Internal Gear Dashboard

Pockets
\( X_{\text{pocket, allowance}} \) Gear Pocket Allowance (%)
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Purpose: Drive Profile Datum.
Logic: The percentage drop from the outer rim circumference to set the chain pitch line.
Type: Manual Entry
Render on Grids: None
Mathematical Formula:
Adjustable User Input
Python Logic: N/A
Live Evaluation: N/A (Input Variable)
\( R_{\text{pocket}} \) Gear Pocket Radius (m) 1.833465
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Purpose: Pitch Circle Radius.
Logic: The exact radial line where the central connector pins physically engage the gear face.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$R_{\text{pocket}} = R_{\text{int}} \times ( 1 - X_{\text{pocket, allowance}} )$$
Python Logic: v["Internal Gear Radius (m)"] * ( 1 - v["Gear Pocket Allowance (%)"] )
Live Evaluation: [ROUTE: STATIC/GLOBAL] 2.0372 * ( 1 - 0.1000 ) = 1.8335
\( X_{\text{chain, pitch}} \) Chain Pitch (m) 0.100000
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Purpose: Link Spacing.
Logic: The distance between individual joint pivots on the articulating chain loop.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$X_{\text{chain, pitch}} = L_{\text{link}}$$
Python Logic: v["Chain link (m)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 0.1000 = 0.1000
\( N_{\text{active, pockets}} \) Number of Active Pockets 57.600000
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Purpose: Load Distribution.
Logic: The total number of pocket indents actively sharing the drive load over a full 180-degree wrap.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$N_{\text{active, pockets}} = ( \pi \times R_{\text{pocket}} ) / X_{\text{chain, pitch}}$$
Python Logic: ( PI() * v["Gear Pocket Radius (m)"] ) / v["Chain Pitch (m)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] ( PI() * 1.8335 ) / 0.1000 = 57.6000
\( F_{\text{pocket}} \) Force per Pocket (N) 3,013.989812
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Purpose: Tooth Contact Pressure.
Logic: The localized physical force hitting a single indent wall during peak operation.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$F_{\text{pocket}} = (\tau_{\text{load}}/ R_{\text{pocket}})/N_{\text{active, pockets}}$$
Python Logic: (v["Gear Torque Load (Nm)"]/ v["Gear Pocket Radius (m)"])/v["Number of Active Pockets"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] (0.0000/ 1.8335)/57.6000 = 3,013.9898
\( A_{\text{req, pocket}} \) Required Pocket Face Area (m2) 0.000006
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Purpose: Material Demand.
Logic: The minimum surface area required per pocket face to prevent localized compression fatigue or bruising over the required service years.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$A_{\text{req, pocket}} = ( F_{\text{pocket}} \times SF_{\text{safety, factor}} ) / ( X_{\text{carbon, fiber}} \times 10^6)$$
Python Logic: ( v["Force per Pocket (N)"] * v["Safety Factor Slider"] ) / ( v["Carbon Fiber Compressive Limit (MPa)"] * 10^6)
Live Evaluation: [ROUTE: STATIC/GLOBAL] ( 3,013.9898 * 2.5000 ) / ( 1,200.0000 * 10**6) = 0.0000
\( W_{\text{target, pocket}} \) Target Pocket Width (m) 0.266504
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Purpose: Flush Core Alignment.
Logic: Forces the pocket track channel to perfectly mirror the full width of the gear face.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$W_{\text{target, pocket}} = W_{\text{predicted}}$$
Python Logic: v["Predicted Gear Width (m)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 0.0000 = 0.2665
\( H_{\text{req, pocket}} \) Required Pocket Depth (m) 0.000024
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Purpose: Minimum Depth Requirement.
Logic: The tiny minimum physical depth needed for strength.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$H_{\text{req, pocket}} = A_{\text{req, pocket}} / W_{\text{target, pocket}}$$
Python Logic: v["Required Pocket Face Area (m2)"] / v["Target Pocket Width (m)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 0.0000 / 0.2665 = 0.0000
\( H_{\text{req, pocket}} \) Required Pocket Depth (mm) 0.002356
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Purpose: Minimum Depth Requirement.
Logic: The tiny minimum physical depth needed for strength.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$H_{\text{req, pocket}} = H_{\text{req, pocket}} \times 100$$
Python Logic: v["Required Pocket Depth (m)"] * 100
Live Evaluation: [ROUTE: STATIC/GLOBAL] 0.0000 * 100 = 0.0024
\( H_{\text{chain, sidewall}} \) Chain Side-Wall Hug Depth (m)
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Purpose: The Deep Hug.
Logic: The physical depth of the structural side flaps that clamp down the gear face to drive the Rigidity Factor.
Type: Manual Entry
Render on Grids: None
Mathematical Formula:
Adjustable User Input
Python Logic: N/A
Live Evaluation: N/A (Input Variable)
Internal Gear Synthesis
\( \tau_{\text{load}} \) Gear Torque Load (Nm) 318,300.172569
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Purpose: Peak Transient Load.
Logic: Captures the absolute maximum rotational torque spike during cycles.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$\tau_{\text{load}} = \text{Max Segment Total GMEG Output Rev 1 Cycle 1 (Nm)}$$
Python Logic: v["Max Segment Total GMEG Output Rev 1 Cycle 1 (Nm)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 0.0000 = 318,300.1726
\( F_{\text{tangential, drive}} \) Tangential Drive Force (N) 173,605.813156
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Purpose: Linear Pull Force.
Logic: The raw directional force applied by the chain link to the gear radius interface.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$F_{\text{tangential, drive}} = \tau_{\text{load}}/ R_{\text{pocket}}$$
Python Logic: v["Gear Torque Load (Nm)"]/ v["Gear Pocket Radius (m)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 318,300.1726/ 1.8335 = 173,605.8132
\( \tau_{\text{req, shear}} \) Required Shear Rigidity (N/m) 2,314,744,175.411171
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Purpose: System Stiffness Target.
Logic: The baseline rigidity required to guarantee the entire system stays perfectly under the strict precision limit.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$\tau_{\text{req, shear}} = F_{\text{tangential, drive}}/ \Delta_{\text{max, allowable}}$$
Python Logic: v["Tangential Drive Force (N)"]/ v["Max Allowable Deflection (m)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 173,605.8132/ 0.0001 = 2,314,744,175.4112
\( X_{\text{coupled, rigidity}} \) Coupled Rigidity Factor 1.732000
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Purpose: Dynamic Bracing Bonus.
Logic: Automatically scales up system support based on how far the U-chain flaps extend down the wheel face.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$X_{\text{coupled, rigidity}} = 1+(H_{\text{chain, sidewall}} \times 3.66)$$
Python Logic: 1+(v["Chain Side-Wall Hug Depth (m)"]*3.66)
Live Evaluation: [ROUTE: STATIC/GLOBAL] 1+(0.2000*3.66) = 1.7320
\( X_{\text{revised, stiffness}} \) Revised Stiffness Demand 1,336,457,376.103447
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Purpose: Net Gear Target.
Logic: The remaining structural stiffness that the gear body itself must independently provide after the chain bracing is accounted for.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$X_{\text{revised, stiffness}} = \tau_{\text{req, shear}} / X_{\text{coupled, rigidity}}$$
Python Logic: v["Required Shear Rigidity (N/m)"] / v["Coupled Rigidity Factor"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 2,314,744,175.4112 / 1.7320 = 1,336,457,376.1034
\( W_{\text{predicted}} \) Predicted Gear Width (m) 0.266504
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Purpose: The Spine Thickness.
Logic: The calculated axial width of the gear face required to resist bending and twisting under peak load.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$W_{\text{predicted}} = \sqrt{ ( 6 \times F_{\text{tangential, drive}} \times R_{\text{int}} ) / ( ( X_{\text{carbon, fibre}} \times 10^9 \times \Delta_{\text{max, allowable}} ) \times X_{\text{coupled, rigidity}} ) )$$
Python Logic: SQRT( ( 6 * v["Tangential Drive Force (N)"] * v["Internal Gear Radius (m)"] ) / ( ( v["Carbon Fibre Modulus (GPa)"] * 10^9 * v["Max Allowable Deflection (m)"] ) * v["Coupled Rigidity Factor"] ) )
Live Evaluation: [ROUTE: STATIC/GLOBAL] SQRT( ( 6 * 173,605.8132 * 2.0372 ) / ( ( 230.0000 * 10**9 * 0.0001 ) * 1.7320 ) ) = 0.2665
Internal Gear Safety Audit
\( \sigma_{\text{bending, stress}} \) Bending Stress (MPa) 0.033309
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Purpose: Flexural Stress Verification.
Logic: Quantifies the structural stress experienced across the active gear teeth to ensure safe material operation.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$\sigma_{\text{bending, stress}} = ( ( 6 \times F_{\text{tangential, drive}} \times R_{\text{int}} ) / ( N_{\text{active, pockets}} \times W_{\text{target, pocket}} \times ( R_{\text{int}}^2 ) ) ) / 10^6$$
Python Logic: ( ( 6 * v["Tangential Drive Force (N)"] * v["Internal Gear Radius (m)"] ) / ( v["Number of Active Pockets"] * v["Target Pocket Width (m)"] * ( v["Internal Gear Radius (m)"]^2 ) ) ) / 10^6
Live Evaluation: [ROUTE: STATIC/GLOBAL] ( ( 6 * 173,605.8132 * 2.0372 ) / ( 57.6000 * 0.2665 * ( 2.0372**2 ) ) ) / 10**6 = 0.0333
\( \sigma_{\text{material, yield}} \) Material Yield Strength (MPa) 598.000000
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Purpose: Elastic Limit Threshold.
Logic: The calculated maximum stress the material can endure before reaching permanent, catastrophic deformation.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$\sigma_{\text{material, yield}} = X_{\text{carbon, fibre}} \times 1000 \times \varepsilon_{\text{strain, limit}}$$
Python Logic: v["Carbon Fibre Modulus (GPa)"]*1000*v["Strain Limit (%)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 230.0000*1000*0.0026 = 598.0000
\( SF_{\text{impact, safety}} \) Impact Safety Factor 0.000000
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Purpose: Structural Safety Margin.
Logic: The ratio between the material yield limit and the actual operating stress.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
$$SF_{\text{impact, safety}} = \sigma_{\text{material, yield}} / \sigma_{\text{impact, stress}}$$
Python Logic: v["Material Yield Strength (MPa)"] / v["Impact Stress (MPa)"]
Live Evaluation: [ROUTE: STATIC/GLOBAL] 598.0000 / 0.0000 = 0.0000
\( X_{\text{int, safety}} \) Internal Gear Safety Blueprint Validation 0.000000
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Purpose: Status Validation.
Logic: Confirms that the newly synthesized gear geometry satisfies all prescribed structural safety protocols.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
IF ( Impact Safety Factor >= Safety Factor Slider , "ULTRA-SAFE" , "REVISE" )
Python Logic: N/A
Live Evaluation: N/A
Internal Gear Mass
\( M_{\text{solid, disk}} \) Solid Disk Mass Equivalent (kg) 0.000000
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Purpose: Theoretical Maximum Mass.
Logic: The weight of a single gear if it were manufactured as a solid, un-spoked cylindrical block of carbon fiber.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
( PI() * ( Gear_Radius^2 ) * Predicted_Gear_Width ) * CF_Density
Python Logic: N/A
Live Evaluation: N/A
\( M_{\text{lightweight, spoke}} \) Lightweight Spoke Factor (%)
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Purpose: Volumetric Efficiency Core.
Logic: Defines the percentage of solid material retained after structural weight reduction (web profiling/spoking).
Type: Manual Entry
Render on Grids: None
Mathematical Formula:
Adjustable User Input
Python Logic: N/A
Live Evaluation: N/A (Input Variable)
\( M \) Mass Per Gear (kg) 0.000000
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Purpose: Single Wheel Mass.
Logic: The calculated physical weight of an individual webbed carbon fiber gear.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
Solid_Disk_Mass * Lightweight_Spoke_Factor
Python Logic: N/A
Live Evaluation: N/A
\( \Sigma X_{\text{gears}} \) Total Gears
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Purpose: Component Population.
Logic: The absolute quantity of large gear hubs integrated within a single track drive loop.
Type: Manual Entry
Render on Grids: None
Mathematical Formula:
Adjustable User Input
Python Logic: N/A
Live Evaluation: N/A (Input Variable)
\( \Sigma M \) Total Gear Weight (kg) 0.000000
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Purpose: Rotary Drive Hub Mass.
Logic: The total combined weight of all rotating gear wheels in the tracking loop.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
Mass Per Gear (kg) * Total Gears
Python Logic: N/A
Live Evaluation: N/A
\( X_{\text{carbon, fiber}} \) Carbon Fiber Cost (£/kg) 0.000000
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Purpose: Material Rate.
Logic: The baseline cost per kilogram of industrial-grade carbon fiber used to estimate structural investment.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
Stand Carbon Fibre Cost (kg)
Python Logic: N/A
Live Evaluation: N/A
\( X_{\text{bom, track}} \) BOM Track Cost (£) 0.000000
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Purpose: Bill of Materials (BOM).
Logic: The primary material expenditure required to manufacture the rotating carbon fiber assemblies.
Type: Calculated (Math Output)
Render on Grids: None
Mathematical Formula:
Total Track Weight * Carbon Fibre Cost (kg)
Python Logic: N/A
Live Evaluation: N/A