【正文】 Manufacture 42 (2020) 467–472When balls are packed by excitation in the cavity ofthe model structure, the maximum packing ratio and thepacked arrangement depend on the ball size. This dependency of the packing ratio affects the damping capacityetc. In this experiment, when the ball size is less than 10mm and the maximum packing ratio is high, the naturalfrequency bees low as a result that the glass ballsand square pipe are united so the equivalent massincreases. Thus, when small balls are packed, it is considered that the friction between the balls and the innersurface of the square pipe is the main factor in the damping capacity generation. On the other hand, when theball size is larger than 12 mm, the movements of ballsin a cavity have a considerable degree of freedom. Thepacked balls in vibration repeat the collisions with aninner surface of the square pipe, and are almost in asuspended state. For this reason, it is considered thatmass of balls contributes little to the equivalent mass,and the natural frequency does not bee low. Whenlarge balls are packed, the energy dissipation by the collision between balls and an inner surface of the squarepipe is considered to be the main factor in dampingcapacity improvement. Both collisions are inelastic.. Effect of repulsion coefficient on dampingcapacityFrom the previous considerations, it can be forecastthat in the damping capacity improvement by packedballs the collision characteristics between balls and theinner surface of the square pipe (., the repulsioncoefficient) also affects the damping capacity improvement. Thus, Fig. 10 shows the result of repulsion coefficient measurement which was obtained by droppingglass balls of different size on the center of square pipewithout ball packing (L=500 mm), stainless steel column(25 mm25 mm500 mm) and glass column (25 mm25mm500 mm) supported at the location of the nodes ofFig. 10. Effect of ball size on repulsion coefficient.the fundamental vibration mode. The repulsion coefficient against stainless steel column and glass columnhas the same changing tendency for the increase in ballsize and decreases with the increase in size of ballslarger than d=10 mm. The repulsion coefficient againstthe glass column is larger than that against the stainlesssteel column for all tested ball sizes. However, thecharacteristics differ considerably against square pipewithout ball packing. The repulsion coefficient decreaseswith the increase of ball size in smaller region than d=10mm, and again increases for balls larger than d=10 mm.The change of repulsion coefficient in this case is plex in parison with the case of the stainless steelor glass column. This is considered to be induced by theparticular crosssection shape (hollow structure).Such results obtained against three objects affect thedamping characteristics of the model structures. It isclarified that the changing tendency of the repulsioncoefficient against square pipe with ball size shows arather plex behavior corresponding to the collisionbetween the packed balls and the inner surface of thesquare pipe in damping vibration experiments. Thus, therepulsion coefficient is the main parameter stronglyaffecting the damping characteristics.Hence, the result of the damping ratio z on both vertical and horizontal excitations is arranged in regard tothe repulsion coefficient against square pipe as shownFig. 11. Numbers written near the plots show the ballsize. In the figure, two types of z–e relations areobserved for both excitations. There are two ball sizeregions, from 1 to 10 mm, and from 12 to 20 mm. Ineither region, the damping ratio is improved with thedecrease in the repulsion coefficient. But, when the ballsize is less than 10 mm, the repulsion coefficient haslittle effect on the damping ratio. When a ball is small,a large number balls are packed together, and many ofthem do not e into collision with the inner surfaceof the structure. A few balls near the surface collide withFig. 11. Effect of repulsion coefficient on damping ratio.472 Y. Wakasawa et al. / International Journal of Machine Tools amp。 Manufacture 42 (2020) 467–472Fig. 3. Effect of length of square pipe o