Abstract
The vibrational resonance (VR) phenomenon has received a great deal of research attention over the two decades since its introduction. The wide range of theoretical and experimental results obtained has, however, been confined to VR in systems with constant mass. We now extend the VR formalism to encompass systems with position-dependent mass (PDM). We consider a generalized classical counterpart of the quantum mechanical nonlinear oscillator with PDM. By developing a theoretical framework for determining the response amplitude of PDM systems, we examine and analyse their VR phenomenona, obtain conditions for the occurrence of resonances, show that the role played by PDM can be both inductive and contributory, and suggest that PDM effects could usefully be explored to maximize the efficiency of devices being operated in VR modes. Our analysis suggests new directions for the investigation of VR in a general class of PDM systems. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.
Highlights
Nonlinear science has attracted global interest on account of its broad applications to a diversity of disciplines
This nonlinearity can impact on the dynamics of the system, including the occurrence of resonance which we demonstrate for the first time below
To validate the analytic results, the theoretical response amplitude Q given by Eq (3.19) was compared with the numerical Q computed from the Fourier spectrum of the solution of the main position-dependent mass (PDM)-Duffing equation (Eq (2.9)) expressed as coupled first-order autonomous ordinary differential equations (ODEs) of the form: Q
Summary
Nonlinear science has attracted global interest on account of its broad applications to a diversity of disciplines. In many physical systems, the associated inertial mass is neither constant, nor stochastically varying, nor dynamically changing with time but, rather, is explicitly position-dependent. The classical equation of motion of PDM systems contains an extra non-conservative generalized force term of quadratic order in the modified Newton’s equation, a term which is nonlinear in velocity and linearly proportional to the mass gradient [18,19,20] This nonlinearity can impact on the dynamics of the system, including the occurrence of resonance which we demonstrate for the first time below. We consider a simple, but general, PDM system with a regular mass function consisting of a constant mass (mass amplitude) and a quadratic spatial nonlinearity, modelled by a bistable potential [14,18,20].
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