﻿ Step 10.4: Approximate and Exact Straight-Line Kinematic-Chains

# Step 10.4: Straight-Line kinematic-chains

History

It is not possible to 'draw' a straight-line with simple tools.

Historically, it was difficult to manufacture linear guide-ways with low friction. Also, after it was manufactured, it was impossible to measure its straightness.

However, it is relatively easy to make Parts with the same length. It is also relatively easy to measure their relative length.

Engineers, mathematicians and inventors searched for mechanism designs with Pin-Joints only and could guide a Point, or Part, along a straight-line.

Engineers found mechanisms by trial and error, while mathematicians searched for mechanisms from analytical geometry.

Straight-Line Mechanisms are classified as:

 • Approximate Straight-Line Mechanisms
 • Exact Straight-Line Mechanisms

### Approximate Straight Line Mechanisms

#### The James Watt Straight-Line Mechanism (Linkage)

The Watt Straight-Line mechanism was invented by James Watt in about 1773.

The historical contextual: A steam engine condenses steam, or applies steam pressure, to the two sides of a working piston. It can 'huff' and 'puff' in the two directions.

However, before the 'Watt Straight Line Mechanism', piston steam engines would do 'work' efficiently only in one direction. This was because the piston rod had to pass through a gland. The gland had to seal the working steam volume from the atmosphere. If the rod moved sideways, the gland would usually leak. To make an seal that was trouble, efficient and leak-free, it was advantageous to be able to guide the piston rod in a straight line through the gland.

The Watt Straight-Line Mechanism was one of the first kinematic-chains used in this contextual. He used the mechanism to guide the piston-rod along a straight-line.

Watt also used a pantograph mechanism to increase the length of the straight line. He called the pantograph the Parallel Motion. See YouTube Video:

The video shows a Watt kinematic-chain in MechDesigner.

Input: is a Rocker – it rotates by 30º with a Rise-Dwell-Return-Dwell motion.

Output: is a Point on the coupler Part that moves vertically in a straight line - approximately.

 In reality, you would connect the Piston Rod to the point that moves in a straight line.

To assemble the Watt's Straight-Line Mechanism

 STEP 1: Edit the Base-Part to add the geometry for the Grounded Pin-Joints.

Add two Lines. One vertical and one horizontal.  Lines have Points. We will use these for Pin-Joints.

Step 1 is complete.

The horizontal distance between the Points is 200mm. Each Part that is joined to the Base-Part will be 100mm long. The vertical distance is 60mm. The coupler will be 60mm long.

Do Tutorial 1 again, or the Summary Steps of Step 10.1 in 'The Scotch-Yoke' Mechanism.

Here, the Part is 100mm long and the Base-Value of the Motion-Dimension is 90º.

Step 2 is complete

Dyads ALWAYS have 2 Parts and 3 Joints.

This Dyad is easy for you. If not, review Tutorial 2.

 1 Add two more Parts – one Part 100mm long, the other Part 60mm long.
 2 Add three Pin-Joints to assemble a four-bar mechanism, as shown to the left.

Step 3 is complete

Add a Point to the short Part

 STEP 4: Add a Point to the coupler Part.

Here, I have added a Circle.

I used a Mid-Point constraint between the CAD-Line and the centre-Point of the Circle.

[You may need to Zoom-In to see the end of the CAD-Line in an Added-Part.

If you click a Point and then the X-axis, and not the CAD-Line, then Add Mid-Point does not work].

Step 4 is complete

To show the straight line on the Mechanism-Plane, use a Trace-Point.

 STEP 6: Click the centre-Point of the Circle
 STEP 7: Click OK in the Command-Manager

 STEP 8: Edit the Base-Value of the Motion-Dimension to 75º

Double-click the Motion-Dimension FB icon to open and edit the Motion-Dimension dialog-box.

Step 8 is complete

You will notice that the Trace-Point is not straight at its top part.

This is because the input Rocker rotates too far.

 STEP 9: Edit the Motion to rotate the input Rocker by 30º

The Motion-Dimension moves 15º on both sides of the horizontal position.

Step 9 is complete.

Measure the vertical motion of the Trace-Point.

 STEP 10: Add a Measurement FB

Click the vertical Line in the sketch (at the left) and the centre-Point of the Circle on the coupler.

Step 10 is complete

The Rocker is at its most down position. The 'error' from 100 is 0.05mm.

At its most up position the distance is 100.05. Again the error is 0.05mm.

A percentage error of ±0.05% is good for such a simple mechanism, that is also very easy to remember.

Explore the ratio of the Parts to see if you can improve the straightness.

### Exact Straight Line kinematic-chains

#### The Peaucellier Straight Line kinematic-chain

The Peaucellier Cell kinematic-chain is the 'daddy' of 'Exact Straight-Line Mechanisms'. It gives a Point that draws an exact straight line for a range of movement of an input 'Rocker'.

Note: The 'Select Elements dialog-box' has been introduced since we wrote this part of the Tutorial. It helps you select the correct elements when there is ambiguity.

The video shows a Peaucellier Cell kinematic-chain in MechDesigner.

The 'Input': Is a Rocker – it oscillates 140º.

The 'Output': Is a Point that moves vertically.

 There are eight Parts (N), including the Base-Part, and ten Joints (J). [You may think there are only six Pin-Joints. However, there are four Points that each have two Pin-Joints.] The Gruebler Equation gives 1 degree-of-freedom: DOF = 3 * (N-1) – 2J = 1. It has a Mobility of Zero when we add a Motion-Dimension.

Add a Part for the input.

It is convenient to use the right-hand of the two Parts connected to the Base-Part. It rotates 140º in my kinematic-chain.

 STEP 1: Add the 'Input' Part – a Rocker.

Do Tutorial 1 again, or the Summary Steps of Step 1 in 'The Scotch-Yoke' kinematic-chain.

While we assemble the kinematic-chain, it is convenient to position the 'Rocker' to be horizontal.

Note 1: The Peaucellier Cell requires that length of this Part is equal to the distance between the Pin-Joints attached to the Base-Part. I have made the length of the Line and the Part equal to 100mm.

This Dyad is easy for you. If not, review Tutorial 2.

Add two Parts – one approximately 280mm long, the other approximately 120mm long. Edit the lengths if required.

Add three Pin-Joints to assemble a standard four-bar kinematic-chain as shown to the left.

You must add a Pin-Joint 'over the top' of an existing Pin-Joint.

When you use Add Pin-Joint, select the Point in the Free Part first, and the a Point in the existing Pin-Joint.

The Select Elements dialog-box opens.

The Point you select first is at the top of the list of Points in the dialog-box.

In the Select Elements dialog-box: CTRL + Select the Point at the end of the Free-Part and then a different Point.

Edit the lengths of the two new Parts to be the same as the equivalent Parts of the first Dyad.

The kinematic-chain should now look like the image to the left.

The second RRR Dyad is in the second closure of the first RRR Dyad.

As above, you add a Pin-Joint 'over the top' of Pin-Joint.

As before, click the Point in the Free Part first, and then the Point in the existing Pin-Joint.

Edit the lengths of the two new Parts in the third Dyad to be equal to the shorter Parts of the first two Dyads.

Four of these Parts should form a 'Diamond' shape. Each Part has the same length – 120mm if you have chosen the same lengths as above.

 STEP 5: Edit the Base-Value of the Motion-Dimension FB.

Double-click the Motion-Dimension FB to open the Motion-Dimension dialog-box.

Change the Base-Value to be 130º

The kinematic-chain should now look like the image to the left.