Vibration of granular materials

Citation

Wassgren, Carl R. (1997) Vibration of granular materials. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QMSQ-QJ88. https://resolver.caltech.edu/CaltechETD:etd-11112004-104123

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

This thesis examines the fundamental behavior of a granular material subject to external vibrations. Experiments were designed to investigate the phenomena that appear in a container filled with glass spheres subject to vertical, sinusoidal oscillations. In addition, a discrete element computer simulation code was written to supplement the experimental program.

Experiments and simulations reveal that the behavior of the particle bed can be classified into two regimes known as shallow and deep beds. For example, when a shallow bed consisting of less than six layers of glass spheres is subjected to oscillations with acceleration amplitudes greater than approximately 2.Og where g is the acceleration due to gravity, the particles in the container are fluidized and do not display coordinated movement. However, when more than six particle layers are used, the particles move coherently and the deep bed behaves as a single, completely inelastic mass.

In the shallow bed regime three distinct sub-states are observed that differ in the degree of coherency in the particle motions. Each appears depending upon the number of particle layers in the bed and on the acceleration amplitude of the oscillations. The transitions between the states are gradual and not well-defined.

The transition from the deep bed to the shallow bed state is characterized by a sudden expansion of the bed that occurs at a critical acceleration amplitude for a fixed bed depth and particle type. Simulations indicate that when the particle fluctuating kinetic energy is dissipated completely each oscillation cycle, the bed remains in the deep bed state. If the energy is not completely dissipated, a shallow bed state results. A simple model consisting of an inelastic ball bouncing on a sinusoidally oscillating table reproduces the sudden expansion.

In the deep bed regime, phenomena such as side wall convection, surface waves, kinks, and kink convection cells appear depending primarily on the acceleration amplitude of the oscillations and, to a lesser degree, the number of particle layers in the bed. Phase maps of when these behaviors occur were constructed using both experimental and simulation data.

When the acceleration amplitude is greater than approximately lg, side wall convection cells appear at the vertical wall boundaries of the container. Particles move down along the vertical walls of the container and up within the bulk of the bed. Simulations indicate that the convection cells are the result of the frictional contact between particles and the walls and the asymmetry of the particle/wall collision rate over an oscillation cycle. Using the simulations, the width of the boundary layer next to the walls, the height of the convection cell center from the container base, and the particle flux in the boundary layer were measured as functions of the vibration parameters and particle properties. The results from the simulation compare well with experimental measurements. The simulation indicates that the boundary layer width is proportional to the container width when the bed aspect ratio, defined as the bed depth to the bed width, is greater than approximately 0.2. For beds with aspect ratios less than 0.2, however, the boundary layer width remains constant. Simulation results also demonstrate that the convection cell height is proportional to the bed depth and that the flux of particles in the boundary layer increases with increasing particle/wall friction and decreases for coefficients of restitution near one.

At acceleration amplitudes between approximately 2.Og and 3.5g, standing waves appear on the top free surface of the bed. These waves form at half the forcing oscillation frequency and are referred to as [...]/2 waves. A second set of standing waves appears when the acceleration amplitude is greater than approximately 5.Og and persist up to at least 7.0g. These waves form at one-quarter the forcing frequency and are known as [...]/4 waves. Experimental measurements indicate that the wave amplitude expressed as a Froude number increases with increasing acceleration amplitude for the [...]/2 waves but remains constant for the [...]/4 waves. Additionally, measurements of the wavelength suggest that the waves have a dispersion relation similar to that for deep fluid gravity waves where the wavelength is proportional to the square of the inverse frequency.

Kinks and kink convection cells appear in the particle bed after a period doubling bifurcation occurs in the flight dynamics of the bed. The formation of kinks can be explained using a simple model consisting of a completely inelastic ball on a sinusoidally oscillating table. Experimental measurements indicate a minimum allowable distance between nodes that is a function of the bed depth and acceleration amplitude. The convection cells bracketing each kink are shown to be the result of the out-of-phase motion of the bed sections and the interaction between fluidized and solidified regions of the bed.

The effect of vertical, sinusoidal vibrations on a discharging wedge-shaped hopper was also investigated. When the hopper exit is closed, side wall convection cells appear with particles moving up at the inclined container boundaries and down at the centerline of the bed. The same mechanism that causes downward convection at vertical walls can also explain the upward motion at inclined walls. Experimental measurements also indicate that the discharge rate from the vibrating hopper scales with the oscillation velocity amplitude. At low velocity amplitudes, the discharge rate from the hopper is slightly greater than the non-vibrating hopper discharge rate. At high velocity amplitudes, however, the discharge rate decreases significantly. A simple model accounting for the change in the effective gravity acting on the particle bed throughout the oscillation cycle and the impact velocity of the bed with the hopper predicts the observed trend.

The experiments and simulations conducted in the present work suggest that the boundary conditions and the fluid- and solid-like nature of granular materials are significant factors affecting the response of a granular bed. Additionally, this work demonstrates the value of discrete element computer simulations as a tool for complementing experimental observations.

Thesis Files