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BELL SYSTEM TECHNICAL JOURNAL 



ment curve consisting of two straight line segments. The load displacement 

 function is (see Fig. 1.4.4) 



P = koXo ^ X-2 ^ (Is 



P = kbX2 — (^6 — kn)ds X2 ^ d, 



(1.4.4) 



It is useful especially in situations where very abrupt bottoming is possible. 



Class E — Hyperbolic Tangent Elasticity. When the mechanism of the 



cushioning is such as to hmit the maximum force that can be transmitted 



over a considerable displacement range, the load-displacement function 



P = Po tanh 



ko X2 



(1.4.5) 



is useful. Po is the asymptotic value of the force and ko is the initial spring 

 rate (see Fig. 1.4.5). 



Fig. 1.4.5 



Fig. 1.4.4 — Bi-linear elasticit}'. Class D. 

 Fig. 1.4.5 — Hyperbolic tangent elasticity. Class E. 



Class F — Anomalous Elasticity. In occasional instances the load-dis- 

 placement curve of the cushioning cannot be matched accurately enough 

 by any of the five preceding functions. In such cases a numerical integra- 

 tion procedure can be used, as described in Section 1.15. 



1.5 Cushioning with Cubic Elasticity (Class B) 

 Substituting (1.4.2) in (1.2.15) and performing the integration, we have: 





Now, let 



V 



2W2h 



ko 



(1.5.1) 



(1.5.2) 



