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Research Papers

A Generalized Approach for Reconstructing the Three-Dimensional Shape of Slender Structures Including the Effects of Curvature, Shear, Torsion, and Elongation

[+] Author and Article Information
Mayank Chadha

Department of Structural Engineering,
University of California San Diego,
9500 Gilman Drive 0085,
La Jolla, CA 92093-0085
e-mail: machadha@ucsd.edu

Michael D. Todd

Department of Structural Engineering,
University of California San Diego,
9500 Gilman Drive 0085,
La Jolla, CA 92093-0085
e-mail: mdtodd@ucsd.edu

1Corresponding author.

Manuscript received September 27, 2016; final manuscript received January 16, 2017; published online February 9, 2017. Assoc. Editor: George Kardomateas.

J. Appl. Mech 84(4), 041003 (Feb 09, 2017) (11 pages) Paper No: JAM-16-1479; doi: 10.1115/1.4035785 History: Received September 27, 2016; Revised January 16, 2017

This paper extends the approach for determining the three-dimensional global displaced shape of slender structures from a limited set of scalar surface strain measurements. It is an exhaustive approach that captures the effect of curvature, shear, torsion, and elongation. The theory developed provides both a determination of the uniaxial strain (in a given direction) anywhere in the structure and the deformed shape, given a set of strain values. The approach utilizes Cosserat rod theory and exploits a localized linearization approach that helps to obtain a local basis function set for the displacement solution in the Cosserat frame. For the assumed deformed shape (both the midcurve and the cross-sectional orientation), the uniaxial value of strain in any given direction is obtained analytically, and this strain model is the basis used to predict the shape via an approximate local linearized solution strategy. Error analysis due to noise in measured strain values and in uncertainty in the proximal boundary condition is performed showing uniform convergence with increased sensor count.

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References

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Figures

Grahic Jump Location
Fig. 1

Deformed configuration of the beam and material-adapted frame (left) and relationship between {d1, d2, d3} and {T, V, H} material frame of reference (right)

Grahic Jump Location
Fig. 6

Comparison of simulation 2 centerlines of exact (gray solid line) and reconstructed via strain (dashed lines) (top plots) and exact directors (d1, d2, and d3 are represented by solid medium, light, and dark gray vectors, respectively) and the predicted directors (black-dashed directors) (bottom plots) of the object

Grahic Jump Location
Fig. 7

Comparison of simulation 2 exact (black line) position vector components (top plots) and director d1 components (bottom plots) with the predicted components for ten strain gauge locations (gray dots) and 50 strain gauge locations (black dots), where x-axis represents ξ1

Grahic Jump Location
Fig. 8

Root mean square error for the position vector for simulation 2 with no noise, considering all the deformations (solid line), ignoring shear (large dashed line), ignoring shear and torsion (medium dashed line), and ignoring shear, torsion, and axial strain (dotted line)

Grahic Jump Location
Fig. 9

Comparison of simulation 3 centerlines of exact (gray line) and reconstructed via strain (dashed line) (top plots) and exact directors (d1, d2, and d3 are represented by solid medium, light, and dark gray vectors, respectively) and the predicted directors (black-dashed directors) (bottom plots) of the object

Grahic Jump Location
Fig. 10

Comparison of simulation 3 exact (black line) position vector components for 50 strain gauge locations (gray dots) and 100 strain gauge locations (black dots), where x-axis represents ξ1

Grahic Jump Location
Fig. 2

Comparison of simulation 1 centerlines of exact (light-gray solid line) and reconstructed via strain (dashed lines) (top plots) and exact directors (d1, d2, and d3 are represented by solid medium, light, and dark gray vectors, respectively) and the predicted directors (black-dashed directors) (bottom plots) of the object

Grahic Jump Location
Fig. 3

Comparison of simulation 1 exact (black line) position vector components (top plots) and director d1 components (bottom plots) with the predicted components for ten strain gauge locations (gray dots) and 50 strain gauge locations (black dots), where x-axis represents ξ1

Grahic Jump Location
Fig. 4

Left plot showing root mean square error for the directors—d1 (solid line), d2 (dashed line), and d3 (dotted line) for no noise case and the right plot showing the rms error for the position vector for various noise level—no noise (solid line), [−5,5] microstrain uniform noise (dashed line), and [−50,50] microstrain uniform noise (dotted line)

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Fig. 5

Average root mean square error in the shape reconstruction for simulation 1 (solid line) and simulation 2 (dashed line) as a function of the uncertainty level in the initial displacement conditions at the proximal end

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