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Research Papers

# An Accurate and Robust Geometrically Exact Curved Beam Formulation for Multibody Dynamic Analysis

[+] Author and Article Information
H. Ren, W. Fan

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China

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,
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 February 19, 2017; final manuscript received July 22, 2017; published online September 26, 2017. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 140(1), 011012 (Sep 26, 2017) (13 pages) Paper No: VIB-17-1069; doi: 10.1115/1.4037513 History: Received February 19, 2017; Revised July 22, 2017

## Abstract

An accurate and robust geometrically exact beam formulation (GEBF) is developed to simulate the dynamics of a beam with large deformations and large rotations. The undeformed configuration of the centroid line of the beam can be either straight or curved, and cross sections of the beam can be either uniform or nonuniform with arbitrary shapes. The beam is described by the position of the centroid line and a local frame of a cross section, and a rotation vector is used to characterize the rotation of the cross section. The elastic potential energy of the beam is derived using continuum mechanics with the small-strain assumption and linear constitutive relation, and a factor naturally arises in the elastic potential energy, which can resolve a drawback of the traditional GEBF. Shape functions of the position vector and rotation vector are carefully chosen, and numerical incompatibility due to independent discretization of the position vector and rotation vector is resolved, which can avoid the shear locking problem. Numerical singularity of the rotation vector with its norm equal to zero is eliminated by Taylor polynomials. A rescaling strategy is adopted to resolve the singularity problem with its norm equal to $2mπ$, where m is a nonzero integer. The current formulation can be used to handle linear and nonlinear dynamics of beams under arbitrary concentrated and distributed loads. Several benchmark problems are simulated using the current formulation to validate its accuracy, adaptiveness, and robustness.

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## Figures

Fig. 1

Geometric description of a particle on a beam in the GEBF

Fig. 2

Static equilibria of the rolling cantilever beam, calculated using 15 elements

Fig. 3

Relative errors of displacements and stretch and shear strains at the free end of the cantilever beam versus the number of elements in logarithmic scales

Fig. 4

First four in-plane mode shapes of the elastic ring

Fig. 5

First four out-of-plane mode shapes of the elastic ring

Fig. 6

Undeformed configuration of the spring

Fig. 7

Setups of the spring under (a) fixed–fixed and (b) spherical–spherical joints

Fig. 8

Calculated constitutive relations of the spring

Fig. 9

Schematic of a straight rotating beam

Fig. 10

(a) Longitudinal and (b) transverse tip deflections of the straight rotating beam during spin-up maneuver

Fig. 11

(a) Upwind, (b) downwind, and (c) out-of-plane configurations of the undeformed beam

Fig. 12

Values of c(x) along the curved beam with different numbers of (a) low-order and (b) high-order elements

Fig. 13

Tip deflections of the curved beam with (a) upwind, (b) downwind, and (c) out-of-plane undeformed configurations during spin-up maneuver

## Errata

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