*A 



kx ex 



x±i 



Fit) 



Figure 5. Definitive sketch, single degree of 



freedom, mechanical vibration system. 



Ideal damping models produce results which satisfactorily predict the response 

 when the damping force is formulated as being proportional to the velocity. 



F^ = c 



3x 

 3t 



(15) 



where c is a constant of proportionality. Equation (14) thus becomes 



3 2 x 3x 



+ c — + kx = F(t) 



3t 2 3t 



(16) 



If F(t) = 0, the homogeneous differential equation results whose solution 

 corresponds physically to that of damped free vibration. However, when the 

 system is subjected to excitation by a harmonic force (e.g., F sin tot), the 

 system becomes one of forced vibration with linear damping. 



3 z x 3x 



+ c — + kx 



3t 2 3t 



(17) 



where F is the magnitude of the periodic forcing function, and 

 frequency of its application. 



is the 



The solution of equation (17) consists of two parts — the complimentary 

 solution, which is the solution of the homogeneous equation corresponding to 

 the case of damped free vibration, and the particular solution of interest 

 representing forced linearly damped vibration. The particular solution is a 

 steady-state oscillation of the same frequency u> as that of the excitation, 

 and can be assumed to be of the form 



x = X sin(wt 



8) 



(18) 



where X is the amplitude of oscillation, and 6 the phase of the displace- 

 ment with respect to the exciting force. The amplitude and phase of equation 

 (18) can be found by substitution into the differential equation (17). In 

 harmonic motion the phases of the velocity and acceleration are ahead of the 

 displacement by 90° and 180° , respectively; hence, it can be easily determined 

 that (Thomson, 1972) 



35 



