Abstract
Constraints work in the simplest way among a variety of methods of modeling the mechanical behaviors on the interface between bodies. The constraints within a persistent point contact usually fall into the following three categories: geometry-dependent constraints due to non-penetration limitation between the two rigid bodies; velocity- or force-dependent constraints due to the vanishing of tangential velocity or sliding friction engaged in tangential interaction. Though those constraints may be intuitively obtained for some simple problems, they are essentially associated with the evolution of location parameters denoting the temporal position of the contact point. Focusing on a multibody system subject to a persistent point contact, we propose a uniform and programmable procedure to formulate the constraint equations. Kinematic analysis along the procedure can clearly expose the dependence of the constraint equations on the location parameters, unveil the reason why the velocity-dependent constraints may become nonholonomic, and exhibit the fulfillment of the Appell–Chetaev’s rule naturally. Furthermore, we employ d’Alembert–Lagrangian principle to yield the dynamical equations of the system via the method of Lagrange’s multipliers. The dynamical equations so obtained are then compared with those derived from a quite different method that characterizes the contact interplay as a pair of contact force vectors. Accordingly, the correlations between the Lagrange multipliers and the components of the real contact force can be clarified. The clarification enables us to correctly embed the force-dependent constraints into the dynamical equations. A classical example of a thin disk contacting a horizontal rough surface is provided to demonstrate the validation of the proposed theory and method.
Similar content being viewed by others
References
Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts, vol. 421. Springer, Berlin (2000)
Wang, J., Liu, C., Zhao, Z.: Nonsmooth dynamics of a 3D rigid body on a vibrating plate. Multibody Syst. Dyn. 32(2), 217–239 (2014)
Brogliato, B.: Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst. Dyn. 32(2), 175–216 (2014)
Goldstein, H.: Classical Mechanics, 6th edn. Addison-Wesley, Reading (1959)
Chen, B.: Analytical Dynamics, 2nd edn. Press of Peking University, Beijing (2012) (in Chinese)
Neimark, Ju.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems, vol. 33. Am. Math. Soc., Ann Arbor (1972)
Cronstrim, C., Raita, T.: On nonholonomic systems and variational principles. J. Math. Phys. 50(4), 042901 (2009)
Flannery, M.R.: The elusive d’Alembert–Lagrange dynamics of nonholonomic systems. Am. J. Phys. 79(9), 932–944 (2011)
Flannery, M.R.: D’Alembert–Lagrange analytical dynamics for nonholonomic systems. J. Math. Phys. 52(3), 032705 (2011)
Wang, J., Liu, C., Ma, D.: Experimental study of transport of a dimer on a vertically oscillating plate. Proc. R. Soc. A, Math. Phys. Eng. Sci. 470(2171), 20140439 (2014)
Wang, J., Liu, C., Jia, Y.-B., Ma, D.: Ratchet rotation of a 3D dimer on a vibrating plate. Eur. Phys. J. E 37(1), 1–13 (2014)
Pfeiffer, F.: Unilateral problems of dynamics. Arch. Appl. Mech. 69(8), 503–527 (1999)
Pars, L.A.: A Treatise on Analytical Dynamics. Wiley, New York (1968)
Zhao, Z., Liu, C., Chen, B., Brogliato, B.: Asymptotic analysis of Painlevé’s paradox. Multibody Syst. Dyn. 33(1), 1–21 (2015)
Montana, D.J.: The kinematics of contact and grasp. Int. J. Robot. Res. 7(3), 17–31 (1988)
Jia, Y.-B., Erdmann, M.: Pose and motion from contact. Int. J. Robot. Res. 18(5), 466–487 (1996)
Li, Z., Canny, J.: Motion of two rigid bodies with rolling constraint. IEEE Trans. Robot. Autom. 6(1), 62–72 (1990)
Bicchi, A., Kumma, V.: Robotic grasping and contact: a review. In: IEEE International Conference on Robotics and Automation, vol. 1, pp. 348–353 (2000)
Cai, C., Roth, B.: On the planar motion of rigid bodies with point contact. Mech. Mach. Theory 21(6), 453–466 (1986)
Sarkar, N., Kuma, V., Yun, X.: Velocity and acceleration analysis of contact between three-dimensional rigid bodies. J. Appl. Mech. 63, 974–984 (1996)
Marigo, A., Bicchi, A.: Rolling bodies with regular surface: controllability theory and applications. IEEE Trans. Autom. Control 45(9), 1586–1599 (2000)
Cui, L., Dai, J.S.: A Darboux-frame-based formulation of spin-rolling motion of rigid objects with point contact. IEEE Trans. Robot. 26(2), 383–388 (2010)
Struik, D.J.: Lectures on Classical Differential Geometry. Am. Math. Soc., Ann Arbor (1950)
Acknowledgements
The authors would like to thank to the supports of the NSFC project (11172019, 11132001, 11472011).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, Z., Liu, C. Contact constraints and dynamical equations in Lagrangian systems. Multibody Syst Dyn 38, 77–99 (2016). https://doi.org/10.1007/s11044-016-9503-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11044-016-9503-1