700 MECHANICAL DESIGN AND PACKAGING 



If the single mass and spring of Fig. 13-7 were to be replaced with a more 



complex system, the single resonant "mode" of Fig. 13-8 would be replaced 



by a number of resonant modes equal in number to the degrees of freedom 



possessed by the system. Each mode of vibration would be characterized 



by a resonant frequency and a ^. It can be seen that for complex electronic 



equipment in an environment of sinusoidal vibration, the most damaging 



structural effects will occur at the resonant frequencies of the various 



modes. 



If the excitation of the system (i.e. the motion of the structure) of 



Fig. 13-7 is random, the response of the mass will necessarily be random; 



its time history, however, will look quite unlike that of the excitation. For 



a single degree of freedom resonant 



system with little damping, the time 



plot of the response is rather similar 



to that of the excitation as viewed 



through a narrow-band filter. A 



brief sample of the acceleration 



response of such a system is shown in 



T? no Mu ^- f c- 1 !-> Fig- 13-9. It can be seen that the 



l^iG. 13-9 Vibration of Single-Degree- ° 



of-Freedom Resonant System Excited curve looks much like a sine wave 

 by Random Vibration. with the resonant frequency of the 



system, but with a continually 

 changing amplitude and phase. 



If the acceleration density of the structure motion is known in the region 

 of the resonant frequency, the rms acceleration of the mass may be ob- 

 tained. 



Let aM = rms random acceleration of the mass, g 



G ^ acceleration density of structure, ^^/cps 



Q = undamped natural frequency of the system of Fig. 13-7 (or of 

 a normal mode of a more complex system), rad/sec 



^ = maximum transmissibility for the corresponding mode. 



Then um = h^JG^. 



To sum up, the response of a resonant system to random vibration is a 

 motion very similar to a sine wave at the resonant frequency of the system. 

 The rms acceleration of the response is directly proportional to the square 

 roots of the acceleration density of the excitation, the ^ of the system, and 

 the undamped natural frequency. 



It was stated in the introduction to this discussion that most fractures 

 resulting from vibration are/aligue failures. The term faligue, by common 

 engineering usage, refers to the behavior of materials under the action of 



