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Continuum Mechanics and Thermodynamics

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17-11-2023 | Original Article

Equivalent analytical formulation-based multibody elastic system analysis using one-dimensional finite elements

Authors: Sorin Vlase, Marin Marin, Andreas Öchsner, Omar El Moutea

Published in: Continuum Mechanics and Thermodynamics | Issue 1/2024

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Abstract

For the particular case of an elastic multibody system (MBS) that can be modeled using one-dimensional finite elements, the main methods offered by analytical mechanics in its classical form for analysis are presented in a unitary description. The aim of the work is to present in a unitary form the main methods offered by classical mechanics for the analysis of solid systems. There is also a review of the literature that uses and highlights these methods, which need to be reconsidered considering the progress of the industry and the complexity of the studied systems. Thus, the kinematics of a finite element is described for the calculation of the main quantities used in the modeling of multibody systems and in analytical mechanics. The main methods used in the research of MBS systems are presented and analyzed. Thus, Lagrange’s equations, Gibbs–Appell equations, Maggi’s formalism, Kane’s equations and Hamilton’s equations are studied in turn. This presentation is determined by the advantages that alternatives to Lagrange’s equations can offer, which currently represent the method most used by researchers.

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Sklar, L.: From virtual work to Lagrange’s equation. In: Vol. Philosophy and the Foundations of Dynamics, pp. 96–101 (2013)

Fabien, B.C.: Lagrange’s equation of motion. In: Vol. Analytical System Dynamics: Modeling and Simulation, pp. 109–160 (2009)

Bianchini, S.: On the Euler–Lagrange equation for a variational problem. Differ. Equ. Chaos Var. Probl. 75, 61–77 (2008)MathSciNet

Sun, W.: Dynamic iteration method for Lagrange’s equations of multibody systems. In: Proceedings of the 31st Chinese Control and Decision Conference (CCDC-2019), Nanchang, China, pp. 571–575 (2019)

Malvezzi, F., Orsino, R.M.M., Coelho, T.A.H.: Lagrange’s, Maggi’s and Kane’s equations applied to the dynamic modelling of serial manipulator. In: Proceedings of the 17th International Symposium on Dynamic Problems of Mechanics (DIINAME 2017), Sao Sebastiao, Brazil, pp. 291–304 (2017). https://doi.org/10.1007/978-3-319-91217-2_20

Craifaleanu, A., Stroe, I.: Study of vibrations of a robotic arm, using the lagrange equations with respect to a non-inertial reference frame. In: Accoustics and Vibration of Mechanical Structures (AVMS-2017), vol. 198, pp. 67–73 (2018). https://doi.org/10.1007/978-3-319-69823-6_8

Li, D.Q., Hong, H.J., Jiang, X.L.: Dynamics modeling, control system design and simulation of manipulator based on Lagrange equation. Mech. Mach. Sci. 408, 1129–1141 (2017). https://doi.org/10.1007/978-981-10-2875-5_91CrossRef

Gans, F.R.: Engineering Dynamics: from the Lagrangian to Simulation. Springer, New York (2013)CrossRef

Shi, Z., Meacci, M., Meli, E., Wang, K.Y., Rindi, A.: Validation of a finite element multibody system model for vehicle-slab track application. In: 26th Symposium of the International Association of Vehicle System Dynamics (IAVSD), Gothenburg, Sweden, August 12–16, 2019. In Advanced in Dynamics of Vehicle on Roads and Tracks, IAVSD 2019, Proceedings Paper, in Book Series, Lecture Notes in Mechanical Engineering, pp. 407–414 (2020). https://doi.org/10.1007/978-3-030-38077-9_48

10.

Tokarczyk, J.: Migration of computational models in virtual prototyping of complex mechanical systems. In: Book Group Author IAENG. World Congress on Engineering and Computer Science, WCECS 2012, San Francisco, CA, 2012, VOL II, Proceedings Paper, Book Series: Lecture Notes in Engineering and Computer Science, pp. 1334–1337 (2012)

11.

Marce-Nogue, J., Klodowski, A., Sanchez, M., Gil, L.: Coupling finite element analysis and multibody system dynamics for biological research. Palaeontol. Electron. 18(2), 1–14 (2015)

12.

Ding, J.G., Dai, Y.W., Qiao, Z., Huang, H.J., Zhuang, W.: Analysis of the response of a frame structure during an earthquake using the transfer matrix method of a multibody system. J. Eng. Mech. 14(8), 04015020 (2015). https://doi.org/10.1061/(ASCE)EM.1943-7889.0000919CrossRef

13.

Wallrapp, O., Sachau, D.: Space flight dynamic simulations using finite element results in multibody system codes. In: 2nd International Conference on Computational Structures Technology, Athens, Greece, 1994, Proceedings Paper. In Advanced in Computational Mechanics, pp. 149–158 (1994)

14.

Scutaru, M.L., Chircan, E., Marin, M., Grif, H.S.: Liaison forces eliminating and assembling of the motion equation in the study of multibody system with elastic elements. In: 13th International Conference Interdisciplinarity in Engineering (Inter-Eng. 2019), Targu Mures, ROMANIA, 2019, Proceedings Paper, Book Series: Procedia Manufacturing, vol. 46, pp. 78–86 (2020). https://doi.org/10.1016/j.promfg.2020.03.013

15.

Shabana, A.A.: On the integration of large deformation finite element and multibody system algorithms. In: Proceedings of the International Conference on Mechanical Engineering and Mechanics 2005, Vols. 1 and 2, pp. 63–70 (2005)

16.

Shabana, A.A., Bauchau, O.A., Hulbert, G.M.: Integration of large deformation finite element and multibody system algorithms. J. Comput. Nonlinear Dyn. 2(4), 351–359 (2007). https://doi.org/10.1115/1.2756075CrossRef

17.

Rui, X., Rong, B., Wang, G.: New method for dynamics modeling and simulation of flexible multibody system. In: Proceedings of the third International Conference on Mechanical Engineering and Mechanics, Beijing, China, 2009, Proceedings Vols. 1 and 2, pp. 17–23 (2009)

18.

Witteveen, W., Stefan, P., Pichler, F.: On the projection of a flexible bodies modal coordinates onto another finite element model with local modifications. J. Comput. Nonlinear Dyn. 14(7), 074501 (2019). https://doi.org/10.1115/1.4043524CrossRef

19.

Öchsner, A.: Computational Statics and Dynamics, An Introduction Based on the Finite Element Method. Springer, Singapore (2020)CrossRef

20.

Liang, Y.T., McPhee, J.: Symbolic integration of multibody system dynamics with the finite element method. Multibody Syst.Dyn. 43(4), 387–405 (2018). https://doi.org/10.1007/s11044-018-9627-6MathSciNetCrossRef

21.

Wallrapp, O.: Standardization of flexible body modeling in multibody system codes. 1. Definition of standard input data. J. Struct. Mach. 22(3), 283–304 (1994). https://doi.org/10.1080/08905459408905214CrossRef

22.

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. J. Multibody Dyn. 230(1), 69–84 (2016). https://doi.org/10.1177/1464419315574641CrossRef

23.

Zhang, J.H., Jiang, S.S.: Rigid-flexible coupling model and dynamic analysis of rocket sled. In: International Conference on Sustainable Construction Materials and Computer Engineering (ICSCMCE 2011). Sustainable Construction Materials and Computer Engineering, Kunming, China, 2011, Proceedings Paper, Book Series: Advanced Materials Research, vol. 346, pp. 447–454 (2012). https://doi.org/10.4028/www.scientific.net/AMR.346.447

24.

Lu, H.J., Rui, X.T., Ding, Y.Y., Chang, Y., Chen, Y.H., Ding, J.G., Zhang, X.P.: A hybrid numerical method for vibration analysis of linear multibody systems with flexible components. Appl. Math. Modell. 101, 748–771 (2022). https://doi.org/10.1016/j.apm.2021.09.015MathSciNetCrossRef

25.

You, T.W., Gong, D., Zhou, J.S., Sun, Y., Chen, J.X.: Frequency response function-based model updating of flexible vehicle body using experiment modal parameter. Veh. Syst. Dyn. (2021). https://doi.org/10.1080/00423114.2021.1983182CrossRef

26.

Costa, J.N., Antunes, P., Magalhaes, H., Pombo, J., Ambrosio, J.: A finite element methodology to model flexible tracks with arbitrary geometry for railway dynamics applications. Comput. Struct. 254, 106519 (2021). https://doi.org/10.1016/j.compstruc.2021.106519CrossRef

27.

Cammarata, A.: Global modes for the reduction of flexible multibody systems Methodology and complexity. Multibody Sys.Dyn. 53(1), 59–83 (2021). https://doi.org/10.1007/s11044-021-09790-0MathSciNetCrossRef

28.

Manca, A.G., Pappalardo, C.M.: Topology optimization procedure of aircraft mechanical components based on computer-aided design, multibody dynamics and finite element analysis. In: 3rd International Conference on Design, Simulation, Manufacturing - (DSMIE), Kharkiv, Ukraine, 2020, Advances in Design, Simulation and Manufacturing III: Mechanical and Chemical Engineering, Vol. 2, Book Series: Lecture Notes in Mechanical Engineering, pp. 159-168 (2020). https://doi.org/10.1007/978-3-030-50491-5_16

29.

Lu, H.J., Rui, X.T., Zhang, X.P.: A computationally efficient modeling method for the vibration analyses of two-dimensional system structures using reduce transfer matrix method for multibody system. J. Sound Vib. 502, 116096 (2021). https://doi.org/10.1016/j.jsv.2021.116096CrossRef

30.

Liu, X., Sun, C.L., Banerjee, J.R., Dan, H.C., Chang, L.: An exact dynamic stiffness method for multibody systems consisting of beams and rigid-bodies. Mech. Syst. Signal Process. 150, 107264 (2021). https://doi.org/10.1016/j.ymssp.2020.107264CrossRef

31.

Raoofian, A., Taghvaeipour, A., Kamali, E.A.: Elastodynamic analysis of multibody systems and parametric mass matrix derivation. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1815211CrossRef

32.

Marin, M.: Some estimates on vibrations in thermoelasticity of dipolar bodies. J. Vib. Control 16(1), 33–47 (2010)MathSciNetCrossRef

33.

Wang, G., Qi, Z.H., Xu, J.S.: A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems. Comput. Methods Appl. Mech. Eng. 360, 112701 (2020). https://doi.org/10.1016/j.cma.2019.112701MathSciNetCrossRefADS

34.

Hou, Y.S., Liu, C., Hu, H.Y.: Component-level proper orthogonal decomposition for flexible multibody systems. Comput. Methods Appl. Mech. Eng. 361, 112690 (2020). https://doi.org/10.1016/j.cma.2019.11269MathSciNetCrossRefADS

35.

Bagci, C.: Elastodynamic response of mechanical systems using matrix exponential mode uncoupling and incremental forcing techniques with finite element method. In: Proceedings of the Sixth Word Congress on Theory of Machines and Mechanisms, India, p. 472 (1983)

36.

Bahgat, B.M., Willmert, K.D.: Finite element vibrational analysis of planar mechanisms. Mech. Mach. Theory 11, 47 (1976)CrossRef

37.

Cleghorn, W.L., Fenton, E.G., Tabarrok, K.B.: Finite element analysis of high-speed flexible mechanisms. Mech. Mach. Theory 16, 407 (1981)CrossRef

38.

Vlase, S., Dănăşel, C., Scutaru, M.L., Mihălcică, M.: Finite element analysis of a two-dimensional linear elastic systems with a plane ’ ’rigid motion. Rom. Journ. Phys. 59(5–6), 476–487 (2014)

39.

Deü, J.-F., Galucio, A.C., Ohayon, R.: Dynamic responses of flexible-link mechanisms with passive/active damping treatment. Comput. Struct. 86(3–5), 258–265 (2008)CrossRef

40.

Othman, M.I.A., Fekry, M., Marin, M.: Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating. Struct. Eng. Mech. 73(6), 621–629 (2020)

41.

Neto, M.A., Ambrósio, J.A.C., Leal, R.P.: Composite materials in flexible multibody systems. Comput. Methods Appl. Mech. Eng. 195(50–51), 6860–6873 (2006)CrossRefADS

42.

Piras, G., Cleghorn, W.L., Mills, J.K.: Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links. Mech. Mach. Theory 40(7), 849–862 (2005)CrossRef

43.

Abo-Dahab, S.M., Abouelregal, A.E., Marin, M.: Generalized thermoelastic functionally graded on a thin slim strip non-Gaussian laser beam. Symmetry 12(7), 1094 (2020)CrossRefADS

44.

Zhang, X., Erdman, A.G.: Dynamic responses of flexible linkage mechanisms with viscoelastic constrained layer damping treatment. Comput. Struct. 79(13), 1265–1274 (2001)CrossRef

45.

Vlase, S., Negrean, I., Marin, M., Scutaru, M.L.: Energy of accelerations used to obtain the motion equations of a three- dimensional finite element. Symmetry 12(2), 321 (2020). https://doi.org/10.3390/sym12020321CrossRefADS

46.

Trapp, M., Öchsner, A.: One-dimensional continuum approach. In: Computational Plasticity for Finite Elements: A Fortran-Based Introduction, pp. 7–18 (2018)

47.

Öchsner, A.: Special numerical techniques to joint design. In: Handbook of Adhesion Technology, vols. 1 and 2, pp. 661–688 (2011)

48.

Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite-element method for conservation-laws. 3. One-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)MathSciNetCrossRefADS

49.

Adjerid, S., Flaherty, J.E.: A moving finite-element method with error estimation and refinement for one-dimensional time-dependent partial-differential equations. SIAM J. Numer. Anal. 23(4), 778–796 (1986)MathSciNetCrossRefADS

50.

Babuska, I., Rheinboldt, W.C.: A posteriori error analysis of finite-element solutions for one-dimensional problems. SIAM J. Numer. Anal. 18(3), 565–589 (1981)MathSciNetCrossRefADS

51.

Zhou, W.J.J., Ichchou, M.N., Bareille, O.: Finite element techniques for calculations of wave modes in one-dimensional structural waveguides. Struct. Control Health Monit. 18(7), 737–751 (2011)CrossRef

52.

Hui, Y., Giunta, G., Belouettar, S., Huang, Q., Hu, H., Carrera, E.: A free vibration analysis of three-dimensional sandwich beams using hierarchical one-dimensional finite elements. Compos. Part B Eng. 110, 7–19 (2017)CrossRef

53.

Freund, M., Ihlemann, J.: Generalization of one-dimensional material models for the finite element method. ZAMM-Z. Agnew. Math. Mech. 90(5), 399–417 (2010)MathSciNetCrossRef

54.

Carrera, E., Filippi, M.: Variable kinematic one-dimensional finite elements for the analysis of rotors made of composite materials. J. Eng. Gas Turbines Power Trans. ASME 136(9), 092501 (2014)CrossRef

55.

Carrera, E., Zappino, E.: One-dimensional finite element formulation with node-dependent kinematics. Comput. Struct. 192, 114–125 (2017)CrossRef

56.

Öchsner, A., Merkel, M.: One-Dimensional Finite Elements, An Introduction to the FE Method. Springer, Cham (2018)CrossRef

57.

Ursu-Fisher, N.: Elements of Analytical Mechanics. House of Science Book Press, C-Napoca (2015)

58.

Gibbs, J.W.: On the fundamental formulae of dynamics. Am. J. Math. 2, 49–64 (1879)MathSciNetCrossRef

59.

Appell, P.: Sur une forme générale des equations de la dynamique. C.R. Acad. Sci. Paris, vol. 129 (1899)

60.

Negrean, I., Crisan, A., Serdean, F., Vlase, S.: New formulations on kinetic energy and acceleration energies in applied mechanics of systems. Symmetry-Basel 14(5), 896 (2022)CrossRefADS

61.

Mirtaheri, S.M., Zohoor, H.: The explicit Gibbs–Appell equations of motion for rigid-body constrained mechanical system. In: Book Series: RSI International Conference on Robotics and Mechatronics ICRoM, pp. 304–309 (2018)

62.

Korayem, M.H., Dehkordi, S.F.: Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation. Appl. Math. Model. 65, 443–463 (2019)MathSciNetCrossRef

63.

Shafei, A.M., Shafei, H.R.: A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment. Multibody Syst. Dyn. 38(1), 21–42 (2017)MathSciNetCrossRef

64.

Korayem, M.H., Dehkordi, S.F.: Derivation of dynamic equation of viscoelastic manipulator with revolute-prismatic joint using recursive Gibbs–Appell formulation. Nonlinear Dyn. 89(3), 2041–2064 (2017)CrossRef

65.

Marin, M., Seadawy, A., Vlase, S., Chirila, A.: On mixed problem in thermos-elasticity of type III for Cosserat media. J. Taibah Univ. Sci. 16(1), 1264–1274 (2022)CrossRef

66.

Cheng, Y.D., Wang, Z.X.: A new discontinuous Galerkin finite element method for directly solving the Hamilton–Jacobi equations. J. Comput. Phys. 268, 134–153 (2014). https://doi.org/10.1016/j.jcp.2014.02.041MathSciNetCrossRefADS

67.

Abbas, I., Hobiny, A., Marin, M.: Photo-thermal interactions in a semi-conductor material with cylindrical cavities and variable thermal conductivity. J. Taibah Univ. Sci. 14(1), 1369–1376 (2020)CrossRef

68.

Anguelov, R., Lubuma, J.M.S., Minani, F.: A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method. Math. Methods Appl. Sci. 33(1), 41–48 (2010). https://doi.org/10.1002/mma.1148MathSciNetCrossRefADS

69.

Liu, H.L., Pollack, M., Saran, H.: Alternating evolution schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 35(1), 122–149 (2013). https://doi.org/10.1137/120862806MathSciNetCrossRef

70.

Rong, B., Rui, X.T., Tao, L., Wang, G.P.: Theoretical modeling and numerical solution methods for flexible multibody system dynamics. Nonlinear Dyn. 98(2), 1519–1553 (2019). https://doi.org/10.1007/s11071-019-05191-3CrossRef

71.

Vlase, S., Marin, M., Scutaru, M.L.: Maggi’s equations used in the finite element analysis of the multibody systems with elastic elements. Mathematics 8(3), 399 (2020). https://doi.org/10.3390/math8030399CrossRef

72.

Bratu, P., Nitu, M.C., Tonciu, O.: Effect of vibration transmission in the case of the vibratory roller compactor. Romanian J. Acoust. Vib. 20(1), 67–72 (2023)

73.

Vlase, S., Negrean, I., Marin, M., Nastac, S.: Kane’s method-based simulation and modeling robots with elastic elements, using finite element method. Mathematics 8(5), 805 (2020). https://doi.org/10.3390/math8050805CrossRef

74.

Mitu, G.L., Chircan, E., Scutaru, M.L., Vlase, S.: Kane’s formalism used to the vibration analysis of a wind water pump. Symmetry-Basel 12(6), 1030 (2020). https://doi.org/10.3390/sym12061030CrossRefADS

75.

Bratu, P., Vlase, S., Dragan, N., Vasile, O., Itu, C., Nitu, M.C.: Modal analysis of the inertial platform of the laser ELI-NP facility in Magurele–Bucharest. Romanian J. Acoust. Vib. 19(2), 112–120 (2022)

76.

Haug, E.J.: Extension of Maggi and Kane equations to holonomic dynamic systems. J. Comput. Nonlinear Dyn. 13(6), 121003 (2018)CrossRef

77.

Noorani, M.R.S.: Hybrid dynamical model of a gait training robot using Maggi’s method for constrained motions. In: 6th RSI International Conference on Robotics and Mechatronics IcRoM, pp. 183–188 (2018)

78.

Amengonu, Y.H., Kakad, Y.P.: Dynamics and control for constrained multibody systems modeled with Maggi’s equation: application to differential mobile robots Part II. In: Proceedings of the 27th International Conference on CADCAM, Robotics and Factories of the Future 2014, London, UK, 22–24 (July 2014)

79.

de Jalon, J.G., Callejo, A., Hidalgo, A.F.: Efficient solution of Maggi’s equations. J. Comput. Nonlinear Dyn. 7(2) (2012)

80.

Chen, Y.H.: Equations of motion of mechanical systems under servo constraints: the Maggi approach. Mechatronics 18(4), 208–217 (2008)CrossRef

81.

Byachkov, A.B., Suslonov, V.M.: Maggi’s equations in terms of quasi-coordinates. Regular Chaot. Dyn. 7(3), 269–279 (2002)MathSciNetCrossRefADS

82.

Codarcea-Munteanu, L., Marin, M., Vlase, S.: The study of vibrations in the context of porous micropolar media thermoelasticity and the absence of energy dissipation. J. Comput. Appl. Mech. 54(3), 437–454 (2023)

83.

Marin, M., et al.: Some results on eigenvalue problems in the theory of piezoelectric porous dipolar bodies. Contin. Mech. Thermodyn. 35, 1969–1979 (2023)MathSciNetCrossRefADS

84.

Marin, M., Öchsner, A.: The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity. Contin. Mech. Thermodyn. 29, 1365–1374 (2017)MathSciNetCrossRefADS

85.

Ghavamian, A., Öchsner, A.: Numerical modeling of eigenmodes and eigenfrequencies of single- and multi-walled carbon nanotubes under the influence of atomic defects. Comput. Mater. Sci. 72, 42–48 (2013)CrossRef

86.

Groza, G., Khan, S.M.A., Pop, N.: Approximate solutions of boundary value problems for ODEs using Newton interpolating series. Carpathian J. Math. 25(1), 73–81 (2009)MathSciNet

87.

Groza, G., Pop, N.: A numerical method for solving of the boundary value problems for ordinary differential equations. Results Math. 53(3/4), 295–302 (2009)MathSciNetCrossRef

88.

Bhatti, M.M., Ellahi, R.: Numerical investigation of non-Darcian nanofluid flow across a stretchy elastic medium with velocity and thermal slips. Numer. Heat Transf. B: Fundam. 83(5), 323–343 (2023)CrossRefADS

89.

Bhatti, M.M., Oztop, H.F., Ellahi, R.: Study of the magnetized hybrid nanofluid flow through a flat elastic surface with applications in solar energy. Materials 15(12), 7507 (2022)CrossRefADS

Title: Equivalent analytical formulation-based multibody elastic system analysis using one-dimensional finite elements
Authors: Sorin Vlase
Marin Marin
Andreas Öchsner
Omar El Moutea
Publication date: 17-11-2023
Publisher: Springer Berlin Heidelberg
Published in: Continuum Mechanics and Thermodynamics / Issue 1/2024
Print ISSN: 0935-1175
Electronic ISSN: 1432-0959
DOI: https://doi.org/10.1007/s00161-023-01270-4

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