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0593_C15_fm Page 534 Tuesday, May 7, 2002 7:05 AM

534 Dynamics of Mechanical Systems

15.11 Closure
Our analysis shows that if a system is out of balance it can create undesirable forces at
the bearings and supports. If the balance is relatively small, it can often be significantly
reduced or even eliminated by judicious placing of balancing weights.
Perhaps the most widespread application of balancing principles is with the balancing
of internal-combustion engines and with similar large systems. Because such systems have
a number of moving parts, complete balance is generally not possible. Designers of such
systems usually attempt to minimize the unbalance while at the same time making com-
promises or tradeoffs with other design objectives.
We saw an example of such a tradeoff in the balancing of an eight-cylinder engine: the
engine could be approximately balanced if the cylinders were all in a line. This arrange-
ment, however, creates a relatively long engine, not practical for many engine compart-
ments. An alternative is to divide the engine into two banks of four cylinders, inclined
relative to each other (the V-8 engine); however, the engine is then out of balance in yaw
moments, requiring damping at the engine mounts to reduce harmful vibration.
Optimal design of large engines thus generally involves a number of issues that must be
resolved for each individual machine. While there are no specific procedures for such
optimal design, the procedures outlined herein, together with information available in the
references, should enable designers and analysts to reach toward optimal design objectives.

References

15.1. Wilson, C. E., Sadler, J. P., and Michaels, W. J., Kinematics and Dynamics of Machinery, Harper
& Row, New York, 1983, pp. 609–632.
15.2. Wowk, V., Machine Vibrations, McGraw-Hill, New York, 1991, pp. 128–134.
15.3. Paul, B., Kinematics and Dynamics of Planar Machinery, Prentice Hall, Englewood Cliffs, NJ,
1979, chap. 13.
15.4. Mabie, H. H., and Reinholtz, C. F., Mechanisms and Dynamics of Machinery, Wiley, New York,
1987, chap. 10.
15.5. Sneck, H. J., Machine Dynamics, Prentice Hall, Englewood Cliffs, NJ, 1991, pp. 211–227.
15.6. Swight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1057, p. 1.
15.7. Shigley, J. E., and Uicker, J. J., Jr., Theory of Machines and Mechanisms, McGraw-Hill, New York,
1980, p. 499.
15.8. Martin, G. H., Kinematics and Dynamics of Machines, McGraw-Hill, New York, 1982, p. 419.
15.9. Taylor, C. F., The Internal-Combustion Engine in Theory and Practice, Vol. II: Combustion, Fuels,
Materials, Design, MIT Press, Cambridge, MA, 1985, pp. 240–305.

Problems

Section 15.2 Static Balancing

P15.2.1: A 125-lb flywheel in the form of a thin circular disk with radius 1.0 ft and thickness
1.0 in. is mounted on a light (low-weight) shaft which in turn is supported by nearly