DYNAMICS OF PACKAGE CUSHIONING 451 



4.3 Effect of Distributed Mass and Elasticity, of an Element 

 OF THE Packaged Article, on the Amplification Factor for 

 A Half-Sine -Wave Pulse Acceleration 



In this section we shall determine the contribution of the higher modes of 

 vibration of a structural element to its total response to a half-sine-wave 

 pulse acceleration. For the shape of the element, we choose a prismatic 

 bar because this leads to the simplest mathematical formulation of the 

 problem and such a bar is also a common structural element. Other con- 

 siderations influence the choice of direction of acceleration with respect to 

 the axis of the bar. The transverse direction (cantilever) is the most 

 practical from a physical standpoint, but, for purposes of comparison with 

 the one-degree-of-freedom system, the parallel (axial) direction of accelera- 

 tion is the more logical. Both problems lead to solutions in the form of 

 infinite series, but, in the latter case, the expression for the strain at a fixed 



-V 



/ 



Fig. 4.3.1 — The system studied in Section 4.3 depicted at the instant of contact with 



the floor. 



end can be summed in terms of elementary functions without difficulty. 

 Since it is necessary to determine maximum values of strain over a wide 

 range of frequency ratios for the plotting of an amplification factor curve, an 

 enormous reduction in the time required for accurate computations is 

 obtained by choosing the axial case. Furthermore, the axial case appears to 

 contain the essential features which might result in differences between the 

 response of a one-degree-of-freedom system and a continuous one. 



The complete system to be studied is illustrated in Fig. 4.3.1. To the 

 mass W2 , supported on massless cushioning of constant spring rate ^2 , 

 is attached one end of an elastic prismatic bar, of length /, cross sectional 

 area A, modulus of elasticity E, and density p, with its axis oriented verti- 

 cally. The system is dropped from a height h so that its velocity is v 

 at the instant of contact of the cushioning with the floor. The mass of the 

 bar is supposed to be small in comparison with nh and perfect rebound is 

 assumed, so that the motion of mo during contact is a half-sine wave of 

 frequency 



