In order to analyze the grinding process of vibration milling, a solution of the fundamental integro-differential equation of batch grinding, based upon the mass balance, was derived by referring to the experimental data for the selection function and distribution function of vibration milling. The transition of particle size distribution during milling was examined and was compared with the derived solution. The results obtained were summarized as follows: (1) A solution of fundamental integro-diffarential equation is derived as R(y, t)=R(y, o)exp(-kyt) where K is constant. This analytical solution is similar to Chujyo's formula, and it agreed with Alyadvin's equation the index of which equals 1. (2) Experimental results of transition of particle size distribution against grinding time could be explained by the derived solution described in (1). (3) The optimum ball diameter for grinding efficiency is about 2cm for a porcelain ball. (4) Grinding rate is proportional to the 1.5 power of the amplitude of vibration mill.

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The effects of vibration amplitude and frequancy on the grinding efficiency of a vibration mill is studied by using test equipment.
Although the impact force (the product of vibration amplitude and the square of frequency) must be increased to improve the grinding efficiency of the mill, the power requirements are to be kept low.
In this paper, the behavior of the particle-size distribution, the mean particle diameter, and the power consumption is studied under conditions of, (i) increasing the vibration amplitude at a constant frequency, (ii) increasing the frequency at a constant vibration amplitude, and (iii) varying the vibration amplitude and the frequency at a constant impact force.
From the results obtained, it is concluded that the increase in the vibration amplitude rather than the frequency results in effective particle-size reduction with low power consumption.

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The grindability of various types of ore and coal was studied and a new classification of iron ores is proposed on the basis of the ore-forming temperature of the deposits. The work indices of magnetite and hematite minerals ranged between 12.5-24.1kWh/t and 6.5-23.8kWh/t, respectively, and, in both cases, increased with the ore-forming temperature. It was also found that iron ores were mainly ground according to Bond's law, and coals were ground according to Rittinger's law in batch-type ball mill tests. It is proposed that the grindability of iron ores is characterized by the ore-forming temperature of deposits and that of coal, by its coking ability.

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A vertical annular type testing apparatus was developed based on the incompressible flow model of a packed granule bed, and an attempt was made to confirm the general rule that controls the surface abrasion and grinding performances between the same granule particles.
Ultra-fine products less than a few microns were constantly generated by the attrition effect of particles when an almina ball sample was ground by this tester.
The theoretical expression on the rate of new surface area generation was derived from an incompressible flow model of a packed granule bed, and its validity was confirmed by the results of the experiment.
Furthermore, it was shown from the experiments that the energy efficiency of the attrition grinding is dominated by Rittinger's law, and it is changed by the influence of the environmental atmosphere surrounding the raw material. The particle size of fine grain generated by attrition is dominated by the intensity of the intergranule contact force.

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The production process of a fine size fraction of silicate sand by ball milling is studied in this paper, by analyzing the size distribution of products with the Rosin-Rammler formula and the rate process of cumulative size fractions with Alyavdin-Chujyo's formula. The size distribution is found to be divided into three regions, a coarser part influenced by feed size, an intermediate part, where the size distribution shows a clear straight line relationship on RRS-chart, and the finest part with the ultimate limit of fineness by ball milling. Alyavdin-Chujyo's relationship is found to be valid over a very wide range of milling conditions, and the straight lines on its relationship with different feed size, intersect each other making another straight line on the same chart, which can be defined as “the limit line of the persistent component region”. The rate process of the final stage of grinding is also formulated into a relatively simple equation, in which almost more of the coefficients are influenced by milling conditions.

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