On application of Newton’s law to mechanical systems with motion constraints
Tóm tắt
This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.
Tài liệu tham khảo
Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, Berlin (1989)
Bauchau, O.A.: Flexible Multibody Dynamics. Springer, London (2011)
Bayo, E., Ledesma, R.: Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dyn. 9, 113–130 (1996)
Bertram, W.: Differential geometry, Lie groups and symmetric spaces over general base fields and rings. Mem. Am. Math. Soc. 192, 1–202 (2008)
Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, New York (2003)
Bowen, R.M., Wang, C.-C.: Introduction to Vectors and Tensors, 2nd edn. Dover, New York (2008)
Brüls, O., Cardona, A., Arnold, M.: Lie group generalized-α time integration of constrained flexible multibody systems. Mech. Mach. Theory 48, 121–137 (2012)
Crouch, P., Grossman, R.: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci. 3, 1–33 (1993)
Essen, H.: On the geometry of nonholonomic dynamics. J. Appl. Mech. 61, 689–694 (1994)
Frankel, T.: The Geometry of Physics: An Introduction. Cambridge University Press, New York (1997)
Geradin, M., Cardona, A.: Flexible Multibody Dynamics. Wiley, New York (2001)
Goudas, I., Stavrakis, I., Natsiavas, S.: Dynamics of slider-crank mechanisms with flexible supports and non-ideal forcing. Nonlinear Dyn. 35, 205–227 (2004)
Guo, Y.-X., Wang, Y., Chee, G.Y., Mei, F.-X.: Nonholonomic versus vakonomic dynamics on a Riemann–Cartan manifold. J. Math. Phys. 46, 0692902 (2005)
Greenwood, D.T.: Principles of Dynamics. Prentice-Hall, Englewood Cliffs (1988)
Kane, T.R., Levinson, D.A.: Dynamics: Theory and Applications. McGraw-Hill, New York (1985)
Kurdila, A.J., Junkins, J.L., Hsu, S.: Lyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamics. Nonlinear Dyn. 4, 51–82 (1993)
Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1952)
Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, New York (1999)
Muller, A.: Motion equations in redundant coordinates with application to inverse dynamics of constrained mechanical systems. Nonlinear Dyn. 67, 2527–2541 (2012)
Munthe-Kaas, H.: Runge–Kutta methods on Lie groups. BIT Numer. Math. 38, 92–111 (1998)
Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robot Manipulation. CRC Press, Boca Raton (1994)
Negrut, D., Haug, E.J., German, H.C.: An implicit Runge–Kutta method for the integration of differential–algebraic equations of multibody dynamics. Multibody Syst. Dyn. 9, 121–142 (2003)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley-Interscience, New York (1995)
Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Englewood Cliffs (1988)
Papastavridis, J.G.: Tensor Calculus and Analytical Dynamics. CRC Press, Boca Raton (1999)
Paraskevopoulos, E., Natsiavas, S.: A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory. Int. J. Solids Struct. 50, 57–72 (2013)
Sattinger, D.H., Weaver, O.L.: Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Springer, New York (1986)
Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, New York (2005)
Shabanov, S.V.: Constrained systems and analytical mechanics in spaces with torsion. J. Phys. A, Math. Gen. 31, 5177–5190 (1998)
Udwadia, F.E., Kalaba, R.E., Phohomsiri, P.: Mechanical systems with nonideal constraints: explicit equations without the use of generalized inverses. J. Appl. Mech. 71, 615–621 (2004)