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Research Papers

A New Locking-Free Formulation of a Three-Dimensional Shear-Deformable Beam

[+] Author and Article Information
W. Fan

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering, University of Maryland, Baltimore,
1000 Hilltop Circle,
Baltimore, MD 21250

W. D. Zhu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 28, 2016; final manuscript received February 14, 2017; published online May 26, 2017. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 139(5), 051001 (May 26, 2017) (13 pages) Paper No: VIB-16-1519; doi: 10.1115/1.4036210 History: Received October 28, 2016; Revised February 14, 2017

Abstract

A new locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross section and stretch and shear strains. The rotation is parameterized by a rotation vector, which has a clear and intuitive physical meaning. Taylor polynomials are used for certain terms that have zero denominators to avoid singularity in numerical implementation. Since the rotation vector can have singular points when its norm equals $2mπ$, where $m$ is a nonzero integer, a rescaling strategy is adopted to resolve the singularity problem when there is only one singular point at a time instant, which is the case for most engineering applications. Governing equations of the beam are obtained using Lagrange's equations for systems with constraints, and several benchmark problems are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear and Poisson locking problems that the absolute nodal coordinate formulation (ANCF) can have. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement with each other, which verifies accuracy of the current formulation. Results from the current formulation are compared with those from commercial software abaqus and recurdyn, and they are in good agreement with each other; the current formulation uses much fewer numbers of elements to yield converged results.

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Figures

Fig. 1

Geometrical description of a three-dimensional shear-deformable beam

Fig. 2

Configurations of the cantilever beam under concentrated bending moments, where dots indicate nodes of elements of the beam

Fig. 3

Schematic of a cantilever beam with (a) one and (b) two concentrated forces at its free end

Fig. 4

Vertical displacements from different formulations with different numbers of elements when α=0.1 and SL=10−4

Fig. 5

(a) X and (b) Y displacements at the tip of the beam

Fig. 6

(a) Longitudinal and (b) transverse deflections at the tip of the beam

Fig. 7

(a) Stretch and (b) shear strains at the midpoint of the centroid line of the beam

Fig. 8

Schematic of a rotating beam, where O-XY is the global coordinate system and O-xy is a rotating frame

Fig. 9

(a) Longitudinal and (b) transverse deflections at the tip of the rotating beam

Fig. 10

Longitudinal deflection at the tip of the rotating beam with material damping

Fig. 11

(a) X, (b) Y, and (c) Z displacements at the tip of the rotating beam under the gravity effect

Fig. 12

Norm of the rotation vector at the pinned end of the rotating beam under the gravity effect

Fig. 13

Angular velocities of the cross section at the tip of the beam

Fig. 14

Kinematic description of a planar extensible and shear-deformable beam

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