. The Bell System technical journal . p = 2 1 1 V Ao ^ b d V ^ ^ >^ ^ ^v sa^ s a ^^ < \^ ^\ -^3 .5 Fig. —Acceleration-time curves for cushioning with bi-Unear ,/do = See equations () and (). As before, if the package does not rebound, the acceleration shown is mir-rored in the time after each half cycle, to form a vibration of period 2x2 • It is useful to know the duration of the complete pulse {aa in Fig. )and also the duration of bottoming {bb in Fig. ). Calling the formerT2 and the latter tb , we have, from equatio
. The Bell System technical journal . p = 2 1 1 V Ao ^ b d V ^ ^ >^ ^ ^v sa^ s a ^^ < \^ ^\ -^3 .5 Fig. —Acceleration-time curves for cushioning with bi-Unear ,/do = See equations () and (). As before, if the package does not rebound, the acceleration shown is mir-rored in the time after each half cycle, to form a vibration of period 2x2 • It is useful to know the duration of the complete pulse {aa in Fig. )and also the duration of bottoming {bb in Fig. ). Calling the formerT2 and the latter tb , we have, from equations () and () DYNAMICS OF PACKAGE CUSHIONING 415 -X, ^ /- \ K h Tn — Z O .^ iT \ t/. / r-^ < ^C :X / --^ <; h^ y z V r ./ < ^^ / ^ V ^ ^ \ ^ ,oJ / \ ^ k^._,. — ^ s \ ^ ^/ D \ A s So |k» % 0 doFig. —Pulse durations for cushioning with bi-linear elasticity. See equations - = - sm - + To TT Co To () and () ^6 1 1 tan. () () These two equations are plotted in Fig. for several values of ^b/^o • Acceleratiox-Tme Relation for Hyperbolic Tangent ElasticityThe relation between acceleration and time for hyperbolic tangentelasticity is found by the same procedure that was used for tangent elasticity 416 BELL SYSTEM TECHNICAL JOURNAL in Section The system considered is that shown in Fig. and theload displacement curve of the cushioning is given by P=Potanh^^ Substituting the above expression for P in the energy equation (), wefind the velocity to be ^2 = A/lgh - ^ log cosh ^-^. () Then, as before, r^ dx2t = — () and the half period (72) of the motion is twice the time required for X2 to increase from 0 to dm , or C^* dx2T2 = 2 —, () Jo ^2 where, from Section , The radian frequency of the acceleration is defined as IT 032 = — T2 an J this is to be compared with the frequency TT coo = — = To Y W2 t
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