Energy exchange between coupled mechanical oscillators: linear regimes

Abstract

Nonlinear mechanisms are frequently invoked to explain unexpected observations during processes of energy exchange in mechanical systems. However, whether the same phenomena could be observed in a purely linear system is seldom considered. In this paper, we revisit the problem of two linearly coupled, damped, harmonic oscillators, with emphasis on the dynamics of their mutual exchange of energy. A novel criterion is established to discern between two well-differentiated regimes of energy exchange under ringdown conditions, when all external excitations are turned off and the oscillatory motion is left to decay by dissipation. The two regimes are induced by different sets of initial conditions, and correspond to extremal values of the total net energy transferred during ringdown. Although the problem is in principle fully solvable, its algebraic involvedness requires in practice to resort to approximated expressions for oscillation frequencies and decay rates. We explicitly provide such approximations in the limit of high quality factors and weak coupling. This limit is relevant to current experiments on micro-and nanomechanical oscillators, whose physical behavior is usually explained by extensive allusion to energy exchange. Our results should help to discern between observations that are genuinely due to nonlinear effects, and those that can be explained in terms of linear mechanisms only.

Dynamics of 2D material membranes

“…Another point to note is that in equation ( 5) it is assumed that the mass and stiffness of the base are infinitely much larger than the mass and stiffness of the 2D membrane, such that the membrane motion does not affect the motion of the base. If this assumption does not hold anymore, the combined membrane-base systems needs to be analyzed using coupled equations of motion for base and membrane [64,95].…”

Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances

“…Several groups [32,[123][124][125][126][127][128] have developed mathematical models to explain and explore the energy transfer among vibration modes using lumped-mass parameter systems or distributed parameter models using for instance the Euler-Bernoulli beam equation [32,123]. For example, Zanette [125][126][127] performed in-depth theoretical analysis on the modal coupling and energy transfer among vibration modes of micro and nanomechanical systems.…”

Tunable Strong Coupling of Mechanical Resonance between Spatially Separated FePS3 Nanodrums

“…We apply the corresponding V g to the drums in order to match their ω 1,2 values and measure the avoided crossing of the resonance frequencies in both configurations of lasers. By solving eq 1 , we find amplitudes for the two configurations of lasers as 35 where A 1 is the oscillation amplitude with lasers on the same drum, A 2 the amplitude with lasers on different drums, the coupling strength coefficient, and the detuning. In Figure 3 b the measured amplitudes A 1,2 at V g,2 = 30 V are compared to simulations based on the continuum mechanics model (see section SI 1 ) as well as eqs 1 and 2 .…”