【正文】 pulsive force F=50–300 N. Thesame results are obtained for other model structurelengths. Accordingly, the subsequent experiments werecarried out under the impulsive force F=150 N as thebasic condition so as to obtain the easiest excitation.The effect of the model structure length L on dampingratio z of the fundamental mode is shown in Fig. 3.Damping ratio z increases with the increase of L. In ourmodel structures having the crosssectional dimensionindicated in Table 1, harmonic ponents of a higherorder bee remarkable at LH11350700 mm. The initialamplitude immediately after excitation was extremelylarge at L=300 mm pared to the subsequent amplitudes. Accordingly, the model structures of L=500 mmwere used in the following experiments.Fig. 2. Effect of impulsive force on damping ratio (L=500 mm).469Y. Wakasawa et al. / International Journal of Machine Tools amp。 Manufacture 42 (2020) 467–472region dH1134910 mm is small. But, in the range dH1135012 mm,this natural frequency agrees approximately with themodel structure without ball packing.Next, the effect of the ball size d on the damping ratioz in both vertical and horizontal excitation is shown inFig. 7. The value at d=0 mm indicates the damping ratioof the model structure without ball packing. In the figure,the difference is not recognized between the dampingratio in both directions. The damping ratio z at d=1mmis slightly larger than that for the model structure withoutball packing. And the damping capacity is improved alittle. And z for d=3, 5, 7, 10 mm ball packing is a littlelarger than without the ball packing case. A large damping capacity improvement is achieved when glass ballsof dH1102210 mm are packed. The damping ratio beesmaximum in the neighborhood of d=12 mm. When ballsize bees large beyond this values, z decreases rapidly.. Packed arrangementThe effect of packed arrangement of glass balls in themodel structure on the damping ratio and the natural frequency are considered here. For these investigations, therelation between the ball size d and the packing ratio bis obtained by experiments and is given in Fig. 8. Thepacking ratio is the ratio of the total volume of packedglass balls to the cavity volume in the model structure.Fig. 8 shows the results of close packing of glass ballsby excitation (excitation packing). Thus, as many glassballs as possible are packed in the cavity of the modelstructure. The packing ratio corresponding to this stateis maximum, and is called the maximum packing ratioin the following discussion. When small spherical particles of a single size pared to the inner size of themodel structure are packed by excitation packing,approximately 60% of the volume is occupied by parFig. 7. Effect of ball size on damping ratio.Fig. 8. Effect of ball size on packing ratio in closepacked structure.ticles and the remaining 40% is the void [8]. Themaximum packing ratio is approximately 60% withind=5 mm, and it decreases for dH113507 mm.Fig. 9 shows the results of observation of the packingarrangements of glass balls in the model structure cavity.The measurement is carried out to examine in detail theball size effect on the damping ratio and the natural frequency. Balls are packed in the form of the facecenteredcubic lattice for the size smaller than d=5 mm and ofthe bodycentered cubic lattice at d=7, 10 mm. A pairof balls is packed at d=12 mm alternately. Balls arepacked in a row for dH1135015 mm. As seen from the above,the maximum packing ratio differs by the ball sizebecause of the different packed arrangements.Fig. 9. Packing arrangements of glass balls.471Y. Wakasawa et al. / International Journal of Machine Tools amp。 Manufacture 42 (2020) 467–472the inner surface, but the dissipation of vibration energyby this collision is small.On the other hand, when the ball size is more than 12mm, the effect of the repulsion coefficient on the damping ratio is large. In this case, a small number of ballsare packed in, leaving much between the packed balls,and the inner surface of the structure. For this reason,collisions between packed balls and the inner surfaceoccur easily. As a result thought to vary, the dampingratio of the str