1920.] 
Jenkinson.—Design of Laminated Springs. 
19 
Again, . 
1 
x 2 x 3 
/y* 2 /y» 3 
*a/ 
y BH Ww + 1 V w + 1 2 6 Ww V w 2 6 
the constants of integration vanishing, and putting x = l n 
_ 1 Jw (i k 2 k 3 
EI i + 1 V w + 1 2 6 / 3 
( 1 ) 
Similarly the deflection of the (w-f-l)th leaf at its extremity is given by 
W n + 1 fin + 1 ^«) 3 
yn +1 — yn d - i^n + 1 ^ d - 
EI 
Substituting, we have 
ln +1 ^n 
i -/t.-p jl -iif j ( I 7 ^n t (^w + 1 ^n) 
yn-{-l yn V - ” w + ly^ + 1 bz. ^ d~ ^ 
3 
l 2 
•)- w -y} "( 2 ) 
Further, by substituting for y n in (2) or by integrating as follows for 
section Y-Y equation (3) is derived:— 
1 
R 
d 2 y 1 f 
dY 2 = EI ( 
W n + 1 
i^n + 1 *^) 
dy_ 
dx 
~'W B+1 ; In 1 ' ' - 
dy 
W l 2 
VV 
since has the value already indicated above when x = l, 
and 
6 
since y has the value already indicated above when x = 1, 
So that when x = l n + i 
yn + 1 — 
= t. 1 w 
l 
EI 
n + 1 
’n + 1 
- w. 
% + 1 In 
b n 
6 
( 3 ) 
Continued application of these three equations give successive values 
of W l5 W 2 , ...... &c., for any values of l lt l 2 , ..&c., and hence the 
stress in successive plates can be arrived at. 
In designing a spring by the aid of these equations the following results 
are naturally aimed at : (1) that the maximum stress in each plate is equal ; 
(2) that each plate is stressed to the maximum allowable amount through 
its whole length. The first consideration gives a spring of maximum strength 
and the second gives one of maximum flexibility. It soon becomes evident 
that neither of these requirements can be fulfilled by square-ended flat 
