つるまきばねを伝わる弾性波について

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Helical springs have vast applications in mechanical and electromechanical systems, e. g. shock absorbing and shock isolating constructions, mechanical filters and delay lines. It is, therefore, of fundamental importance to understand the wave propagation along a helical spring. Yoshimura and Murata [5] investigated the propagation characteristics on an infinite helical spring, which was analitically treated but with the approximation of zero helix angle, though they mentioned the effect of the helix angle. The assumption of zero helix angle decouples the longitudinal motion of the spring from the torsional motion with respect to the axis of the coil. The purpose of the present paper is to investigate the wave propagation along an infinite helical spring with a helix angle, and to see its influence on the phase velocities and the displacement characteristics based on the complete set of the equations. The characteristic equation developed in terms of the frequency parameter shows six possible waves. The propagation velocities at different frequency parameters are numerically computed for the helix angle α=0゜, 5゜ and 10゜ with the ratio of the coil radius to the rod radius R_r=10. The dispersion curves are shown in Figs. 2(α=0゜), 3(α=5゜) and 4(α=10゜). The relative displacements B, V^- and W^- are also shown in Tables 1(α=0. 5゜), 2(α=5゜) and 3(α=10゜). Small value of α(=0. 5) is chosen for this case, instead of α=0, because V^- and W^- are separated or uncoupled when α is zero. The relative displacements indicate the nature of the vibration and the degree of the coupling. Refering to [6] we may calculate from the static deformation, the normalized propagation velocities of the longitudinal and the torsional motion at zero frequency, that is, C_<pt>=0. 043 and C_<pt>=0. 05 for the zero helix angle when R_r=10. They are indicated in Fig. 2, and fairly meet the two higher velocities. From the relative displacements we find that these two velocities in the lower frequency range correspond to the torsional and the longitudinal motion of the spring. However this distinction of torsional and longitudinal nature is completely lost at the frequency parameter √<K>=0. 342 and √<K>=0. 308, where the half wavelength equals to the circumference of the coil. This is consistent with the fact that Kagawa's result [6] for finite springs showed no transmission beyond these frequencies. Since Kagawa was paying attention only to the torsional and the longitudinal waves, the absence of these waves in a distinct manner must be interpreted as non-transmission in his case. When α is not zero, the above two kinds of modes are not independent at all but are coupled. As seen from Fig. 3 and 4, the higher two branches of velocities in the lower frequency range are shifted from the static ones. when the helix angle is small (α=0. 5゜) as in Table 1 (K=0. 03), V^-/W^- =-4. 81×10^<-2> for C_p=4. 75×10^<-2> while V^-/W^- =26. 1 for C_p=4. 121×10^<-2>. The first case is predominently torsional and the second case predominently longitudinal where the coupling between both modes is small. For α=5゜ in Table 2, however, we can not say that the vibration is predominently torsional for the first case. This tendency is more pronounced for larger α. Except higher two branches of velocity in the lower frequency range discussed above, most modes are involves with the bending of the rod coupling with the torsion. Some modes in the figures are inclined at about 45゜degrees throughout the frequency range. That is, the propagation velocities are almost proportional to the frequency. There are also some other modes whose velocities are not much effected by frequency. It is seen from the figures and the tables that these two type of modes are coupled, at a certain frequency parameter around √<K>=1.
社団法人日本音響学会の論文
1970-12-10

つるまきばねを伝わる弾性波について

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