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Recursive formulations for multibody systems with frictional joints based on the interaction between bodies. (English) Zbl 1376.70016
Summary: In practice, the clearances of joints in a great number of mechanical systems are well under control. In these cases, some of the existing methods become unpractical because of the little differences in the order of magnitude between relative movements and computational errors. Assuming that the effects of impacts are negligible, we proved that both locations and forces of contacts in joints can be fully determined by parts of joint reaction forces. Based on this fact, a method particularly suited for multibody systems possessing frictional joints with tiny clearances is presented. In order to improve the efficiency of computation, recursive formulations are proposed based on the interactions between bodies. The proposed recursive formulations can improve the computation of joint reaction forces. With the methodology presented in this paper, not only the motion of bodies in a multibody system but also the details about the contacts in joints, such as forces of contacts and locations of contact points, can be obtained. Even with the assumption of impact free, the instants of possible impacts can be detected without relying upon any ambiguous parameters, as indicated by numerical examples in this paper.

##### MSC:
 70E55 Dynamics of multibody systems 70B15 Kinematics of mechanisms and robots
##### Keywords:
multibody systems; frictional joints; recursive formulation
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##### References:
 [1] Flores, P., Ambrósio, J., Claro Pimenta, J., Lankarani, H.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies. Springer, Dordrecht (2008) · Zbl 1142.70001 [2] Gilardi, G., Sharf, I.: Literature survey of contact dynamics modeling. Mech. Mach. Theory 37(10), 1213–1239 (2002) · Zbl 1062.70553 [3] Glocker, C., Pfeiffer, F.: Dynamical systems with unilateral contacts. Nonlinear Dyn. 3(4), 245–259 (1992) [4] Glocker, C., Pfeiffer, F.: Multiple impacts with friction in rigid multibody systems. Nonlinear Dyn. 7(4), 471–497 (1995) [5] Glocker, C., Pfeiffer, F.: Complementarity problems in multibody systems with planar friction. Arch. Appl. Mech. 63(7), 452–463 (1993) · Zbl 0782.70008 [6] Glocker, C., Studer, C.: Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics. Multibody Syst. Dyn. 13(4), 447–463 (2005) · Zbl 1114.70008 [7] Leine, R.I., van Campen, D.H., Glocker, C.: Nonlinear dynamics and modeling of various wooden toys with impact and friction. J. Vib. Control 9(1–2), 25–78 (2003) · Zbl 1045.70008 [8] Glocker, C.: Concepts for modeling impacts without friction. Acta Mech. 168(1–2), 1–19 (2004) · Zbl 1063.74075 [9] Djerassi, S.: Collision with friction; Part A: Newton’s hypothesis. Multibody Syst. Dyn. 21(1), 37–54 (2009) · Zbl 1163.70007 [10] Djerassi, S.: Collision with friction; Part B: Poisson’s and Stornge’s hypotheses. Multibody Syst. Dyn. 21(1), 55–70 (2009) · Zbl 1163.70008 [11] Najafabadi, S.A.M., Kövecses, J., Angeles, J.: Impacts in multibody systems: modeling and experiments. Multibody Syst. Dyn. 20(2), 163–176 (2008) · Zbl 1347.70019 [12] Schiehlen, W., Seifried, R.: Three approaches for elastodynamic contact in multibody systems. Multibody Syst. Dyn. 12(1), 1–16 (2004) · Zbl 1174.70312 [13] Pang, J., Trinkle, J.C.: Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction. Math. Program. 73(2), 199–226 (1996) · Zbl 0854.70008 [14] Trinkle, J.C., Tzitzoutis, J., Pang, J.S.: Dynamic multi-rigid-body systems with concurrent distributed contacts: theory and examples. Philos. Trans. Math. Phys. Eng. Sci. Ser. A 359(1789), 2575–2593 (2001) · Zbl 1014.70007 [15] Pfeiffer, F., Foerg, M., Ulbrich, H.: Numerical aspects of non-smooth multibody dynamics. Comput. Methods Appl. Mech. Eng. 195(50–51), 6891–6908 (2006) · Zbl 1120.70305 [16] Förg, M., Pfeiffer, F., Ulbrich, H.: Simulation of unilateral constrained systems with many bodies. Multibody Syst. Dyn. 14(2), 137–154 (2005) · Zbl 1146.70317 [17] Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S.: A study on dynamics of mechanical systems including joints with clearance and lubrication. Mech. Mach. Theory 41(3), 247–261 (2006) · Zbl 1143.70319 [18] Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Influence of the contact-impact force model on the dynamic response of multibody systems. Proc. Inst. Mech. Eng., Part-K J. Multi-body Dyn. 220(1), 21–34 (2006) [19] Gonthier, Y., McPhee, J., Lange, C., Piedboeuf, J.C.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11(3), 209–233 (2004) · Zbl 1143.74344 [20] Bing, S., Ye, J.: Dynamic analysis of the reheat-stop-valve mechanism with revolute clearance joint in consideration of thermal effect. Mech. Mach. Theory 43(12), 1625–1638 (2008) · Zbl 1193.70007 [21] Srivastava, N., Haque, I.: Clearance and friction-induced dynamics of chain CVT drives. Multibody Syst. Dyn. 19(3), 255–280 (2008) · Zbl 1336.70017 [22] Orden, J.C.G.: Analysis of joint clearances in multibody systems. Multibody Syst. Dyn. 13(4), 401–420 (2005) · Zbl 1284.70011 [23] Liu, C.S., Zhang, K., Yang, L.: Normal force-displacement relationship of spherical joints with clearances. J. Comput. Nonlinear Dyn. 1(2), 160–167 (2006) [24] Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Translational joints with clearance in rigid multibody systems. J. Comput. Nonlinear Dyn. 3(1), 0110071-10 (2008) [25] Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Dynamics of multibody systems with spherical clearance joints. J. Comput. Nonlinear Dyn. 1(3), 240–247 (2006) · Zbl 1143.70319 [26] Flores, P., Ambrósio, J.: Revolute joints with clearance in multibody systems. Comput. Struct. 82(17–19), 1359–1369 (2004) · Zbl 1174.70307 [27] Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Spatial revolute joints with clearance for dynamic analysis of multibody systems. Proc. Inst. Mech. Eng., Part-K J. Multi-body Dyn. 220(4), 257–271 (2006) · Zbl 1143.70319 [28] Inna, S., Yuning, Z.: A contact force solution for non-colliding contact dynamics simulation. Multibody Syst. Dyn. 16(3), 263–290 (2006) · Zbl 1207.70006 [29] Paul, B.: Kinematics and Dynamics of Planar Machinery. Prentice-Hall, Englewood Cliffs (1979) [30] Hall, A.S.: Notes on Mechanism Analysis. Waveland Press Inc., Long Grove (1986) [31] Haug, E.J., Wu, S.C., Yang, S.M.: Dynamics of mechanical systems with Coulomb friction stiction, impact and constraint addition-deletion–I and II. Mech. Mach. Theory 21(5), 401–425 (1986) [32] Bae, D.S., Haug, E.J.: A recursive formulation for constrained mechanical system dynamics. I. Open loop systems. Mech. Struct. Mach. 15(3), 359–382 (1987) [33] Bae, D.S, Haug, E.J.: A recursive formulation for constrained mechanical system dynamics: Part II. Closed loop systems. Mech. Des. Struct. Mach. 15(4), 481–506 (1987) [34] Kim, S.S., Haug, E.J.: A recursive formulation for flexible multibody dynamics. I. Open loop systems. Comput. Methods Appl. Mech. Eng. 71(3), 293–314 (1988) · Zbl 0679.73045 [35] Kim, S.S., Haug, E.J.: A recursive formulation for flexible multibody dynamics. II. Closed-loop systems. Comput. Methods Appl. Mech. Eng. 74(3), 251–269 (1989) · Zbl 0724.73293 [36] Hwang, Y.L.: Dynamic recursive decoupling method for closed-loop flexible mechanical systems. Int. J. Non-Linear Mech. 41(10), 1181–1190 (2006) [37] Featherstone, R.: The calculation of robot dynamics using articulated-body inertias. Int. J. Robot. Res. 2(1), 13–30 (1983) [38] Wittenburg, J.: Dynamics of Multibody Systems: Dynamics of Systems of Rigid Bodies. Springer, Berlin (2007) · Zbl 1131.70001 [39] Featherstone, R.: Robot Dynamics Algorithms. Kluwer Academic, Dordrecht (1987) [40] Leine, R.I., Glocker, Ch.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech.-A/Solids 22(2), 193–216 (2003) · Zbl 1038.74513 [41] Leine, R.I., van de Wouw, N.: Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact. Nonlinear Dyn. 51(4), 551–583 (2008) · Zbl 1170.70340 [42] Pennestri, E., Valentini, P.P., Vita, L.: Multibody dynamics simulation of planar linkages with Dahl friction. Multibody Syst. Dyn. 17(4), 321–347 (2007) · Zbl 1111.70005 [43] Rooney, G.T., Deravi, P.: Coulomb friction in mechanism sliding joints. Mech. Mach. Theory 17(3), 207–211 (1982)
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