The development of the Ko displacement theory for predictions of structure deformed shapes was motivated in 2003 by the Helios flying wing, which had a 247-ft (75-m) wing span with wingtip deflections reaching 40 ft (12 m). The Helios flying wing failed in midair in June 2003, creating the need to develop new technology to predict in-flight deformed shapes of unmanned aircraft wings for visual display before the ground-based pilots.

Any types of strain sensors installed on a structure can only sense the surface strains, but are incapable to sense the overall deformed shapes of structures. After the invention of the Ko displacement theory, predictions of structure deformed shapes could be achieved by feeding the measured surface strains into the Ko displacement transfer functions for the calculations of out-of-plane deflections and cross sectional rotations at multiple locations for mapping out overall deformed shapes of the structures. The new Ko displacement theory combined with a strain-sensing system thus created a revolutionary new structure-shape-sensing technology.

The formulation of the Ko displacement theory stemmed from the integrations of the beam curvature equation (second order differential equation). The beamlike structure (wing) was first discretized into multiple small domains so that beam depth and surface strain distributions could be represented with piece-wise linear functions. This discretization approach enabled piecewise integrations of the beam curvature equation in closed forms to yield slope and deflection equations for each domain in recursion formats. The final deflection equations in summation forms (called Ko displacement transfer functions), which contain no structural properties (such as bending stiffness), were then expressed in terms of domain length, beam depth factor, and surface bending strains at the domain junctures. In fact, the effect of the structural properties is absorbed by surface strains.

For flying wing structures, the two-line strain-sensing system is a powerful method for simultaneously monitoring the bending and cross sectional rotations. The two-line strain-sensing system eliminates the need for installing the shear strain sensors to measure the surface distortions through which the wing structure cross sectional rotations could be determined.

The Ko displacement theory combined with onboard fiber-optic strain-sensing system forms a powerful tool for in-flight deformed shape monitoring of flexible wings and tails, such as those often employed on unmanned flight vehicles by the ground-based pilot for maintaining safe flights. In addition, the real-time wing shape monitored could then be input to the aircraft control system for aero-elastic wing-shape control.

This work was done by William L. Ko of Dryden Flight Research Center. For more information, download the Technical Support Package (free white paper) at under the Mechanics/Machinery category. DRC-006-024

NASA Tech Briefs Magazine

This article first appeared in the December, 2010 issue of NASA Tech Briefs Magazine.

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