dynamic effects of the interference fit of motor rotor on the

DYNAMIC EFFECTS OF THE INTERFERENCE FIT OF MOTOR ROTOR ON THESTIFFNESS OF A HIGH SPEED ROTATING SHAFT

Shin-Yong Chen1, Chieh Kung2, Te-Tan Liao3, Yen-Hsien Chen3

1Department of Automation and Control Engineering, Far East University, Taiwan2

Department of Computer Application Engineering, Far East University, Taiwan3

Department of Mechanical Engineering, Far East University, Taiwan

E-mail: [email protected]; [email protected]; [email protected]

Received October 2009, Accepted April 2010

No. 09-CSME-68, E.I.C. Accession 3154

ABSTRACT

Developing a motor-built-in high speed spindle is an important key technology for domesticprecision manufacturing industry. The dynamic analysis of the rotating shaft is the major issuein the analysis for a motor-built-in high speed spindle. One of the major concerns is how themotor rotor is mounted on the shaft, by interference (shrink) fit or else. In this study, dynamicalanalyses are carried out on a motor-built-in high speed spindle. The motor rotor is mounted onthe spindle shaft by means of interference fit. Modal testing and numerical finite elementanalyses are conducted to evaluate the dynamical characteristics of the spindle. The stiffness ofthe shaft accounting for the interference fit is investigated for the finite element model of thespindle. This study also proposes an analysis procedure to dynamically characterize the highspeed spindle with a built-in motor. Based on the results of modal testing and the numericalanalyses, it may conclude that the proposed procedure is feasible for the spindle and is effectivefor other similar applications.

EFFETS DYNAMIQUES DE L’AJUSTEMENT DE L’INTERFERENCE D’UN ROTORDE MOTEUR SUR LA RAIDEUR D’UN ARBRE DE ROTATION A GRANDE

VITESSE

RESUME

Le developpement d’electrobroches a haute vitesse avec moteur integre est une technologie clepour l’industrie domestique de fabrication d’outils de precision. L’analyse dynamique de l’arbrede rotation est le point principal dans l’analyse d’electrobroches a haute vitesse avec moteurintegre. Une des preoccupations majeures est la facon que le rotor du moteur est monte surl’arbre de rotation, soit par ajustement d’interference ou autrement. Nous avons procede a desanalyses dynamiques sur des electrobroches a haute vitesse avec moteur integre. Le rotor dumoteur est monte sur la tige de l’electrobroches par ajustement d’interference. Des tests sur unmodele et des analyses numeriques des elements finis sont fait pour evaluer le caracteristiquesdynamiques de l’electrobroches. La raideur de l’arbre de rotation qui est importante dansl’ajustement de l’interference est etudiee et optimisee pour le modele d’elements finis del’electrobroches. La proposition presente egalement une procedure d’analyse pour definir ladynamique de l’electrobroche a grande vitesse avec moteur integre. En conclusion, en nousbasant sur les resultats obtenus sur le modele et sur l’analyse numerique, la procedure preposeepourrait servir pour l’electrobroche, et etre utiles dans des applications similaires.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2, 2010 243

1. INTRODUCTION

Recently rotary shaft assemblies have become popular in engineering applications rangingfrom high-tech products such as jet engines and computer hard disk drives (HDD), tohousehold appliances, such as washing machines and refrigerator compressors. A rotary shaftassembly consists of a rotating part (rotor), a stationary part (stator), and multiple bearingsconnecting the rotor and the stator. The rotor varies in its geometry for various applications.One application is that the rotor made of stacked steel sheet annuli is mounted on a hollowshaft by means of interference fit. This unique application posts a challenge to engineers ondetermination of the stiffness of the rotor-shaft assembly. Chen, et al. [1] suggested the stiffnessof a rotor-solid shaft assembly is proportional to the interference fit. However, the stiffness of arotor-hollow shaft assembly is yet to be studied. The stiffness of a rotor-hollow shaft assemblyplays a major role in the subsequent analyses of dynamical characterization of its final form inwhich other accessories including bearings are added. Thus, it is crucial to determine thestiffness of the rotor-hollow shaft assembly as a priori.

In the past many efforts had been devoted to the study of rotary machines. Perhaps Prohl [2]was the first who applied the FEA to the analysis of a rotor-bearing system. Ruhl and Booker[3] applied the finite element method to analyzing the steady-state of turbo-rotor systems. Intheir study the influences of the rotary inertia, gyroscopic moment, bending, shear deformation,axial load and internal damping were neglected to simplify the model. On the other hand,Nelson and McVaugh [4] introduced Rayleigh beam elements to their rotating shaft model andderived the motion equations for the shaft. The effects of translational and rotary inertia, axialload, and gyroscopic moments were considered. In addition, numerous finite element modelsand procedures have been proposed, such as by Nelson [5] and Ozguven and Ozkan [6], in aneffort to generalize and optimize the stability of the rotor-shaft system. Studies on revisingnumerical models cooperated with the modal testing were also suggested. The revision strategieswere based on the homogenous motion equations and orthogonality of the rotary shaft system.For example, Chen [7] presented a direct identification procedure based on the modal testing.The procedure enables the mass and stiffness matrices to be obtained from calculatedeigenvalues and eigenvectors incurred from the test data. The procedure was based on thetheory of matrix perturbation in which the correct mass and stiffness matrices are expanded interms of analytical values and a modification matrix. In 1996, Chen et al. [8] developed afrequency-domain method to estimate the mass, stiffness and damping matrices of the model ofa structure. Furthermore, Chen and Tsuei [9], Ibrahim and Fullekrug [10] extracted thenormalized mode shapes as the basis of a non-damped oscillation experiment. There weredeviations reported in the studies due the model elements and the structural parameters such asthe Young’s modulus, material density, and the assembling conditions. Chu [11] considered ashaft model with a regional enlarged diameter to account for the interference fit between the

Nomenclature

M, mass matrixK, stiffness matrixf(t), force vector in time domainf(v), force vector in frequency domainH(v), normal frequency response function

matrix

x(t), response vector in time domain fordamped system

rQj, the j-th component of the r-th modeshape vector

v, frequency of excitation�vvr, the r-th natural frequency of the system[.]21, inverse of a matrix


rotor and the shaft. The model with the regional enlarged diameter of the shaft results in amismatch of the mass and mass moment with those of the test shaft. On the other hand, in thestudy by Huang [12], the Young’s modulus was modified locally to account for the interferencefit of the rotor. The Young’s modulus was fine-tuned so that the natural frequencies andfrequency response functions by the FEA were matched with those obtained from the modaltesting. There was little difference reported in the results. More recently, Altintas and Cao [13]employed a general finite element method to predict the static and dynamic behavior of spindlesystems. The spindle and housing were modeled by Timoshenko beam elements. Theirsimulation showed that the rotational speed of the spindle shaft had a larger influence on thelower natural frequencies. Erturka, et al. [14–16] presented an analytical method that usedTimoshenko beam theory for calculating the tool point frequency response function (FRF) of aspindle–holder–tool combination by using the receptance coupling and structural modificationmethods. They proposed a mathematical model, as well as the details of obtaining the systemcomponent (spindle, holder and tool) dynamics and coupling them to obtain the tool pointFRF. The model could be used in predicting and following the changes in the tool point FRFdue to possible variations in tool and holder types and/or tool length very quickly and in a verypractical way. Also, through the model, the stability diagram for an application could bemodified in a predictable manner in order to maximize the chatter-free material removal rate byselecting favorable system parameters. Ozasahin, et al. [17] presented a new method foridentifying contact dynamics in spindle–holder–tool assemblies from experimental measure-ments. They extended a previously developed elastic receptance coupling equations to give thecomplex stiffness matrix at the holder–tool and spindle–holder interfaces in a closed-formmanner.

In this study, dynamical analyses are carried out on a motor-built-in high speed spindle. Themotor rotor is made of stacked steel sheet annuli and is mounted on a hollow shaft by means ofinterference fit. Modal testing and numerical finite element analyses are conducted to evaluatethe dynamical characteristics of the spindle. A simplified finite element model is proposed. Theequivalent stiffness of finite element model representing the rotor-shaft assembly accounting forthe interference fit is studied. The proposed model is verified with modal test results.

2. MODAL ANALYSIS

In this study, the frequency response function (FRF) and modal parameters obtainedthrough a modal testing are used as a benchmark to establish a finite element analysis (FEA)model for subsequent dynamical analyses. In general, the FRF is derived for an un-dampeddynamic system. Consider an un-damped dynamic system, the governing equation of themotion of the system is

M€xx tð ÞzKx tð Þ~f tð Þ ð1Þ

where M, K[Rn6n are the mass matrix and the stiffness matrix, respectively, of the dynamicsystem, and x, f[Rn are, respectively, the vectors of displacement responses and externallyapplied excitations. Taking Laplace transformation of Eq. (1) with zero initial conditions, oneobtains

s2MX sð ÞzKX sð Þ~F sð Þ ð2Þ


the motion of the system can be evaluated in frequency domain by letting s 5 jv. Thus, Eq.(2)becomes

{v2MzK� �

X vð Þ~F vð Þ ð3Þ

or

X vð Þ~H vð ÞF vð Þ ð4Þ

where

H vð Þ~ K{v2M� �{1 ð5Þ

is denoted as the normal FRF. Through variable transformation and orthogonal equation, weobtain

hjk vð Þ~XN

r~1

rQj

� �rQkð Þ

�vv2r{v2

ð6Þ

in which hjk is the FRF of j-th node corresponding to the excitation at the k-th node, rQj is thej-th component of the r-th mode shape vector, �vvr is the r-th natural frequency of the system,and N is number of the total natural frequencies considered in calculating the response. In thisstudy, both rQj and �vvr are obtained through FEA. The frequency response functions aredetermined using Eq. (6) and are compared with those through the modal testing.

The FEA involve a modal analysis whose main purpose is to characterize the vibrationcharacters of mechanical elements of the shaft. The FEA model is posted with a boundarycondition of ‘‘soft suspension’’, which is an idealized free-free beam boundary condition. Theparameters of interest of the modal analysis include the natural frequencies, mode shapes andthe damping ratio of the shaft system. Prior to the FEA, a modal testing is performed for theshaft structure. The test apparatus consists of an excitation source, a signal acquisition device, asignal analyzer, and frequency response function generator. To comply with the soft-suspensionboundary condition, the shaft is suspended with rubber bands during the modal testing. Fig. 1shows the diagram of experiment apparatus.

For a high speed rotating shaft with the motor rotor interference fitting to the shaft, thefitting condition has a significant influence on the stiffness of the shaft[1]. An improperinterference fit will result in an unpredictable stiffness of the shaft; thus, the natural frequencymight be within the proximity of the operating frequency. In this study, the effect of the fittingamount on the stiffness of the spindle is not known as a priori. To best qualify the effect for theFEA model, the modal testing offers a measure to characterize the interference fit for the shaftand to verify the FEA model. The flow chart shown in Fig. 2 illustrates the procedures ofconstruction and modification of the FEA model based on the results of the modal testing.

To begin with the procedure, a modal testing is performed on a bare hollow shaft. The resultsof the modal parameters are recorded. Meanwhile, an FEA model using PIPE16 line elementsrepresenting the bare shaft is created and the modal analysis is carried out. The purpose of thisstep is to confirm the dimensions and the material properties for the FEM model. Next, the


modal testing is performed on the shaft with a fitted motor rotor. An amount of 0.022 mminterference fit is designated. The resulting modal parameters are then recorded. An FEA modelof the rotor-shaft assembly is also created concurrently. This FEA model employs PIPE16elements as a simplified model representing the rotor-shaft assembly. To account for theinterference fit, a segment on the model is partitioned wherein a localized Young’s modulus isobtained as the local stiffness of the line model; the value of the Young’s modulus of the rest

Fig. 1. Schematic of the experiment apparatus.

Fig. 2. The flow of creating and modifying the finite element model.


segments is the one confirmed in the first step. In this step, the effects of interference fit on thelocal stiffness are also investigated. Finally, the FEA model is expanded to model the fullyequipped spindle assembly by adding additional MASS21 point mass elements representing allnecessary accessory components. Dynamical analyses are then conducted on the fully equippedspindle assembly model.

3. ILLUSTRATION STUDY EXAMPLE

In this study, numerical and experimental modal analyses are carried out on a motor built-in high speed spindle featured with 4.6 KW/30000 rpm and an automatic tool changer(ATC). Finite element analysis software, ANSYSTM, is employed to perform the numericalanalysis. The study consists of three stages of evaluations, that is, evaluations of a barehollow shaft model, a hollow shaft model with an interference fit rotor (rotor-shaft) model,and a fully equipped rotor-shaft model in which other accessory components are mounted.Fig. 3 shows the longitudinal cross section of the hollow shaft. The segmental length, innerradius and outer radius along the shaft are listed in Table 1. The rotor is composed ofstacked steel sheet annuli with outer diameter 39.5 mm and inner diameter 23.982 mm. Its

Fig. 3. Schematic of the hollow shaft.

Table 1. Segmental dimension along the shaft. (unit: mm).


longitudinal length is 80 mm. The location of the rotor mounted on the shaft is shown inFig. 4.

3. 1. The hollow shaft modeling and modificationIn the first stage of this study, a bare hollow shaft without a motor rotor is considered and the

analysis is described in this section.

First, a modal testing is conducted on the bare hollow shaft. The shaft is suspended at its twoends with rubber bands. A hammer is used to create excitation at each of the 11 locations shownin Fig. 3. At each location, five FRFs are measured and are averaged. The first two naturalfrequencies of the shaft are then recorded as 1535.35 Hz and 3720.38 Hz, respectively. Next, anumerical modal analysis is performed. Because the shaft (290 mm long) is a bare hollow tubeas shown in Fig. 3, a simplified finite element model with line elements, PIPE16, is thereforeconstructed using the ANSYSTM. PIPE16 is a uni-axial element with tension-compression,torsion, and bending capabilities. The element has six degrees of freedom at two nodes:translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes.This element is simplified due to its symmetry and standard pipe geometry. The segmentallength, inner radius and outer radius along the shaft as seen in Fig. 3 are listed in Table 1. Thestructural nodes along the finite element model are created in a manner of 1 mm spacing tocomply with the aforementioned excitation locations. For the segment where tapering exists, thesegment is further evenly divided with two additional intermediate nodes. Different realconstants, required by the ANSYSTM, are assigned to account for the variation of segmentradius as well as tapering along the shaft. The material considered for the shaft has a density of7950 kg/m3, Young’s modulus of 210GPa, and Poisson’s ratio of 0.3. The model has a free-freeend boundary condition. The numerical results show that the first two significant frequenciesare 1587.07 Hz and 3844.01 Hz, respectively.

Comparing the simulated frequency results with the experimental ones, one may see that bothsimulated frequencies are about 3.3% higher. It might suggest that the deviations be resultedfrom the simplified finite element model.

It is acknowledged that parameters that influence the natural frequencies of a structureinclude density, Poisson’s ratio, and Young’s modulus of the material and the boundaryconditions of the structure. In this study, the density is calculated based on the real mass andvolume of the shaft; therefore, the density of the shaft is considered constant. The boundaryconditions as described previously are kept in a free-free end condition. Thus, it may beappropriate to evaluate the effects of Poisson’s ratio and/or Young’s modulus to account for thedeviations of the natural frequencies. Our results show that the effects of the Poisson’s ratio onnatural frequency are insignificant. On the other hand, by fine tuning the Young’s modulusfrom original 210 GPa to 192.69 GPa, we obtain an optimal condition where both simulated

Fig. 4. Schematic of the rotor-shaft assembly.


frequencies fall into a region of the least deviation from the experimental natural frequencies.Table 2 lists the numerical values of the density, spindle length, Poisson’s ratio and the Young’smodulus of the shaft before and after the modification of the Young’s modulus. Noted that theYoung’s modulus in Table 2 is intentionally denoted as E1 so as to distinguish itself from E2which represents the local Young’s modulus accounting for the interference. The resultsaccounting for the modification are summarized in Table 3. The results in Table 3 indicate thatboth simulated frequencies are closed to experimental ones less than 0.2%. Comparisons offrequency response functions (FRF) h1–11and h10–11due to the modification of Young’smodulus are shown in Figs. 5(a) and 6, respectively. Fig. 5(b) is the callout of Fig. 5(a) showingfinite FRF values near zero frequency. Comparisons of the experimental mode shapes due totuning the Young’s modulus are shown in Figs. 7 and 8, for the 1st and the 2nd naturalfrequencies, respectively. It can be seen that these figures reveal closeness of the FRF curves andmode shapes for both experimental results and modified numerical model results, the Young’smodulus of the shaft will then be considered as 1.926961011 N/m2 (E1) in the subsequentanalyses of the study.

3. 2. The rotor-shaft modeling and modificationIn the second stage of the study, the hollow shaft with an interference fit motor rotor is

considered. The relative position of the rotor on the shaft has been shown in Fig. 4. Previousstudy [1] has indicated that although the use of a point mass element helps simplify constructinga finite element model with interference fit components, the effect of the components on thestiffness of the shaft should also be involved. The local stiffness of a shaft due to an interference

Table 2. Numerical values of the shaft parameters before and after modification of the Young’s modulus.

Table 3. Comparison of natural frequency results before and after modifying finite element model ofthe shaft.


Fig. 5. (a). Comparison of the experimental FRF h1–11 results with that of numerical results beforeand after modifying the Young’s modulus for the bare hollow shaft, (b). The callout of Fig. 5 (a)showing finite values of FRF near zero frequency for the bare hollow shaft.


fit rotor tends to be higher. To account for the increase of the stiffness, one should introduceboth mass and mass inertia values to the finite element model. In addition, the study [11]suggests that to account for the increase of the bending stiffness due to an additional

Fig. 6. Comparison of the experimental FRF h10–11 results with that of numerical results before andafter modifying the Young’s modulus for the bare hollow shaft.

Fig. 7. Comparison of the experimental mode shape (frequency 1535.35 Hz) with that of numericalresults before and after tuning the Young’s modulus E1 for the bare hollow shaft.


interference fit rotor, the shaft should be ‘‘enlarged’’ locally by increasing the radius of thesegment where the rotor is fitted. Nevertheless, their results show that there is a deviation of 6%;that is, errors exist in the mass and mass inertia. In our study, an equivalent Young’s modulus isproposed as the local stiffness to account for the additive interference fit rotor.

First, a modal testing is carried out on the rotor-shaft structure. The amount of interference is0.022 mm. The procedure of obtaining the averaged FRFs is the same as described in the firststage. The fundamental natural frequency is found to increases from 1535.35 Hz for the bareshaft to 1640.96 Hz, an addition of 105.61 Hz, and the second natural frequency becomes3371.92 Hz.

A numerical modal simulation then follows. The finite element model consists of PIPE16 lineelements and point mass elements, MASS21, to account for the interference fit rotor. The lineelements are modeled with E151.926961011 N/m2 as the Young’s modulus. The mass elementsare characterized by the mass and the mass inertia. A free-free end boundary condition is postedon the model. The fundamental natural frequency from the modal analysis becomes 1249.07 Hz,a decrease of 392 Hz comparing with that of 1640.96 Hz. To compensate the deviation, anequivalent local Young’s modulus is proposed to account for the increase of local stiffness dueto the interference fit rotor. It is emphasized that the line elements carrying no rotor areremained with E151.926961011 N/m2 as the Young’s modulus. The equivalent Young’smodulus is E254.9050 6 1011 N/m2, and the first two natural frequencies become 1640.05 Hz,and 3372.86 Hz, respectively. These two natural frequencies are less than 0.1% off theexperimental natural frequencies (1640.96 Hz and 3371.92 Hz, respectively). Comparisons offrequency response functions h1–11 and h10–11 based on the equivalent local Young’s modulus areshown in Figs. 9 and 10, respectively. Comparisons of the experimental mode shapes due to theequivalent local Young’s modulus are shown in Figs. 11 and 12, for the 1st and the 2nd naturalfrequencies, respectively. It is seen that these figures reveal the closeness of the FRF curves andmode shapes for both simulated and experimental results, it may conclude that the proposedequivalent Young’s modulus to account for the interference fit rotor is advisable to accuratelymodel the rotor-shaft assembly.

It is noted that the amount of interference due to the interference fit also contribute thevariation to the stiffness of the shaft [1]. In this study, the influence of the amount ofinterference fit on the dynamical characters of the shaft is also investigated. We consider theamount of interference fit to be: 0.011 mm, 0.022 mm, 0.026 mm, and 0.03 mm, respectively.The frequency results associated with 0.022 mm interference have been stated above.

Fig. 8. Comparison of the experimental mode shape (frequency 3720.38 Hz) with that of numericalresults before and after tuning the Young’s modulus E1 for the bare hollow shaft.


Summarized in Table 4 are the influences of the interference fit on the natural frequencies. Thevalues under the column E2 represent the local Young’s modulus accounting for the amount ofinterference fit. To ease comparison, these values are graphed versus the amount of interferencein Fig. 13. The curve shown in Fig. 13 suggests that for the rotor-hollow shaft consideredherein, the equivalent Young’s modulus increase with the amount of interference, indicating anincrease of stiffness with the interference amount; however, the stiffness decreases as theamount of interference become greater, in our case around 0.018 mm.

3. 3. The fully-equipped shaft modelIn the third stage of the study, the fully equipped shaft is considered. The approach described

above is now applied to the shaft herein. The fully equipped shaft consists of the hollow shaft,the motor rotor with a 0.022 mm interference fit, and other accessory components including thepress ring, the front spacer, and the inner spacer of the frontal bearing, the inner ring of thefrontal/rear bearing, the press ring of the rear bearing, and the precision tight nuts. Theschematic of the fully equipped shaft is illustrated in Fig. 14. As these accessory components aresliding-fitting with the shaft, it is reasonably to consider they contribute no effects to thestiffness of the shaft. Thus, in the numerical modal analysis, these components are modeled withpoint mass elements, MASS21.

Fig. 9. Comparison of the experimental FRF h1–11 results with that of numerical results before andafter modifying local Young’s modulus due to interference fit of the rotor.


Prior to the numerical modal analysis, a modal testing is conducted. The experimentprocedure has been described in previous sections except that the fully equipped shaft isconsidered. The fundamental natural frequency is found to be 1249.70 Hz and the second

Fig. 10. Comparison of the experimental FRF h10–11 results with that of numerical results beforeand after the modifying local Young’s modulus due to an interference fit of the rotor.

Fig. 11. Comparison of the experimental mode shape (frequency 1640.05 Hz) with that of numericalresults before and after modifying the local Young’s modulus E2 of the rotor-shaft assembly.


natural frequency 2886.49 Hz. Comparing this value with the one obtained in the last section,one may find the frequencies decrease from 1640.05 Hz to 1249.70 Hz, and 3372.86 Hzto 2886.49 Hz. These drops are expected as additional mass representing the accessorycomponents is added on the shaft assembly.

Following the modal testing, the numerical modal analysis is accomplished. The numericalmodel contains the model described in Section 3.2 and additional mass points to account for theaccessory components. The input to the mass points is the mass only. The first naturalfrequency from the numerical analysis is 1225.29 Hz, an error of 1.95 %, and the second naturalfrequency of 2846.96 Hz shows an error of 1.37%. Comparisons of frequency responsefunctions h1–11 and h10–11 based on the results are shown in Figs. 15 and 16, respectively.

Fig. 12. Comparison of the experimental mode shape (frequency 3372.86 Hz) with that of numericalresults before and after modifying the local Young’s modulus E2 of the rotor-shaft assembly.

Table 4. Variation of the natural frequencies corresponding to the modified Young’s modulus.


Fig. 15(b) is the callout of Fig. 15(a) showing finite FRF values near zero frequency.Comparisons of the experimental mode shapes are shown in Figs. 17 and 18, for the 1st and the2nd natural frequencies, respectively. It is seen that in these figures, the curves representing testresults and simulation results coincide closely. It may therefore conclude again that theproposed local equivalent Young’s modulus to account for the interference fit rotor is advisableto accurately model the rotor-shaft assembly.

4. CONCLUSIONS

In this study, an approach of local equivalent Young’s modulus is proposed to enable betterfinite element predictions of the dynamical characters of a motor built-in high speed hollowrotating shaft. Both numerical and experimental modal analyses are carried out on the shaftfeatured with 4.6 kw/30000 rpm and an automatic tool changer (ATC). The effects ofinterference fit are also investigate. Some conclusions are drawn as below:

Fig. 14. Schematic of the fully-equipped shaft.

Fig. 13. The effects of the amount of interference fit on the equivalent Young’s modulus.


Fig. 15. (a). Comparison of the test FRF h1–11 with the numerical FRF of the fully-equipped shaft,(b). The callout of Fig. 15(a) showing finite values of FRF near zero frequency FRF of the fully-equipped shaft.


1. The bending stiffness of a shaft mounted with a rotor via hot-fitting tends to be increaseddue to intereference. To account for the increase of stiffness, the approach of localequivalent Young’s modulus proposed herein offers better prediction on the naturalfrequencies of the shaft (within 1%) and enables facilitating finite element modeling.

2. The dynamical stiffness of a shaft equipped with an interference fit rotor increases with theamount of the interference. The stiffness of the shaft decrease when the amount ofinterference is greater than, in this study, about 0.018 mm. It is emphasized that there founda peak stiffness suggests that an optimal amount of interference fit should sought for theapplication of rotor-shaft assembly.

3. The results have indicates that a motor-built-in spindle presents higher natural frequenciesthan a bare shaft because the local stiffness of the spindle is increased due to the rotor. Thus,

Fig. 16. Comparison of the test FRF h10-11 with the numerical FRF of the fully-equipped shaft.

Fig. 17. Comparison of the experimental mode shape (frequency 1225.29 Hz) with that of numericalresults for the fully equipped rotor-shaft assembly.


without considering the rotor, the characterizing the dynamical effect of a spindle isincomplete and the design will become too conservative.

4. A simplified finite element model where 1-D PIPE16 elements are employed is established inthis research. This simplified model could enable the designers to quickly obtain the firstengineering estimation on dynamical characteristics at the beginning stage of spindle design.

5. The study suggests that the dynmical stiffness of a shaft can be increased with added a longdynamic balancing ring based on the evidence that the natural frequency decreases as moreaccessory components are added.

ACKNOWLEDGEMENTS

The authors are grateful to the assistance by Parfaite Company on offering the drawings,parts/components, and working assemblies.

REFERENCES

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Fig. 18. Comparison of the experimental mode shape (frequency 2846.96 Hz) with that of numericalresults for the fully equipped rotor-shaft assembly.


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dynamic effects of the interference fit of motor rotor on the

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