Objective Function
To guarantee that the mass distribution is properly scaled, frequencies and mode shapes of the reduced model must match the scaled frequencies and mode shapes of the full-size wing.
For the frequencies similarity, the absolute value of the difference between the target frequency and the model frequency is used. To match the mode shapes, we resort to the COMAC, since this method has proven to provide better results (see Section 4.5.1). So, the objective function is the one presented in Equation 3.6.
Initial Design Variables
The design variables for optimization process 2 are the radius and positions of the structural masses added to the reduced model.
Nonlinear Constraints
For the nonlinear constraints, the same option described in methodology 1 is considered, which cor- responds to Equation 3.9.
After the problem is formulated, static and dynamic analysis can be done.
For static analysis, the system of equation takes the form:
[k]{d}={F}, (3.13)
yielding that the potential energy is stationary. By solving this equation we can obtain the displacements field.
For free vibration analysis, it is necessary to solve Equation (??) consideringF = 0, meaning that the system is free of external forces:
[k]{d}+ [M]{d}¨ = 0 (3.14)
The solution for the free vibration problem can be assumed as:
d=φexp(iωt), (3.15)
where φis the amplitude of the nodal displacement, w is the (angular) frequency of the free vibration, andt is the time.
With the objective of enabling the model with the capability of incorporating geometric non-linearities, a few changes to both static and dynamic analyses need to be performed. To do so, the stiffness matrix becomes dependent of the deformed state, and an iterative method is required. The one implemented in the computational tool used is Newton-Raphson method [38], which aims at reaching the following equilibrium:
t+∆t{F} −t+∆t{R}i={0}, (3.16) where the applied load, {F}, does not depend on the deformation, but the restoring force {R} does, withibeing the current iteration. The restoring forces are the internal structural nodal forces, which are computed from the previous iteration stiffness matrix as follows:
t+∆t{R}i=t+∆t{K}(i−1){d}i (3.17)
To treat large displacements and large rotations, NASTRANTMmethodology [39] was followed, which consists in computing the net displacements and using the gimbal angle approach to treat, respectively, large displacements and large rotations.
Since gravity is important when studying high aspect-ratio wings due to the flexibility effects, the element weight is also considered in the model, and the gravity load, {Fg} is computed by multiplying the mass matrix by the gravity acceleration vector, {g}[35]:
{Fg}= [M]{g} (3.18)
3.3.2 Aerodynamic Model
Panel methods are numerical schemes for solving the Prandtl-Glauert equation for linear, inviscid, ir- rotational, and incompressible flow about aircraft flying at subsonic or supersonic speeds [40]. The Prandtl-Glauert equation is the simplest form of the fluid-flow equations that contain compressibility effects, but neglecting all the viscous and heat transfer terms, assuming that the flow is irrotational, and discarding all nonlinear terms. Physically, these restrictions mean that important flow behaviour such as separation, skin-friction drag, and transonic shocks are not predicted with panel methods [40].
The formulation consists in the potential flow model implemented in the nonlinear aeroelastic frame- work [35] is based in the Katz and Plotkin formulation [41].
3.3.3 Aeroelastic Model
To properly couple the aerodynamic and structural solvers, in order to perform an aeroelastic analysis, the framework used to analyse the complex wing has a Fluid-Structure Interaction (FSI) algorithm capable of performing steady and unsteady calculations.
The FSI methodology can be classified as: (1) monolithic and (2) loosely coupled [42]. It is responsible for transferring the loads from the aerodynamic panels to the structure and transferring the structural displacements to the aerodynamic panels. Since the meshes are not coincidental (for fluid dynamics modelling a detailed mesh is required, as for structural dynamics modelling a relatively coarse mesh suffices [43]), a proper load and displacement transfer method is necessary.
The FSI is an iterative process which converges when equilibrium is reached, i.e. when both displace- ments and loads have converged. The convergence criteria is given by
1 N N
N N
X
i=1
| dni −dn−1i
dni |≤1×10−7, (3.19)
whereN Nanddiare, respectively, the number of structural nodes and nodal displacement (displacements and rotations)i,nis the current load step andn−1 the previous load step.
For the static aeroelastic calculations, a steady FSI algorithm is employed [35], which calculates the linear or non-linear static equilibrium position.
For the dynamic aeroelastic analyses, two methodologies were implemented [35]: (1) frequency domain, in which the flutter boundary is calculated at the linear or nonlinear equilibrium state (computed with the steady FSI algorithm) using the p-k method [44]; and (2) time domain, in which the unsteady FSI [35] is used to compute the nonlinear aeroelastic deformed state in time.
Chapter 4
Primary Quantities
The choice of the primary quantities can be influenced by different aspects. Depending on the type of experimental test that the model is intended for (wind-tunnel test or flight test), and on the test characteristics, some constraints will be added to the scaled model: span (due to the limited dimensions of the wind tunnel), wind speed, air density (it can be altered in wind-tunnel test but may prove expensive;
for flight test the altitude can be defined based on this parameter but it can be challenging to achieve the exact value), thicknesses of the inner structure (structures that are too thin become more difficult to build and less resistant), among others.
With the objective of analysing the influence of the primary quantities choice in the scaling procedure, five different sets will be defined and applied to a simple model.