ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 117 



So that the differential equation of motion (of the center of each section will be — 



where ay can be taken from under the integral sign only where both a and y are 

 cimstant throughout the length. 



If now we succeed in finding the solution for y, from this equation, in ascend- 

 ing powers of x, we can readily find the period of the vibration. 



Let us assume a value of y^ such that it satisfies the end conditions. For in- 

 stance, let 



y = A -\- Bx -f Cx^ + Dx' + Ex\ 



At the free end, where x = o, we have y = 3'i (let this amplitude of the end be 

 any) ; then A^ 3/1 (by making, in above equation, x = o); also cfy/dx'' = o (because 

 the bending moment is here = o) ; and d^y/dx^ = o (because the shear at the free 

 end is = o) ; hence, C = o and D = o. On the other hand, at the fixed end, where 

 x = l,we have y = o and dy/dx = o (the elastic line is tangent to its neutral posi- 

 tion) ; so that 



B=-±y^ and 5-=^ 

 So that the assumed equation, corrected for the end conditions, becomes 



{Important note : The reader's attention is called first to the manner of handling the end conditions, 

 which is of greatest importance in our last chapter; and, second, in the fact that the degree of the assumed 

 equation must be fourth ; it is easy enough to see that nothing else would answer ; the other examples of Pro- 

 fessor Morrow's method call for different degrees of the assumed expression for y, such as fifth or sixth.) 



Since, then, we now have the general expression of the ordinate 3; (or 3/^) in func- 

 tion of any arbitrary ordinate 34 (say the amplitude of the free end), let us substi- 

 tute it for 3'^ in equation (4). 



dx-~ EJ^'K \ 3/3 I'l^ ^ 





_ ^y 



'^ 



Integrating twice, each time adjusting the constants of integration to suit the end 

 condition, we have — 



^y=^y\ /. 08194 /^ -. 1 12698/^-1-. 0416 X^-. 01 ^4-. 000198^) (6) 



which is approximately the vibration curve and, for the free end, gives 



- yi = . 08194 -^—3;;' (7) 



