Research Papers

Analytical and Numerical Validation of a Moving Modes Method for Traveling Interaction on Long Structures

[+] Author and Article Information
Antonio M. Recuero

Department of Mechanical
and Industrial Engineering,
University of Illinois at Chicago,
842 West Taylor Street,
Chicago, IL 60607
e-mail: amrecuero85@gmail.com

José L. Escalona

Department of Mechanical
and Materials Engineering,
University of Seville,
Camino de los descubrimientos s/n,
Seville 41092, Spain
e-mail: escalona@us.es

Manuscript received May 4, 2015; final manuscript received December 2, 2015; published online February 3, 2016. Assoc. Editor: Jozsef Kovecses.

J. Comput. Nonlinear Dynam 11(5), 051002 (Feb 03, 2016) (14 pages) Paper No: CND-15-1120; doi: 10.1115/1.4032247 History: Received May 04, 2015; Revised December 02, 2015

This work is devoted to the validation of a computational dynamics approach previously developed by the authors for the simulation of moving loads interacting with flexible bodies through arbitrary contact modeling. The method has been applied to the modeling and simulation of the coupled dynamics of railroad vehicles moving on deformable tracks with arbitrary undeformed geometry. The procedure presented makes use of a fully arbitrary Lagrangian–Eulerian (ALE) description of the long flexible solid (track) whose mechanical properties may be captured using a dynamics-preserving selection of modes, e.g., via a Padé approximation of a transfer function. The modes accompany the contact interaction rather than being referred to a fixed frame, as it occurs in the finite-element floating frame of reference formulation. In the method discussed in this paper, the mesh, which moves through the long flexible solid, is defined in the trajectory coordinate system (TCS) used to describe the dynamics of the set of bodies (vehicle) that interact with the long flexible structure. For this reason, the selection of modes can be focused on the preservation of the dynamics of the structure instead of having to ensure the structure's static displacement convergence due to the motion of the load. In this paper, the validation of the so-called trajectory coordinate system/moving modes (TCS/MM) method is performed in four different aspects: (a) the analytical mechanics approach is used to obtain the equations of motion in a nonmaterial volume, (b) the resulting equations of motion are compared to the classical discretization procedures of partial differential equations (PDE), (c) the suitability of the moving modes (MM) to describe deformation due to variable-velocity moving loads, and (d) the capability of the finite nonmaterial volume to describe the dynamics of an infinitely long flexible body. Validation (a) is completely general. However, the particular example of a moving load applied to a straight beam resting on a Winkler foundation, with known semi-analytical solution, is used to perform validations (b), (c), and (d).

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Shabana, A. A. , 1997, “ Flexible Multibody Dynamics: Review of Past and Recent Developments,” Multibody Syst. Dyn., 1(2), pp. 189–222. [CrossRef]
Gerstmayr, J. , Sugiyama, H. , and Mikkola, A. , 2013, “ Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 8(3), p. 031016. [CrossRef]
Kalker, J. , 1982, “ A Fast Algorithm for the Simplified Theory of Rolling Contact,” Veh. Syst. Dyn., 11(1), pp. 1–13. [CrossRef]
Grassie, S. , Gregory, R. , Harrison, D. , and Johnson, K. , 1982, “ The Dynamic Response of Railway Track to High Frequency Vertical Excitation,” J. Mech. Eng. Sci., 24(2), pp. 77–90. [CrossRef]
Galvín, P. , and Romero, A. , 2013, “ A 3D Time Domain Numerical Model Based on Half-Space Green's Function for Soil–Structure Interaction Analysis,” Comput. Mech., 53(5), pp. 1073–1085. [CrossRef]
Clouteau, D. , Cottereau, R. , and Lombaert, G. , 2013, “ Dynamics of Structures Coupled With Elastic Media—A Review of Numerical Models and Methods,” J. Sound Vib., 332(10), pp. 2415–2436. [CrossRef]
François, S. , Schevenels, M. , Galvín, P. , Lombaert, G. , and Degrande, G. , 2010, “ A 2.5D Coupled FE–BE Methodology for the Dynamic Interaction Between Longitudinally Invariant Structures and a Layered Halfspace,” Comput. Methods Appl. Mech. Eng., 199, pp. 1536–1548. [CrossRef]
Chamorro, R. , Escalona, J. L. , and González, M. , 2011, “ An Approach for Modeling Long Flexible Bodies With Application to Railroad Dynamics,” Multibody Syst. Dyn., 26(2), pp. 135–152. [CrossRef]
Pechstein, A. , and Gerstmayr, J. , 2013, “ A Lagrange–Eulerian Formulation for an Axially Moving Beam Based on the Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 30(3), pp. 343–358. [CrossRef]
Lehner, M. , and Eberhard, P. , 2006, “ On the Use of Moment-Matching to Build Reduced Order Models in Flexible Multibody Dynamics,” Multibody Syst. Dyn., 16(2), pp. 191–211. [CrossRef]
Fehr, J. , and Eberhard, P. , 2011, “ Simulation Process of Flexible Multibody Systems With Non-Modal Model Order Reduction Techniques,” Multibody Syst. Dyn., 25(3), pp. 313–334. [CrossRef]
Sherif, K. , Irschik, H. , and Witteveen, W. , 2012, “ Transformation of Arbitrary Elastic Mode Shapes Into Pseudo-Free-Surface and Rigid Body Modes for Multibody Dynamic Systems,” ASME J. Comput. Nonlinear Dyn., 7(2), p. 021008. [CrossRef]
Rieker, J. R. , Lin, Y.-H. , and Trethewey, M. W. , 1996, “ Discretization Considerations in Moving Load Finite Element Beam Models,” Finite Elem. Anal. Des., 21(3), pp. 129–144. [CrossRef]
Koh, C. , Ong, J. , Chua, D. , and Feng, J. , 2003, “ Moving Element Method for Train-Track Dynamics,” Int. J. Numer. Methods Eng., 56(11), pp. 1549–1567. [CrossRef]
Recuero, A. M. , and Escalona, J. L. , 2013, “ Application of the Trajectory Coordinate System and the Moving Modes Method Approach to Railroad Dynamics Using Krylov Subspaces,” J. Sound Vib., 332(20), pp. 5177–5191. [CrossRef]
Recuero, A. M. , and Escalona, J. L. , 2014, “ Dynamics of the Coupled Railway Vehicle–Flexible Track System With Irregularities Using a Multibody Approach With Moving Modes,” Veh. Syst. Dyn., 52(1), pp. 45–67. [CrossRef]
Tamarozzi, T. , Ziegler, P. , Eberhard, P. , and Desmet, W. , 2013, “ On the Applicability of Static Modes Switching in Gear Contact Applications,” Multibody Syst. Dyn., 30(2), pp. 209–219. [CrossRef]
Tamarozzi, T. , Heirman, G. , and Desmet, W. , 2014, “ An On-Line Time Dependent Parametric Model Order Reduction Scheme With Focus on Dynamic Stress Recovery,” Comput. Methods Appl. Mech. Eng., 268, pp. 336–358. [CrossRef]
Recuero, A. M. , 2012, “ Simulation of Coupled Railroad Vehicle–Flexible Track Dynamics Using Moving Modes and Krylov Subspace Techniques,” Ph.D. thesis, University of Seville, Seville, Spain.
Shabana, A. A. , 2005, Dynamics of Multibody Systems, Cambridge University Press, Cambridge, UK.
Kawamoto, A. , Krenk, S. , Suzuki, A. , and Inagaki, M. , 2010, “ Flexible Body Dynamics in a Local Frame With Explicitly Predicted Motion,” Int. J. Numer. Methods Eng., 81(2), pp. 246–268.
Chamorro, R. , Escalona, J. L. , and Recuero, A. M. , 2014, “ Stability Analysis of Multibody Systems With Long Flexible Bodies Using the Moving Modes Method and Its Application to Railroad Dynamics,” ASME J. Comput. Nonlinear Dyn., 9(1), p. 011005.
Recuero, A. M. , Aceituno, J. , Escalona, J. , and Shabana, A. , “ A Nonlinear Approach for Modeling Rail Flexibility Using the Absolute Nodal Coordinate Formulation,” Nonlinear Dyn. 83(1), pp. 463–481.
Hong, D. , and Ren, G. , 2011, “ A Modeling of Sliding Joint on One-Dimensional Flexible Medium,” Multibody Syst. Dyn., 26(1), pp. 91–110. [CrossRef]
Hyldahl, P. , Mikkola, A. , and Balling, O. , 2013, “ A Thin Plate Element Based on the Combined Arbitrary Lagrange–Euler and Absolute Nodal Coordinate Formulations,” Proc. Inst. Mech. Eng., Part K, 227(3), pp. 211–219. [CrossRef]
Escalona, J. L. , 2012, “ Modeling Hoisting Machines With the Arbitrary Lagrangian–Eulerian Absolute Nodal Coordinate Formulation,” 2nd International Conference on Multibody System Dynamics, Stuttgart, Germany, May 29–June 1.
Irschik, H. , and Holl, H. , 2002, “ The Equations of Lagrange Written for a Non-Material Volume,” Acta Mech., 153(3), pp. 231–248. [CrossRef]
Shabana, A. A. , Zaazaa, K. E. , and Sugiyama, H. , 2008, Railroad Vehicle Dynamics. A Computational Approach, CRC Press, Boca Raton, FL.
Popp, K. , and Schiehlen, W. , 2010, Ground Vehicle Dynamics, Springer-Verlag, Berlin.
Recuero, A. M. , Escalona, J. L. , and Chamorro, R. , 2012, “ A Trajectory Frame-Based Dynamic Formulation for Railroad Vehicles Simulation,” Int. J. Railw. Technol., 1(2), pp. 21–44. [CrossRef]
Donea, J. , Huerta, A. , Ponthot, J.-P. , Rodríguez-Ferran, A. , 2004, “ Arbitrary Lagrangian–Eulerian Methods,” Encyclopedia of Computational Mechanics, Volume 1: Fundamentals, Wiley, Chichester, UK, pp. 1–25.
Schilders, W. H. , van der Vorst, H. A. , and Rommes, J. , eds., 2008, Model Order Reduction: Theory, Research Aspects and Applications, Springer-Verlag, Berlin.
Feldmann, P. , and Freund, R. , 1995, “ Efficient Linear Circuit Analysis by Padé Approximation Via the Lanczos Process,” IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 14(5), pp. 639–649. [CrossRef]
Koutsovasilis, P. , and Beitelschmidt, M. , 2008, “ Comparison of Model Reduction Techniques for Large Mechanical Systems,” Multibody Syst. Dyn., 20(2), pp. 111–128. [CrossRef]
Liang, Y. , Lee, H. , Lim, S. , Lin, W. , Lee, K. , and Wu, C. , 2002, “ Proper Orthogonal Decomposition and Its Applications—Part I: Theory,” J. Sound Vib., 252(3), pp. 527–544. [CrossRef]
Goldstein, H. , Poole, C. , and Safko, J. , 2001, Classical Mechanics, 3rd ed., Addison-Wesley, New York.
Giménez, G. , 1978, “ Cálculo y Optimización del Comportamiento Dinámico de Vehículos Ferroviarios,” Ph.D. thesis, Universidad de Navarra, Navarra, Spain.
Mallik, C. S. , and Singh, A. K. , 2006, “ Steady-State Response of an Elastically Supported Infinite Beam to a Moving Load,” J. Sound Vib., 291, pp. 1148–1169. [CrossRef]
Shabana, A. , 1997, Vibration of Discrete and Continuous Systems, Springer, New York.
Frýba, L. , 1999, Vibration of Solids and Structures Under Moving Loads, 3rd ed., Thomas Telford Ltd., Prague, Czech Republic.
Vostroukhov, A. , and Metrikine, A. , 2003, “ Periodically Supported Beam on a Visco-Elastic Layer as a Model for Dynamic Analysis of a High-Speed Railway Track,” Int. J. Solids Struct., 40(21), pp. 5723–5752. [CrossRef]


Grahic Jump Location
Fig. 3

Sketch of a moving nonmaterial volume with a load-adaptive mesh. The moving mesh captures small, linear deformation of an arbitrarily shaped structure.

Grahic Jump Location
Fig. 2

Description of structure deformation using a TCS. In this figure, Rt denotes the location of the TCS with respect to the inertial frame, u0P is a vector that describes the structure's arbitrary nonlinear undeformed geometry, u¯fP refers to small deformation described in the moving frame, and rP denotes the deformed material points of the structure in the inertial frame.

Grahic Jump Location
Fig. 4

Infinite beam on a Winkler foundation under a moving load

Grahic Jump Location
Fig. 5

FRF versus forward velocity

Grahic Jump Location
Fig. 6

Orthonormal FRMs obtained with V = 0 m/s. (a)–(e) The FRMs at the load frequencies ωk = 0, 0.5ωn, 1.0ωn, 1.5ωn, and 2.0ωn for k = 1,…, 5, respectively.

Grahic Jump Location
Fig. 7

Orthonormal FRMs obtained with V = 100 m/s. (a)–(e) The FRMs at the load frequencies ωk = 0, 0.5ωn, 1.0ωn, 1.5ωn, and 2.0ωn for k = 1,…, 5, respectively.

Grahic Jump Location
Fig. 8

Approximation of the FRF with reduced-order models. (a) A comparison of the FRF of exact model and that obtained using PDE–FRM. (b) The FRFs of the exact model compared to that obtained using PDE–FRM with a different load velocity V = 0 m/s.

Grahic Jump Location
Fig. 9

Transient deformation of infinite beam after removing the load for three load velocities: (a) V = 0 m/s, (b) V = 20 m/s, and (c) V = 100 m/s

Grahic Jump Location
Fig. 10

Deformed shapes at different instants. (a)–(c) The influence of the FRMs used in the ROM, which have been calculated at a load velocity V = 0 m/s, V = 20 m/s, and V = 60 m/s, respectively.

Grahic Jump Location
Fig. 11

History of deformation at different sections. (a)–(c) The time domain evolution at three sections of the beam: ξ = 0 m, ξ = –2 m, and ξ = 2 m, respectively.

Grahic Jump Location
Fig. 12

Deformation results for CLE extended to infinity (circle), CLE extended to a nonmaterial volume of 4 m (square), and nonmaterial Lagrange equations for a nonmaterial volume of 4 m (triangle). The time instants follow the code: solid line—1 ms, dotted line—3 ms, and dashed line—6 ms.

Grahic Jump Location
Fig. 13

Total energy during the simulation: CLE—solid line and nonmaterial Lagrange equations—dashed line



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In