🌀 Blade Harmonics Visualizer

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Forcing Function: \(F(\psi) = \sum_{n=1}^{6} A_n \sin(n\psi)\)

Adjust amplitude coefficients for each harmonic (0 = disabled):

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Forcing Function Applied to Each Blade: \(F(\psi + \Delta\psi + \frac{2\pi b}{B})\)
Combined Load in Inertially-Fixed Reference Frame (Tower/Hub Load)
Output Spectrum: Rotating vs Fixed Frame Comparison

Theory

Forcing Function: This visualizer allows you to create a custom forcing function composed of harmonics from 1P to 6P:

$$F(\psi) = A_1 \sin(\psi) + A_2 \sin(2\psi) + A_3 \sin(3\psi) + A_4 \sin(4\psi) + A_5 \sin(5\psi) + A_6 \sin(6\psi)$$

Blade Response: Each blade experiences this forcing function with a spatial phase shift based on its azimuthal position. Note that the phase shift is the same for all harmonics:

$$F_b(\psi) = \sum_{n=1}^{6} A_n \sin\left(n\psi + \Delta\psi + \frac{2\pi b}{B}\right)$$

Rotating vs Fixed Frame:

  • Rotating Frame (Blue): \(B \times\) input amplitudes, where \(B\) is the number of blades. Each blade experiences the same forcing function, so the total harmonic content in the rotating frame is simply \(B\) times the input.
  • Fixed Frame (Red): Load projected onto a fixed direction in space:
    $$\sum_{b=0}^{B-1} F_b(\psi) \cos\left(\psi + \Delta\psi + \frac{2\pi b}{B}\right)$$
    This is what the tower experiences.

Key Insight: For a \(B\)-bladed turbine, harmonics at integer multiples of \(B\) reinforce in the rotating frame but may transform to different frequencies in the fixed frame. The comparison chart shows which harmonics survive in each reference frame.

Practical Application: This helps understand how different frequency components in wind loads (tower shadow, yaw error, wind shear, etc.) affect both rotating (hub) and non-rotating (tower) structure loads.