D'Alembert's principle is manipulated in the presence of nonholonomic constraints to derive the principle of virtual power in nonholonomic form, and Lagrange's equations for nonholonomic systems. The Lagrangian equations had been expressed previously for conservative systems, derived by variational methods. The D'Alembert derivation confirms the roles of constrained and unconstrained Lagrangians directly by the presence of constrained and unconstrained velocities in D'Alembert's principle. The constrained form of nonconservative generalized forces is also determined for both particles and rigid bodies. An example is a rolling disk.
Volume Subject Area:Design Engineering
Keywords:Lagrange's equations, D'Alembert's principle, principle of virtual power, nonholonomic systems, constraints
Dimarogonas, A. D. and S. Haddad, 1992, Vibration for Engineers, Prentice Hall, Englewood Cliffs.
Dai, Nianzu 1983, Ancient China’s technology and science, compiled by the Institute of the History of Natural Sciences, Chinese Academy of Sciences. Beijing: Foreign Languages Press.
Baruh, H. 1999, Analytical Dynamics, WCB-McGraw-Hill, Boston.
Dugas, R. 1955, A History of Mechanics, Editions du Griffon, Neuchatel.
Whittaker, E. T. 1944, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover, New York.
Neimark, Ju. I. and N. A. Fufaev, 1972, Dynamics of Nonholonomic Systems, American Mathematical Society, Providence, Rhode Island. Translated from the Russian text (1967) by J. R. Barbour.
A Panoramic Overview of the Principles and Equations of Motion of Advanced Engineering Dynamics,”
Applied Mechanics Reviews
Applied Mechanics Reviews
Nonholonomic mechanical systems with symmetry,”
Archives of Rational Mechanics and Analysis
Cendra, H., Marsden, J. E., and Ratiu, T. S., 2001, “Geometric mechanics, Lagrangian reduction, and nonholonomic systems,” in Mathematics Unlimimted—2001 and Beyond, B. Engquist and W. Schmid, eds., Springer, Berlin, 221–274.
Pars, L. A. 1965, A Treatise on Analytical Dynamics, John Wiley and Sons, New York.
Moon, F. C. 1998, Applied Dynamics, Wiley—Interscience, New York.
Kane, T. R. and D. A. Levinson, 1985, Dynamics: Theory and Applications, McGraw-Hill, New York.
Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss Principle,”
Journal of Applied Mechanics
Equations of motion for Mechanical Systems: A unified Approach,”
International Journal of Non-Linear Mechanics
Martin de Diego
On the Geometry of Nonholonomic Lagrangian Systems,”
Journal of Mathematical Physics,
The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems,”
Reports in Mathematical Physics
Reduction of Some Classical Nonholomic Systems with Symmetry,”
Archives of Rational Mechanics Analysis
A New Formulation of the Equations of Dynamics,”
Foundations of Physics Letters
Greenwood, D. T. 1988, Principles of Dynamics, Second Edition, Prentice Hall, Upper Saddle River, NJ.
Rand, R. H. 1985, Topics in Nonlinear Dynamics with Computer Algebra, Gordon and Breach Science Publishers, Langhorne, Pennsylvania.
The Dynamics of Rolling Disks and Sliding Disks,”
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