Miles' evolution equations for axisymmetric Shells: Simple strange attractors in structural dynamics

Maewal, A.

doi:10.1016/0020-7462(86)90039-9

Cited by 20 publications

(6 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…M any of the im portant features of the solutions for the string system are found for every m em ber of the family. W orks of Miles (19846, c), Maewal (1986, 1987), and Funakoshi & Inoue (1988 have shown th a t both Rdssler and Lorenz type chaotic a ttra c to rs exist throughout the family.…”

Section: (G) Connections To Similar Systemsmentioning

confidence: 99%

“…In recent years m any investigators, including Miles (1984a, c), Maewal (1986Maewal ( , 1987, Johnson & Bajaj (1989) and Tousi & Bajaj (1985), have found cascades of period-doubling bifurcations (or torus-doubling bifurcations, if the averaged equations are cast in the form of equations ( 29)) leading to chaos in the averaged system . This implies a series of torus-doubling bifurcations in the corresponding nonautonom ous systems.…”

Section: Resonant Motion In Stretched Strings (B)averaging Revisitedmentioning

confidence: 99%

“…1 II was pointed out by Miles (1084a-c) and Maewal (1086Maewal ( , 1987 th a t the equations describing the am plitude dynamics of the single-mode truncation of a stiiim are ar1q0J m SC rC W6akly nonlinear r««>nant motions of. a spherical pendulum 088 1900? '…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the amplitude dynamics and crisis in resonant motion of stretched strings

Bajaj

Johnson

1992

Phil. Trans. R. Soc. Lond. A

View full text Add to dashboard Cite

An N -mode truncation of the equations governing the resonantly forced nonlinear motions of a stretched string is studied. The external forcing is restricted to a plane, and is harmonic with the frequency near a linear natural frequency of the string. The method of averaging is used to investigate the weakly nonlinear dynamics. By using the amplitude equations, which are a function of the damping and the frequency of excitation, it is shown that to O(e) , only the resonantly forced mode has non-zero amplitude. Both planar (i.e. lying in the planar of forcing) and non-planar constant solutions are studied and amplitude-frequency curves are determined. For small enough damping, solutions in the non-planar branch become unstable via a Hopf bifurcation and give rise to a branch of periodic solutions in the amplitude-frequency plane. This branch exhibits several period-doubling bifurcations, but does not directly result in the formation of a chaotic attractor. At lower values of damping, many other branches of periodic solutions exist. A series of bifurcations leads to the formation of chaotic attractors in some of these solution branches. Various types of interactions between the different solutions are found to result in many interesting phenomena including, the formation of a homoclinic orbit and chaos quenching. These results are discussed in the context of the Sil’nikov mechanism near a homoclinic orbit for a saddle-focus. Results from the investigations of the averaged system are interpreted for the truncated string system using the averaging theory and the theory of integral manifolds. Numerical investigations with the single mode truncation of the non-autonomous string system show that there is a good correspondence even between chaotic solutions of the averaged system and those of the original system.

show abstract

Section: (G) Connections To Similar Systemsmentioning

confidence: 99%

Section: Resonant Motion In Stretched Strings (B)averaging Revisitedmentioning

confidence: 99%

See 1 more Smart Citation

On the amplitude dynamics and crisis in resonant motion of stretched strings

Bajaj

Johnson

1992

Phil. Trans. R. Soc. Lond. A

View full text Add to dashboard Cite

show abstract

“…The equations of motions of the fluid-filled thinwalled structure are derived by Amabili and co-authors [15].The bifurcation behavior of the travelling waves is analyzed. Maewal [16] analyzed the large-amplitude oscillations of cylindrical shells using the Sanders-Koiter nonlinear theory. Nayfeh, Raouf [17] studied the nonlinear vibrations of cylindrical shells by using the perturbation analysis and the planestrain shell theory.…”

Section: Introductionmentioning

confidence: 99%

Free nonlinear oscillations of cantilever circular cylindrical shells

Аврамов

2021

Z Angew Math Mech

View full text Add to dashboard Cite

show abstract

“…modal equations to four first-order differential equations representing the slow-time evolution of amplitudes of harmonic motion of the two modes. These amplitude or averaged equations are a generalization of those that describe the motion of square plates [12] and membranes [10] and, when the additional restriction of circular symmetry is imposed, they have arisen in the study of resonant motions of a spherical pendulum [4], a stretched string [11], and forced response of axisymmetric shells [15] and beams [9]. The amplitude equations for the rectangular plate depend on three nonlinear coefficients, in contrast to the two independent nonlinear coefficients found in the above mentioned studies.…”

Section: Introductionmentioning

confidence: 99%

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

1993

View full text Add to dashboard Cite

Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1:1 resonance in the (1, 2) and (3, 1) plate modes.

show abstract

Miles' evolution equations for axisymmetric Shells: Simple strange attractors in structural dynamics

Cited by 20 publications

References 3 publications

On the amplitude dynamics and crisis in resonant motion of stretched strings

On the amplitude dynamics and crisis in resonant motion of stretched strings

Free nonlinear oscillations of cantilever circular cylindrical shells

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

Contact Info

Product

Resources

About