||Study on whirling motion and torsional vibration of the power-transmission shaft systems
||Department of Systems and Naval Mechatronic Engineering
equivalent mass moment of inertia.
The vibration analyses of the ship propulsion system, including axial vibration, torsional vibration, transverse vibration and whirling motion etc., are the critical issues for its design, and all the ship-design engineers must face this problem carefully. In which, both the torsional vibration and whirling motion are associated with the shaft rotation, it is obvious that the gyroscopic moment on the propeller, flywheel and flanges etc. will vary periodically, when a torsional vibration occurs in the ship propulsion system, and any initial-displacement-induced non-synchronous precession could be critical.
In the existing literature, the dynamic characteristics of whirling motions of the shaft-disk systems are usually analyzed by using the TMM or FEM. The results of the last methods are the approximate solutions and the exact solution concerned is rare. For this reason, this thesis presents an analytical method for yielding the exact whirling speeds and mode shapes of a distributed-mass shaft mounted by arbitrary rigid disks. In theory, the whirling motion of a shaft-disk system is three-dimensional, however, if the transverse displacement in the xy-plane and that in the xz-plane are combined by using a complex number, and the effects of each rigid disk on the shaft are replaced by a lumped mass together with a frequency-dependent equivalent mass moment of inertia, then the whirling motion of a rotating shaft-disk system will be similar to the two-dimensional transverse free vibration of a stationary beam. Therefore, after the foregoing manipulations, some techniques for the free vibrations of a stationary beam carrying various concentrated elements will be available for the whirling motions of a rotating shaft mounted by arbitrary rigid disks. Based on the above theory, some practical examples are illustrated. Numerical results reveal that when a shaft-disk system is put into a spinning motion about its longitudinal axis, each of its natural frequencies of stationary transverse free vibrations will split into forward whirling speed and backward one, furthermore, the characteristics of whirling motions are significantly dependent on the slopes of the associated natural mode shapes at the positions where the rigid disks located. In order to confirm the reliability of the presented theory and the developed computer programs, most of the results obtained from the presented method are compared with those obtained from the existing literature or the FEM and good agreements are achieved.
Table of Contents
Abstract in Chinese (摘要) I
Table of Contents V
List of Figures VIII
List of Tables XIII
Chapter 1 Introduction 1
1.1 Motive and objective 1
1.2 Literature reviews 3
1.2.1 Whirling motion 3
1.2.2 Torsional vibration 5
1.3 Structure of the thesis 8
Chapter 2 Whirling motion of the Shaft System 11
2.1 Equation of motion and displacement function for a shaft segment 11
2.2 Differential equations for bending moments and shearing forces 13
2.3 Equilibrium equations for forces and moments on rigid disk 15
2.4 Continuity of deformations and equilibrium of forces (and moments) at each intermediate node i 18
2.5 Boundary conditions at two ends of the entire shaft 20
2.5.1 Shaft with rigid ball-bearing supports 20
2.5.2 Shaft with rigid sliding-bearing supports 20
2.5.3 Shaft with two ends overhanging 21
2.5.4 Shaft with elastic bearing supports 22
2.6 Determination of Whirling speeds and mode shapes 23
2.7 Numerical results and discussions 27
2.7.1 Comparisons with existing literature 27
188.8.131.52 Whirling speeds of an overhanging shaft-disk system 28
184.108.40.206 Whirling motion of a uniform P-P shaft carrying a rigid disk 34
2.7.2 Free vibrations and whirling motions of a P-P shaft carrying multiple disks 38
220.127.116.11 A P-P shaft carrying one central rigid disk 39
18.104.22.168 A P-P shaft carrying two identical rigid disks 45
22.214.171.124 A P-P shaft carrying three identical rigid disks 50
2.8 Experimental results and discussions 55
2.8.1 The arrangement of experimental apparatus 55
2.8.2 The calculation of whirling speed 57
2.8.3 The measuring method and results 57
2.8.4 Comparison of measured data and calculated data 62
Chapter 3 Torsional Vibration 64
3.1 Holzer method for torsional vibration 64
3.2 Transfer matrix method 66
3.3 Finite element method 67
3.4 Numerical assembly method 68
3.4.1 Equation of motion and displacement function for a shaft segment 69
3.4.2 Coefficient matrix for the i-th node point 71
3.4.3 Boundary conditions at two ends of the entire shaft 72
3.4.4 Overall coefficient matrix and frequency equation 74
3.5 Numerical examples 74
3.5.1 A free-free shaft carrying one central rigid disk 75
3.5.2 A free-free shaft carrying two identical rigid disks 78
3.5.3 A free-free shaft carrying three identical rigid disks 82
Chapter 4 Conclusions 84
Curriculum Vita 91
Publication List 92
 S. Dunkerley, “On the whirling and vibration of shafts”, Philosophical transactions of the royal society of London, Vol. 185, pp. 279-360, 1894.
 R.L. Eshleman and R.A. Eubanks, “On the critical speeds of a continuous shaft-disk system”, ASME Journal of Engineering for Industry, Vol. 89, pp. 645-652, 1967.
 R.L. Eshleman and R.A. Eubanks, “On the critical speeds of a continuous rotor”, ASME Journal of Engineering for Industry, Vol. 92, pp. 1180-1188, 1969.
 H. D. Nelson and J. M. McVaugh, “The dynamics of rotor-bearing systems using finite elements”, ASME Journal of Engineering for Industry, Vol. 98, pp. 593-600, 1976.
 R.B. Green, “Gyroscopic effects on the critical speeds of flexible rotors”, ASME Journal of Applied Mechanics, Vol. 15, pp. 369-376, 1948.
 M. A. Prohl, “A general method for calculating critical speeds of flexible rotors”, ASME Journal of Applied Mechanics, Vol. 12, pp. 142-148, 1945.
 H. Holzer, “Analysis of Torsional Vibration”, Springer, Berlin, 1921.
 N. O. Myklestad, “A new method of calculating natural modes of uncoupled bending vibration of airplane wings and other types of beams”, Journal of the Aeronautical Science, Vol. 11, pp. 153-162, 1944.
 T. N. Shiau and J. L. Hwang, “A new approach to the dynamic characteristic of undamped rotor-bearing systems”, Journal of Vibration, Stress, and Reliability in Design, Vol. 111, pp. 379-385, 1989.
 R. Firoozian and H. Zhu, “A hybrid method for the vibration analysis of rotor-bearing systems”, Journal of Mechanical Engineering Science (Part C), Vol. 25, pp. 131-137, 1991.
 M. Aleyaasin, M. Ebrahimi and R. Whalley, “Multivariable hybrid models for rotor-bearing systems”, Journal of Sound and Vibration, Vol. 233, pp. 835-856, 2000.
 Jong-Shyong Wu and Foung-Tang Lin, “Study on the analytical solutions of whirling frequencies and mode shapes of a continuous shaft with multiple disks”, NSC 99-2221-E-006-241, 2012.
 W. Z. Zu and P. S. Han, “Natural frequencies and normal modes of spinning Timoshenko beam with general boundary conditions”, ASME Journal of Applied Mechanics, Vol. 59, pp. 197-204, 1992.
 Jong-Shyong Wu, Foung-Tang Lin and Heiu-Jou Shaw, “Analytical solution for whirling speeds and mode shapes of a distributed-mass shaft with arbitrary rigid disks”, ASME Journal of applied Mechanics, Vol. 81(3), / 034503-1~10, 2014.
 Jong-Shyong Wu and Foung-Tang Lin, “Study on the exact solutions of whirling frequencies and mode shapes of a continuous Timoshenko shaft with multiple rigid disks”, NSC 100-2221-E-006-012, 2013.
 H. D. Nelson, “A finite rotating shaft element using Timoshenko beam theory. Journal of Mechanical Design”, Vol. 102, pp. 793-803, 1980.
 W. H. Liu and C.C. Huang, “Vibrations of a constrained beam carrying a heavy tip body”, Journal of Sound and Vibration, Vol. 123, pp. 15-29, 1988.
 Jong-shyong Wu and H. M. Chou, “A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying and number of sprung masses”, Journal of Sound and Vibration, Vol. 220(3), pp. 451-468, 1999.
 H. Y. Lin, “On the natural frequencies and mode shapes of a multi-span and multi-step beam carrying a number of concentrated elements”, Structural Engineering and Mechanics, Vol. 29, pp. 531-550, 2008.
 J. S. Przemieniecki, “Theory of Matrix Structural Analysis”, McGraw-Hill, New York, 1968.
 E.C. Pestel and F.A. Leckie, “Matrix methods in Elastomechanics”, McGraw-Hill, New York, 1963.
 Jong-shyong Wu, “Torsional Vibration of Propulsion System”, Sound and vibration technical conference, pp. 11-40, 1987.
 S. Sankar, “On the torsional vibration of branched system using extended transfer matrix method”, ASME Journal of Mechanical design, Vol. 101, pp. 546-553, 1979.
 S. Doughty and G. Vafaee, “Transfer matrix eigensolutions for damped torsional system”, ASEM Journal of vibration, acoustic, stress, and realibility in design, Vol. 107. pp128-132, 1985.
 Yuan Mao Huang and C.D. Horng, “Analysis of torsional vibration systems by the extended transfer matrix method”, ASME Journal of Vibration and Acoustics, Vol. 121, pp. 250-255, 1999.
 Chin-Tzu Chen, “Free vibration analyses of uniform or non-uniform beams carrying various concentrated elements with various supporting conditions using transfer matrix method”, Thesis for the doctorate, National Cheng Kung University, 2007.
 Jong-Shyong Wu, “Analytical and numerical methods for vibration analyses”, John Wiley & Sons Singapore Pte. Ltd., 2013.
 Jong-shyong Wu and Der-Wei Chen, “Free vibration analysis of a Timoshenko beam carrying multiple spring-mass system by using the numerical assembly technique”, Internation Journal for Numerical Methods in Engineering”, Vol. 50, pp. 1039-1058, 2001.
 D.W. Chen and J.S. Wu, “The exact solutions for the natural frequencies and mode shapes of non-uniform beam with multiple spring-mass systems”, ASME Journal of Sound and Vibration, Vol. 255(2), pp. 299-322, 2002.
 Der-Wei Chen, “An exact solution for free torsional vibration of a uniform circular shaft carrying multiple concentrated elements”, ELSEVIER Journal of Sound and Vibration, Vol. 291, pp. 627-643, 2006.
 J.P. Chopade and R.B. Barjibhe, “Free vibration analysis of fixed beam with theoretical and numerical approach”, International Journal of Innovations in engineering and Technology, Vol. 2 (1), pp. 352-356, 2013.
 Jong-Shyong Wu, Foung-Tang Lin and Heiu-Jou Shaw, “Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements”, ELSEVER Applied Mathematical Modelling, Vol. 37, pp. 7588-7610, 2013.
 Jia-Jang Wu, “Torsional vibrations of a conic shaft with opposite tapers carrying arbitrary concentrated elements”, Methematical Problems in Engineering, Vol. 2013, 491062, 2013.
 Alexander Hrennikoff, “Solution of Problems of Elasticity by the Frame-Work Method”, ASME Journal of Applied Mechanics Vol.8, pp.619–715, 1941.
 Kang Feng, “Difference schemes based on variational principle”, Journal of applied mathematics, Vol. 2（4）, pp. 238-262, 1965.
 P. Schwibinger and R. Nordmann, “Torsional vibrations in turbogenerators due to network disturbances”, Vol. 112, pp. 312-320, 1990.
 Jong-shyong Wu and C. H. Chen, “Torsional vibration analysis of gear-branched systems by finite element method”, Journal of Sound and Vibration, Vol. (2001) 240(1), pp. 159-182, 2001.
 B. Carnahan, H.A. Luther and J.O. Wilkes, “Applied Numerical Methods”, John Wiley & Sons, New York, 1969.
 Jong-Shyong Wu and Y.C. Chen, “Out-of-plane free vibrations of a horizontal circular curved beam carrying arbitrary sets of concentrated elements”, ASCE Journal of Structural Engineering, Vol. 137, pp. 220-241, 2011.
 W. T. Thomson, “Theory of Vibration with Applications”, 2nd Ed. Prentice Hall, New Jersey, 1981.
 M. F. Spotts, “Design of Machine Elements”, 4th Ed. Prentice-Hall, New Jersey, 1978.