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Research Papers

Development of an Active Curved Beam Model—Part I: Kinematics and Integrability Conditions

[+] Author and Article Information
Hidenori Murakami

Department of Mechanical and
Aerospace Engineering,
University of California,
San Diego, La Jolla, CA 92093-0411
e-mail: hmurakami@ucsd.edu

Manuscript received December 16, 2016; final manuscript received March 18, 2017; published online April 18, 2017. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 84(6), 061002 (Apr 18, 2017) (16 pages) Paper No: JAM-16-1605; doi: 10.1115/1.4036308 History: Received December 16, 2016; Revised March 18, 2017

In order to develop an active nonlinear beam model, the beam's kinematics is examined in this paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in Part II, starting from the three-dimensional Hamilton's principle and the d'Alembert's principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

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Figures

Grahic Jump Location
Fig. 1

A curved beam at the reference configuration B(0) and the current configuration B(t)

Grahic Jump Location
Fig. 2

Push forward of a vector

Grahic Jump Location
Fig. 3

Piola transformation: φt*ibvol3=iBVOL3

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