292 



THEORY OF SEAKEEPING 



widely in deriving a differential equation for the free 

 transverse or flexural \'ibrations of uniform slender bars, 

 the differential equation being 



d'y 



d-'/j 



^^ .^4 + "' ?^ = 



d.r 



dt- 



(10) 



where 



E = Young's modulus 



/ = moment of inertia of area of section with respect 



to its neutral axis. 



ij = displacement in vertical plane 



.V = co-ordinate along axis of bar 



?n = mass of bar per unit length 



t = time 



"The solution of this equation for a uniform bar with 

 free ends yields the formula for the natural frec[uencies: 



COs 



EI 

 ml* 



(11) 



where 



Un = circular freciuency 



L = length 



an 'characteristic numbers' arising in the solution of this 



differential equation with these specific 



boundary conditions. 



"As given by Rayleigh in his 'Theory of Sound' the 

 characteristic numbers fall closely in the ratios of the 

 odd numbers starting with 3; that is, 3, 5, 7, 9, etc. The 

 first three characteristic numbers are 4.73; 7.853; and 

 10.996." 



The foregoing expressions were based on the considera- 

 tion of pure flexural deflections neglecting the inertial 

 effects involved in the inclination (or rotation) of beam 

 sections and neglecting damping. Further development 

 consists of introducing shear deflections, rotary inertia, 

 and damping. The damping is composed of the internal 

 damping of the structure and external damping caused 

 by surrounding water. This latter appears to be small, 

 as will be shown later. The following expression is ab- 

 stracted from Kumai (1958), neglecting the external 

 damping considered by him: 



EI[l + 



Edt 



1 + 



Gdtj 



^^v 



-"^{k^V 



Edt 



+ '-M 1 + ^ ^ 



Gdt 



I d.r-dt- 



+m- 1 + 7, ^ 



Gdt 



5^ 

 dt- 



-\- m- 



k'AG 



^=0 



(12) 



where 



G = shear modulus 

 k'AG = effective shear rigidity of hull .section 



r = radius of gyration of mass moment of inertia 

 ^ = coefficient of normal viscosity 

 ■q = coefficient of tangential viscosity 



The solution of equation (12) with suitable boundary 

 conditions gives the freciuencies of various vibration 

 modes. 



.AIcGoldrick et al (1953) cite three DTAIB reports=^ 

 pertaining to theoretical and experimental investigation 

 of vertical vibration characteristics of USS Niagara. The 

 results of digital computations, based on the vibration 

 theory, are given in the following quotation: ". . . The 

 principal facts disclosed were the following: The calcula- 

 tion based on bending only is in fair agreement for the first 

 vertical mode but becomes progressively too high beyond 

 the first mode ; the calculation liased on shear deflection 

 only is quite high for the first mode but becomes progres- 

 sively nearer the true value as the order of the mode in- 

 crea.ses; in the case of USS Niagara the inclusion of ro- 

 tary inertia had a negligible effect on the results. 



"As will be seen from the tabulation in [Mathewson, 

 194:9], the calculations based on shear and bending with 

 rotary inertia neglected check the experimental values 

 up to the sixth mode within 5 percent with the exception 

 of the fundamental mode. 



"From the profile of this vessel it can be seen that its 

 island or superstructure comprising three decks extends 

 for about 30 percent of the length of the hull. When 

 the moment of inertia of this island was added to the 

 moment of inertia previously computed up to the weather 

 deck and the calculation repeated, it was found that the 

 first mode checked within 1 percent but that the re- 

 maining frequencies were all too high. It thus appeared 

 that the stiffening effect of a superstructure of such pro- 

 portions cannot be neglected in the first mode but that it 

 has little effect beyond the first mode. This does not 

 seem at all unreasonable as the first mode is the only 

 one in which bending predominates and the superstruc- 

 ture probably adds very little to the shear stiffness." 



There exists a vast amount of literature on the subject 

 of free vibratif)n of ships. Some of the references are 

 listed in the bil)liography at the end of this chapter and 

 some will be found in AIcGoldrick et al (1953), Csupor 

 (1957), and Lewis and Gerard (1958). Detailed dis- 

 cussion of the subject is outside the scope of the present 

 monograph and the foregoing brief outline was presented 

 merely in order to bring out the salient features of a 

 ship's response to slamming to be discussed fiu'ther in the 

 following sections. 



5.52 Forced and transient vibrations. The term 

 "forced vibrations" is used in the present exposition for 

 the continuously acting excitation such as is caused in 

 ships by propellers and machinery. Equations (10) and 

 (12) apply in this case, provided the time-dependent 

 force P{t) is inserted on the right-hand side in place of 

 zero. This is usually a sinusoidal function. While all 

 vibration modes are excited in principle, the response of a 

 particular mode, with the natural freciuency nearest to 

 that of the exciting cause, strongly predominates. A 

 pi-actical engineering problem is concentrated, therefore, 

 on avoidance of synchronism between any one of the 



"Jasper (1948) and Mathewson (1949, 1950). 



