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Research Papers

A Simple Absolute Nodal Coordinate Formulation for Thin Beams With Large Deformations and Large Rotations

[+] Author and Article Information
Hui Ren

MSC Software Corporation,
201 Depot Street, Suite 100,
Ann Arbor, MI 48104
e-mail: hui.ren@mscsoftware.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 9, 2014; final manuscript received September 17, 2014; published online April 9, 2015. Assoc. Editor: José L. Escalona.

J. Comput. Nonlinear Dynam 10(6), 061005 (Nov 01, 2015) (10 pages) Paper No: CND-14-1095; doi: 10.1115/1.4028610 History: Received April 09, 2014; Revised September 17, 2014; Online April 09, 2015

A simple but effective formulation of beams with large deformation and large rotation is derived from the principles of continuum mechanics. Proper assumptions are imposed, and the beam strain tensors are formulated from the Green strain tensors. The mass matrix is constant, and the elastic forces and the stiffness matrix entries are polynomials of the generalized coordinates, so numerical quadratures are not required in each time step of simulation, which makes the current approach much more superior in numerical efficiency than other formulations. The shape of the cross sections can be arbitrary, either uniform or nonuniform, and the beam can be either straight or curved. The generalized free of traction assumption ensures the strains in the cross section and the beam strains are independent, which resolves the Possion's locking issue and renders this approach can be accurately applied to general composite material beams. The elastic line approach (ELA) in the absolute nodal coordinate formulation (ANCF) can be derived from the current formulation.

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References

Figures

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Fig. 1

The static equilibria of the rolling cantilever beam, calculated using (a) 8 C1 elements and (b) 30 C0 elements

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Fig. 2

The (a) x and (b) y tip displacements of the cantilever beam under a concentrated force at the free end, calculated using 32 C0 elements (dotted), 10 C1 elements (dashed), 35 elements from the ELA (dashed–dotted), and ten elements from the GEBF (solid), as well as ten elements from the planar ANCF approach (light dotted), respectively

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Fig. 3

The (a) x, (b) y, and (c) z displacements of the cantilever beam under a concentrated torque at the free end, calculated using 10 C0 elements (dotted), 4 C1 elements (dashed), ten elements from the ELA (dashed–dotted), and four elements from the GEBF (solid), respectively

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Fig. 4

The first six in-plane mode shapes for the elastic ring, and the calculated frequencies are in good agreement with those calculated from the analytic formula in Eq. (52)

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Fig. 5

The first four out-of-plane mode shapes for the elastic ring, and the calculated frequencies are compared with those from the approximated formula in Eq. (53)

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Fig. 6

Schematic of the rotating beam

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Fig. 7

The (a) longitudinal and (b) lateral deflections at the tip of the spin-up maneuver, calculated using 12 C0 elements (dotted), 6 C1 elements (dashed), 12 elements from the ELA (dashed–dotted), and six elements from the GEBF (solid), respectively

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Fig. 8

The (a) upwind, (b) downwind, and (c) out-of-plane configuration of the spin-up maneuver of curved beams

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Fig. 9

Tip deflections of the (a) upwind, (b) downwind, and (c) out-of-plane spin-up maneuver of curved beams, calculated using 16 C0 elements (dotted), 8 C1 elements (dashed), 16 elements from the ELA (dashed–dotted), and eight elements from the GEBF (solid), respectively

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