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Rack and pinion
Rack and pinion
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Animation of a rack and pinion

A rack and pinion is a type of linear actuator that comprises a circular gear (the pinion) engaging a linear gear (the rack).[1] Together, they convert between rotational motion and linear motion: rotating the pinion causes the rack to be driven in a line. Conversely, moving the rack linearly will cause the pinion to rotate.

The rack and pinion mechanism is used in rack railways, where the pinion mounted on a locomotive or a railroad car engages a rack usually placed between the rails, and helps to move the train up a steep gradient. It is also used in arbor presses and drill presses, where the pinion is connected to a lever and displaces a vertical rack (the ram). In pipelines and other industrial piping systems, a rack displaced by a linear actuator turns a pinion to open or close a valve. Stairlifts, lock gates, electric gates, and the mechanical steering mechanism of cars are other notable applications.

The term "rack and pinion" may be used also when the rack is not straight but arcuate (bent), namely just a section of a large gear.[2]

A single pinion can simultaneously drive two racks, parallel but opposite; which will always be displaced by the same distance, only in opposite directions. Conversely, by applying opposite forces to the two racks one can obtain pure torque on the pinion, without any force component. This double rack and pinion mechanism can be used, for example, with a pair of pneumatic actuators to operate a valve with minimum stress.[3]

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History

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The time and place of the invention of the rack-and-pinion mechanism are unknown, but it presumably was not long after the invention of gears. The south-pointing chariot from China and the Antikythera mechanism are evidence of these being well-known already a couple of centuries BC.[citation needed]

In 1598, firearms designer Zhao Shizhen developed the Xuanyuan arquebus (軒轅銃), featuring a rack-and-pinion matchlock mechanism derived from an Ottoman Turkish matchlock design.[4] The Wu Pei Chih (1621) later described Ottoman Turkish muskets that used a rack-and-pinion mechanism.[5][6]

The use of a variable rack (still using a normal pinion) was invented by Arthur Ernest Bishop[7] in the 1970s, so as to improve vehicle response and steering "feel", especially at high speeds. He also created a low cost press forging process to manufacture the racks, eliminating the need to machine the gear teeth.

Comparison with Worm gear

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A rack and pinion has roughly the same purpose as a worm gear with a rack replacing the gear, in that both convert torque to linear force. However the rack and pinion generally provides higher linear speed — since a full turn of the pinion displaces the rack by an amount equal to the pinion's pitch circle whereas a full rotation of the worm screw only displaces the rack by one tooth width. By the same token, a rack and pinion mechanism yields a smaller linear force than a worm gear, for the same input torque. Also, a rack and pinion pair can be used in the opposite way, to turn linear force into torque; whereas a worm drive can be used in only one way.[8]

Geometry

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The teeth of a rack and pinion pair may be either straight (parallel to the rotation axis, as in a spur gear) or helical. On the pinion, the profile of the working tooth surfaces is usually an arc of involute, as in most gears. On the rack, on the other hand, the matching working surfaces are flat. One may interpret them as involute tooth faces for a gear with infinite radius. In both parts the teeth are typically formed with a gear cutter (a hob).[1]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia
from Grokipedia
A rack and pinion is a fundamental mechanical system consisting of a linear gear bar, known as the rack, with straight teeth along one edge, and a small circular gear, called the , whose teeth mesh with those of the rack to convert rotational motion into or vice versa. This mechanism operates on the principle of gear meshing, where rotating the causes the rack to move linearly along its length, providing precise control over position and speed based on the gear ratio and input . Rack and pinion systems are widely applied in automotive , where the pinion connects to the and the rack to the front wheels, enabling responsive directional control in vehicles from compact cars to heavy machinery. Beyond steering, they serve as linear actuators in industrial automation, machine tools for precise positioning, and even in wave energy converters to transform oscillatory motion into rotational energy. The rack and pinion concept traces its origins to early gear innovations, with foundational principles in gear systems attributed to ancient engineers like , though practical implementations evolved over centuries in clocks, mills, and lifting devices. In the modern era, the rack and pinion steering mechanism was patented by Gustave Dumont in 1922, marking a significant advancement for automotive applications due to its compact design and efficiency. It gained prominence in production vehicles with the Citroën Traction Avant in 1936, and was featured in the Citroën 2CV starting in 1948, with subsequent innovations, such as variable-ratio designs by Arthur Ernest Bishop in the 1970s, further enhancing steering precision and safety.

Principles of Operation

Basic Mechanism

A is a type of that consists of a circular gear that meshes with a linear rack gear, enabling the conversion of rotational motion into or vice versa. The is typically a small cylindrical gear with teeth around its circumference, while the rack is a straight bar featuring teeth along one edge, functioning as a gear with an infinite radius. This setup forms a simple gear pair where the meshing teeth ensure synchronized movement between the two components. In operation, the teeth of the rotating engage with the corresponding teeth on the rack, causing the rack to move linearly along its length as the pinion turns. This engagement allows the pinion to push or pull the rack depending on the direction of , with the teeth distributing the load across multiple contact points for smooth transfer. Force transmission occurs through the contact between the pinion's teeth and the rack's teeth, where applied on the pinion generates a linear on the rack, proportional to the gear's . The mechanism supports two primary types of motion conversion: rotary-to-linear and linear-to-rotary. These conversions are bidirectional, allowing the system to function in either direction based on the input. Rack and pinion systems offer basic advantages including direct and efficient power transfer due to their straightforward design, which minimizes energy loss, and the potential for minimal backlash when properly manufactured and assembled. Their simplicity also contributes to high , typically ranging from 98% to 99.5%, making them suitable for applications requiring precise linear control.

Kinematics and Dynamics

The kinematics of a rack and pinion system describe the relationship between the rotational motion of the and the of the rack. The vv of the rack is directly proportional to the ω\omega of the , given by the equation v=rωv = r \omega, where rr is the pitch radius of the . This relation arises because the point of contact between the rack teeth and moves at the same tangential speed as the rack's . Similarly, the linear displacement ss of the rack corresponds to the θ\theta of the via s=rθs = r \theta, with θ\theta measured in radians. These equations assume ideal meshing without slippage, providing a foundational understanding of motion conversion in the system. In dynamic analysis, the forces transmitted through the rack and pinion determine the system's ability to handle loads and . The tangential FtF_t applied to the rack equals the TT on the pinion divided by the pinion's pitch , expressed as Ft=T/rF_t = T / r. At the contact point between teeth, additional forces include the normal , which acts to the tooth surface and supports the load without contributing to motion, and frictional forces that oppose sliding along the contact line. at this point arises from the relative sliding during meshing, influencing dissipation and requiring to minimize . Backlash in a rack and pinion refers to the clearance or play between the mating of the rack and pinion when no load is applied, allowing slight relative movement. This play primarily results from tolerances, such as variations in thickness or spacing during cutting or grinding processes, which prevent perfect fit without excessive . Backlash adversely impacts precision by introducing lost motion in reversible systems, leading to inaccuracies in positioning and potential vibrations under cyclic loading. Rack and pinion systems typically achieve high , up to 97%, owing to the predominantly rolling contact between teeth that minimizes sliding losses. losses primarily stem from at the tooth contact points and any sliding components, though proper and precise manufacturing can maintain values near 97% in well-designed setups. These factors make rack and pinion mechanisms suitable for applications requiring reliable with low energy dissipation.

Design and Geometry

Gear Profiles and Meshing

The standard tooth profile for rack and pinion systems employs curves on the gear, which with straight-sided teeth on the rack to ensure a constant during operation. This conjugate profile design allows the pinion's curved flanks to roll smoothly against the rack's linear teeth, maintaining uniform motion without slippage. The shape on the pinion is generated from the unwinding of a around a base circle, providing inherent advantages in manufacturability and tolerance to center distance variations. The , typically 20° in modern designs but occasionally 14.5° or 17.5° for specific applications, defines the inclination of the flanks and the direction of force transmission during meshing. This angle determines the orientation of the , which is the path along which the contact point travels and is to the rack's straight flanks while to the pinion's base circle. The can be expressed geometrically as the common normal to the profiles at the contact point, inclined at the α\alpha relative to the at the pitch point, ensuring the force vector aligns properly for efficient power transfer. Meshing in a rack and pinion occurs through point contact that progresses along the pitch line, with the and straight profiles generating conjugate action to produce smooth, continuous from the pinion's . At any instant, the contact point lies on the , and the common normal to both profiles passes through the fixed pitch point, guaranteeing a constant speed regardless of tooth position. This point contact minimizes sliding friction compared to line contact in other gear types, though it requires precise alignment to avoid undercutting or interference. The rack's tooth geometry features straight flanks inclined at the , with and dedendum heights standardized to match the pinion's module mm, typically set as ha=mh_a = m and dedendum hf=1.25mh_f = 1.25m to provide clearance and strength. These dimensions ensure full-depth teeth that align perfectly with the pinion's profile, promoting even load distribution across the contact line. The module mm serves as the fundamental scaling parameter, defining the pitch and tooth proportions for interchangeable components. Special profiles, such as cycloidal teeth, can be used on the to with modified rack flanks, offering advantages like reduced and due to broader contact areas and smoother paths compared to systems. Cycloidal profiles achieve lower operating in high-speed applications by minimizing impact during entry and exit. Additionally, helical racks with angled teeth provide reduced backlash through gradual multi-tooth and axial thrust that preloads the , enhancing precision in positioning tasks without split gearing. These variants maintain the core conjugate principles but are selected for environments demanding quieter or more accurate operation.

Sizing and Performance Calculations

Sizing rack and pinion systems begins with the selection of the module mm, a fundamental metric parameter in gear design that defines the size scale. The circular pitch pp, which is the distance along the pitch circle between corresponding points on adjacent teeth, is calculated as p=πmp = \pi m. Standard module values typically range from 1 mm to 10 mm for most industrial applications, allowing designers to match the system's linear travel requirements and load capacities to available manufacturing capabilities. The pinion's pitch diameter dd is determined by the module and the number of teeth zz, given by d=mzd = m z. To prevent undercutting—where the gear cutter removes material from the root during —the minimum number of pinion teeth is typically 18 for a standard 20° full-depth profile when meshing with a rack. Performance calculations for load capacity focus on the bending strength of the teeth, primarily evaluated using the Lewis formula for the , as the rack's straight teeth experience distributed loading. The bending stress σ\sigma is computed as σ=Ftbmy\sigma = \frac{F_t}{b m y}, where FtF_t is the tangential load, bb is the face width, and yy is the Lewis form factor dependent on tooth geometry and . This stress must remain below the material's allowable bending strength to avoid tooth failure under static or dynamic loads. For and resistance, particularly pitting on the surfaces, the Hertzian contact stress is assessed to predict surface . The maximum contact stress σH\sigma_H for a rack and is derived from Hertzian theory for line contact between the curved and the straight rack flank, approximated as σH=FtEπbr\sigma_H = \sqrt{\frac{F_t E^*}{\pi b r}}
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