1916]
MEMBRANES, BARS, AND PLATES
429
sin ny, and is of course in the same direction as the displacement along the same line. When n — 1, both forces and displacements are in a fixed direction. It will be of interest to examine what happens when the force is concentrated at a single point on the line x = 0, instead of being distributed over the whole of it between y — 0 and y — IT. But for this purpose it may be well to simplify the problem by supposing I infinite.
On the analogy of (7) we take
w = %An(l + nx) e~nx sin wy, ..................... (23)
making, when as = 0,
d*w/da? = 2XwM n sin ny ......................... (24)
If, then, Zi represent the force operative upon dy, analysable by Fourier's theorem into
Z = Z± sin y + Z* sin 2y + Za sin By +• . . ., ............ , . .(25)
we have
2 f*" 2
Zn = ~ Z sin ny dy = - Z, sin ny, .................. (26)
ifj Q ' tr
if the force is concentrated at y = y. Hence by (24)
A -Z* sinm? jlM~ TT n* '
so that
w. ft aq^yr.?.) :?«*"(»+ a e-»«(i + M)> ......... (28)
'2tir n
where n=l,2, S, etc. It will be understood that a constant factor, depending upon the elastic constants and the thickness of the plate, but not upon n, has been omitted.
The series in (28) becomes more tractable when differentiated. We have dw xZ cosn(y-r))-CQ8n(y + ri) . .
—j — — — -jr - £/ - ~ — -- V ) ......... \&O I
dx 2-7T n '
and the summations to be considered are of the form
Sn-1 Gosn/3e-nx ............................ (30)
This may be considered as the real part of
2n-le-*<«-W, .............................. (31)
that is, of
-log(l-e-«»-*5>) ............................ (32)
Thus, if we take
........................ (33)
so that
(34)tart from x = Z where w = 0.suppose that the plate is clamped at x= ± I, instead of merely supported. The conditions are of course w = 0, dw/dx= 0. They give