Vibration of Continuous Systems

Vibration of Continuous Systems

Volumn , Issue , 2007, Pages 1-720

  Rao, Singiresu S ^a  

^a University of Miami College of Engineering   (United States)

Author keywords

[No Author keywords available]

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EID: 84889302076     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9780470117866     Document Type: Book

References (339)

1. D. C. Miller, Anecdotal History of the Science of Sound, Macmillan, New York, 1935.
2. N. F. Rieger, The quest for √k/m: notes on the development of vibration analysis, Part I, Genius awakening, Vibrations, Vol. 3, No. 3-4, pp. 3-10, 1987.
3. Chinese Academy of Sciences, Ancient China's Technology and Science, Foreign Languages Press, Beijing, 1983.
4. R. Taton, Ed., Ancient and Medieval Science: From the Beginnings to 1450, translated by A. J. Pomerans, Basic Books, New York, 1957.
5. S. P. Timoshenko, History of Strength of Materials, McGraw-Hill, New York, 1953.
6. R. B. Lindsay, The story of acoustics, Journal of the Acoustical Society of America, Vol. 39, No. 4, pp. 629-644, 1966.
7. J. T. Cannon and S. Dostrovsky, The Evolution of Dynamics: Vibration Theory from 1687 to 1742, Springer-Verlag, New York, 1981.
8. L. L. Bucciarelli and N. Dworsky, Sophie Germain: An Essay in the History of the Theory of Elasticity, D. Reidel, Dordrecht, The Netherlands, 1980.
9. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
10. E. B. Magrab, Vibrations of Elastic Structural Members, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1979.
11. L. Fryba, Vibration of Solids and Structures Under Moving Loads, Noordhoff International Publishing, Groningen, The Netherlands, 1972.
12. W. Nowacki, Dynamics of Elastic Systems, translated by H. Zorski, Wiley, New York, 1963.
13. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.
14. S. K. Clark, Dynamics of Continuous Elements, Prentice-Hall, Englewood Cliffs, NJ, 1972.
15. A. W. Leissa, Vibration of Plates, NASA SP-160, National Aeronautics and Space Administration, Washington, DC, 1969.
16. A. W. Leissa, Vibration of Shells, NASA SP-288, National Aeronautics and Space Administration, Washington, DC, 1973.
17. C. M. Harris and A. G. Piersol, Eds, Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill, New York, 2002.
18. D. I. G. Jones, Handbook of Viscoelastic Vibration Damping, Wiley, Chichester, West Sussex, England, 2001.
19. S. G. Braun, D. J. Ewans, and S. S. Rao, Eds., Encyclopedia of Vibration, 3 vol., Academic Press, San Diego, CA, 2002.
20. A. J. Pretlove, BASIC Mechanical Vibrations, Butterworths, London, 1985.
21. International Organization for Standardization, Acoustics, Vibration and Shock: Handbook of International Standards for Acoustics, Mechanical Vibration and Shock, Standards Handbook 4, ISO, Geneva, Switzerland, 1980.
22. R. D. Blevins, Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold, New York, 1979.
23. I. A. Karnovsky and O. I. Lebed, Free Vibrations of Beams and Frames: Eigenvalues and Eigenfunctions, McGraw-Hill, New York, 2004.
24. I. A. Karnovsky and O. I. Lebed, Non-classical Vibrations of Arches and Beams: Eigenvalues and Eigenfunctions, McGraw-Hill, New York, 2004.
25. J. Wijker, Mechanical Vibrations in Spacecraft Design, Springer-Verlag, Berlin, 2004.
26. R. D. Blevins, Flow-Induced Vibration, 2nd ed., Krieger, Melbourne, FL, 2001.
27. H. S. Tzou and L. A. Bergman, Eds., Dynamics and Control of Distributed Systems, Cambridge University Press, Cambridge, 1998.
28. J. P.Wolf and A. J. Deaks, Foundation Vibration Analysis: A Strength of Materials Approach, Elsevier, Amsterdam, 2004.
29. J. D. Smith, Gears and Their Vibration: A Basic Approach to Understanding Gear Noise, Marcel Dekker, New York, 1983.
30. J. D. Smith, Vibration Measurement Analysis, Butterworths, London, 1989.
31. G. Lipovszky, K. Solyomvari, and G. Varga, Vibration Testing of Machines and Their Maintenance, Elsevier, Amsterdam, 1990.
32. S. Korablev, V. Shapin, and Y. Filatov, in Vibration Diagnostics in Precision Instruments, Engl. ed., E. Rivin, Ed., Hemisphere Publishing, New York, 1989.
33. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
34. L. Meirovitch, Fundamentals of Vibrations, McGraw-Hill, New York, 2001.
35. D. J. Inman, Engineering Vibration, 2nd ed., Prentice Hall, Upper Saddle River, NJ, 2001.
36. A. K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice Hall, Englewood Cliffs, NJ, 1995.
37. J. R. Anderson and A. A. Ferri, Behavior of a single-degree-of-freedom system with a generalized friction law, Journal of Sound and Vibration, Vol. 140, No. 2, pp. 287-304, 1990.
38. R. E. D. Bishop, A. G. Parkinson, and J. W. Pendered, Linear analysis of transient vibration, Journal of Sound and Vibration, Vol. 9, No. 2, pp. 313-337, 1969.
39. A. Soom and M. Lee, Optimal design of linear and nonlinear vibration absorbers for damped system, Journal of Vibrations, Acoustics, Stress and Reliability in Design, Vol. 105, No. 1, pp. 112-119, 1983.
40. J. Shah, S. W. Shah, and A. G. Haddow, On the response of the nonlinear vibration absorber, Journal of Non-linear Mechanics, Vol. 24, pp. 281-293, 1989.
41. R. J. Guyan, Reduction of stiffness and mass matrices, AIAA Journal, Vol. 3, No. 2, p. 380, 1965.
42. R. Lim and Y. Xia, A new eigensolution of structures via dynamic condensation, Journal of Sound and Vibration, Vol. 266, No. 1, pp. 93-106, 2003.
43. G. Prater and R. Singh, Quantification of the extent of non-proportional viscous damping in discrete vibratory systems, Journal of Sound and Vibration, Vol. 104, No. 1, pp. 109-125, 1986.
44. R. Lauden and B. Akesson, Damped second-order Rayleigh-Timoshenko beam vibration in space: an exact complex dynamic member stiffness matrix, International Journal for Numerical Methods in Engineering, Vol. 19, No. 3, pp. 431-449, 1983.
45. T. K. Caughey and M. E. J. O'Kelly, Classical normal modes in damped systems, Journal of Applied Mechanics, Vol. 27, pp. 269-271, 1960.
46. A. S. Phani, On the necessary and sufficient conditions for the existence of classical normal modes in damped linear dynamic systems, Journal of Sound and Vibration, Vol. 264, No. 3, pp. 741-745, 2002.
47. A. D. S. Ross and D. J. Inman, Settling time of underdamped linear lumped parameter systems, Journal of Sound and Vibration, Vol. 140, No. 1, pp. 117-127, 1990.
48. F. P. Beer, E. R. Johnston, Jr., and J. T. DeWolf, Mechanics of Materials, 3rd ed., McGraw-Hill, New York, 2002.
49. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
50. A. C. Ugural, Stresses in Plates and Shells, McGraw-Hill, New York, 1981.
51. S. S. Rao, The Finite Element Method in Engineering, 3rd ed., Butterworth-Heinemann, Boston, 1999.
52. E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York, 2001.
53. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed, Dover, New York, 1944.
54. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959.
55. G. R. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastishen Scheibe, Journal fuer die Reine und Angewandte Mathematik, Vol. 40, pp. 51-88, 1850.
56. Lord Kelvin and P. G. Tait, Treatise on Natural Philosophy, Vol. 1, Clarendon Press, Oxford, 1883.
57. W. Flügge, Stresses in Shells, 2nd ed., Springer-Verlag, New York, 1973.
58. S. S. Rao, Theory of vibration: variational methods, in Encyclopedia of Vibration, S. Braun, D. Ewins, and S. S. Rao, Eds., Academic Press, London, 2002, Vol. 2, pp. 1344-1360.
59. J. S. Rao, Advanced Theory of Vibration, Wiley Eastern, New Delhi, India, 1992.
60. K. M. Liew, C. M. Wang, Y. Xiang, and S. Kitipornchai, Vibration of Mindlin Plates: Programming the p-Version Ritz Method, Elsevier Science, Oxford, 1998.
61. R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, Vol. 18, No. 1, pp. 31-38, 1951.
62. R. Weinstock, Calculus of Variations: With Applications to Physics and Engineering, McGraw-Hill, New York, 1952.
63. R. R. Schechter, The Variational Method in Engineering, McGraw-Hill, New York, 1967.
64. Y. C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1965.
65. J. N. Reddy, Energy and Variational Methods in Applied Mechanics,Wiley, New York, 1984.
66. S. S. Rao, Engineering Optimization: Theory and Practice, 3rd ed., Wiley, New York, 1996.
67. K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford, 1968.
68. R. Nagem, B. A. Rhodes, and G. V. H. Sandri, Hamiltonian mechanics of the damped harmonic oscillator, Journal of Sound and Vibration, Vol. 144, No. 3, pp. 536-538, 1991.
69. K. Sato, On the governing equations for vibration and stability of a Timoshenko beam: Hamilton's principle," Journal of Sound and Vibration, Vol. 145, No. 2, pp. 338-340, 1991.
70. J. P. Singh and S. S. Dev, Variational finite difference method for the vibration of sector plates, Journal of Sound and Vibration, Vol. 136, No. 1, pp. 91-104, 1990.
71. G. M. L. Gladwell and G. Zimmermann, On energy and complementary energy formulations of acoustic and structural vibration problems, Journal of Sound and Vibration, Vol. 3, No. 3, pp. 233-241, 1966.
72. J. Kondo, Integral Equations, Kodansha, Tokyo, and Clarendon Press, Oxford, 1991.
73. S. S. Rao, Applied Numerical Methods for Engineers and Scientists, Prentice Hall, Upper Saddle River, NJ, 2002.
74. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
75. F. G. Tricomi, Integral Equations, Interscience, New York, 1957.
76. J. A. Cochran, The Analysis of Linear Integral Equations, McGraw-Hill, New York, 1972.
77. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.
78. S. Kukla, Application of Green functions in frequency analysis of Timoshenko beams with oscillators, Journal of Sound and Vibration, Vol. 205, No. 3, pp. 355-363, 1997.
79. M. Abu-Hilal, Forced vibration of Euler-Bernoulli beams by means of dynamic Green functions, Journal of Sound and Vibration, Vol. 267, No. 2, pp. 191-207, 2003.
80. L. A. Bergman and D. M. McFarland, On the vibration of a point-supported linear distributed system, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 110, No. 4, pp. 485-492, 1988.
81. J. P. Spence and C. O. Horgan, Bounds on natural frequencies of composite circular membranes: integral equation methods, Journal of Sound and Vibration, Vol. 87, pp. 71-81, 1983.
82. R. H. Gutierrez and P. A. A. Laura, Analysis of transverse vibrations of composite membranes using the integral equation method and Rayleigh's optimization suggestion, Journal of Sound and Vibration, Vol. 147, No. 3, pp. 515-518, 1991.
83. W. B. Bickford and S. Y. Wu, Bounds on natural frequencies of composite circular plates: integral equation methods, Journal of Sound and Vibration, Vol. 126, No. 1, pp. 19-36, 1988.
84. P. A. A. Laura and R. H. Gutierrez, Rayleigh's optimization approach for the approximate solution of integral equations for vibration problems, Journal of Sound and Vibration, Vol. 139, No. 1, pp. 63-70, 1990.
85. E. Kreyszig, Advanced Engineering Mathematics, 8th ed., Wiley, New York, 1999.
86. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd ed., Wiley, New York, 1989.
87. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.
88. R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, Vol. I, 1937, Vol. II, 1962.
89. F. Y. Chen, On modeling and direct solution of certain free vibration systems, Journal of Sound and Vibration, Vol. 14, No. 1, pp. 57-79, 1971.
90. G. L. Anderson and C. R. Thomas, A forced vibration problem involving time derivatives in the boundary conditions, Journal of Sound and Vibration, Vol. 14, No. 2, pp. 193-214, 1971.
91. G. L. Anderson, On the forced vibrations of elastic bodies in contact, Journal of Sound and Vibration, Vol. 16, No. 4, pp. 533-549, 1971.
92. B. Yang and X. Wu, Transient response of one-dimensional distributed systems: a closed form eigenfunction expansion realization, Journal of Sound and Vibration, Vol. 208, No. 5, pp. 763-776, 1997.
93. B. Yang, Integral formulas for non-self-adjoint distributed dynamic systems, AIAA Journal, Vol. 34, No. 10, pp. 2132-2139, 1996.
94. L. Gavric, U. Carlsson, and L. Feng, Measurement of structural intensity using a normal mode approach, Journal of Sound and Vibration, Vol. 206, No. 1, pp. 87-101, 1997.
95. O. Heaviside, Electromagnetic Theory, 1899; reprint by Dover, New York, 1950.
96. T. Bromwich, Normal coordinates in dynamical systems, Proceedings of the London Mathematical Society, Ser. 2, Vol. 15, pp. 401-448, 1916.
97. K. W. Wagner, Über eine Formel Von Heaviside zur Berechnung von Einschaltvorgangen, Archiv fuver Elektrotechnik, Vol. 4, pp. 159-193, 1916.
98. J. R. Carson, On a general expansion theorem for the transient oscillations of a connected system, Physics Review, Ser. 2, Vol. 10, pp. 217-225, 1917.
99. B. Van der Pol, A simple proof and extension of Heaviside's operational calculus for invariable systems, Philosophical Magazine, Ser. 7, pp. 1153-1162, 1929.
100. L. Debnath, Integral Transforms and Their Applications, CRC Press, Boca Raton, FL, 1995.
101. I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951.
102. M. R. Spiegel, Theory and Problems of Laplace Transforms, Schaum, New York, 1965.
103. W. T. Thomson, Laplace Transformation, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1960.
104. C. J. Tranter, Integral Transforms in Mathematical Physics, Methuen, London, 1959.
105. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, New York, 1948.
106. W. Nowacki, Dynamics of Elastic Systems, translated by H. Zorski, Wiley, New York, 1963.
107. J. W. Cooley, P. A. W. Lewis, and P. D. Welch, The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine and Laplace transforms, Journal of Sound and Vibration, Vol. 12, No. 3, pp. 315-337, 1970.
108. M. H. Cobble and P. C. Fang, Finite transform solution of the damped cantilever beam equation having distributed load, elastic support, and the wall edge elastically restrained against rotation, Journal of Sound and Vibration, Vol. 6, No. 2, pp. 187-198, 1967.
109. G. R. Sharp, Finite transform solution of the vibrating annular membrane, Journal of Sound and Vibration, Vol. 6, No. 1, pp. 117-128, 1967.
110. G. R. Sharp, Finite transform solution of the symmetrically vibrating annular membrane, Journal of Sound and Vibration, Vol. 5, No. 1, pp. 1-8, 1967.
111. G. L. Anderson, On the determination of finite integral transforms for forced vibrations of circular plates, Journal of Sound and Vibration, Vol. 9, No. 1, pp. 126-144, 1969.
112. G. L. Anderson, On Gegenbauer transforms and forced torsional vibrations of thin spherical shells, Journal of Sound and Vibration, Vol. 12, No. 3, pp. 265-275, 1970.
113. S. C. Tsai, E. C. Ong, B. P. Tan, and P. H. Wong, Application of the z-transform method to the solution of the wave equation, Journal of Sound and Vibration, Vol. 19, No. 1, pp. 17-20, 1971.
114. W. Nowacki, Dynamics of Elastic Systems, translated by H. Zorski, Wiley, New York, 1963.
115. N. W. McLachlan, Theory of Vibrations, Dover, New York, 1951.
116. S. Timoshenko, D. H. Young, and W. Weaver, Jr, Vibration Problems in Engineering, 4th ed., Wiley, New York, 1974.
117. J. E. Bolwell, The flexible string's neglected term, Journal of Sound and Vibration, Vol. 206, No. 4, pp. 618-623, 1997.
118. M. S. Triantafyllou, Dynamics of cables and chains, Shock and Vibration Digest, Vol. 19, pp. 3-5 1987.
119. M. S. Triantafyllou and G. S. Triantafyllou, The paradox of the hanging string: an explanation using singular perturbations, Journal of Sound and Vibration, Vol. 148, No. 2, pp. 343-351, 1991.
120. D. H. Griffel, The dynamics of plucking, Journal of Sound and Vibration, Vol. 175, No. 3, pp. 289-297, 1994.
121. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, Springer-Verlag, New York, 1991.
122. A. Simpson, On the oscillatory motions of translating elastic cables, Journal of Sound and Vibration, Vol. 20, No. 2, pp. 177-189, 1972.
123. T. W. Barrett, On vibrating strings and information theory, Journal of Sound and Vibration, Vol. 20, No. 3, pp. 407-412, 1972.
124. B. Beddoe, Vibration of a sectionally uniform string from an initial state, Journal of Sound and Vibration, Vol. 4, No. 2, pp. 215-223, 1966.
125. R. Narasimha, Non-linear vibration of an elastic string, Journal of Sound and Vibration, Vol. 8, No. 1, pp. 134-146, 1968.
126. S. M. Shahruz, Vibration of wires used in electro-discharge machining, Journal of Sound and Vibration, Vol. 266, No. 5, pp. 1109-1116, 2003.
127. H. M. Irvine, Cable Structures, MIT Press, Cambridge, MA, 1981.
128. S. S. Sergev and W. D. Iwan, The natural frequencies and mode shapes of cables with attached masses, Journal of Energy Resources Technology, Vol. 103, pp. 237-242, 1981.
129. H. M. Irvine and T. K. Caughey, The linear theory of free vibrations of a suspended cable, Proceedings of the Royal Society, London, Vol. A-341, pp. 299-315, 1974.
130. P. Yu, Explicit vibration solutions of a cable under complicated loads, Journal of Applied Mechanics, Vol. 64, No. 4, pp. 957-964, 1997.
131. S.-P. Cheng and N. C. Perkins, Theoretical and experimental analysis of the forced response of sagged cable/mass suspension, Journal of Applied Mechanics, Vol. 61, No. 4, pp. 944-948, 1994.
132. N. C. Perkins, Linear dynamics of a translating string on an elastic foundation, Journal of Vibration and Acoustics, Vol. 112, No. 1, pp. 2-7, 1990.
133. R. E. D. Bishop, Longitudinal waves in beams, Aeronautical Quarterly, Vol. 3, No. 2, pp. 280-293, 1952.
134. J. W. S. Rayleigh, The Theory of Sound, Dover, New York, 1945.
135. J. S. Rao, Advanced Theory of Vibration, Wiley Eastern, New Delhi, India, 1992.
136. A. E. H. Love, A Treatize on the Mathematical Theory of Elasticity, 4th ed., Dover, New York, 1944.
137. S. Timoshenko, D. H. Young, and W. Weaven, Jr., Vibration Problems in Engineering, 4th ed., Wiley, New York, 1974.
138. E. Volterra and E. C. Zachmanoglou, Dynamics of Vibrations, Charles E. Merrill, Columbus, OH, 1965.
139. S. K. Clark, Dynamics of Continuous Elements, Prentice-Hall, Englewood Cliffs, NJ, 1972.
140. R. Kohoutek, Natural longitudinal frequencies of a uniform rod with a tip mass or spring, Journal of Sound and Vibration, Vol. 77, No. 1, pp. 147-148, 1981.
141. A. Raj and R. I. Sujith, Closed-form solutions for the free longitudinal vibration of inhomogeneous rods, Journal of Sound and Vibration, Vol. 283, No. 3-5, pp. 1015-1030, 2005.
142. H. Matsuda, T. Sakiyama, C. Morita, and M. Kawakami, Longitudinal impulsive response analysis of variable cross-section bars, Journal of Sound and Vibration, Vol. 181, No. 3, pp. 541-551, 1995.
143. Q. S. Li, Exact solutions for free longitudinal vibration of non-uniform rods, Journal of Sound and Vibration, Vol. 234, No. 1, pp. 1-19, 2000.
144. B. M. Kumar and R. I. Sujith, Exact solutions for the longitudinal vibration of non-uniform rods, Journal of Sound and Vibration, Vol. 207, No. 5, pp. 721-729, 1997.
145. N. Kounadis, On the derivation of equation of motion for a vibrating Timoshenko column, Journal of Sound and Vibration, Vol. 73, No. 2, pp. 174-184, 1980.
146. A. V. Ramesh and S. Utku, A comparison of approximate solutions for the axial free vibrations of uniform rods, Journal of Sound and Vibration, Vol. 139, No. 3, pp. 407-424, 1990.
147. S. Kukla, J. Przybylski, and L. Tomski, Longitudinal vibration of rods coupled by translational springs, Journal of Sound and Vibration, Vol. 185, No. 4, pp. 717-722, 1995.
148. W. F. Riley, L. D. Sturges, and D. H. Morris, Mechanics of Materials, 5th ed., Wiley, New York, 1999.
149. S. Timoshenko, D. H. Young, and W. Weaver, Jr., Vibration Problems in Engineering, 4th ed., Wiley, New York, 1974.
150. W. B. Bickford, Advanced Mechanics of Materials, Addison-Wesley, Reading, MA, 1998.
151. J. S. Rao, Advanced Theory of Vibration, Wiley, New York, 1992.
152. S. K. Clark, Dynamics of Continuous Elements, Prentice-Hall, Englewood Cliffs, NJ, 1972.
153. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
154. J. H. Faupel and F. E. Fisher, Engineering Design, 2nd ed., Wiley-Interscience, New York, 1981.
155. M. Vet, Torsional vibration of beams having rectangular cross sections, Journal of the Acoustical Society of America, Vol. 34, p. 1570, 1962.
156. J. M. Vance, Rotordynamics of Turbomachinery, Wiley, New York, 1988.
157. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Dover, New York, 1944.
158. J. M. Gere, Torsional vibrations of beams of thin walled open section, Journal of Applied Mechanics, Vol. 21, p. 381, 1954.
159. H. Lo and M. Goulard, Torsion with warping restraint from Hamilton's principle, Proceedings of the 2nd Midwestern Conference on Solid Mechanics, 1955, p. 68.
160. A. K. Darpe, K. Gupta, and A. Chawla, Coupled bending, longitudinal and torsional vibration of a cracked rotor, Journal of Sound and Vibration, Vol. 269, No. 1-2, pp. 33-60, 2004.
161. I. Y. Shen, W. Guo, and Y. C. Pao, Torsional vibration control of a shaft through active constrained layer damping treatments, Journal of Vibration and Acoustics, Vol. 119, No. 4, pp. 504-511, 1997.
162. L. Ran, R. Yacamini, and K. S. Smith, Torsional vibrations in electrical induction motor drives during start up, Journal of Vibration and Acoustics, Vol. 118, No. 2, pp. 242-251, 1996.
163. C. P. Filipich and M. B. Rosales, Free flexural-torsional vibrations of a uniform spinning beam, Journal of Sound and Vibration, Vol. 141, No. 3, pp. 375-387, 1990.
164. S. P. Timoshenko, Theory of bending, torsion and buckling of thin-walled member of open cross-section, Journal of the Franklin Institute, Vol. 239, pp. 201, 249, and 343, 1945.
165. L. Fryba, Vibration of Solids and Structures Under Moving Loads, Noordhoff International Publishing, Groningen, The Netherlands, 1972.
166. J. E. Shigley and L. D. Mitchell, Mechanical Engineering Design, 4th ed., McGraw-Hill, New York, 1983.
167. W. B. Bickford, Advanced Mechanics of Materials, Addison-Wesley, Reading, MA, 1998.
168. S. S. Rao, Natural vibrations of systems of elastically connected Timoshenko beams, Journal of the Acoustical Society of America, Vol. 55, pp. 1232-1237, 1974.
169. C. F. Garland, Normal modes of vibration of beams having noncollinear elastic and mass axes, Journal of Applied Mechanics, Vol. 62, p. 97, 1940.
170. M. S. Ewing, Another second order beam vibration theory: explicit bending, warping flexibility and restraint, Journal of Sound and Vibration, Vol. 137, No. 1, pp. 43-51, 1990.
171. M. I. Qaisi, Normal modes of a continuous system with quadratic and cubic nonlinearities, Journal of Sound and Vibration, Vol. 265, No. 2, pp. 329-335, 2003.
172. T. M. Wang, Natural frequencies of continuous Timoshenko beams, Journal of Sound and Vibration, Vol. 13, No. 3, pp. 409-414, 1970.
173. R.-T. Wang, Vibration of multi-span Timoshenko beams to a moving force, Journal of Sound and Vibration, Vol. 207, No. 5, pp. 731-742, 1997.
174. A. H. Nayfeh and M. Abdel-Rohman, Analysis of galloping responses in cantilever beams, Journal of Sound and Vibration, Vol. 144, No.1, pp. 87-93, 1991.
175. H. Q. Wang, J. C. Miao, J. H. Luo, F. Huang, and L. G. Wang, The free vibration of long-span transmission line conductors with dampers, Journal of Sound and Vibration, Vol. 208, No. 4, pp. 501-516, 1997.
176. D. Zhou, The vibrations of a cantilever beam carrying a heavy tip mass with elastic supports, Journal of Sound and Vibration, Vol. 206, No. 2, pp. 275-279, 1997.
177. S. Ilanko, The vibration behaviour of initially imperfect simply supported beams subject to axial loading, Journal of Sound and Vibration, Vol. 142, No.2, pp. 355-359, 1990.
178. A. Bokaian, Natural frequencies of beams under tensile axial loads, Journal of Sound and Vibration, Vol. 142, No. 3, pp. 481-498, 1990.
179. R. H. Gutierrez and P. A. A. Laura, Vibrations of a beam of non-uniform cross-section traversed by a time varying concentrated force, Journal of Sound and Vibration, Vol. 207, No. 3, pp. 419-425, 1997.
180. C. T. Hugin, A physical description of the response of coupled beams, Journal of Sound and Vibration, Vol. 203, No. 4, pp. 563-580, 1997.
181. N. Popplewell and D. Chang, Free vibrations of a stepped, spinning Timoshenko beam, Journal of Sound and Vibration, Vol. 203, No. 4, pp. 717-722, 1997.
182. A. P. Gupta and N. Sharma, Effect of transverse shear and rotatory inertia on the forced motion of a stepped rectangular beam, Journal of Sound and Vibration, Vol. 209, No. 5, pp. 811-820, 1998.
183. P.-O. Larsson, Determination of Young's and shear moduli from flexural vibrations of beams, Journal of Sound and Vibration, Vol. 146, No. 1, pp. 111-123, 1991.
184. Y. Yaman, Vibrations of open-section channels: a coupled flexural and torsional wave analysis, Journal of Sound and Vibration, Vol. 204, No. 1, pp. 131-158, 1997.
185. A. N. Bercin and M. Tanaka, Coupled flexural-torsional vibrations of Timoshenko beams, Journal of Sound and Vibration, Vol. 207, No. 1, pp. 47-59, 1997.
186. M. Wang and T. A. Kinsman, Vibrations of frame structures according to the Timoshenko theory, Journal of Sound and Vibration, Vol. 14, No. 2, pp. 215-227, 1971.
187. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
188. W. T. Thomson, Laplace Transformation, Second Edition, Prentice-Hall, Englewood Cliffs, NJ, 1960.
189. L. L. Philipson, On the role of extension in the flexural vibrations of rings, Journal of Applied Mechanics, Vol. 23 (Transactions of ASME, Vol. 78), p. 364, 1956.
190. B. S. Seidel and E. A. Erdelyi, On the vibration of a thick ring in its own plane, Journal of Engineering for Industry (Transactions of ASME, Ser. B), Vol. 86, p. 240, 1964.
191. S. Timoshenko, On the correction for shear of the differential equation for transverse vibration of prismatic bars, Philosophical Magazine, Ser. 6, Vol. 41, p. 744, 1921.
192. J. P. Den Hartog, Vibrations of frames of electrical machines, Journal of Applied Mechanics, Transactions of ASME, Vol. 50, APM-50-11, 1928.
193. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Dover, New York, 1944.
194. S. S. Rao and V. Sundararajan, Inplane flexural vibrations of circular rings, Journal of Applied Mechanics, Vol. 36, No. 3, pp. 620-625, 1969.
195. S. S. Rao, Effects of transverse shear and rotatory inertia on the coupled twist-bending vibrations of circular rings, Journal of Sound and Vibration, Vol. 16, No. 4, pp. 551-566, 1971.
196. S. S. Rao, On the natural vibrations of systems of elastically connected concentric thick rings, Journal of Sound and Vibration. Vol. 32, No. 3, pp. 467-479, 1974.
197. S. S. Rao, Three dimensional vibration of a ring on an elastic foundation, Aeronautical Journal of the Royal Aeronautical Society, London, Vol. 75, pp. 417-419, 1971.
198. R. E. Peterson, An experimental investigation of ring vibration in connection with the study of gear-noise, Transactions of the American Society of Mechanical Engineers, Vol. 52, APM-52-1, 1930.
199. E. R. Kaiser, Acoustical vibrations of rings, Journal of the Acoustical Society of America, Vol. 25, p. 617, 1953.
200. L. T. Stavridis and G. T. Michaltsos, Eigenfrequency analysis of thin-walled girders curved in plan, Journal of Sound and Vibration, Vol. 227, No. 2, pp. 383-396, 1999.
201. Y. Wasserman, Spatial symmetrical vibrations and stability of circular arches with flexibly supported ends, Journal of Sound and Vibration, Vol. 59, No. 2, pp. 181-194, 1978.
202. W. B. Bickford and S. P. Maganty, On the out-of-plane vibrations of thick rings, Journal of Sound and Vibration, Vol. 108, No. 3, pp. 503-507, 1986.
203. C. G. Culver and D. J. Oestel, Natural frequencies of multispan curved beams, Journal of Sound and Vibration, Vol. 10, No. 3, pp. 380-389, 1969.
204. Y. Kagawa, On the dynamical properties of helical springs of finite length with small pitch, Journal of Sound and Vibration, Vol. 8, No. 1, pp. 1-15, 1968.
205. R. P. Tanna and T. C. Lim, Modal frequency deviations in estimating ring gear modes using smooth ring solutions, Journal of Sound and Vibration, Vol. 269, No. 3-5, pp. 1099-1110, 2004.
206. E. Erdelyi and G. Horvay, Vibration modes of stators of induction motors, Journal of Applied Mechanics, Vol. 24 (Transactions of ASME, Vol. 79), p. 39, 1957.
207. S. F. Borg and J. J. Gennaro, Modern Structural Analysis, Van Nostrand Reinhold, New York, 1969.
208. J. Henrych, The Dynamics of Arches and Frames, Elsevier, Amsterdam, 1981.
209. S. J. Oh, B. K. Lee, and I. W. Lee, Natural frequencies of non-circular arches with rotatory inertia and shear deformation, Journal of Sound and Vibration, Vol. 219, No. 1, pp. 23-33, 1999.
210. B. K. Lee and J. F. Wilson, Free vibrations of arches with variable curvature, Journal of Sound and Vibration, Vol. 136, No. 1, pp. 75-89, 1989.
211. K. F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH, 1975.
212. K. F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH, 1975.
213. N. W. McLachlan, Bessel Functions for Engineers, Oxford University Press, New York, 1934.
214. G. N. Watson, Theory of Bessel Functions, Cambridge University Press, London, 1922.
215. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.
216. S. Timoshenko, D. H. Young, and W. Weaver, Jr, Vibration Problems in Engineering, 4th ed., Wiley, New York, 1974.
217. E. Volterra and E. C. Zachmanoglou, Dynamics of Vibrations, Charles E. Merrill, Columbus, OH, 1965.
218. S. K. Clark, Dynamics of Continuous Elements, Prentice-Hall, Englewood Cliffs, NJ, 1972.
219. Lord Rayleigh, The Theory of Sound, 2nd ed., Vol. 1, Dover, New York, 1945.
220. A. Jeffrey, Advanced Engineering Mathematics, Harcourt/Academic Press, San Diego, CA, 2002.
221. P. M. Morse, Vibration and Sound, McGraw-Hill, New York, 1936.
222. J. P. Spence and C. O. Horgan, Bounds on natural frequencies of composite circular membranes: integral equation methods, Journal of Sound and Vibration, Vol. 87, No. 1, pp. 71-81, 1983.
223. G. Yamada, Y. Kobayashi, and H. Hamaya, Transient response of a hanging curtain, Journal of Sound and Vibration, Vol. 130, No. 2, pp. 223-235, 1989.
224. N. Y. Olcer, General solution to the equation of the vibrating membrane, Journal of Sound and Vibration, Vol. 6, No. 3, pp. 365-374, 1967.
225. A. W. Leissa and A. Ghamat-Rezaei, Vibrations of rectangular membranes subjected to shear and nonuniform tensile stresses, Journal of the Acoustical Society of America, Vol. 88, No. 1, pp. 231-238, 1990.
226. G. R. Sharp, Finite transform solution of the symmetrically vibrating annular membrane, Journal of Sound and Vibration, Vol. 5, No. 1, pp. 1-8, 1967.
227. S. De, Vibrations of a loaded kettledrum, Journal of Sound and Vibration, Vol. 20, No. 1, pp. 79-92, 1972.
228. P. J. Torvik and F. E. Eastep, A method for improving the estimation of membrane frequencies, Journal of Sound and Vibration, Vol. 21, No. 3, pp. 285-294, 1972.
229. C. Y. Wang, Some exact solutions of the vibration of non-homogeneous membranes, Journal of Sound and Vibration, Vol. 210, No. 4, pp. 555-558, 1998.
230. A. W. Leissa, Vibration of Plates, NASA SP-160, National Aeronautics and Space Administration, Washington, DC, 1969.
231. K. F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH, 1975.
232. R. D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic plates, Journal of Applied Mechanics, Vol. 18, pp. 31-38, 1951.
233. R. D. Mindlin and H. Deresiewicz, Thickness-shear and flexural vibrations of a circular disk, Journal of Applied Physics, Vol. 25, pp. 1320-1332, 1954.
234. E. H. Mansfield, The Bending and Stretching of Plates, Macmillan, New York, 1964.
235. R. Szilard, Theory and Analysis of Plates: Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974.
236. K. M. Liew, C. M. Wang, Y. Xiang, and S. Kitipornchai, Vibration of Mindlin Plates: Programming the p-Version Ritz Method, Elsevier, Amsterdam, 1998.
237. E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York, 2001.
238. E. B. Magrab, Vibrations of Elastic Structural Members, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1979.
239. E. Skudrzyk, Simple and Complex Vibratory Systems, Pennsylvania State University Press, University Park, PA, 1968.
240. S. S. Rao and A. S. Prasad, Vibrations of annular plates including the effects of rotatory inertia and transverse shear deformation, Journal of Sound and Vibration, Vol. 42, No. 3, pp. 305-324, 1975.
241. H. D. Conway, E. C. H. Becker, and J. F. Dubil, Vibration frequencies of tapered bars and circular plates, Journal of Applied Mechanics, Vol. 31, No. 2, pp. 329-331, 1964.
242. K. G. Korenev, Bessel Functions and Their Applications, Taylor & Francis, London, 2002.
243. I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951.
244. W. Nowacki, Dynamics of Elastic Systems, translated by H. Zorski, Wiley, New York, 1963.
245. R. C. Colwell and H. C. Hardy, The frequencies and nodal systems of circular plates, Philosophical Magazine, Ser. 7, Vol. 24, No. 165, pp. 1041-1055, 1937.
246. G. Herrmann, The influence of initial stress on the dynamic behavior of elastic and viscoelastic plates, Publications of the International Association for Bridge and Structural Engineering, Vol. 16, pp. 275-294, 1956.
247. A. Jeffrey, Advanced Engineering Mathematics, Harcourt/Academic Press, San Diego, CA, 2002.
248. T. Wah, Vibration of circular plates, Journal of the Acoustical Society of America, Vol. 34, No. 3, pp. 275-281, 1962.
249. R. D. Mindlin, A. Schacknow, and H. Deresiewicz, Flexural vibrations of rectangular plates, Journal of Applied Mechanics, Vol. 23, No. 3, pp. 430-436, 1956.
250. D. J. Gorman, Free Vibration Analysis of Rectangular Plates, Elsevier/North-Holland, New York, 1982.
251. A. Cote, N. Atalla, and J. Nicolas, Effects of shear deformation and rotary inertia on the free vibration of a rotating annular plate, Journal of Vibration and Acoustics, Vol. 119, No. 4, pp. 641-643, 1997.
252. C. S. Kim and S. M. Dickinson, The flexural vibration of rectangular plates with point supports, Journal of Sound and Vibration, Vol. 117, No. 2, pp. 249-261, 1987.
253. S. A. Vera, M. Febbo, C. G. Mendez, and R. Paz, Vibrations of a plate with an attached two degree of freedom system, Journal of Sound and Vibration, Vol. 285, No. 1-2, pp. 457-466, 2005.
254. D. J. Gorman, A comprehensive study of the free vibration of rectangular plates resting on symmetrically distributed uniform elastic edge supports, Journal of Applied Mechanics, Vol. 56, No. 4 pp. 893-899, 1989.
255. Y. Xiang, K. M. Liew, and S. Kitipornchai, Vibration analysis of rectangular Mindlin plates resting on elastic edge supports, Journal of Sound and Vibration, Vol. 204, No. 1, pp. 1-16, 1997.
256. A. P. Gupta and N. Sharma, Forced motion of a stepped semi-infinite plate, Journal of Sound and Vibration, Vol. 203, No. 4, pp. 697-705, 1997.
257. J. Wang, Generalized power series solutions of the vibration of classical circular plates with variable thickness, Journal of Sound and Vibration, Vol. 202, No. 4, pp. 593-599, 1997.
258. J. S. Horvath, Subgrade models for soil-structure interaction analysis, in Foundation Engineering: Current Principles and Practices, American Society of Civil Engineers, Reston, VA, 1989 Vol. 1, pp. 599-612.
259. K. K. Raju and G. V. Rao, Effect of elastic foundation on the mode shapes in stability and vibration problems of simply supported rectangular plates, Journal of Sound and Vibration, Vol. 139, No. 1, pp. 170-173, 1990.
260. J. S. Smaill, Dynamic response of circular plates on elastic foundations: linear and non-linear deflection, Journal of Sound and Vibration, Vol. 139, No. 3, pp. 487-502, 1990.
261. K. K. V. Devarakonda and C. W. Bert, Flexural vibration of rectangular plates subjected to sinusoidally distributed compressive loading on two opposite sides, Journal of Sound and Vibration, Vol. 283, No. 3-5, pp. 749-763, 2005.
262. S. M. Dickinson, Lateral vibration of rectangular plates subject to in-plane forces, Journal of Sound and Vibration, Vol. 16, No. 4, pp. 465-472, 1971.
263. N. G. Stephen, Mindlin plate theory: best shear coefficient and higher spectra validity, Journal of Sound and Vibration, Vol. 202, No. 4, pp. 539-553, 1997.
264. K. M. Liew, J. B. Han and Z. M. Xiao, Vibration analysis of circular Mindlin plates using the differential quadrature method, Journal of Sound and Vibration, Vol. 205, No. 5, pp. 617-630, 1997.
265. G. Martincek, The determination of Poisson's ratio and the dynamic modulus of elasticity from the frequencies of natural vibration in thick circular plates, Journal of Sound and Vibration, Vol. 2, No. 2, pp. 116-127, 1965.
266. C. S. Huang, O. G. McGee and A. W. Leissa, Exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges, International Journal of Solids and Structures, Vol. 31, No. 11, pp. 1609-1631, 1994.
267. W. H.Wittrick, Analytical, three-dimensional elasticity solutions to some plate problems and some observations on Mindlin plate theory, International Journal of Solids and Structures, Vol. 23, No. 4, pp. 441-464, 1987.
268. E. Kreyszig, Advanced Engineering Mathematics, 8th ed., Wiley, New York, 1999.
269. R. Solecki and R. J. Conant, Advanced Mechanics of Materials, Oxford University Press, New York, 2003.
270. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Dover, New York, 1944.
271. W. Flügge, Stresses in Shells, Springer-Verlag, Berlin, 1962.
272. A. L. Goldenveizer, Theory of Thin Shells, translated by G. Herrmann, Pergamon Press, New York, 1961.
273. H. Kraus, Thin Elastic Shells, Wiley, New York, 1967.
274. A. W. Leissa, Vibration of Shells, NASA SP-288, National Aeronautics and Space Administration, Washington, D C, 1973.
275. W. Soedel, Vibration of Shells and Plates, 2nd ed., Marcel Dekker, New York, 1993.
276. A. C. Ugural, Stresses in Plates and Shells, McGraw-Hill, New York, 1981.
277. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959.
278. E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York, 2001.
279. V. V. Novozhilov, Thin Shell Theory, translated by P. G. Lowe, Noordhoff, Groningen, The Netherlands, 1964.
280. L. H. Donnell, Beams, Plates, and Shells, McGraw-Hill, New York, 1976.
281. V. Z. Vlasov, General Theory of Shells and Its Applications in Engineering, NASA-TT-F99, National Aeronautics and Space Administration, Washington, DC, 1964.
282. N. A. Kilchevskiy, Fundamentals of the Analytical Mechanics of Shells, NASA-TT-F292, National Aeronautics and Space Administration, Washington, DC, 1965.
283. W. Soedel, On the vibration of shells with Timoshenko-Mindlin type shear deflections and rotatory inertia, Journal of Sound and Vibration, Vol. 83, No. 1, pp. 67-79, 1982.
284. P. M. Naghdi and R. M. Cooper, Propagation of elastic waves in cylindrical shells, including the effects of transverse shear and rotatory inertia, Journal of the Acoustical Society of America, Vol. 28, No. 1, pp. 56-63, 1956.
285. T. C. Lin and G. W. Morgan, A study of axisymmetric vibrations of cylindrical shells as affected by rotary inertia and transverse shear, Journal of Applied Mechanics, Vol. 23, No. 2, pp. 255-261, 1956.
286. I. Mirsky and G. Herrmann, Nonaxially symmetric motions of cylindrical shells, Journal of the Acoustical Society of America, Vol. 29, No. 10, pp. 1116-1123, 1957.
287. I. Mirsky, Vibrations of orthotropic, thick, cylindrical shells, Journal of the Acoustical Society of America, Vol. 36, No. 1, pp. 41-51, 1964.
288. W. D. Pilkey, Mechanically and/or thermally generated dynamic response of thick spherical and cylindrical shells with variable material properties, Journal of Sound and Vibration, Vol. 6, No. 1, pp. 105-109, 1967.
289. T. Wah, Flexural vibrations of ring-stiffened cylindrical shells, Journal of Sound and Vibration, Vol. 3, No.3 pp. 242-251, 1966.
290. A. E. Armenakas, D. C. Gazis, and G. Herrmann, Free Vibrations of Circular Cylindrical Shells, Pergamon Press, Oxford, 1969.
291. K. P. Soldatos and V. P. Hadjigeorgiou, Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels, Journal of Sound and Vibration, Vol. 137, No. 3, pp. 369-384, 1990.
292. G. Prathap and K. A. V. Pandalai, The role of median surface curvature in large amplitude flexural vibrations of thin shells, Journal of Sound and Vibration, Vol. 60, No. 1, pp. 119-131, 1978.
293. K. R. Sivadas and N. Ganesan, Free vibration of circular cylindrical shells with axially varying thickness, Journal of Sound and Vibration, Vol. 147, No.1, pp. 73-85, 1991.
294. K. Suzuki and A. W. Leissa, Free vibrations of non-circular cylindrical shells having circumferentially varying thickness, Journal of Applied Mechanics, Vol. 52, No. 1, pp. 149-154, 1985.
295. W. Soedel, Shells, in Encyclopedia of Vibration, S. Braun, D. Ewins, and S. S. Rao, Eds., Academic Press, San Diego, CA, 2002, Vol. 3, pp. 1155-1167.
296. G. B. Warburton and A. M. J. Al-Najafi, Free vibration of thin cylindrical shells with a discontinuity in the thickness, Journal of Sound and Vibration, Vol. 9, No. 3, pp. 373-382, 1969.
297. C. B. Sharma and D. J. Johns, Natural frequencies of clamped-free circular cylindrical shells, Journal of Sound and Vibration, Vol. 21, No. 3, pp. 317-318, 1972.
298. H. Matsunaga, Free vibration of thick circular cylindrical shells subjected to axial stresses, Journal of Sound and Vibration, Vol. 211, No. 1, pp. 1-17, 1998.
299. S. Liang, H. L. Chen, and T. X. Liang, An analytical investigation of free vibration for a thin-walled regular polygonal prismatic shell with simply supported odd/even number of sides, Journal of Sound and Vibration, Vol. 284, No. 1-2, pp. 520-530, 2005.
300. A. Bedford and D. S. Drumheller, Introduction to Elastic Wave Propagation, Wiley, Chichester, West Jeessey, England, 1994.
301. S. K.Clark, Dynamics of Continuous Elements, Prentice-Hall, Englewood Cliffs, NJ, 1972.
302. E.Volterra and E. C. Zachmanoglou, Dynamics of Vibrations, Charles E. Merrill, Columbus, OH, 1965.
303. K. F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH 1975.
304. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967.
305. H. J. Pain, The Physics of Vibrations and Waves, 4th ed, Wiley, Chichester, West Jeessey, England, 1993.
306. J. W. S. Rayleigh, On waves propagated along the plane surface of an elastic solid, Proceedings of the London Mathematical Society, Vol. 17, pp. 4-11, 1887.
307. A. Knopoff, On Rayleigh wave velocities, Bulletin of the Seismological Society of America, Vol. 42, pp. 307-308, 1952.
308. H. Kolsky, Stress Waves in Solids, Dover, New York, 1963.
309. Beddone, Propagation of elastic stress waves in a necked rod, Journal of Sound and Vibration, Vol. 2, No. 2, pp. 150-164, 1965.
310. S. Y. Lee and W. F. Yeen, Free coupled longitudinal and flexural waves of a periodically supported beam, Journal of Sound and Vibration, Vol. 142, No. 2, pp. 203-211, 1990.
311. G. Sen Gupta, Propagation of flexural waves in doubly-periodic structures, Journal of Sound and Vibration, Vol. 20, No. 1, pp. 39-49, 1972.
312. S. H. Ju, Finite element analysis of wave propagations due to high-speed train across bridges, International Journal for Numerical Methods in Engineering, Vol. 54, No. 9, pp. 1391-1408, 2002.
313. R. S. Langley and K. H. Heron, Elastic wave transmission through plate/beam junctions, Journal of Sound and Vibration, Vol. 143, No. 2, pp. 241-253, 1990.
314. R. S. Langley, Analysis of beam and plate vibrations by using the wave equation, Journal of Sound and Vibration, Vol. 150, No. 1, pp. 47-65, 1991.
315. C. R. Halkyard and B. R. Mace, A Fourier series approach to the measurement of flexural wave intensity in plates, Journal of Sound and Vibration, Vol. 203, No. 1, pp. 101-126, 1997.
316. A. Freedman, The variation, with the Poisson ratio, of Lamb modes in a free plate, I: General spectra, Journal of Sound and Vibration, Vol. 137, No. 2, pp. 209-230, 1990.
317. S. H. Crandall, Engineering Analysis: A Survey of Numerical Procedures, McGraw-Hill, New York, 1956.
318. S. S. Rao, Applied Numerical Methods for Engineers and Scientists, Prentice Hall, Upper Saddle River, NJ, 2002.
319. S. S. Rao, The Finite Element Method in Engineering, 3rd ed., Butterworth-Heinemann, Boston, 1999.
320. S. S. Rao, Mechanical Vibrations, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2004.
321. J. N. Reddy, Energy and Variational Methods in Applied Mechanics,Wiley, New York, 1984.
322. C. Levy, An iterative technique based on the Dunkerley method for determining the natural frequencies of vibrating systems, Journal of Sound and Vibration, Vol. 150, No. 1, pp. 111-118, 1991.
323. F. Badrakhan, On the application of Rayleigh's method to an unconstrained system, Journal of Sound and Vibration, Vol. 162, No.1, pp. 190-194, 1993.
324. G. Radhakrishnan, M. K. Sundaresan, and B. Nageswara Rao, Fundamental frequency of thin elastic plates, Journal of Sound and Vibration, Vol. 209, No. 2, pp. 373-376, 1998.
325. C. W. Bert, Application of a version of the Rayleigh technique to problems of bars, beams, columns, membranes and plates, Journal of Sound and Vibration, Vol. 119, No. 2, pp. 317-326, 1987.
326. V. T. Nagaraj, Relationship between fundamental natural frequency and maximum static deflection for rotating Timoshenko beams, Journal of Sound and Vibration, Vol. 201, No. 3, pp. 404-406, 1997.
327. R. B. Bhat, Nature of stationarity of the natural frequencies at the natural modes in the Rayleigh-Ritz method, Journal of Sound and Vibration, Vol. 203, No. 2, pp. 251-263, 1997.
328. K. H. Low, Frequencies of beams carrying multiple masses: Rayleigh estimation versus eigenanalysis solutions, Journal of Sound and Vibration, Vol. 268, No. 4, pp. 843-853, 2003.
329. G. M. L. Gladwell, The vibration of frames, Journal of Sound and Vibration, Vol. 1, No. 4, pp. 402-425, 1964.
330. K. Takahashi, Dynamic stability of cables subjected to an axial periodic load, Journal of Sound and Vibration, Vol. 144, No. 2, pp. 323-330, 1991.
331. G. Yamada, Y. Kobayashi, and H. Hamaya, Transient response of a hanging curtain, Journal of Sound and Vibration, Vol. 130, No. 2, pp. 223-235, 1989.
332. K. M. Liew and K. Y. Lam, Application of two-dimensional orthogonal plate function to flexural vibration of skew plates, Journal of Sound and Vibration, Vol. 139, No. 2, pp. 241-252, 1990.
333. H. T. Saliba, Transverse free vibration of fully clamped symmetrical trapezoidal plates, Journal of Sound and Vibration, Vol. 126, No. 2, pp. 237-247, 1988.
334. H. T. Saliba, Transverse free vibration of simply supported right triangular thin plates: a highly accurate simplified solution, Journal of Sound and Vibration, Vol. 139, No. 2, pp. 289-297, 1990.
335. D. J. Gorman, A general solution for the free vibration of rectangular plates resting on uniform elastic edge supports, Journal of Sound and Vibration, Vol. 139, No. 2, pp. 325-335, 1990.
336. D. Avalos, P. A. A. Laura, and H. A. Larrondo, Vibrating circular plates with stepped thickness over a concentric circular region: a general approximate solution, Journal of the Acoustical Society of America, Vol. 84, No. 4, pp. 1181-1185, 1988.
337. M. Salari, C. W. Bert, and A. G. Striz, Free vibrations of a solid circular plate free at its edge and attached to a Winkler foundation, Journal of Sound and Vibration, Vol. 118, No. 1, pp. 188-191, 1987.
338. P. A. A. Laura, R. H. Gutierrez, R. Carnicer, and H. C. Sanzi, Free vibrations of a solid circular plate of linearly varying thickness and attached to a Winkler foundation, Journal of Sound and Vibration, Vol. 144, No. 1, pp. 149-161, 1991.
339. P. A. A. Laura, R. H. Gutierrez, A. Bergmann, R. Carnicer, and H. C. Sanzi, Vibrations of circular plates with discontinuous variation of the thickness in a non-concentric fashion, Journal of Sound and Vibration, Vol. 144, No. 1, pp. 1-8, 1991.