incompressible filtered Navier-Stokes
equations. A detailed description is given by Kato et al [7, 8].
The computed polygon motor model sealed with casing and glass is shown in figure 1. It is composed of 6
mirror faces with a tip diameter Dt = 42.7 mm that rotates at 63,000 min-1, resulting in a tip speed of Ut = 141
m/s. The Reynolds number based on this tip speed Ut and the representative diameter Dt is Re = 3.98✕105.
The computational region that is composed of a combination of rotational and stationary frames of reference
is depicted with the computational mesh as shown in figure 2. Each mesh includes appropriate margins ofoverlap with its neighboring mesh upstream and downstream. At every time step, the velocity components
and static pressure within such a margin are the values interpolated in the computational mesh of the corresponding neighbor. The number of grid points is approximately 1 million for a mirror face and 5 million intotal. The time increment ∆t of the computation was set such that 4,000 time steps corresponded to one
revolution of the polygon motor ( ∆t = 7.85✕10-4 [-]).

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Fig. 3 Computed instantaneous distributions of tangential velocity
in the polygon motor’fs mid-height and x=0 planes

Fig. 4 Computed instantaneous streamlines on θ = 10, 28, 57
degree planes grey-scaled by vorticity ωθFig. 5 Computed instantaneous vortical structures around the mirror ( ∇2 p =1000)

Numerical Results Figure 3 shows the computed instantaneous distributions of tangential velocity. Thisconfirms that flow velocity in downstream of a tip is locally fast ( uθ / Ut = 1.53) due to the recirculating flow
downstream of the tip. The detail view near a tip is shown in figure 3 with Mach number contours. The Mach
number in the reverse flow region is 0.4 or above and reaches to a local maximum value of 0.64. However,
the effect of compressibility is presumably ignorable because such a high Mach number region is localizedwith a small area and the entire computational domain is also comparatively small.
Figure 4 shows the computed instantaneous streamlines on θ = (a) 10 deg, (b) 28 deg and (c) 57 deg
planes. Streamlines are plotted together with tangential direction component vorticity ωθ indicated by grayscale. In the (a) 10 degree plane, flows separated at the edge of the upper and lower surfaces come into theabove-mentioned recirculating zone and form intense vortices there. In the (b) 28 degree plane, some vortices
are still generated, but most of the vorticites generated upstream have been and are being destroyed without
spreading to radial direction. In the (c) 57 degree plane, all the intense vortices have been destroyed. It can besaid that vortical motion has no apparent effects on the downstream mirror.

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Fig. 6 Computed instantaneous pressure distributions in themid-height and θ = constant planes

Fig. 7 Computed time-averaged ( upper ) and fluctuating( bottom ) pressure on a mirror faceFig. 8 Computed power spectra of pressure fluctuations

Figure 5 shows the computed instantaneous vortical structures around the mirror with iso-surface of∇ p=1000. As already mentioned, three-dimension vortical structures generated behind a tip are destroyed
without spreading to the axial and radial directions. The turbulent vortical motion thus has a limited effectbehind a tip and no effect on the other place such as inner casing or downstream mirror faces.
Figure 6 shows the computed instantaneous pressure distributions. Negative pressure regions (relatively
low level of pressure region) lie behind a tip, expand to the radial direction and reach to the inner casing
surface. When it is viewed from the stationary frame, the pressure difference between the front and rear sideof a tip generates fluctuating component at nz frequency on the inner surface of the casing.
To focus on the mirror face, better understanding of the pressure field can be obtained by viewing it fromthe rotating frame. Figure 7 depicts the computed time-averaged and fluctuating pressure on a mirror face.
These statistical values are obtained by using flow fields on the six mirror faces during the last three rotations.
The three lines show the upper, center and lower distributions against the m-axis shown in figure 6. In thetime-averaged pressure Cp, the negative pressure region behind a tip ( m/Dt < 0.15) corresponds to the216

KONICA MINOLTA TECHNOLOGY REPORT VOL.4（2007）

reverse flow region ( M ≥ 0.4 in figure 3). In the fluctuating pressure C’p , the maximum point located atupper and lower position of m/Dt = 0.09 is identical to the θ =10 degree place shown in figures 4 and 5. This
indicates that those vortices originating from the separated flows at the upper and lower surfaces cause theintense pressure fluctuation around m/Dt = 0.09 shown in figure 7. Figure 8 (i) depicts the computed power
spectra of pressure fluctuation at θ = 10, 28 and 57 degree points of the mirror face. The rotational frequency
n and its harmonics have a prominent peak. Note here that 6n, 12n components are not dominant as comparedwith other harmonics components. At θ =10 degree point, where vortical motion is most activated as mentioned above, the computed power spectrum of the fluctuating pressure in the range St = 3 - 5 closely obeys
what is expected for the inertial subrange of turbulence. This implies that the important vortical motion isappropriately computed by the present LES.
Figure 8 (ii) plots the computed power spectra of pressure fluctuations at point (1) - (3) in the casing innersurface ( in the stationary reference frame). The frequency characteristic of sound source has a prominent
peak at nz and 2nz components. Note here that nz component is greater than 2nz one. The feature is identified
not only at these three points, but also at other points.

WAVE PROPAGATION ANALYSIS IN A CASING
Dynamic Explicit Finite Element Method Elastic wave propagation in the casing is computed by NEXSTImpact [9], which is a structural analysis code based on a dynamic explicit finite element method. The
structural analysis treats computed pressure fluctuations mentioned in the preceding chapter as excitation
forces. Equation of motion for dynamic elastic vibrations of structure is expressed as,(1)(2)
where,and
are the displacement, velocity and acceleration vectors, respectively. [M] , [C] and[K] are the mass, damping and stiffness matrices, respectively, andis the load vector. Assuming that
Rayleigh damping is applicable to the casing, the damping matrix [C] is expressed as Eq. (2) where Cm isevaluated through an impact test and the corresponding computation, while Ck is assumed to be zero.
Computational Conditions Pressure fluctuations computed by the LES feed the load matrix
in a formof the time history data [3]. The connection and contact boundary conditions for pairing the flange andbasement surface are identified such that the computed natural frequency agrees with the measured one.
Figure 9 shows the computational mesh for the structural analysis together with the mapped force boundary
condition that is indicated by the gray scale. The aluminum casing model with an outer diameter of 64 mm
and wall thickness of 7 mm is composed of approximately 50,000 tetrahedral elements, and the glass part fortransmitting laser beam is ignored.
Numerical Results Figure 10 shows the distribution of computed normal velocity at 2nz (12.6 kHz). The
deflection behavior shown in figure 10 is the same as one of the natural modes computed by an eigen-value
analysis although it is not shown here. Figure 11 shows a measured deflection shape of the top of casing at2nz in operation. The measured deflection shape is very similar to the computed one (figure 10). This is thefirst membrane oscillatory mode that efficiently radiates acoustic noise, in general.

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Fig. 9 Computational mesh with force boundary
conditions mapped for structural analysis

Fig. 10 Distribution of computed normal
velocity at 2nz (12.6 kHz)

Fig. 11 Measured deflection of the casing
top at 2nz (12.6 kHz) in operation

Figure 12 shows a comparison of the normal velocity at the top of casing where Power Spectrum Density(PSD) of the measured and computed velocity are plotted. Both computed and measured velocity have thesame trend in which magnitude of 2nz component is greater than nz one. This trend is apparently the results ofthe propagating characteristics of the structure at 12.0 kHz. Note that the experimental results have several
other peaks such as rotational frequency n (1050 Hz) and its harmonics components generated due to theimbalance of the rotor weighted, and 8n (8.4 kHz) component excited by the motor coil that are not considered in the present computational model. Although the viscous damping is most likely to be dependent on theplace as well as on the frequency, it is treated as a single constant Cm in the present study. A more sophisticated model of the viscous damping will perhaps improve the prediction accuracy in the future.

ACOUSTICAL ANALYSIS OF SOUND FIELD
Numerical Method and Computational Conditions Sound fields are predicted in the frequency domain
by a commercial acoustical analysis code SYSNOISE using a boundary element method for the Helmholtz
equations [10,11,12]. By applying the particle velocity computed on the external surface of the casing by thestructural analysis, is computed at an arbitrary monitor point in a semi-sphere. The computational mesh for theacoustical computation is composed of approximately 5,000 three-node triangle elements with a relatively
high resolution where six or more elements are used per wave length of the sound. All reflection condition isgiven on the z = 0 plane in order to model sound radiation in a semi-anechoic room where the measurement
was made. The velocity distributions computed by the structural analysis are transferred to the acousticalmesh shown in figure 13. SPL is computed on a hemisphere at r = 1.0 m distance.
Numerical Results Figure 14 compares the predicted and measured SPL at monitor point 10 (r = 1.0 m
point in the z-direction). Note here that the predicted SPL has a different frequency resolution from that in themeasurement and therefore its level is adjusted to compensate this difference. As already mentioned, polygon
motor noise was measured in a semi-anechoic room with a set of non-directivity microphones in conformity
with ISO3746. The present computation successfully predicts that 2nz component is greater than nz component and dominates the overall noise level although the computed absolute values of these two peak components are 10 - 20 dB lower than the measured equivalents. Regarding the broadband component, the slope
around the 2nz peak (at 12.6 kHz), which is mainly the results of the structural characteristics, are quantitatively predicted by the present computation. As already mentioned, other peak components are produced by18KONICA MINOLTA TECHNOLOGY REPORT VOL.4（2007）

Fig. 12 Comparison of normal velocity at the top of casing

Fig. 14 Comparison of sound pressure level at monitorpoint 10 ( case top- direction, 1.0 m )Fig. 13 Computational mesh used for acousticalanalysis based on BEM

Fig. 15 Comparison of sound pressure level at several
measurement points ( 2nz:12.6 kHz )what is not modeled in the present study and thus not predicted. Figure 15 shows comparisons of the SPL at
2nz component at all the monitor points. Note that the measured and computed SPL refer to a different axis,
respectively to the left and right axes. Although the computed SPLs are by around 20 dB lower than themeasured ones, the directivity of the sound radiation where intense sound is radiated to the upward direction
is reasonably well predicted.
The principal mechanism of sound radiation is then the 1st membrane oscillatory mode of the casing thatis excited by the 2nd harmonics of the fluctuating pressure generated by the rotating polygon mirror. Bothstructural and acoustical analyses quantitatively predict the broadband component while they underpredictthe peak components as shown in figure 12 and 14. The discrepancy is, therefore, attributed to the flow fieldcomputation and/or the structural analysis. The structural analysis, however, is more likely to be the cause
because it dose not predict those peaks at 3.4 kHz and 5.3 kHz that are confirmed by a hammering test.
Finally, we comment on the acoustical mode inside the casing that was ignored in the present analysis.
From the inner diameter 48.4 mm of the casing, the frequency of the fundamental mode is estimated to bearound 7.0 kHz. But no strong peak exist neither in the measured vibration velocity nor in the measured
sound pressure. Therefore, the internal acoustical mode can be safely neglected in this case. However, thismay not be the case for a different number of mirror faces and/or rational speed for which the frequencies ofthe source fluctuation are different.

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CONCLUSIONSThis paper proposes a fluid-structure-acoustic coupling analysis method that is capable of predicting
aeroacoustic noise generated from a polygon motor. By comparisons with measurements, accuracy of theproposed method is verified in terms of velocity spectra on the casing and spectrum of SPL of the radiatedsound. The method thus seems a promising engineering tool that can be applied to the prediction and reduction of noise generated through a mechanism similar to here in presented.
The following issues should perhaps be considered for possible improvement in the prediction accuracyof the proposed method: 1) a more sophisticated model for viscous damping used in the structural analysisand 2) inclusions of the effects of the acoustical models that may develop inside the casing.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge the benefit of advice and cooperation from Yoshinobu Yamade of Mizuho
Information & Research Institute, Inc. and Dr. Yu Yan Jiang of AdvanceSoft Co., LTD. The computation
codes used in this paper, Front Flow-Blue and NEXST-Impact, have been developed by ‘Frontier Simulation
Software for Industrial Science’ project sponsored by the Ministry of Education, Culture, Sports, Scienceand Technology (MEXT).

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