Four-bar linkage

A planar four-bar linkage (Watt linkage) used as a train suspension.

A Bennett spatial four-bar linkage.

A four-bar linkage, also called a four-bar, is the simplest movable closed chain linkage. It consists of four bodies, called bars or links, connected in a loop by four joints. Generally, the joints are configured so the links move in parallel planes, and the assembly is called a planar four-bar linkage.

If the linkage has four hinged joints with axes angled to intersect in a single point, then the links move on concentric spheres and the assembly is called a spherical four-bar linkage. Bennett's linkage is a spatial four-bar linkage with hinged joints that have their axes angled in a particular way that makes the system movable.

Planar four-bar linkages

Planar four-bar linkages are important mechanisms found in machines. The kinematics and dynamics of planar four-bar linkages are important topics in mechanical engineering.

Planar four-bar linkages are constructed from four links connected in a loop by four one degree of freedom joints. A joint may be either a revolute, that is a hinged joint, denoted by R, or a prismatic, as sliding joint, denoted by P. The planar quadrilateral linkage is formed by four links and four revolute joints, denoted RRRR. The slider-crank linkage is constructed from four links connected by three revolute and one prismatic joint, or RRRP. The double slider is a PRRP linkage.

Planar four-bar linkages can be designed to guide a wide variety of movements.

planar quadrilateral linkage

Planar quadrilateral linkage, RRRR or 4R linkages have four rotating joints. One link of the chain is usually fixed, and is called the ground link, fixed link, or the frame. The two links connected to the frame are called the grounded links and are generally the input and output links of the system, sometimes called the input link and output link. The last link is the floating link, which is also called a coupler or connecting rod because it connects an input to the output.

Assuming the frame is horizontal there are four possibilities for the input and output links:

• A crank: can rotate a full 360 degrees
• A rocker: can rotate through a limited range of angles which does not include 0° or 180°
• A 0-rocker: can rotate through a limited range of angles which includes 0° but not 180°
• A π-rocker: can rotate through a limited range of angles which includes 180° but not 0°

Some authors do not distinguish between the types of rocker.

Grashof condition

The Grashof condition for a four-bar linkage states: If the sum of the shortest and longest link of a planar quadrilateral linkage is less than or equal to the sum of the remaining two links, then the shortest link can rotate fully with respect to a neighboring link. In other words, the condition is satisfied if S+L ≤ P+Q where S is the shortest link, L is the longest, and P and Q are the other links.

Classification

The types of linkage can be further classified into eight cases. Let $a$, $b$, $g$, $h$ be the lengths of the input link, output link, ground link and coupler respectively. If $T_1=g+h-a-b$, $T_2=b+g-a-h$, $T_3=b+h-a-g$ then the eight types are classified according to the signs of $T_1$, $T_2$, $T_3$.

$T_1$	$T_2$	$T_3$	Grashof condition	Input link	Output link
-	-	+	Grashof	Crank	Crank
+	+	+	Grashof	Crank	Rocker
+	-	-	Grashof	Rocker	Crank
-	+	-	Grashof	Rocker	Rocker
-	-	-	Non-Grashof	0-Rocker	0-Rocker
-	+	+	Non-Grashof	π-Rocker	π-Rocker
+	-	+	Non-Grashof	π-Rocker	0-Rocker
+	+	-	Non-Grashof	0-Rocker	π-Rocker

The figure shows examples of the various cases for a planar quadrilateral linkage.

Types of four-bar linkages, s = shortest link, l = longest link

Examples

A four-bar linkage used as the suspension for a bicycle. If we count the two bars that form the shock absorber attached to the output link, then this is a Watt II six-bar linkage

• Pantograph (four-bar, two degrees of freedom, i.e., only one pivot joint is fixed.)
• Crank-slider, (four-bar, one degree of freedom)
• Double wishbone suspension
• Watt's linkage and Chebyshev linkage (linkages that approximate straight line motion)
• Biological linkages
• Bicycle suspension

See also

• Burmester theory
• Cognate linkage
• Glider (furniture)

External links

• The four-bar linkages in the collection of Reuleaux models at Cornell University
• mechanisms101.com – Flash Four-bar Linkages simulator
• softintegration.com – Animated GIF Four-bar linkage with triangular coupler link
• 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.