Abstract
The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.
Highlights
Aerial ropeways are transportation systems which are becoming increasingly popular in recent years, in mountain regions with ski resorts and in sightseeing areas, and in urban environments [1]
Some works [4,5,6] investigated the dynamic behavior of carrying hauling ropes, dealing with the study of the effects of moving loads in an existing ropeway system, while [7] investigated the nonlinear coupling between the motion of the hauling cable and the swaying dynamics of the cabins in bi-cable circulating gondola ropeway systems
By defining the relative phase γm(t2) = σt2 − θm(t2), with σ being a detuning parameter, it turns out that the fixed points of the dynamic system are obtained when ∂2am = 0 and ∂2γm = 0; the detuning can be calculated from Equation (21) as σ = Γm a2m and the frequency of the mth nonlinear normal mode can be expressed in terms of the real modal amplitude am as: ωmnl = ωm + Γm a2m
Summary
Aerial ropeways are transportation systems which are becoming increasingly popular in recent years, in mountain regions with ski resorts and in sightseeing areas, and in urban environments [1]. The present work aims at filling the lack of knowledge regarding the characterization of the nonlinear dynamic features of roller battery systems and to investigate the effects of the periodic excitations induced by the moving cable that may give rise to resonance phenomena and nonlinear modal interactions These phenomena were studied in the past in different mechanical systems, such as cranes [16,17,18,19], tethering [20], pendulum systems [21,22], piezoelectric beams [23,24], and plates [25,26], or, more generally, in slender structures possessing strong geometric nonlinearities [27,28,29]. A thorough bifurcation analysis, carried out via path-following of the fixed points of the modulation equations shed light onto several interesting features of the roller battery singleand coupled-mode responses
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