Linkage (mechanical)

Mechanical linkages are a series of rigid links connected with joints to form a closed chain, or a series of closed chains. Each link will have two or more joints, and the joints will have various degrees of freedom to allow motion between the links. A linkage is called a mechanism if two or more links are movable with respect to a fixed link. Mechanical linkages are usually designed to take an input and produce a different output, altering the motion, velocity, acceleration, and applying mechanical advantage.

A linkage designed for no motion is called a structure or truss. See this article for more information.

History

Mechanical linkages are a fundamental part of machine design, and yet many simple linkages were not well understood, or invented until the 19^th century. Consider a stick; it has six degrees of freedom, it can be moved any which way. Once nudged between a boulder and fulcrum it is constrained to a particular motion, to act as a lever to move the boulder. When more links are added and joined in varius ways their collective motion can be further defined. Very complicated and precise motions can be designed into a linkage with only a few parts.

The industrial revolution was the golden age of mechanical linkages. Mathematical, engineering and manufacturing advances provided both the need and the ability to create new mechanisms. Many simple mechanisms that seem obvious today required some of the greatest minds of the era to create. Leonard Euler was one of the first mathematicians to study linkage synthesis, James Watt worked very hard to invent the Watt linkage to support his steam engine's piston. Chebyshev worked on mechanical linkage design for over thirty years, which led to his work on polynomials². New linkage inventions, designed by need, were instrumental in cloth making, power conversion and speed regulation. Even the ability of a mechanism to produce accurate linear motion, without a reference guide way, took years to solve.

Electronic technology has replaced many linkage applications taken for granted today, such as mechanical computation, typewriting and machining. Modern linkage design continues to advance, and designs that used to occupy an engineer for days are now optimized with a computer in seconds.

Theory

The most common linkages will have one degree of freedom, meaning that there is one input motion that produces one output motion. Most linkages are also planar, meaning all the motion takes place in one plane. Spatial linkages (non-planar) are more difficult to design and therefore not as common.

Gruebler's equation is used to calculate the degrees of freedom of planar, closed linkages. Degrees of freedom of a linkage is also called mobility;

= mobility = degrees of freedom

= number of links (including a single ground link)

= number of one degree of freedom joints (pin or slider joints)

The mobility of hydraulic machinery can easily be identified by counting the number of independently controlled hydraulic cylinders.

Types of common joints:

• Pin, one DOF rotation. Examples are; bushings, bearings, bolted joints, rivets and hinges.
• Slider, one or two DOF linear motion. Examples are; linear bearings, hydraulic cylinders, rollers and pistons.
• Ball and socket, three DOF rotation, usually restricted to one DOF by other joints in the mechanism.

Designers will synthesize a linkage by starting with the required output motion, mechanical advantage, velocity and acceleration. A type of linkage is chosen and modified to deliver the required performance.

Each link is treated as a vector and the vectors can be combined into a system of equations because they form a loop. The matrix is solved to create a closed form equation that relates input motion to output motion. The same is done for mechanical advantage, or any other important quantity. The equations of motion are differentiated with respect to time to find velocity and acceleration of the mechanism parts.

Types of linkages

Four bar linkages are the simplest closed loop kinematic linkage. They perform a wide variety of motions with a few simple parts. They were also popular in the past due to the ease of calculations, prior to computers, compared to more complicated mechanisms.

Grashof's law is applied to pinned linkages and states; The sum of the shortest and longest link of a planar four bar linkage cannot be greater than the sum of remaining two linkages if there is to be continuous relative motion between the links. Below are the possible types of pinned, four-bar linkages;

Other notable types of linkages;

• Pantograph (four-bar, two DOF)
• Crank-slider, (four bar, one DOF)
• Five bar linkages often have meshing gears for two of the links, creating a one DOF linkage. They can provide greater power transmission with more design flexibility than four bar linkages.
• Six bar, single DOF linkages offer greater design flexibility than four bar linkages, but require more parts and are more difficult to design. ³ ;
  • Watt kinematic chain
  • Watt I, II
  • Stephenson kinematic chain
  • Stephenson I, II, III
• Peaucellier-Lipkin linkage, the first linkage to create a straight line output from rotary input; eight-bar, one DOF.

Uses

Linkages are primarily used as machine components and tools. Typical examples are automotive suspensions and bolt cutters. The internal combustion engine's piston/rod/crank is a classic four-bar linkage with one degree of freedom. Linkages are often the simplest, least expensive and most efficient mechanism to perform complicated motions.

One highly visible application is the windshield wiper; a four bar linkage changes the motor's rotary motion to oscillation, some wipers also have a second set of four bar linkages keep the wiper blades oriented correctly as they sweep. Another visible application is heavy equipment which makes extensive use of four and six bar linkages.

Spatial linkages are becoming more common due to computer aided design.

References

1. Erdman, Arthur G.; Sandor, George N. (1984). Mechanism Design: Analysis and Synthesis. Prentice-Hall. ISBN 0-13-572396-5.
2. How to Draw a Straight Line, historical discussion of linkage design
3. What is a Watt I Linkage?

See also

• Engineering mechanics
• Machine
• Three-point hitch
• Lever
• Parallel motion

External links

• Linkage calculations
• Java animated linkages
• Gif animated linkages
• MIT Open Course ware, Matlab code for four bar linkages
• MIT Open Course ware, linkage spreadsheets
• Introductory Linkage Lecture, 2MB PDF

See also: Linkage (mechanical), Bolt cutters, Boulder, Centrifugal governor, Chebyshev polynomials, Computer aided design, Degrees of freedom, Derivative, Engineering mechanics