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in mathematics, a fundamental concept of combinatorial topology. A chain is a generalization, to the case of n dimensions, of the concept of an oriented polygonal line.
a series of identical segments (links) joined together and movable relative to one another. It is known that chains were used for several millennia before the Common Era; for example, chains were used in simple mechanisms in the third century B.C. in Greece. In the first century B.C. the Roman architect Vitruvius described a water pump that used buckets attached to an endless pintle chain. Leonardo da Vinci left drawings of chains with link plates of approximately the same shape as those of modern chains. I. I. Polzunov developed a pintle chain for driving machines that was similar to the chain patented 65 years later (in 1829) by the French engraver A. Galle.
The chains used in technology may be classified according to their purpose in the following categories: drive chains (in drives for machines, used to transmit motion from a drive shaft to one or more driven shafts), hauling chains (in conveyors, elevators, escalators, paternosters, and similar machines in which the working members that directly move loads and people are attached to the chains), hoisting chains (in load-lifting machines and water-engineering works, used to suspend, raise, and lower loads), anchor chains (on ships, used to tie an anchor to the ship), chains used to bind groups of logs into rafts, antiskid chains (used to increase the grip of motor vehicle wheels on icy roads and in similar slippery conditions), saw chains (for motor-driven saws used in the lumber industry), and holing and cutting chains (used in machines for extracting and processing minerals and for digging trenches).
Chains are also classified according to the method used in fabricating the links. The categories include ordinary round-link chains with circular cross sections, which may be welded or cast; pintle chains with link plates connected by pins or bushings; open-hook chains having forged or cast links with hooks at one end; detachable chains with hot-forged or cold-forged links; and block chains with links in the form of thick bars.
Round-link chains are used for hauling, lifting, attaching anchors, and tying logs and as antiskid chains. Pintle chains are the most common type. Those used in hoists that operate only occasionally and for short periods at velocities up to 0.25 m per second and in slow-moving conveyors for removing manure or distributing feed are made with simple articulated chains with twin-plate or multiplate links. In order to reduce unit loads on the joints and joint wear, pintle chains used for driving and hauling are equipped with bushings. Rollers may be fitted over the bushings to reduce wear on the bushings and sprocket teeth. Such single-and multistrand roller chains represent the most developed designs and have been used extensively. For handling large dynamic loads that are applied in high-frequency cycles, as in the case of excavator mechanisms, the drive chains used have offset sidebars that reduce the dynamic load on the mechanisms and on the elements of the chain itself. In order to improve engagement with the sprockets, reduce noise, and increase reliability, inverted-tooth silent chains with multiplate links are supplied for high-power, high-speed drives. With the advent of precision, multistrand roller chains that are simpler and cheaper to fabricate, the use of inverted-tooth silent chains has declined and is now limited primarily to older types of machines.
In the 1970’s open-link twin-pin chains came to be used (together with round-link chains and detachable chains with open links) for operations in powdery, corrosive, and chemically-active media that cause a loss of mobility in the joints of roller chains and pintle chains equipped with bushings. The new design also differs from roller chains in having joint parts with better impact and fatigue resistance, in requiring less labor to produce, and in being lighter and cheaper. Open-hook and block chains are used for both driving and hauling. Detachable chains are widely used for hauling.
The ends of chains are connected by special links. In order to reduce wear on the joint parts or, in the case of round-link chains, on the chain as a whole, chains may be subjected to heat treatment or chemical case hardening; they may also be lubricated while in operation.
The primary geometric parameters of chains are the pitch (average length) and, for round-link chains, the gauge (cross-sectional diameter of the links). In order to achieve compact drives and mechanisms, the pitches of driving and hoisting chains are kept to a minimum (4–300 mm); hauling chains, which are very long, are designed with a large pitch (up to 1,400 mm), which permits substantial reductions in weight and cost.
The principal types of chains produced in the USSR are standardized, and their production is specialized.
REFERENCESVorob’ev, N. V. Tsepnye peredachi, 4th ed. Moscow, 1968.
Reshetov, D. N. Detail mashin, 3rd ed. Moscow, 1974.
I. I. IVASHKOV and A. A. PARKHOMENKO
Compare with the more modern "subshell".
x0 <= x1 <= x2 ...
A chain satisfies:
for all x,y in S, x <= y \/ y <= x.
I.e. any two elements of a chain are related.
("<=" is written in LaTeX as \sqsubseteq).
chain(1) See chain printer.
(2) An abbreviation of "blockchain." A major chain such as Bitcoin or Ethereum is known as the "main chain," "mainnet" or "Layer 1 network." Auxiliary chains that handle transactions that are eventually summarized onto the "main chain" are known as "sidechains" or "Layer 2 chains." See blockchain, sidechain, Layer 2 blockchain and mainnet.