Modeling viscoelastic behavior in flexible multibody systems. (English) Zbl 1466.70008
Summary: Viscoelasticity plays an important role in the dynamic response of flexible multibody systems. First, single degree-of-freedom joints, such as revolute and prismatic joints, are often equipped with elastomeric components that require complex models to capture their nonlinear behavior under the expected large relative motions found at these joints. Second, flexible joints, often called force or bushing elements, present similar challenges and involve up to six degrees of freedom. Finally, flexible components such as beams, plates, and shells also exhibit viscoelastic behavior. This paper presents a number of viscoelastic models that are suitable for these three types of applications. For single degree-of-freedom joints, models that capture their nonlinear, frequency-dependent, and frequency-independent behavior are necessary. The generalized Maxwell model is a classical model of linear viscoelasticity that can be extended easily to flexible joints. This paper also shows how existing viscoelastic models can be applied to geometrically exact beams, based on a three-dimensional representation of the quasi-static strain field in those structures. The paper presents a number of numerical examples for three types of applications. The shortcomings of the Kelvin-Voigt model, which is often used for flexible multibody systems, are underlined.
MSC:
| 70E55 | Dynamics of multibody systems |
| 74D05 | Linear constitutive equations for materials with memory |
References:
| [1] | Flügge, W., Viscoelasticity (1975), New York: Springer, New York · Zbl 0352.73033 |
| [2] | Christensen, R. M., Theory of Viscoelasticity. An Introduction (1982), New York: Academic Press, New York |
| [3] | Lazan, B. J., Damping of Materials and Members in Structural Mechanics (1968), Oxford: Pergamon, Oxford |
| [4] | Banks, H. T.; Inman, D. J., On damping mechanisms in beams, J. Appl. Mech., 58, 3, 716-723 (1991) · Zbl 0741.73030 |
| [5] | Simo, J. C.; Hjelmstad, K. D.; Taylor, R. L., Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear, Comput. Methods Appl. Mech. Eng., 42, 3, 301-330 (1984) · Zbl 0517.73074 |
| [6] | Mata, P.; Oller, S.; Barbat, A. H., Dynamic analysis of beam structures considering geometric and constitutive nonlinearity, Comput. Methods Appl. Mech. Eng., 197, 6-8, 857-878 (2008) · Zbl 1169.74462 |
| [7] | Simo, J. C., A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput. Methods Appl. Mech. Eng., 49, 1, 55-70 (1985) · Zbl 0583.73037 |
| [8] | Linn, J.; Lang, H.; Tuganov, A., Geometrically exact Cosserat rods with Kelvin-Voigt type viscous damping, Mech. Sci., 4, 79-96 (2013) |
| [9] | Abdel-Nasser, A. M.; Shabana, A. A., A nonlinear viscoelastic constitutive model for large rotation finite element formulations, Multibody Syst. Dyn., 26, 3, 57-79 (2011) · Zbl 1287.74041 |
| [10] | Antman, S. S., Invariant dissipative mechanisms for the spatial motion of rods suggested by artificial viscosity, J. Elast., 70, 55-64 (2003) · Zbl 1046.74025 |
| [11] | Lang, H.; Linn, J.; Arnold, M., Multibody dynamics simulation of geometrically exact Cosserat rods, Multibody Syst. Dyn., 25, 3, 285-312 (2011) · Zbl 1271.74264 |
| [12] | Schulze, M.; Dietz, S.; Burgermeister, B.; Tuganov, A.; Lang, H.; Linn, J.; Arnold, M., Integration of nonlinear models of flexible body deformation in multibody system dynamics, J. Comput. Nonlinear Dyn., 9, 1 (2013) |
| [13] | Lang, H.; Leyendecker, S.; Linn, J.; Terze, b. Z.; Vrdoljak, M., Numerical experiments for viscoelastic Cosserat rods with Kelvin-Voigt damping, Proceedings of the ECCOMAS Thematic Conference Multibody Dynamics, 453-462 (2013) |
| [14] | Malvern, L. E., Introduction to the Mechanics of a Continuous Medium (1969), Englewood Cliffs: Prentice Hall, Englewood Cliffs |
| [15] | Simo, J. C.; Hughes, T. J.R., Computational Inelasticity (1998), New York: Springer, New York · Zbl 0934.74003 |
| [16] | Bauchau, O. A., Flexible Multibody Dynamics (2011), New-York: Springer, New-York · Zbl 1209.70001 |
| [17] | Haupt, P.; Sedlan, K., Viscoplasticity of elastomeric materials: experimental facts and constitutive modeling, Arch. Appl. Mech., 71, 5, 89-109 (2001) · Zbl 1007.74027 |
| [18] | Höfer, P.; Lion, A., Modeling of frequency- and amplitude-dependent material properties of filler-reinforced rubber, J. Mech. Phys. Solids, 57, 4, 500-520 (2009) · Zbl 1273.74065 |
| [19] | Arnold, M.; Brüls, O., Convergence of the generalized-\( \alpha\) scheme for constrained mechanical systems, Multibody Syst. Dyn., 18, 2, 185-202 (2007) · Zbl 1121.70003 |
| [20] | Arnold, M.; Brüls, O.; Cardona, A., Error analysis of generalized-\( \alpha\) Lie group time integration methods for constrained mechanical systems, Numer. Math., 129, 1, 149-179 (2015) · Zbl 1309.70016 |
| [21] | Welsh, W. A., Simulation and correlation of a helicopter air-oil strut dynamic response, American Helicopter Society 43rd Annual Forum Proceedings (1987) |
| [22] | Welsh, W. A., Dynamic modeling of a helicopter lubrication system, American Helicopter Society 44th Annual Forum Proceedings (1988) |
| [23] | Borri, M.; Bottasso, C. L., An intrinsic beam model based on a helicoidal approximation. Part I: formulation. Part II: linearization and finite element implementation, Int. J. Numer. Methods Eng., 37, 2267-2309 (1994) · Zbl 0806.73028 |
| [24] | McRobie, F. A.; Lasenby, J., Simo-Vu-Quoc rods using Clifford algebra, Int. J. Numer. Methods Eng., 45, 4, 377-398 (1999) · Zbl 0940.74081 |
| [25] | Merlini, T.; Morandini, M., The helicoidal modeling in computational finite elasticity. Part I: variational formulation. Part II: multiplicative interpolation, Int. J. Solids Struct., 41, 18-19, 5351-5409 (2004) · Zbl 1075.74015 |
| [26] | Sander, O., Geodesic finite elements for Cosserat rods, Int. J. Numer. Methods Eng., 82, 13, 1645-1670 (2010) · Zbl 1193.74157 |
| [27] | Sonneville, V.; Cardona, A.; Brüls, O., Geometrically exact beam finite element formulated on the special Euclidean group SE(3), Comput. Methods Appl. Mech. Eng., 268, 1, 451-474 (2014) · Zbl 1295.74050 |
| [28] | Demoures, F.; Gay-Balmaz, F.; Kobilarov, M.; Ratiu, T. S., Multisymplectic Lie group variational integrator for a geometrically exact beam in R^3, Commun. Nonlinear Sci. Numer. Simul., 19, 10, 3492-3512 (2014) · Zbl 1473.70029 |
| [29] | Murray, R. M.; Li, Z.; Sastry, S. S., A Mathematical Introduction to Robotic Manipulation (1994), Boca Raton: CRC Press, Boca Raton · Zbl 0858.70001 |
| [30] | Selig, J. M., Geometric Fundamentals of Robotics (2005), New York: Springer, New York · Zbl 1062.93002 |
| [31] | Borri, M.; Trainelli, L.; Bottasso, C. L., On representations and parameterizations of motion, Multibody Syst. Dyn., 4, 129-193 (2000) · Zbl 0980.70004 |
| [32] | Study, E., Geometrie der Dynamen (1903), Leipzig: Teubner, Leipzig · JFM 34.0741.10 |
| [33] | Martinez, J. M.R.; Duffy, J., The principle of transference: history, statement and proof, Mech. Mach. Theory, 28, 1, 165-177 (1993) |
| [34] | Han, S. L.; Bauchau, O. A., Manipulation of motion via dual entities, Nonlinear Dyn., 85, 1, 509-524 (2016) · Zbl 1349.70011 |
| [35] | Bauchau, O. A.; Choi, J. Y., The vector parameterization of motion, Nonlinear Dyn., 33, 2, 165-188 (2003) · Zbl 1038.70002 |
| [36] | Dimentberg, F.M.: The screw calculus and its applications. Technical Report AD 680993, Clearinghouse for Federal and Scientific Technical Information, Virginia, USA (1968) |
| [37] | Lin, Q.; Burdick, J. W., Objective and frame-invariant kinematic metric functions for rigid bodies, Int. J. Robot. Res., 19, 6, 612-625 (2000) |
| [38] | Bauchau, O. A.; Han, S. L., Three-dimensional beam theory for flexible multibody dynamics, J. Comput. Nonlinear Dyn., 9, 4 (2014) |
| [39] | Han, S. L.; Bauchau, O. A., Nonlinear three-dimensional beam theory for flexible multibody dynamics, Multibody Syst. Dyn., 34, 3, 211-242 (2015) · Zbl 1329.70031 |
| [40] | Christensen, R. M., Mechanics of Composite Materials (1979), New York: Wiley, New York |
| [41] | Tsai, S. W.; Hahn, H. T., Introduction to Composite Materials (1980), Westport: Technomic Publishing Co., Inc., Westport |
| [42] | Han, S. L.; Bauchau, O. A., On Saint-Venant’s problem for helicoidal beams, J. Appl. Mech., 83, 2 (2016) |
| [43] | Han, S. L.; Bauchau, O. A., Spectral formulation for geometrically exact beams: a motion interpolation based approach, AIAA J., 57, 10, 4278-4290 (2019) |
| [44] | Han, S. L.; Bauchau, O. A., Nonlinear, three-dimensional beam theory for dynamic analysis, Multibody Syst. Dyn., 41, 2, 173-200 (2017) · Zbl 1431.74064 |
| [45] | Bauchau, O. A.; Betsch, P.; Cardona, A.; Gerstmayr, J.; Jonker, B.; Masarati, P.; Sonneville, V., Validation of flexible multibody dynamics beam formulations using benchmark problems, Multibody Syst. Dyn., 37, 1, 29-48 (2016) · Zbl 1359.70060 |
| [46] | Giavotto, V.; Borri, M.; Mantegazza, P.; Ghiringhelli, G.; Carmaschi, V.; Maffioli, G. C.; Mussi, F., Anisotropic beam theory and applications, Comput. Struct., 16, 1-4, 403-413 (1983) · Zbl 0499.73066 |
| [47] | Hodges, D. H., Nonlinear Composite Beam Theory (2006), Reston: AIAA, Reston |
| [48] | Simo, J. C.; Vu-Quoc, L., On the dynamics in space of rods undergoing large motions - a geometrically exact approach, Comput. Methods Appl. Mech. Eng., 66, 1, 125-161 (1988) · Zbl 0618.73100 |
| [49] | Borri, M.; Merlini, T., A large displacement formulation for anisotropic beam analysis, Meccanica, 21, 30-37 (1986) · Zbl 0593.73047 |
| [50] | Danielson, D. A.; Hodges, D. H., A beam theory for large global rotation, moderate local rotation, and small strain, J. Appl. Mech., 55, 1, 179-184 (1988) |
| [51] | Volovoi, V. V.; Hodges, D. H.; Berdichevsky, V. L.; Sutyrin, V. G., Dynamic dispersion curves for non-homogeneous, anisotropic beams with cross sections of arbitrary geometry, J. Sound Vib., 215, 1101-1120 (1998) |
| [52] | Bauchau, O. A.; Craig, J. I., Structural Analysis with Application to Aerospace Structures (2009), New-York: Springer, New-York |
| [53] | Bauchau, O. A.; Wang, J. L., Stability analysis of complex multibody systems, J. Comput. Nonlinear Dyn., 1, 1, 71-80 (2006) |
| [54] | Bauchau, O. A.; Wang, J. L., Stability evaluation and system identification of flexible multibody systems, Multibody Syst. Dyn., 18, 1, 95-106 (2007) · Zbl 1180.70011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
