ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 137 



which is still nearer the exact solution: — 



The other root would give the first overtone, but with less accuracy. If greater 

 precision is desired for harmonics, it is necessary to take larger values for n, whereby 

 more unknowns to find and a higher order of the corresponding determinant. 



4. Hull as a Free-free Bar. — Applying Ritz's method to the problem of find- 

 ing the fundamental tone of free vibrations of vessels let us consider the hull as a 

 free-free bar of variable cross-section, irregularly loaded. 



Let both the curve of moments of inertia and the curve of loads be parabolic 

 curves, symmetrical with the middle of the ship, which will be our origin ; let the 

 length be = 2/. The moments of inertia will be given by — 



Also the load curve will be 



F = Foil— ex'), 



where b and c are suitable constants. 



The tentative substitution will be of the same form : — 



:y2 = «i ■^.- + «2 4'2 , 



limiting ourselves to but two terms ; the form of the functions ypi will be taken as 

 follows : — 



cosli ki cos —y- + cos ki cosli -y- 



^.•= ; — 



V cos^ ki -\- cosli ^ ki 



where the transcendental equation for hi will be 



tan k + tanh /e = o 



(thus ^p, will be recognized as an even normal function for the case of a free-free 

 prismatic bar.) 



These functions, ;/',, will be easily seen to satisfy the end conditions — 



ii^A =0 and {^ij =0. 



In order to simplify writing let us assume — 



o o 



also 



J fi - dx") xPi" )/'/ dx = a,j; f{i-co(?)xPi i^j dx = pij 



so that the partial derivatives — 



^ = 0; ^ = 0; etc., 

 dai d az 



