gear_design_assignment.md

layout	default
title	MEE342 Gear Design Activity 2026

MEE 342 Gear Design Activity

Image source: "Gear Train" by Aliva Sahoo, via Wikimedia Commons, licensed CC BY-SA 4.0.

Objective

Design a spreadsheet-based spur gear design tool and use it for two related tasks:

1. Single-stage design: size one spur gear–pinion pair from motor power and speed requirements using Shigley's standard spur gear design workflow.
2. Two-speed gearbox design: replace the single-stage reduction with a shiftable two-speed gearbox, choose first gear and second gear ratios that minimize 0–60 mph time, choose the best shift speed, and then run the gear design from Step 1 for each selected ratio.

The final deliverable is an interactive spreadsheet where a user can enter power, speed, safety factors, life, and packaging limits and receive a feasible gear design (or a clear warning that the design is infeasible).

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Recommended benchmark vehicle and assumptions

Use a Tesla Model 3 rear-wheel-drive benchmark.

Baseline vehicle assumptions

Use these values as defaults in the spreadsheet:

• Vehicle type: Tesla Model 3 RWD benchmark
• Peak motor power: 194 kW
• Peak motor torque: 440 N·m
• Vehicle mass: 1760 kg
• Existing single-speed final drive ratio benchmark: 9.04:1
• Approximate tire diameter: 0.669 m
• Drag coefficient: 0.219
• Frontal area: 2.22 m² (reasonable approximation from vehicle dimensions)
• Air density: 1.225 kg/m³
• Rolling resistance coefficient: 0.010
• Tire-road friction coefficient for hard launch estimate: 1.0

These are design-study values, not production-certified Tesla internal values. The spreadsheet should expose them as editable assumptions.

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Step 1 — Single-stage spur gear and pinion design

Goal

Design one spur gear–pinion pair that meets a required speed reduction, transmits the required power and torque safely, and has the smallest packaging size.

What the spreadsheet must accept as user inputs

At minimum, include these inputs on a sheet called Inputs:

Powertrain and duty inputs

• Input power, horsepower or kW
• Input speed, rpm
• Output speed, rpm
• Or gear ratio as an alternative input
• Design life, hours or cycles
• Desired reliability

Gear-design inputs

• Pressure angle, default 20°
• Gear type: standard full-depth involute spur gears
• Minimum acceptable bending factor of safety
• Minimum acceptable contact factor of safety
• Preferred gear material options and heat treatment options
• Gear quality number to be used in the dynamic factor model
• Target face-width range: 3p to 5p, where p is the circular pitch.

Optional but useful inputs

• User-selected minimum pinion tooth count
• User-selected diametral pitch list or module list to search
• User-selected material library entry
• User-selected efficiency assumption

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Required outputs for the single-stage gear design tool

Create a sheet called SingleStage_Results that reports:

• Selected pinion teeth, N_p
• Selected gear teeth, N_g
• Actual ratio, i = N_g / N_p
• Diametral pitch P_d or module m
• Pitch diameters
• Base diameter, addendum, dedendum, whole depth
• Center distance
• Face width
• Pitch-line velocity
• Transmitted tangential force W_t
• Radial force W_r
• Bending stress for pinion and gear
• Contact stress
• Bending safety factor for pinion and gear
• Contact safety factor
• Predicted efficiency
• Pass/fail flag for packaging, stress, and safety-factor constraints

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Detailed Shigley-based design workflow

The spreadsheet should follow this sequence.

1. Determine the target ratio

If the user provides input and output speed,

i_target = n_in / n_out

If the user provides one speed and the ratio, compute the missing speed.

2. Compute the transmitted torque

For shaft torque,

T_in = 9550 * P_kW / n_in      (N·m)

or in U.S. customary form,

T_in = 63025 * HP / rpm        (lb·in)

Then compute the tangential tooth load once a tentative pitch diameter is selected:

W_t = 2 * T_in / d_p

3. Choose an initial pinion tooth count

Use a practical starting range such as:

• N_p = 17 to 28 for 20° full-depth involute gears

Then compute:

N_g = round(i_target * N_p)
i_actual = N_g / N_p

Keep the actual ratio close to the target ratio.

4. Search feasible diametral pitch or module values

For each candidate P_d or m, compute:

d_p = N_p / P_d
d_g = N_g / P_d
C = (d_p + d_g)/2

or metric equivalent:

d_p = m * N_p
d_g = m * N_g
C = (d_p + d_g)/2

Reject any design exceeding package or center-distance limits.

5. Estimate pitch-line velocity

V = pi * d_p * n_p / 12        (ft/min, if d_p in inches)

or metric equivalent.

6. Compute AGMA/Shigley stress quantities

Use Shigley's spur gear framework and calculate the standard modifiers.

Bending stress

sigma_b = W_t * K_o * K_v * K_s * P_d * K_m * K_B / (F * J)

Contact stress

sigma_c = C_p * sqrt((W_t * K_o * K_v * K_s * K_m) / (F * I * d_p))

The spreadsheet should include the following factors explicitly:

• K_o overload factor
• K_v dynamic factor
• K_s size factor
• K_m load-distribution factor
• K_B rim-thickness factor
• J bending geometry factor
• I pitting resistance geometry factor
• C_p elastic coefficient

For a course spreadsheet, it is acceptable to implement J and I using:

• table lookup,
• or user-entered values if exact AGMA geometry tables are unavailable.

The spreadsheet must clearly label any factor that is approximate or user-supplied.

7. Compute allowable stresses and safety factors

From material and hardness selection, determine:

• allowable bending stress number
• allowable contact stress number
• life factor
• temperature factor
• reliability factor

Then compute design safety factors:

n_b = sigma_allow_b / sigma_b
n_c = sigma_allow_c / sigma_c

8. Iterate until a feasible design is found

Iterate over:

• pinion tooth count
• diametral pitch or module
• face width
• material / heat treatment

until all of the following are satisfied:

• ratio error is acceptable
• center distance is acceptable
• bending safety factors exceed targets
• contact safety factor exceeds target
• face width is practical
• pinion tooth count avoids undercut and poor manufacturability

9. Rank feasible designs

The spreadsheet should rank feasible solutions by center distance.

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Recommended spreadsheet structure for Step 1

Use these tabs:

Inputs

All user-entered quantities, units, dropdowns, and design targets.

Materials

Material library containing:

• material name
• heat treatment
• Brinell hardness or surface hardness
• allowable bending stress number
• allowable contact stress number
• elastic modulus / Poisson ratio if needed
• density
• cost multiplier, optional

AGMA_Factors

Lookup tables or user-entry areas for:

• K_o
• K_v
• K_s
• K_m
• K_B
• J
• I
• life factors
• reliability factors
• temperature factors

SingleStage_Search

Candidate-design search table with one row per design candidate.

SingleStage_Results

Clean presentation of the chosen design.

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Step 2 — Shiftable two-speed gearbox optimized for fastest 0–60 mph

Goal

Replace the single-stage reduction with a shiftable two-speed gearbox. The spreadsheet must choose:

• a high gear ratio G_high for launch,
• a low gear ratio G_low for acceleration after the shift,
• and the shift speed (or shift motor speed)

so that the vehicle reaches 60 mph in the shortest possible time.

After the optimal ratios are chosen, design the actual gear pairs for:

• First gear
• Second gear

using the same Shigley-based spur gear workflow from Step 1.

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Vehicle longitudinal model to use

Create the sheet called Acceleration_Model and compute acceleration from:

F_net = F_drive - F_drag - F_roll

where

F_drag = 0.5 * rho * C_d * A * v^2
F_roll = C_rr * m_vehicle * g

and the drive force is limited by both motor capability and traction:

F_drive = min(F_motor_wheels, F_traction_limit)

Traction limit

F_traction_limit = mu * m_vehicle * g

Wheel and motor speed relationship

For the currently engaged gear ratio G,

omega_wheel = v / r_w
omega_motor = G * omega_wheel

Simple motor model

Use a Tesla-inspired piecewise motor model:

• constant torque up to base speed
• constant power above base speed

omega_base = P_peak / T_peak

Then:

T_motor = T_peak                      if omega_motor <= omega_base
T_motor = P_peak / omega_motor        if omega_motor > omega_base

Wheel force in the currently engaged gear:

F_motor_wheels = eta_G * G * T_motor / r_w

Acceleration:

a = F_net / m_vehicle

Integrate forward in time until v = 60 mph.

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Shifted 0–60 simulation

The spreadsheet must evaluate a two-segment acceleration problem.

Segment 1 — First gear

Accelerate from rest using G_high until the chosen shift point is reached.

Shift event

At the shift point:

• vehicle speed is unchanged during the shift,
• motor speed drops according to the ratio change.

Segment 2 — Second gear

Continue acceleration with G_low until 60 mph is reached.

Total time is:

t_0_60 = t_first_gear + t_second_gear

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Optimization variables for Step 2

The spreadsheet should vary:

• G_high = first gear ratio
• G_low = second gear ratio
• v_shift or omega_shift = shift point

with the constraints:

G_high > G_low

and both must be practical for a two-speed EV gearbox.

Practical design preferences

• First gear should maximize launch force without causing excessive traction saturation far beyond what the tires can use.
• Second gear should keep the motor in a favorable region after the shift and still allow reaching 60 mph without another shift.
• The shift should occur near the speed where staying in first gear becomes less beneficial than shifting to second gear.

A very useful rule in the spreadsheet is to compare wheel force in each gear at the same vehicle speed. The force-optimal shift is near the speed where:

F_wheel,1(v) ≈ F_wheel,2(v).

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Recommended computational method

Create a sheet called TwoSpeed_Search. Each row should contain one candidate design with:

• G_high
• G_low
• v_shift
• predicted t_0_60
• peak motor speed in first gear by 60 mph
• post-shift motor speed
• traction-limited flag
• feasibility flag

A simple search strategy is:

1. Loop G_high over a practical range.
2. Loop G_low over a practical range with G_high > G_low.
3. For each ratio pair, loop over shift speed or shift motor speed.
4. Run the two-segment acceleration simulation.
5. Save the shortest 0–60 time among feasible cases.

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Suggested default ratios for the spreadsheet

Use these as default initial guesses, then let the spreadsheet optimize around them:

• G_high = 14.0
• G_low = 8.0

The actual best values should be determined by the spreadsheet search, not hard-coded.

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How to choose actual tooth counts for the two-speed gearbox

Once the optimizer identifies G_high and G_low, choose integer tooth counts for each selectable gear pair.

First gear

G_high = N_g1 / N_p1

Second gear

G_low = N_g2 / N_p2

Use practical pinion tooth counts and round gear tooth counts to match the target ratios while still respecting:

• undercut avoidance
• center-distance limits
• bending and contact stress limits
• manufacturability

Example near-target combinations

If the optimizer lands near:

G_low  ≈ 14.0
G_high ≈ 8.0

then possible starter tooth-count choices could be:

• First gear: N_p1 = 18, N_g1 = 252 → G_low = 14.00
• Second gear: N_p2 = 20, N_g2 = 160 → G_high = 8.00

Those exact values may be too large for packaging in a direct single-pair implementation. Therefore, the spreadsheet should treat them only as ratio examples, not final geometry. In practice, if package size becomes excessive, you should either:

• allow the user to choose a larger pinion and smaller ratio compromise,
• add a separate fixed final drive outside the shiftable gearbox,
• or explicitly note that a one-pair-per-gear architecture may not fit the package and that a compound transmission architecture would be more realistic.

For this assignment, the important point is that the spreadsheet converts optimized ratios into actual integer tooth counts and then checks feasibility.

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How to run the gear design after optimization

After the acceleration model chooses the optimal two-speed gearbox parameters, do the following:

1. Select the best G_high, G_low, and v_shift.
2. Design first gear as a spur gear pair using the Step 1 method.
3. Design second gear as a second spur gear pair using the Step 1 method.
4. Evaluate each gear pair at the torque and speed it sees in service.
5. Report both gear designs and the overall gearbox performance.

Gear-specific loading cases

First gear loading case

First gear should be checked at the most severe launch condition, typically near peak motor torque and low vehicle speed:

n_1,in = n_motor during launch
T_1,in = T_motor during launch

Because first gear multiplies torque the most, it is often the critical strength case.

Second gear loading case

Second gear should be checked at the worst torque-speed combination it experiences after the shift:

n_2,in = motor speed immediately after shift and through the rest of the run
T_2,in = corresponding motor torque

Second gear usually carries lower multiplication than first gear, but it may still be critical if it operates at higher pitch-line velocity or if packaging drives smaller teeth.

Efficiency handling

Use per-gear efficiency assumptions such as:

eta_1 = 0.97
eta_2 = 0.97

Then the active wheel force uses the efficiency of the currently engaged gear only.

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Required outputs for the two-speed tool

Create a sheet called TwoStage_Results or rename it to TwoSpeed_Results. It should report:

• optimized first gear ratio for minimum 0–60 time
• optimized second gear ratio for minimum 0–60 time
• optimal shift speed or motor shift speed
• assumed shift delay
• estimated 0–60 time for the original single-speed baseline
• estimated 0–60 time for the optimized two-speed gearbox
• first-gear tooth counts, pitch diameters, center distance, face width, stresses, safety factors
• second-gear tooth counts, pitch diameters, center distance, face width, stresses, safety factors
• post-shift motor speed
• peak motor speed reached by 60 mph
• warning if either gear pair fails stress or package limits

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Recommended spreadsheet structure for Step 2

The following tabs are recommended:

Acceleration_Model

Contains the vehicle model, drag, rolling resistance, traction limit, motor model, and time integration.

TwoSpeed_Search

Contains candidate combinations of G_low, G_high, and v_shift with predicted 0–60 times.

TwoSpeed_Results

Clean presentation of the selected ratio pair, shift point, performance comparison, and actual gear design summary.

If you want to keep the old tab name TwoStage_Results for consistency, clearly label it as a two-speed shiftable gearbox result sheet, not a fixed two-stage reduction sheet.

Deliverable requirements

The final spreadsheet should contain at least these sheets:

1. Inputs
2. Materials
3. AGMA_Factors
4. SingleStage_Search
5. SingleStage_Results
6. Acceleration_Model
7. TwoSpeed_Search
8. TwoSpeed_Results

Optional but recommended:

9. Charts
10. Units_and_Assumptions

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Minimum charts to include

Include plots for:

• motor torque vs motor speed
• motor power vs motor speed
• tractive force vs vehicle speed for several candidate total ratios
• net acceleration vs vehicle speed
• cumulative 0–60 time vs candidate total ratio
• bending and contact safety factors for the selected design