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Research Papers

A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters

[+] 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 County,
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 County
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

H. Ren

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250;
MSC Software Corporation,
201 Depot Street, Suite 100,
Ann Arbor, MI 48105

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 14, 2015; final manuscript received September 15, 2015; published online January 4, 2016. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 11(4), 041013 (Jan 04, 2016) (13 pages) Paper No: CND-15-1217; doi: 10.1115/1.4031769 History: Received July 14, 2015; Revised September 15, 2015

In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.

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Figures

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Fig. 1

Geometrical description of a three-dimensional Euler–Bernoulli beam

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Fig. 2

Static equilibria of a rolling cantilever beam calculated with two elements in the current formulation; circles indicate nodes of the beam

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Fig. 3

Static equilibria of a cantilever beam under different compressive forces that are over the critical load, calculated with 20 elements in the current formulation; θ represents the rotation angle of the free end of the beam

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Fig. 4

Relative errors of rotations of the free end of the beam under different compressive loads: (a) P=1.015 Pcr and (b) P=9.116 Pcr

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Fig. 5

The first four (a) in-plane and (b) out-of-plane mode shapes of an elastic ring; dashed lines correspond to the undeformed ring

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Fig. 6

Nonlinear dynamic responses at the free end of a cantilever beam under end moments Mx=50 sin(πt) and Mz=100 sin(πt/2) from the current formulation and ADAMS

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Fig. 7

(a) Angular velocity of a falling pinned-free beam under gravity and (b) its transverse response at its free end

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Fig. 8

Evolution of various energies of the beam during its motions

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Fig. 9

Violation of the normalization constraint of Euler parameters at the free end of the beam

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Fig. 10

Schematic of a double elastic pendulum falling under gravity

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Fig. 11

The (a) X, (b) Y, and (c) Z displacements at the free end of the beam B2 from the current formulation and recurdyn

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