130 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



work of inertia forces to the increase of the potential energy. This will lead to the 

 desired variational equation, which it will be our immediate aim to reduce to the or- 

 dinary problem of maxima and minima. 



5. Two Propositions from Calculus; both are no doubt well familiar to the 

 reader. 



(a) The minima or maxima values of a function, u, of many independent 

 variables, x,y, 2, .... is found in rather the same manner as when there is only one 

 such variable, x. We first find the partial derivatives with respect to each indepen- 

 dent variable, and equate them to zero. 



du du du 



and, further, ascertain whether or not the second derivatives — 



d'^'u d^u d'^u 

 ~d^^' ~df' ^' ' 



all have the same sign. If they are all negative, with the values of x,y,z, ... found 

 from the above system of first derivatives substituted in them, we have a maximum ; 

 otherwise, we have the minimum value of the function u. 



(b) According to Leibnitz's well-known rule, if we have a definite integral of a 

 function involving an arbitrary parameter, a, in order to find the derivative of the 

 integral with respect to a we simply diflferentiate under the integral sign. Thus — 



da J J da 



a a 



This only applies to the case when the limits are not functions of a, and even then 

 not without certain restrictions, in which,, however, we are not here immediately in- 

 terested. 



6. End Conditions. — It will be remembered, from the Strength of Materials, 

 that, if X is the abscissa (along the length of the beam) and y the ordinate of the 

 elastic line, in other words, the corresponding deflection, then — 



(o) dy/dx is the tangent of the very small angle of the elastic line with the 

 original axis of the beam. 



(&) d^y/dx" is a quantity proportional to the bending moment 



(since M= EJ/p or = EJ ^). 



(c) d^y/dx^ is proportional to the shear; and 



(d) d^y/d^r* is proportional to the unit load (if distributed), (c) and (d) are 

 called Schwedler's theorems. 



Now in all problems on rod vibrations we shall make constant use of these rela- 

 tions, in adjusting the end conditions to each case, and thereby in determining the 

 required constants of integration, etc. We generally put matters as follows : — At 



one end (say x = 0) we have at the other end (where, say, ;r = /) we 



have 



