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We present a study of the behavior of a ball under the influence of gravity on a platform. A propagating surface wave travels on the surface of the platform while the platform remains motionless. This is a modification of the classical bouncing ball problem and describes the transport of particles by surface waves. Phase and velocity maps cannot be expressed in the explicit form due to implicit formulations, and no formal analytical analyses is possible. Numerical analysis shows that the transition to chaos is produced via a period doubling routewhich is a common property for classical bouncers. These numerical analysis have been carried out for the conservative and for the viscous cases and also for elastic and for inelastic collisions. The bouncing process can be sensitive to the initial conditions and can be useful for control techniques which can dramatically increase the effectiveness of particle transport in practical applications. Finally, we also consider the mechanical model of a particle sliding on a surface which is also important because it has important physical implications such as the transportation of thin films in biomedical applications, among others.

A particle falling down, in a constant gravitational field, on a moving platform is called a bouncing ball problem, or a bouncer. This model was suggested more than thirty years ago [

The bouncer model can be briefly characterized by the following basic statements. (i) Maps derived for the bouncer model can be exactly iterated for any time function describing the moving platform [

In this paper we assume that a particle is falling down in a constant gravitational field on a stationary platform. A propagating surface wave travels on the surface of the platform while the platform remains motionless. Such a model can be used to describe the transport of particles by propagating surface waves, which is an important problem with numerous applications. Powder transport by piezoelectrically excited ultrasonic surface waves [

This paper is organized as follows. In Section

We consider the two-dimensional system shown in Figure

Schematic diagram illustrating the collision between the particle and the surface.

Explicitly, the longitudinal and transverse displacements of the medium at the surface of flat boundary with travelling Rayleigh wave can be expressed like [

Rayleigh waves are dispersive due to a dependence of the wave’s speed on its wavelength. Typical example is Rayleigh waves in the Earth where waves with a higher frequency travel more slowly than those with a lower frequency. Rayleigh waves thus often appear spread out on seismograms recorded at distant earthquake recording stations [

Whenever a traveling nondispersive Rayleigh surface wave occurs in a medium, it can be characterized by a retrograde elliptic motion of the particles of that medium:

The coordinates of the particle are denoted as

In other words, the instantaneous shape of the surface cannot be described by an explicit function. Nevertheless, the tangent to the surface at the point with abscise

The governing equations of motion of a particle in a free flight mode are

The free flight stage continues until the particle collides with the surface. Unfortunately, it is impossible to determine the explicit time moment of the collision due to the fact that the instantaneous shape of the surface cannot be expressed by an explicit function. Instead, one has to use iterative numerical techniques in order to determine the exact moment of the bounce.

Localization of the root (the time moment of the collision) is performed using a time marching technique starting from the initial conditions until

When the root

Such iterative method of determination of the collision moment leads us to the important conclusion that phase and velocity maps cannot be expressed in an explicit form, and no formal analytical analysis is possible.

Nevertheless, the geometrical coordinates of the point of collision are

Projections of the particle’s velocities just before the collision to the normal and to the tangent to the surface at the contact point can be expressed in the following form:

Analogously, projections of velocities of the point of the surface in contact with the particle to the normal and to the tangent take the following form:

Then, the velocities of the particle just after the collision (in the normal and tangent directions) are

The free flight stage starts over again immediately after the collision, and the initial conditions are

The presented model is a modification of the classic bouncer model which can be derived assuming

We will demonstrate that the modified bouncer model possesses such an inherent feature as chaotic dynamics. Moreover, we will show that the sensitivity to initial conditions can be exploited for the control of the process of conveyance. We will show these results for the conservative or nonviscous case (

We take

The dynamics of a bouncing particle on a surface of a propagating wave is very sensitive to the initial conditions if the dynamics is Hamiltonian. Apparently, it is possible to find such a set of initial conditions which lead to regular and periodic dynamics. This is illustrated in Figure

Collision heights

Period

We plot the trajectory of the particle in

Chaotic trajectory of a bouncing particle for

For this case, we assume that collisions are completely elastic (

Transport of particles at increasing wave speeds (elastic collisions, viscous media over the surface). Reduced impact representation (a) shows the transition to chaos via a period doubling route. Note that impact heights are distributed in the interval

A phenomenological model could be used to exemplify the bifurcation diagram presented in Figure

An important parameter characterizing the effectiveness of the transport is the average longitudinal velocity of the particle

It is interesting to observe that the particle is transported with the average velocity of the traveling wave until the period 3 bouncing mode after a cascade of period doubling bifurcations (see Figure

Figures

Transient processes for parameter values as follows:

The situation becomes different when collisions are inelastic, as shown in [

In this section we thoroughly analyze the case in which the particle is sliding on the surface instead of the case in which it is falling down on it, analyzed previously. Our motivation is the following. Conveyance of particles and bodies by propagating waves is an important scientific and engineering problem with numerous applications. Manipulation of bioparticles and gene expression profiling using traveling wave dielectrophoresis [

We now describe, as in the bouncer model, the equations of motions of our sliding particle model.

It is assumed that a mass particle is in contact with the deformed surface at a point

A geometric scheme of the dynamical system showing the particle sliding on the surface.

The condition that the particle is located on the surface leads to the following constraint:

Though the instantaneous shape of the oscillating surface cannot be described by an explicit function, the tangent to the surface at the point

Instantaneous velocities of the surface's point

When a mass particle slides on the surface, it does not necessarily move in contact with one point of the surface. Therefore

The condition that the mass particle continuously slides on the surface brings another constraint into force (the relative velocity in the normal direction to the surface at the contact point must be zero):

Equation (

Differentiation of (

Then the relative sliding velocity of the particle on the surface

The condition of dynamic equilibrium leads to the following system of equations:

It is assumed that the friction force is linear. Thus

Finally, the governing equation of motion can be derived from (

A major obstacle is eliminated, and direct numerical time marching techniques can be used for integration of (

But before proceeding with the analysis of dynamic equilibrium the following observation can be done. If kinematic relationships describing a traveling Rayleigh wave are in force, the change of variables

An important conclusion can be done. Dynamics of a particle sliding on the surface of a propagating Rayleigh wave cannot be chaotic. This is due to the fact that the governing equation of motion is a second-order autonomous ordinary differential equation with smooth parameter functions [

Equation (

The term

As mentioned earlier, the explicit governing equation is formulated in terms of

The described computational technique is used to construct basin boundaries of the system’s attractors (Figure

Basin boundaries at

A special attention should be paid to dashed line intervals on basin boundaries. Equation (

Conveyance of a particle by a propagating Rayleigh wave is a nonlinear problem, so such effects as the coexistence of stable attractors should not be astonishing. Stable equilibrium point type attractor in Figure

Illustration of the attractor control strategy: limit cycle is represented as a periodic trajectory in frame

It can be noted that the up-mentioned control strategy can be implemented only when the stable equilibrium point and the stable limit cycle coexist. Thus, it would be impossible to transport a sand particle with the velocity of the propagating wave by an acoustic surface Rayleigh wave. Nevertheless, such attractor control strategies could be implemented for transportation of biomedical objects on the surface of an undulation film [

Transport of particles by surface waves is an important scientific and engineering problem, with numerous practical applications, including MEMS (micro-electro-mechanical systems) used to manipulate objects like particles or cells. We show that this problem is a modification of the classical bouncer model which is considered as a paradigm model in nonlinear physics. The formulations of our model are implicit, thus phase and velocity maps cannot be expressed in explicit form.

Chaotic dynamics of a conveyed particle is not an unexpected fact due to the complexity of the constitutive model. More surprising is the rich dynamical behavior in models comprising dissipative dynamics, elastic and inelastic collisions. It appears that the transition to chaos via a period doubling route is a universal property for bouncers and is observed in our model of particles transport in both, conservative and viscous media. Moreover, the sensitivity to initial conditions can be useful for control techniques which can dramatically increase the effectiveness of particles transport by surface waves. These results are relevant in the sense that we have also found the sensitivity to the initial conditions for the sliding particle model, which may have important applications in practical implementations as powder transport by piezoelectrically excited ultrasonic waves, transport of sand particles, among others.

Though the numerical analysis was concentrated on the dimensionless system only, theoretical and experimental investigation of dry particle conveyance and its control is a definite object for future research.

This work was supported by the Spanish Ministry of Education and Science under project number FIS2006-08525 and by Universidad Rey Juan Carlos and Comunidad de Madrid under project number URJC-CM-2007-CET-1601.