A study of the Numerical Convergence of Rayleigh-Ritz Method for the Free Vibrations of Cantilever Beam of Variable Cross-Section with Tip Mass
A numerical study on the convergence properties of the Rayleigh-Ritz method is presented, for the dynamic analysis of beams with continuously varying cross-section. The beam is assumed to be slender, the Euler-Bernoulli hypotheses are accepted, and some particular cases are considered, for which a closed-form solution is available in terms of Bessel functions. The comparisons between exact and approximate results can give some hint about the usefulness of the approximate method in more complex situations, for which the exact solution is not attainable.
References
H.H. MABIE and C.B. ROGERS, Transverse vibrations of tapered cantilever beams with end support, J. Acoustical Society of America, 44, 1739-1741, 1968.
H.H. MABIE and C.B. ROGERS, Transverse vibrations of double-tapered cantilever beams with end support and with end mass, J. Acoustical Society of America, 55, 986-991, 1974.
L. KLEIN, Transverse vibrations of non-uniform beams, J. Sound and Vibration, 37, 491-505, 1974.
G.V. SANKARAN, K. KANAKA RAJU and G. VENKATESWARA RAO, Vibration frequencies of a tapered beam with one end spring-hinged and carrying a mass at the other free end, J. Applied Mechanics, 42, 740-741, 1975.
P.A.A. LAURA, M.J. MAURIZI and J.L. POMBO, A note on the dynamic analysis of an elastically restrained-free beam with a mass at the free end, J. Sound and Vibration, 41, 397-405, 1975.
R.P. GOEL, Transverse vibrations of tapered beam, J. Sound and Vibration, 47, 1-7, 1976.
T.W. LEE, Transverse vibrations of a tapered beam carrying a concentrated mass, J. Applied Mechanics, 43, 366-367, 1976.
K. SATO, Transverse vibrations of linearly tapered beams with ends restrained elastically against notation subjected axial force, Intern. J. Mech. Science, 22, 109-115, 1980.
L. MEIROVITCH, Computational methods in structural dynamics, Sijthoff & Noordhoff Int. Publishers, B.V. Alphen aan den Rijn, The Netherlands 1980.
R. SCHMIDT, Estimation of buckling loads and other eigenvalues via a modification of the Rayleigh-Ritz method, J. Applied Mechanics, 49, 639-640, 1982.
C.W.S. TO, Vibration of a cantilever beam with a base excitation and tip mass, J. Sound and Vibration, 83, 445-460, 1982.
J.S. VANDERGRAFT, Introduction to numerical computations, II Edition, Academic Press, 1983.
R.B. BHAT, Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method, J. Sound and Vibration, 102, 493-499, 1985.
P.A.A. LAURA and R.H. GUTIERREZ, Vibrations of an elastically restrained cantilever beam of varying cross-section with tip mass of finite length, J. Sound and Vibration, 108, 123-131, 1986.
C.W. BERT, Application of a version of the Rayleigh technique to problems of bars, beams, columns, membranes and plates, J. Sound and Vibration, 119, 317-326, 1987.
W.H. LIU and C.-C. HUANG, Vibrations of a constrained beam carrying a heavy tip body, J. Sound and Vibration, 123, 15-29, 1988.
M.J. MAURIZI, P. BELLES and M. ROSALES, A note on fee vibrations of a constrained cantilever beam with a tip mass of finite length, J. Sound and Vibration, 138, 170-172, 1990.
R.O. GROSSI, A. ARANDA and R.B. BHAT, Vibration of tapered beams with one end spring hinged and the other end with tip mass, J. Sound and Vibration, 147, 174-178, 1993.
W.L. CRAVER jr. and P. JAMPALA, Transverse vibrations of a linearly tapered cantilever beam with constraining springs, J. Sound and Vibration, 166, 521-529, 1993.
S. NAGULESWARAN, A direct solution for the transverse vibration of Euler-Bernoulli wedge and cone beams, J. Sound and Vibration, 172, 289-304, 1994.
N.M. AUCIELLO, A comment on "A note on vibrating trapered beams", J. Sound and Vibration, 187, 721-726, 1995.
N.M. AUCIELLO, Transverse vibrations of a linearly tapered cantilever beam with tip mass of finite length, J. Sound and Vibration, 194, 25-34, 1996.
R.L. BURDEN and J.D. FAIRES, Numerical analysis, IV Edition, PWS-KENT P.C. Boston 1981.
C.A. TAN and W. KUANG, Distributed transfer function analysis of cone and wedge beams, J. Sound and Vibration, 170, 557-566, 1994.