698 



MECHANICAL DESIGN AND PACKAGIIsfG 



Structure 



Fig. 



13-7 Mechanical System with 

 Single Degree of Freedom. 



force is applied to a linear mechani- 

 cal system, it will vibrate (in the 

 steady state) at the frequency of the 

 applied force, with an amplitude de- 

 pending upon the magnitude and 

 frequency of the force, and upon the 

 characteristics of the system. Let 

 Fig. 13-7 represent a single degree of 

 freedom translational system con- 

 sisting of a mass mounted upon a 

 structure through a spring and a 

 viscous damper. The motion of the 

 structure provides the excitation for 

 the system. 



In the figure, let 



M = mass, slugs or lb sec^/ft 



c = damping coefficient, lb sec /ft 



K = spring rate, lb /ft 



Cc. = 2^1 KM = critical damping coefficient, lb sec /ft 



^ = ^]KM = undamped natural frequency, rad/sec. 



The response of the mass M to a steady sinusoidal motion of the sup- 

 porting structure is shown in Fig. 13-8 as a function of frequency and of 

 the ratio of the given damping to the critical damping. The ordinate of the 

 figure represents the displacement transmissibility, that is, the ratio of the 

 maximum displacement of the mass M to the maximum displacement of 

 the structure. It can also be taken to represent /orcd" transmissibility, the 

 ratio of the maximum transmitted force to the maximum applied force. 

 The abscissa is the ratio of the forcing frequency to the undamped natural 

 frequency of the mass spring system (obtained by assuming infinite 

 structural mass). This figure shows that at low frequencies the spring acts 

 in a rigid manner, causing the mass nearly to reproduce the motion of the 

 structure. At high frequencies the spring more or less isolates the mass from 

 the excitation. At intermediate frequencies, close to the undamped natural 

 frequency, the maximum displacement reaches values large compared to 

 the excitation. This condition is known as resonance; the frequency at 

 which maximum displacement is reached, incident to a sinusoidal force of 

 constant amplitude, is known as the resonant frequency . When the damping 

 is small, as is usually the case with mechanical vibrations, the resonant 

 frequency nearly equals the undamped natural frequency. 



