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

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1 DYNAMIC EFFECTS OF THE INTERFERENCE FIT OF MOTOR ROTOR ON THE STIFFNESS OF A HIGH SPEED ROTATING SHAFT Shin-Yong Chen 1, Chieh Kung 2, Te-Tan Liao 3, Yen-Hsien Chen 3 1 Department of Automation and Control Engineering, Far East University, Taiwan 2 Department of Computer Application Engineering, Far East University, Taiwan 3 Department of Mechanical Engineering, Far East University, Taiwan sychen88@cc.feu.edu.tw; julius@cc.feu.edu.tw; ttliao@cc.feu.edu.tw 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 domestic precision manufacturing industry. The dynamic analysis of the rotating shaft is the major issue in the analysis for a motor-built-in high speed spindle. One of the major concerns is how the motor rotor is mounted on the shaft, by interference (shrink) fit or else. In this study, dynamical analyses are carried out on a motor-built-in high speed spindle. The motor rotor is mounted on the spindle shaft by means of interference fit. Modal testing and numerical finite element analyses are conducted to evaluate the dynamical characteristics of the spindle. The stiffness of the shaft accounting for the interference fit is investigated for the finite element model of the spindle. This study also proposes an analysis procedure to dynamically characterize the high speed spindle with a built-in motor. Based on the results of modal testing and the numerical analyses, it may conclude that the proposed procedure is feasible for the spindle and is effective for other similar applications. EFFETS DYNAMIQUES DE L AJUSTEMENT DE L INTERFÉRENCE D UN ROTOR DE MOTEUR SUR LA RAIDEUR D UN ARBRE DE ROTATION À GRANDE VITESSE RÉSUMÉ Le développement d électrobroches à haute vitesse avec moteur intégré est une technologie clé pour l industrie domestique de fabrication d outils de précision. L analyse dynamique de l arbre de rotation est le point principal dans l analyse d électrobroches à haute vitesse avec moteur intégré. Une des préoccupations majeures est la façon que le rotor du moteur est monté sur l arbre de rotation, soit par ajustement d interférence ou autrement. Nous avons procédé à des analyses dynamiques sur des électrobroches à haute vitesse avec moteur intégré. Le rotor du moteur est monté sur la tige de l électrobroches par ajustement d interférence. Des tests sur un modèle et des analyses numériques des éléments finis sont fait pour évaluer le caractéristiques dynamiques de l électrobroches. La raideur de l arbre de rotation qui est importante dans l ajustement de l interférence est étudiée et optimisée pour le modèle d éléments finis de l électrobroches. La proposition présenté également une procédure d analyse pour définir la dynamique de l électrobroche à grande vitesse avec moteur intégré. En conclusion, en nous basant sur les résultats obtenus sur le modèle et sur l analyse numérique, la procédure préposée pourrait servir pour l électrobroche, et être utiles dans des applications similaires. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

2 Nomenclature M, mass matrix K, stiffness matrix f(t), force vector in time domain f(v), force vector in frequency domain H(v), normal frequency response function matrix x(t), response vector in time domain for damped system rq j, the j-th component of the r-th mode shape vector v, frequency of excitation v r, ther-th natural frequency of the system [. ] 21, inverse of a matrix 1. INTRODUCTION Recently rotary shaft assemblies have become popular in engineering applications ranging from high-tech products such as jet engines and computer hard disk drives (HDD), to household appliances, such as washing machines and refrigerator compressors. A rotary shaft assembly consists of a rotating part (rotor), a stationary part (stator), and multiple bearings connecting 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 hollow shaft by means of interference fit. This unique application posts a challenge to engineers on determination of the stiffness of the rotor-shaft assembly. Chen, et al. [1] suggested the stiffness of a rotor-solid shaft assembly is proportional to the interference fit. However, the stiffness of a rotor-hollow shaft assembly is yet to be studied. The stiffness of a rotor-hollow shaft assembly plays a major role in the subsequent analyses of dynamical characterization of its final form in which other accessories including bearings are added. Thus, it is crucial to determine the stiffness 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. In their 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 and derived the motion equations for the shaft. The effects of translational and rotary inertia, axial load, and gyroscopic moments were considered. In addition, numerous finite element models and procedures have been proposed, such as by Nelson [5] and Özgüven and Özkan [6], in an effort to generalize and optimize the stability of the rotor-shaft system. Studies on revising numerical models cooperated with the modal testing were also suggested. The revision strategies were 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 calculated eigenvalues and eigenvectors incurred from the test data. The procedure was based on the theory of matrix perturbation in which the correct mass and stiffness matrices are expanded in terms of analytical values and a modification matrix. In 1996, Chen et al. [8] developed a frequency-domain method to estimate the mass, stiffness and damping matrices of the model of a structure. Furthermore, Chen and Tsuei [9], Ibrahim and Fullekrug [10] extracted the normalized mode shapes as the basis of a non-damped oscillation experiment. There were deviations reported in the studies due the model elements and the structural parameters such as the Young s modulus, material density, and the assembling conditions. Chu [11] considered a shaft model with a regional enlarged diameter to account for the interference fit between the Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

3 rotor and the shaft. The model with the regional enlarged diameter of the shaft results in a mismatch of the mass and mass moment with those of the test shaft. On the other hand, in the study by Huang [12], the Young s modulus was modified locally to account for the interference fit of the rotor. The Young s modulus was fine-tuned so that the natural frequencies and frequency response functions by the FEA were matched with those obtained from the modal testing. 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 spindle systems. The spindle and housing were modeled by Timoshenko beam elements. Their simulation showed that the rotational speed of the spindle shaft had a larger influence on the lower natural frequencies. Erturka, et al. [14 16] presented an analytical method that used Timoshenko beam theory for calculating the tool point frequency response function (FRF) of a spindle holder tool combination by using the receptance coupling and structural modification methods. They proposed a mathematical model, as well as the details of obtaining the system component (spindle, holder and tool) dynamics and coupling them to obtain the tool point FRF. The model could be used in predicting and following the changes in the tool point FRF due to possible variations in tool and holder types and/or tool length very quickly and in a very practical way. Also, through the model, the stability diagram for an application could be modified in a predictable manner in order to maximize the chatter-free material removal rate by selecting favorable system parameters. Ozasahin, et al. [17] presented a new method for identifying contact dynamics in spindle holder tool assemblies from experimental measurements. They extended a previously developed elastic receptance coupling equations to give the complex stiffness matrix at the holder tool and spindle holder interfaces in a closed-form manner. In this study, dynamical analyses are carried out on a motor-built-in high speed spindle. The motor rotor is made of stacked steel sheet annuli and is mounted on a hollow shaft by means of interference fit. Modal testing and numerical finite element analyses are conducted to evaluate the dynamical characteristics of the spindle. A simplified finite element model is proposed. The equivalent stiffness of finite element model representing the rotor-shaft assembly accounting for the 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 obtained through 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-damped dynamic system. Consider an un-damped dynamic system, the governing equation of the motion of the system is M x ðþzkx t ðþ~f t ðþ t ð1þ where M, K[R n6n are the mass matrix and the stiffness matrix, respectively, of the dynamic system, and x, f[r n are, respectively, the vectors of displacement responses and externally applied excitations. Taking Laplace transformation of Eq. (1) with zero initial conditions, one obtains s 2 MXðÞzKX s ðþ~f s ðþ s ð2þ Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

4 the motion of the system can be evaluated in frequency domain by letting s 5 jv. Thus, Eq.(2) becomes {v 2 MzK XðvÞ~FðvÞ ð3þ or XðvÞ~HðvÞFðvÞ ð4þ where HðvÞ~ K{v 2 {1 M ð5þ is denoted as the normal FRF. Through variable transformation and orthogonal equation, we obtain h jk ðvþ~ XN r~1 rq j ðrq k v 2 r {v2 Þ in which h jk is the FRF of j-th node corresponding to the excitation at the k-th node, r Q j is the j-th component of the r-th mode shape vector, v r is the r-th natural frequency of the system, and N is number of the total natural frequencies considered in calculating the response. In this study, both r Q j and v r are obtained through FEA. The frequency response functions are determined 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 vibration characters of mechanical elements of the shaft. The FEA model is posted with a boundary condition of soft suspension, which is an idealized free-free beam boundary condition. The parameters of interest of the modal analysis include the natural frequencies, mode shapes and the damping ratio of the shaft system. Prior to the FEA, a modal testing is performed for the shaft structure. The test apparatus consists of an excitation source, a signal acquisition device, a signal analyzer, and frequency response function generator. To comply with the soft-suspension boundary condition, the shaft is suspended with rubber bands during the modal testing. Fig. 1 shows the diagram of experiment apparatus. For a high speed rotating shaft with the motor rotor interference fitting to the shaft, the fitting condition has a significant influence on the stiffness of the shaft[1]. An improper interference fit will result in an unpredictable stiffness of the shaft; thus, the natural frequency might be within the proximity of the operating frequency. In this study, the effect of the fitting amount on the stiffness of the spindle is not known as a priori. To best qualify the effect for the FEA model, the modal testing offers a measure to characterize the interference fit for the shaft and to verify the FEA model. The flow chart shown in Fig. 2 illustrates the procedures of construction 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 results of the modal parameters are recorded. Meanwhile, an FEA model using PIPE16 line elements representing the bare shaft is created and the modal analysis is carried out. The purpose of this step is to confirm the dimensions and the material properties for the FEM model. Next, the ð6þ Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

5 Fig. 1. Schematic of the experiment apparatus. modal testing is performed on the shaft with a fitted motor rotor. An amount of mm interference fit is designated. The resulting modal parameters are then recorded. An FEA model of the rotor-shaft assembly is also created concurrently. This FEA model employs PIPE16 elements as a simplified model representing the rotor-shaft assembly. To account for the interference fit, a segment on the model is partitioned wherein a localized Young s modulus is obtained as the local stiffness of the line model; the value of the Young s modulus of the rest Fig. 2. The flow of creating and modifying the finite element model. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

6 segments is the one confirmed in the first step. In this step, the effects of interference fit on the local stiffness are also investigated. Finally, the FEA model is expanded to model the fully equipped spindle assembly by adding additional MASS21 point mass elements representing all necessary accessory components. Dynamical analyses are then conducted on the fully equipped spindle assembly model. 3. ILLUSTRATION STUDY EXAMPLE Fig. 3. Schematic of the hollow shaft. In this study, numerical and experimental modal analyses are carried out on a motor builtin high speed spindle featured with 4.6 KW/30000 rpm and an automatic tool changer (ATC). Finite element analysis software, ANSYS TM,isemployedtoperformthenumerical analysis. The study consists of three stages of evaluations, that is, evaluations of a bare hollow 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, inner radius and outer radius along the shaft are listed in Table 1. The rotor is composed of stacked steel sheet annuli with outer diameter 39.5 mm and inner diameter mm. Its Table 1. Segmental dimension along the shaft. (unit: mm). Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

7 Fig. 4. Schematic of the rotor-shaft assembly. longitudinal length is 80 mm. The location of the rotor mounted on the shaft is shown in Fig The hollow shaft modeling and modification In 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 two ends with rubber bands. A hammer is used to create excitation at each of the 11 locations shown in Fig. 3. At each location, five FRFs are measured and are averaged. The first two natural frequencies of the shaft are then recorded as Hz and Hz, respectively. Next, a numerical modal analysis is performed. Because the shaft (290 mm long) is a bare hollow tube as shown in Fig. 3, a simplified finite element model with line elements, PIPE16, is therefore constructed using the ANSYS TM. 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 segmental length, inner radius and outer radius along the shaft as seen in Fig. 3 are listed in Table 1. The structural nodes along the finite element model are created in a manner of 1 mm spacing to comply with the aforementioned excitation locations. For the segment where tapering exists, the segment is further evenly divided with two additional intermediate nodes. Different real constants, required by the ANSYS TM, are assigned to account for the variation of segment radius as well as tapering along the shaft. The material considered for the shaft has a density of 7950 kg/m 3, Young s modulus of 210GPa, and Poisson s ratio of 0.3. The model has a free-free end boundary condition. The numerical results show that the first two significant frequencies are Hz and Hz, respectively. Comparing the simulated frequency results with the experimental ones, one may see that both simulated frequencies are about 3.3% higher. It might suggest that the deviations be resulted from the simplified finite element model. It is acknowledged that parameters that influence the natural frequencies of a structure include density, Poisson s ratio, and Young s modulus of the material and the boundary conditions of the structure. In this study, the density is calculated based on the real mass and volume of the shaft; therefore, the density of the shaft is considered constant. The boundary conditions as described previously are kept in a free-free end condition. Thus, it may be appropriate to evaluate the effects of Poisson s ratio and/or Young s modulus to account for the deviations of the natural frequencies. Our results show that the effects of the Poisson s ratio on natural frequency are insignificant. On the other hand, by fine tuning the Young s modulus from original 210 GPa to GPa, we obtain an optimal condition where both simulated Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

8 Table 2. Numerical values of the shaft parameters before and after modification of the Young s modulus. 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 s modulus of the shaft before and after the modification of the Young s modulus. Noted that the Young s modulus in Table 2 is intentionally denoted as E1 so as to distinguish itself from E2 which represents the local Young s modulus accounting for the interference. The results accounting for the modification are summarized in Table 3. The results in Table 3 indicate that both simulated frequencies are closed to experimental ones less than 0.2%. Comparisons of frequency response functions (FRF) h 1 11 and h due to the modification of Young s modulus are shown in Figs. 5(a) and 6, respectively. Fig. 5(b) is the callout of Fig. 5(a) showing finite FRF values near zero frequency. Comparisons of the experimental mode shapes due to tuning the Young s modulus are shown in Figs. 7 and 8, for the 1st and the 2nd natural frequencies, respectively. It can be seen that these figures reveal closeness of the FRF curves and mode shapes for both experimental results and modified numerical model results, the Young s modulus of the shaft will then be considered as N/m 2 (E1) in the subsequent analyses of the study The rotor-shaft modeling and modification In 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. Previous study [1] has indicated that although the use of a point mass element helps simplify constructing a finite element model with interference fit components, the effect of the components on the stiffness of the shaft should also be involved. The local stiffness of a shaft due to an interference Table 3. Comparison of natural frequency results before and after modifying finite element model of the shaft. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

9 Fig. 5. (a). Comparison of the experimental FRF h 1 11 results with that of numerical results before and 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. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

10 Fig. 6. Comparison of the experimental FRF h results with that of numerical results before and after modifying the Young s modulus for the bare hollow shaft. fit rotor tends to be higher. To account for the increase of the stiffness, one should introduce both 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. 7. Comparison of the experimental mode shape (frequency Hz) with that of numerical results before and after tuning the Young s modulus E1 for the bare hollow shaft. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

11 Fig. 8. Comparison of the experimental mode shape (frequency Hz) with that of numerical results 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 the segment 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 is proposed 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 is mm. The procedure of obtaining the averaged FRFs is the same as described in the first stage. The fundamental natural frequency is found to increases from Hz for the bare shaft to Hz, an addition of Hz, and the second natural frequency becomes Hz. A numerical modal simulation then follows. The finite element model consists of PIPE16 line elements and point mass elements, MASS21, to account for the interference fit rotor. The line elements are modeled with E N/m 2 as the Young s modulus. The mass elements are characterized by the mass and the mass inertia. A free-free end boundary condition is posted on the model. The fundamental natural frequency from the modal analysis becomes Hz, a decrease of 392 Hz comparing with that of Hz. To compensate the deviation, an equivalent local Young s modulus is proposed to account for the increase of local stiffness due to the interference fit rotor. It is emphasized that the line elements carrying no rotor are remained with E N/m 2 as the Young s modulus. The equivalent Young s modulus is E N/m 2, and the first two natural frequencies become Hz, and Hz, respectively. These two natural frequencies are less than 0.1% off the experimental natural frequencies ( Hz and Hz, respectively). Comparisons of frequency response functions h 1 11 and h based on the equivalent local Young s modulus are shown in Figs. 9 and 10, respectively. Comparisons of the experimental mode shapes due to the equivalent local Young s modulus are shown in Figs. 11 and 12, for the 1st and the 2nd natural frequencies, respectively. It is seen that these figures reveal the closeness of the FRF curves and mode shapes for both simulated and experimental results, it may conclude that the proposed equivalent Young s modulus to account for the interference fit rotor is advisable to accurately model the rotor-shaft assembly. It is noted that the amount of interference due to the interference fit also contribute the variation to the stiffness of the shaft [1]. In this study, the influence of the amount of interference fit on the dynamical characters of the shaft is also investigated. We consider the amount of interference fit to be: mm, mm, mm, and 0.03 mm, respectively. The frequency results associated with mm interference have been stated above. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

12 Fig. 9. Comparison of the experimental FRF h 1 11 results with that of numerical results before and after modifying local Young s modulus due to interference fit of the rotor. Summarized in Table 4 are the influences of the interference fit on the natural frequencies. The values under the column E2 represent the local Young s modulus accounting for the amount of interference fit. To ease comparison, these values are graphed versus the amount of interference in Fig. 13. The curve shown in Fig. 13 suggests that for the rotor-hollow shaft considered herein, the equivalent Young s modulus increase with the amount of interference, indicating an increase of stiffness with the interference amount; however, the stiffness decreases as the amount of interference become greater, in our case around mm The fully-equipped shaft model In 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 mm interference fit, and other accessory components including the press ring, the front spacer, and the inner spacer of the frontal bearing, the inner ring of the frontal/rear bearing, the press ring of the rear bearing, and the precision tight nuts. The schematic of the fully equipped shaft is illustrated in Fig. 14. As these accessory components are sliding-fitting with the shaft, it is reasonably to consider they contribute no effects to the stiffness of the shaft. Thus, in the numerical modal analysis, these components are modeled with point mass elements, MASS21. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

13 Fig. 10. Comparison of the experimental FRF h results with that of numerical results before and after the modifying local Young s modulus due to an interference fit of the rotor. Prior to the numerical modal analysis, a modal testing is conducted. The experiment procedure has been described in previous sections except that the fully equipped shaft is considered. The fundamental natural frequency is found to be Hz and the second Fig. 11. Comparison of the experimental mode shape (frequency Hz) with that of numerical results before and after modifying the local Young s modulus E2 of the rotor-shaft assembly. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

14 Fig. 12. Comparison of the experimental mode shape (frequency Hz) with that of numerical results before and after modifying the local Young s modulus E2 of the rotor-shaft assembly. natural frequency Hz. Comparing this value with the one obtained in the last section, one may find the frequencies decrease from Hz to Hz, and Hz to Hz. These drops are expected as additional mass representing the accessory components is added on the shaft assembly. Following the modal testing, the numerical modal analysis is accomplished. The numerical model contains the model described in Section 3.2 and additional mass points to account for the accessory components. The input to the mass points is the mass only. The first natural frequency from the numerical analysis is Hz, an error of 1.95 %, and the second natural frequency of Hz shows an error of 1.37%. Comparisons of frequency response functions h 1 11 and h based on the results are shown in Figs. 15 and 16, respectively. Table 4. Variation of the natural frequencies corresponding to the modified Young s modulus. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

15 Fig. 13. The effects of the amount of interference fit on the equivalent 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 the 2nd natural frequencies, respectively. It is seen that in these figures, the curves representing test results and simulation results coincide closely. It may therefore conclude again that the proposed local equivalent Young s modulus to account for the interference fit rotor is advisable to accurately model the rotor-shaft assembly. 4. CONCLUSIONS In this study, an approach of local equivalent Young s modulus is proposed to enable better finite element predictions of the dynamical characters of a motor built-in high speed hollow rotating shaft. Both numerical and experimental modal analyses are carried out on the shaft featured with 4.6 kw/30000 rpm and an automatic tool changer (ATC). The effects of interference fit are also investigate. Some conclusions are drawn as below: Fig. 14. Schematic of the fully-equipped shaft. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

16 Fig. 15. (a). Comparison of the test FRF h 1 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 fullyequipped shaft. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

17 Fig. 16. Comparison of the test FRF h with the numerical FRF of the fully-equipped shaft. 1. The bending stiffness of a shaft mounted with a rotor via hot-fitting tends to be increased due to intereference. To account for the increase of stiffness, the approach of local equivalent Young s modulus proposed herein offers better prediction on the natural frequencies 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 the amount of the interference. The stiffness of the shaft decrease when the amount of interference is greater than, in this study, about mm. It is emphasized that there found a peak stiffness suggests that an optimal amount of interference fit should sought for the application of rotor-shaft assembly. 3. The results have indicates that a motor-built-in spindle presents higher natural frequencies than a bare shaft because the local stiffness of the spindle is increased due to the rotor. Thus, Fig. 17. Comparison of the experimental mode shape (frequency Hz) with that of numerical results for the fully equipped rotor-shaft assembly. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

18 Fig. 18. Comparison of the experimental mode shape (frequency Hz) with that of numerical results for the fully equipped rotor-shaft assembly. without considering the rotor, the characterizing the dynamical effect of a spindle is incomplete and the design will become too conservative. 4. A simplified finite element model where 1-D PIPE16 elements are employed is established in this research. This simplified model could enable the designers to quickly obtain the first engineering 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 long dynamic balancing ring based on the evidence that the natural frequency decreases as more accessory components are added. ACKNOWLEDGEMENTS The authors are grateful to the assistance by Parfaite Company on offering the drawings, parts/components, and working assemblies. REFERENCES 1. Chen, S.Y., Huang, R.H. and Yang, S.K., The Investigation of Interference fitting Effect of Motor Rotor and its Finite Element Modeling for a High Speed Spindle, Proc. of the 22nd CSME National conference on Mechanical Engineering, 25,26 November, Taiwan, pp , Prohl, M.A., A General Method for Calculating Critical Speeds of Flexible Rotors, ASME J. of Applied Mechanics, Vol. 12, No. 3, pp , Ruhl, R.L. and Booker, J.F., A Finite Element Models for Distributed Parameter Turborotor Systems, ASME J. of Engineering for Industry, Vol. 94, pp , Nelson, H.D. and McVaugh, J.M., The Dynamics of Rotor-bearing Systems Using Finite Elements, ASME J.of Engineering for Industry, Vol. 98, No. 2, pp , Nelson, H.D., A Finite Rotating Shaft Element Using Timoshenko Beam Theory, ASME J. of Mechanical Design, Vol. 102, pp , Özgüve, H.N. and Özkan, Z.L. Whirl Speeds and Unbalance Response of Multibearing Rotors Using Finite Elements, ASME J. of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 106, pp , Chen, J.C., Direct Structural Parameter Identification by Modal Test Results, AIAA Journal, Vol. 9, No. 8, p. 1481,1983. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,

19 8. Chen, S.Y., Ju, M.S. and Tsuei, Y.G., Estimation of Mass, Stiffness and Damping Matrices from Frequency Response Functions, Transaction of ASME, Journal of Vibration and Acoustics, Vol. 118, pp , Chen, S.Y., and Tsuei, Y.G., Estimation of System Matrices by Dynamic Condensation and Application to Structural Modification, AIAA Journal, Vol 33, No. 11, pp , Ibrahim, S.R. and Fullekrug, U., Investigation into Exact Normalization of Incomplete Complex Modes by Decomposition Transformation, Proc. of the 8th International Modal Analysis Conference, Kissimmee, FL, pp , Chu, C.C., The Modification of Finite Element Model and Modal Analysis for Motor Rotor- Bearing System, M.S. dissertation (in Chinese), Chung Yuan Christian University, Chung Li, Taiwan, Huang, R.H., The Modification of the Finite Element Model of a Motor Built-in Speed-Booster Spindle Using the Modal Testing Data, M.S. dissertation (in Chinese), Far-East University, Tainan, Taiwan, Altintas, Y. and Cao, Y., Virtual Design and Optimization of. Machine Tool Spindles, Annals of the CIRP, pp. M16, Erturk, A., Ozguven, H.N., Budak, E., Analytical modeling of spindle tool dynamics on machine tools using Timoshenko beam model and receptance coupling for the prediction of tool point FRF, Int. J. of Machine Tools and Manufacture, Vol. 46, pp , Erturk, A., Ozguven, H.N., Budak, E., Effect analysis of bearing and interface dynamics on tool point FRF for chatter stability in machine tools by using a new analytical model for spindle tool assemblies, Int. J. of Machine Tools and Manufacture, Vol. 47, pp , Erturk, A., Budak, E., Ozguven, H.N., Selection of design and operational parameters in spindle holder tool assemblies for maximum chatter stability by using a new analytical model, Int. J. of Machine Tools and Manufacture, Vol. 47, pp , Ozsahin, O., Erturk, A., Ozguven, H.N., Budak, E., A closed-form approach for identification of dynamical contact parameters in spindle holder tool assemblies, Int. J. of Machine Tools & Manufacture, Vol. 49, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2,