538 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



8. Vibrations of Cylindrical Shell. 



21. As a further example, suppose the middle-surface of the shell cylindrical ; and, 

 to fix ideas, suppose there is a rigid disc at one end, and at a distance c from it a free 

 edge bounded by a circle. 



Let a be the radius of the circular section of the cylinder, and a, z, </> cylindrical 

 coordinates of a point on the middle-surface, the origin being at the centre of the 

 rigid disc. 



In the equations of motion of Art. 13, we have to put 



h l =l, h. 2 = I/a, l/ Pl = 0, I/A, = I/a. 



Taking u, v, w proportional to &>*, and * 3 ?i = pa z p, these equations become 



3%** ^ 3*K 3m TO /f) 2 ? . 1 3'0 \ . m n 1 8w 



a,* T * u ~r ,.2 a .5 T 



m + n \02 J a oz dq>/ m + n a 



a~ + ~ V + " - -f- - -7- '- H - ) 4- 4 - - = (87) 



ck 2 a 2 a 3 3$ 2 m -f- n \a s 3< 2 a vzo<j>/ m + n a? c<f> 



H? 4m 1 / . 3 \ . i TO 1 3w 



-w=- - ( w + ^- )+ 2 - -5-. (88) 



a* m + TO a 2 \ o0/ m + TO a 82 



Put 4mj8/(m + n) = /c 2 4w/(w + n), then (88) is 

 and (86), (87) give 



4:WZ- G U , /C .X (J"U OtTt ~~ ft/ J. C/"y . ^ 77t ' u, I \y v wwi _ 



5I"rn ~r To ' ( ^, 52 "T ^^ o^ / == ^> 



_ _i I 2 I 2 Ci J.2 I , i ., d M &JL ' > 



7/t "t~ * cs tt a O(p T/I' ~r ?& a oz dp flo ?/t 



9 3 v 2 4w 1 9 2 v 3m w 1 9 2 w 1 4m / 3' 3 w , 3 2 v \ 



4 . I -L. I - / I \ fi 



^ o i o " i o K io "i *> "i i o/^ I "^ r> *N r n 10 / v > 



O-2 fl Til -\- H d VW W "- r ' ?r ''" /.*H iw. -4- T). \ n?r ri/ri r)/r- / 



or 



1 9 9 



