Abstract
The interaction of a mechanical oscillator with the operation actions and environment fields will give rise to the splitting of normal modes. In this study, we investigate the normal mode splitting behaviour of a moving-particles-pumped single-span clamped-hinged elastic homogeneous beam based on the proposed multi-octave modal parametric oscillation model. Numerical experiments show the entangled and squeezed oscillations of the phase-conjugated waves near the resonance tongues. Theoretical analysis predicts the occurrence of multiple simple resonances and the squeezing of the twin waves modal oscillation in the region with a low speed of movement for the studied system.
Highlights
Where ρA, EI(x), L, and w(x, t) denote the mass per unit length, flexural rigidity, span length, and flexural deformation response of the supporting structure; x is the longitudinal coordinate of the supporting structure, with the origin at the left end-support of the structure; δ is the Dirac delta function; m, qi(t), v, and ti = t − (i − 1)T denote the mass, transverse displacement, travelling speed, and travelling time, respectively, of the i-th particle on the supporting structure;T = when lm = L are discussed
Calculated modal response phase trajectories based on FEM structure modelling around the principal resonance tongue of the fundamental mode under initial disturbance of w1(0) = 1, w 1(0)/ω1 = 1 when μ = 0.1
For μ = 0.1 and β = 1.8, which is found in the principal resonance tongue of the fundamental mode, the computed phase trajectories based on the FEM structure modelling considering a stream of 50 particles passing the beam verify that the amplitude-amplified wave dominates the modal oscillation when w1(0) = 0, w 1(0)/ω1 = 1 and the amplitude-suppressed wave dominates the oscillation when w1(0) = 1, w 1(0)/ω1 = 0
Summary
The interaction of a mechanical oscillator with the operation actions and environment fields will give rise to the splitting of normal modes. We investigate the moving-particles-pumped normal mode splitting and twin waves characteristics of a single-span clamped-hinged elastic beam around the parametric resonance tongues based on this model. For μ = 0.1 and β = 1.8, which is found in the principal resonance tongue of the fundamental mode, the computed phase trajectories based on the FEM structure modelling (as shown in Fig. 1D) considering a stream of 50 particles passing the beam verify that the amplitude-amplified wave dominates the modal oscillation when w1(0) = 0, w 1(0)/ω1 = 1 and the amplitude-suppressed wave dominates the oscillation when w1(0) = 1, w 1(0)/ω1 = 0. If multiple structural modes considering geometric configuration, higher dimension effects, supporting conditions, particle moving routes, and environmental disturbance patterns are important, the mode splitting and multi-wave oscillation phenomena with varying patterns around the simple and combination resonance tongues will be ubiquitous and need to be carefully discriminated
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