where 



and a = (Iq when t = 0. 















Bi 



«! 





% 







- e 



2m 



2 



4 1. 63 



% 



+ 



«1 

 2 



















_e 2m 



1 - 



41/63 







^0 





3n- 4^6, 



3ff 



Setting 6 = we obtain the usual solution to the linear damped equation 



a = ^ e 2/71 



The decay curves for free heaving oscillations of the model under test have been calcu- 

 lated for several initial displacements, with and without quadratic damping. The curves were 

 computed at 1/ = 6.1, the natural frequency in heave, using the measured damping coefficients. 



In Figure 17 the amplitudes of successive peaks have been plotted for 1-, 2-, and 3-inch 

 initial displacements. The dashed curves include the quadratic damping while the solid curves 

 do not. It can be seen that the neglect of nonlinear damping becomes more serious as the 

 initial amplitude increases. After the fourth peak, when all the amplitudes are small, all 

 curves are essentially parallel. This indicates that when the motion is small enough they 

 have the same logarithmic decrement or ratio of the amplitude of successive peaks. The rate 

 of decay is then governed by the linear damping coefficient and the quadratic damping influ- 

 ence is negligible. 



For this particular model, the peak of the linear damping curve occurs at about the 

 natural frequency in heave. This would tend to minimize the influence of nonlinear damping. 

 No general conclusions should be drawn since this may be coincidental and other forms may 

 show greater nonlinear effects at the frequency of free oscillation. 



For forced oscillations, the method of "equivalent linearization" can be used. The 

 nonlinear damping term 6 (sgn 3) 3^ is replaced by a linear term such that the work dissipated 

 per cycle by each term is the same; that is. 



(B/z) 3dt= (BJ 1 3 1) idt 



Jo Jo -^ 



For pure harmonic heaving motion, s = 2 sin cot, the above equality can be replaced by 



^2jr rn/2 



z^ oBg cos^ cot d(cj t) = 4 63^0 ^^ cos^ at d((ot) 



Jo •'0 



24 



