The Chebyschev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.

It was invented by the 19th century mathematician Pafnuty Chebyschev who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight line motion. This was also studied by James Watt in his improvements to the steam engine. (Read more info about Watt Straight-line Mechanism)

The straight-line linkage of Chebyschev confines the point P — the midpoint on the link AB — on a straight line at the two extremes and at the center of travel. Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are

Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt to constrain the movement of a steam engine piston in a straight line. The idea of its genesis using links is contained in a letter he wrote to Matthew Boulton in June 1784.

"I have got a glimpse of a method of causing a piston rod to move up and down perpendicularly by only fixing it to a piece of iron upon the beam, without chains or perpendicular guides [...] and one of the most ingenious simple pieces of mechanics I have invented."

This linkage does not generate a true straight line motion, and indeed Watt did not claim it did so.

Watt's straight-line mechanism is used in the rear axle of some car suspensions. It intends to prevent relative sideways motion between the axle and body of the car. Watt’s linkage approximates a vertical straight line motion more closely, and does so while locating the center of the axle rather than toward one side of the vehicle.

It consists of two horizontal rods of equal length mounted at each side of the chassis. In between these two rods, a short vertical bar is connected. The center of this short vertical rod – the point which is constrained in a straight line motion - is mounted to the center of the axle. All pivoting points are free to rotate in a vertical plane.

Many engineering applications require things move in a linear fashion or "straight-line motion". We can use a linear motion guide that can guide a device accurately along a straight line. Manufacturing know-how of most linear guide manufacturers has let us keep expanding the range of linear guidance. The picture shown here is an example of commercially available linear guides from THK. This Linear Ball Slide is a lightweight, compact, limited stroke linear guide unit that operates with very low sliding resistance. It excels in high-speed responsive performance due to its very small frictional factor and low inertia.

In the late seventeenth century, before the development of the milling machine, it was extremely difficult to machine straight, flat surfaces. For this reason, good prismatic pairs without backlash were not easy to make. During that era, much thought was given to the problem of attaining a straight-line motion as a part of the coupler curve of a linkage having only revolute connection. Probably the best-known result of this search is the straight line mechanism development by Watt for guiding the piston of early steam engines. Although it does not generate an exact straight line, a good approximation is achieved over a considerable distance of travel.

In this post, we show approximated straight-line mechanism discovered by Richard Roberts (1789-1864). He discovered the Roberts' Straight-line mechanism.

Lengths:
O_{2}A = 100
O_{4}B = 100
AB = 100
AC = 100
BC = 100
O_{2}O_{4} = 200

We can find from the following video clip that point C moves as an approximated straight line. Though it is not an exact straight-line motion, but it's good as a starting point. In later post, we will explore more straight-line mechanisms that can give better straight-line approximation.

Another quick way to create and test Roberts straight-line mechanism is to use design wizard in SAM 7.0 The Ultimate Mechanism Designer. Length of each link can be changed easily and it can display the path of desired node. Watch the following video...

We can make a quick motion simulation using "animate dimension" command in Unigraphics (UG) NX4 sketch. Just draw lines as per a sketch and add one driving dimension as shown below. Then use animate dimension command to set the lower and upper limits, for this case they're minimum and maximum angles.

In previous posts, the fixed pivot points were determined from the moving pivot points. We can get result that can't be fitted in our design due to space limit. The principle of inversion can be applied to solve this problem. The first step is to find the three positions of the ground plan that correspond to the three desired coupler positions.

We start with our desired positions of fixed pivot points.

1) Draw desired fixed pivots (O_{2} and O_{4}) and moving pivot points. Red lines are three desired positions of links (moving pivots).

2) Draw lines to make fixed relations between the ground plane (O_{2}O_{4}) and the second coupler position.

3) Transfer the ground position to the first coupler position using same relations developed in previous step as shown in dashed lines. Name new ground positions as O'_{2} and O'_{4} respectively.

4) Draw lines to make fixed relations between the ground plane (O_{2}O_{4}) and the third coupler position (A_{3}B_{3}).

5) Transfer the ground position to the first coupler position using same relations developed in previous step as shown in dashed lines. Name new ground positions as O''_{2} and O''_{4} respectively.

6) Draw lines O_{2}O'_{2} and O'_{2}O''_{2}, bisect both lines and extend the perpendicular bisectors until they intersect. Label the intersection G.

6) Draw lines O_{4}O'_{4} and O'_{4}O''_{4}, bisect both lines and extend the perpendicular bisectors until they intersect. Label the intersection H.

7) Draw O_{2}G, GH and O_{4}H. Now we get G and H as inverted fixed pivot points of moving link O_{2}O_{4}.

8) Re-invert the linkage to return to the original arrangement.

Example [3-Position Motion Generation Synthesis with Alternate Moving Pivots using Unigraphics NX4 Sketch] shows how to synthesize four-bar linkages according to required moving pivots. By doing this, we first define desired locations of moving pivots then get positions of fixed pivots O_{2} and O_{4}. However, sometimes it may be more useful to define the location of fixed pivots O_{2} and O_{4 }first, then find other linkages that can move according to 3 desired positions of moving pivots.

In the previous post [3-Position Motion Generation Four-Bar Linkage Synthesis using Unigraphics NX4 Sketch], the locations of fixed pivots O_{2} and O_{4} are fixed due to the fixed locations of moving pivots A and B. Sometimes, the location of O_{2} and O_{4} are undesirable with respect to your design constraints. More flexible method to get desirable locations of O_{2} and O_{4} will be shown in this post.

1) Draw link AB in its three design positions A_{1}B_{1}, A_{2}B_{2}, and A_{3}B_{3} as shown above.

2) Draw new attachment points C_{1} and D_{1} and other lines to form the rigid link ABDC. Do the same for position 2 and 3. Use "Constraint" command in Unigraphics NX4 sketch to set the equal length constraint to all relevant lines e.g. A_{1}C_{1}, A_{2}C_{2}, and A_{3}C_{3} have the same length, but don't need to specify the fixed value for it. Once we complete these settings at all positions, it means we have set the fixed relationship between our desired line AB and other moving pivots CD.

3) Draw construction lines from point C_{1} to C_{2} and C_{2} to C_{3}

4) Bisect line C_{1}C_{2} and line C_{2}C_{2} and extend their perpendicular bisectors until they intersect with each other. Label the intersection O_{2}.

5) Draw line O_{2}C_{1}. It's link 2.

6) Repeat the same for another end of the link. Draw construction lines from point D_{1} to D_{2} and D_{2} to D_{3}

7) Bisect line D_{1}D_{2} and line D_{2}D_{3} and extend their perpendicular bisectors until they intersect with each other. Label the intersection O_{4}.
8) Draw line O_{4}D_{1}. It's link 4.

9) Draw line O_{2}M and set the same length as O_{2}C_{1}.
10) Draw line O_{4}N and set the same length as O_{4}D_{1}.
11) Draw line MN and set the same length as C_{1}D_{1}. Do the same for remaining lines, set constraint so that it form the same rigid link as ABDC.
12) Set angular dimension between O_{4}D_{1} and O_{4}N to any desired value e.g. 20 degrees.

13) Select "Animate Dimension" Command in Unigraphics NX4 sketch and set lower limt to 0 and Upper limit to 60 (or any other value) and Steps/Cycle to 150 (the more steps, the smoother simulation).

See the results of four-bar linkage from three-position motion generation synthesis with alternate moving pivots in the following video clip.

In the real design problem, it will be more practical with 3 specified positions of a line in sequential order than 2-position synthesis. We can use a logical extension of the linkage synthesis technique described in [Four-bar linkage Synthesis using Unigraphics NX4 Sketch] to do the linkage synthesis with 3 positions.

Assume that we have to design the mechanism that move the link AB through 3 positions as shown below. The rectangle represents the limit of link AB, so link AB cannot interfere with the rectangle.

1) Draw link AB in its three design positions A_{1}B_{1}, A_{2}B_{2}, and A_{3}B_{3 }as shown above.
2) Draw construction lines from point A_{1} to A_{2} and A_{2 }to A_{3}

3) Bisect line A_{1}A_{2} and line A_{2}A_{3} and extend their perpendicular bisectors until they intersect with each other. Label the intersection O_{2}.
4) Draw line O_{2}A_{1}. It's link 2.
5) Repeat the same for another end of the link. Draw construction lines from point B_{1} to B_{2} and B_{2 }to B_{3}

6) Bisect line B_{1}B_{2} and line B_{2}B_{3} and extend their perpendicular bisectors until they intersect with each other. Label the intersection O_{4}.

7) Draw line O_{4}B_{1}. It's link 4.

8) Draw line O_{2}M and set the same length as O_{2}A_{1}.

9) Draw line O_{4}N and set the same length as O_{4}B_{1}.

10) Draw line MN and set the same length as A_{1}B_{1}.

11) Set angular dimension between O_{4}B_{1} and O_{4}N to any desired value e.g. 15 degrees.

12) Select "Animate Dimension" Command in Unigraphics NX4 sketch and set lower limt to 0 and Upper limit to 56.355 (or any other value) and Steps/Cycle to 150 (the more steps, the smoother simulation).

See the results of four-bar linkage from three-position motion generation synthesis in the following video clip.

In most design problems, we need to design the mechanism that moves between required positions. In this post, we will show the motion generation synthesis method to control a line in the plane to move in sequential positions. Any CAD software can be used to do this. But here we use Unigraphics NX4 sketch to draw and use "Animate Dimension" command to simulate the movement of all concerned links.

Assume that we need to design the mechanism to move line AB to A'B'. A simple and most commonly used four-bar linkage will be used here. Let's see how to find the fixed pivoting point at the ground.

1) Draw the link AB and A'B' in its desired positions in sketch of Unigraphics NX4 sketch (or any other CAD software) as shown in the above figure.

2) Draw construction lines from point A to A' and bisect line AA' and extend the perpendicular bisectors in convenience direction. (The choice is yours).

3) Select any convenient point on bisector as the fixed pivot O_{2}. For this example, we use length O_{2}A = 500 mm.
4) Draw line O_{2}A as link 1.
5) Do the same for point B and B'. Draw construction lines from point B to B' and bisect line BB' and extend the perpendicular bisectors in convenient direction.

6) Select any convenient point on bisector as the fixed pivot O_{4}. For this example, we select 60^{o} angle between O_{4}B and O_{4}B'
7) Draw line O_{4}B as link 2.
8) Draw line O_{2}M and set the same length as O_{2}A.
9) Draw line O_{4}N and set the same length as O_{4}B.
10) Draw line MN and set the same length as AB.
11) Set angular dimension between O_{4}B and O_{4}N to any desired value e.g. 45 degrees.

12) Select "Animate Dimension" Command in Unigraphics NX4 sketch and set lower limt to 0 and Upper limit to 60 and Steps/Cycle to 150 (the more steps, the smoother simulation).

See the results of four-bar linkage synthesis in the following video clip.