Research Article  Open Access
Modal Analysis of a ThickDisk Rotor with Interference Fit Using Finite Element Method
Abstract
This paper is concerned with the modal analysis of a thickdisk rotor, which consists of an elastic shaft with a rigid thick disk assembled by interference fit, and the width of the thick disk is not negligible. Firstly, the friction moment on the contact surface of disk and shaft is deduced in terms of elastic theory, and a new enhanced coefficient of bending stiffness of assembly body is proposed and calculated for the first time. Secondly, the effect of the width of thick disk on diametrical moment of inertia, as well as the enhanced coefficient of bending stiffness of interferencefit part between disk and shaft, is included in the motion equations of thickdisk rotor, which are established based on finite element method, and the natural frequencies of rotor are obtained by solving the motion equations. Then the modal analysis is performed to get the natural frequencies in ANSYS Workbench, in which the friction coefficient and interference fit are set to be the same as those of the finite element calculation method. At last the modal experiment is done to verify the accuracy of calculation and simulation. The results show that the calculation values using enhanced stiffness of assembly part are in good agreement with those of ANSYS Workbench and experiment, and the percent errors of the first natural frequency and the second natural frequency are down to about 0.32% and 0.83%, respectively.
1. Introduction
Rotor is the key component of rotating machinery, which is widely used in mechanical devices including gas turbines, aeroengines, industrial compressors, and motors etc., and the accurate prediction of natural frequencies of rotor is important for the safe operation of rotating machineries. There are many studies on dynamic analysis of rotor system for the solution of critical speeds and natural frequencies. In the studies by Nelson and Zorzi et al. [1, 2], dynamic analysis of rotor system was investigated based on finite element method, and the critical speeds were calculated, which included the effect of various parameters such as rotary inertia, gyroscopic moment, and internal damping. Ku et al. [3] developed a Timoshenko beam finite element model and studied the combined effects of shear deformations and internal damping on forward and backward whirl speeds. Kalita et al. [4] and Chouksey et al. [5] analyzed the effect of fluid film bearing on whirl speeds of rotorbearing system and found that a half whirl exists in the rotation caused by hydrodynamic bearings. In addition, Lee et al. [6] investigated two different approaches (Floquet theory and coordinate transformation) to develop the complex modal analysis for periodically timevarying linear rotor systems and found that the results of coordinate transformation are much efficient. Khulief et al. [7] investigated the modal characteristics of complex rotorbearing systems using finite element method and came to the conclusion that the results including the rotational effects were more accurate. Forrai [8] dealt with the stability analysis of selfexcited bending vibrations of linear symmetrical rotorbearing systems with internal damping.
In the establishment of the above finite element model of singledisk rotor or complex rotor system, the width (or thickness) of rotor disk is usually neglected, and the bending stiffness of interferencefit part between disk and shaft equals the stiffness of shaft. In fact, the width of thick disk of rotor has an obvious effect on the diametrical moment of inertia of disk and bending stiffness of interferencefit part, which will change the calculation accuracy of natural frequencies of rotor system.
Liu et al. [9] analyzed the transient vibration of a single thickdisk rotor, whose radial moment of inertia of disk was bigger than the axial one. Pan et al. [10] studied the influence of quality and position of disk on critical speeds of rotor system, which was verified by theoretical calculation and experiment. Petrescu et al. [11] concluded the mass moments of inertia corresponding to different geometric shapes, objects, and profiles. Chen et al. [12] studied the influence of a hotfit rotor on the local stiffness of the hollow shaft and obtained the accurate natural frequencies using contact elements. Huang et al. [13] optimized the contact stiffness factor of flexible rotor system of magnetic levitation motors and obtained the very close results of natural frequencies by ANSYS Workbench and modal experiment. Kim et al. [14] analyzed the forced vibration of a Timoshenko beam subjected to stationary and moving loads using the modal analysis method. Jalali et al. [15] performed the dynamic analysis of a high speed rotorbearing system and got the natural frequencies and mode shapes of the rotor at rest under freefree boundary conditions. Giannopoulos et al. [16] analyzed the local flexibility of a shaft with a breathing crack, which is reciprocal of stiffness. In addition, the influences of normal contact stiffness on vibration modes of rod fastening rotors were studied and analyzed in [17, 18].
In this study, the modal characteristics of a thickdisk rotor with interference fit is analyzed, whose radial moment of inertia of disk is smaller than the axial one. The friction moment of contact surfaces between disk and shaft is firstly deduced based on elastic theory, and the enhanced coefficient of bending stiffness of thick disk on shaft is proposed and calculated. Then the finite element motion equation of the thickdisk rotor is established, and the enhanced coefficient of stiffness and effect of thickdisk width on diametrical moment of inertia are included in the solution of motion equation for natural frequencies. In addition, the modal analysis is performed on ANSYS Workbench for the verification of calculation results, and the friction coefficient is set to be in accordance with that of finite element calculation. At last the modal experiment is carried out by the application of LMS Test.Lab, and the experiment results are in good agreement with those of theoretical calculation and ANSYS Workbench containing enhanced stiffness of assembly part. The research results not only provide a theoretical basis for the modal analysis of thickdisk rotor with interference fit, but also are suitable for the accurate calculation of natural frequencies of other rotors with interference fit.
2. Calculation of Friction Moment and Enhanced Coefficient of Bending Stiffness
2.1. Friction Moment
The mechanical model of diskshaft assembly body is shown in Figure 1, in which the disk and shaft are assembled by interference fit. When the assembly body is subject to external bending moment, the bending of assembly body will produce friction moment on the contact surfaces because of the relative sliding of disk against shaft. In Figure 1, is the bending angle of assembly body, is the external bending moment, is half of the width of assembly body, is the internal diameter of hollow disk, which is equal to the diameter of shaft, R_{0} is the radius of shaft (d=2R_{0}), and D and R_{2} are the external diameter and radius of hollow disk (D=2R_{2}), respectively.
(a)
(b)
For the convenience of calculation and analysis, the dimensionless parameters are supposed to be as follows:where is the magnitude of interference fit along diameter direction, is a coordinate point of assembly body in xaxis direction, and is the dimensionless parameter corresponding to .
The bending stiffness () of assembly body is expressed aswhere is the elastic modulus of disk and shaft material, is the equivalent section moment of inertia of assembly body, is the section moment of inertia of shaft, and is the enhanced coefficient of bending stiffness, which depends on the dimensionless parameters .
According to Lame equation [19, 20], the interference fit between disk and shaft can be expressed aswhere is the compressive stress on the contact surfaces of disk and shaft, which is produced by interference fit.
Substituting Eq. (1) into Eq. (3), the compressive stress can be obtained as
The pressure of upper and lower contact surfaces between shaft and disk are expressed aswhere is the variation of interference fit in radius direction as the assembly body is bent under external moment and , are the pressure of the upper and lower contact surface, respectively. Multiplication coefficient represents and which are the contact pressure component of y direction and the contact pressure component of x direction, which can cancel each other out.
Figure 2 shows the friction moment on the contact surfaces. In Figure 2, represents unit area of shaft surface, and the multiplication coefficient represents the unit area perpendicular to the pressures and . When the assembly body of disk and shaft is bent under external bending moment, there will be the trend of relative sliding on the contact surfaces, and it is supposed that is the friction coefficient of contact surfaces, so and are the produced friction forces on the unit area of upper or lower contact surfaces, and they can be expressed as
From the integral of Eqs. (7) and (8), the friction forces can be expressed as follows:
So the friction moment can be written
Substituting Eqs. (9) and (10) into Eq. (11) and solving the integral of Eq. (11) along axis, the friction moment can be obtained as follows:
2.2. Enhanced Coefficient of Bending Stiffness
In Figure 1, the bending angle of shaft is expressed as
The maximum external bending moment of shaft occurs when the stress of cross section reaches yield stresswhere is the maximum external bending moment, is the radius of shaft, and is the yield stress of shaft material.
Then the external bending moment can be expressed aswhere the coefficient satisfies the condition 0<α⩽1.
To bend the assembly body of disk and shaft, extra external bending moment is needed to overcome the friction moment, so Eq. (13) can be expressed as
According to Eqs. (12), (15), and (16), the enhanced coefficient of bending stiffness can be obtained as follows: where the enhanced coefficient depends on the parameters such as friction coefficient, yield stress, Young’s modulus, , , etc.
3. Motion Equation of Rotor System
The mathematical model of rotor system is composed of single rigid thick disk, flexible shaft, and bearings, which is illustrated in Figure 3.
3.1. Rigid Thick Disk
In Figure 3, the displacement vector of disk is given by
And the motion equation of the rigid disk is expressed aswhere is the spin angular velocity of disk, is the vector of generalized forces, and and are mass matrix and gyroscopic matrix, respectively, which are defined aswhere is the mass of disk and and are diametrical and polar moments of inertia of disk, respectively.
When the width of thick disk is included in the calculation of moment of inertia as in Figure 4, the diametrical and polar moments of inertia of thick disk can be expressed as follows: [11]where , are the external and inner radius of thick disk, respectively, and is the width of thick disk.
3.2. Shaft Finite Element
The shaft finite element has eight degrees of freedom, and it is defined by two nodes, as shown in Figure 5. Its parameters are denoted as follows: is the length of the shaft element, is the area of the cross section of the shaft element, and are diametrical and polar moments of inertia of shaft element, and , are the Young’s modulus and density of the shaft material, respectively.
The nodal displacement vector of the th shaft element is given by
When the rotor shaft is discretized into “N” finite elements, the total number of nodes becomes “N+1”. And the motion equation of the shaft finite element without internal viscous damping can be expressed as [1, 2, 5]:where , , are the mobile mass matrix, rotating mass matrix, and moment of inertia, respectively, and is the stiffness of shaft element, which is written as
The enhanced stiffness of the assembly part of shaft will be enhanced for the shrink fit of thick disk; according to Eqs. (2) and (17), the enhanced stiffness can be got as follows:
3.3. Bearings
The linear isotropic damped bearings can be modeled by the equationwhere is the displacement vector at the bearing location, is the bearing force vector, and subscript “” represents the bearing. And the damping and stiffness matrices of bearings are defined aswhere subscript “xx” represents x direction, “yy” represents y direction, and “xy” and “yx” represent the couple direction of x and y.
3.4. Motion Equation of Rotor System
Based on Eqs. (19), (25), and (28), the motion equation of the rotorbearing system can be written as follows:
For the solution of natural frequencies and mode shapes of rotor, the external force and damping can be ignored, and Eq. (30) becomes the free motion equation of thickdisk rotor as follows:where the stiffness matrix has included the enhanced stiffness.
It is supposed that the solution of Eq. (31) is as follows: , and the characteristic equation can be got as follows:
In Eq. (32), as not all amplitudes are equal to zero for free vibration rotor, the only condition for nonzero solution is that the coefficient of the determinant is equal to zero.
Then the characteristic values of Eq. (33) are the natural frequencies of rotor, and the nonzero vectors of Eq. (32) respond to the corresponding vibration modes.
4. Calculation of Natural Frequencies and Modal Analysis
4.1. Calculation of Natural Frequencies
The calculation model of a thickdisk rotor is made up of disk and shaft via an interference fit, and the structural parameters of the thickdisk rotor are shown in Table 1.

The finite element structure of the thickdisk rotor is shown in Figure 6. As shown in Figure 6(a), there are 9 nodes and the mass of disk acts on node 5; the stiffness matrix of thickdisk rotor is equal to the stiffness of shaft, so the model solution is not affected by the enhanced coefficient of stiffness, and the results will be the same with those of conventional thindisk rotor. In Figure 6(b), the rotor is discretized into 11 nodes, the mass of disk acts on node 6, and the enhanced coefficient of stiffness is taken into account in the model establishment, so the stiffness of shaft between nodes 5 and 7 should multiply the enhanced coefficient for the solution of natural frequencies according to Eq. (27).
(a) Finite element structure of 9 nodes
(b) Finite element structure of 11 nodes
By solving Eq. (33), the natural frequencies of thick rotor are obtained using the parameters in Table 1, and the calculation results are shown in Table 2. Comparing the natural frequencies of 9node rotor with those of 11 nodes, it can be seen that the natural frequencies are greatly influenced by the enhanced coefficient of stiffness, and the percent error of first frequency is about 5.22%, and the second is about 3.15%.

4.2. Modal Analysis of ThickDisk Rotor in ANSYS Workbench
The 3D finite element solid model of thickdisk rotor is established in ANSYS Workbench, as shown in Figure 7. The finite element grid of rotor model is automatically generated, and the calculating model contains 2908 nodes and 526 units. By contrast, the natural frequencies obtained in ANSYS Workbench will be more accurate than those by finite element calculation.
The influence of interference fit of thick disk and shaft on stiffness is included in the ANSYS Workbench. So the contact mode is set to be of frictional type in ANSYS Workbench, and the magnitude of interference fit and friction coefficient are set to be in accordance with those of theoretical finite element calculation as in Table 1. In addition, the contact stiffness should be considered in ANSYS Workbench, and the conventional value is from 0.01 to 0.1 for bending analysis. When the contact stiffness is set to be 0.01, solving the model in ANSYS Workbench, the modal shape of the first two natural frequencies of the thickdisk rotor is obtained as shown in Figure 8. It can be seen that the first two natural frequencies of the rotor are =199.47Hz, =494.99Hz, respectively. When the contact stiffness is equal to 0.1, the first two natural frequencies are =203.31Hz, =499.39Hz. All the results obtained in ANSYS Workbench are very close to those of theoretical finite element calculation with enhanced stiffness.
(a) First natural frequency (199.47Hz)
(b) Second natural frequency (494.99Hz)
To compare the enhancement of thick disk on the bending stiffness of shaft in ANSYS Workbench, another 3D model is established, as shown in Figure 9. The mass of thick disk is simplified to a point, which is applied at the middle point of the rotating shaft, and the mass and moment of inertia of this point are exactly equal to those of the thick disk. Compared to the solid model in Figure 8, the modal solution is not subject to the influence of the enhanced stiffness of assembly part.
Solving the rotor model in ANSYS Workbench, the modal shape of the first two natural frequencies of rotor is obtained as shown in Figure 10. It can be seen that the natural frequencies are=189.03Hz, =482.74Hz, respectively. The results are very close to those of finite element calculation without enhanced stiffness, which verifies that the calculation results of finite element method are consistent with the results of ANSYS Workbench without considering the influence of the enhanced stiffness.
(a) First natural frequency (189.03Hz)
(b) Second natural frequency (482.74Hz)
4.3. Modal Experiment and Results Analysis
In order to accurately test the natural frequencies of the thickdisk rotor, the modal experiment with hammering method is done by using LMS Test.Lab. The modal experiment system is shown in Figure 11. The rotor is suspended by a set of bungees to simulate the freefree boundary conditions of the rotor, and the acceleration sensors are fixed on the different positions of the rotor surface (total 30 points). In order to ensure the validity of the amplitude signal and reduce the measurement error, each point is hammered three times, and the average value is adopted to obtain the excitation spectrum of the rotor.
The natural frequencies of modal experiment of the thickdisk rotor are shown in Figure 12, the horizontal coordinate represents the frequency range, the longitudinal coordinate represents the response amplitude, and o, f, v, and d represent the response points of the excitation. In Figure 12, o indicates the mechanical and electrical instability, f indicates the frequency stability, v indicates the pole vector stability, d indicates the damping and frequency stability, and s indicates that the three parameters are stable. And the first two natural frequencies of the rotor obtained by modal experiment are =198.83Hz and =503.58Hz.
The first two natural frequencies of thickdisk rotor with interference fit obtained by different methods are shown in Table 3. Comparing the natural frequencies of different methods, the influence of enhanced stiffness on natural frequencies of thickdisk rotor is very obvious, and the percent errors of natural frequencies with or without enhanced stiffness are about 5.32% and 3.16% in finite element calculation. The results without enhanced stiffness by finite element calculation with 9 nodes are very close to those of rotor with shaft and single mass point in ANSYS Workbench. The percent errors of the first two natural frequencies are down to 0.02% and 0.77%. The results with enhanced stiffness by finite element calculation with 11 nodes are close to those of 3D model in ANSYS Workbench with contact stiffness 0.01. When the contact stiffness is 0.1, the first two natural frequencies are =203.31Hz, =499.39Hz. The results of modal experiment are close to the results with enhanced stiffness. The first two natural frequencies approach the results of 3D solid model in ANSYS Workbench, the contact stiffness of which is 0.1, 0.01, respectively. And the error percent is down to 0.32% and 0.84%. The variation of contact stiffness is caused by the deformation of rotor under different modes.

5. Conclusions
In this paper, the enhanced coefficient of stiffness for thickdisk rotor with interference fit is firstly put forward, which is deduced based on elastic theory, and the natural frequencies of thickdisk rotor with enhanced stiffness are calculated using finite element method. In addition, the modal analysis of thick disk is performed by ANSYS Workbench and experiment to verify the accuracy of calculation results, and the following conclusions are obtained.
(i) For the thickdisk rotor with interference fit, the width of thick disk has obvious effect on the stiffness of shaft, and the enhanced coefficient of stiffness can be deduced according to elastic theory, which depends on the parameters such as friction coefficient, yield stress, Young’s modulus, , , , etc.
(ii) The influence of enhanced stiffness on natural frequencies of thickdisk rotor is very obvious, and the percent errors of natural frequencies with or without enhanced stiffness are about 5.32% and 3.16% in finite element calculation.
(iii) The finite element calculation results without enhanced stiffness are very close to those of model composed of shaft and single mass point in ANSYS Workbench, and the results with enhanced stiffness are consistent with the 3D model in ANSYS Workbench and experiment.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research has been supported by China Postdoctoral Science Foundation (2015M582112), Key Research and Development Project of Shandong Province (2017GGX203005), and Qingdao Postdoctoral Application Research Project (2014). The support is gratefully acknowledged.
References
 H. D. Nelson and J. M. McVaugh, “The dynamics of rotorbearing systems using finite elements,” Journal of Engineering for Industry, vol. 98, no. 2, pp. 593–600, 1976. View at: Publisher Site  Google Scholar
 E. S. Zorzi and H. D. Nelson, “Finite element simulation of rotorbearing systems with internal damping,” Journal of Engineering for Gas Turbines and Power, vol. 99, no. 1, pp. 71–76, 1977. View at: Google Scholar
 D.M. Ku, “Finite element analysis of whirl speeds for rotorbearing systems with internal damping,” Mechanical Systems and Signal Processing, vol. 12, no. 5, pp. 599–609, 1998. View at: Publisher Site  Google Scholar
 M. Kalita and S. K. Kakoty, “Analysis of whirl speeds for rotorbearing systems supported on fluid film bearings,” Mechanical Systems and Signal Processing, vol. 18, no. 6, pp. 1369–1380, 2004. View at: Publisher Site  Google Scholar
 M. Chouksey, J. K. Dutt, and S. V. Modak, “Modal analysis of rotorshaft system under the influence of rotorshaft material damping and fluid film forces,” Mechanism and Machine Theory, vol. 48, no. 1, pp. 81–93, 2012. View at: Publisher Site  Google Scholar
 C.W. Lee, D.J. Han, J.H. Suh, and S.W. Hong, “Modal analysis of periodically timevarying linear rotor systems,” Journal of Sound and Vibration, vol. 303, no. 35, pp. 553–574, 2007. View at: Publisher Site  Google Scholar
 Y. A. Khulief, F. A. AlSulaiman, A. Arif, R. BenMansour, A. AlQutub, and M. Anis, “Computational Tradeoff in Modal Characteristics of Complex Rotor Systems Using FEM,” Arabian Journal for Science and Engineering, vol. 37, no. 6, pp. 1653–1664, 2012. View at: Publisher Site  Google Scholar
 L. Forrai, “A finite element model for stability analysis of symmetrical rotor systems with internal damping,” Journal of Computational & Applied Mechanics, vol. 1, 2000. View at: Google Scholar
 Z. Liu and J. Wang, “Analysis of transient vibration of singlethickdisk rotor crossing two orders of critical speeds,” Beijing Hangkong Hangtian Daxue Xuebao/Journal of Beijing University of Aeronautics and Astronautics, vol. 41, no. 11, pp. 2144–2157, 2015. View at: Google Scholar
 H. Pan, H. Yuan, T. Zhao, and W. Yang, “Sensitivity Analysis of Wheel Quality and Location on Rotor Critical Speed,” Zhendong Ceshi Yu Zhenduan/Journal of Vibration, Measurement and Diagnosis, vol. 37, no. 3, pp. 532–538, 2017. View at: Google Scholar
 F. I. T. Petrescu, A. Apicella, A. Raffaella et al., “Something about the mechanical moment of inertia,” American Journal of Applied Sciences, vol. 13, no. 11, pp. 1085–1090, 2016. View at: Publisher Site  Google Scholar
 S.Y. Chen, C. Kung, and J.C. Hsu, “Dynamic analysis of a rotary hollow shaft with hotfit part using contact elements with friction,” Transactions of the Canadian Society for Mechanical Engineering, vol. 35, no. 3, pp. 461–474, 2011. View at: Publisher Site  Google Scholar
 Z. Huang, B. Han, and Y. Zhou, “Modal analysis of the flexible rotor system of magnetic levitation motors under nonlinear contact,” Zhongguo Dianji Gongcheng Xuebao/Proceedings of the Chinese Society of Electrical Engineering, vol. 34, no. 15, pp. 2438–2444, 2014. View at: Google Scholar
 T. Kim, I. Park, and U. Lee, “Forced Vibration of a Timoshenko Beam Subjected to Stationary and Moving Loads Using the Modal Analysis Method,” Shock and Vibration, vol. 2017, 2017. View at: Google Scholar
 M. H. Jalali, M. Ghayour, S. ZiaeiRad, and B. Shahriari, “Dynamic analysis of a high speed rotorbearing system,” Measurement, vol. 53, pp. 1–9, 2014. View at: Publisher Site  Google Scholar
 G. I. Giannopoulos, S. K. Georgantzinos, and N. K. Anifantis, “Coupled vibration response of a shaft with a breathing crack,” Journal of Sound and Vibration, vol. 336, pp. 191–206, 2015. View at: Publisher Site  Google Scholar
 Y. T. Ai, X. Zhai, Z. Wang, and Y. L. Qiao, “Influences of normal contact stiffness on an assembly's vibration modes,” Journal of Vibration & Shock, 2012. View at: Google Scholar
 Y. Zhang, Z. Du, L. Shi, and S. Liu, “Determination of Contact Stiffness of RodFastened Rotors Based on Modal Test and Finite Element Analysis,” Journal of Engineering for Gas Turbines and Power, vol. 132, no. 9, p. 094501, 2010. View at: Publisher Site  Google Scholar
 H. Ding and L. Xiao, “The strength analysis of rotor sleeve and PM for highspeed electrical machine,” Advanced Materials Research, vol. 338, pp. 477–480, 2011. View at: Publisher Site  Google Scholar
 A. Özel, Ş. Temiz, M. D. Aydin, and S. Şen, “Stress analysis of shrinkfitted joints for various fit forms via finite element method,” Materials and Corrosion, vol. 26, no. 4, pp. 281–289, 2005. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2018 Yuhua Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.