【正文】 are 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 on damping ratio (F=150 N).The response acceleration wave of the square pipewith F=150 N and L=500 mm is shown in Fig. 4(a). Fig.4(b) gives the frequency response of Fig. 4(a). The natural frequency of the fundamental mode was approx. 650Hz, and the damping ratio obtained by the halfpowermethod was approx. .. Effect of ball size and excitation direction ondamping characteristicsThe effect of the ball size d on the maximum acceleration amaxboth in vertical and horizontal excitation isFig. 4. Damping characteristics of square pipe without glass ballpacking (F=150 N, L=500 mm). (a) Response acceleration, (b) frequency response.Fig. 5. Effect of ball size on maximum acceleration.shown in Fig. 5. The horizontal broken line shows themaximum acceleration in square pipe without ball packing. In the vertical excitation, amaxfor ball diameterssmaller than d=10 mm is nearly equal to 60% of thevalue for the square pipe without ball packing, and isnot affected by the ball diameter. However, in the caseof dH1135012 mm, amaxis larger than that for dH1134910 mm. Asa result, the dynamic rigidity of model structures packedwith glass balls of dH1135012 mm is not improved due to thelarge acceleration response. On the other hand, in thehorizontal excitation, the value of amaxwithin the regiondH1134917 mm is evidently smaller than the value for thesquare pipe without ball packing.The effect of the ball size d on the natural frequencyf of the fundamental mode both in vertical and horizontalexcitation is shown in Fig. 6. The horizontal broken lineshows the natural frequency of the square pipe withoutball packing. In the figure, the effect of the excitationdirection is not clear. The natural frequency within theFig.