A New Appraisal of Strip Theory 



w = Zo + "o^- (26) 



w = ■z^+ u^S. (27) 



Substituting Eqs. (26) and (27) in Eqs. (23) and (24), calling q = ^ and q = and 

 rearranging terms in Eqs. (17) and (18), we finally obtain the coupled pitching 

 and heaving equations of motion in the form: 



(m-Z.)z^ + Z,z„ - Z^^z„ - (Z.+mxQ)^- (Z^ + u^Z.)^ 



- (Zg + u„Z^)0 = Z^(h,h,h,t) , (28) 



- Mz Zq = M^(h,h,h,t) . (29) 



If the above set were developed on the basis of Eqs. (19) and (20), i.e,, for the 

 origin of the 'TDody" axes at the center of gravity, then the resulting equations 

 would differ in form from Eqs. (28) and (29) only in that the mx^ term would be 

 missing in the coefficients of pitch and heave accelerations in Eqs. (28) and (29). 



CALCULATION OF THE COEFFICIENTS 



The solutions to Eqs. (28) and (29) are in principle easily obtained provided 

 that the twelve coefficients and the two forcing functions are known. Although, 

 rational treatment of the problem does provide a precise identification of each 

 term, the present state of art permits only rough approximations on the basis of 

 theory and it is to this end that resource will be made to strip techniques. De- 

 ferring the discussion of the exciting terms until the next section, attention is 

 here focused on the forces and moments brought about by the ship's oscillatory 

 motion in calm water, and which are identified as the terms on the left hand side 

 of Eqs. (28) and (29). 



In order to exhibit the relationship between the new approach and that of 

 Korvin-Kroukovsky, we shall, without loss of generality, substitute for each co- 

 efficient of Eqs. (28) and (29) a letter and thus obtain the set: 



a(^e) z'o + b(^e)^o + cz„ + d(aj^)0 + e(oj^)'d + f0 = Z^(h,h,h,t) (30) 



A(c^^)0 + B(a^^)0 + C0 + D(a;J z^ + E(co^) z^ + Fz^ = M^(h,h,h,t) (31) 



where co^ is the frequency of encounter with sinusoidal exciting waves and hence 

 the frequency of the forced responses also. 



In the interest of brevity and since the assumptions and steps to be followed 

 in the computations by strip theory will be similar for all coefficients, we shall 

 next examine in detail the manner in which one of the coefficients of the equation 



331 



