Abstract
Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-of-freedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincaré projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Tseng, W.Y., Dugundji, J.: Nonlinear vibrations of a buckled beam under harmonic excitation. ASME J. Appl. Mech. 38, 467–476 (1971)
Yamaki, N., Mori, A.: Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement, part I: theory. J. Sound Vib. 71(3), 333–346 (1980)
Yamaki, N., Otomo, K., Mori, A.: Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement, part II: experiment. J. Sound Vib. 71(3), 347–360 (1980)
Holmes, P.J.: A nonlinear oscillator with a strange attractor. Philos. Trans. R. Soc. Lond. A 292, 419–448 (1979)
Moon, F.C., Holmes, P.J.: The magnetoelastic strange attractor. J. Sound Vib. 65(2), 276–296 (1979)
Pezeshki, C., Dowell, E.H.: Generation and analysis of Lyapunov exponents for the buckled beam. Int. J. Non-Linear Mech. 24(2), 79–97 (1989)
Nagai, K.: Nonlinear vibrations of a shallow arch under periodic lateral force (Theory). Trans. Jpn. Soc. Mech. Eng., C 51(471), 2820–2827 (1985) (in Japanese)
Nagai, K.: Nonlinear vibrations of a shallow arch under periodic lateral force (2nd Report, Experiment). Trans. Jpn. Soc. Mech. Eng., C 52(484), 3047–3054 (1986) (in Japanese)
Nagai, K.: Experimental study of chaotic vibration of a clamped beam subjected to periodic lateral forces. Trans. Jpn. Soc. Mech. Eng., C 56(525), 1171–1177 (1990) (in Japanese)
Nagai, K., Yamaguchi, T.: Chaotic vibrations of a post-buckled beam carrying a concentrated mass (1st Report, Experiment). Trans. Jpn. Soc. Mech. Eng., C 60(579), 3733–3740 (1994) (in Japanese)
Yamaguchi, T., Nagai, K.: Chaotic vibrations of a post-buckled beam carrying a concentrated mass (2nd Report, Theoretical analysis). Trans. Jpn. Soc. Mech. Eng., C 60(579), 3741–3748 (1994) (in Japanese)
Yamaguchi, T., Nagai, K.: Chaotic oscillations of a shallow arch with variable cross section subjected to periodic excitation. Trans. Jpn. Soc. Mech. Eng., C 61(583), 799–807 (1995) (in Japanese)
Maruyama, S., Nagai, K., Yamaguchi, T., Hoshi, K.: Contribution of multiple vibration modes to chaotic vibrations of a post-buckled beam with an axial elastic constraint. J. Syst. Des. Dyn. 2(3), 738–749 (2008)
Yanagisawa, D., Nagai, K., Maruyama, S.: Chaotic vibrations of a clamped-supported beam with a concentrated mass subjected to static axial compression and periodic lateral acceleration. J. Syst. Des. Dyn. 2(3), 762–773 (2008)
Nagai, K., Maruyama, S., Sakaimoto, K., Yamaguchi, T.: Experiments on chaotic vibrations of a post-buckled beam with an axial elastic constraint. J. Sound Vib. 304, 541–555 (2007)
Wang, F., Bajaj, A.K.: Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure. Nonlinear Dyn. 47, 25–47 (2007)
Warminski, J., Cartmell, M.P., Bochenski, M., Ivanov, I.: Analytical and experimental investigations of an autoparametric beam structure. J. Sound Vib. 315, 486–508 (2008)
Nayfeh, A.H., Zavodney, L.D.: Experimental observation of amplitude- and phase-modulated responses of two internally coupled oscillators to a harmonic excitation. ASME J. Appl. Mech. 55, 706–710 (1988)
Nayfeh, A.H., Balachandran, B., Colbert, M.A., Nayfeh, M.A.: An experimental investigation of complicated responses of a two-degree-of-freedom structure. ASME J. Appl. Mech. 56, 960–967 (1989)
Nagai, K., Arai, N., Nagaya, K., Takeda, S.: A flexural vibration analysis of a cantilevered beam carrying a concentrated mass by mode shape function approach. Trans. Jpn. Soc. Mech. Eng., C 55(516), 1941–1947 (1989) (in Japanese)
Nagai, K., Nagaya, K., Takeda, S., Arai, N.: A free vibration of beams carrying a concentrated mass under distributed axial forces. Trans. Jpn. Soc. Mech. Eng., C 54(497), 39–46 (1988) (in Japanese)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica 16D, 285–317 (1985)
Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamic systems. Prog. Theor. Phys. 61, 1605–1616 (1979)
Kaplan, J., Yorke, J.: The Lyapunov dimension of strange attractors. J. Differ. Equ. 49, 185–207 (1983)
Loève, M.M.: Probability Theory. Van Nostrand, Princeton (1955)
Feeny, B.F., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211(4), 607–616 (1998)
Yamaguchi, T., Nagai, K., Maruyama, S.: Identification of spatial modes in chaotic vibration involving dynamic snap-through using KL method. Trans. Jpn. Soc. Mech. Eng., C 69(687), 2937–2942 (2003) (in Japanese)
Azeez, M.F.A., Vakakis, A.F.: Proper orthogonal decomposition of a class of vibroimpact oscillations. J. Sound Vib. 240, 859–889 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Onozato, N., Nagai, Ki., Maruyama, S. et al. Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. Nonlinear Dyn 67, 2363–2379 (2012). https://doi.org/10.1007/s11071-011-0151-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0151-y