A mechanical linkage is an assembly of bodies connected to manage forces and
movement. The movement of a body, or link, is studied using geometry so the
link is considered to be rigid. The connections between links are modeled as
providing ideal movement, pure rotation or sliding for example, and are called
joints. A linkage modeled as a network of rigid links and ideal joints is
called a kinematic chain. Linkages may be constructed from open
chains, closed chains, or a combination of open and closed chains. Each link in
a chain is connected by a joint to one or more other links. Thus, a kinematic
chain can be modeled as a graph in which the links are paths and the joints are
vertices, which is called a linkage graph.
The movement of an ideal joint is generally associated with a subgroup of
the group of Euclidean displacements. The number of parameters in the subgroup
is called the degrees of freedom of the joint. Mechanical linkages are usually
designed to transform a given input force and movement into a desired output
force and movement. The ratio of the output force to the input force is known
as the mechanical advantage of the linkage, while the ratio of the input
speed to the output speed is known as the speed ratio. The speed ratio and
mechanical advantage are defined so they yield the same number in an ideal
linkage. A kinematic chain, in which one link is
fixed or stationary, is called a mechanism, and a linkage designed to be
stationary is called a structure. Uses
Perhaps the simplest linkage is the lever, which is a link that pivots
around a fulcrum attached to ground, or a fixed point. As a force rotates the
lever, points far from the fulcrum have a greater velocity than points near the
fulcrum. Because power into the lever equals the power out, a small force
applied at a point far from the fulcrum equals a larger force applied at a point
near the fulcrum. The amount the force is amplified is called mechanical
advantage. This is the law of the lever. Two levers connected by a rod so that a
force applied to one is transmitted to the second is known as a four-bar
linkage. The levers are called cranks, and the fulcrums are called pivots. The
connecting rod is also called the coupler. The fourth bar in this assembly
is the ground, or frame, on which the cranks are mounted.
Linkages are important components of machines and tools. Examples range from
the four-bar linkage used to amplify force in a bolt cutter or to provide
independent suspension in an automobile, to complex linkage systems in robotic
arms and walking machines. The internal combustion engine uses a slider-crank
four-bar linkage formed from its piston, connecting rod, and crankshaft to
transform power from expanding burning gases into rotary power. Relatively
simple linkages are often used to perform complicated tasks.
Interesting examples of linkages include the windshield wiper, the bicycle
suspension, and hydraulic actuators for heavy equipment. In these examples the
components in the linkage move in parallel planes and are called "planar
linkages." A linkage with at least one link that moves in three-dimensional
space is called a "spatial linkage." The skeletons of robotic systems are
examples of spatial linkages. The geometric design of these systems relies
on modern computer aided design software.
The 4-bar linkage is an adapted mechanical linkage used on bicycles.
With a normal full-suspension bike the back wheel moves in a very tight arc
shape. This means that more power is lost when going uphill. With a bike
fitted with a 4-bar linkage, the wheel moves in such a large arc that it is
moving almost vertically. This way the power loss is reduced by up to 30%.
History Archimedes applied geometry to the study
of the lever. Into the 1500s the work of Archimedes and Hero of Alexandria were
the primary sources of machine theory. It was Leonardo da Vinci who brought an
inventive energy to machines and mechanism.
In the mid-1700s the steam engine was of growing importance, and James Watt
realized that efficiency could be increased by using different cylinders
for expansion and condensation of the steam. This drove his search for a
linkage that could transform rotation of a crank into a linear slide, and
resulted in his discovery of what is called Watt's linkage. This led to the
study of linkages that could generate straight lines, even if only
approximately; and inspired the mathematician J. J. Sylvester, who
lectured on the Peaucellier linkage, which generates an exact straight line
from a rotating crank. The work of Sylvester inspired A. B.
Kempe, who showed that linkages for addition and multiplication could be
assembled into a system that traced a given algebraic curve. Kempe's design
procedure has inspired research at the intersection of geometry and computer
science. In the late 1800s F. Reuleaux, A. B. W.
Kennedy, and L. Burmester formalized the analysis and synthesis of linkage
systems using descriptive geometry, and P.L.Chebyshev introduced analytical
techniques for the study and invention of linkages.
In the mid-1900s F. Freudenstein and G. N. Sandor used the newly developed
digital computer to solve the loop equations of a linkage and determine its
dimensions for a desired function, initiating the computer-aided design of
linkages. Within two decades these computer techniques were integral to the
analysis of complex machine systems and the control of robot manipulators.
R.E.Kaufman combined the computer’s ability to rapidly compute the roots of
polynomial equations with a graphical user interface to unite Freudenstein’s
techniques with the geometrical methods of Reuleaux and Burmester and form
KINSYN, an interactive computer graphics system for linkage design
The modern study of linkages includes the analysis and design of articulated
systems that appear in robots, machine tools, and cable driven and tensegrity
systems. These techniques are also being applied to biological systems and even
the study of proteins. Mobility
The configuration of a system of rigid links connected by ideal joints is
defined by a set of configuration parameters, such as the angles around a
revolute joint and the slides along prismatic joints measured between
adjacent links. The geometric constraints of the linkage allow
calculation of all of the configuration parameters in terms of a minimum set,
which are the input parameters. The number of input parameters is called the
mobility, or degree of freedom, of the linkage system.
A system of n rigid bodies moving in space has 6n degrees of freedom measured
relative to a fixed frame. Include this frame in the count of bodies, so that
mobility is independent of the choice of the fixed frame, then we have M=6(N-1),
where N=n+1 is the number of moving bodies plus the fixed body.
Joints that connect bodies in this system remove degrees of freedom and
reduce mobility. Specifically, hinges and sliders each impose five constraints
and therefore remove five degrees of freedom. It is convenient to define the
number of constraints c that a joint imposes in terms of the joint's freedom
f, where c=6-f. In the case of a hinge or slider, which are one degree of
freedom joints, we have f=1 and therefore c=6-1=5.
Thus, the mobility of a linkage system formed from n moving links and j joints
each with fi, i=1, ..., j, degrees of freedom can be computed as,
where N includes the fixed link. This is known as Kutzbach-Gruebler's equation
There are two important special cases: a simple open chain, and a simple closed
chain. A simple open chain consists of n moving links connected end to end by j
joints, with one end connected to a ground link. Thus, in this case N=j+1
and the mobility of the chain is For a simple closed chain, n moving
links are connected end-to-end by n+1 joints such that the two ends are
connected to the ground link forming a loop. In this case, we have N=j and the
mobility of the chain is An example of a simple open chain is a
serial robot manipulator. These robotic systems are constructed from a series of
links connected by six one degree-of-freedom revolute or prismatic
joints, so the system has six degrees of freedom.
An example of a simple closed chain is the RSSR spatial four-bar linkage. The
sum of the freedom of these joints is eight, so the mobility of the linkage is
two, where one of the degrees of freedom is the rotation of the coupler around
the line joining the two S joints. = Planar and spherical movement=
It is common practice to design the linkage system so that the movement of
all of the bodies are constrained to lie on parallel planes, to form what is
known as a planar linkage. It is also possible to construct the linkage system
so that all of the bodies move on concentric spheres, forming a spherical
linkage. In both cases, the degrees of freedom of the link is now three rather
than six, and the constraints imposed by joints are now c=3-f.
In this case, the mobility formula is given by
and we have the special cases, planar or spherical simple open chain,
planar or spherical simple closed chain, An example of a planar simple closed
chain is the planar four-bar linkage, which is a four-bar loop with four one
degree-of-freedom joints and therefore has mobility M=1.
Joints The most familiar joints for linkage
systems are the revolute, or hinged, joint denoted by an R, and the
prismatic, or sliding, joint denoted by a P. Most other joints used for spatial
linkages are modeled as combinations of revolute and prismatic joints. For
example, the cylindric joint consists of an RP or
PR serial chain constructed so that the axes of the revolute and prismatic
joints are parallel, the spherical joint consists of an RRR
serial chain for which each of the hinged joint axes intersect in the same
point; the planar joint can be constructed
either as a planar RRR, RPR, and PPR serial chain that has three
degrees-of-freedom. Analysis and synthesis of linkages
The primary mathematical tool for the analysis of a linkage is known as the
kinematics equations of the system. This is a sequence of rigid body
transformation along a serial chain within the linkage that locates a
floating link relative to the ground frame. Each serial chain within the
linkage that connects this floating link to ground provides a set of equations
that must be satisfied by the configuration parameters of the system.
The result is a set of non-linear equations that define the configuration
parameters of the system for a set of values for the input parameters.
Freudenstein introduced a method to use these equations for the design of a
planar four-bar linkage to achieve a specified relation between the input
parameters and the configuration of the linkage. Another approach to planar
four-bar linkage design was introduced by L. Burmester, and is called Burmester
theory. Planar one degree-of-freedom linkages
The mobility formula provides a way to determine the number of links and joints
in a planar linkage that yields a one degree-of-freedom linkage. If we require
the mobility of a planar linkage to be M=1 and fi=1, the result is
or This formula shows that the linkage must
have an even number of links, so we have N=2, j=1: this is a two-bar linkage
known as the lever; N=4, j=4: this is the four-bar linkage;
N=6, j=7: this is a six-bar linkage [ it has two links that have three joints,
called ternary links, and there are two topologies of this linkage depending how
these links are connected. In the Watt topology, the two ternary links are
connected by a joint. In the Stephenson topology the two ternary links are
connected by binary links; N=8, j=10: the eight-bar linkage has 16
different topologies; N=10, j=13: the 10-bar linkage has 230
different topologies, N=12, j=16: the 12-bar has 6856
topologies. See Sunkari and Schmidt for the number
of 14- and 16-bar topologies, as well as the number of linkages that have two,
three and four degrees-of-freedom. The planar four-bar linkage is probably
the simplest and most common linkage. It is a one degree-of-freedom system that
transforms an input crank rotation or slider displacement into an output
rotation or slide. Examples of four-bar linkages are:
the crank-rocker, in which the input crank fully rotates and the output link
rocks back and forth; the slider-crank, in which the input
crank rotates and the output slide moves back and forth;
drag-link mechanisms, in which the input crank fully rotates and drags the output
crank in a fully rotational movement. Other interesting linkages
Pantograph 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. Jansen's linkage is a twelve-bar Leg
mechanism that was invented by kinetic sculptor Theo Jansen.
Klann linkage is a six-bar linkage that forms the leg of a walking mechanism;
Toggle mechanisms are four-bar linkages that are dimensioned so that they can
fold and lock. The toggle positions are determined by the colinearity of two of
the moving links. The linkage is dimensioned so that the linkage reaches
a toggle position just before it folds. The high mechanical advantage allows the
input crank to deform the linkage just enough to push it beyond the toggle
position. This locks the input in place. Toggle mechanisms are used as clamps.
= Straight line mechanisms= James Watt's parallel motion and Watt's
linkage Peaucellier–Lipkin linkage, the first
planar linkage to create a perfect straight line output from rotary input;
eight-bar, one DOF. A Scott Russell linkage, which converts
linear motion, to linear motion in a line perpendicular to the input.
Chebyshev linkage, which provides nearly straight motion of a point with a
four-bar linkage. Hoekens linkage, which provides nearly
straight motion of a point with a four-bar linkage.
Sarrus linkage, which provides motion of one surface in a direction normal to
another. Hart's inversor, which provides a
perfect straight line motion without sliding guides.
= Biological linkages= Linkage systems are widely distributed
in animals. The most thorough overview of the different types of linkages in
animals has been provided by M. Muller, who also designed a new classification
system which is especially well suited for biological systems. A well-known
example is the cruciate ligaments of the knee.
An important difference between biological and engineering linkages is
that revolving bars are rare in biology and that usually only a small range of
the theoretically possible is possible due to additional mechanical
constraints. Biological linkages frequently are compliant. Often one or
more bars are formed by ligaments, and often the linkages are
three-dimensional. Coupled linkage systems are known, as well as five-,
six-, and even seven-bar linkages. Four-bar linkages are by far the most
common though. Linkages can be found in joints, such as
the knee of tetrapods, the hock of sheep, and the cranial mechanism of
birds and reptiles. The latter is responsible for the upward motion of the
upper bill in many birds. Linkage mechanisms are especially
frequent and manifold in the head of bony fishes, such as wrasses, which have
evolved many specialized feeding mechanisms. Especially advanced are the
linkage mechanisms of jaw protrusion. For suction feeding a system of linked
four-bar linkages is responsible for the coordinated opening of the mouth and 3-D
expansion of the buccal cavity. Other linkages are responsible for protrusion
of the premaxilla. Linkages are also present as locking
mechanisms, such as in the knee of the horse, which enables the animal to sleep
standing, without active muscle contraction. In pivot feeding, used by
certain bony fishes, a four-bar linkage at first locks the head in a ventrally
bent position by the alignment of two bars. The release of the locking
mechanism jets the head up and moves the mouth toward the prey within 5-10 ms.
Image gallery See also
References Further reading
External links Kinematic Models for Design Digital
Library — Major web resource for kinematics. Movies and photos of
hundreds of working mechanical-systems models in the Reuleaux Collection of
Mechanisms and Machines at Cornell University, plus 5 other major
collections. Includes an e-book library of dozens of classic texts on mechanical
design and engineering. Includes CAD models and stereolithographic files for
selected mechanisms. Digital Mechanism and Gear Library —
Online library about linkages and cams Linkage calculations
Introductory linkage lecture Virtual Mechanisms Animated by Java
Linkage-based Drawing Apparatus by Robert Howsare
(ASOM) Analysis, synthesis and optimization of multibar linkages
Linkage animations on mechanicaldesign101.com include planar
and spherical four-bar and six-bar linkages.
Animations of planar and spherical four-bar linkages.
Animation of Bennett's linkage. Example of a six-bar function generator
that computes the elevation angle for a given range.
Animations of six-bar linkage for a bicycle suspension.
A variety of six-bar linkage designs. Introduction to Linkages