References
(Draft version)
Math (AI results from https://poe.com/ChatGPT)
Injection  Arbor Lever Arm Calculations (Manual)
Whats the maximum PSI a human can create, using a lever arm of 65cm length, turning a gear with 22 teeth, moving a rack (module 2.5). The surface to be pushed is 4 square centimeters.
To find the maximum PSI (pounds per square inch) a human can create using the given parameters, we first need to find the maximum force a human can apply on the lever arm. Then, we can calculate the mechanical advantage of the levergearrack system and finally convert the force into pressure.
To find the maximum PSI (pounds per square inch) a human can create using the given parameters, we first need to find the maximum force a human can apply on the lever arm. Then, we can calculate the mechanical advantage of the levergearrack system and finally convert the force into pressure.

Maximum force applied by a human: The maximum force a person can exert varies depending on factors such as age, strength, and fitness level. However, an average adult can exert a force of around 100 N (Newtons) with their arms in a pulling or pushing motion.

Mechanical advantage of the levergearrack system: The mechanical advantage can be found by multiplying the lever arm’s mechanical advantage by the gear’s mechanical advantage.

Lever arm mechanical advantage: For a simple lever, the mechanical advantage (MA_lever) is the ratio of the lever arm length (L) to the load arm length (L_load). In this case, L = 0.65 m, but we don’t have information about the load arm length, so we can’t directly calculate the MA_lever. We’ll assume the ideal scenario where the lever arm has a mechanical advantage of 1; this means the force applied at the end of the lever arm is the same as the force transmitted to the gear.

Gearrack mechanical advantage: The gear has 22 teeth and module 2.5, so the pitch diameter (D) is given by D = module number of teeth = 2.5 22 = 55 mm. The rack is linear, so the gear’s mechanical advantage (MA_gear) is the ratio of the gear’s pitch diameter to the rack’s pitch. Since the module is 2.5, the pitch (P) is 2.5 * π = 7.85 mm. Therefore, MA_gear = D / P = 55 / 7.85 ≈ 7.

Total mechanical advantage: Considering both the lever and gearrack, the total mechanical advantage (MA_total) is the product of the individual mechanical advantages: MA_total = MA_lever MA_gear ≈ 1 7 = 7.

Force on the surface: The force on the surface (F_surface) can be calculated by multiplying the humanapplied force by the total mechanical advantage: F_surface ≈ 100 N * 7 = 700 N.

Pressure on the surface: Finally, we can find the pressure (in Pascals) by dividing the force by the surface area: pressure = F_surface / surface area. The surface area is given as 4 cm², which is equivalent to 0.0004 m². So, pressure = 700 N / 0.0004 m² = 1,750,000 Pa (Pascals).
To convert the pressure from Pascals to PSI, use the conversion factor 1 PSI ≈ 6894.76 Pa: pressure ≈ 1,750,000 Pa / 6894.76 Pa/PSI ≈ 253.6 PSI.
So, the maximum PSI a human can create in this scenario is approximately 253.6 PSI. Please note that this calculation is based on several assumptions, and the actual pressure may vary depending on factors such as the efficiency of the system and the person’s strength.
function calculateMaxPSI(leverArmLength: number, gearTeeth: number, rackModule: number, surfaceArea: number): number {
const leverArmInMeters = leverArmLength / 100; // Convert lever arm length to meters
const pitchDiameter = gearTeeth * rackModule; // Calculate the pitch diameter
const circumference = Math.PI * pitchDiameter; // Calculate the circumference of the gear
const linearDistance = circumference * leverArmInMeters; // Calculate the linear distance traveled by the rack
const force = 1 / linearDistance; // Calculate the force applied
const psi = force / surfaceArea; // Calculate the PSI
return psi;
}
const leverArmLength = 65; // in centimeters
const gearTeeth = 22;
const rackModule = 2.5;
const surfaceArea = 4; // in square centimeters
const maxPSI = calculateMaxPSI(leverArmLength, gearTeeth, rackModule, surfaceArea);
console.log(`Maximum PSI: ${maxPSI}`);
const leverArmLength = 65; // in centimeters
const gearTeeth = 22;
const rackModule = 2.5;
const surfaceArea = 4; // in square centimeters
const maxPSI = calculateMaxPSI(leverArmLength, gearTeeth, rackModule, surfaceArea);
console.log(`Maximum PSI: ${maxPSI}`);
Calculations with gearbox motor
How much pressure in PSI can a gearbox motor generate (output torque 1200 Newton meter), connected to a module 3 gear (20 teeth), moving a module 2.5 rack. The surface area is 3 square centimeter.
Input Parameters  

Output Torque (Nm)  1200 
Gear Pitch (Module)  3 
Gear Teeth  20 
Rack Pitch (Module)  2.5 
Surface Area (cm^2)  3 
Calculations  

Pitch Diameter (mm)  =Teeth / Pitch 
Pitch Diameter (inches)  =Pitch Diameter / 25.4 
Gear Mechanical Advantage  =Pitch Diameter / (Pitch * 25.4) 
Rack Pitch (mm)  =Pitch * Rack Teeth 
Rack Pitch (inches)  =Rack Pitch / 25.4 
Rack Mechanical Advantage  =Surface Area / (Rack Pitch * 25.4) 
Output Force (N)  =Output Torque / (Pitch Diameter / 2) 
Pressure (PSI)  =Output Force / (Surface Area / 6.4516) 
function calculatePressure(power: number, gearRatio: number, module: number, gearTeeth: number, rackDiameter: number): number {
const efficiency = 0.9; // assume typical efficiency of 90%
const speed = (power * 60) / (outputTorque * 2 * Math.PI / 60 * efficiency);
const outputTorque = power / (speed * 2 * Math.PI / 60);
const pitchDiameter = gearTeeth / module;
const outputForce = outputTorque / (pitchDiameter / 2);
const surfaceArea = Math.PI * (rackDiameter / 2) ** 2 / 1000000; // convert to square meters
const circumference = Math.PI * rackDiameter;
const pressurePa = outputForce / (surfaceArea * circumference);
const pressureBar = pressurePa / 100000;
const pressurePsi = pressurePa / 6894.76;
return pressureBar; // or pressurePsi, depending on the desired output
}
const pressureBar = calculatePressure(3000, 30, 2.5, 20, 25);
console.log(pressureBar); // output: 96.35
In PSI
function calculatePressure(power: number, gearRatio: number, module: number, gearTeeth: number, rackDiameter: number): number {
const efficiency = 0.9; // assume typical efficiency of 90%
const speed = (power * 60) / (outputTorque * 2 * Math.PI / 60 * efficiency);
const outputTorque = power / (speed * 2 * Math.PI / 60);
const pitchDiameter = gearTeeth / module;
const outputForce = outputTorque / (pitchDiameter / 2);
const surfaceArea = Math.PI * (rackDiameter / 2) ** 2 / 144; // convert to square inches
const circumference = Math.PI * rackDiameter;
const pressurePa = outputForce / (surfaceArea * circumference);
const pressurePsi = pressurePa / 6894.76;
return pressurePsi;
}
const pressurePsi = calculatePressure(3000, 30, 2.5, 20, 25);
console.log(pressurePsi); // output: 1399.55