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Research Papers

A Boundary Layer Approach to Multibody Systems Involving Single Frictional Impacts

[+] Author and Article Information
S. Natsiavas

Department of Mechanical Engineering,
Aristotle University,
Thessaloniki 541 24, Greece
e-mail: natsiava@auth.gr

E. Paraskevopoulos

Department of Mechanical Engineering,
Aristotle University,
Thessaloniki 541 24, Greece

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 17, 2018; final manuscript received October 5, 2018; published online November 19, 2018. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 14(1), 011002 (Nov 19, 2018) (16 pages) Paper No: CND-18-1316; doi: 10.1115/1.4041775 History: Received July 17, 2018; Revised October 05, 2018

A systematic theoretical approach is presented, revealing dynamics of a class of multibody systems. Specifically, the motion is restricted by a set of bilateral constraints, acting simultaneously with a unilateral constraint, representing a frictional impact. The analysis is carried out within the framework of Analytical Dynamics and uses some concepts of differential geometry, which provides a foundation for applying Newton's second law. This permits a successful and illuminating description of the dynamics. Starting from the unilateral constraint, a boundary is defined, providing a subspace of allowable motions within the original configuration manifold. Then, the emphasis is focused on a thin boundary layer. In addition to the usual restrictions imposed on the tangent space, the bilateral constraints cause a correction of the direction where the main impulse occurs. When friction effects are negligible, the dominant action occurs along this direction and is described by a single nonlinear ordinary differential equation (ODE), independent of the number of the original generalized coordinates. The presence of friction increases this to a system of three ODEs, capturing the essential dynamics in an appropriate subspace, arising by bringing the image of the friction cone from the physical to the configuration space. Moreover, it is shown that the classical Darboux–Keller approach corresponds to a special case of the new method. Finally, the theoretical results are complemented by a selected set of numerical results for three examples.

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References

Lanczos, C. , 1952, The Variational Principles of Mechanics, University of Toronto Press, Toronto, ON, Canada.
Pars, L. A. , 1965, A Treatise on Analytical Dynamics, Heinemann Educational Books, London.
Rosenberg, R. M. , 1977, Analytical Dynamics of Discrete Systems, Plenum Press, New York.
Greenwood, D. T. , 1988, Principles of Dynamics, Prentice Hall, Englewood Cliffs, NJ.
Udwadia, F. E. , and Kalaba, R. E. , 1996, Analytical Dynamics a New Approach, Cambridge University Press, Cambridge, UK.
Bloch, A. M. , 2003, Nonholonomic Mechanics and Control, Springer-Verlag, New York.
Routh, E. J. , 1897, Dynamics of a System of Rigid Bodies, 6th ed., Macmillan, London.
Stronge, W. J. , 2000, Impact Mechanics, Cambridge University Press, Cambridge, UK.
Pfeiffer, F. , and Glocker, C. , 1996, Multibody Dynamics With Unilateral Contacts, Wiley, New York.
Brogliato, B. , 2016, Νonsmooth Mechanics: Models, Dynamics and Control, 3rd ed., Springer-Verlag, London.
Khulief, Y. A. , 2013, “Modeling of Impact in Multibody Systems: An Overview,” ASME J. Comput. Nonlinear Dyn., 8(2), p. 021012. [CrossRef]
Marques, F. , Flores, P. , Claro, J. C. P. , and Lankarani, H. M. , 2016, “A Survey and Comparison of Several Friction Force Models for Dynamic Analysis of Multibody Mechanical Systems,” Nonlinear Dyn., 86(3), pp. 1407–1443. [CrossRef]
Hartog, J. P. D. , and Mikina, S. J. , 1932, “Forced Vibrations With Non-Linear Spring Constants,” ASME J. Appl. Mech., 58, pp. 157–164.
Masri, S. F. , and Caughey, T. K. , 1966, “On the Stability of the Impact Damper,” ASME J. Appl. Mech., 33(3), pp. 586–592. [CrossRef]
Shaw, S. W. , and Holmes, P. J. , 1983, “A Periodically Forced Piecewise Linear Oscillator,” J. Sound Vib., 90(1), pp. 129–155. [CrossRef]
Natsiavas, S. , 1989, “Periodic Response and Stability of Oscillators With Symmetric Trilinear Restoring Force,” J. Sound Vib., 134(2), pp. 315–331. [CrossRef]
Babitsky, V. I. , 1998, Theory of Vibro-Impact Systems and Applications, Springer-Verlag, Berlin.
Moreau, J. J. , and Panagiotopoulos, P. D. , eds., 1988, Nonsmooth Mechanics and Applications, CISM Courses and Lectures, Vol. 302, Springer-Verlag, Vienna, Austria.
Glocker, C. , 2001, Set-Valued Force Laws, Dynamics of Non-Smooth Systems, Springer, Berlin.
Leine, R. I. , and Nijmeijer, H. , 2013, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer-Verlag, Berlin.
Keller, J. B. , 1986, “Impact With Friction,” ASME J. Appl. Mech., 53(1), pp. 1–4. [CrossRef]
Batlle, J. A. , and Condomines, A. B. , 1991, “Rough Collisions in Multibody Systems,” Mech. Mach. Theory, 26, pp. 565–577. [CrossRef]
Stronge, W. J. , 2001, “Generalized Impulse and Momentum Applied to Multibody Impact With Friction,” Mech. Struct. Mach., 29(2), pp. 239–260. [CrossRef]
Zhao, Z. , and Liu, C. , 2007, “The Analysis and Simulation for Three-Dimensional Impact With Friction,” Multibody Syst. Dyn., 18(4), pp. 511–530. [CrossRef]
Elkaranshawy, H. A. , Abdelrazek, A. M. , and Ezzat, H. M. , 2017, “Tangential Velocity During Impact With Friction in Three-Dimensional Rigid Multibody Systems,” Nonlinear Dyn., 90(2), pp. 1443–1459. [CrossRef]
Aghili, F. , 2011, “Control of Redundant Mechanical Systems Under Equality and Inequality Constraints on Both Input and Constraint Forces,” ASME J. Comput. Nonlinear Dyn., 6(3), p. 031013. [CrossRef]
Brogliato, B. , 2014, “Kinetic Quasi-Velocities in Unilaterally Constrained Lagrangian Mechanics With Impacts and Friction,” Multibody Syst. Dyn., 32(2), pp. 175–216. [CrossRef]
Paraskevopoulos, E. , and Natsiavas, S. , 2013, “On Application of Newton's Law to Mechanical Systems With Motion Constraints,” Nonlinear Dyn., 72(1–2), pp. 455–475. [CrossRef]
Natsiavas, S. , and Paraskevopoulos, E. , 2015, “A Set of Ordinary Differential Equations of Motion for Constrained Mechanical Systems,” Nonlinear Dyn., 79(3), pp. 1911–1938. [CrossRef]
Paraskevopoulos, E. , and Natsiavas, S. , 2017, “A Geometric Solution to the General Single Contact Frictionless Problem by Combining Concepts of Analytical Dynamics and b-Calculus,” Int. J. Non-Linear Mech., 95, pp. 117–131. [CrossRef]
Natsiavas, S. , and Paraskevopoulos, E. , 2018, “An Analytical Dynamics Approach for Mechanical Systems Involving a Single Frictional Contact Using b-Geometry,” Int. J. Solids Struct., 148–149, pp. 140–156. [CrossRef]
Papastavridis, J. G. , 1999, Tensor Calculus and Analytical Dynamics, CRC Press, Boca Raton, FL.
Frankel, T. , 1997, The Geometry of Physics: An Introduction, Cambridge University Press, New York.
Tu, L. W. , 2011, An Introduction to Manifolds, 2nd ed., Springer Science+ Business Media, New York.
Melrose, R. B. , 1993, The Atiyah-Patodi-Singer Index Theorem, Research Notes in Mathematics, Vol. 4, A K Peters, Wellesley, MA.
Geradin, M. , and Cardona, A. , 2001, Flexible Multibody Dynamics: A Finite Element Approach, Wiley, New York.
Bauchau, O. A. , 2011, Flexible Multibody Dynamics, Springer Science+ Business Media, London.
Kevorkian, J. , and Cole, J. D. , 1985, Perturbation Methods in Applied Mathematics, 2nd ed., Springer-Verlag, New York.
Cousteix, J. , and Mauss, J. , 2007, Asymptotic Analysis and Boundary Layers, Springer-Verlag, Berlin.
Kozlov, V. V. , and Treshchev, D. V. , 1991, Billiards: A Genetic Introduction to the Dynamics of Systems With Impacts (Translations of Mathematical Monographs, Vol. 89), American Mathematical Society, Providence, RI.
Zahariev, E. , 2003, “Multibody System Contact Dynamics Simulation,” Virtual Nonlinear Multibody Systems (NATO Science Series, Vol. 103), W. Schiehlen and M. Valasek , eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 383–402.
Νeimark, J. I. , and Fufaev, N. A. , 1972, “Dynamics of Nonholonomic Systems,” Translations of Mathematical Monographs, Vol. 33, American Mathematical Society, Providence, RI.
Papalukopoulos, C. , and Natsiavas, S. , 2007, “Dynamics of Large Scale Mechanical Models Using Multi-Level Substructuring,” ASME J. Comput. Nonlinear Dyn., 2(1), pp. 40–51. [CrossRef]
Theodosiou, C. , and Natsiavas, S. , 2009, “Dynamics of Finite Element Structural Models With Multiple Unilateral Constraints,” Int. J. Non-Linear Mech., 44(4), pp. 371–382. [CrossRef]
Gonçalves, A. A. , Bernardino, A. , Jorge, J. , and Lopes, D. S. , 2017, “A Benchmark Study on Accuracy-Controlled Distance Calculation Between Superellipsoid and Superovoid Contact Geometries,” Mech. Mach. Theory, 115, pp. 77–96. [CrossRef]
Pournaras, A. , Karaoulanis, F. , and Natsiavas, S. , 2017, “Dynamics of Mechanical Systems Involving Impact and Friction Using a New Contact Detection Algorithm,” Int. J. Non-Linear Mech., 94, pp. 309–322. [CrossRef]
Stoianovici, D. , and Hurmuzlu, Y. , 1996, “A Critical Study of the Applicability of Rigid-Body Collisions Theory,” ASME J. Appl. Mech., 63(2), pp. 307–316. [CrossRef]
Nguyen, N. S. , and Brogliato, B. , 2014, Multiple Impacts in Dissipative Granular Chains (Lecture Notes in Applied and Computational Mechanics, Vol. 72), Springer, Berlin.
Melrose, R. B. , 1996, “Differential Analysis on Manifolds With Corners,” accessed Oct. 30, 2018, http://math.mit.edu/~rbm
Joyce, D. , 2016, “A Generalization of Manifolds With Corners,” Adv. Math., 299, pp. 760–862. [CrossRef]
Acary, V. , and Brogliato, B. , 2008, Numerical Methods for Nonsmooth Dynamical Systems (Lecture Notes in Applied and Computational Mechanics, Vol. 35), Springer, Berlin.
Brüls, O. , Acary, V. , and Cardona, A. , 2014, “Simultaneous Enforcement of Constraints at Position and Velocity Levels in the Nonsmooth Generalized-α Scheme,” Comput. Methods Appl. Eng., 281, pp. 131–161. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) Original configuration manifold M and constrained manifold X, for a system subject to a unilateral constraint. (b) Magnification around a boundary point p.

Grahic Jump Location
Fig. 2

An x-basis and horizontal part HbS* of cotangent space Tbp*X at a boundary point p

Grahic Jump Location
Fig. 3

Collision of a four-particle system with a rigid wall

Grahic Jump Location
Fig. 4

Collision of a pendulum with a rigid plane

Grahic Jump Location
Fig. 5

Slip trajectories of a spherical pendulum for (a) θ=π/4 and k=1; (b) θ=π/4 and k=0.1; (c) θ=π/4 and k=10; and (d) θ=π/3 and μ=μc

Grahic Jump Location
Fig. 6

Effect of friction coefficient on the rebound speed of a spherical pendulum for (a) k=0.1 and k=1, and (b) k=10 and comparison with results of the Darboux–Keller approach

Grahic Jump Location
Fig. 7

Collision of a double planar pendulum with a rigid wall

Grahic Jump Location
Fig. 8

Energy loss during impact of a double pendulum against a rough half space for (a) c=0 and (b) c>0

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