inertial correction. Phase angles at 1 knot are not presented for this reason and also be- 

 cause the moment signal was unsteady in the neighborhood of &j = 4 rad/sec at 1 knot. 



Applying Equation [4b] to these data yields the coefficients d and e. In Figure 12a, 

 the product doi^ (which is proportional to the moment on the model) has been plotted in order 

 to better present the data at high frequencies. In Figure 12b a nondimensional moment co- 

 efficient c?'= ^ has been used. The nondimensional coefficient e '= — V q/L is given in 

 AL ^ A ' 



Figure 13. Unlike the coefficient d^ the value of e' at 1 knot is given because it is insensitive 

 to errors in phase angle. 



SECOND HARMONICS IN THE LIFT FORCE 



It is commonly assumed in ship-motion problems that the hydrodynamic damping varies 

 as the first power of the velocity. This is true of bodies moving in a fluid at relatively low 

 velocities but at larger velocities a dependence on the second power of velocity more nearly 

 describes the physical phenomenon. Such an assumption destroys the linearity of the system 

 and necessarily complicates the solution of the problem. Therefore, it would be of interest to 

 examine the lift data in these experiments in order to determine the validity of the usual 

 linearizing assumptions. 



The existence of square law damping would be indicated by significant second harmonic 

 content in the lift force trace. This can be shown as follows: assume a damping force of the 

 form 6- (ds/dt)^. Then, if the ship is constrained to perform heaving oscillations s = Sq sin 

 cot, the measured damping force will be 



.„ , „ 1 + cos 2cot 



Thus the observed amplitude of second harmonic in the lift force will be 



B2 '0 -' 



Unfortunately, the present experiments do not lend themselves readily to such meas- 

 urements. The second harmonic is largely masked by the inertia, buoyancy, and fundamental 

 of the damping force. It only becomes visually apparent on the records at the frequency where 

 the inertia and buoyancy forces cancel one another. If one were to redesign the experiment 

 to accurately determine this component, the fundamental should be filtered-out and the sec- 

 ond harmonic amplified greatly. 



All lift curves at frequencies above 7 rad/sec and at speeds of 0, 1, and 2 knots were 

 harmonically analyzed in order to obtain accurate phase lags at the fundamental frequency. 

 This analysis also yielded the amplitude of the second harmonic. In addition a few tests at 



17 



