Vibrational Transportation of Deformable Axisymmetric Particles

TL;DR

The paper tackles vibrational transportation of a deformable axisymmetric particle on a substrate driven by a traveling Rayleigh wave, going beyond traditional point-mass models. It develops a deformable contact framework based on Hertz–Mindlin theory extended through the Method of Memory Diagrams (MMD) to handle arbitrary loading histories, and nondimensionalizes the problem to reveal two key parameters, $m^*$ and $A_y^*$, governing dynamics. The study maps out vertical and horizontal motion regimes, identifying mechanisms such as asymmetric sliding and synchronous jumping, including their regimes of stability, drift directions, and resonance effects; it contrasts these with the dynamics of a material point, showing substantial qualitative differences. The results yield phase diagrams and actionable insights for vibrational transport and dust-cleaning applications, demonstrating that deformability enables diverse, controllable drift regimes that are not captured by rigid-particle models.

Abstract

A particle on a substrate supporting a surface acoustic wave can experience horizontal drift excited by the dry friction force. The effect is referred to as vibrational transportation, or as a surface acoustic wave motor. A traditional theory of vibrational transportation considers a particle as a material point moving on a rigid substrate. A more realistic representation is a contact model based on Cattaneo-Mindlin (also called Hertz-Mindlin) mechanics applicable to an axisymmetric deformable particle. A recent semi-analytical extension of the Cattaneo-Mindlin solution called the Method of Memory Diagrams allows one to compute the hysteretic friction force for an arbitrary loading history in terms of contact displacements, and, subsequently, to numerically solve the equations of motion. Depending on the materials' and excitation parameters, the particle can stay in permanent contact with the substrate or experience multiple jumps. In the former case, the particle can slide along the surface, during each wave period, advancing and receding with different efficiencies, which finally results in a drift. The drift can occur both in the wave propagation direction and against it. In the regime of multiple jumps, directed horizontal motion is also possible. It is based on synchronization between the wave period and rebounding events. A rebound occurs once per period and consistently at the same phase. At the beginning of the process, the particle moves with an acceleration that decreases and finally disappears. Exactly the same type of motion against the wave has been observed in our preliminary experiments. We demonstrate that a point mass behaves differently: in a regime of permanent contact, negative and positive sliding are equilibrated, which produces no drift, whereas multiple rebounds of a point mass are always chaotic, at least for fully conservative collisions.

Figures (14)

• Figure 1: Geometry of a deformable particle moving on a substrate in which a Rayleigh wave is excited. $x$ and $y$ are the coordinates of the particle in the laboratory reference frame, $b$ and $a$ are its displacements relative to the moving substrate.
• Figure 2: Phase diagrams for (a) deformable particle (Hertzian sphere) and (b) material point with domains corresponding to two modes of vertical motion: jumping and permanent contact. The boundary between the domains (black curve) is obtained numerically in (a) and analytically in (b), following verma_particle_2013. Fundamental, subharmonic and superharmonic resonances are indicated by horizontal dashed lines in (a); in (b) there are no resonances. The inclined dashed line in (a) corresponds to the black line in (b) and is included for comparison.
• Figure 3: Dimensionless tangential velocity (see Eq. \ref{['eq:colormap']}) of a deformable Hertz-Mindlin sphere (a) and a material point (b) in the functional space $(A_y^*,m^*)$, averaged over 2000 wave periods. Red and blue colors represent motion in the direction of wave propagation and against it, respectively. Points with $|v_x^*|<v^*_0$ (Eq. \ref{['eq:colormap']}) are shown in white. The solid line separates permanent contact motion from hopping. Here, $\mu=0.3$. Numbers 1–10 indicate different motion types discussed below.
• Figure 4: Drift against the direction of wave propagation for a deformable particle in permanent contact (example for $A_y^*=10^{-0.3}$, $m^*=10^{1}$; point 1 in Fig. \ref{['fig:velocity_MMD']}). (a) Normal displacements $a^*(t^*)$ and $y^*(t^*)$. (b) Tangential displacements $b^*(t^*)$ and $x^*(t^*)$. (c) Fragments of the above curves. The advancing stage corresponds to total sliding at low compression, while during the receding stage the particle remains stuck at least partially. (d) Normal and tangential forces $N^*(t^*)$, $T^*(t^*)$, and horizontal velocity $\dot{b^*}(t^*)$ relative to the substrate. During total sliding, both $T^*$ and $\dot{b^*}$ are negative, resulting in negative drift direction.
• Figure 5: Drift in the wave propagation direction for a deformable particle in permanent contact with a substrate (example for $A_y^*=10^{1}$, $m^*=10^{-2.7}$ ; point 2 in Fig. \ref{['fig:velocity_MMD']})). (a) Normal displacements $a^*(t^*)$ and $y^*(t^*)$. (b) Tangential displacements $b^*(t^*)$ and $x^*(t^*)$. (c) Fragments of the above curves. Advancing in the positive direction occurs during the low compression phase (local minimum of $a^*(t^*)$) under total sliding. (d) Normal and tangential forces $N^*(t^*)$, $T^*(t^*)$, and horizontal velocity $\dot{b^*}(t^*)$ relative to the substrate. During total sliding, both $T^*$ and $\dot{b^*}$ are positive, resulting in positive drift direction.