Ratchet due to broken friction symmetry

The Notched Stick, an ancient vibrot example

An intriguing simple toy, commonly known as the Notched Stick, is discussed as an example of a “vibrot”, a device designed and built to yield conversion of mechanical vibrations into a rotational motion. The toy, that can be briefly described as a propeller fixed on a stick by means of a nail and free to rotate around it, is investigated from both an experimental and a numerical point of view, under various conditions and settings, to investigate the basic working principles of the device.

The conversion efficiency from vibration to rotational motion turns out to be very small, or even not detectable at all, whenever the propeller is tightly connected to the stick nail and perfectly axisymmetrical with respect to the nail axis; the small effects possibly observed can be ascribed to friction forces. In contrast, the device succeeds in converting vibrations into rotations when the propeller center of mass is not aligned with the nail axis, a condition occurring when either the nail-propeller coupling is not tight or the propeller is not completely axisymmetrical relative to the nail axis. The propeller rotation may be induced by a process of parametric resonance for purely vertical oscillations of the nail, by ordinary resonance if the nail only oscillates horizontally or, finally, by a combination of both processes when nail oscillations take place in an intermediate direction. Parametric resonance explains the onset of rotations also when the weight of the propeller is negligible. In contrast with what is commonly claimed in the literature, the possible elliptical motion of the nail, due to a composition of two harmonic motions of the same frequency imposed along orthogonal directions, seems unnecessary to determine the propeller rotation.

Ratchet effect for two-dimensional nanoparticle motion in a corrugated oscillating channel

The motion of a single rigid or elastic particle inside a corrugated narrow channel is investigated by means of Brownian dynamics simulations. Periodic oscillations of one of the asymmetric channel surfaces induce directed particle transport. For different surface structures of the resting channel surface, we determine optimal transport properties in terms of the driving frequency, particle size, and corrugation amplitude. The transport direction is changed when switching from perpendicular motion of the oscillating surface to parallel motion with respect to the resting surface, which can be rationalized by a transition from a flashing to a pushing ratchet effect. We also study the diffusion behavior and find strongly enhanced diffusion for parallel oscillatory motion with a diffusivity significantly larger than for free diffusion. Elastic large particles exhibit suppressed transport with increasing rigidity. In contrast, for small particles, increasing rigidity enhances the particle transport both in terms of particle velocity and diffusivity.

Vibrot, a simple device for the conversion of vibration into rotation mediated by friction: preliminary evaluation

While “vibrational noise” induced by rotating components of machinery is a common problem constantly faced by engineers, the controlled conversion of translational into rotational motion or vice-versa is a desirable goal in many scenarios ranging from internal combustion engines to ultrasonic motors. In this work, we describe the underlying physics after isolating a single degree of freedom, focusing on devices that convert a vibration along the vertical axis into a rotation around this axis. A typical Vibrot (as we label these devices) consists of a rigid body with three or more cantilevered elastic legs attached to its bottom at an angle. We show that these legs are capable of transforming vibration into rotation by a “ratchet effect”, which is caused by the anisotropic stick-slip-flight motion of the leg tips against the ground. Drawing an analogy with the Froude number used to classify the locomotion dynamics of legged animals, we discuss the walking regime of these robots. We are able to control the rotation frequency of the Vibrot by manipulating the shaking amplitude, frequency or waveform. Furthermore, we have been able to excite Vibrots with acoustic waves, which allows speculating about the possibility of reducing the size of the devices so they can perform tasks into the human body, excited by ultrasound waves from the outside.

Active particles with broken symmetry

We discuss and analyze the driving a polar active particle with a head-tail asymmetry based on the dynamics of an internal motor variable driven by an energy depot and a broken symmetry of friction with respect to the internal degree of freedom. We show that such a driving may be advantageous for driving large masses with small energy uptake from the environment and exhibits interesting properties such as resonance-driven optimal propulsion.

SCREW MOTION OF DNA DUPLEX DURING TRANSLOCATION THROUGH PORE I: INTRODUCTION OF THE COARSE-GRAINED MODEL

Based upon the structural properties of DNA duplexes and their counterion-water surrounding in solution, we have introduced here a screw model which may describe translocation of DNA duplexes through artificial nanopores of the proper diameter (where the DNA counterion–hydration shell can be intact) in a qualitatively correct way. This model represents DNA as a kind of "screw," whereas the counterion-hydration shell is a kind of "nut." Mathematical conditions for stable dynamics of the DNA screw model are investigated in detail. When an electrical potential is applied across an artificial membrane with a nanopore, the "screw" and "nut" begin to move with respect to each other, so that their mutual rotation is coupled with their mutual translation. As a result, there are peaks of electrical current connected with the mutual translocation of DNA and its counterion–hydration shell, if DNA is possessed of some non-regular base-pair sequence. The calculated peaks of current strongly resemble those observed in the pertinent experiments. An analogous model could in principle be applied to DNA translocation in natural DNA–protein complexes of biological interest, where the role of "nut" would be played by protein-tailored "channels." In such cases, the DNA screw model is capable of qualitatively explaining chemical-to-mechanical energy conversion in DNA–protein molecular machines via symmetry breaking in DNA–protein friction.

Botanical ratchets

Ratcheting surfaces are a common motif in nature and appear in plant awns and grasses. They are known to proffer selective advantages for seed dispersion and burial. In two simple model experiments, we show that these anisotropically toothed surfaces naturally serve as motion rectifiers and generically move in a unidirectional manner, when subjected to temporally and spatially symmetric excitations of various origins. Using a combination of theory and experiment, we show that a linear relationship between awn length and ratchet efficiency holds under biologically relevant conditions. Grass awns can thus efficiently transform non-equilibrium environmental stresses from such sources as humidity variations into useful work and directed motion using their length as a fluctuation amplifier, yielding a selective advantage to these organelles in many plant species.

Ratchet effect of a dimer with broken friction symmetry in a symmetric potential

The one-dimensional overdamped Brownian motion of a dimer consisting of two harmonically interacting components is considered. Both components are coupled to the same heat bath and feel the same spatially periodic symmetric potential, whose amplitude is modulated periodically in time. The friction coefficients may differ between dimer components, thus breaking the dynamical symmetry of the system. In the absence of any external bias, a ratchet effect (directed transport) arises generically. Two accurate approximations for the dimer's velocity and diffusion coefficient are obtained for weak and strong couplings. The velocity of the system can be maximized for each direction by adding an optimal amount of noise and by tuning the driving frequency to an optimal value. Furthermore, there exist two optimal coupling strengths at which the velocity is the largest.

Active elastic dimers: self-propulsion and current reversal on a featureless track

We present a Brownian inchworm model of a self-propelled elastic dimer in the absence of an external potential. Nonequilibrium noise together with a stretch-dependent damping form the propulsion mechanism. Our model connects three key nonequilibrium features -- position-velocity correlations, a nonzero mean internal force, and a drift velocity. Our analytical results, including striking current reversals, compare very well with numerical simulations. The model unifies the propulsion mechanisms of DNA helicases, polar rods on a vibrated surface, crawling keratocytes and Myosin VI. We suggest experimental realizations and tests of the model.

Stability analysis of dynamic ratcheting in elastoplastic systems

This paper deals with the stability and the dynamics of a harmonically excited elastic perfectly plastic asymmetrical oscillator. The hysteretical system is written as a nonsmooth, forced autonomous system. The dimension of the phase space can be reduced using adapted variables. It is shown that asymmetry of boundary conditions (forcing term) and material asymmetry lead to an equivalent system for this simple structural case. The forced vibration of such an oscillator is treated by a numerical approach by using time locating techniques. Stability of the limit cycles is analytically investigated with a perturbation approach. The boundary between elastoplastic shakedown and dynamic ratcheting is given in closed form. It is shown that the divergence rate is strongly correlated to the internal asymmetry of the oscillator.