MEMBRANES, BARS, AND PLATES
Along the perforated portion AS the couple required to produce the one-dimensional bending fails. The actual deformation accordingly differs from the one-dimensional bending by the deformation that would be produced by a couple over AB acting upon the plate, as clamped along GA, BD, but otherwise free from force. This deformation is evidently symmetrical with change of sign upon the two sides of GD, w being positive on the left, negative on the right, and vanishing on AB itself. Thus upon the whole a downward force acting on the left gives rise to an upward motion on the right, in opposition to the general rule proposed for examination.
If we suppose a load attached at the place where the force acts, but that otherwise the plate is devoid of mass, we see that a clamped plate vibrating freely in its gravest mode may have internal nodes in the sense that w is there evanescent, but of course not in the full sense of places which behave as if they were clamped.
In the case of a plate whose boundary is merely supported, i.e. acted upon by a force (without couple) constraining w to remain zero*, it is still easier to recognize that a part of the plate may move in the direction opposite to that of an applied force. We may contemplate the arrangement of fig. 2, where, however, the partition GD is now merely supported and not clamped. Along the unperforated parts GA, BD the plate must be supposed cut through so that no couple is transmitted. And in the same way we infer that internal nodes are possible when a supported plate vibrates freely in its gravest mode.
But although a movement opposite to that of the impressed force may be possible in a plate whose boundary is clamped or supported, ib would seem that this occurs only in rather extreme cases when the boundary is strongly re-entrant. One may suspect that such a contrary movement is excluded when the boundary forms an oval curve, i.e. a curve whose curvature never changes sign. A rectangular plate comes under this description; but according to M. Mesnagerf, "M. J. R6sal a montre" qu'en applicant une charge au centre d'une plaque rectangulaire de proportions convenables, on produit tres probable-ment le soulevement de certaines regions de la plaque." I understand that the boundary is supposed to be " supported " and that suitable proportions , are attained when one side of the rectangle is relatively long. It seems therefore desirable to inquire more closely into this question.
The general differential equation for the equilibrium of a uniform elastic plate under an impressed transverse force proportional to #isj
=Z. .......... . .......... (3)
* It may be remarked that the substitution of a supported for a clamped boundary is equivalent to the abolition of a constraint, and is in consequence attended by a fall in the frequency of free vibrations.
f C. JR. t. CLXII. p. 826 (1916).
$ Theory of Sound, §§ 215, 225 ; Love's Mathematical Theory of Elasticity, Chapter xxn.iv. p. 88.