Page 188 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.2 Reciprocity Theorem 175

Figure 6.1-5. Demonstration gyro­
scope. {Courtesy of UCSB Mech­
anical Engineering Undergraduate
Laboratory.)

The equation presented here could offer a basis for solving the problem of
the gyroscopic whirl of a spinning wheel fixed to the end of an overhanging shaft. P
and M in this case would be replaced by the inertia force and the gyroscopic
moment of the spinning wheel. By including the flexibility of the supporting
bearing, a still more general problem can be examined (see Prob. 6-41).
Figure 6.1-5 shows a demonstration gyroscope in gimbals. The mass distribu­
tion of the wheel is adjustable to obtain general moment of inertia configuration
other than that of the symmetric wheel resulting in the simple inertia force P and
the gyroscopic moment M shown in Fig. 6.1-4.

6.2 RECIPROCITY THEOREM

The reciprocity theorem states that in a linear system, = aj¿. For the proof of
this theorem, we consider the work done by forces /, and /y, where the order of
loading is i followed by j and then by its reverse. Reciprocity results when we
recognize that the work done is independent of the order of loading.
By applying /•, the work done is By applying /y, the work done by
is {f/cijj. However, i undergoes further displacement, and the additional
work done by /, becomes Thus, the total work done is

^ = iLa.i + + a ,/,/,