374 BELL SYSTEM TECHNICAL JOURNAL 



and 



, = iL+4^' - 1, (1.7.U) 



1 — 

 equation (1.7.9) becomes 



P = K((z+i\. (1.7.12) 



It is seen, by comparison with (1.4.2) that this is Class B cushioning 

 (cubic elasticity). A' is the initial spring rate and c determines the rate of 

 increase of stiffness with displacement. With the notation ^o , '' of Section 

 1.5, we see that 



h = K (1.7.13) 



. = ^p. (1.7.14) 



Hence equations (1.5.6) and (1.5.4) may again be used to calculate maximum 

 acceleration and displacement. B has the same meaning as before (Eq. 

 1.5.3). 



To predict the performance, in the vertical direction (Fig. 1.7.1), of an 

 existing tension spring package the same procedure as outlined in Section 

 1.6 may be used, except that it is not necessary to have a load-displacement 

 curve for calculating y^o and r. Instead, these parameters may be calculated 

 directly from equations (1.7.10), (1.7.11), (1.7.13) and (1.7.14). The 

 remainder of the procedure is the same as in Section 1.6(d). 



To predict the performance perpendicular to another face, say AEHD 

 of Fig. 1.7.1, it is only necessary, in the calculation of ^o and r, to substitute 

 x'q for .vo , (' for ( (see Fig. 1.7.1) and, in place of b: 



b' = \ - j,{\ - b). (1.7.15) 



The initial spring rate A' for any direction of acceleration may be calcu- 

 lated from the initial spring rates Ai , A2 , A'3 in the three directions normal 

 to the faces of the frame by using the relation 



1 s- t^ u 



W'^Kl^Kl^ A3 



2 -I- -2 -I- -2, (1.7.16) 



where 5, /, u are the direction cosines of the acceleration direction with 

 respect to the normals to the faces of the frame. It is seen, from (1.7.16), 

 that the spring rate is given by the radius to the surface of an ellipsoid whose 

 principal semi-axes are A'l , A2 , A3 . 



