for determining Frequencies of Lateral Vibration. 423 



tit a completely free end (where the resultant shear, as well 

 as the resultant bending-moment on the section, is zero) 

 we have 



M=0; §r =0: • • • • • (8) 



and at an end which is " clamped " (as by a fixed bearing in 

 the problem of the whirling shaft) we have 



y=°-' i=° <»> 



It will be convenient at this point to introduce a new 

 •quantity M, defined by the equation 



M + »^ = 0, (10) 



— so that M is equal to M, simply, in the problem of the 

 whirling shaft, and to M sin 27rTi.t in the problem of the vi- 

 brating bar. Then we can express the foregoing conditions 

 of constraint in terms of Y and M, as follows : — 



At a " simply supported " end, Y = M =0 ; 

 at a completely free end, M= -=— =0 ; 



dY 



and at a clamped end, Y= — — =0. 



a oc 



(ii) 



Corresponding relations can be written down to represent 



the conditions imposed by other types of constraint, e.g., the 



effects of large concentrated masses, spinnino- masses which 



introduce gyroscopic couples, etc. For present purposes it 



is only necessary to add that the existence of " simple 



supports " at intermediate sections of a continuous rod 



dY 

 requires that the values of Y, of -=— ? and of M shall be 



continuous at such sections. 



Whatever hd the specified conditions in the problem under 

 consideration, it will be found that a solution of (6) is ob- 

 tainable by graphical methods, when we have assumed any 

 value for the frequency n, which satisfies all but one of them ; 

 but that they cannot all be satisfied simultaneously unless 11 has 

 •certain definite values, known as the " natural frequencies " 

 of whirling or of vibration. It is these frequencies with 

 which we are primarily concerned in practice, and the 



2 F 2 



