Keywords: vibration, Mindlin plates, modal stress-resultants, analytical methods, Ritz method, least squares curve fitting.

The Kirchhoff analogy between the equilibrium of a 3D originally straight uniform rod and the dynamics of a spinning top is well known and used extensively. The rod is described by spatial coordinate and the spinning top is described by time coordinate. The extended Kirchhoff analogy between the spatial equilibrium of a 3D force-free buckled elastica and the temporal dynamics of the torque-free gyrostat is less well known and hardly used. The extended Kirchhoff analogy and the Melnikov integral are used to determine analytically the conditions for the possible onset of spatial chaos in the elastica by exploring the Hamiltonian structure of the rotational motion of a perturbed gyrostat. The analytical results are cross-checked by the 7-8th order Runge-Kutta algorithm to numerically integrate the governing equations of the 3D equilibrium of the elastica. Interesting spatial buckling patterns are depicted the first time. Apprehension of the complex deformations of the elastica under different load conditions is of both theoretical and practical interest. The simulation results show that there exist homoclinic/heteroclinic bifurcations to chaos in the equilibrium of the elastica under the appropriate load conditions, equivalently, boundary conditions.

Based upon the extended Kirchhoff analogy between the temporal dynamics of the gyrostat and the spatial equilibrium of the elastica, chaotic dynamics of the modern nonlinear dynamical system are employed to investigate computationally and analytically the 3-D complex deformations of the centerline of the elastica structure with spatially periodic varying bending modulii and torsion modulus, and naturally initial curves under stress-free conditions. The elastica structures may be used to model mathematically the super coiling DNA in biology, videotape, the satellite tether and cable dynamics. The complex static configuration of the disturbed elastica under appropriate load conditions is yielded. According to the Melnikov theory of chaotic dynamics, the existence of the simple zeros with respect to the initial space coordinate s₀ of the vanishing Melnikov's integral means that the existence of the transversal intersections of the stable and unstable manifolds of the disturbed elastica, one can deduce that the chaotic deformations are possible in the investigated elastica structures. For the elastica with small, initial stress- free curvatures and twist, the Melnikov integral based upon the heteroclinic orbits of the single spinning rigid-body under torque-free motions is applied to be the criterion for the judgment of the onset of the chaotic deformations of the elastica under force-free conditions at each section along the elastica. For the elastica with large, initial stress-free curvatures and twist, the Melnikov integral based upon the homoclinic orbits of the spinning gyrostat under torque-free motions is used to be the criterion for the judgment of the occurrences of the chaotic deformations of the disturbed elastica under force-free conditions at each section along the elastica. The numerical simulation results agree with the claims made by the use of the Melnikov integral.

The issue on the stability/instability of these static solutions is more than academic. Under the action of the disturbance torques or forces, these deformation patterns leading to much more comprehensive contact between one part of the elastica with another part of it through loop deformations make the stability/instability problem much more sophisticated. El Nachie and Kapitaniak established some connections between the purely spatial chaos of loop and envelope soliton localization in long planar elastic strings in the application of molecular biology. Shi and Hearst consider the equilibrium of the elastica under the most general self-contact force in terms of the nonlinear Schrödinger equation. In addition, a few comments are made as follows.

Comment 1:

the governing equations of equilibrium of the elastica are pure Hamiltonian systems with zero dissipation and do not possess any attractor. Thus, ordinary integration algorithms sometimes introduce artificial damping no matter how much the step-size is selected when the length of the centerline of the elastica is greater than certain lengths. Thus, some numerical results may not be always reliable (Parker and Chua; Chen and Leung). This is the reason why one needs to develop the Melnikov integral as a potential criterion to check the chaotic deformations of the disturbed elastica. In this work, the elastica cross-sectional moment is also checked.

Comment 2:

The nominal physical parameters corresponding to the disturbance-free case (cf. ) are selected so that the homoclinic / heteroclinic orbits are obtained reasonably. This is the prerequisite for one to use the Melnikov integral properly.

Comment 3:

The Melnikov integral is established for detecting successfully the transversal intersections of the stable and unstable manifolds of the force- free elastica. The existence of the real zeros of the vanishing Melnikov's integral with respect to the initial distance s₀ means the existence of the chaotic deformations in the sense of Smale's horseshoes. This is a necessary condition for the chaotic deformations to take place.

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