Simulation of Elastic Tree-Like Dynamic Systems in Presence of External Holonomic Constraints
Authors: Gevorkian H.A. | Published: 22.04.2020 |
Published in issue: #2(89)/2020 | |
DOI: 10.18698/1812-3368-2020-2-4-24 | |
Category: Mathematics and Mechanics | Chapter: Differential Equations and Mathematical Physics | |
Keywords: Newton --- Euler method, dynamic system, elastic tree-like systems, Lagrange multipliers, slider-crank mechanism, elastic connecting rod, deformations, holonomic constraint |
Modern scientific literature pays close attention to the problems of optimal modeling of elastic dynamic systems. The symbolic-recursive model of Newton --- Euler method with the provision of computational algorithms with a degree of complexity proportional to the dimension of these systems, i.e., O(n), has been adapted for dynamic systems with open kinematic chains, in particular, for elastic manipulators. If dynamic systems have closed kinematic chains, it is extremely difficult to propagate the strategy of numerical analysis without the mass matrix inversion. Consequently, the task of optimal modeling of tree-like dynamic systems is reduced to the search for combined strategies that use the procedures of strategies with and without inversion of mass matrices simultaneously. The paper introduces a method of numerical dynamic analysis of elastic tree-like multilink systems, the method combining the procedure of inverting the mass matrix with the procedure of effective kinematic calculation, borrowed from the generalized Newton --- Euler method. An approximate dynamic analysis technique is proposed that fully reproduces the recursive procedures of the generalized Newton --- Euler method. The technique is confirmed to the extent that the period of inversion of these systems is less than the time of their full functioning, and the range of displacements during one cycle is less than the complete revolution of the mechanism. The use of the approximate method for the dynamic analysis of elastic mechanisms is considered using the example of a numerical dynamic calculation of a slider-crank mechanism with an elastic connecting rod
References
[1] Luh J.Y.S., Walker M.W., Paul R.P.C. On-line computational scheme for mechanical manipulators. J. Dyn. Sys., Meas., Control., 1980, vol. 102, iss. 2, pp. 69--76. DOI: https://doi.org/10.1115/1.3149599
[2] Featherstone R. The calculation of robot dynamics using articulated-body inertias. Int. J. Robot. Res., 1983, vol. 2, iss. 1, pp. 13--30. DOI: https://doi.org/10.1177%2F027836498300200102
[3] Dombre E., Khalil W. Modelisation et commande des robots. Paris, Hermes, 1988.
[4] Dombre E., Khalil W. Modelisation, identification et commande des robots. Paris, Hermes, 1999.
[5] Verlinden O., Dehombreux P., Conti C., et al. A new formulation for the direct dynamic simulation of flexible mechanisms based on the Newton --- Euler inverse method. Int. J. Numer. Meth. Eng., 1994, vol. 37, iss. 19, pp. 3363--3387. DOI: https://doi.org/10.1002/nme.1620371910
[6] Fisette P., Samin J.-C. Symbolic generation of large multibody system dynamic equations using a new semi-explicite Newton/Euler recursive scheme. Arch. Appl. Mech., 1996, vol. 66, iss. 3, pp. 187--199. DOI: https://doi.org/10.1007/BF00795220
[7] Boyer F., Coiffet P. Generalization of Newton --- Euler model for flexible manipulators. Int. J. Robot. Syst., 1996, vol. 13, iss. 1, pp. 11--24. DOI: https://doi.org/10.1002/(SICI)1097-4563(199601)13:1%3C11::AID-ROB2%3E3.0.CO;2-Y
[8] Boyer F., Coiffet P. Symbolic modeling of a flexible manipulator via assembling of its generalized Newton --- Euler model. Mech. Mach. Theory, 1996, vol. 31, no. 1, pp. 45--56. DOI: https://doi.org/10.1016/0094-114X(95)00038-Z
[9] Boyer F., Khalil W. An efficient calculation of flexible manipulators inverse dynamics. Int. J. Robot. Res., 1998, vol. 17, iss. 3, pp. 282--293. DOI: https://doi.org/10.1177%2F027836499801700305
[10] Sarkisyan Yu.L., Stepanyan K.G., Azuz N., et al. Dynamic analysis of flexible manipulators using the generalized Newton --- Euler method. Izv. NAN RA i GIUA. Ser. TN [Proceedings of the NAS RA and SEUA: Technical Sciences], 2004, vol. 57, no. 1, pp. 3--10 (in Russ.).
[11] Sarkisyan Yu.L., Stepanyan K.G., Gevorkian H.A. Dynamic analysis of flexible tree-like mechanical systems without external constraints. Izv. NAN RA i GIUA. Ser. TN [Proceedings of the NAS RA and SEUA: Technical Sciences], 2006, vol. 59, no. 1, pp. 3--9 (in Russ.).
[12] Gevorkian H.A. Dynamic analysis of elastic tree-like mechanical systems with external holonomic constraints. Information Technologies and Management, 2004, no. 4, pp. 36--43 (in Russ.).
[13] Gevorkian H.A. Dynamic modeling of mechanisms with elastic links of variable length. Vestnik GIUA. Seriya: Mekhanika, Mashinovedenie, Mashinostroenie [Proceedings of National Polytechnic University of Armenia --- Mechanics, Machine Science, Machine Building], 2014, no. 1, pp. 34--41 (in Russ.).
[14] Gevorkian H.A. An application of the generalized Newton --- Euler method in the optimal control problems of flexible mechanisms. Izv. NAN RA i GIUA. Ser. TN [Proceedings of the NAS RA and SEUA: Technical Sciences], 2010, vol. 63, no. 2, pp. 133--138 (in Russ.).
[15] Gevorkian H.A. [On a kind of dynamical analysis of manipulators by means of their mass matrix inversion]. Mater. V mezhdunar. konf. "Aktual’nye problemy mekhaniki sploshnoy sredy" [Proc. V Int. Conf. Actual Problems of Continuum Mechanics]. Tsakhkadzor, NUASA Publ., pp. 61--62 (in Russ.).
[16] Gevorkian H.A. [Dynamical modelling of four-bar linkage mechanism with elastic piston rod]. Sb. tr. VIII mezhdunar. konf. "Problemy dinamiki vzaimodeystviya deformiruemykh sred" [Proc. VIII Int. Conf. Problems of Interaction Dynamics of Deformable Medium]. Goris-Stepanakert, 2014, pp. 143--147 (in Russ.).
[17] Gevorkian H.A. [Frequency analysis of piston rod elastic displacements in motion process of slider-crank mechanism without taking dissipative forces into account]. Sb. tr. VI mezhdunar. konf. "Aktual’nye problemy mekhaniki sploshnoy sredy" [Proc. VI Int. Conf. Actual Problems of Continuum Mechanics]. Dilizhan, 2019, pp. 111--114 (in Russ.).
[18] Gofron M., Shabana A.A. Control structure interaction in the nonlinear analysis of flexible mechanical systems. Nonlinear Dyn., 1993, vol. 4, iss. 2, pp. 183--206. DOI: https://doi.org/10.1007/BF00045253