Exact Closed-Form Solution for Vibration Analysis of Truncated Conical and Tapered Beams Carrying Multiple Concentrated Masses

Document Type: Research Paper


1 Department Mechanical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran

2 Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan, Iran

3 Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

4 Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Iran


In this paper, an exact closed-form solution is presented for free vibration analysis of Euler-Bernoulli conical and tapered beams carrying any desired number of attached masses. The concentrated masses are modeled by Dirac’s delta functions which creates no need for implementation of compatibility conditions. The proposed technique explicitly provides frequency equation and corresponding mode as functions with only two integration constants which leads to solution of a two by two eigenvalue problem for any number of attached masses. Using Basic functions which are made of the appropriate linear composition of Bessel functions leads to make implementation of boundary conditions much easier. The proposed technique is employed to study effect of quantity, position and translational inertia of the concentrated masses on the natural frequencies and corresponding modes of conical and tapered beams for all standard boundary conditions. Unlike many of previous exact approaches, presented solution has no limitation in number of concentrated masses. In other words, by increase in number of attached masses, there is no considerable increase in computational effort.


[1] Chen Y., 1963, On the vibration of beams or rods carrying a concentrated mass, Journal of Applied Mechanics 30: 310-311.
[2] Low K.H., 2000, A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses, International Journal of Mechanical Sciences 42: 1287-1305.
[3] Laura P.A.A., Pombo J.L., Susemihl E.L., 1974, A note on the vibration of a clamped–free beam with a mass at the free end, Journal of Sound and Vibration 37: 161-168.
[4] Dowell E.H., 1979, On some general properties of combined dynamical systems, ASME Journal of Applied Mechanics 46: 206-209.
[5] Laura P.A.A., Irassar P.L., Ficcadenti G.M., 1983, A note of transverse vibration of continuous beams subjected to an axial force and carrying concentrated masses, Journal of Sound and Vibration 86: 279-284.
[6] Gurgoze M., 1984, A note on the vibrations of restrained beams and rods with point masses, Journal of Sound and Vibration 96: 461-468.
[7] Gurgoze M., 1985, On the vibration of restrained beams and rods with heavy masses, Journal of Sound and Vibration 100: 588-589.
[8] Liu W.H., Wu J.R., Huang C.C., 1988, Free vibrations of beams with elastically restrained edges and intermediate concentrated masses, Journal of Sound and Vibration 122: 193-207.
[9] Torabi K., Afshari H., Najafi H., 2013, Vibration analysis of multi-step Bernoulli-Euler and Timoshenko beams carrying concentrated masses, Journal of Solid Mechanics 5: 336-349.
[10] Farghaly S.H., El-Sayed T.A., 2016, Exact free vibration of multi-step Timoshenko beam system with several attachments, Mechanical Systems and Signal Processing 72-73: 525-546.
[11] Cranch E.T., Adler A.A., 1956, Bending vibration of variable section beams, Journal of Applied Mechanics 23: 103-108.
[12] Conway H.D., Dubil J.F., 1965, Vibration frequencies of truncated-cone and wedge beams, Journal of Applied Mechanics 32: 932-934.
[13] Mabie H.H., Rogers C.B., 1968, Transverse vibration of tapered cantilever beams with end support, Journal of Acoustics Society of America 44: 1739-1741.
[14] Heidebrecht A.C., 1967, Vibration of non-uniform simply supported beams, Journal of the Engineering Mechanics Division 93: 1-15.
[15] Mabie H.H., Rogers C.B., 1972, Transverse vibration of double-tapered cantilever beams, Journal of Acoustics Society of America 5: 1771-1775.
[16] Goel R.P., 1976, Transverse vibrations of tapered beams, Journal of Sound and Vibration 47: 1-7.
[17] Downs B., 1977, Transverse vibration of cantilever beams having unequal breadth and depth tapers, Journal of Applied Mechanics 44: 737-742.
[18] Bailey C.D., 1978, Direct analytical solution to non-uniform beam problems, Journal of Sound and Vibration 56: 501-507.
[19] Gupta A.K., 1985, Vibration of tapered beams, Journal of Structural Engineering 111: 19-36.
[20] Naguleswaran S., 1992, Vibration of an Euler–Bernoulli beam of constant depth and with linearly varying breadth, Journal of Sound and Vibration 153: 509-522.
[21] Naguleswaran S., 1994, A direct solution for the transverse vibration of Euler–Bernoulli wedge and cone beams, Journal of Sound and Vibration 172: 289-304.
[22] Abrate S., 1995, Vibration of non-uniform rods and beams, Journal of Sound and Vibration 185: 703-716.
[23] Laura P.A.A., 1996, Gutierrez R.H., Rossi R.E., Free vibration of beams of bi-linearly varying thickness, Ocean Engineering 23: 1-6.
[24] Datta A.K., Sil S.N., 1996, An analysis of free undamped vibration of beams of varying cross-section, Computers & Structures 59: 479-483.
[25] Hoffmann J.A., Wertheimer T., 2000, Cantilever beam vibration, Journal of Sound and Vibration 229: 1269-1276.
[26] Genta G., Gugliotta A., 1988, A conical element for finite element rotor dynamics. Journal of Sound and Vibration 120(l): 175-182.
[27] Attarnejad R., Manavi N., Farsad A., 2006, Exact solution for the free vibration of tapered bam with elastic end rotational restraints, Chapter Computational Methods 1993-2003.
[28] Torabi K., Afshari H., Zafari E., 2012, Transverse Vibration of Non-uniform Euler-Bernoulli Beam, Using Differential Transform Method (DTM), Applied Mechanics and Materials 110-116: 2400-2405.
[29] Yan W., Kan Q., Kergrene K., Kang G., Feng X.Q., Rajan R., 2013, A truncated conical beam model for analysis of the vibration of rat whiskers, Journal of Biomechanics 46: 1987-1995.
[30] Boiangiu M., Ceausu V., Untaroiu C.D., 2014, A transfer matrix method for free vibration analysis of Euler-Bernoulli beams with variable cross section, Journal of Vibration and Control 22: 2591-2602.
[31] Lau J.H., 1984, Vibration frequencies for a non-uniform beam with end mass, Journal of Sound and Vibration 97: 513-521.
[32] Grossi R.O., Aranda A., 1993, Vibration of tapered beams with one end spring hinged and the other end with tip mass, Journal of Sound and Vibration 160: 175-178.
[33] Auciello N.M., 1996, Transverse vibration of a linearly tapered cantilever beam with tip mass of rotatory inertia and eccentricity, Journal of Sound and Vibration 194: 25-34.
[34] Wu J.S., Chen C.T., 2005, An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia, Journal of Sound and Vibration 286: 549-568.
[35] Auciello N.M., Maurizi M.J., 1997, On the natural vibration of tapered beams with attached inertia elements, Journal of Sound and Vibration 199: 522-530.
[36] Wu J.S., Hsieh M., 2000, Free vibration analysis of a non-uniform beam with multiple point masses, Structural Engineering and Mechanics 9: 449-467.
[37] Wu J.S., Lin T.L., 1990, Free vibration analysis of a uniform cantilever beam with point masses by an analytical-and numerical- combined method, Journal of Sound and Vibration 136: 201-213.
[38] Wu J.S., Chen D.W., 2003, Bending vibrations of wedge beams with any number of point masses, Journal of Sound and Vibration 262: 1073-1090.
[39] Caddemi S., Calio I., 2008, Exact solution of the multi-cracked Euler–Bernoulli column, International Journal of Solids and Structures 45: 1332-1351.
[40] Caddemi S., Calio I., 2009, Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks, Journal of Sound and Vibration 327: 473-489.
[41] De Silva C.W., 2000, Vibration: Fundamentals and Practice, CRC Press.
[42] Karman T.V., Biot M.A., 1940, Mathematical Methods in Engineering, McGraw-Hill, New York.
[43] Lighthill M.J., 1958, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, London.
[44] Colombeau J.F., 1984, New Generalized Functions and Multiplication of Distribution, North-Holland, Amsterdam.