Page 358 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

11

Mode-Summation Procedures

for Continuous Systems

Structures made up of beams are common in engineering.^ They constitute systems
of an infinite number of degrees of freedom, and the mode-summation methods
make possible their analysis as systems of a finite number of degrees of freedom.
The effect of rotary inertia and shear deformation is sometimes of interest in beam
problems. Constraints are often found as additional supports of the structure, and
they alter the normal modes of the system. In the use of the mode-summation
method, convergence of the series is of importance, and the mode-acceleration
method offers a varied approach. The modes used in representing the deflection of
a system need not always be orthogonal. The synthesis of a system using
nonorthogonal functions is illustrated.
Large structures such as space stations are generally composed of continuous
sections which can be analyzed by the mode participation methods. Shown in Fig.
11.1-1 is one such design, parts of which offer opportunities for challenging
analysis.

11.1 MODE-SUMMATION METHOD
In Sec. 6.8, the equations of motion were decoupled by the modal matrix to obtain
the solution of forced vibration in terms of the normal coordinates of the system.
In this section, we apply a similar technique to continuous systems by expanding
the deflection in terms of the normal modes of the system.

^The Olympus Satellite, shown in Figure 11.1-1, is one of several configurations proposed for
deployment in space. The large panels of solar cells are deployed by lightweight booms of glass fiber,
ingeniously designed to extend the panels for hundreds of feet.

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