Stability of rigid body motion through an extended intermediate axis theorem: application to rockfall simulation. (English) Zbl 1483.70017
Summary: The stability properties of a freely rotating rigid body are governed by the intermediate axis theorem, i.e., rotation around the major and minor principal axes is stable whereas rotation around the intermediate axis is unstable. The stability of the principal axes is of importance for the prediction of rockfall. Current numerical schemes for 3D rockfall simulation, however, are not able to correctly represent these stability properties. In this paper an extended intermediate axis theorem is presented, which not only involves the angular momentum equations but also the orientation of the body, and we prove the theorem using Lyapunov’s direct method. Based on the stability proof, we present a novel scheme which respects the stability properties of a freely rotating body and which can be incorporated in numerical schemes for the simulation of rigid bodies with frictional unilateral constraints. In particular, we show how this scheme is incorporated in an existing 3D rockfall simulation code. Simulations results reveal that the stability properties of rotating rocks play an essential role in the run-out length and lateral spreading of rocks.
MSC:
| 70E50 | Stability problems in rigid body dynamics |
Keywords:
intermediate axis theorem; Lyapunov stability; Euler equations: 3D rockfall simulation; energy-momentum preserving schemeReferences:
| [1] | Acary, V.; Brogliato, B., Numerical Methods for Nonsmooth Dynamical Systems; Applications in Mechanics and Electronics (2008), Berlin: Springer, Berlin · Zbl 1173.74001 · doi:10.1007/978-3-540-75392-6 |
| [2] | Andrew, R., Hume, H., Bartingale, R., Rock, A., Zhang, R.: CRSP-3D USER’S MANUAL Colorado Rockfall Simulation Program. Federal Highway Administration (2012). Publication No. FHWA-CFL/TD-12-007 |
| [3] | Betsch, P.; Siebert, R., Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration, Int. J. Numer. Methods Eng., 79, 4, 444-473 (2009) · Zbl 1171.70300 · doi:10.1002/nme.2586 |
| [4] | Caviezel, A.; Demmel, S.; Ringenbach, A.; Bühler, Y.; Lu, G.; Christen, M.; Dinneen, C.; Eberhard, L.; Von Rickenbach, D.; Bartelt, P., Reconstruction of four-dimensional rockfall trajectories using remote sensing and rock-based accelerometers and gyroscopes, Earth Surf. Dyn., 7, 1, 199-210 (2019) · doi:10.5194/esurf-7-199-2019 |
| [5] | Christen, M.; Bühler, Y.; Bartelt, P.; Leine, R.; Glover, J.; Schweizer, A.; Graf, C.; McArdell, B.; Gerber, W.; Deubelbeiss, Y.; Feistl, T.; Volkwein, A.; Koboltschnig, G.; Hübl, J.; Braun, J., Integral hazard management using a unified software environment: numerical simulation tool “RAMMS” for gravitational natural hazards, Proceedings 12th Congress INTERPRAEVENT, 77-86 (2012) |
| [6] | Dimnet, E.; Fremond, M.; Gormaz, R.; San Martin, J.; Bathe, K., Collisions of rigid bodies, deformable bodies and fluids, Computational Fluid and Solid Mechanics 2003, vols. 1 and 2, Proceedings, 1317-1321 (2003), Amsterdam: Elsevier, Amsterdam · doi:10.1016/B978-008044046-0.50324-9 |
| [7] | Erismann, T. H.; Abele, G., Dynamics of Rockslides and Rockfalls (2001), Berlin: Springer, Berlin · doi:10.1007/978-3-662-04639-5 |
| [8] | Grammel, R., Der Kreisel. Seine Theorie und seine Anwendungen (1920), Wiesbaden: Vieweg, Wiesbaden · JFM 47.0727.05 |
| [9] | Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I; Nonstiff Problems (2000), Berlin: Springer, Berlin · Zbl 1185.65115 |
| [10] | Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2002), Berlin: Springer, Berlin · Zbl 0994.65135 · doi:10.1007/978-3-662-05018-7 |
| [11] | Holm, D.; Marsden, J.; Ratiu, T.; Weinstein, A. J., Stability of rigid body motion using the energy-Casimir method, Contemp. Math., 28, 15-23 (1984) · Zbl 0526.70025 · doi:10.1090/conm/028/751972 |
| [12] | Krishnaprasad, P. S.; Marsden, J. E., Hamiltonian structures and stability for rigid bodies with flexible attachments, Arch. Ration. Mech. Anal., 98, 1, 71-93 (1987) · Zbl 0624.58010 · doi:10.1007/BF00279963 |
| [13] | Le Hir, C.: Forêt et chutes de blocs: méthodologie de modéelisation spatialisée du rôle de protection. PhD thesis, Cemagref/Université de Marne-La-Vallée (2005) |
| [14] | Le Hir, C.; Dimnet, E.; Berger, F.; Dorren, L., TIN-based 3D deterministic simulation of rockfall taking trees into account, Geophys. Res. Abstr., 8 (2006) |
| [15] | Leine, R. I.; Nijmeijer, H., Dynamics and Bifurcations of Non-smooth Mechanical Systems (2004), Berlin: Springer, Berlin · Zbl 1068.70003 · doi:10.1007/978-3-540-44398-8 |
| [16] | Leine, R. I.; Schweizer, A.; Christen, M.; Glover, J.; Bartelt, P.; Gerber, W., Simulation of rockfall trajectories with consideration of rock shape, Multibody Syst. Dyn., 32, 241-271 (2014) · Zbl 1351.70002 · doi:10.1007/s11044-013-9393-4 |
| [17] | Lu, G.; Caviezel, A.; Christen, M.; Demmel, S. E.; Ringenbach, A.; Bühler, Y.; Dinneen, C. E.; Gerber, W.; Bartelt, P., Modelling rockfall impact with scarring in compactable soils, Landslides, 16, 2353-2367 (2019) · doi:10.1007/s10346-019-01238-z |
| [18] | Lu, G.; Ringenbach, A.; Caviezel, A.; Sanchez, M.; Christen, M.; Bartelt, P., Mitigation effects of trees on rockfall hazards: does rock shape matter?, Landslides, 18, 59-77 (2021) · doi:10.1007/s10346-020-01418-2 |
| [19] | Magnus, K., Kreisel; Theorie und Anwendungen (1971), Berlin: Springer, Berlin · Zbl 0228.70005 |
| [20] | McLachlan, R. I.; Zanna, A., The discrete Moser-Veselov algorithm for the free rigid body, revisited, Found. Comput. Math., 5, 1, 87-123 (2004) · Zbl 1123.70012 · doi:10.1007/s10208-004-0118-6 |
| [21] | Moreau, J. J., Unilateral contact and dry friction in finite freedom dynamics, Non-smooth Mechanics and Applications, 1-82 (1988), Berlin: Springer, Berlin · Zbl 0703.73070 · doi:10.1007/978-3-7091-2624-0 |
| [22] | Poinsot, M., Théorie nouvelle des la rotation des corps (1834) |
| [23] | Rowenhorst, D.; Rollett, A.; Rohrer, G.; Groeber, M.; Jackson, M.; Konijnenberg, P. J.; De Graef, M., Consistent representations of and conversions between 3D rotations, Model. Simul. Mater. Sci. Eng., 23, 8 (2015) · doi:10.1088/0965-0393/23/8/083501 |
| [24] | Schweizer, A.: Ein nichtglattes mechanisches Modell für Steinschlag. PhD thesis, ETH Zurich (2015) |
| [25] | Shepperd, S., Quaternion from rotation matrix, J. Guid. Control, 1, 3, 223-224 (1978) · Zbl 0403.15019 · doi:10.2514/3.55767b |
| [26] | Simo, J. C.; Wong, K. K., Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum, Int. J. Numer. Methods Eng., 31, 1, 19-52 (1991) · Zbl 0825.73960 · doi:10.1002/nme.1620310103 |
| [27] | Volkwein, A.; Schellenberg, K.; Labiouse, V.; Agliardi, F.; Berger, F.; Bourrier, F.; Dorren, L. K.A.; Gerber, W.; Jaboyedoff, M., Rockfall characterisation and structural protection – a review, Nat. Hazards Earth Syst. Sci., 11, 9, 2617-2651 (2011) · doi:10.5194/nhess-11-2617-2011 |
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.
