A slender wing behaves structurally as a beam, but a laminated box is not a simple one. Off-axis plies couple bending to twist, and that coupling is the mechanism of aeroelastic tailoring, by which a designer steers divergence and flutter speeds through the layup (Shirk et al., 1986). The modern treatment separates the problem into a two-dimensional cross-sectional analysis, which yields a fully populated stiffness matrix for the section, and a one-dimensional geometrically exact beam analysis that is valid for arbitrarily large displacement and rotation provided strains stay small (Hodges, 2006). This split is itself a model reduction, and an accurate one for wings whose chord is small beside their span.

Flutter analysis needs the aerodynamic forces produced by wing motion, including the lag caused by the shed wake. For a thin section in incompressible flow Theodorsen (1935) gave these in the frequency domain. Lifting surface methods, of which the doublet-lattice method is the workhorse, extend the idea to finite wings in subsonic flow (Albano and Rodden, 1969). Both deliver forces as tabulated functions of reduced frequency. To place them in a time-domain or state-space model they are fitted with rational functions, at the cost of added lag states, and the minimum-state formulation keeps that cost small (Karpel, 1982). Finite-state inflow theory reaches the same end directly from the governing equations (Peters et al., 1995). The unsteady vortex-lattice method works in the time domain, follows a deforming wake and can itself be linearised and reduced (Murua et al., 2012). Where shocks matter, reduced models are identified from computational fluid dynamics by proper orthogonal decomposition or by Volterra and related system identification methods (Hall et al., 2000; Silva, 2005).

With structure and aerodynamics in compatible form, flutter is found as the speed at which a root of the coupled system crosses into the unstable half-plane. Frequency-domain aerodynamics lead to iterative schemes such as the p-k method, in which frequency and damping are found by repeated solution at matched reduced frequency. State-space aerodynamics give an ordinary eigenvalue problem. Both routes are described in the standard text of Hodges and Pierce (2011).

The analyst records the planform, the spanwise distribution of section geometry and laminate, the non-structural masses including fuel states, and the range of speed, altitude and load factor to be cleared. The boundary condition at the root is fixed at this point: a cantilever is adequate for component studies, while a very flexible aircraft may need the rigid-body degrees of freedom retained, since its lowest elastic frequencies can approach those of the flight dynamic modes (Patil et al., 2001).

A cross-sectional analysis is run at a set of spanwise stations to obtain, for each, the six-by-six stiffness matrix, the mass per unit length, the mass moments of inertia and the positions of the shear centre and centre of mass. The off-diagonal stiffness terms must be kept. Discarding them reduces the wing to an isotropic equivalent and removes exactly the tailoring effect a composite design relies on. The output of this step is a table of sectional properties against span.

For each speed, altitude and load factor, the geometrically exact beam is loaded with steady aerodynamic forces and gravity, and the equilibrium is found iteratively, updating the aerodynamic loads as the wing deforms. A nonlinear lifting-line or steady vortex-lattice model on the deformed geometry is sufficient in attached subsonic flow. The result is the deflected shape with its internal forces. Both are needed, since prestress alters stiffness as well as geometry.

Tangent stiffness and mass matrices are formed at the equilibrium and the natural modes computed. On a strongly bent wing these modes differ from the jig-shape modes: torsion and edgewise bending, which are independent on a straight wing, combine. A small number of modes, enough to span the frequency range of interest with margin, forms the structural basis. When stores or control surfaces are attached, component mode synthesis offers a consistent way of assembling the basis from substructures. The basis must be recomputed, or at least checked, at each trim state and not carried over from the undeformed wing.

The aerodynamic reduction is chosen to suit the regime, and Table 1 compares the main options. For low-speed, very high aspect ratio wings, strip theory with finite-state inflow is cheap, already in state-space form and easily evaluated on a curved wing. Where three-dimensional effects or wake geometry matter, a linearised unsteady vortex-lattice model about the deformed shape, reduced by balanced truncation or modal projection, is the natural choice. Doublet-lattice forces with a rational fit remain convenient when an industrial linear toolchain already exists, though the lattice must then follow the deformed surface. In transonic flow a model identified from a nonlinear flow solver is required.

Structural and aerodynamic states are combined into one first-order system at each flight condition, and its eigenvalues computed. Damping and frequency are followed from condition to condition, with roots matched by eigenvector correlation and not by sorting on frequency, since modes cross and exchange character as the wing bends. Flutter onset is the lowest speed at which a root reaches zero damping along a path of physically consistent trim states. The path matters. Increasing speed at constant load factor and increasing load factor at constant speed approach the boundary through different shapes.

A reduced model earns trust by comparison. At a few conditions spread across the envelope, including the most deformed, the trim shape, the modal frequencies and the damping trend should be compared with a higher-order solution, for example a shell finite element model coupled to the unreduced aerodynamic method. Disagreement points to the step at fault: a trim error to the beam properties or the steady loads, a frequency error to the basis, a damping error to the aerodynamic reduction.