Vibration of Bars treated simply. 579 



a wider range of students who are concerned rather with 

 the physical than with the mathematical aspect of the subject. 

 With this object in view the following method has been found 

 useful. It avoids the calculus of variations ; moreover the 

 simplifications made finally by Lord Rayleigh when pro- 

 ceeding to a solution are here made at the outset, and the 

 treatment of the terminal conditions is also simplified. 

 Further, the solutions of the transcendental equations leading 

 to the frequencies of the tones possible to any bar are 

 exhibited graphically. Of course the treatment of the ro- 

 tatory energy as negligible becomes nearer the truth as the 

 bar is made thinner ; but the experiments on a bar half 

 an inch thick with nodes only 10 cms. apart, given at the end, 

 show a fairly close agreement with theory. 



Assumptions. — Consider the case of a bar of uniform cross- 

 section, not subject to tension and straight when left at rest 

 free from external forces. 



Suppose further that the curvatures are so small as to be 

 representable by d^y/dx' 2 , and to leave the lengths along the 

 bar and along the axis of x practically equal. Finally let 

 the rotatory inertia of the elements of the bar be neglected 

 in comparison with the inertia of translation. That this 

 holds closer and closer as the bar is made thinner is seen 

 from the fact that the energy of translation varies as the 

 thickness simply, while that of rotation varies as the cube 

 of the thickness, since it involves the moment of inertia of 

 the section. 



Bending Moment. — In attacking the problem we need an 

 expression for the bending moment or couple required to bend 

 a bar to a given curvature of radius R say. Let the bar be 

 bent in the plane of xy, then there is in it, perpendicular to 

 this plane, a neutral surface which is neither extended nor 

 contracted. Outside this surface extension occurs, inside it 

 contraction. Consider an element of the bar of length x, 

 distant 97 radially from the neutral surface. Then, by the 

 bending of the bar to the radius R, the length of this element 

 becomes x + hx, where 



x + Bx _ x _ Bx 

 R + 77 ~~ R ~~ V 



Thus, the fractional elongation of the element is given by 

 Bx/x = 7)111. Hence, by definition of Young's modulus, which 

 we denote by g, the tensile force on this element of cross- 

 section dm is qrj dco/R. The bending moment is, of course, 

 found by multiplying this expression by 77 (the distance of 

 the force's point of application from the neutral surface) and 



