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270 Dynamics of Mechanical Systems

Finally, if D is spinning (or pivoting) at a constant rate in a vertical position, we have:

˙
θ = 0, φ = constant, ψ = constant (8.13.23)
These values of θ , φ ˙ O , and ψ O are also seen to be solutions of Eqs. (8.13.16), (8.13.17),
O
and (8.13.18).
In Chapter 13, we will investigate the stability of these special case solutions.

8.14 Closure
Newton’s laws, together with their modification to form d’Alembert’s principle, are
undoubtedly the most widely used of all dynamics principles — particularly in the anal-
ysis of elementary problems. The reason for this widespread use and popularity is that
Newton’s laws and d’Alembert’s principle are always applicable and, in principle, they
always produce the governing equations of motion. For many mechanical systems, how-
ever, Newton’s laws and d’Alembert’s principle are not especially convenient. Their suc-
cessful and efficient application often requires insight not available to many analysts. In
the following chapters, we will consider other dynamics principles which, while being
more specialized, are more efficient and easier to use.

References
8.1. Levinson, D. A., paper presented at the Seminar on Dynamics, University of Cincinnati, 1990.
8.2. Newton, I., Philosophiae Naturalis Principia Mathematics, First Ed., London, 1687.
8.3. Temple, R., The Genius of China, Simon & Schuster, New York, 1987, p. 161.
8.4. d’Alembert, J. L., Traite’ de Dynamique, Paris, 1743.
8.5. Hamilton, W. R., Second essay on a general method of dynamics, Philosophical Transactions of
the Royal Society of London, 1835.
8.6. Gibbs, J. W., On the fundamental formulae of dynamics, Am. J. Math., Vol. II, pp. 49–64, 1879.
8.7. Appell, P., Sur une Forme Generale Eqs. de Dynamique, Journal fur die Reine und Angewandte
Mathematic, Vol. 121, pp. 310–319, 1900.
8.8. Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge
University Press, London, 1937.
8.9. Brand, L., Vectorial Mechanics, Wiley, New York, 1947.
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8.11. Halfman, R. L., Dynamics, Addison-Wesley, Reading, MA, 1959.
8.12. Housner, G. W., and Hudson, D. E., Applied Mechanics: Dynamics, D. van Nostrand, Princeton,
NJ, 1959.
8.13. McCaskey, S. W., An Introduction to Advanced Dynamics, Addison-Wesley, Reading, MA, 1959.
8.14. Yeh, H., and Abrams, J. I., Principles of Mechanics of Solids and Fluids, Vol. 1, McGraw-Hill, New
York, 1960.
8.15. Kane, T. R., Analytical Elements of Mechanics, Vol. 1, Academic Press, New York, 1961.
8.16. Kane, T. R., Dynamics of nonholonomic systems, J. Appl. Mech., 28, 574–578, 1961.
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8.19. Meirovitch, L., Methods of Analytical Dynamics, McGraw-Hill, New York, 1970.
8.20. Tuma, J. J., Dynamics, Quantum Publishers, New York, 1974.