Document Type: Research Paper

Authors

1 Department of Mechanical Engineering, Pulchowk Campus, Institute of Engineering, Tribhuvan University

2 Department of Mechanical Engineering, Pulchowk Campus , Institute of Engineering, Tribhuvan University

Abstract

Performance and reliability of any rotating machine can be studied by proper dynamic analysis of the machine. In this regard, this paper presents the method to study the dynamic response of the shaft of a Pelton turbine due to the impact of water jet. Equations of motion for the bending vibration of Pelton turbine assembly, in two transverse directions, is developed by using Lagrange equation of motion with the help of assumed modes method. The Pelton wheel is assumed as a rigid disk attached on Euler-Bernoulli shaft. The impact provided by the water jet is represented in the form of Fourier series. Critical speeds of the system are determined by performing free vibration analysis and presented in the form of Campbell diagram. The response plots due to impact of water are generated by performing forced response analysis. Both free and forced analyses are carried out by considering first three modes of vibration.

Graphical Abstract

Keywords

Main Subjects

[1] M. Sabuncu and A. Kacar, "Critical Speeds of Continuous Shaft-Disc System," Vibration and Wear in High Speed Rotating Machinery , pp. 241-251, 1990. 

 

[2] C. Rajalingham, R. B. Bhat, and G. D. Xistris, "Influence of External Damping on the Stability and Response of a Horizontal Rotor with Anisotropic Bending Stiffness," Tribology Transactions, pp. 393-398, 1993. 

 

[3] C. W. Lee and J. S. Yun, "Dynamic Analysis of Flexible Rotors Subjected to Torque and Force," Journal of Sound and Vibration , pp. 439-452, 1996. 

 

[4] N. Khader, A. Atoum, and A. Al-Qaisia, "Theoretical and Experimental Modal Analysis of Multiple Flexible Disk-Flexible Shaft System," 2007. 

 

[5] T.A.N. Silva and N.M.M. Maia, "Modelling a Rotating Shaft as an Elastically Restrained Bernoulli-Euler Beam," Experimental Techniques, 2011. 

 

[6] C. Chattoraj, S.N. Sengupta, and M. C. Majumder, "Analysis of Dynamic Behaviour a Rotating Shaft with Central Mono-Disk," International Journal of Engineering & Technology Research, pp. 77-88, 2013. 

 

[7] O. Gundogdu, K. Alnefaie, and H. Diken, "Modelling and Analysis of a Jeffcott Rotor as a Continuous Cantilever Beam and an Unbalanced Disk System," Gazi University Journal of Science Part A: Engineering And Innovation, pp. 77-85, 2014. 

 

[8] M Shahgholi and S.E. Khadem, "Analysis of Stability and Bifurcation of an Asymmetrical Rotor," in Proceedings of the ASME 2015 Dynamic Systems and Control Conference, Columbus, Ohio, USA, 2015. 

 

[9] X. Lin, R. Zhou, and N. Xiao, "Influence Characteristics of Shaft and Disk Models on Natural Frequency of Single-Rotor System," Applied Mechanics and Materials, pp. 490-494, 2014. 

 

[10] D. J. Han, "Complex Harmonic Modal Analysis of Rotor Systems," Journal of Mechanical Science and Technology, pp. 2735-2746, 2015. 

 

[11] D. G. Huang, "Characteristics of Torsional Vibration of a Shaft with Unbalance," Journal of Sound and Vibration, pp. 692-698, 2007. 

 

[12] C. O. Chang and J. W. Cheng, "Non-linear Dynamical and Instability of a Rotating Shaft-Disk System," Journal of Sound and Vibration , pp. 433-454, 1993. 

 

[13] G. Genta and C. Delprete, "Acceleration Through Critical Speeds of an Anisotropic, Non-Linear Torsionally Stiff Rotor with Many Degrees of Freedom," Journal of Sound and Vibration , pp. 369-386, 1995. 

 

[14] T. Inoue and Y. Ishida, "Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems," Journal of Vibration and Acoustics, Transactions of the ASME, pp. 156-169, 2006. 

 

[15] H. Diken and I. G. Tadjbakhsh, "Non-Linear Vibration Analysis and Subharmonic Whirl Frequencies of the Jeffcott Rotor Model," Journal of Sound and Vibration, pp. 117-125, 2001. 

 

[16] M. R. Shad, G. Michon, and A Berlioz, "Modelling and analysis of nonlinear rotordynamics due to higher order deformations in bending," Applied Mathematical Modelling, pp. 2145-2159, 2011. 

 

[17] H. P. Phadatare and B. Pratiher, "Nonlinear Frequencies and Unbalanced Response Analysis of High Speed Rotor-Bearing Systems," Procedia Engineering, pp. 801-809, 2016. 

 

[18] B. O. Al-Bedoor, "Modelling the coupled torsional and lateral vibrations of unbalanced rotors," Comput. Methods Appl. Mech. Engrg., pp. 5999-6008, 2001. 

 

[19] L. Xiang and S. Yang, "Analysis of Flexural and Torsional Vibration for Turbogenerator Shafts on Power Impact," Advanced Materials Research, pp. 2498-2501, 2010. 

 

[20] K. Alnefaie, "Lateral and Longitudinal Vibration of a Rotating Flexible Beam Coupled with Torsional Vibration of a Flexible Shaft," World Academy of Science, Engineering and Technology, pp. 317-324, 2013. 

 

[21] H. Kojima and K. Nagaya, "Nonlinear Torsional Vibrations of a Rotating Shaft System with a Magnet Couplling," Bulletin of the JSME, pp. 1258-1263, 1984. 

 

[22] A. Muszynska and P. Goldman, "Chaotic Responses of Unbalanced Rotor/Bearing/Stator Systems with Looseness or Rubs," Chaos, Solitons & Fractals, pp. 1683-1704, 1995. 


[23] M. Behzad and M. Asayesh, "Vibration Analysis of Rotating Shaft with Loose Disk," IJE Transactions B: Applications, pp. 385-393, 2002. 

 

[24] M. F. A. Azeez and A. F. Vakakis, "Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts," International Journal of Non-Linear Mechanics, pp. 415-435, 1999. 

 

[25] X. Shen, J. Jia, and M. Zhao, "Numerical Analysis of a Rub-impact Rotor-bearing System with Mass Unbalance," Journal of Vibration and Control, pp. 1819-1834, 2007. 

 

[26] C. W. C. Jian and C. K. Chen, "Chaos of rub–impact rotor supported by bearings with nonlinear suspension," Tribology International, pp. 426-439, 2009. 

 

[27] H.M. Khanlo, M. Ghayour, and S. Ziaei-Rad, "Chaotic vibration analysis of rotating, flexible, continuous shaft-disk system with a rub-impact between the disk and the stator," Commun Nonlinear Sci Numer Simulat, pp. 566-582, 2011. 

 

[28] H.M. Khanlo, M. Ghayour, and S. Ziaei-Rad, "The effects of lateral–torsional coupling on the nonlinear dynamic behaviour of a rotating continuous flexible shaft–disk system with rub–impact," Commun Nonlinear Sci Numer Simulat, pp. 1524-1538, 2013. 

 

[29] W. Jiao, Q. Yuan, and Y. Chang, "Study on the coupled bending-torsional vibration of unbalanced rotor system with external excitations," IEEE International Conference on Computer Science and Automation Engineering (CSAE), 2012. 

 

[30] H. Ma, X. Y. Tai, H. L. Yi, S. Lv, and B. C. Wen, "Nonlinear dynamic characteristics of a flexible rotor system with local rub-impact," Journal of Physics: Conference Series, 2013. 

 

[31] X. Tai, H. Ma, F. Liu, Y. Liu, and B. Wen, "Stability and steady-state response analysis of a single rub-impact rotor system," Arch Appl Mech, pp. 133-148, 2013. 

 

[32] A. M. Wahab, Z. A. Rasid, and A. Abu, "Parametric Instability of Static Shafts-Disk System Using Finite Element Method," in 5th Asia Conference on Mechanical and Materials Engineering (ACMME 2017), 2017. 

 

[33] L. Chen, J. Wang, Q. Han, and F. Chu, "Nonlinear dynamic modelling of a simple flexible rotor system subjected to time-variable base motions," Journal of Sound and Vibration, vol. 404, pp. 58-83, 2017. 

 

[34] S Karki, M C Luintel, and L Poudel, "Dynamic Response of Pelton Turbine Unit for Forced Vibration," in IOE Gradauate Conference, Kathamndu, 2017. 

CAPTCHA Image