Summary
Periodic structures with cyclic symmetry are often used as idealized models of physical systems and one such model structure is considered. It consists ofn identical particles, arranged in a ring, interconnected by extensional springs with nonlinear stiffness characteristics, and hinged to the ground individually by nonlinear torsional springs. These cyclic structures that, in their linear approximations, are known to possess pairwise double degenerate natural frequencies with orthogonal normal modes, are studied for their forced response when nonlinearities are taken into account. The method of averaging is used to study the nonlinear interactions between the pairs of modes with identical natural frequencies. The external harmonic excitation is spatially distributed like one of the two modes and is orthogonal to the other mode. A careful bifurcation analysis of the amplitude equations is undertaken in the case of resonant forcing. The response of the structure is dependent on the amplitude of forcing, the excitation frequency, and the damping present. For sufficiently large forcing, the response does not remain restricted to the directly excited mode, as both the directly excited and the orthogonal modes participate in it. These coupled-mode responses arise due to pitchfork bifurcations from the single-mode responses and represent traveling wave solutions for the structure. Depending on the amount of damping, the coupled-mode responses can undergo Hopf bifurcations leading to complicated amplitude-modulated motions of the structure. The amplitude-modulated motions exhibit period-doubling bifurcations to chaotic amplitude-modulations, multiple chaotic attractors as well as “crisis”. The existence of chaotic amplitude dynamics is related to the presence of Sil'nikov-type conditions for the averaged equations.
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References
Nayfeh, A. H., Balachandran, B.: Modal interaction in dynamical and structural systems. Appl. Mech. Rev.42, 175–201 (1989).
Tousi, S., Bajaj, A. K.: Period-doubling bifurcations and modulated motions in forced mechanical systems. ASME J. Appl. Mech.52, 446–452 (1985).
Miles, J. W.: Resonant motion of a spherical pendulum. Physica D11, 309–323 (1984).
Miles, J. W.: Resonant, nonplanar motion of a stretched string. J. Acous. Soc. Am.15, 1505–1510 (1984).
Johnson, J. M., Bajaj, A. K.: Amplitude modulated and chaotic dynamics in resonant motion of strings. J. Sound Vib.128, 87–107 (1989).
Maewal, A.: Chaos in a harmonically excited elastic beam. ASME J. Appl. Mech.53, 625–632 (1986).
Yang, X. L., Sethna, P. R.: Nonlinear phenomena in forced vibrations of a nearly square plate-antisymmetric case. J. Sound Vib.155, 413–441 (1992).
Ewins, D. J.: Vibration characteristics in bladed disk assemblies. J. Mech. Eng. Sci.15, 165–186 (1973).
Thomas, D. L.: Standing waves in rotationally periodic structures. J. Sound Vib.37, 288–290 (1974).
Wildheim, S. J.: Excitation of rotationally periodic structures. ASME J. Appl. Mech.46, 878–882 (1979).
Muszynska, A., Jones, D. I. G.: A parametric study of dynamic response of a discrete model of turbomachinery bladed disk. ASME J. Vib. Acous. Str. Rel. Des.105, 434–443 (1983).
Wei, S.-T., Pierre, C.: Localization phenomena in mistuned assemblies with cyclic symmetry. Part I: free vibration. ASME J. VIB. Acous. Str. Rel. Des.110, 429–438 (1988).
Bendiksen, O. O.: Mode localization phenomena in large space structures. AIAA J.25, 1241–1248 (1987).
Tobias, S. A., Arnold, R. N.: The influence of dynamical imperfection on the vibration of rotating disks. Proc. Inst. Mech. Eng.171, 669–690 (1957).
Tobias, S. A.: Free undamped non-linear vibrations of imperfect circular disks. Proc. Inst. Mech. Eng.171, 691–7021 (1957).
Vakakis, A. F.: Dynamics of a nonlinear periodic structure with cyclic symmetry. Acta Mech.95, 197–226 (1992).
Bajaj, A. K., Johnson, J. M.: On the amplitude dynamics and crisis in resonant motion of stretched strings. Phil. Trans. Roy. Soc. London Ser.338, 1–41 (1992).
Doedel, E.: AUTO: Software for continuation and bifurcation problems in ordinary differential equations. Report Dep. Appl. Math. Cal. Inst. Tech. (1986).
Miles, J. W.: Resonantly forced surface waves in a circular cylinder. J. Fluid Mech.149, 15–31 (1984).
Maewal, A.: Miles' evolution equations for axisymmetric shells: simple strange attractors in structural dynamics. Int. J. Nonlin. Mech.21, 433–438 (1987).
Nayfeh, A. H., Mook, D. T.: Nonlinear oscillations. New York: Wiley-Interscience 1979.
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vectorfields. New York: Springer 1983.
Grebogi, C., Ott, E., Yorke, J. A.: Crises, sudden changes in chaotic attractors, and transient chaos. Physica D7, 181–200 (1983).
Benettin, G., Galgani, L., Giorgilli, A., Streleyn, J. M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Meccanica15, 9–30 (1986).
Kaplan, J. L., Yorke, J. A.: Chaotic behavior in multidimensional difference equations. In: Lect. Notes in Math.730, (Peitgen, H. O., Walther, H. O., eds.), pp. 228–237. New York: Springer 1979.
O'Reilly, O. M.: Global bifurcations in the forced response of a damped string. Int. J. Nonlinear Mech.28, 337–351 (1993).
Funakoshi, M., Inone, S.: Surface waves due to resonant horizontal oscillation. J. Fluid Mech.192, 219–247 (1988).
Knobloch, E., Weiss, N. O.: Bifurcations in a model of magnetoconvection. Physica D9, 379–407 (1983).
Sil'nikov, L. P.: A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type. Math. USSR Sbornik10, 91–102 (1970).
Glendinning, P., Sparrow, C.: Local and global behavior near homoclinic orbits. J. Stat. Phys.35, 645–696 (1984).
Wiggins, S.: Global bifurcations and chaos. New York: Springer 1988.
Glendinning, P.: Bifurcations near homoclinic orbits with symmetry. Phys. Lett.103A, 163–166 (1984).
Gambaudo, J. M., Glendinning, P.A., Tresser, C.: Stable cycles with complicated structure. J. Physique Lett.46, 653–657 (1985).
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Samaranayake, S., Bajaj, A.K. & Nwokah, O.D.I. Amplitude modulated dynamics and bifurcations in the resonant response of a structure with cyclic symmetry. Acta Mechanica 109, 101–125 (1995). https://doi.org/10.1007/BF01176819
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DOI: https://doi.org/10.1007/BF01176819