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T E C H N I C A L P A P E R

Design and construction of an in situ inspection machine

for ensuring the dynamic performance of fluid bearings

in data storage devices

Paul C.-P. ChaoÆ Chi-Wei Chiu Æ J.-S. Huang Æ Chien-Ming Chen

Received: 31 July 2007 / Accepted: 30 January 2009 / Published online: 21 February 2009 Ó Springer-Verlag 2009

Abstract The objective of the study is to design and construct an in situ inspection machine that would be used to evaluate the dynamic performance of the small-scaled hydrodynamic journal bearings used often for data-storage devices. To this end, novel design ideas of using a high-stiffness air bearing and suspending the test bearing by the specially-designed fixture frame are first proposed, which are accompanied by the special requirements in the man-ufacturing precisions for various dimensions and geometry characteristics of the components of the experiment sys-tem. The capacitor-type sensors, including two custom-mades are next utilized to measure the motions of the spindle and bearing. Finally, a calculating algorithm for deriving performance parameters of the test bearing are proposed, which is intended primarily to estimate rotor stiffness coefficients, load capacity, and eccentricity ratio to determine if the test bearing meet the dynamic perfor-mance expected by the original design.

1 Introduction

The data storage capacity of hard disk drives is increased in an amazing speed as a result of several key technology advancements in recording data on a disk with very high

areal density. Recently, the areal density of hard disk drives has been increasing at a growth rate of 100%. The early-day limit of areal density of 40 GB/in has been broken. It is expected that the areal density will soon reach the level of 100 GB/in Therefore it is anticipated that the allowable nonrepeatable run out (NRRO) of spindle motors has to be suppressed to below 1 lm, which is required in high track density recording. In additions, the hard disk drives need also to work faster, quieter and more reliable. All of these mentioned performances necessitate further improvements for the spindle motor.

It is known that the performance of a spindle motor largely depends on the performance of its bearings. Therefore, the bearing system becomes a key technology for the advancement of the spindle motor. In recent years, the fluid-film bearing or namely hydrodynamic journal bearings (HJB) becomes the popular choice for the high-speed spindle motors owing to its dynamic characteristics of lower non-repeatable runouts, lower acoustic noise, higher stiffness and higher damping coefficients, as com-pared to conventional ball bearings (Zhang et al.1999). To estimate the dynamic coefficients, several methods were proposed in the past study, such as selective vibration orbits (Brockwell and Dmochowski1989) etc. The method, however, estimates the coefficients by collecting the data in the discrete time, which may lead to the error in the measurements under the high speed operation. To meet this demand of fast and precise developments in varied types of HJB, an in situ and in-lab inspection machine for evalu-ating the dynamic performance of fluid bearings in data storage devices, including hard and optical disk drives, is successfully designed and constructed in this study.

To determine the rotordynamic coefficients of an HJB, an appropriate data analysis method must be selected based on the experimental setup and excitation type used.

P. C.-P. Chao (&)  C.-W. Chiu

Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan e-mail: pchao@mail.nctu.edu.tw

J.-S. Huang C.-M. Chen

Department of Mechanical Engineering, R&D Center for Membrane Technology, Chung-Yuan Christian University, Chung-Li 320, Taiwan

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Identification methods can be divided into two general categories: time domain methods and frequency domain methods. One example of a time domain used for bearing parameter identification is the Kalman Filter (Fritzen and Seibold 1990; Mohanmmad and Burdess 1990). Even though time domain methods do not require transformation of the data to the frequency domain, they are generally nonlinear problems, which must be solved by iterative processes. The analysis used by Brockwell and Dmo-chowski (1989), for straight line orbit excitation, may also be classified as a time domain method. The coefficients are determined from measurements at discrete times when only one velocity or displacement is non-zero. This method tends to results in large errors, since only selected points are used to exact the coefficients, instead of the entire time histories. Furthermore, inertia coefficients cannot be found by this method. On the other hand, frequency domain methods are well suited for the analysis of data obtained from multiple-frequency excitation methods. For linear systems, the multi-frequency excitation and response data are expressed in the frequency domain as a series of dis-crete frequency responses. The rotordynamic coefficients can be extracted directly from frequency domain data by rewriting the equations in such a way that the coefficients appear in a vector, and then using a least squares formu-lation to estimate the coefficient values (Burrows et al.

1987; Nordmann and Massmann1984). This method gives more insight into the bearing characteristics, and provides valuable information in analyzed data. In addition to the general frequency method, the excitation input wave form has to be determined to calculated low-noised coefficients. The main criteria in selecting an excitation waveform should be to achieve a response sufficiently higher than the noise, but small enough that it does not violate the linearity assumption. With multiple frequency excitation methods, satisfactory signal to noise (S/N) ratios are difficult to achieve without averaging. For deterministic signals, time domain averaging is used (Nordmann and Massmann1984) (or equivalently averaging of the Fourier transforms), while for random signals, power spectral densities must be used (Yasuda et al.1986). However, Rouvas et al. (1992) have shown that power spectral density methods can be used to averaging deterministic signals. Power spectral density methods are shown to eliminate errors caused by variations in the deterministic excitation signals used for averaging some amount of variation if vibration being inadvertently present in any measured signal. Following the aforemen-tioned idea if identification, Childs and Hale (1994) synthesize an identification scheme to derive low-noised estimation of rotordynamic coefficients of a testing rig for HJBs. This scheme is followed by the current study to design/build an in situ bearing inspection machine for performing coefficient estimation.

The proposed inspection machine consists mainly of a driving motor, a flexible coupling, an air bearing, posi-tioning fixtures, a newly-designed suspension for the test bearing, a shaker to exert exciting forces on the test bearing and two capacitor-type sensors. The key, the novel idea of the inspection machine design lies in the use of the high-stiffness air bearing to maintain radial run-outs of the spindle below 1 lm, and the suspension in a specially-designed fixture structure to allow the test bearing case in relatively-free motions to the spindle with much higher stiffness induced by the lubrication oil of the bearing. The air bearing for the inspection machine are selected to the special requirements in the manufacturing precisions for various dimensions and geometry characteristics of the components of the inspection machine. The capacitor-type sensors, includ-ing two custom-made ones, are also utilized to measure the motions of the spindle and bearing. Applying pre-specified external force on the test fluid bearing by the shaker in desired frequency contents and acquiring rel-ative radial motion between the test bearing case and the spindle based on the measurements from capacitor-type sensors, the dynamic coefficients of the test fluid-film bearing can then be calculated by the algorithm forged for deriving performance parameters of the test bearing. The algorithm is designed primarily to estimate rotor stiffness coefficients, load capacity, and eccentricity ratio to determine if the test bearing meet expected dynamic performance. In this algorithm, the rotordynamic coeffi-cients are extracted directly from frequency domain data by rewriting the equations in a way that the coefficients appear in a vector, and using a least squares formulation to estimate the coefficient values (Burrows; Nordmann). Finally, the experimentally-estimated dynamic coeffi-cients of the bearing are compared to theoretical estimates (Kawabata et al.1989; Chao and Haung2005), demonstrating favorable performance of the inspection machine.

This paper is organized as follows. Section2 states the design ideas and detailed description on the experiment system. Section3 derives the dynamics model of the experiment system, along with methods of identification. Section4gives results and discussion. Section5concludes this study and present future works.

2 The inspection machine

The design idea and detailed description on the in situ inspection machine are provided in this section for the further employment of the experimental identification algorithm presented in the next section to estimate the dynamic coefficients of the bearings.

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2.1 Theoretical basis for experiment

The configuration of the in situ inspection machine is depicted in Fig.1a, while Fig.1b presents a picture of the realistic in situ inspection machine in a entire experimental setup for testing the performance of a hydrodynamic journal bearing (HJB). It is seen from Fig.1a or b that the designed in situ inspection machine consists of three parts: (1) a motor frame to support the driving motor; (2) an high-stiffness air bearing to offers relative stillness of the test bearing; (3) a suspension frame to support the test bearing. Between the motor frame and the air bearing is a flexible coupling to isolate the vibration of the driving motor to the rest of the inspection machine. Theoretical basis for iden-tification originates from Child and Hale (1994). In identification, the external excitation forces and measure-ments are applied and recorded, respectively, at the test HJB and spindle. The measurements will be used to explore HJB performance. Focusing on the subsystem of bearing, sus-pension, and spindle, the third part of Figs.1a and 2

illustrates the theoretical basis of the experiment identifi-cation. Figure2in fact displays free body diagrams of two rigid bodies—the spindle and the HJB for testing. In this figure, {fx, fy} are x and y components of the input excitation

forces exerted by shaker. As the test bearing in vibration, there are bearing reaction forces, denoted by {fxb, fyb},

acting on the spindle as shown in Fig. 2a, while becoming {-fxb, fyb} at the bearing inner surface in Fig.2b. Based on

the free body diagram in Fig. 2, the equations of motion for the stator (bearing) mass Msin Fig.2can be written by

Ms € xs € ys   ¼ fx fy   þ fxb fyb   ; ð1Þ

where f€xs; €ysg the (measured) components of the stator’s

acceleration, bearing reaction forces {fxb, fyb} are, based on

simple dynamics,  fbx fby   ¼ Kxx Kxy Kyx Kyy   Dx Dy   þ Cxx Cxy Cyx Cyy   D _x D _y   þ Mxx Mxy Myx Myy   D€x D€y   ; ð2Þ

Combining Eqs. (1) and (2) gives fx Ms€xs fy Ms€ys   ¼ Kxx Kxy Kyx Kyy   Dx Dy   þ Cxx Cxy Cyx Cyy   D _x D _y   þ Mxx Mxy Myx Myy   D€x D€y   ; ð3Þ where DxðtÞ ¼ x  xsand DyðtÞ ¼ y  ys: ð4Þ

with known Ms and {x, y, xs, ys}, the motions of spindle

and bearing, measurable from sensors, the dynamic

Motor Frame (A) Air Bearing Frame(B) Suspension Frame(C) Motor

Air Bearing Suspension Flexible Coupling Adjusted plane Test Bearing Test HJB Spindle (a) (b)

Fig. 1 (a) The experimental diagram. (b) The experimental setup including the in situ inspection machine

xb f yb f

Rotor

x f y f xb fyb f

Bearing

(a) (b)

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coefficients M’s, C’s, and K’s can be identified based on Eq. (3) via a calculating algorithm to be developed in the next section.

In order to obtain satisfactory identification results, the generation of Dx(t) and Dy(t) via Eq. (4) into Eq. (3) ought to render large sensor-to-noise (S/N) ratio, which leads to one of experiment design directives that the spindle is supported by a high-stiffness air bearing to contain the radial runouts of the spindle {x, y} within small ranges, as compared to those radial vibrations of the bearing {xs, ys},

which are only constrained by the specially-designed sus-pension frame with relatively soft stiffness. From another point of view, on the contrary to the practical operation of a fluid-film bearing where the bearing is fixed while the spindle exerts radial vibration, in this experiment design the spindle is relatively motionless relatively to the bear-ing. Note also from Eq. (3) that beside the measurements of {x, y, xs, ys}, certain types of forces {fx, fy} need to be

applied at the bearing wall to generate nonzero measure-ments of {x, y, xs, ys} for identifying rotordynamic

coefficients. Thus, this generation of {fx, fy} is part of

experiment design. The general directives of determining {fx, fy} are that {fx, fy} have to be the dynamic loads with

rich frequency content in almost-equal levels to each other of measurement in displacements, velocities and accelera-tion of {x, y, xs, ys}. In this way, as applying Eq. (3) for

identifying M’s, C’s, and K’s, the S/N ratios accompanied with estimated M’s, C’s, and K’s, are commensurate. With the measurement of {x, y, xs, ys} rich in frequency content,

a calculating algorithm is developed in the next section to obtain satisfactory estimates of the rotordynamic coefficients.

Some further simplification steps are adopted to obtain preliminary estimates on rotordynamic coefficients. Considering the fact of thin fluid film for the bearing used for storage device and that no lubrication oil pumped in or out of the bearing during operation, the inertia effects on the dynamics of the system in Eq. (3) are small as com-pared to those of dampings and stiffness. Henceforth, the inertial terms in Eq. (3) is ignored in section four of experiment conduction. In addition, one can only applied static load other than dynamic ones to bearing wall, in which way all characteristics indices of the bearing except for the damping coefficients can be obtained. In the experiment conduction of this study, only the static load is applied to render preliminary estimation on bearing stiffness. Other than the application of the dynamic load is provided by shakers, the static load can be realized, as illustrated in Fig.3 and photographed in Fig.1b, by screwing a string at the bearing wall, stretching it through a pulley, and suspending with a designated weight. Note that the line of action of the static load (directed along the string and through the center of the bearing) is

designed herein to have an angle of 45 degree relative to the horizontal, which makes easy the setup of the sensor system.

2.2 Required specifications on components 2.2.1 Driving motor and fixture

A DC motor made by NICHIBO Corp. is utilized in this study, which can rotate the rotor above 18,000 rpm without loading. A supporting frame is designed and manufactured as shown in Fig.4. Note that the vibrations induced by the DC motor are not particularly required to be small since

Fig. 3 Static loading system

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they are expected to be damped out by the flexible coupling connected to spindle of the motor.

2.2.2 Flexible coupling

A flexible coupling in a particular geometric design is shown in Fig.5. This function of this coupling is provide enough flexibility to damp put the vibrations induced by the power source, the motor, before the rotational spindle enters into the high-stiffness air bearing. Note that during experiment, the tightness in assembling two pieces of the coupling and interconnecting blue piece in Fig.5 are adjusted through trials to render small spindle runouts at bearing location.

2.2.3 Air bearing

An air bearing made by, as shown in Fig.6, is utilized in this study to exert high stiffness on the rotational spindle,

resulting in a small spindle runouts around 1 lm at bearing location. The set of the air bearing consists of the bearing itself, a compressor providing air, and set of air purifiers to keep the inner compartment of the air bearing free of contamination. Other than a right choice of the air bearing, the supporting frame of the air bearing, as shown in Fig.7, needs also to be designed stiff enough with natural frequencies above 1.0E ? 6 N/um and manufac-tured in high precision to accomplish the goal of providing high stiffness. Note that he verticalness and flatness of the vertical contact planar surface of the frame to the air bearing are two critical manufacturing qualities required.

2.2.4 The suspension frame

The suspension frame supporting the test bearing is designed in particular geometry, as shown in Fig.8a. This particular geometry would be able to provide much softer stiffness in radial direction as opposed to those in axial of the spindle. As results, the radial stiffness during experi-ment is mostly contributed from bearing stiffness. To ensure the stiffness of the suspension frame as desired, finite element analysis is conducted, as shown in Fig.8b, to determine various dimensions of the suspension frame. It is shown from Fig. 8b that the radial stiffness of the sus-pension frames is 2.8 N/m which is much smaller than those of commercial bearings, which are within the ranges of 3.0E ? 07 N/m. It is worthy to point out at this point that the design of the supporting frame also suits well for the needs of exchanging the bearings in various sizes for frequent tests.

Fig. 5 The flexible coupling

Fig. 6 The air bearing

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2.3 Sensing system

1. For spindle runouts, {x, y}: Two custom-made capacitor-type sensors as shown in Fig.9a are used for measuring the runouts of the small-size spindle of 3 mm in diameter.

2. For bearing vibrations, {xs, ys}: Two common

capac-itor-type sensors as shown in Fig.9b are used for measuring the vibrations of the bearing wall in X and Y directions.

3. For rotor speed, a common fiber-optic-type non-contact sensor is used to report the rotational speed of the rotor. The measuring range is from 10 to 99999 rpm.

4. The Cantilever-beam-type fixtures employed for the four capacitor-type sensors are required to be strong enough to sustain environmental vibrations. Pre-calibrations have been done to assists the selection

of the fixtures in the purpose of ensuring the stiffnesses high enough to avoid extra commensurate vibration.

2.4 Experiment steps

The experiment steps for the case of applying static loads to identify various parameters of bearing performance from measurements are detailed in the followings.

1. Place the desired weight to the end of suspending string

2. Choose the test speeds as 2,000, 3,000, 4,000, 4,500 rpm, which, in experiments, are achieved by applying 1.32, 1.74, 2.2, 5.5 V to the DC motor, respectively.

3. Turn on the power supply and adjust the input volt to render the rotation speed steadily at the desired rpms, which is assured by the optic-fiber speed sensor. 4. Record measurement data in specified time period of

5 s from four capacitor-type sensors for spindle runouts and bearing vibration via an A/D card and data-acquisition software, while the rotation speed is steady on applied voltage. Within the 5 s, 5,000 data are taken from each of four capacitor-type sensors. 5. Take average of one oscillating period on data for

first-level noise reduction.

6. Incorporate the averaged data into the algorithms developed for calculating various performance factors of the test bearing.

Fig. 8 (a) The suspension frame; (b) The finite element analysis

Fig. 9 (a) The custom-made capacitor-type sensors. (b) The common capacitor-type sensors

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3 Identification of dynamic coefficients

The algorithm for calculating rotordynamic coefficient is developed herein, which is originated from Childs and Hale (1994).

3.1 Equations for identification

The rotordynamic coefficients are determined in the fre-quency domain via the Fourier Transform = on Eq. (3), which yields Fx MsAs Fy MsAs   ¼ Hxx Hxy Hyx Hyy   Dx Dy   ; ð5Þ where Fk¼ =ðfkÞ; Ak¼ =ð €kkÞ; Dk¼ =ðDkÞ;

and k could be x or y. The elements of the frequency-response function (FRF) H are related to the coefficients defined in Eq. (3) by

Hij¼ ðKij x2MijÞ þ jðxCijÞ; ð6Þ

where x is the excitation frequency and j¼pffiffiffiffiffiffiffi1. It should be note that Eq. (5) only provides two complex equations for the four unknown Hij’s. To successfully

estimate all rotordynamic coefficients, two independent orthogonal dynamic loads are applied to the stator of the test bearing by alternately exciting along the X and Y directions in Fig.2. Each of these ‘‘pseudo-random’’ dynamic loads contains 41 sinusoids. The superposition of these sinusoids is optimized to provide a composite loading that has a high= spectral-line energy to high S/N ratio. The loadings are periodic and have energy only at 10 Hz frequency increments in the range from 100 to 500 Hz. The two independent loadings result in the four complex equations, Fxx MsAxx Fxy MsAxy Fyx MsAyx Fyy MsAyy " # ¼ Hxx Hxy Hyx Hyy " # Dxx Dxy Dyx Dyy " # ð7Þ

where the subscripts iy of the force and displacement matrices correspond to response in the i direction as shown in Fig.2. Childs and Hale (1994) provides com-plete details of using the power-spectral density method by Bendat and Peirsol (1986) to calculate the Hij

func-tions. Spectral density averaging techniques are next used to obtain the frequency-response functions (FRF) of Eq. (7) for damping out noises, which defines a multi-input multi-output (MIMO) system. The details are stated in Appendix1.

4 Results and discussion

Conducting the experiment steps as given in Sect.2.4, one can obtain sensor outputs for the motions of the spindle and bearing, and their differences for later analysis. The varied parameters of the test HJB are listed in Table 1. Figure10

show the measured motion difference between the spindle and bearing, i.e., Dx(t) = x -xsand Dy(t) = y -ys, where

the measurements in three spindle-rotating periods at the steady state are shown. It can be seen from these figures that the motions of the spindle and bearing should be both periodic and harmonic-like with low noises in one rotation of the spindle. Therefore, the results obtained from the computation procedure based on Eq. (7) are reliable to identify the rotordynamic coefficients of the test HJB.

With experimental data in hand by conducting those steps in Sect. 2.4, three related characteristics indices of the test bearing are calculated. They are load capacity, attitude angle, and two stifnesses coefficients. Figure11

shows the load capacity with respect to various rotor speeds of 2,000, 3,000, 4,000, and 4,500 rpm, and also theoretical predictions obtained based on the model and computation procedure established in (Kawabata et al.

1989; Chao and Haung2005) for rotor speed of 3,500 rpm. It is seen from this figure that the load capacity increases slowly as the eccentricity is low until the eccentricity reaches around 0.6. It reflects the fact that the bearing has higher capacity as the rotor exhibits higher eccentricity, which conforms to common performance of a bearing. On the other hand, the theoretical capacities for 3,500 rpm are in the vicinity of corresponding experimental data. In particular, the theoretical capacity at the eccentricity of 0.71 is between experimental counterparts of 3,000 and 4,000 rpms, showing the legitimacy of the in situ inspec-tion machine built in this study. Figure12 shows the attitude angle with respect to various rotor speeds, where the attitude angle keeps constant around within the range of low eccentricity and then moderately decreases approxi-mately above the eccentricity of 0.7. The change in the

Table 1 The bearing specifications

Bearing diameter 6 mm L/D ratio 1.5 Radial clearance 2 lm Bearing mass 10.577 g Groove depth 5 lm Groove angle 21° Groove values 8 Fluid viscosity 0.0297 N s/m2 Operating speed 2,000–4,500 rpm Static load 35–350 g

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attitude angle found herein also conforms to common understanding on an HJB (Kawabata et al.1989; Chao and Haung2005). It is seen from this figure that the theoretical angles for 3,500 rpm are between experimental counter-parts of 3,000 and 4,000 rpms, validating again the performance of the built inspection machine. Figures13

and 14 show, respectively, the bearing stiffnesses of kyy

(direct stiffness) and kyx (cross-coupled stiffness) as

opposed to rotor eccentricity and for various rotor speeds. The identified kxxand kxyare not shown herein due to their

similarity to kyy and kyx, respectively. It is seen from

Figs.13and14that kyyincreases almost exponentially as

the rotor eccentricity increases, while kyx, as all negative,

decreases almost exponentially as the rotor eccentricity

increases. It should be noted at this point that as rotor speed increases, kyydecreases slightly, which is due to the

mag-nification in the inertial effects. Also seen in Figs.13,14is the closeness between the previous theoretical predictions documented in the past studies and the identified counter-parts presented in this study. The identified stiffnesses at 3,500 rpms are all between those counterparts of 3,000 and 4,000 rpms.

5 Conclusion and future works

An in situ inspection machine has been successfully designed and constructed successfully in the laboratory for

Fig. 10 Measured motion difference between the spindle and bearing

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testing the performance of a small-scaled hydrodynamic journal bearing (HJB) frequently used in data-storage drives, such as hard disc drives and optical disc ones. The key component of the machine is the high stiffness air bearing that keeps the spindle runouts within 1 um. The performance indices of the bearing identified from the experimental data have reflected realistic dynamic char-acteristics of a small-scaled HJB documented in the past studies, validating the effectiveness of the constructed machine. In the future, in order to improve the S/N ratio of the raw time-domain experimental data, the assembling tightness of the flexible coupling should be re-determined systematically based on theoretical analysis in order to render the spindle runouts as small as possible.

Appendix 1

Technique for reducing data noise

Spectral density averaging techniques are used to obtain the frequency-response functions (FRF) of Eq. (6) for damping out noises, which defines a input multi-output (MIMO) system. The inputs are Dxx, Dxy, Dyx, and

Dyy, and the outputs are Fxx- MsAxx, Fxy- MsAxy, Fyx

-MsAyx, and Fyy- MsAyy. Before calculating the FRF for

this MIMO system, first consider the best solution for the FRF of a single-input single-output (SISO) system. A SISO system with a clean input a and a noise contaminated output b will be examined. According to Bendat and Peirsol (1986), an unbiased estimate H of the FRF for this type of SISO system is given by,

H¼Gab Gaa ð8Þ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Eccentricity Ratio yti c a p a C d a o L d e zil a n oi s n e mi d n o N By Chao et al. 2000rpm by current exp. 3000rpm by current exp. 4000rpm by current exp. 4500rpm by current exp.

Fig. 11 The load capacity with respect to eccentricity ratio in various rotor speeds 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 10 20 30 40 50 60 70 80 90 Eccentricity Ratio ) g e d( el g n A e d uti tt A By Kawabata et al. 2000rpm by current exp. 3000rpm by current exp. 4000rpm by current exp. 4500rpm by current exp.

Fig. 12 The attitude angle versus eccentricity ratio

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 x 10 6 Eccentricity Ratio ) m/ N( y y K , s s e nff it S g nir a e B By Kawabata et al. 2000rpm by current exp. 3000rpm by current exp. 4000rpm by current exp. 4500rpm by current exp.

Fig. 13 The bearing stiffnesses kyyversus eccentricity ratio

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 x 10 6 Eccentricity Ratio ) m/ N( x y K , s s e nf fit S g nir a e B By Kawabata et al. 2000rpm by current exp. 3000rpm by current exp. 4000rpm by current exp. 4500rpm by current exp.

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with Gab being the cross spectral density and Gaa the

auto spectral density. For discrete data (which we have) an FFT algorithm is used to calculate the Discrete Fourier Transforms (DFT’s) of Eq. (7). In particular, the spectral densities are calculated by,

Gap ¼ 2 ndNDt Xnd i¼1 Ai ðxÞPiðxÞ ð9Þ

where ndis the number of statistically independent blocks

of data, N is the number of data points in each data block, Dt is the time increment between sampling of each data point, A*(x) = the complex conjugate of ={a(t)} and P(x) =={p(t)} = is used to represent the DFT here).

We wish to apply the unbiased estimator for the SISO system described above to the MIMO system of Eq. (8). First, recall that we have 32 statistically independent blocks of data (nd= 32), each block containing 1,024

(N = 1,024) data points for each of the dynamic loads, accelerations, and relative motions whose= appear in Eq. (8). Next, we identify each independent dynamic load of Eq. (8) (Fxx and Fyy) as a clean input a, and each of the

resulting elements of the force and displacement matrices as noise contaminated outputs b. Eq. (8) is then applied separately to each individual element of the force and displacement matrices, yielding an unbiased estimate for each input and output element contained in these matrices, The resulting equation,

1 Ms Gfxx axx Gfxx fxx Gfyyfxy Gfyyfyy Ms Gfyy axy Gfyyfyy Gfxx fyx Gfxx fxx  Ms Gfxx ayx Gfxx fxx 1 Ms Gfyy ayy Gfyy fyy 2 4 3 5 ¼ Hxx Hxy Hyy Hyx   Gfxxdxx Gfxx fxx Ms Gfyy dxy Gfyyfyy Gfxxdyx Gfxx fxx Gfyy dyy Gfyyfyy 2 4 3 5; ð10Þ

yields improved estimates for the FRF when solved.

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數據

Fig. 1 (a) The experimental diagram. (b) The experimental setup including the in situ inspection machine
Fig. 4 The motor frame
Fig. 5 The flexible coupling
Fig. 8 (a) The suspension frame; (b) The finite element analysis
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