A strain-based geometrically-nonlinear beam formulation has previously been developed and applied in analyzing slender wings of very flexible aircraft. This formulation features strains and curvatures of the beam reference line as the independent variables in the solution. In this study, the deformation of beam cross-sections (warping) is allowed in the modeling process. The warping field of the beam is represented by the Ritz approximation functions at each cross-section, as well as their spatial variations along the beam span. This treatment enhances the one-dimensional beam formulation associated with large-scale beam bending/twist curvatures, with the additional elastic degrees of freedom that consider smallscale local deformations of the cross-sections. It has the potential to provide an efficient solution base for preliminary design and time-domain analysis of flexible aircraft with compliant airfoils, as well as the shape and load control of very flexible aircraft with active actuations.
Nomenclature
B= body reference frame B F , B M = influence matrices for distributed forces and moments b = positions and orientations of B frame, as time integral of β D = constitutive matrix E n = inertia sub-matrix consisting of influence from finite-section modes ψ n F n = inertia sub-matrix consisting of self-influence from finite-section modes ψ n F dist , F pt = distributed and point forces G = global (inertial) reference frame g = gravity acceleration column vector, m/s 2 H hb = matrix consisting of influence from Jacobian J hb and body angular velocities ω B h = width of the strip section, m h = absolute positions and rotations of beam nodes h r = relative positions and rotations of beam nodes J = Jacobian matrix K , n q K , n q K , n n q q K = components of element stiffness matrix k = cross-sectional stiffness matrix L n = gravity sub-matrix consisting of influence from finite-section modes ψ n l = beam/wing span, m M, C, K = discrete mass, damping, and stiffness matrices of whole system M dist , M pt = distributed and point moments N = influence matrix for gravity force N q = numer of finite-section modes p B , θ B = position and orientation of B frame, as time integral of v B and ω B , respectively p w = position of w frame with respect to B frame q n = magnitude of finite-section modes 1 Assistant Professor (suw@eng.ua.edu), Dept. of Aerospace Engineering and Mechanics, Senior Member AIAA. Downloaded by KUNGLIGA TEKNISKA HOGSKOLEN KTH on July 29, 2015 | http://arc.aiaa.org | 2 b R , n q R , R = rigid-body, camber, and flexible components of generalized load vector r = warping field r = residual warping field s = beam curvilinear coordinate, m t = thickness of the strip section, m U = total strain energy of the beam v B , ω B = linear and angular velocities of B frame, resolved in B frame itself W ext , W int = external and internal virtual work, respectively w = local beam reference frame defined at each node along beam reference line β = body velocities, with translational and angular components, re...