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Research Papers

# An Accurate Singularity-Free Formulation of a Three-Dimensional Curved Euler–Bernoulli Beam for Flexible Multibody Dynamic Analysis

[+] 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

Professor
ASME Fellow
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

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 7, 2015; final manuscript received March 24, 2016; published online May 25, 2016. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 138(5), 051001 (May 25, 2016) (14 pages) Paper No: VIB-15-1420; doi: 10.1115/1.4033269 History: Received October 07, 2015; Revised March 24, 2016

## Abstract

An accurate singularity-free formulation of a three-dimensional curved Euler–Bernoulli beam with large deformations and large rotations is developed for flexible multibody dynamic analysis. Euler parameters are used to characterize orientations of cross sections of the beam, which can resolve the singularity problem caused by Euler angles. The position of the centroid line of the beam is integrated from its slope, and position vectors of nodes of beam elements are no longer used as generalized coordinates. Hence, the number of generalized coordinates for each node is minimized. Euler parameters instead of position vectors are interpolated in the current formulation, and a new C1-continuous interpolation function is developed, which can greatly reduce the number of elements. Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by the generalized- $α$ method for resulting differential-algebraic equations (DAEs). The current formulation can be used to calculate static and dynamic problems of straight and curved Euler–Bernoulli beams under arbitrary, concentrated and distributed forces. The stiffness matrix and generalized force in the current formulation are much simpler than those in the geometrically exact beam formulation (GEBF) and absolute node coordinate formulation (ANCF), which makes it more suitable for static equilibrium problems. Numerical simulations show that the current formulation can achieve the same accuracy as the GEBF and ANCF with much fewer elements and generalized coordinates.

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## Figures

Fig. 1

Schematic of a three-dimensional Euler–Bernoulli beam under deformation

Fig. 2

Illustration of generalized coordinates of the kth element

Fig. 3

Deformed shapes of the curved cantilever beam under different forces at its free end

Fig. 4

(a) Undeformed three-dimensional configuration of the spring and (b) its projection on the XY plane

Fig. 5

Schematic of the spring stretched by a force F

Fig. 6

Relationship between the displacement and force for the spring

Fig. 7

Schematic of a cantilever beam subjected to the vertical concentrated force at its free end

Fig. 8

Tip displacements of the cantilever beam in the (a) X and (b) Y directions under the vertical concentrated force at its free end

Fig. 9

Schematic of a spin-up maneuver of a beam: (a) the initial configuration and global frame O–XYZ and (b) a rotating frame O–xyZ

Fig. 10

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

Fig. 11

Relative errors of transverse deflections at the tip of the beam at time t=7s versus the number of elements on logarithmic scales

Fig. 12

Initial configuration of a curved beam under a spin-up maneuver

Fig. 13

(a) Tangential and (b) normal deflections at the tip of the curved beam during the spin-up maneuver

Fig. 14

(a) Transverse deflections at the end of the beam and (b) their relative errors at t=2.5s versus the number of elements on logarithmic scales

Fig. 15

(a) Curvatures at the midpoint of the beam and (b) their relative errors at t=2.5s versus the number of elements on logarithmic scales

Fig. 16

Schematic of a rotating shaft about the global X axis under an initial impulse along the global Y axis

Fig. 17

The Y and Z displacements at the right end of the shaft with the rotating speed N=5.983 r/s

Fig. 18

The Y and Z displacements at the right end of the shaft with the rotating speed N = 6.383 r/s

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