【正文】 6. Effect of ball size on natural frequency.470 Y. Wakasawa et al. / International Journal of Machine Tools amp。 Damping energy。International Journal of Machine Tools amp。 Vibration。 Manufacture 42 (2020) 467–472Fig. 1. Experimental apparatus.the relation between the shape and size of the modelstructure and the packed balls. The ball arrangement andpacking ratio were obtained for various ball sizes. Theeffects of these factors on damping characteristics werealso examined, along with the effect of the excitationdirection on the damping characteristics.2. Experimental apparatus and methodThe experimental apparatus is shown in Fig. 1. Themodel structure used consisted of drawn square pipes ofstainless steel (SUS304 in JIS) with outer side 25 mm,thickness mm (inner side b=22 mm), and lengthL=300–1100 mm. Glass balls of various diameters dwere closely packed by hammer excitation. Both ends ofthe model structure were sealed tightly using methacrylicplastic plates 5 mm in thickness.Model structures were suspended by stainless steelwires at the position of nodes of the fundamentalvibration mode, and the center was impacted by animpulse hammer. The impulsive force and the outputsignal from a small and light accelerometer bonded atthe position of 10 mm from the end were transmitted toan FFT analyzer through a charge amplifier and bandpass filter. The impulsive force and the time history ofacceleration, the calculated damping ratio from frequency response curve etc. were examined.The experiments were carried out for various valuesof L=300–1100 mm, d=1–20 mm and F=50–300 N asshown in Table 1. Damping characteristics in two exciTable 1Model structure and impulsive forceSquare pipe Size (inner) 2525 (2222) mm(SUS304 in JIS) Thickness t=Length L=300–1100 mmPacked material Glass ball diameter d=1–20 mmand sizeImpulsive force F=150 Ntation directions (vertical and horizontal) were estimatedby the damping ratio, which was obtained by the halfpower method, using the frequency response of the fundamental mode. The following experimental results arethe average value of approximately ten data.3. Experimental results and considerations. Damping characteristics of square pipe withoutball packingWe first examined the damping characteristics of thesquare pipe without the ball packing by changing themagnitude of the impulsive force F and the length the case of square pipe without ball packing, only theresults of the vertical excitation were given, since thedamping characteristics are not affected by the excitation direction.The effect of the impulsive forces F on damping ratioz is shown in Fig. 2 in the case of L=500 mm. Dampingratio z increases slightly with the increase in the impulsive force F. However, the effect of F on z was notobserved within the impulsive 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) 467–472the 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