Rack and Pinion Travel, Speed, and Force

Rack-and-pinion travel and force - the rotary-to-linear drive on a CNC gantry, sliding gate, steering rack, or linear actuator. Pinion pitch diameter PD = N/Pd; the rack advances one pitch circumference per turn, so travel per revolution = pi PD and per tooth = pi/Pd; linear speed = pi PD x rpm; rack push force F = 2 T/PD (torque over the pitch radius). A 20-tooth, 10-pitch pinion (PD 2 in) at 100 in-lbf and 500 rpm moves the rack 6.28 in/rev at 262 ft/min and pushes 100 lbf; double the pinion to 40 teeth and it moves twice as fast but pushes half as hard - speed for force. Tooth stress, backlash, mesh efficiency, and inertia are separate. A design aid; Machinery Handbook and the drive maker govern.

PD = N/Pd; travel per rev = pi PD; travel per tooth = pi/Pd; linear speed = pi PD x rpm; rack force F = 2 T/PD.

The rack-and-pinion kinematics (linear travel = pi x pitch diameter per revolution) and the tangential force F = 2 T/PD (Machinery's Handbook), by name.

The rack-and-pinion travel and force relations are standard published gear-drive results; the tooth count, diametral pitch, torque, and speed are the user's inputs.

Estimate. AHJ and licensed professional govern.

Field names used by the API: pinion_teeth, diametral_pitch_1_in, pinion_torque_in_lb, pinion_rpm, pitch_diameter_in, travel_per_rev_in, rack_force_lbf

• Kinematics travel per rev = pi PD, per tooth = pi/Pd; linear speed = pi PD rpmMachinery's Handbook
• Force F = 2 T/PD, tangential push from pinion torqueMachinery's Handbook
• Scope ideal kinematics/static force; tooth stress, backlash, inertia are separatescope of this tile