428 ' ON VIBRATIONS AND DEFLEXIONS OF [411
As before, A, C, D may be expressed in terms of B. For shortness we may set 5 = 1, and write
~ - °—~ ' n.........................(19)
We find
-nA + I = 2nWe-n-nl/H,
D = (2nl + 1- e~*nl) e~2nl/ff,
nC+D = - e--ni . 2n2Z2/F. Thus
— = sin ny [e~nx (-nA + l- nx) + enx (nC+D + nDx)\
CLOG
= H'1 sin ny . e~nx [2n2l-e~2nl -nas{l + e~2nl (2nl - 1)}] + H-1 sin ny . en<^ [- ZnH2 + nx [Znl + 1 - e~znl}], vanishing when oc = 0, and when x = l. This may be put into the form
sin ny [2n2l (I - so) e~*nl (enx - e~nx) nl(l-e-*nl}(en(l-x]-e-nt-;K])~\, ............... (20)
in which the square bracket is positive from x — 0 to x — I.
It is easy to see that H also is positive. When nl is small, (19) is positive,
and it cannot vanish, since
ezm > i > i _ Znl.
It remains to show that the sign of w follows that of sin ny when so = 0. In this case
w-(A +0) sin TOT/; ........................... (21)
and
n(A + C)ff = l- e~2nl (2 + 4>n2l2) + e~inl
= e-Znl (#nl + e-2nl _ 2 _ 4n2£2) ................ (22)*
The bracket on the right of (22) is positive, since
Oinl _L O—*M _ y I 1 _i_ .-•_:.:. i _____ _L.
4!
\
... J.
We see then that for any value of y, the sign* of dw/dx over the whole range from x = 0 to x = I is the opposite of the sign of w when x = Of; and since w = 0 when x = I, it follows that it cannot vanish anywhere between. When n = 1, w retains the same sign at x = 0 whatever be the value of y, and therefore 'also at every point of the whole plate. No more in this case than when the edges at x — ± I are merely supported, can there be anywhere a deflexion in the reverse direction.
In both the cases just discussed the force operative at x •=• 0 to which the deflexion is due is, as in (8), proportional simply to dsw/dx8, and therefore to
* [Some corrections have been made in this equation. W. F. S.] f This follows at once if we start 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