【正文】 mprovement of machine structures by closepacking with ballsYasunori Wakasawaa,*, Masatoshi Hashimotoa, Etsuo MaruibaDepartment of Mechanical Engineering, Toyota National College of Technology, 21 Eiseicho, Toyotashi 4718525, JapanbFaculty of Engineering, Gifu University, 11 Yanagido, Gifushi 5011193, JapanReceived 12 February 2020。 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. 6. Effect of ball size on natural frequency.470 Y. Wakasawa et al. / International Journal of Machine Tools amp。 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 avera