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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).

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Figures

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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.

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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.

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Fig. 4

Infinite beam on a Winkler foundation under a moving load

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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.

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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.

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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.

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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

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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.

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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.

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Fig. 13

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

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Fig. 5

FRF versus forward velocity

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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.

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