In the study ofmechanisms, afour-bar linkage, also called afour-bar, is the simplest closed-chain movablelinkage. It consists of fourbodies, calledbars orlinks, connected in a loop by fourjoints. Generally, the joints are configured so the links move in parallel planes, and the assembly is called aplanar four-bar linkage. Spherical and spatial four-bar linkages also exist and are used in practice.[1]
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 arevolute joint – also known as a pin joint or hinged joint – denoted by R, or aprismatic joint – also known as a sliding pair – denoted by P.[Note 1]
A link that is fixed in place relative to the viewer is called aground link.
A link connecting to the ground by a revolute joint that can perform a complete revolution is called acrank link.
A link connecting to the ground by a revolute joint that cannot perform a complete revolution is called arocker link.
A link connecting to a ground line by a prismatic joint is called a slider. Sliders are sometimes considered to be cranks that have a hinged pivot at an infinitely long distance away perpendicular to the travel of the slider.
A link connecting to two other links is called afloating link orcoupler.A coupler connecting a crank and a slider in a single slider crank mechanism is often called aconnecting rod, however, it has also been used to refer to any type of coupler.
There are three basic types of planar four-bar linkage, depending on the use of revolute or prismatic joints:
Planar four-bar linkages can be designed to guide a wide variety of movements, and are often the basemechanisms found in manymachines. Because of this, thekinematics anddynamics of planar four-bar linkages are also important topics inmechanical engineering.
Planar quadrilateral linkage, RRRR or 4R linkages have four rotating joints. One link of the chain is usually fixed, and is called theground link,fixed link, or theframe. The two links connected to the frame are called thegrounded links and are generally the input and output links of the system, sometimes called theinput link andoutput link. The last link is thefloating link, which is also called acoupler orconnecting rod because it connects an input to the output.
Assuming the frame is horizontal there are four possibilities for the input and output links:[2]
Some authors do not distinguish between the types of rocker.
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 ifS +L ≤P +Q, whereS is the shortest link,L is the longest, andP andQ are the other links.
The movement of a quadrilateral linkage can be classified into eight cases based on the dimensions of its four links. Let a, b, g and h denote the lengths of the input crank, the output crank, the ground link and floating link, respectively. Then, we can construct the three terms:
The movement of a quadrilateral linkage can be classified into eight types based on the positive and negative values for these three terms, T1, T2, and T3.[2]
| 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 cases of T1 = 0, T2 = 0, and T3 = 0 are interesting because the linkages fold. If we distinguish folding quadrilateral linkage, then there are27 different cases.[3]
The figure shows examples of the various cases for a planar quadrilateral linkage.[4]
The configuration of a quadrilateral linkage may be classified into three types: convex, concave, and crossing. In the convex and concave cases no two links cross over each other. In the crossing linkage two links cross over each other. In the convex case all four internal angles are less than 180 degrees, and in the concave configuration one internal angle is greater than 180 degrees. There exists a simple geometrical relationship between the lengths of the two diagonals of the quadrilateral. For convex and crossing linkages, the length of one diagonal increases if and only if the other decreases. On the other hand, for nonconvex non-crossing linkages, the opposite is the case; one diagonal increases if and only if the other also increases.[5]
The synthesis, or design, of four-barmechanisms is important when aiming to produce a desired output motion for a specific input motion. In order to minimize cost and maximize efficiency, a designer will choose the simplest mechanism possible to accomplish the desired motion. When selecting a mechanism type to be designed, link lengths must be determined by a process called dimensional synthesis. Dimensional synthesis involves aniterate-and-analyze methodology which in certain circumstances can be an inefficient process; however, in unique scenarios, exact and detailed procedures to design an accurate mechanism may not exist.[6]
The time ratio (Q) of a four-bar mechanism is a measure of its quick return and is defined as follows:[6]
With four-bar mechanisms there are twostrokes, the forward and return, which when added together create a cycle. Each stroke may be identical or have different average speeds. The time ratio numerically defines how fast the forward stroke is compared to the quicker return stroke. The total cycle time (Δtcycle) for a mechanism is:[6]
Most four-bar mechanisms are driven by a rotational actuator, or crank, that requires a specific constant speed. This required speed (ωcrank)is related to the cycle time as follows:[6]
Some mechanisms that produce reciprocating, or repeating, motion are designed to produce symmetrical motion. That is, the forward stroke of the machine moves at the same pace as the return stroke. These mechanisms, which are often referred to asin-line design, usually do work in both directions, as they exert the same force in both directions.[6]
Examples of symmetrical motion mechanisms include:
Other applications require that the mechanism-to-be-designed has a faster average speed in one direction than the other. This category of mechanism is most desired for design when work is only required to operate in one direction. The speed at which this one stroke operates is also very important in certain machine applications. In general, the return and work-non-intensive stroke should be accomplished as fast as possible. This is so the majority of time in each cycle is allotted for the work-intensive stroke. Thesequick-return mechanisms are often referred to asoffset.[6]
Examples of offset mechanisms include:
With offset mechanisms, it is very important to understand how and to what degree the offset affects the time ratio. To relate the geometry of a specific linkage to the timing of the stroke, an imbalance angle (β) is used. This angle is related to the time ratio,Q, as follows:[6]
Through simple algebraic rearrangement, this equation can be rewritten to solve forβ:[6]
Timing charts are often used to synchronize themotion between two or more mechanisms. They graphically display information showing where and when each mechanism is stationary or performing its forward and return strokes. Timing charts allow designers to qualitatively describe the requiredkinematic behavior of a mechanism.[6]
These charts are also used to estimate the velocities and accelerations of certain four-bar links.The velocity of a link is the time rate at which its position is changing, while the link's acceleration is the time rate at which its velocity is changing. Both velocity and acceleration arevector quantities, in that they have bothmagnitude anddirection; however, only their magnitudes are used in timing charts. When used with two mechanisms, timing charts assumeconstant acceleration. This assumption producespolynomial equations for velocity as a function of time. Constant acceleration allows for the velocity vs. time graph to appear as straight lines, thus designating a relationship betweendisplacement (ΔR), maximum velocity (vpeak), acceleration (a), and time(Δt). The following equations show this.[6][7]
Given the displacement and time, both the maximum velocity and acceleration of each mechanism in a given pair can be calculated.[6]
A slider-crank linkage is a four-bar linkage with three revolute joints and one prismatic, or sliding, joint. The rotation of thecrank drives the linear movement the slider, or the expansion of gases against a slidingpiston in a cylinder can drive the rotation of the crank.
There are two types of slider-cranks: in-line and offset.
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 aspherical four-bar linkage. The input-output equations of a spherical four-bar linkage can be applied to spatial four-bar linkages when the variables are replaced bydual numbers.[8]Note that the cited conference paper incorrectly conflatesMoore-Penrose pseudoinverses withone-sided inverses of matrices, falsely claiming that the latter are unique whenever they exist. This is contradicted by the fact that admits the set of matrices as all itsleft inverses.
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.[9][10][2]
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