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The goal of this project is to optimize the wing plant form and structural design of a single engine aircraft to maximize range and minimize time and material cost. It will be considered incompressible regime and low subsonic.
Problem Definition
Problem Definition
The plane considered is a single engine aircraft with 4 seats, it will be assumed a taper wing with one part rectangular and another part trapezoidal like it's shown in the picture.
Design variables will be the chord at the basis (Cr), span (b), the narrowing in the trapezoidal part (λ), and the relationship between the trapezoidal and rectangular spans (λb).
Block Diagram
Block Diagram
To find the optimal solution a block diagram is needed to combine both the aerodynamic and inertial models in a specific way while adding structural restrictions. This will help understand the optimization process and how the final solution will be computed.
Variable Relationships
Variable Relationships
The table on the left shows the different variables involved in the problem divided in design variables, dependant variables, parameters and objective functions.
Simulator Development
Simulator Development
As it has been mentioned, the goal of the project is to combine an aerodynamic and structural/inertial modules to come up with the best solution for the main system. Before applying the block diagram, both modules need to be studied individually.
Inertial and Structural Module
Inertial and Structural Module
The structural model will be based on square rib inside the wing that will hold all the forces of the airfoil like shown in the figure.
Structural constraint will be base don the interial momentum developed at the wing base and it will be computed with the equation shown. The bending momentof the wing will also be computed.
Aerodynamic Module
Aerodynamic Module
The aerodynamic properties of the wing will be computed using the software MATLAB Tornado. This software implements a Vortex Lattice Method VLM.
This model analyzes the wing as a combination of panels from which individual whirlwinds are developed just like shown in the picture. By doing this, the lift variations can be studied along the chord and the span from one panel to another.
Objective Function
Objective Function
The two functions that need to be optimized are the ones attached to the table shown above which are the range, R, and wing cost, C.
The range implies more complexity since it has several equations attached and considers various parameters like weight, lift and drag coefficients and efficiency among others. This equation is shown on the right.
Wing cost function is an inmediate calculation that multiplies the inner structure size by the material.
Feasibility
Feasibility
Applying the design variables shown at the begining, the wing geometry is shown on the picture on the right along with the reference point and point center mass.
Simulator Validation
Simulator Validation
To validate the simulator the results from the software have been tested and compared along with the analytic simple formulations.
The analytic process was based on the basic drag coefficient and wing elongation. Stablishing a specifi Oswald coefficient as well as the weights, velocities and other parameters the final equation came to be like the one shown below.
Experiment Design
Experiment Design
A statistical technique called hypercube was performed to analyze the design environment. This technique generates values from arbitrary parameters in a multidimensional space.
This allows to visualize the whole space where optimization takes place validating there are no discontinuities or infinite apporaches that can mess with the final result.
Optimization Algorithm
Optimization Algorithm
To optimize the system, the Optimization Toolbox from Matlab will be used, specifically the function fmincon. This function allows the optimization without restrictions of five different algorithms:
Interior-point
Trust-region-refelctive
Active-set
SQP
SQP-Legacy
Results
Results
Sensitivity Analysis
Sensitivity Analysis
From the optimal solution extracted from the optimization process a sensitivity analysis of the different variables in the system was performed. The variables considered were:
Chord
Wing span
Wing span parameter
Narrowing parameter
Heuristic Optimization
Heuristic Optimization
The algorithm selection was based on the own problem structure. The variable design vector was selected for fiberglass since it is required a good initial approximation for convergence.
Conclusions
Conclusions
After the final multiobjective optimization and pareto front estimations the optimal solution for the system was the one shown on the right.
Although several algorithms were applied the results were very close from one another.
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