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Unformatted text preview: Wing Bending Calculations Lab 10 Lecture Notes Nomenclature L y spanwise coordinate q net beam loading S shear M bending moment θ deﬂection angle (= dw/dx ) w deﬂection κ local beam curvature ′ lift/span distribution ′ S η normalized spanwise coordinate (= 2 y/b ) c local wing chord wing wing area b wing span λ taper ratio E Young’s modulus δ tip deﬂection N load factor m wing mass/span distribution I bending inertia i spanwise station index n last station index at tip L lift W weight g gravitational acceleration () o quantity at wing root Loading and Deﬂection Relations The net wing beam load distribution along the span is given by ′ ′ q ( y ) = L ( y ) − N g m ( y ) (1) where m ′ ( y ) is the local mass/span of the wing, and N is the load factor. In steady level ﬂight we have N = 1. The net loading q ( y ) produces shear S ( y ) and bending moment M ( y ) in the beam structure. This resultant distribution produces a deﬂection angle θ ( y ), and deﬂection w ( y ) of the beam, as sketched in Figure 1. net loading q(y) = L’(y) − N g m’(y) L’(y) − N g m’(y) aero loading θ y gravity + inertial w loading q S M y y y y y 0 b/ 2 Figure 1: Aerodynamic and mass loadings, and resulting structural loads and deﬂection. 1 The standard differential equations derived via simple Bernoulli-Euler beam model, with the primary structural axis along the y direction, relate the loads and deﬂections to the loading q ( y ) and the bending stiffness EI ( y ). d S = q dy (2) dM = S dy (3) M dθ = dy (4) EI dw = θ dy (5) To allow integration of these equations, it’s necessary to impose four boundary conditions. For a cantilevered wing beam, they are y = b/ 2 : S = 0 (6) y = b/ 2 : M = 0 (7) y = 0 : θ = 0 (8) y = 0 : w = 0 (9) Load Distribution The lift distribution L ′ ( y ) needed to define q ( y ) depends on the induced angle α i ( y ) and hence the overall wing shape in a complicated manner. hence the overall wing shape in a complicated manner....
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- Fall '05
- Trigraph, Second moment of area, Boundary value problem, Wing, dy