Vibration of Bars treated simply. 



583 



This equation is identical with that obtained by Lord Rayleigh 

 after discarding from the complete equation the terms 

 expressing the rotatory inertia. Thus, inspecting (5) and 

 glancing back over the considerations which led to it, we see 

 that the argument may be expressed briefly as follows. The 

 acceleration of an element of the bar is proportional to the 

 first derivative of the shearing force or the second of the 

 bending moment with respect to x. But the bending moment 

 is itself the second derivative of the displacement with respect 

 to x. Hence the second derivative of the displacement with 

 respect to time is proportional to its fourth derivative with 

 respect to x. 



Thus equation (5) expressing this relation with the proper 

 coefficients must be satisfied at every point of the bar and at 

 every instant of time. But we need in addition other equations 

 for the ends of the bar which depend on the special conditions 

 there obtaining. 



Terminal Conditions. — The ends may be fixed (i. e. clamped), 

 free, or "supported." At a fixed end it is clear that we have, 

 for all values of t, displacement and slope each zero. At a 

 free end, on the other hand, these quantities are arbitrary. 

 But, as there is nothing beyond to produce a couple or 

 transmit force, there can be at the end neither bending 

 moment nor shearing force. By a " supported " end is to be 

 understood an end at which no displacement is allowed but 

 any slope may be assumed. Thus the only external force 

 added by the constraint is applied at the end, and conse- 

 quently can have no moment there. Accordingly there is no 

 curvature possible. 



Hence for the three conditions we have the following- 

 scheme of equations, shown in Table I. 



Table I. — Terminal Conditions. 



State of End. 



Quantities equated to Zero. 



y- 



dijjdx. 



d 2 y/dx-. 



d*yjdx\ 



Nos. of 

 Equations. 



Fixed 



= 



= 







.... 



(6) 



Free 



— 



— 



= 



= 



(V 



Supported . . 



= 



— 



= 



— 



(8) 



