Nonlinear stochastic analysis of subharmonic response of a shallow cable

Abstract

The paper deals with the subharmonic response of a shallow cable due to time variations of the chord length of the equilibrium suspension, caused by time varying support point motions. Initially, the capability of a simple nonlinear two-degree-of-freedom model for the prediction of chaotic and stochastic subharmonic response is demonstrated upon comparison with a more involved model based on a spatial finite difference discretization of the full nonlinear partial differential equations of the cable. Since the stochastic response quantities are obtained by Monte Carlo simulation, which is extremely time-consuming for the finite difference model, most of the results are next based on the reduced model. Under harmonical varying support point motions the stable subharmonic motion consists of a harmonically varying component in the equilibrium plane and a large subharmonic out-ofplane component, producing a trajectory at the midpoint of shape as an infinity sign. However, when the harmonical variation of the chordwise elongation is replaced by a narrow-banded Gaussian excitation with the same standard deviation and a centre frequency equal to the circular frequency of the harmonic excitation, the slowly varying phase of the excitation implies that the phase difference between the in-plane and outof-plane displacement components is not locked at a fixed value. In turn this implies that the trajectory of the displacement components is slowly rotating around the chord line. Hence, a large subharmonic response component is also present in the static equilibrium plane. Further, the time variation of the envelope process of the narrow-banded chordwise elongation process tends to enhance chaotic behaviour of the subharmonic response, which is detectable via extreme sensitivity on the initial conditions, or via the sign of a numerical calculated Lyapunov exponent. These effects have been further investigated based on periodic varying chord elongations with the same frequency and standard deviation as the harmonic excitation, for which the amplitude varies in a well-defined way between two levels within each period. Depending on the relative magnitude of the high and low amplitude phase and their relative duration the onset of chaotic vibrations has been verified.

Stochastic and chaotic sub- and superharmonic response of shallow cables due to chord elongations

“…As seen, the non-dimensional chord elongation e(t) is exposing the system both to external and parametric excitation. Based on simulations with a full non-linear finite difference model, it was demonstrated by Zhou et al [12] that the indicated two degree-of-freedom model was adequate in predicting to a high degree of accuracy the qualitative, and even quantitative, dynamic responses and chaotic behavior of the cable as seen from the midpoint trajectories under stochastic excitation, as shown in Fig. 3, and from the Poincare-maps of the in-plane modal coordinate shown in Fig.…”

“…A theory for determining the probability of occupying either of these modes of vibration was derived based on a continuous time two-state Markov chain model. The corresponding stochastic resonance of order 2:1 was investigated by Zhou et al [12] based on analytical solutions for the deterministic ordered response, and the subharmonic stochastic response was analyzed by Monte Carlo simulations. It was found that the stochastic variations of the chord elongation enhanced the tendency to chaotic response relative to the comparable harmonic excitation, and that the coupled mode of vibration only exists for bandwidths below a certain critical value.…”

A generalized frequency detuning method for multidegree-of-freedom oscillators with nonlinear stiffness

“…option N2, (27), must be applied). From β 1 it can be seen that the resonant terms are [1,2], [1,4], [1,5], [1,9], [1,12] and [1,15] for mode 1 and [2, 1], [2,6], [2,7], [2,11], [2,18] and [2,19] for mode 2. Applying option N2 to these terms gives the transformed equations of motionü…”

“…These expressions have come from terms [1,1] and [1,11] in β 1 for mode 1 (note that terms [1,6], [1,7], [1,18] and [1,19] also result in a response at 3Ω however the corresponding terms in n * u1 are zero) and terms [2,2], [2,4], [2,5], [2,9], [2,12] and [2,15] for mode 2. The resulting amplitudes of these sinusoidal responses are X 1,3Ω and X 2,Ω respectively, where to calculate X 2,Ω (54) is used.…”

“…In terms of practical motivation, resonances between primary and/or secondary resonant frequencies are important for a wide range of physical applications -see for example [1][2][3][4][5][6][7]. Typically models of the these type of systems are in the form of weakly nonlinear oscillators.…”