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1 KiB
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32 lines
No EOL
1 KiB
Text
Desmos Graph for Corrected Reynolds/Hermite Activation Bridge
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Copy these equations into Desmos (https://www.desmos.com/calculator):
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1. Normalized activation curve:
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A(x) = 3x^2 - 2x^3
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2. Offset physical bridge:
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f_A(x) = 0.0278 + 0.012(3x^2 - 2x^3)
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3. Reynolds number to x conversion:
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x(Re) = clamp((Re - 2300) / 1700, 0, 1)
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4. Direct Reynolds to activation:
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A_Re(Re) = 3 * clamp((Re - 2300) / 1700, 0, 1)^2 - 2 * clamp((Re - 2300) / 1700, 0, 1)^3
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5. Direct Reynolds to physical bridge:
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f_A_Re(Re) = 0.0278 + 0.012 * (3 * clamp((Re - 2300) / 1700, 0, 1)^2 - 2 * clamp((Re - 2300) / 1700, 0, 1)^3)
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Set x-axis range: 0 to 1 (for x) or 2000 to 4500 (for Re)
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Set y-axis range: 0 to 0.05
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Key points:
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- Re = 2300 → x = 0 → A(0) = 0 → f_A(0) = 0.0278
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- Re = 4000 → x = 1 → A(1) = 1 → f_A(1) = 0.0398
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- Re = 3150 → x = 0.5 → A(0.5) = 0.5 → f_A(0.5) = 0.0338
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Properties:
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- A(x) is monotone increasing on [0,1]
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- A'(x) = 6x(1-x) ≥ 0 for 0 ≤ x ≤ 1
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- A'(0) = 0, A'(1) = 0 (smooth endpoints)
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- f_A(x) maps [0,1] to [0.0278, 0.0398] |