ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



By N. W. AKfMOFF, Esq., Visitor. 



[Read at the twenty-sixth general meeting of the Society of Naval Architects and Marine Engineers, held in 



Philadelphia, November 14 and IS, 1918.] 



INTRODUCTION. 



The object of this paper is twofold: — 



1. To present a succinct review of the fundamental results of the investigations 

 of the best known problems of vibratory motion of rods. 



2. To revive the interest of research engineers in Ritz's* remarkable method, 

 published some nine or ten years ago, but comparatively unknown in many engi- 

 neering circles, especially in this country. 



I. SYNOPSIS OF KNOWN RESULTS. 



I. Pendulum. — Everyone knows the formula, 7^= 27t ^|_ ^ where g = ^2.16; I 



is the length (in feet) of the string or weighless rod, supporting the bob; T is the 

 complete period (double) of the oscillations in seconds. This formula holds good 

 only for small oscillations not over 4 degrees or 5 degrees each way. The free 

 period of a loaded spring is expressed by the same formula, except that here instead 

 of / we have 5, the deflection of the spring under its load (in feet). It should not be 

 thought that the similarity of these formulae is due to the fact that 8 corresponds 

 to / ; the true reason lies in the fact that both formulae derive from the same differ- 

 ential equation, except that for the loaded spring this equation is simply written 

 down ; whereas for the simple pendulum it is obtained only after we have agreed to 

 limit ourselves to small oscillations, in view of which assumption the original equa- 

 tion has been simplified ; otherwise it could not be solved in elementary functions. A 

 very simple method has been proposed by the present writer (American Machinist, 

 August 24, 1916) whereby the frequency of a loaded spring can be found in one set- 

 ting of the slide rule. For a compound or physical pendulum, such as a swinging 



rod, we have a different formula : 7 = 27t V— , where k is the radius of gyration 

 about the axis of oscillation, and h is the distance between that axis and the center of 



*Walther Ritz, an illustrious Swiss physicist, died in 1909 at the age of thirty-one. His works have 

 been collected by the Swiss Physical Society and published by the well-known firm, Gauthier-Villars (Paris, 

 1911). They comprise a great variety of papers, all original research, relative mostly to optics and electro- 

 dynamics. Three papers are on the subject of vibratory motion ; they are very difficult, but the sensation 

 created by them was immense, and they immediately took root in the field of modern theories of elasticity. 

 Our exposition of these methods will necessarily be of the most elementary character, 



