SOLVING DIFFERENTIAL ALGEBRAIC EQUATIONS FOR SLIDER-CRANK MECHANISMS
Keywords:
DAE, ODE, Euler-Lagrange systems, dynamic modeling, multibody systemsAbstract
The paper deals with the simulation of dynamical models of constrained multibody systems which can be formulated as a set of Differential Algebraic Equations (DAEs). We consider the general dynamical model resulting from Euler-Lagrange formulation. We investigate the solution with the index reduction method. We illustrate our analysis by the example of the slider-crank mechanism. Among many possible representations, we derive a dynamical model based on two variables offering an easier analysis and implementation. We solve the DAE problem with a Matlab function (ode15s) which is dedicated to solve stiff Ordinary Differential Equations (ODEs). Concordant simulation results have been obtained in comparison to other methods proving an acceptable stability and accuracy of the used method for solving this problem.
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References
Sandier, B.Z., "ROBOTICS-Designing the Mechanisms for Automated Machinery", Second Edition, ACADEMIC PRESS, (1999).
Bei, Y. and Fregly, B. J., “Multibody dynamic simulation of knee contact mechanics,” Medical Engineering and Physics 26, pp. 777-789, (2004).
Meriam, J. L. and Kraige, L. G., "Engineering Mechanics, Vol. 2, Dynamics", 3. edition John Wiley & Sons, inc., (1993).
Boudon, B., "Méthodologie de modélisation des systèmes mécatroniques complexes à partir du multi-bond graph : application à la liaison BTP-fuselage d’un hélicoptère" , Doctoral thesis, École Nationale Supérieure d'Arts et Métiers, Décembre (2014).
Ascher, U. and Petzold, L., “Stability of computational methods for constrained dynamics systems,” SIAM J. Sci. Comput., vol. 14, pp. 95-120, (1993).
Pfeiffer, F,et al , "Numerical aspects of non-smooth multibody dynamics", Comput.Methods Appl. Mech. Eng. 195(50–51), 6891–6908 (2006).
Benhammouda, B. and Vazquez-Leal, H. , “Analytical Solution of a Nonlinear Index-Three DAEs System Modelling a Slider-Crank Mechanism,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 206473, 14 pages, (2015).
Horváth Zs.,and Molnárka Gy., " The Dynamic Model of the Slider-crank Mechanism", Acta Technica Jaurinensis Series Transitus, Vol. 6. No. 3. (2013).
Haa, J.L et al, "Dynamic modeling and identification of a slider-crank mechanism", Journal of Sound and Vibration 289 , 1019–1044, (2006)
Zhou, W. et al ; "Symbolic Computation Sequences and Numerical Analytic Geometry Applied to Multibody Dynamical Systems"; Symbolic-Numeric Computation; D. Wang and L. Zhi, Eds. Trends in Mathematics, , , Birkha¨user Verlag Basel/Switzerland, 335–347, (2007).
Fox , B. et al, "Numerical Computation of Differential-Algebraic Equations for Nonlinear Dynamics of Multibody Android Systems in Automobile Crash Simulation', IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 10, October (1999)
[ 12] Shampine, L. F. ," Numerical Solution of Ordinary Differential Equations", Chapman & Hall, New York, (1994).
Gear C.W., " Numerical initial value problems in Ordinary Differential Equations". Prentice Hall,New Jersey, 1971.
Shampine L. F. and. Reichelt M. W. "The MATLAB ODE Suite". SIAM J. Sci. Comput.,18(1):1–22, (1997).
Celaya, E A., et al, "Implementation of an Adaptive BDF2 Formula and Comparison with the MATLAB Ode15s"; CCS 2014. 14th International Conference on Computational Science; Procedia Computer Science; Volume 29, Pages 1014–1026, (2014).
Kurz.T. et al, "Systems with constraint equations in the symbolic multibody simulation software NEWEUL-M2; Multibody Dynamics", 2011, ECCOMAS Thematic Conference, J.C. Samin, P. Fisette (eds.) Brussels, Belgium, 4-7 July (2011).
