Planar non-linear oscillations of elastic cables under subharmonic resonance conditions

Numerical simulations of chaotic dynamics in a model of an elastic cable

“…Thus, for the cable, due to the high values of the nonlinear coefficients, chaos develops at values of the forcing amplitude notably lower than those in [7], which is of practical interest. As the forcing amplitude increases, the region of period 1 response extends notably to the right, contrary to the prediction of the second-order subharmonic perturbation solution [6] and consistent with the fourth-order primary perturbation solution [5], which predicts more and more pronounced bending to the right of the frequency-response curve with increasing excitation amplitude. Close to the right neighbourhood of the chaotic region, a band of parameter values where regular motions with period other than 2 occur is found.…”

“…Regions of existence or non-existence of finite-amplitude stable subharmonic oscillations with period 2 were obtained in the parameter space (f/, p) of the excitation. For prestressed cables subjected to uniform forcing and vibrating with the first symmetric mode, which is the first mode of a cable with sag-to-span ratio up to about 1/20 and technical values of EA/H (~500), those regions are plotted in Figure 2 In the case of a sagged cable that exhibits a frequency-response curve of the softening type, the approximate solution shows possible existence of a finite response with period 2 at frequencies notably less than the subharmonic-resonance value, a physically unrealistic behaviour which may be due to the order considered in the perturbation analysis [6]. Indeed, some numerical integrations of equation (4) showed that, when decreasing [1 with p =/~ and fixed initial conditions (q = c~ = 0), the response is actually of period 2 up to a certain frequency (fZ ~ 1.7) and then it becomes of period 1, as one would expect since the fundamental harmonic firmly prevails in the response right of the primary resonance condition.…”

“…Moreover, using just the same nondimensionalization for the displacement as considered herein, we obtained [6] approximate steady-state solutions to equation (4) in the neighbourhood of the subharmonic resonance of order one half (f~ ~ 2) by a second-order perturbation approach. Regions of existence or non-existence of finite-amplitude stable subharmonic oscillations with period 2 were obtained in the parameter space (f/, p) of the excitation.…”

“…Most of them deal with a single cable for which simple models, with one or two degrees-of-freedom, have been developed and used to obtain an analytical solution of the problem and an effective description of the nonlinear dynamic behaviour. Free planar and nonplanar motions were studied widely [1][2][3], while relatively few works were devoted to forced motions [4][5][6]. The main features of the relevant dynamical problems are associated with the presence of both quadratic and cubic nonlinearities in the equations of motion, the former is due to the cable initial curvature and the latter is due to stretching of the cable axis.…”

Coupled Nonlinear Dynamics of Geometrically Imperfect Shear Deformable Extensible Microbeams

“…5. As seen, due to increased amplitude of the initial imperfection, the nonlinear resonant response of the system is a softening-hardening type, showing the simultaneous effect of the quadratic and cubic nonlinear terms in the equation of motion-such a behavior has already been reported by researchers for macrosystems, for instance by Benedettini and Rega [37,38] for suspended cables under primary and subharmonic excitations, showing a qualitative behavior similar to that of the system examined here. The initial softening and the followed hardening behaviors are due to the presence of quadratic and cubic nonlinear terms, respectively, in the equations of motion.…”