Abstract
In this paper, a new analytical approach suitable for the stability analysis of multibody mechanical systems is introduced in the framework of Lagrangian mechanics. The approach developed in this work is based on the direct linearization of the index-three form of the differential-algebraic dynamic equations that describe the motion of mechanical systems subjected to nonlinear constraints. One of the distinguishing features of the proposed method is that it can handle general sets of nonlinear holonomic and/or nonholonomic constraints without altering the original mathematical structure of the equations of motion. While the typical state-space dynamic description associated with multibody systems leads to the definition of a standard eigenproblem, which is impractical, if not impossible, to implement in the case of complex systems, the method developed in this paper involves a generalized state-space representation of the dynamic equations and allows for the formulation of a generalized eigenvalue problem that extends the scope of applicability of the stability analysis to complex mechanical systems. As demonstrated in this investigation employing simple numerical examples, the proposed methodology can be readily implemented in general-purpose multibody computer programs and compares favorably with several other reference computational approaches already available in the multibody literature.
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References
Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1(2), 149–188 (1997)
Tian, Q., Flores, P., Lankarani, H.M.: A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints. Mech. Mach. Theory 122, 1–57 (2018)
Ebrahimi, S., Eberhard, P.: Rigid-elastic modeling of meshing gear wheels in multibody systems. Multibody Syst. Dyn. 16(1), 55–71 (2006)
Meli, E., Malvezzi, M., Papini, S., Pugi, L., Rinchi, M., Rindi, A.: A railway vehicle multibody model for real-time applications. Veh. Syst. Dyn. 46(12), 1083–1105 (2008)
Villecco, F.: On the evaluation of errors in the virtual design of mechanical systems. Machines 6(3), 36 (2018)
Villecco, F., Pellegrino, A.: Evaluation of uncertainties in the design process of complex mechanical systems. Entropy 19(9), 475 (2017)
Villecco, F., Pellegrino, A.: Entropic measure of epistemic uncertainties in multibody system models by axiomatic design. Entropy 19(7), 291 (2017)
Escalona, J.L., Orzechowski, G., Mikkola, A.M.: Flexible multibody modeling of reeving systems including transverse vibrations. Multibody Syst. Dyn. 44(2), 107–133 (2018)
Patel, M., Orzechowski, G., Tian, Q., Shabana, A.A.: A new multibody system approach for tire modeling using ANCF finite elements. Proc. Inst. Mech. Eng. K J. Multi-body Dyn. 230(1), 69–84 (2016)
Rahikainen, J., Mikkola, A., Sopanen, J., Gerstmayr, J.: Combined semi-recursive formulation and lumped fluid method for monolithic simulation of multibody and hydraulic dynamics. Multibody Syst. Dyn. 44(3), 293–311 (2018)
Palomba, I., Richiedei, D., Trevisani, A.: Kinematic state estimation for rigid-link multibody systems by means of nonlinear constraint equations. Multibody Syst. Dyn. 40(1), 1–22 (2017)
Lyapunov, A.M.: The General Problem of the Stability of Motion. Taylor and Francis, London (1992)
Eberhard, P., Schiehlen, W.: Computational dynamics of multibody systems: history, formalisms, and applications. J. Comput. Nonlinear Dyn. 1(1), 3–12 (2006)
Huston, R.L.: Multibody dynamics—modeling and analysis methods. Appl. Mech. Rev. 44(3), 109–117 (1991)
Lot, R., Massaro, M.: A symbolic approach to the multibody modeling of road vehicles. Int. J. Appl. Mech. 9(05), 1750068 (2017)
Maddio, P.D., Meschini, A., Sinatra, R., Cammarata, A.: An optimized form-finding method of an asymmetric large deployable reflector. Eng. Struct. 181, 27–34 (2019)
Leine, R.I.: The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability. Nonlinear Dyn. 59(1–2), 173–182 (2010)
Vukić, Z., Kuljača, L.,Đonlagić, D., Tešnjak, S.: Nonlinear Control Systems. Marcel Dekker, New York (2003)
Jain, A.: Multibody graph transformations and analysis: part I: tree topology systems. Nonlinear Dyn. 67(4), 2779–2797 (2012)
Jain, A.: Multibody graph transformations and analysis: part II: closed-chain constraint embedding. Nonlinear Dyn. 67(4), 2153–2170 (2012)
Ripepi, M., Masarati, P.: Reduced order models using generalized eigenanalysis. Proc. Inst. Mech. Eng. K J. Multi-body Dyn. 225, 52–65 (2011)
Lehner, M., Eberhard, P.: A two-step approach for model reduction in flexible multibody dynamics. Multibody Syst. Dyn. 17, 157–176 (2007)
Koutsovasilis, P., Beitelschmidt, M.: Comparison of model reduction techniques for large mechanical systems: AA study on an elastic rod. Multibody Syst. Dyn. 20, 111–128 (2008)
Nikravesh, P.E., Gim, G.: Ride and stability analysis of a sports car using multibody dynamic simulation. Math. Comput. Model. 14, 953–958 (1990)
Kim, J.K., Han, J.H.: A multibody approach for 6-DOF flight dynamics and stability analysis of the hawkmoth Manduca sexta. Bioinspir. Biomim. 9(1), 016011 (2014). https://doi.org/10.1088/1748-3182/9/1/016011
Sun, M., Wang, J., Xiong, Y.: Dynamic flight stability of hovering insects. Acta Mech. Sin. Xuebao 23, 231–246 (2007)
Bauchau, O.A., Wang, J.: Stability analysis of complex multibody systems. J. Comput. Nonlinear Dyn. 1, 71–80 (2006)
Cuadrado, J., Vilela, D., Iglesias, I., Martín, A., Peña, A.: A multibody model to assess the effect of automotive motor in-wheel configuration on vehicle stability and comfort. Proc. ECCOMAS Themat. Conf. Multibody Dyn. 2013, 1083–1092 (2013)
Escalona, J.L., Chamorro, R.: Stability analysis of vehicles on circular motions using multibody dynamics. Nonlinear Dyn. 53, 237–250 (2008)
Quaranta, G., Mantegazza, P., Masarati, P.: Assessing the local stability of periodic motions for large multibody non-linear systems using proper orthogonal decomposition. J. Sound Vib. 271, 1015–1038 (2004)
Masarati, P.: Direct eigenanalysis of constrained system dynamics. Proc. Inst. Mech. Eng. K J. Multi-body Dyn. 223, 335–342 (2010)
Negrut, D., Ortiz, J.L.: A practical approach for the linearization of the constrained multibody dynamics equations. J. Comput. Nonlinear Dyn. 1, 230–239 (2006)
Ortiz, J.L., Negrut, D.: Exact linearization of multibody systems using user-defined coordinates. SAE Tech Pap. (2006)
Nichkawde, C., Harish, P.M., Ananthkrishnan, N.: Stability analysis of a multibody system model for coupled slosh-vehicle dynamics. J. Sound. Vib. 275, 1069–1083 (2004)
Bencsik, L., Kovács, L.L., Zelei, A.: Stabilization of internal dynamics of underactuated systems by periodic servo-constraints. Int. J. Struct. Stab. Dyn. 17, 1–14 (2017)
Zenkov, D.V., Bloch, A.M., Marsden, J.E.: The energy-momentum method for the stability of non-holonomic systems. Dyn. Stab. Syst. 13, 123–165 (1998)
Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems, vol. 33. American Mathematical Society, Philadelphia (2004)
Ruina, A.: Nonholonomic stability aspects of piecewise holonomic systems. Rep. Math. Phys. 42, 91–100 (1998)
Pollard, B., Fedonyuk, V., Tallapragada, P.: Swimming on limit cycles with nonholonomic constraints. Nonlinear Dyn. 97, 2453–2468 (2019)
Zhang, C., Li, Y., Qi, G., Sheng, A.: Distributed finite-time control for coordinated circumnavigation with multiple non-holonomic robots. Nonlinear Dyn. 98, 573–588 (2019)
Pappalardo, C.M., Guida, D.: On the dynamics and control of underactuated nonholonomic mechanical systems and applications to mobile robots. Arch. Appl. Mech. 89(4), 669–698 (2019)
Pappalardo, C.M., Guida, D.: Forward and inverse dynamics of a unicycle-like mobile robot. Machines 7(1), 5 (2019)
Laulusa, A., Bauchau, O.A.: Review of classical approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011004 (2008)
Bauchau, O.A., Laulusa, A.: Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011005 (2008)
Cheli, F., Pennestrí, E.: Cinematica e Dinamica dei Sistemi Multibody, Volume 1, CEA Casa Editrice Ambrosiana (2009)
Cheli, F., Pennestrí, E.: Cinematica e Dinamica dei Sistemi Multibody, Volume 2, CEA Casa Editrice Ambrosiana (2009)
Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, New York (2003)
Meirovitch, L.: Methods of Analytical Dynamics, Courier Corporation (2010)
Haug, E.J.: Computer Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989)
Flannery, M.R.: The enigma of nonholonomic constraints. Am. J. Phys. 73(3), 265–272 (2005)
Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Englewood Cliffs (1988)
Flannery, M.R.: D’Alembert–Lagrange analytical dynamics for nonholonomic systems. J. Math. Phys. 52(3), 032705 (2011)
Flores, P.: Concepts and Formulations for Spatial Multibody Dynamics. Springer, Berlin (2015)
Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (2013)
Pappalardo, C.M., Guida, D.: On the Lagrange multipliers of the intrinsic constraint equations of rigid multibody mechanical systems. Arch. Appl. Mech. 88(3), 419–451 (2018)
Pappalardo, C.M., Guida, D.: On the use of two-dimensional Euler parameters for the dynamic simulation of planar rigid multibody systems. Arch. Appl. Mech. 87(10), 1647–1665 (2017)
Lanczos, C.: The Variational Principles of Mechanics, Courier Corporation (2012)
Shabana, A.A., Sany, J.R.: An augmented formulation for mechanical systems with non-generalized coordinates: application to rigid body contact problems. Nonlinear Dyn. 24(2), 183–204 (2001)
Nikravesh, P.E.: Planar Multibody Dynamics: Formulation, Programming and Applications. CRC Press, Boca Raton (2007)
Rabier, P.J., Rheinboldt, W.C.: Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint. SIAM, Philadelphia (2000)
Cuadrado, J., Cardenal, J., Bayo, E.: Modeling and solution methods for efficient real-time simulation of multibody dynamics. Multibody Syst. Dyn. 1(3), 259–280 (1997)
Mariti, L., Belfiore, N.P., Pennestrí, E., Valentini, P.P.: Comparison of solution strategies for multibody dynamics equations. Int. J. Numer. Methods Eng. 88(7), 637–656 (2011)
Wehage, K.T., Wehage, R.A., Ravani, B.: Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring. Mech. Mach. Theory 92, 464–483 (2015)
Pappalardo, C.M.: A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems. Nonlinear Dyn. 81(4), 1841–1869 (2015)
Marques, F., Souto, A.P., Flores, P.: On the constraints violation in forward dynamics of multibody systems. Multibody Syst. Dyn. 39(4), 385–419 (2017)
Cossalter, V., Lot, R., Massaro, M.: An advanced multibody code for handling and stability analysis of motorcycles. Meccanica 46(5), 943–958 (2011)
Cheli, F., Diana, G.: Advanced Dynamics of Mechanical Systems. Springer, Berlin (2015)
Shabana, A.A.: Computational Continuum Mechanics. Wiley, New York (2018)
Shabana, A.A., Zaazaa, K.E., Sugiyama, H.: Railroad Vehicle Dynamics: A Computational Approach. CRC Press, Boca Raton (2007)
Shi, P., McPhee, J.: Dynamics of flexible multibody systems using virtual work and linear graph theory. Multibody Syst. Dyn. 4(4), 355–381 (2000)
Pappalardo, C.M., Guida, D.: A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems. Arch. Appl. Mech. 88(12), 2153–2177 (2018)
Shabana, A.A.: Computational Dynamics. Wiley, New York (2009)
Garcia De Jalon, J.G., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer, Berlin (2012)
Pennestrí, E., Vita, L.: Strategies for the numerical integration of DAE systems in multibody dynamics. Comput. Appl. Eng. Educ. 12(2), 106–116 (2004)
Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (2007)
Udwadia, F.E., Wanichanon, T.: On general nonlinear constrained mechanical systems. Numer. Algebra Control Optim. 3(3), 425–443 (2013)
Shabana, A.A.: Euler parameters kinetic singularity. Proc. Inst. Mech. Eng. K J. Multi-body Dyn. 228(3), 307–313 (2014)
Udwadia, F.E., Kalaba, R.E.: A new perspective on constrained motion. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 439(1906), 407–410 (1992)
Udwadia, F.E., Kalaba, R.E.: On the foundations of analytical dynamics. Int. J. Non-linear Mech. 37(6), 1079–1090 (2002)
Shabana, A.A.: Vibration of Discrete and Continuous Systems, 3rd edn. Springer, New York (2019)
Meirovitch, L.: Fundamentals of Vibrations. Waveland Press, New York (2010)
Amirouche, F.: Fundamentals of Multibody Dynamics: Theory and Applications. Springer, Berlin (2007)
Hogben, L.: Handbook of Linear Algebra. Chapman and Hall/CRC, London (2013)
Golub, G.H., Van Der Vorst, H.A.: Eigenvalue computation in the 20th century. J. Comput. Appl. Math. 123(1–2), 35–65 (2000)
Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997)
Ashino, R., Nagase, M., Vaillancourt, R.: Behind and beyond the MATLAB ODE suite. Comput. Math. Appl. 40(4–5), 491–512 (2000)
Pappalardo, C.M., De Simone, M.C., Guida, D.: Multibody modeling and nonlinear control of the pantograph/catenary system. Arch. Appl. Mech. 89(8), 1589–1626 (2019)
Guida, R., De Simone, M.C., Dasic, P., Guida, D.: Modeling techniques for kinematic analysis of a six-axis robotic arm. IOP Conf. Ser. Mater. Sci. Eng. 568(1), 012115 (2019)
Rivera, Z.B., De Simone, M.C., Guida, D.: Unmanned ground vehicle modelling in Gazebo/ROS-based environments. Machines 7(2), 42 (2019)
Colucci, F., De Simone, M.C., Guida, D.: TLD design and development for vibration mitigation in structures. Lect. Notes Netw. Syst. 76, 59–72 (2019)
De Simone, M., Rivera, Z., Guida, D.: Obstacle avoidance system for unmanned ground vehicles by using ultrasonic sensors. Machines 6(2), 18 (2018)
De Simone, M.C., Guida, D.: Control design for an under-actuated UAV model. FME Trans. 46(4), 443–452 (2018)
De Simone, M.C., Guida, D.: Identification and control of a unmanned ground vehicle by using Arduino. UPB Sci. Bull. Ser. D Mech. Eng. 80(1), 141–154 (2018)
De Simone, M.C., Guida, D.: Modal coupling in presence of dry friction. Machines 6(1), 8 (2018)
De Simone, M.C., Rivera, Z.B., Guida, D.: Finite element analysis on squeal-noise in railway applications. FME Trans. 46(1), 93–100 (2018)
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This research paper was principally developed by the first author (Carmine Maria Pappalardo). A great support was provided by the second author (Antonio Lettieri). The detailed review carried out by the third author (Domenico Guida) considerably improved the quality of the work.
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Appendix A
Appendix A
In this appendix, the main equations describing four alternative computational algorithms for the determination of the generalized acceleration vector of a multibody system are synthetically reported.
1.1 A.1 Embedding technique
1.2 A.2 Amalgamated formulation
1.3 A.3 Projection method
1.4 A.4 Fundamental equations of constrained motion: Udwadia–Kalaba equations
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Pappalardo, C.M., Lettieri, A. & Guida, D. Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints. Arch Appl Mech 90, 1961–2005 (2020). https://doi.org/10.1007/s00419-020-01706-2
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DOI: https://doi.org/10.1007/s00419-020-01706-2