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0593_C13_fm Page 466 Monday, May 6, 2002 3:21 PM

466 Dynamics of Mechanical Systems

where as before T is the period. Solving Eq. (13.11.20) for T we obtain:

T = 2π l g 1 − A ( [ 2 16 )] (13.10.21)
0
For a measure of the accuracy of the approximation, consider again the example of the
foregoing section where the pendulum is displaced through 60° and released from rest.
Then, A is π/3, and T becomes:
0
[ { 2 ]}
π
T = 2π l g 1 − ( ) 3 16 = 6 7455 l g (13.10.22)
.
Recall that for the linearized equation we obtained (see Eq. (13.3.18)):

T = 2π l g = 6 2832 l g (13.10.23)
.
Recall also that the “exact” solution, obtained with elliptic integrals, is (see Eq. (13.9.24)):

T = 6 7432 l g (13.10.24)
.

13.11 Closure
Vibration analysis is based primarily upon the solution of differential equations; therefore,
vibration analyses are largely dependent upon available methods for solving differential
equations. This means that most analyses are confined to systems that can be modeled by
linear equations. Linear models, however, may fail to adequately represent a mechanical
system if displacements are large or if damping is not proportional to the velocity. Thus,
most vibration analyses ultimately involve approximate numerical and experimental pro-
cedures. Research in vibration is ongoing, with an emphasis upon experimental techniques
for determining natural frequencies, mode shapes, and damping characteristics. In the
following chapter, we will use some of our fundamental results in examining stability of
mechanical systems.

References

13.1. Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, New York, 1975.
13.2. Newland, D. E., Mechanical Vibration Analysis and Computation, Wiley, New York, 1989.
13.3. Thompson, W. T., Theory of Vibration with Application, Prentice Hall, Englewood Cliffs, NJ, 1988.
13.4. Tse, F. S., Morse, I. E., and Hinkle, R. T., Mechanical Vibration Theory and Applications, Allyn &
Bacon, Boston, MA, 1978.
13.5. Steidel, R. F., An Introduction to Mechanical Vibration, Wiley, New York, 1989.
13.6. Weaver, W., Timoshenko, S. P., and Young, D. H., Vibration Problems in Engineering, John Wiley
& Sons, New York, 1990.