Page 480 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 14.3 Stability of Equilibrium 467

singular point (x^, y^), we obtain for Q{x, y)

Q{x, y) = Q {x„y,) + V + (14.3-5)
du^

and a similar equation for P{x, y). Because Q{x^, y^) is zero and (dQ/du)^ and
(dQ/du)^ are constants, Eq. (14.3-1) in the region of the singularity becomes
dv cu + ev
(14.3-6)
du au + bv
where the higher-order derivatives of P and Q have been omitted. Thus, a study
of the singularity at (x^, y^) is possible by studying Eq. (14.3-6) for small u and v.
Returning to Eq. (14.3-3) and taking note of Eqs. (14.3-4) and (14.3-5),
Eq. (14.3-6) is seen to be equivalent to
du
-jj = au + bv
(14.3-7)
dv _
I t ~ cu + ev

which can be rewritten in matrix form:
(14.3-8)

It was shown in Sec. 6.7 that if the eigenvalues and eigenvectors of a matrix
equation such as Eq. (14.3-8) are known, a transformation

(14.3-9)

where [P] is a modal matrix of the eigenvector columns, will decouple the equation
to the form
0
[A] (14.3-10)

Because Eq. (14.3-10) has the solution

(14.3-11)
7] =
the solution for u and v are

(14.3-12)
+ v.e'"