Page 371 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

358 Mode-Summation Procedures for Continuous Systems Chap. 11

which can be simplified to

(1 ~ A) + ^ 9/ X ^ , 2/ X

K
2 ( 1 - A )
\Mo)\ kp^ \ M(i)\ I kp

^ I j k L ' h « ) + \ [ v M ) - a<p\{a)Y\ = 0
Mo)
A number of special cases of the preceding equation are of interest, and we
mention one of these. If AT= 0, the frequency equation simplifies to

A^ - 1 + 1 + — + ip\{a) A +
Mo)] P \Mo)\
Here x = a might be taken negatively so that the missile is hanging by a spring.

11.4 MODE-ACCELERATION METHOD

One of the difficulties encountered in any mode-summation method has to do with
the convergence of the procedure. If this convergence is poor, a large number of
modes must be used, thereby increasing the order of the frequency determinant.
The mode-acceleration method tends to overcome this difficulty by improving the
convergence so that a fewer number of normal modes is needed.
The mode-acceleration method starts with the same differential equation for
the generalized coordinate but rearranged in order. For example, we can start
with Eq. (11.3-4) and write it in the order

^ f ( a , Q y , ( f l ) M(a,t)(p'i(a) _ g, ( Q
(11.4-1)
M.o,? M.o.? o)?
Substituting this into Eq. (11.3-1), we obtain

Q,iOv>,{x)
(11.4-2)