The main objective of the project was to develop a model for the motion of granular media under vibration in a tub vibrator. For such a system, it was decided that a discrete element method (DEM) was the most appropriate tool to model bulk velocity and circulation of media. In the first phase of the work, a vibratory finisher was modified to introduce planar vibration into a single layer of particles. The motion of the tub was measured using accelerometers and the corresponding granular media behavior was determined by video recording. A discrete element model, based on Cundall’s approach to contact, was developed to model granular flow in different vibratory beds, and the results were compared with experimental measurements of bulk flow velocity and bed expansion for the tub finisher. The sensitivity of the model predictions to the contact parameters was considered and the parameters were optimized with respect to the experimental results. After optimization, the difference between the model predictions of the bulk flow velocity and the measurements was less than 20% at four locations in media beds of two depths. The average bulk density of the vibrating beds was also predicted to be within 20% of the measured values. In the next phase, a two-dimensional discrete element model was developed to model single-cell circulation in vibratory beds that had both vertical and horizontal components of motion. The model predictions were compared with experimental measurements of the onset and growth of circulation in beds of steel and glass spheres as a function of bed depth, inter-particle and wall friction coefficients, and the amplitude of vibration. While the values from the DEM showed an error of up to 50% in the predicted circulation strength, depending on the type of the media and system conditions, the trends predicted by the model closely matched those in the experiments. Finally, a physical model was developed to describe the relationship between the onset and direction of circulation with the vibration of the container. A similar model was used to describe the experimental results as well as the transition in circulation patterns in terms of the resultant shear forces at the vibrating container walls and the interlocking of media close to the container walls. It was also demonstrated that a two-dimensional DEM could model a granular flow in which the media had three-dimensional contact and freedom of movement, but that was driven by vibrations in a plane. In summary, it was found that the linear optimization procedure for the contact parameters is an efficient way to improve the results from DEM. Additionally, the circulation in a tub-vibrator increased with the depth of the particulate media in the container, and with the magnitude of the wall-particle and particle-particle friction coefficients. The strength of circulation also increased with the amplitude of vibration. A strong correlation existed between the total shear force along the vibrating container walls and the circulation behavior. Bulk circulation increased sharply when increasing bed depth increased the pressure and the shear forces at the walls and between particle layers. It was also concluded that dimensionless bed depth (the ratio of bed depth to particle diameter) was not a proper dimensionless group when discussing the circulation behavior and it should act in conjunction with other parameters.

The main objective of the project was to develop a model for the motion of granular media under vibration in a tub vibrator. For such a system, it was decided that a discrete element method (DEM) was the most appropriate tool to model bulk velocity and circulation of media. In the first phase of the work, a vibratory finisher was modified to introduce planar vibration into a single layer of particles. The motion of the tub was measured using accelerometers and the corresponding granular media behavior was determined by video recording. A discrete element model, based on Cundall’s approach to contact, was developed to model granular flow in different vibratory beds, and the results were compared with experimental measurements of bulk flow velocity and bed expansion for the tub finisher. The sensitivity of the model predictions to the contact parameters was considered and the parameters were optimized with respect to the experimental results. After optimization, the difference between the model predictions of the bulk flow velocity and the measurements was less than 20% at four locations in media beds of two depths. The average bulk density of the vibrating beds was also predicted to be within 20% of the measured values. In the next phase, a two-dimensional discrete element model was developed to model single-cell circulation in vibratory beds that had both vertical and horizontal components of motion. The model predictions were compared with experimental measurements of the onset and growth of circulation in beds of steel and glass spheres as a function of bed depth, inter-particle and wall friction coefficients, and the amplitude of vibration. While the values from the DEM showed an error of up to 50% in the predicted circulation strength, depending on the type of the media and system conditions, the trends predicted by the model closely matched those in the experiments. Finally, a physical model was developed to describe the relationship between the onset and direction of circulation with the vibration of the container. A similar model was used to describe the experimental results as well as the transition in circulation patterns in terms of the resultant shear forces at the vibrating container walls and the interlocking of media close to the container walls. It was also demonstrated that a two-dimensional DEM could model a granular flow in which the media had three-dimensional contact and freedom of movement, but that was driven by vibrations in a plane. In summary, it was found that the linear optimization procedure for the contact parameters is an efficient way to improve the results from DEM. Additionally, the circulation in a tub-vibrator increased with the depth of the particulate media in the container, and with the magnitude of the wall-particle and particle-particle friction coefficients. The strength of circulation also increased with the amplitude of vibration. A strong correlation existed between the total shear force along the vibrating container walls and the circulation behavior. Bulk circulation increased sharply when increasing bed depth increased the pressure and the shear forces at the walls and between particle layers. It was also concluded that dimensionless bed depth (the ratio of bed depth to particle diameter) was not a proper dimensionless group when discussing the circulation behavior and it should act in conjunction with other parameters.

The surface-normal impact velocity distributions, impact frequencies and impact power per unit area were measured using a force sensor in a vibratory finisher for two types of spherical media. These parameters control the degree, rate and character of plastic deformation of a workpiece surface in vibratory finishing. The force sensor was also used to quantify the effect of media type, finisher amplitude, and location within the finisher on the probability distribution of the particle impact velocity normal to the workpiece. It was found that reducing the total media mass in the finisher and moving closer to the wall resulted in a more aggressive process. It was also found that contacts occured periodically within time periods that corresponded to the finisher’s driving frequency. The Almen system was adapted to a vibratory finishing process to characterize the effect of varying process parameters for the purposes of process development and control. Saturation curves for two types of aluminum Almen strips were obtained by finishing at two distinct conditions. Comparison with the normal contact forces and effective impact velocities, measured for both these conditions, provided insight into the mechanics of the vibratory finishing process. An electromagnetic apparatus was constructed to simulate the normal impacts in the vibratory finisher. It was found that surface-normal impacts at velocities comparable to the higher range in the vibratory finisher produced Almen saturation curves similar to those created in the vibratory finisher. This provided support for the modeling approximation of treating all contact events in a vibratory finisher as effective surface-normal impacts, and the accuracy of the effective impact velocity measurement. A model of the process by which Almen strips were plastically deformed by media impacts in vibratory finishing was presented. The motivation was to extend the use of Almen strip measurements as a means of characterizing vibratory finishing through an improved understanding of the process parameters that controlled time-dependent curvature development. Two thicknesses of Almen strip were tested for two finishing conditions. The quantitative agreement between the model saturation curves and the experimental curves was fair, although the overall trends were predicted very well.

The surface-normal impact velocity distributions, impact frequencies and impact power per unit area were measured using a force sensor in a vibratory finisher for two types of spherical media. These parameters control the degree, rate and character of plastic deformation of a workpiece surface in vibratory finishing. The force sensor was also used to quantify the effect of media type, finisher amplitude, and location within the finisher on the probability distribution of the particle impact velocity normal to the workpiece. It was found that reducing the total media mass in the finisher and moving closer to the wall resulted in a more aggressive process. It was also found that contacts occured periodically within time periods that corresponded to the finisher’s driving frequency. The Almen system was adapted to a vibratory finishing process to characterize the effect of varying process parameters for the purposes of process development and control. Saturation curves for two types of aluminum Almen strips were obtained by finishing at two distinct conditions. Comparison with the normal contact forces and effective impact velocities, measured for both these conditions, provided insight into the mechanics of the vibratory finishing process. An electromagnetic apparatus was constructed to simulate the normal impacts in the vibratory finisher. It was found that surface-normal impacts at velocities comparable to the higher range in the vibratory finisher produced Almen saturation curves similar to those created in the vibratory finisher. This provided support for the modeling approximation of treating all contact events in a vibratory finisher as effective surface-normal impacts, and the accuracy of the effective impact velocity measurement. A model of the process by which Almen strips were plastically deformed by media impacts in vibratory finishing was presented. The motivation was to extend the use of Almen strip measurements as a means of characterizing vibratory finishing through an improved understanding of the process parameters that controlled time-dependent curvature development. Two thicknesses of Almen strip were tested for two finishing conditions. The quantitative agreement between the model saturation curves and the experimental curves was fair, although the overall trends were predicted very well.