Four-bar linkage

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.

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:

${\displaystyle T_{1}=g+h-a-b}$;

${\displaystyle T_{2}=b+g-a-h}$;

${\displaystyle T_{3}=b+h-a-g}$.

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]

${\displaystyle T_{1}}$	${\displaystyle T_{2}}$	${\displaystyle T_{3}}$	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 cases of T₁ = 0, T₂ = 0, and T₃ = 0 are interesting because the linkages fold. If we distinguish folding quadrilateral linkage, then there are 27 different cases.[3]

The figure shows examples of the various cases for a planar quadrilateral linkage.[4]

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

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 time ratio (Q) of a four-bar mechanism is a measure of its quick return and is defined as follows:[6]

${\displaystyle Q={\frac {\text{Time of slower stroke}}{\text{Time of quicker stroke}}}\geq 1}$

With four-bar mechanisms there are two strokes, 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 (Δt_cycle) for a mechanism is:[6]

${\displaystyle \Delta t_{\text{cycle}}={\text{Time of slower stroke}}+{\text{Time of quicker stroke}}}$

Most four-bar mechanisms are driven by a rotational actuator, or crank, that requires a specific constant speed. This required speed (ω_crank)is related to the cycle time as follows:[6]

${\displaystyle \omega _{\text{crank}}=(\Delta t_{\text{cycle}})^{-1}}$

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 as in-line design, usually do work in both directions, as they exert the same force in both directions.[6]

Examples of symmetrical motion mechanisms include:

• Windshield wipers
• Engine mechanisms or pistons
• Automobile window crank

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. These quick-return mechanisms are often referred to as offset.[6]

Examples of offset mechanisms include:

• Cutting machines
• Package-moving devices

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]

${\displaystyle Q={\frac {180^{\circ }+\beta }{180^{\circ }-\beta }}}$

Through simple algebraic rearrangement, this equation can be rewritten to solve for β:[6]

${\displaystyle \beta =180^{\circ }\times {\frac {Q-1}{Q+1}}}$

Timing charts are often used to synchronize the motion 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 required kinematic 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 are vector quantities, in that they have both magnitude and direction; however, only their magnitudes are used in timing charts. When used with two mechanisms, timing charts assume constant acceleration. This assumption produces polynomial 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 between displacement (ΔR), maximum velocity (v_peak), acceleration (a), and time(Δt). The following equations show this.[6][7]

ΔR = 1/2v_peakΔt

ΔR = 1/4a(Δt)²

Given the displacement and time, both the maximum velocity and acceleration of each mechanism in a given pair can be calculated.[6]

• Chebyshev linkage
• Chebyshev lambda linkage
• Evans "Grasshopper" linkage
• Hoecken linkage
• Horse-head linkage
• Pantograph
• Roberts linkage
• Valve gear
• Watt's linkage

• Bicycle suspension
• Biological linkages
• Double wishbone suspension
• Door closer
• Foldable steps and foldable chairs
• Foot operated machines (grindstone, lathe, sewing machine, treadle, etc.)
• Gear shifter
• Glider (furniture)
• Oscillating fan
• Pumpjack
• Step-on trash can
• Windshield wiper

• 
  Oil well pump-jack.
• 
  Train Suspension using Watt's linkage.
• 
  Rear car suspension using Watt's linkage.
• 
  Suspension for VPK-3927 Volk family armored vehicles.
• 
  Door closer mechanism.
• 
  The Howe Machine Co. Sewing machine.
• 
  A pedal-driven grindstone.
• 
  The Brown Bros screwcutting treadle lathe.
• 
  Ford 3-speed Overdrive Transmission
• 
  Gearshift on a 2003 SV650S motorcycle.
• 
  A foldable chair.
• 
  A diagram of the suspension on a Specialized Stumpjumper FSR mountain bike.
• 
  Animation of a Steam locomotive.