Abstract
In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlinear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of multiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequencies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency–amplitude and frequency–response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.
Similar content being viewed by others
References
Woinowsky-Krieger S.: The effect of an axial force on the vibration of hinged bars. ASME J. Appl. Mech. 38, 35–36 (1950)
Srinivasan A.H.: Large amplitude-free oscillations of beams and plates. AIAA J. 3, 1951–1953 (1965)
Wrenn B.G., Mayers J.: Nonlinear beam vibration with variable axial boundary restraint. AIAA J. 8, 1718–1720 (1970)
Nayfeh A.H., Mook D.T.: Nonlinear Oscillations. Willey, New York (1979)
Nayfeh A.H., Nayfeh J.F., Mook D.T.: On methods for continuous systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 3, 145–162 (1979)
Pakdemirli M., Nayfeh A.H.: Nonlinear vibration of beam-spring-mass system. J. Vib. Acoust. 166, 433–438 (1994)
Pakdemirli M.: Vibrations of continuous systems with a general operator notation suitable for perturbative calculations. J. Sound Vib. 246(5), 841–851 (2001)
Özkaya E., Pakdemirli M., Öz H.R.: Nonlinear vibrations of a beam-mass system under different boundary conditions. J. Sound Vib. 199(4), 679–696 (1997)
Özkaya E.: Linear transverse vibrations of a simply supported beam carrying concentrated mass. Math. Comput. Appl. 6(3), 147–151 (2001)
Rao G.V., Saheb K.M., Janardhan G.R.: Fundamental frequency for large amplitude vibrations of uniform timoshenko beams with central point concentrated mass using coupled displacement field method. J. Sound Vib. 298, 221–232 (2006)
Abe A.: On nonlinear analysis of continuous systems with quadratic and cubic nonlinearities. Nonlinear Mech. 41, 873–879 (2006)
Rehfield L.W.: Nonlinear flexural oscillation of shallow arches. Am. Inst. Aeronaut. Astronaut. J. 12, 91–93 (1974)
Singh P.N., Ali S.M.J.: Nonlinear vibration of a moderately thick shallow arches. J. Sound Vib. 41, 275–282 (1975)
Yamaki N., Mori A.: Nonlinear vibrations of a clamped beam with initial deflection and initial axial displacement. J. Sound Vib. 71, 333–346 (1980)
Krishnan A., Suresh Y.J.: A simple cubic linear element for static and free vibration analyses of curved beam. Comput. Struct. 68, 473–489 (1998)
Öz H.R., Pakdemirli M., Özkaya E., Yilmaz M.: Nonlinear vibrations of a slightly curved beam resting on a nonlinear elastic foundation. J. Sound Vib. 212(1), 295–309 (1998)
Lacarbonara W., Yabuno H.: Closed-loop nonlinear control of an initially imperfect beam with non-collocated input. J. Sound Vib. 273, 695–711 (2004)
Nayfeh A.H.: Introduction to Perturbation Techniques. Willey, New York (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Özkaya, E., Sarigül, M. & Boyaci, H. Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass. Acta Mech Sin 25, 871–882 (2009). https://doi.org/10.1007/s10409-009-0275-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-009-0275-1