Optimizing Mechanisms Using CAD Sketch Constraints – Part 2

This article is the second part in a series looking at practical ways to synthesize mechanisms that generate specified paths or motions given complex constraints such as link lengths and joint positions. It also deals with how you can ensure smooth motion by constraining the solution to prevent transmission angles going below a minimum value. Part 1 describes the important concepts, such as transmission angles, toggles and spline constraints. This article gives a complete example of the design of a four-bar linkage with joints in defined areas and free movement ensured by minimum transmission angle constraints.

These articles assume that you already understand the basics of mechanism synthesis using the geometric constraint solver within a parametric CAD program, such as SOLIDWORKS. If you’re not sure about that, then you should start by reading the introductory articles explaining mechanism synthesis in CAD from first principles, adapting traditional graphical methods to CAD and some examples of basic mechanism synthesis problems.

Defining the Problem

Real mechanism design usually has many constraints and objectives, with the optimum design balancing these considerations. The first step is to clearly define the constraints according to your design requirements. The constraints can be classified under the following general headings:

  • The mechanism synthesis requirement, or the type of movement that is required. This might be:
    • Path generation involves moving a single point through a prescribed path, defined as a number of points.
    • Motion generation involves moving a line through a number of prescribed positions, typically defined as a number of lines.
    • Function generation is a catch-all term for any other more specific requirement, such as a specific velocity ration between the input and output link.
  • Force transfer, considering where force will be applied to actuate the mechanism and what transmission angles are required. It was explained in Part 1 that transmission angles determine when a mechanism will lock or toggle, and that the transmission angle is defined by which link is actuated, the driver.
  • Size constraints, such as where the ground link should be located and the overall size of the mechanism.

It’s worth noting here that the standard definition of a transmission angle assumes that the driven link is joined to the ground link. That means any forces applied to this link result in rotary motion and the direction of the force it imparts on the coupler link is always tangential to this arc of motion or, putting it another way, the force is perpendicular to the link.

In some situations, these rules can be difficult to apply. Mechanisms that are not driven by a rotary motor are often actuated by a force being applied directly to what would normally be considered the coupler. In that case, it appears that the force applied may not result in a rotary motion, so the normal rules about transmission angles don’t apply. It’s not really clear which link is the coupler and which is the output, as shown below:

Figure 1: A four-bar linkage with the force applied to an ungrounded link.

However, that is due to the frame of reference. If we consider the motion of the links with respect to one of the links connected to the driven link, then the motion of the driven link becomes rotational, making it is possible to define a transmission angle in the normal way:

Figure 2: Changing the frame of reference so that the transmission angle can be identified. Note that there are two ways of doing this.

Synthesis Example

The mechanism being considered in this article must generate a motion, defined in Figure 1, using three lines. The intent of the design is that the mechanism should move backward from the lower vertical position, without rotating, and then rotate up to the final position. The small 5-degree angle in the middle position has been defined to make this easier to achieve without the mechanism needed to change direction.

Figure 3: Motion generation defined as three lines.

The simplest way to synthesize this motion would be to assume that joints are located at the ends of the lines that define the motion. Since each line is shown in three positions, the three positions of each endpoint fully define a circular arc. Each arc describes the motion of one of the links. Therefore, this fully defines the mechanism with only one possible solution. Arcs drawn through the endpoints have their centers located at the position of the grounded joints. This construction is shown in Figure 2 with the resulting mechanism, shown in blue, constrained to the defining arcs.

Figure 4: Simple mechanism synthesis using two arcs to fully define the mechanism.

Dragging the mechanism through its range of motion reveals that there are some issues with this design. Regardless of which link is selected as the driver, the mechanism moves through a toggle. If the longer link is chosen as the driver, there is a toggle close to the lower position. If the shorter link is driven, the toggle occurs close to the upper position. This is shown in Figure 3.

Figure 5: Simple mechanism shown in a range of positions with the two possible toggle positions in bold.

It will be necessary to move the joint positions away from the endpoints of the motion defining lines and introduce some additional constraints on the transmission angles to prevent the toggles.

Remember how we define the transmission angle by first naming each link:

  • The ground link is the fixed point of reference to which stationary joints are fixed.
  • The driver is the link to which force is applied, causing it to rotate about the ground link. The driver may also be referred to as the input link or, for certain types of linkage, the crank.
  • The output link is the other link that rotates about a fixed point on the ground. It does not have any external forces acting on it. Its motion is due to forces transferred through the other links.
  • The coupler is the moving link that connects the driver to the output.
  • The transmission angle is defined as the acute angle between the coupler and output link.

When the transmission angle is 90 degrees, all of the force transferred by the coupler is applied as a moment to rotate the output link about the ground link. When the transmission angle is zero, all of the force is transferred directly through the grounded joint at the other end of the output link. There is no resulting moment to cause movement of the output link and the mechanism locks. This is called a toggle position.

To define joint locations in each of the three motion positions, sketch two perpendicular lines attached to each end of the motion defining lines. Use perpendicular constraints and equal length constrains to ensure that the joint locations are consistent in each position, as shown in Figure 6.

Figure 6: Lines with perpendicularity and equal length constraints defining consistent joint locations for each motion position.

As before, arcs can now be used to synthesize the mechanism, locating the grounded joints at the centers of these arcs, as shown in Figure 7.

Figure 7: Arcs added to synthesize the mechanism.

Adding some construction lines to the end positions can help identify where toggles have occurred and impose constrains to prevent them. In Figure 8, the shorter grounded link has crossed the longer grounded link, which means that it has moved through a position where the longer link was aligned with the coupler. If the shorter link is the driver, this alignment would have been a toggle position where the mechanism would have locked.

Figure 8: Construction lines added to show link positions, helping to identify where they have moved through a toggle.

To prevent this toggle, we can adjust the joint positions so that the links to no finish is this crossed configuration. In Figure 9, the mechanism has been dragged to a position that satisfies this condition. A dimension has been added to define the minimum transmission angle that is reached at the end of the motion.

Figure 9: Constraint added to prevent toggle occurring during motion.

Continue adding dimensions to fully define the sketch, specifying lengths that correspond with what will produce a practical mechanism, as shown in Figure 9.

Figure 9: Fully constrained sketch giving a practical mechanism.

Be careful that the position of the joints doesn’t invert as you are doing this. It is possible to satisfy all of the sketch constrains without actually having consistent joint locations in all three motion positions, as shown in Figure 10.

Figure 10: Be careful that the position of the lines defining the joint locations doesn’t invert in one of the motion positions. This sketch satisfies all the sketch constraints, but it doesn’t actually represent the motion of a consistent body.

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