https://doi.org/10.1140/epje/s10189-024-00421-y
Regular Article – Soft Matter
A multiple-timing analysis of temporal ratcheting
1
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA
2
Courant Institute, New York University, New York, NY, USA
3
Department of Mathematics, Rice University, Houston, TX, USA
4
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA
Received:
8
February
2024
Accepted:
4
April
2024
Published online:
27
April
2024
We develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies 
and 
, where 
is a rational number. If is a ratio of odd and even integers (e.g., 
), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for 
. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for 
and 
appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems.
© The Author(s) 2024
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