🚁 Turbine Blade Equation Builder

📐 Blade Profile Parameters

Quadratic Term (α): 0.5

Sine Amplitude (β): 0.3

Sine Frequency (γ): 2

Exponential Term (δ): 0.1

Current Blade Equation:
y(x) = 0.5x² + 0.3sin(2πx) + 0.1e^(-2x)

🎯 Volume of Revolution:
V = ∫₀¹ π[y(x)]² dx ≈ 0 cubic units

Blade Profile Shape

This is the 2D profile that gets rotated to form the 3D blade

🔧 Step 1: How I can Understand the Blade Profile

What I am seeing: The curve above represents the cross-section of my own turbine blade. When I rotate this curve around the x-axis, I will get a 3D blade shape.

The equation components:

αx² - Should create the basic curved shape (parabola)⤴️
β sin(γπx) - Should Add oscillations for aerodynamic efficiency🎢
δe^(-2x) - Should Create tapering at the tip

📊 Step 2: Friendly reminder of how I should do volume Calculation

Volume of Revolution Formula:

V = ∫₀¹ π[y(x)]² dx

What this should mean: This is rotating the curve around the x-axis. At each point x, I create a circular disk with radius y(x). The volume should be the sum of all these disks.

Why this is important?: Different blade volumes affect weight, material cost🤑, and performance📈. My optimization will find the best balance!

🎯 Step 3: MY Optimization Setup

Reminder of my optimization problem:

Maximize: Efficiency which = f(α, β, γ, δ)

Subjected to: Volume ≤ V_max, Stress ≤ σ_max

How calculus WILL help: Use partial derivatives to find where ∂(Efficiency)/∂α = 0, ∂(Efficiency)/∂β = 0, etc.