Mechanical Calculator

Mechanical Calculator

This mechanical calculator computes gear ratios, output speed, and torque multiplication for gear systems. Enter the number of teeth on the driving and driven gears, your input speed in RPM, and the input torque to instantly see gear ratio, output speed, output torque, and how power is conserved across the gear pair.

How Gear Ratio Is Calculated

The fundamental gear ratio formula is:

Gear Ratio = Driven Teeth / Driving Teeth

From that single ratio, the calculator derives all other output values:

• Output Speed (RPM) = Input Speed / Gear Ratio
• Output Torque (N-m) = Input Torque x Gear Ratio x Efficiency
• Power (W) = Torque (N-m) x Angular Velocity (rad/s), where Angular Velocity = RPM x (2 x pi / 60)

A gear ratio above 1:1 is a speed reduction — the output shaft spins slower but delivers more torque. A ratio below 1:1 is a speed increase — the output spins faster but with less torque. In an ideal frictionless gear pair, power is conserved exactly. Real gear meshes lose roughly 1-3% per stage through friction and heat, so a two-stage gearbox runs at approximately 94-98% overall efficiency.

Worked Examples

Scenario 1 — Motor driving a conveyor at reduced speed

• Driving gear: 18 teeth, Driven gear: 72 teeth, Input: 1,800 RPM, 10 N-m
• Gear Ratio = 72 / 18 = 4:1
• Output Speed = 1,800 / 4 = 450 RPM
• Output Torque = 10 x 4 x 0.97 = 38.8 N-m (assuming 97% efficiency)
• Result: 450 RPM at 38.8 N-m

Scenario 2 — Speed increaser for a generator

• Driving gear: 60 teeth, Driven gear: 20 teeth, Input: 300 RPM, 50 N-m
• Gear Ratio = 20 / 60 = 0.333 (speed increase)
• Output Speed = 300 / 0.333 = 900 RPM
• Output Torque = 50 x 0.333 x 0.97 = 16.2 N-m
• Result: 900 RPM at 16.2 N-m — useful for small wind turbine generators

Scenario 3 — Two-stage compound gearbox

• Stage 1: 15T driving, 45T driven = 3:1; Stage 2: 20T driving, 60T driven = 3:1
• Total ratio = 3 x 3 = 9:1
• Input: 2,700 RPM, 5 N-m; Output: 300 RPM, 5 x 9 x 0.94 = 42.3 N-m (two stages at 97% each)
• Result: 300 RPM at 42.3 N-m — achieves 9:1 reduction with manageable individual gear sizes

Gear Ratio Reference Table

When to Use This Calculator

• Matching a standard-speed motor (typically 1,750 RPM) to a slow, high-torque output shaft requirement
• Verifying that a compound gearbox meets the output speed specification before ordering
• Checking whether a speed-increaser gear set can bring a low-speed input (wind, water) up to generator speed
• Comparing single-stage versus multi-stage ratios for stress and efficiency trade-offs
• Converting between RPM and rad/s for power calculations in motor sizing problems

Common Mistakes

1. Confusing driving and driven gears. Gear ratio = driven / driving. If you flip the values, you get the inverse ratio — a 3:1 reduction becomes a 1:3 speed increase, and your output speed will be off by a factor of 9.
2. Ignoring efficiency losses when sizing motors. A 3:1 gear at 97% efficiency delivers 29.1 N-m from a 10 N-m input, not 30 N-m. In multi-stage gearboxes the compounding effect matters: three stages at 97% = 91.3% overall, not 97%.
3. Treating compound ratios as addition instead of multiplication. A 3:1 stage followed by a 4:1 stage gives 12:1 total — not 7:1. Always multiply individual stage ratios together.
4. Forgetting angular velocity units in power checks. Torque in N-m times RPM does not equal watts directly. Convert first: Power (W) = Torque (N-m) x RPM x pi / 30.

Real-World Applications

Gear ratios appear across virtually every sector of mechanical engineering. Automotive transmissions use ratios ranging from about 4:1 in first gear (for torque when accelerating) down to 0.7:1 in overdrive (for highway fuel economy). Industrial gearboxes on conveyor belts and mixers typically operate in the 10:1 to 30:1 range to bring 1,750 RPM motors down to the 60-180 RPM range that bulk material handling requires. Bicycle drivetrains use variable ratios from roughly 0.7:1 (large chainring, small sprocket — fast cadence) to about 4:1 (small chainring, large sprocket — climbing). Robot joint actuators often use planetary gearheads with ratios of 50:1 to 200:1, converting high-speed brushless motor torque into the slow, powerful joint rotation needed for precise arm movements. Clockmakers use very high compound ratios — a typical wall clock moves its second hand once per revolution of a gear that turns 3,600 times faster.

Tips

1. For loads requiring more than 10:1 reduction, use two or three gear stages rather than one enormous ratio — it keeps individual gears smaller, cheaper, and more efficient
2. Keep output speed within the driven equipment’s specified RPM range — running a pump at 20% above its rated speed shortens bearing life significantly
3. When selecting gear ratios for motors, aim for the motor to run near its rated RPM for best power and efficiency rather than throttling it back
4. Add 5-10% to your required output torque before selecting a gearbox to leave headroom for startup loads, which can be 2-3x steady-state torque on inertial loads
5. For back-drivability — where the output shaft can turn the input — avoid worm gears with ratios above 30:1 since the helix angle makes them self-locking
6. Always verify that your selected gear ratio produces an output speed your driven equipment can safely handle before committing to a gearbox purchase