Page 254 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.5 Convergence to Higher Modes 241

Thus, after several repetitions of the procedure, we obtain

^ n - l = + • • • + (8.4-4)
(jl) j it) 2 ^3
> > 0)1 > co], the convergence is to the fundamental
mode.

8.5 CONVERGENCE TO HIGHER MODES

In Sec. 8.4, we have shown that when the equation of motion is formulated in
terms of flexibility, the iteration procedure converges to the lowest mode. It is
evident that if the lowest mode is absent in the assumed deflection, the iteration
will converge to the next lowest, or the second, mode. However, because round-off
errors will always reintroduce a small component of (f>^ during each iteration, it
will be necessary to remove this component from each iterated vector in order for
the iteration to converge to (/>2.
To accomplish this removal procedure, we again start with the expansion
theorem:

X —Cj(/)j + ^2^2 ^3^3 “1“ *‘ ‘ "I" (8.5-1)

Next, premultiply this equation by where is the first normal mode, which
was already found:
(¡)\M X = C^(f)\M(f)^ + C2(t>\M(f)2 + C 2,(f) { M (f) 2 + (8.5-2)
Due to orthogonality, all the terms on the right side of this equation except the
first are zero and we have
(l)]MX = c^cf)\M(i) (8.5-3)
We note from Eq. (8.5-1) that if c, = 0, we have a displacement free from
Also because (f)\M(f)^ cannot be zero, with c, = 0, Eq. (8.5-3) is reduced to
(f)\MX = 0 (8.5-4)
which is the constraint equation.
Writing out this equation for a 3 X 3 problem, we have
m.

= + m2^2 ^^2 + = 0

where x[^\ x^2\ and are known, and the x, without the superscript belong to