Page 373 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

360 Mode-Summation Procedures for Continuous Systems Chap. 11

we can eliminate y(a) between the previous two equations, obtaining

- - ^ F{ a ) x { a , a) + (X) ------ r—
ojj
or

T
F(a) = —
^ - a(a,a)

If we now substitute this equation into Eq. (11.3-4) and assume harmonic motion, we
obtain the equation

2 / ^

F{a)(p,{a)
(ft),- -
M,
M. - a{a,a)
Rearranging, we have
2 / \l / 2 2\- ^ ^()^/(^)
[I - moio a(a,a)\{co^ - oj )q, = ----- ---------2^

which represents a set of linear equations in The series represented by the
summation will, however, converge rapidly because of coj in the denominator.
Offsetting this advantage of smaller number of modes is the disadvantage that these
equations are now quartic rather than quadratic in w.

11.5 COMPONENT-MODE SYNTHESIS

We discuss here another mode-summation procedure, in which the deflection of
each structural subcomponent is represented by the sum of polynomials instead of
normal modes. These mode functions themselves need not be orthogonal or satisfy
the junction conditions of displacement and force as long as their combined sum
allows these conditions to be satisfied. Lagrange’s equation, and in particular the
method of superfluous coordinates, forms the basis for the synthesis process.
To present the basic ideas of the method of modal synthesis, we consider a
simple beam with a 90° bend, an example that was used by W. Hurty.^ The
beam, shown in Fig. 11.5-1, is considered to vibrate only in the plane of the paper.

^Walter C. Hurty, “Vibrations of Structural Systems by Component Synthesis,” J. Eng. Mech.
Div., Proc. ASCE (August 1960), pp. 51-69.