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Example text

G. Fliigge (1960)], which need not concern us here; n is the circumferential wavenumber. 7(c). Consider first the case where the outer shell is replaced by a rigid immobile cylinder of inner radius R,, and let v; and wi be the displacement components of the inner shell. g. w , ; ( t ) = w,; exp(if2t). The corresponding velocity potential is - 4 = $(r, e) eiRf. 73b), yielding 39 CONCEPTS. 1 18b) ' The subscript notation i, ni indicates the pressure on cylinder i due to nth mode vibration of cylinder i, whereas 0,ni indicates the pressure on cylinder o due to the same vibration of cylinder i.

This problem has in fact been solved by Chen (1987), but it is not difficult to reproduce the results. One finds 2*= 2, but a new set of boundary conditions for the adjoint problem, namely @(o)= @'(o)= 0, $"(I) + 9$(1)= 0, @"'(l) + p$'(l) = 0. 59) A4 = -A2, A3 = - p / q , B2 = - B1 = 1, + p q sinh q)/(p2 cos p + q2 cosh q), = -p/q, 84 [(9 - p 2 ) sin p - ( p / q ) ( B+ q2>sinh 91 ' [(9 - p 2 ) cos p - (9+ q 2 )cosh q] = -B2. The characteristic equation is P2+ 2h(l + cos pcosh q ) + 9 f i sin p sinh q = 0, and it is the same for both problems; hence, so are the eigenvalues.

23), this is a standard problem. I/&, and the problem is unusual in that the eigenvalue appears in the boundary conditions. Hence, strictly (Friedman 1956), the domain 9 depends upon A. 1 were compared, we have proceeded by blithely ignoring this subtlety (by retaining 9= [0, l]), yet still obtained the correct results. 4. 1 Approximations to the lowest three eigenfrequencies of the modified cantilevered pipe for various N in the case of M, = i m L . 936 52. e. the pipe with the extra mass Me at the free end.

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