Page 497 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

ANALYTICAL EVALUATION OF b,, c,, AND d,, 467

for clamped ends - clearly indeterminate for r = s. Hence, here a method will be
presented for the evaluation of err, which may be written as

(BSc)

(B.5d)

Multiplying each of these by h: and adding them together gives
I
4
4h4ci-r = [h:4:4r + 4:4;11: + 1 [2h:4:4r - Ar(4,) - (@:")'I dt. (B.6)
I2
0
Now, for any 4r = A cos(h&) + B sin(A,.() + C cosh(A&) + D sinh(h,(), it is easy to
verify that the integrand in (B.6) is equal to 2h:[-A2 - B2 + C2 - 0'1. Hence,

4h4crr = [A:&$,. + 4:&!']1: + 2h:[-A2 - B2 + C2 - D2]. 03.7)

For a clamped-clamped pipe, 4,. and 4; are zero at both limits, while 4:(1) = 2h;(-1)r+1,
q(1) = 2h)cJr(-1)r+', 4:(0) = 2h:, &"(O) = -2h)ar, A = -1, B = CJ,, C = 1, D =
-ur, leading to
err = h,a,(2 - h,a,). (B.8)
For a pipe with pinned ends, @r = 1/2 sin Art, with A, = m, the 1/2 factor ensuring
orthonormality. In this case, it follows easily from (B.7) that
2
c, = -Ar. (B.9)