Mathematics-in-Industry Case Studies, Vol 4 (2012)

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A variational approach to modeling aircraft hoses and flexible conduits

Ke Han, Huiyi Hu, Eunkyung Ko, Ahmet Ozkan Ozer, Cory Simon, Changhui Tan

Abstract


Airplanes have thousands of hoses and flexible conduits in conjunction with moving parts. It is essential in the design of the airplane that the conduits do not tangle or kink and also that the minimal amount of material is used to minimize weight. To prevent the necessity of building plywood prototypes and arranging hoses by trial and error, the simulation of the resting states of the conduits serves as a tool for design. Here we model the resting state of a conduit as being a critical point of the elastic energy functional-- the integral of the square curvature-- in neglecting gravity and torsional effects. Using the variational approach, we find the Euler-Lagrange equations for the critical point of the energy functional as well as sufficient conditions for a minimizer. We use both function representations and parametric representations for a curve that models the centerline of the conduit in two and three dimensional space and consider both free length and constrained length problems using the Lagrangian multiplier method. This work serves as a reference for the Euler-Lagrange equations of minimal energy curves with fixed ends and slopes in two and three dimensional space. Further, we outline the collocation method using a quintic B-spline basis to numerically solve the resulting fourth order ODEs. The ultimate goal towards which we step with this paper is to, with a bottoms-up approach, use large-scale simulations in the design of the positions and slopes of the conduits in an airplane.

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Mathematics-in-Industry Case Studies. ISSN: 1913-4967