DYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM FOR FLEXIBLE BEARING SUPPORT CONDITION

International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 7, July 2017, pp. 1785 1792, Article ID: IJMET_08_07_197 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=7 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 IAEME Publication Scopus Indexed DYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM FOR FLEXIBLE BEARING SUPPORT CONDITION Mr. Sandeep Tiwari M.E. Student, Department of Mechanical Engineering, B.I.T. MESRA, Ranchi, India. Dr. Sankha Bhaduri Assistant Professor, Department of Mechanical Engineering, B.I.T. MESRA, Ranchi, India. ABSTRACT Rotating machineries are rotating at a very high speed. It is therefore very important to determine the natural frequencies of the rotating machineries. In this study, a rotor bearing system is considered to detect the natural frequencies of rotating machineries, where two discs are mounted on the rotor. The purpose of this study is to investigate the effect of boundary conditions on the natural frequencies of the rotor bearing system. The different boundary conditions like rigid, isotropic flexible and orthotropic flexible bearings are considered for the analysis. In this study, a double disc rotor bearing system is analyzed. The finite element model of the rotor bearing system is modeled in ANSYS software. The effect of the spin speed of the spindle on natural frequencies is also observed in this study. The Campbell diagram of the model is also drawn in this paper. It is observed that there is a significant change in the natural frequencies of the model with the variation of the spinning speed of the spindle. Key words: Rotating Machineries, Finite Element Model, Modal Analysis, Natural frequencies, Campbell Diagram, Flexible support. Cite this Article: Mr. Sandeep Tiwari and Dr. Sankha Bhaduri Dynamic Analysis of Rotor-Bearing System for Flexible Bearing Support Condition. International Journal of Mechanical Engineering and Technology, 8(7), 2017, pp. 1785 1792. http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=7 1. INTRODUCTION Rotating machineries such as compressor, turbine, pump, are subjected to vibration while they are rotating at high speed. The major parameters of this unbalance excitation are the eccentricity between the mass center and geometric center, the disc position, the support or the bearing characteristics and spinning speed of the spindle. Rotor dynamics analysis have been performed by different authors in the past. 1 Zhang et al presented a paper on the dynamic behavior of rotor bearing-sfd system with bearing inner-race defect and explained the effect http://www.iaeme.com/ijmet/index.asp 1785 editor@iaeme.com

Dynamic Analysis of Rotor-Bearing System for Flexible Bearing Support Condition of SFD on the system stability with the change in speed. 2 Tamrakar and Mittal discussed about the vibration response of cracked rotor with the help of Campbell diagram and the effect of open crack on stiffness and natural frequency of rotor is also discussed in their paper. 3 Navin et al presented a review paper on the static, dynamic and harmonic analysis of rotor bearing system. 5 Jalali et al presented the dynamic behavior of high speed rotor system and suggested the importance of modal analysis in the design and development of rotor system. 6 Fegade et al, with the help of harmonic analysis, identified the frequency of system through critical speed, amplitude and phase angle plot using ANSYS. 7 Surovec et al presented a lateral rotor vibration analysis model, in which they proposed that the lateral vibrations lead to unstable rotor behavior in rotating machinery. 10 Saleem et al presented the paper on detection of unbalance in rotating machines using shaft deflection measurement. 11 Chiu and Chen worked on the coupled vibration in a rotating multi-disc rotor system. 16 Lee and Chun developed an assumed method to investigate the effect of multiple flexible discs on the vibration modes of a flexible rotor system. In this paper, the critical speeds of the rotor have been found out for different types of bearing conditions. Mainly, rigid bearings, flexible isotropic and flexible orthotropic bearing conditions are considered here for the analysis. The natural frequencies of rotor bearing system between different support conditions are compared and the Campbell diagram is plotted for different boundary conditions. 2. METHODOLOGY The methodology consists of two parts, the first part includes Mathematical Modelling, while the second part deals with the Modal Analysis. 2.1. Mathematical Modelling For mathematical modelling of rotor, the simplest rotor model called Jeffcott rotor is considered. It consists of a massless shaft, at the center of which, a fixed rigid circular disc is mounted, which is supported on a pair of rigid bearings. Considering the assumption that the rotor disc does not affect the stiffness of the massless shaft, the stiffness of shaft supported on simple support is given by = 48 where E is the modulus of elasticity of beam, L is length between the two supports, I, is the moment of inertia of shaft and D is the diameter of shaft. = 64 The shaft has a circular cross-section with constant diameter and the disc is mounted with its plane perpendicular to shaft axis. The position of the mass center can be determined from equations: =+Ω (1) =+Ω (2) The equation of motion for the mass center can be derived from Newton s second law (+Ωt) = - ẋ- x (3) (+Ωt) = - "#- " y (4) http://www.iaeme.com/ijmet/index.asp 1786 editor@iaeme.com

Mr. Sandeep Tiwari and Dr. Sankha Bhaduri Equation (3) and (4) can be re-written as $ + # + =Ω % cosωt (5) $ + " # + " =Ω % sinωt (6) where m is the mass of the disc, k and C are the stiffness and damping coefficients of the support respectively, e is the eccentricity and is the angular velocity of rotor. 2.2. Analysis 2.2.1. Selection of elements The following are the element types used in building the ANSYS Model of rotor-bearing system. BEAM-188: BEAM-188 is a 3-dimensional 2-Node beam element having tension, compression, torsion and bending capabilities. The element has six degrees of freedom at each node: translations in the nodal X, Y and Z directions & rotations about the nodal X, Y and Z axes. MASS-21: MASS-21 is a point element and it is defined by a single node. The degrees of freedom of the element can be extended up to six directions. The element can also be reduced to a 2-D capability. If the element has only one mass input, it is assumed that mass acts in all coordinate directions (MX, MY, MZ). COMBI-214: It is a 2-dimensional spring damper bearing element with longitudinal tension and compression capability. It is defined by two nodes and has two degrees of freedom at each node. The element has stiffness (k) and damping characteristics (c). 2.3. Finite element Model The properties of test model are given in the table below. Table 1 Model Properties (Taken from [14]) Model Specifications (Disk and shaft are assumed to be of steel) Shaft Disc Length= 1 m Dia.= 0.5 m Dia.=0.05 m Thickness=0.05m E= 206.82e+09 N/ % Mass= 76.102 kg G= 130e+09 N/ % Id= 1.2168 kg- % Density= 7830.6 kg/ Ip= 2.4020 kg- % ANSYS is an important tool used for finite element analysis of rotor. http://www.iaeme.com/ijmet/index.asp 1787 editor@iaeme.com

Dynamic Analysis of Rotor-Bearing System for Flexible Bearing Support Condition Figure 1 Finite element model of Double Disc rotor 2.3.1. Boundary Conditions The shaft nodes are constrained in axial and torsional directions and the base of the bearing nodes are fixed in all directions. 2.3.2. Modal Analysis A modal analysis of rotor model for flexible bearing support is performed to obtain the mode shapes and corresponding natural frequencies. Flexible Isotropic Bearing support: The shaft is supported on identical undamped isotropic bearings of stiffness = "" =1.75/10 1 3/ Figure 2 First mode shape (flexible support) http://www.iaeme.com/ijmet/index.asp 1788 editor@iaeme.com

Mr. Sandeep Tiwari and Dr. Sankha Bhaduri Flexible Orthotropic Bearing support Figure 3 Second mode shape (flexible support) =1.75/10 1 3/ "" =3.5/10 1 3/ Similarly, the modal analysis for orthotropic bearing support is performed and a comparative result of eigen frequencies for different support conditions are tabulated below: Table 2 Natural Frequencies (Result) Bearing supports Natural Frequencies (Hz) 1 2 7 3 8 Rigid (simple support) 63.63 124.63 186.58 Flexible isotropic 26.55 73.54 124.19 Flexible orthotropic 26.55 73.54 124.19 27.38 79.44 135.52 2.3.3. Campbell Diagram It is a graphical representation between system frequency and excitation frequency as a function of rotational speed. Campbell diagram is usually plotted to determine the critical speed of the rotor-bearing system. In this analysis, the intersection points between the frequency curves and excitation lines are calculated to find out the critical speed. http://www.iaeme.com/ijmet/index.asp 1789 editor@iaeme.com

Dynamic Analysis of Rotor-Bearing System for Flexible Bearing Support Condition Figure 4 Campbell diagram for flexible isotropic support Figure 5 Campbell diagram for flexible orthotropic support Here, the rotational speed of the rotor is plotted along x-axis and the system frequencies are plotted along y-axis. The above campbell diagrams are plotted for runing the rotor at different speeds. Frequency, of circular whirling motion, when occurs in the same direction of the spin motion, is known as Forward whirling, and when it occurs in the opposite direction, it is known as reverse whirling or Backward whirling. In case of forward whirl, the natural frequency is increasing with the increase in rotational speed while in backward whirl, natural frequency is decreasing as the rotational speed is increasing. 3. RESULTS AND DISCUSSION http://www.iaeme.com/ijmet/index.asp 1790 editor@iaeme.com

Mr. Sandeep Tiwari and Dr. Sankha Bhaduri Considering the tabular results of natural frequencies of rotor system supported on flexible bearing, it can be noted that for flexible isotropic and orthotropic conditions, natural frequencies are different due to the difference in stiffness property in either directions. In campbell diagram, it is observed that the frequencies are not constant over the rotational speed range and also each natural frequency of whirl splits into two frequencies forward and backward whirl when rotor is runing above the zero. This behaviour is due to the influence of gyroscopic effect. 4. CONCLUSION From the modal analysis of rotor-bearing system, it is observed that the dynamic behavior of rotor-bearing system is affected by the support conditions i.e. when the support is changed from rigid to flexible isotropic, the natural frequency of the system decreases and also in case of flexible orthotropic bearing condition, there is change in natural frequency of system due to difference in stiffness in two mutually perpendicular directions. From Campbell diagram, the effect of change in spin on critical speed of rotor is observed. So, for stable operation of rotating machines, this type of analysis is important. REFERENCES [1] Zhang, J., Lu, X., Liu, J., Ma, L. and Wang, J. (2017) Dynamic Analysis of a Rotor Bearing-SFD system with the Bearing Inner-race Defect, Shock and Vibration, Vol. 2017, Pp. 13. [2] Tamrakar, R., and Mittal, N.D. (2016) Campbell Diagram Analysis of Open Cracked Rotor, Engineering Solid Mechanics, Vol. 4, Pp. 159-166. [3] Kumar, N.M., Kishore, S., Kumar, R., Sugumar, K., Kumar, S. and Kumar S. (2016) Rotordynamic Analysis of a Rotating System, International Conference on Systems, Science, Control, Communication, Engineering and Technology, Vol. 2, Pp. 161. [4] Yamamoto, G.K., Costa, C. and Sousa, J.S.S. (2016) A smart experimental setup for vibration measurement and imbalance fault detection in rotating machinery, Case Studies in Mechanical Systems and Signal Processing, Vol. 4. [5] Jalali, M.H., Ghayour, M., Ziaei-Rad, S. and Shahriari, B. (2014) Dynamic Analysis of High Speed Rotor-Bearing System, Measurement, Vol. 53, Pp. 1-9. [6] Fegade, R., Patel, V., Nehete, R.S. and Bhandarkar, B.M. (2014) Unbalanced response of rotor using ANSYS parametric design for different bearings, International Journal of Engineering Sciences and Emerging Technologies, Vol. 7, Iss. 1, Pp. 506-515. [7] Surovec, R., Bocko, J. and Sarlosi, J. (2014) Lateral Rotor Analysis Model, American Journal of Mechanical Engineering, Vol. 2 [8] Ranjan, A., Kochupillai, J., Shijukumar, P.K. and Shajan, S. (2014) Dynamic Analysis of Unbalance in Rotating Machinery, International Journal of Scientific and Engineering Research, Vol. 5, Iss.7. [9] Das, S.K. (2013) Study of Flexural Behaviour of Jeffcott Rotor, NIT Rourkela. [10] Saleem, M.A., Diwakar, G. and Satyanarayana, M.R.S. (2012) Detection of Unbalance in Rotating Machines using Shaft Deflection measurement during its operation, IOSR Journal of Mechanical and Civil Engineering, Vol. 3, Iss. 3, Pp. 08-20. [11] Chiu, Y.J. and ZenChen D. (2011) The Coupled Vibration in a Rotating Multi-Disk Rotor System, International Journal of Mechanical Sciences, Vol. 53. [12] YUNLU, L.I. (2007) Modelling and performance investigation of a rotor with dissimilar bearing support system, Shenyang Institute of Technology, China. [13] Penny, J.E.T., Friswell, M.I., Less, A.W. and Garvey, S.D. (2004) A Simple but Versatile Rotor Model, IMechE Conference transactions. http://www.iaeme.com/ijmet/index.asp 1791 editor@iaeme.com

Dynamic Analysis of Rotor-Bearing System for Flexible Bearing Support Condition [14] Gunter, E.J. (2004) Critical Speed Analysis of offset Jeffcott Rotor, RODYN Vibration Inc. [15] Loparo, K.A., Adams, M.L., Lin, W., Magied, M.F and Afshari, N (2000) Fault Detection and Diagnosis of Rotating Machinery, IEEE Transactions on industrial electronics, Vol. 47. [16] Lee, C.W. and Chun, S.B. (1998) Vibration Analysis of a Rotor with Multiple Flexible Disks Using Assumed Modes Method, ASME, Vol.120. [17] ANSYS, Help Documentation, Version 16.0. [18] R. Tiwari, NPTEL Lecture notes, IIT Kharagpur. [19] Arundhati Garad and Prof. V. J. Shinde. A Theoretical Model of Deep Groove Ball Bearing for Predicting the Effect of Localized Defects on Vibrations. International Journal of Mechanical Engineering and Technology, 8(6), 2017, pp.760-769. http://www.iaeme.com/ijmet/index.asp 1792 editor@iaeme.com