I recently wrote about how the constraint-based sketcher within SolidWorks, and many other parametric CAD software, provides a general purpose way to design mechanisms. The basic method I described was to:

- Sketch a link in a number of positions describing the required motion.
- Sketch an approximate mechanism at each of these positions, with the unknown joint positions and link lengths unconstrained.
- Set the grounded joints to be coincident for each position.
- Set the link lengths to be equal for each position.
- Allow the geometric constraint solver, with the CAD’s sketching environment, to solve the mechanism.
- Constrain any remaining unconstrained degrees of freedom in the model to create an optimal mechanism.

This is just a quick summary. If you want to understand that method better, then go back and read my previous article. The methodology I described is versatile and can solve many problems that couldn’t normally be solved graphically. However, it is not always the most efficient approach. Traditional graphical synthesis has been used for 100s of years to create many classic mechanisms in machine design. In this article I will summarize how some of these established techniques can be applied in CAD to more quickly design many common mechanisms.

Traditional graphical methods are, due to the nature of paper, two-dimensional. This means that they are only of use for designing planar mechanisms in which all motion is constrained to a single plane. The general purpose method I described in my previous article can be applied to three-dimensional mechanisms. The traditional graphical methods can be applied using a compass and ruler. They are also much more accurate and efficient when they are applied using CAD software.

### Some Mechanism Design Fundamentals

Mechanism design involves rigid body kinematics. This basically means that a mechanism is made up of links, which are assumed to be rigid bodies that cannot bend or twist in any way. In three-dimensional space, a rigid body has 6 degrees of freedom (DOF): translation in x, y and z; and rotation about each one of these axes. For most mechanism synthesis, however, motion is assumed to be constrained to a single plane, two-dimensional motion. In this case, a rigid body has 3 DOF: translation in x and y; and rotation in the plane of motion. A four-bar planar linkage, involving four rigid bodies, would therefore have 12 DOF if the links were not joined together in any way. This would, of course, not be a mechanism. It would simply be a collection of parts. The links are connected together with joints. In a planar mechanism, these may be pin joints, a full sliding joint that acts like a piston or a half slider that acts like a pin in the slot. A pin joint removes 2 DOF, both translations; full slider also removes 2 DOF, a translation and a rotation; and half slider removes 1 DOF, a translation. Generally, one link is considered to be grounded and acts as a reference frame about which motion takes place. It therefore has no degrees of freedom. These considerations lead to Gruebler’s equation giving the degrees of freedom for a mechanism:

Where *L* is the number of links, including ground, *J _{P}* is the number of pin joints,

*J*is the number of full sliding joints and

_{FS}*J*is the number of half sliding joints.

_{HS}Generally, a mechanism should have 1 DOF. A structure with 0 DOF is a truss, while a structure with a negative DOF is statically indeterminate or preloaded. Working through Gruebler’s equation can be a useful way to understand a complex mechanism. However, it is not entirely reliable due to something known as Gruebler’s paradox. Under certain circumstances, mechanisms may have more degrees of freedom than the equation suggests.

### Types of Mechanism Synthesis

Mechanism synthesis can be divided into three categories, depending on the objective:

attempts to map an input function to an output function.*Function generation*moves a single point through a prescribed path.*Path generation*moves a line through a number of prescribed positions.*Motion generation*

In this article, I provide a few examples of synthesizing a mechanism for motion generation. In subsequent articles I will also demonstrate methods for function and path generation.

### Example 1: Two-position Motion Generation of a Single Pivot Point

One of the simplest graphical methods for synthesizing a mechanism involves finding the pivot position that will transform a line from one known position and orientation to another. To do this in CAD, we will use four sketches. The sketch-1 defines the two positions we want our mechanism to move through. Each position is represented by a line with an arbitrary position and orientation. The two lines must be of equal length since they represent the same link at different positions.

Start by sketching two lines of equal length, defining the required positions.

In sketch-2, create two construction lines connecting the endpoints of the position lines.

In the second sketch, create two construction lines, connecting the endpoints of the lines in the first sketch.

Next, create “perpendicular bisectors” to these construction lines. This means that for each of the previously created construction lines, you place a new construction line that starts at its midpoint and is perpendicular to it.

Create perpendicular bisector lines from each construction line.

The final step in sketch-2 is to extend the perpendicular bisectors so that their endpoints are coincident. This intersection between the construction lines is the point of rotation that will achieve the required motion. The mechanism synthesis problem is solved at this point.

The intersection between the perpendicular bisectors is the point of rotation for the mechanism.

You should now create a third sketch. This will help verify that required motion is obtained. Create two lines, each starting at the rotation point and extending to either end of one of the position lines in sketch-1.

Sketch-3 is used as reference geometry to verify the mechanisms motion.

In the fourth and final sketch, a triangle is drawn with one vertex located at the rotation point and the lengths set to be equal to the reference geometry in sketch-3. This triangle can then be dragged to simulate motion of the mechanism.

Sketch-4 shown with the triangle in two different positions, verifying the motion of the mechanism.

### Example 2: Two-position Motion Generation for a Four-bar Linkage

In the previous example, the pivot position was quite a long way from the body being positioned. This may not be convenient. The required motion could be achieved more compactly using a four-bar linkage. To synthesize this mechanism, we start as before. Sketch-1 contains the two desired positions, defined as lines. Sketch-2 contains construction lines connecting the endpoints and perpendicular bisectors to these. In this case, we do not, however, use the point of intersection between the perpendicular bisectors. Instead, in sketch-3, we draw two lines, from the endpoints of one of the position lines, to the corresponding perpendicular bisector. The length of these lines can be chosen to be convenient and produce a mechanism that moves through a reasonable range of angles. I will look at the importance of transmission angles for mechanism design in a future article.

Sketch-3 defines the joint positions, located on the perpendicular bisectors, as well as the conveniently chosen link lengths.

In sketch-4, three lines are created to represent the three moveable links in the mechanism. Equal length constraints are used to set the link lengths to be equal to those defined in sketch-1 and sketch-3. The grounded pin joints are also defined by making the ends of these lines coincident with the end lines created in sketch-3. This sketch can then be dragged through its range of motion to verify the mechanism.

Dragging sketch-4 to verify the mechanism’s motion.

### Conclusions

This article introduced some fundamental concepts in mechanism synthesis. The examples demonstrated how some of the more basic traditional methods can be applied within CAD. In my next article, examples will include more complex motion generation problems. Over the coming months, articles will explore the full range of modern CAD based approaches to mechanism synthesis.

**About the Author**

Dr. Jody Muelaner’s career began in machine design, working on everything from medical devices to saw mills. He trained in AutoCAD in 1997 and transferred to SolidWorks in 2002. Since 2007 he has also been involved with quality engineering in aerospace manufacturing. This work unifies SPC, MSA and metrology to enable optimized production systems. His interests also include bicycle design. Visit his website for more information: www.muelaner.com