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Research Papers

# Tensorial Deformation Measures for Flexible Joints

[+] Author and Article Information
Olivier A. Bauchau

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332olivier.bauchau@ae.gatech.edu

Leihong Li

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332leihong.li@gatech.edu

Pierangelo Masarati

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, 20133 Milano, Italymasarati@aero.polimi.it

Marco Morandini

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, 20133 Milano, Italymorandini@aero.polimi.it

J. Comput. Nonlinear Dynam 6(3), 031002 (Dec 15, 2010) (8 pages) doi:10.1115/1.4002517 History: Received May 12, 2010; Revised July 15, 2010; Published December 15, 2010; Online December 15, 2010

## Abstract

Flexible joints, sometimes called bushing elements or force elements, are found in all multibody dynamics codes. In their simplest form, flexible joints simply consist of sets of three linear and three torsional springs placed between two nodes of a multibody system. For infinitesimal deformations, the selection of the lumped spring constants is an easy task, which can be based on a numerical simulation of the joint or on experimental measurements. If the joint undergoes finite deformations, the identification of its stiffness characteristics is not so simple, especially if the joint itself is a complex system. When finite deformations occur, the definition of deformation measures becomes a critical issue. Indeed, for finite deformation, the observed nonlinear behavior of materials is partly due to material characteristics and partly due to kinematics. This paper focuses on the determination of the proper finite deformation measures for elastic bodies of finite dimension. In contrast, classical strain measures, such as the Green–Lagrange or Almansi strains, among many others, characterize finite deformations of infinitesimal elements of a body. It is argued that proper finite deformation measures must be of a tensorial nature, i.e., must present specific invariance characteristics. This requirement is satisfied if and only if the deformation measures are parallel to the eigenvector of the motion tensor.

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## Figures

Figure 1

Configuration of the flexible joint

Figure 2

Configuration of the flexible joint for scenario ℓ. For clarity of the figure, the elastic body is not shown.

Figure 3

Configuration of the flexible joint for scenario ℓ. For clarity of the figure, the elastic body is not shown.

Figure 4

Reference configuration of the flexible beam

Figure 5

Joint deformation under a single moment. Top figure: displacement components u1 (○) and u2 (◇) and bottom figure: rotation r3 (△). Exact solution: symbols. Present solution: κ(ϕ)=ϕ, dashed line, κ(ϕ)=4 sin ϕ/4, dotted line, and κ(ϕ)=4 tan ϕ/4, dashed-dotted line.

Figure 6

Joint deformation under two moments. Top figure: displacement components u1 (○), u2 (◇), and u3 (◻) and bottom figure: exponential map components r1 (▽), r2(◁), and r3 (△). Finite element solution: symbols. Present solution: κ(ϕ)=ϕ, dashed line, κ(ϕ)=2 sin ϕ/2, dotted line, and κ(ϕ)=2 tan ϕ/2, dashed-dotted line.

Figure 7

Joint deformation under two forces. Top figure: displacement components u1 (○) and u2 (◇) and bottom figure: exponential map component r3 (△). Finite element solution: symbols. Present solution: κ(ϕ)=ϕ, dashed line, κ(ϕ)=2 sin ϕ/2, dotted line, and κ(ϕ)=2 tan ϕ/2, dashed-dotted line.

Figure 8

Joint deformation under compressive force. Top figure: displacement components u1 (○) and u2 (◇) and bottom figure: exponential map component r3 (△). Finite element solution: symbols. Present solution: κ(ϕ)=ϕ, dashed line, κ(ϕ)=4 sin ϕ/4, dotted line, and κ(ϕ)=4 tan ϕ/4, dashed-dotted line.

Figure 9

Joint deformation under two forces. Top figure: displacement components u1 (○) and u2 (◇) and bottom figure: exponential map component r3 (△). Finite element solution: symbols. Present solution: κ(ϕ)=ϕ, dashed line, κ(ϕ)=2 sin ϕ/2, dotted line, and κ(ϕ)=2 tan ϕ/2, dashed-dotted line.

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