ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 135 



As has been explained above, the fact that a variation of a definite integral is ^ o 

 means that the integral itself must be either maximum or minimum. Therefore our 

 problem is reduced to finding maximum or minimum of the integral — 



observing, of course, the end conditions. 



We shall avoid the methods of the calculus of variations, also secure a direct 

 solution, by again applying Ritz's method, that is, by assuming, tentatively — 



3'» = «i "Ai + «2 ^2 + . • • a„ rpn, 



where ^i, ^2, • • • '^n, are functions of x, and the coefficients, ai, a2, .... a„, are, in 

 broader sense, special coordinates ; by assuming a certain n, we transform the system 

 into one with, as it were, » degrees of freedom. These coefficients, Ui, 02, .... a„, 

 are found from a system of equations — 



each of which is a partial derivative of the integral with the assumed value y„ sub- 

 stituted in it, made equal to zero, in our attempt to find such values of Oi, etc., that 

 the integral 5" will be a maximum or minimum. Here we also start with the assump- 

 tion of a certain form for y, the deflection, as above, but our real object is to find 

 the frequency constants of various modes of vibrations, as will be made clear through 

 the following examples. 



As an example let us take a clamped wedge, whose width is constant and whose 

 depth varies according to the law of an equilateral triangle, the depth initially being 

 = o, and increasing to 2b at the clamped end. Let us take the origin at the left, 

 free end of the beam, whose length is /, the axis, y, of deflection, being, of course, 

 directed downward ; let us introduce the substitution x = %l, so that the integral be- 

 comes, upon substitution of the variable values of the area and the moment of 

 inertia — 



As regards the end conditions, the following remarks can be made. At the left 

 end both the bending moment and the shear vanish in view of the pointed nature of 

 the beam. At the right end the deflection and the angle of the elastic line with the x 

 axis are both = o ; that is — 



( ) =o;f^) = ^(^) =0. 



The tentative expression for y„ may be taken of the same form as above: — 

 Indeed, when |^ i, both y and its derivative vanish. 



