Abstract
This paper addresses the problem of sliding beams or beams with sliding appendages, with special emphasis on situations where the axial motion of the beam or appendage is not prescribed a priori. Hamilton’s variational principle is used to derive the weak and strong forms of governing equations based on the systematic use of Reynolds’ transport theorem. The strong form of the governing equations involve the mechanical and configurational momentum equations, together with the proper boundary conditions. It is shown that the configurational momentum equations are linear combinations of their mechanical counterparts and hence, are redundant. A weak form of the same equations is also developed; the configurational and mechanical momentum equations become independent because they combine in an integral form the strong mechanical and configurational momentum equations with their respective natural boundary conditions. The domain- and boundary-based formulations, stemming from these two forms of the governing equations, respectively, are proposed, and numerical examples are presented to contrast their relative performances. The predictions of both formulations are found to be in good agreement with those obtained from an ABAQUS model using contact pairs. The domain-based formulation presents a higher convergence rate than the boundary-based formulation. Clearly, the proper treatment of the configurational forces impacts the accuracy of the model significantly.
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References
Kirk, D.E.: Optimal Control Theory: An Introduction. Prentice-Hall, Englewood Cliffs (1970)
Hu, H.C.: Variational Principles of Theory of Elasticity with Applications. Science Press, Beijing (1984)
Dems, K., Mróz, Z.: Variational approach by means of adjoint systems to structural optimization and sensitivity analysis–II: structure shape variation. Int. J. Solids Struct. 20(6), 527–552 (1984)
Sokolowski, J., Zolésio, J.P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992)
Delfour, M.C., Zolésio, J.P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. SIAM, Philadelphia (2001)
Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic press, New York, London, Toronto, Sydney, San Francisco (1982)
Vu-Quoc, L., Li, S.: Dynamics of sliding geometrically-exact beams: large angle maneuver and parametric resonance. Comput. Methods Appl. Mech. Eng. 120, 65–118 (1995)
Behdinan, K., Stylianou, M.C., Tabarrok, B.: Dynamics of flexible sliding beams-non-linear analysis part I: formulation. J. Sound Vib. 208(4), 517–539 (1997)
Behdinan, K., Tabarrok, B.: Dynamics of flexible sliding beams - nonlinear analysis part II: transient response. J. Sound Vib. 208(4), 541–565 (1997)
Hong, D., Ren, G.: A modeling of sliding joint on one-dimensional flexible medium. Multibody Syst. Dyn. 26(1), 91–106 (2011)
Pechstein, A., Gerstmayr, J.: A Lagrange-Eulerian formulation of an axially moving beam based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 30(3), 343–358 (2013)
Escalona, J.L.: An arbitrary Lagrangian-Eulerian discretization method for modeling and simulation of reeving systems in multibody dynamics. Mech. Mach. Theory 112, 1–21 (2017)
Humer, A., Steinbrecher, I., Vu-Quoc, L.: General sliding-beam formulation: a non-material description for analysis of sliding structures and axially moving beams. J. Sound Vib. 480, 115341 (2020)
Singh, N., Sharma, I., Gupta, S.S.: Dynamics of variable length geometrically exact beams in three-dimensions. Int. J. Solids Struct. 191–192, 614–627 (2020)
Pennisi, G., Bauchau, O.A.: Variational principles for non-material systems within an Arbitrary Lagrangian Eulerian description of motion. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC) (2020)
Boyer, F., Lebastard, V., Candelier, F., Renda, F.: Extended Hamilton’s principle applied to geometrically exact Kirchhoff sliding rods. J. Sound Vib. 516, 116511 (2022)
Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 244, 87–112 (1951)
Kienzler, R., Herrmann, G.: On material forces in elementary beam theory. J. Appl. Mech. 53(3), 561–564 (1986)
O’Reilly, O.M.: A material momentum balance law for rods. J. Elast. 86, 155–172 (2007)
Hanna, J.A.: Jump conditions for strings and sheets from an action principle. Int. J. Solids Struct. 62, 239–247 (2015)
O’Reilly, O.M.: Modeling Nonlinear Problems in the Mechanics of Strings and Rods: The Role of the Balance Laws. Springer, New York (2017)
Singh, H., Hanna, J.A.: On the planar elastica, stress, and material stress. J. Elast. 136, 87–101 (2019)
Bigoni, D.: Instability of a penetrating blade. J. Mech. Phys. Solids 64, 411–425 (2014)
Bigoni, D., Dal Corso, F., Bosi, F., Misseroni, D.: Eshelby-like forces acting on elastic structures: theoretical and experimental proof. Mech. Mater. 80, 368–374 (2015)
Armanini, C., Dal Corso, F., Misseroni, D., Bigoni, D.: Configurational forces and nonlinear structural dynamics. J. Mech. Phys. Solids 130, 82–100 (2019)
Han, S.L.: Configurational forces and geometrically exact formulation of sliding beams in non–material domains. Comput. Methods Appl. Mech. Eng. 395, 115063 (2022)
Singh, H., Hanna, J.A.: Pseudomomentum: origins and consequences. Z. Angew. Math. Phys. 72, 122 (2021)
Han, S.L., Bauchau, O.A.: Manipulation of motion via dual entities. Nonlinear Dyn. 85(1), 509–524 (2016)
Bauchau, O.A.: Flexible Multibody Dynamics. Solid Mechanics and Its Applications, vol. 176. Springer, Dordrecht (2011)
Lanczos, C.: The Variational Principles of Mechanics. Dover, New York (1970)
Zhong, W.X.: Duality System in Applied Mechanics and Optimal Control. Kluwer Academic, Boston, Dordrecht, New York, London (2004)
Balabukh, L.I., Vulfson, M.N., Mukoseev, B.V., Panovko, Y.G.: On work done by reaction forces of moving supports. Res. Theory Constr. 18, 190–200 (1970)
Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 6th edn. Elsevier, Butterworth Heinemann, Amsterdam (2005)
Bauchau, O.A., Han, S.L.: Interpolation of rotation and motion. Multibody Syst. Dyn. 31(3), 339–370 (2014)
Sonneville, V., Brüls, O., Bauchau, O.A.: Interpolation schemes for geometrically exact beams: a motion approach. Int. J. Numer. Methods Eng. 112(9), 1129–1153 (2017)
Han, S.L., Bauchau, O.A.: On the global interpolation of motion. Comput. Methods Appl. Mech. Eng. 337(10), 352–386 (2018)
Arnold, M., Brüls, O.: Convergence of the generalized-\(\alpha \) scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185–202 (2007)
Brüls, O., Cardona, A.: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dyn. 5(3), 0310021 (2010)
Bauchau, O.A., Han, S.L.: Three-dimensional beam theory for flexible multibody dynamics. J. Comput. Nonlinear Dyn. 9(4), 041011 (2014)
Han, S.L., Bauchau, O.A.: Nonlinear three-dimensional beam theory for flexible multibody dynamics. Multibody Syst. Dyn. 34(3), 211–242 (2015)
Han, S.L., Bauchau, O.A.: On Saint-Venant’s problem for helicoidal beams. J. Appl. Mech. 83(2), 021009 (2016)
Han, S.L.: Sensitivity analysis for sectional stiffness of anisotropic beams: the direct and adjoint methods. Compos. Struct. 285, 115215 (2022)
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Supported by the National Natural Science Foundation of China (Grant No. 12172042).
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Han, S., Bauchau, O.A. Configurational forces in variable-length beams for flexible multibody dynamics. Multibody Syst Dyn 58, 275–298 (2023). https://doi.org/10.1007/s11044-022-09866-5
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DOI: https://doi.org/10.1007/s11044-022-09866-5