. The Bell System technical journal . 106 5 X 103 200 .474 180 5 X 10^ 2,000 .0474 180 5 X 10^ 20,000 .0047 180 In each case the product of AmGm is less than the allowable 200 and, as longas the combination of Gm and the amplification factors for other elementsdoes not exceed the allowable AmGm for those elements, the cushioning issuitable. The precaution to observ^e is that higher-frequency elementsshall not have amplification factors such that A,nGm may be excessive forthem. Amplification Factors for Damped Sinusoidal Acceleration If the outer container of the package is h
. The Bell System technical journal . 106 5 X 103 200 .474 180 5 X 10^ 2,000 .0474 180 5 X 10^ 20,000 .0047 180 In each case the product of AmGm is less than the allowable 200 and, as longas the combination of Gm and the amplification factors for other elementsdoes not exceed the allowable AmGm for those elements, the cushioning issuitable. The precaution to observ^e is that higher-frequency elementsshall not have amplification factors such that A,nGm may be excessive forthem. Amplification Factors for Damped Sinusoidal Acceleration If the outer container of the package is heavy enough (see Section )there will be no rebound and the packaged item will vibrate in the cushion-ing after impact. For linear cushioning with velocity damping, the accelera-tion produced by the vibration will be a damped sinusoid (equation ()and Fig. ). The system to be considered is shown in Fig. determine the efifect of the damped vibration of Wo on the mass nii , 432 BELL SYSTEM TECHNICAL JOURNAL \, 1 (a). Fig. —Idealized system for linear damped cushioning with no rebound.(a) initial position, (b) first contact with floor. 5 3 — — ■ fii = 1 / 1 I/ // ^1 = 5 #/ ^-^^,-— J / vSn/ / / oS* // / \ III ^ III ^i ^ ^ ^ 1 1 L ^ ^ — — y^ ^ = 2 5 s 5 5 = = m 1^ — p ;zf =: ^ = ±J oo = — = .=^ ^~2 -OS 39- i / ml / Sf 1 i 1 / \ // Ol w _ L Fig. —Amplification factors for linear damped cushioning with no = See equations () and (), we note that the equation of motion and initial conditions are identicalwith () except that for the acceleration ^2 we use the damped sinusoid,equation (), instead of the half-sine pulse (). The solution of(), the relative displacement (xi — x^) of nii with respect to tn^,is DYNAMICS OF PACKAGE CUSHIONING 433 ^ = lYy {e-^-^lA sin Ut J^y - 8)+Bsm ic,[t - 7 - f)] /W2«Ji - e-^^^^lA sin (ojo^ + 7 - 5)- 5 sin (wW + 7 + f)]} (
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