134 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



and the maximum bending moment, at least for greater values of y; this will indi- 

 cate that the precision of Ritz's method is sufficient for practical applications even 

 if n is not great. If the moment of inertia at the left end of the beam is not ^ o, a 

 different expression for -^ will have to be chosen, so that the end conditions (the 

 moment and the shear both vanish) : — 







are satisfied. 



Here the form of the functions ^ may be taken as follows : 



ys 



= ai(i-^? + y^*) + «2(?-4lV3$') + «3(r-2?^ + ?«). 



This type may be applied to all cases where the beam depth is varying according to 

 a law of straight line or of parabola, that is when 



h=^h\-\- (ho — hi) ^, 

 or when 



h = hi-\- (ho — hi) ^\ 



Having thus found the necessary coefficients, ai, 02, etc., we can change the indepen- 

 dent variable back to x, and then write down the first answer giving the deflection, y, 

 in function of the abscissa, x. 



3. Vibrations of Beams with Variable Cross-section. — ^The fundamental varia- 

 tional equation of motion can be formed as follows : — 

 Let V be the potential energy. 



/ the length of the beam, 



y the deflection on abscissa x. 



y the weight of i cubic foot of material. 



F the area of cross-section. 



E the modulus of elasticity. 



/ the moment of inertia of the cross-section. 

 Then, expressing the fact that, for any virtual displacement, the work of the inertia 



forces (2 (-»«^) is always equal to the corresponding increase of the potential 

 energy, we can write — 



Remembering the expression of potential energy of bending in terms of y, assuming 

 also that y changes according to some such law as — 



y = const. X cos pt 

 (in other words, introducing the frequency-constant, p) we can reduce this expres- 

 sion to 



