Abstract
In this paper, the nonlinear planar response of a hinged–hinged buckled beam to a primary-resonance excitation of its first vibration mode is computed by a new numerical scheme. The beam is subjected to an axial force beyond the critical load of the first buckling mode and to a transverse harmonic excitation. The nonlinear dynamical problem is solved by deducing directly the discretized equations governing the problem thanks to a new approach, here called DQ based approach, since it is based on the application of the quadrature rules of the DQM. As it will be shown, for the problem here considered, the minimum number of degrees of freedom to be retained to limit the numerical errors is four. Computer simulations of the dynamic behaviour of the discretized system are conducted by means of the IDQ method, a method proposed and recently generalized by the author. A sequence of supercritical period-doubling bifurcations leading to chaos, snapthrough motions and quasi-periodic motions can be observed, similarly to some cases existing in literature.
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References
Tseng, W.Y., Dugundji, J.: Nonlinear vibrations of a buckled beam under harmonic excitation. J. Appl. Mech. 38, 467–476 (1971)
Moon, F.C., Holmes, P.J.: A magnetoelastic strange attractor. J. Sound Vib. 65, 275–296 (1979)
Holmes, P.J.: A nonlinear oscillator with a strange attractor. Phil. Trans. R. Soc. Lond. 292, 419–448 (1979)
Holmes, P., Marsden, J.: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Ration. Mech. Anal. 76, 135–165 (1981)
Holmes, P.J., Moon, F.C.: Strange attractor and chaos in nonlinear mechanics. J. Appl. Mech. 50, 1021–1032 (1983)
Tang, D.M., Dowell, E.H.: On the threshold force for chaotic motions for a forced buckled beam. J. Appl. Mech. 55, 190–196 (1988)
Seydel, R.: From Equilibrium to Chaos — Practical Bifurcation and Stability Analysis. Elsevier, New York (1988)
Hanagaud, S., Abhyankar, N.S., Chander, R.: Studies in chaotic vibrations of buckled beams. Appl. Mech. Rev. 42(11), 100–107 (1989)
Reynolds, T.S., Dowell, E.H.: The role of higher modes in the chaotic motion of the buckled beam-I. Int. J. Non-Linear Mech. 31, 931–939 (1996)
Reynolds, T.S., Dowell, E.H.: The role of higher modes in the chaotic motion of the buckled beam-II. Int. J. Non-Linear Mech. 31, 941–950 (1996)
Abou-Rayan, A.M., Nayfeh, A.H., Mook, D.T.: Nonlinear response of a parametrically excited buckled beam. Nonlinear Dyn. 4, 499–525 (1993)
Nayfeh, A.H., Lacarbonara, W.: On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997)
Kreider, W., Nayfeh, A.H.: Experimental investigation of single-mode responses in a fixed-fixed buckled beam. Nonlinear Dyn. 15, 155–177 (1998)
Emam, S.A., Nayfeh, A.H.: On the nonlinear dynamics of a buckled beam subjected to a primary-resonance excitation. Nonlinear Dyn. 35(1), 1–17 (2004)
Sinha, S.C., Senthilnathan, N.R., Pandiyan, R.: A new numerical technique for the analysis of parametrically excited nonlinear systems. Nonlinear Dyn. 4, 483–498 (1993)
Krishnamurthy, K., Burton, T.D., Zeller, L.: Finite element analysis of nonlinear oscillators. Int. J. Numer. Methods Eng. 21, 409–420 (1985)
Samoilenko, A.M., Ronto, N.I.: Numerical-Analytic Methods of Investigating Periodic Solutions. Mir Publishers, Moscow (1979)
Villadson, J., Michelson, M.L.: Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall, New Jersey (1978)
Tomasiello, S.: A generalization of the IDQ method and a DQ based approach to approximate non-smooth solutions. J. Sound Vib. To appear (2006)
Tomasiello, S.: Simulating non-linear coupled oscillators by an iterative differential quadrature method. J. Sound Vib. 265, 507–525 (2003)
Tomasiello, S.: Stability and accuracy of the iterative differential quadrature method. Int. J. Numer. Methods Eng. 58, 1277–1296 (2003)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1987)
Jang, S.K., Bert, C.W., Striz, A.G.: Application of differential quadrature method to deflection and buckling of structural components. Int. J. Numer. Methods Eng. 28, 561–577 (1989)
Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49(1), 1–28 (1996)
Shu, C., Du, H.: Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates. Int. J. Solid Struct. 34, 819–835 (1997)
Shu, C., Chen, W.: On optimal selection of interior points for applying discretized boundary conditions in DQ vibration analysis of beams and plates. J. Sound Vib. 222(2), 239–257 (1999)
Nayfeh, A.H., Kreider, W., Anderson, T.J.: Investigation of natural frequencies and mode shapes of buckled beams. AIAA J. 33, 1121–1126 (1995)
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Tomasiello, S. A DQ based approach to simulate the vibrations of buckled beams. Nonlinear Dyn 50, 37–48 (2007). https://doi.org/10.1007/s11071-006-9141-x
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DOI: https://doi.org/10.1007/s11071-006-9141-x