Study of Aircraft Behavior Project

Project Overview

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This project focuses on analyzing aircraft wing structures and lift distribution using computational simulations. The MATLAB-based implementation investigates aerodynamic properties using techniques such as Oscar Shrink Lift Distribution and Prandtl's Lifting Line Theory. Our simulations aim to enhance the understanding of aircraft performance and structural efficiency.

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Project Contributors

• Oumarou Moussa Bola
• Nouaili Mariem

Repository Contents

This repository contains:

• PFA.code: Provides insights into the aerodynamic performance analysis.
• Oscar Shrink Code: Implements the Oscar Shrink method for lift distribution calculations.
• Basic Simulation Output: Contains generated plots illustrating key findings from our simulations.

Theoretical Background

Lift Distribution

The lift force $L$ acting on an aircraft wing can be expressed as:

$$L = \frac{1}{2} \rho V^2 S C_L$$

where:

• $\rho$ = air density,
• $V$ = freestream velocity,
• $S$ = wing surface area,
• $C_L$ = lift coefficient.

The Oscar Shrink Lift Distribution method models how lift varies along the wingspan, helping optimize aerodynamic efficiency.

Prandtl's Lifting Line Theory

Prandtl's Lifting Line Theory provides a mathematical framework to analyze lift along a finite wing:

$$\Gamma(y) = \frac{2 V \alpha}{\pi} \int_{-b/2}^{b/2} \frac{C_L(y') dy'}{(y - y')}$$

where:

• $\Gamma(y)$ is the circulation distribution,
• $\alpha$ is the angle of attack,
• $b$ is the wingspan.

This equation helps predict induced drag and optimize wing performance.

Structural Analysis and Calculations

To ensure the structural integrity of the aircraft wing, various engineering methods are used to analyze internal forces and deformations.

Virtual Work Method

The Virtual Work Method is used to determine displacements and deformations of structures under external forces. The principle states that the work done by external forces is equal to the internal virtual work:

$$\delta W_{ext} = \delta W_{int}$$

This method is particularly useful for analyzing deflections in aircraft wing structures.

Shear Force Calculation

The shear force at a given section of the wing is computed using:

$$V(x) = \int_{x}^{L} q(x) dx$$

where:

• $V(x)$ is the shear force at position $x$,
• $q(x)$ is the distributed load (e.g., lift force),
• $L$ is the span of the wing.

Castigliano’s Theorem

Castigliano’s method is a principle used in structural mechanics to determine displacements in a structure subjected to external loads. It states that the partial derivative of the internal strain energy $W_{\text{int}}$, expressed as a function of external forces, with respect to an applied force $P_i$, gives the displacement $\delta_i$ at the point of application of $P_i$:

$$ \frac{\partial W_{\text{int}}}{\partial P_i} = \delta_i $$

Using Castigliano's theorem, the shear force at a point $M$ along the beam can be determined as:

$$ V(M) = \frac{\partial W_{\text{int}}}{\partial F} $$

where $F$ is an applied force at point $M$ with coordinate $x_M$.

Internal Force Distribution

For the segment $[M, L]$, where $x_M < x_G < x_L$:

$$T_{\text{int}, G} = T_{\text{ext}, G}^{+} = \begin{bmatrix} \int_{x_M}^{x_L} p(x) \cdot \mathbf{y} ,dx \\ \int_{x_M}^{x_L} p(x) \cdot (x - x_G) ,dx \cdot \mathbf{z} \end{bmatrix}_G $$

For the segment $[O, M]$, where $0 < x_G < x_M$:

$$T_{\text{int}, G} = T_{\text{ext}, G}^{+} = \begin{pmatrix} \int_{0}^{x_M} p(x) \cdot \mathbf{y} \,dx \\\ \int_{0}^{x_M} p(x) \cdot (x_G - x) \,dx \cdot \mathbf{z} \end{pmatrix}_G + \begin{pmatrix} F \cdot \mathbf{y} \\\ F \cdot (x_M - x_G) \cdot \mathbf{z} \end{pmatrix}_G + \begin{pmatrix} \int_{x_G}^{x_L} p(x) \cdot \mathbf{y} \,dx \\\ \int_{x_G}^{x_L} p(x) \cdot (x - x_G) \,dx \cdot \mathbf{z} \end{pmatrix}_G$$

Thus, the internal force tensor is:

$$T_{\text{int}, G} = \begin{bmatrix} \int_{0}^{x_M} p(x) \cdot \mathbf{y} ,dx + \int_{x_G}^{x_L} p(x) \cdot \mathbf{y} ,dx + F \cdot \mathbf{y} \\ \int_{x_G}^{x_M} p(x) \cdot (x - x_G) ,dx \cdot \mathbf{z} + \int_{x_G}^{x_L} p(x) \cdot (x - x_G) ,dx \cdot \mathbf{z} + F (x_M - x_G) \cdot \mathbf{z} \end{bmatrix}_G$$

Bending Resistance Criterion

The bending resistance criterion can be expressed as follow:

$$ \sigma_{\text{maxi}} \leq Rpe $$

Or equivalently:

$$ \sigma = \frac{M_{fz}}{\left( \frac{IGz}{v} \right)} \leq Rpe $$

With:

$$ \left( \frac{IGz}{v} \right) $$

as the bending modulus.

• ( v ) : Distance between the neutral axis and the farthest fiber ( $v = y_{\text{max}}$ )

Bending Moment Calculation

For the segment $[M, L]$:

$$ M_{fz} = \int_{x_M}^{x_L} p(x) \cdot (x - x_G) ,dx $$

For the segment $[O, M]$:

$$M_{fz} = \int_{x_G}^{x_M} p(x) \cdot (x - x_G) ,dx + \int_{x_G}^{x_L} p(x) \cdot (x - x_G) ,dx + F (x_M - x_G)$$

Under the Navier-Bernoulli hypothesis, the effects of shear force are neglected, leading to:

$$V(M) = \left(\frac{\partial W_{\text{int}}}{\partial F}\right)_{F=0}$$

$$V(M)=\frac{1}{EI}\int_{0}^{x_M}(x_M - x_G)\left[\int_{x_G}^{x_M} p(x) \cdot (x - x_G) ,dx + \int_{x_G}^{x_L} p(x) \cdot (x - x_G) ,dx\right]$$

where:

• $E$ is the Young's modulus of the material,
• $I$ is the moment of inertia of the beam cross-section.

This formulation is crucial for predicting deflections and structural behavior in aircraft wing analysis.

Wing Bending Moment Calculation

The bending moment along the wing is given by:

$$M(x) = \int_{x}^{L} V(x) dx$$

where $M(x)$ represents the internal moment at position $x$. This helps in assessing wing bending stresses.

External Load Considerations

Aircraft wings are subject to:

• Aerodynamic Loads (Lift, Drag)
• Structural Loads (Weight, Fuel, Engine Forces)
• Dynamic Loads (Gusts, Maneuvers)

The combination of these loads determines the stress distribution within the wing.

Case Study : Fi-156

The Fieseler Fi 156 is a German military reconnaissance aircraft of the Second World War, manufactured by the Fieseler firm and designed in 1935. It is nicknamed Storch (stork in German) because of its high-legged landing gear. Equivalent to the American Piper L-4 Grass Hopper or Stimson L-5 Sentinel, it excelled in its observation missions, transport of personalities or equipment, and medical evacuation. From 1935 to 1945, the Luftwaffe used approximately 2,900 Fieseler Fi 156s, on all fronts and throughout the war. Indeed, it only needed 65 m to take off and less than 20 m was enough to land. The stall speed was very low, less than 50 km/h, and the maximum speed was 170 km/h.

Simulation Results

Architectural Plan Fi-156

Table 1: Parameters of the Fi-156

Parameter	Value
Envergure	14.25 m
Longueur	9.90 m
Hauteur	3.05 m
Surface alaire	26 m²
MTOW	1320 kg
$V_{\text{max}}$	175 km/h
Stall speed	46 km/h
Range	385 km

Table 2: Material Properties

Property	Value
Masse volumique	ρ = 2700 kg/m³
Module de Young	E = 65000 MPa
Poisson's ratio	ν = 0.33
elastic resistance	Rₑ = 190 MPa

Below are some visualizations of our analysis:

Lift Distribution

Displacement Over the Wing

Tangential Stress

Getting Started

Prerequisites

To run the MATLAB simulations, ensure you have:

• MATLAB installed (R2020a or later recommended)
• Required MATLAB toolboxes for aerodynamics and simulations

Running the Code

1. Clone this repository:

   git clone https://github.com/oumarou-Bola/Study-Of-Aircraft-Behavior-Project.git

2. Open MATLAB and navigate to the repository directory.
3. Run PFA.code or Oscar Shrink Code to start the analysis.

Contact

For more information, experimental setup or access to the detailed project report, feel free to reach out:
📧 moussabolaoumarou@gmailcom