Solving for Sgjwe obtain 







Oit cos^ 

















n/2 





sin 



6) 



t 





2 



Sin 



Ct> 



t. 









3 







+ 





3 











0)^ sin 2 (i)< 



+ 



2 4 



= i7^o'"^2 



Jacobsen^^ has adopted this linearizing procedure for the case of forced steady vibra- 

 tions with nonlinear damping. The reasoning was intuitive but was substantiated for several 

 cases by comparing the approximate solutions with either experiments or known exact solu- 

 tions, 



Schwesinger^^ has shown that the linearized equation obtained by equating the work 

 done per cycle yields a "best" one term approximation to the solution; that is, the sum of 

 the forces is minimized rather than being exactly zero. 



With this justification, we can replace the equation for the forced nonlinear vibration 

 by a linear one 



iV + (S^ + 6g)3 + kz ^ P sin at 



where 6 = — 2 



In Figure 18 the equivalent damping coefficient B for 1-inch and 1/2-inch oscillations 

 is shown. A band is indicated within which all test points fall. The mean line and the limits 

 correspond to the measured values of S, = 0.11* ^•^^, For comparison purposes an average 

 curve for the linear damping coefficient B. is plotted. It appears that B^ becomes comparable 

 to S, at higher frequencies and for larger amplitudes of oscillation. 



To assess the effect of the nonlinearity on the motion of a ship acted on by a harmonic 

 heaving force, the uncoupled heave equation is considered. Knowing the frequency dependence 

 of the coefficients, the variation of magnification factor with frequency can be computed when 

 the equivalent linear damping term is neglected. This is shown as the solid curve in Figure 

 19. This frequency dependence can then be recomputed for the nonlinear case by using the 

 mean line in Figure 18 as the equivalent damping term. This cannot be done explicitly since 

 B depends on the resultant amplitude, so that successive approximations must be used. The 

 resultant solutions depend on the magnitude of the harmonic force imposed on the system and 

 these have been obtained for forces equivalent to static deflections of 1 and 3 inches. 



It should be noted that the curves in Figure 19 are not the usual magnification factor 

 plots although they superficially resemble them. Usually the variation with frequency is 

 shown with the ratio of the damping to critical damping held fixed. In Figure 19, however, 

 as the frequency varies the virtual mass as well as the damping ratio is changing. 



26 



