Four Bar Linkage Synthesis

MEboost has a four bar linkage synthesis tool for the three types of linkage synthesis.  This article will explain terms, conventions and constraints.  This is the first of four articles on four bar linkage synthesis.  The remaining articles will show specific examples of each synthesis type.

Part 2: Four Bar Path Synthesis

Part 3: Four Bar Function Synthesis

Part 4: Four Bar Motion Synthesis

Starting with a desired path that a point on the linkage must follow, or angles that a certain link must be at, we can design a four bar linkage.  MEboost is not limited to a few points/angles.  This is what makes it so powerful and flexible.

General Information

Let's go over some terms to understand what needs to be synthesized. The four bar linkage below will serve as an example.

The MEboost link numbering convention is that the fixed link (frame) is always link 1.  Link 2 is the driving link.  All link angles are measured from the x axis, positive counterclockwise.

During path and motion synthesis there are nine dimensions required to specify the linkage.  Function synthesis requires five since J12x, J12y, point radius, and point angle are not used.

  • Link 1 length
  • Link 2 length
  • Link 3 length
  • Link 4 length
  • Link 1 angle from x axis, β
  • J12x coordinate from origin
  • J12y coordinate from origin
  • Point radius
  • Point angle, φ

In addition to the nine dimensions, an additional variable is whether the linkage is assembled in the open or crossed position.

With a typical design project, some dimensions must be constrained to a range or a fixed value.  This can be easily handled as we'll see later.

MEboost uses optimization to synthesize the linkage and uses the dimensional constraints to make sure the constraints are satisfied.  It's important to consider that unnecessarily tight constraints may result in a solution that is not the best possible.

Synthesis Difficulty

The difficulty of synthesis varies by type, path shape and timing of the input link.  As a loose rule of thumb the following synthesis operations are ordered in increasing difficulty.

  • Untimed path
  • Function
  • Timed path
  • Untimed motion
  • Timed motion

Timed synthesis adds to the complexity by stipulating that the driving link must be at a specific angle for each precision point.  For untimed synthesis, the driving link can be at any angle and allows for more possible solutions.

Motion adds an additional requirement that the link containing the point must be at a specific angle.  For path synthesis, the link can be at any angle and allows for more possible solutions.

Function synthesis is by definition timed since the input link angle is related to output link angle.  It does not have a path to follow so it falls between untimed path and timed path.

Always bear in mind that a four bar linkage may not be capable of performing the required task.  A different linkage such as a slider crank, or a more complex linkage may be required.

Opening the Four Bar Synthesis Tool

In the MEboost ribbon, click the Linkages button, then the 4 Bar Synthesis button.

The linkage synthesis form will appear.


For each dimension, a minimum and/or maximum constraint can be applied.  In the form below, several constraints are added.

Sometimes you may want to set a dimension at a specific value.  To do this, use the desired dimension in both the minimum and maximum boxes.  For the example above, link 1 length is constrained to 8.

Link 2 length is constrained to a maximum length of 5.

Link 3 length must be at least 3.

Link 4 length must be from 2 to 12.

There are also assembly constraints.  You can allow the software to find the best solution regardless if the assembly is open or crossed.  You can also limit the assembly to just one assembly position.

The link 2 must fully rotate constraint will only find solutions where link 2 (the driver) can fully rotate.  If a closed path is specified, leaving this constraint unchecked will improve performance.  Since the path is closed, a good solution should allow link 2 to fully rotate anyway.  This constraint is most useful when the path is not closed, but you want link 2 to fully rotate (it may be connected to a motor, etc.).

Soft Constraints

Soft constraints can be employed to limit the search space and thereby improve performance.  A soft constraint isn't dictated by the design, but are merely used to put limits on the software when searching for a solution.

For closed paths, the following method can be used.

Consider the path below.  Looking at the path there is a Δx and Δy which is the maximum travel in the x and y directions.  A rule of thumb is to set the maximum link lengths and point radius to MAX [Δx, Δy].  This is only a rule of thumb so it can be adjusted as needed.

The use of soft constraints assumes that J12x and J12y are UNCONSTRAINED.  If either is constrained, the rule of thumb may not allow a satisfactory solution.

For paths that are open, soft constraints should be larger since longer link lengths may be required to trace the path.

Δx = 19.62 and Δy = 19.74.  Since Δy is larger we could round up and use soft constraints of 20 as shown below.  NOTE: If a link or point radius has a design constraint, it should be entered instead of a soft constraint.