Some Problems of Shocks transmitted in Bars and in Plates

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The transmission of shocks in bars and in plates, when the material is visco-elastic, together with the behaviours of such shocks, when there are various kinds of boundaries or conditions, has been studied. The method of analysis in this is, in the first place, to derive the solutions in the form of wave transmission from the modified equations of motion of bars or of plates in the case where the material is visco-elastic, and secondly to apply on these solutions various kinds of boundary conditions appropriate to respective cases. The problems contained in this paper are enumerated as below: Part I Shocks transmitted in bars. A. A note on forced oscillations and free waves. B. Diffusion of shocks from some disturbed origins. C. Reflection and transmission of shocks at some discontinuities. D. Anomalous dispersion of harmonic shocks of high frequency. E. Transmission of shocks in tension and compression bars. F. Remarks on the propagation of the statical deformation and on the failing section of a beam due to shocks. Part II Shocks transmitted in plates. G. Diffusion of shocks in two dimensions from some disturbed origins. H. Shocks transmitted along the edge of a plate. I. Reflection and refraction of shocks at some discontinuities. J. Scattering of shocks due to a circular obstacle. K. Diffraction of shocks from a narrow slit. L. Transmission of shocks of high frequency, the azimuthal variation being taken into account. In Section A it is explained why it is more easy and reasonable, in the problem of shocks in elastic body, to employ the equation of free waves rather than to apply the ordinary method of solving the problem of the forced oscillations. In fact, when the forces act on a few points of the body, the natures of deformation seem to be those in the transmission of shocks in a form of progressive waves from the disturbed portion. This conception is clearly understood if the decaying nature of waves in visco-elastic body is taken into consideration. Section B deals with the behaviours of waves transmitted from the disturbed portion of the initial shocks. Further, it is shown, on the simplest case of the transmission of harmonic disturbances, that the motions are composed of two kinds, one of them being of the transmitted waves and the other of the decaying motion near the origin of shocks. Section C contains some investigations on the reflection of the transmitted waves or on the transmission of these waves at certain discontinuities of the density or of the elasticity. The result, which comes out, answers to the question that what part of the energy of waves will be transmitted over the discontinuity and what kind of waves will be formed at that portion. Thus from this investigation it is found that, of whichever side the material may be rigid, the more different the elasticities of both sides, the more distinct becomes the reflection phenomena, densities of both media being taken the same. Again, from the detailed investigation it appears to be unlikely that the energy of waves generated from the origin is kept constant; for, when the waves are transmitted along very rigid bar towards the discontinuity connected with a soft bar, we can discover a new energy propagated along the soft bar, besides the energy reflected totally at the discontinuity. Section D is devoted to the study of the dispersion of waves having the finite length of a train. In this it is described that waves with limited extent are propagated with the group velocity of this system, keeping the original extent of the disturbance constant, while the oscillatory forms are transmitted with a velocity peculiar to the wave length of the oscillatory part, the oscillatory nature disappearing at the extreme ends of the disturbed portion. The problem discussed in Section E is that of the flexural waves when the bar is compressed or tracted lengthway. The result tells us that a bar under tension is a good transmitter of shocks both in quickness and in the conservation of the original active energy. In Section F the analysis developed in some of the previous sections is adapted to the problem of the failing section of a beam due to shocks. We can determine the breaking-down section, when we consider the resultant of the shock waves and the transmission of the statical deformation. The remaining six sections deal with some complex problems in connection with the shocks in plates. In Section G a few cases similar to Section B have been dealt with. In Section H the author has discovered a new type of flexural waves transmitted along the edge of a plate. Such waves have a constant velocity which is somewhat less than that of plane flexural waves transmitted in a plate. The remaining portion of this section is occupied with the transmission of such waves originated from a point in the interior of the plate. Diverging in two dimensions from the origin, we must acquire at a great distance from the source a continually increasing preponderance of the edge waves given above. In Section I it is contained that how the shocks of plane type are reflected and refracted at the discontinuity line. Though the angles of reflection and refraction are the same as those in the ordinary law, the effect peculiar to the discontinuity region is not without importance. The extreme case of the reflection at the free edge has a close connection with the problem given in Section H. Section J and K give the problems of scattering and diffraction which are interesting from the point of view in physical mechanics. Detailed explanation will be seen in the paper. Section L treats of the transmission of harmonic shocks of very high frequency. We will find in this section many properties of waves which can never be obtained in the ordinary method of attacking the flexural waves. The fact that the energy of short wavelength accumulates on the surface of the plate and that there is a higher limit of velocity of short waves, are all without significance.

Some Problems of Shocks transmitted in Bars and in Plates

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