Mechanical Drives
Program 4bar

1. Introduction

In the mechanical community the drive depicted in Fig. 1 is called a variation of a 4-bar linkage. Among other things it is able to convert rotational motion of one point ( point 1 in the figure ) into a nearly linear motion of another point ( point 4 in the figure ). This type of linkage provides the foundation of what are known in the Stirling engine community as the Ross linkage as well as the bowtie mechanism.

Unfortunately, a vertical, straight line motion of point 4 is achieved only approximately and if the various lengths r, a, b, c, d, e, and f are not chosen wisely the entire linkage is rather useless. Care and some patience are needed to find a satisfactory configuration. A first goal one might want to achieve in selecting values for the seven parameters is to obtain a high ratio of the achieved stroke to the lateral motion of point 4. We address this ratio as the quality of your drive. This quality can go as high as one hundred which makes it possible to have point 4 directly connected to a piston. A warning though, a high value for the quality puts a high demand on the manufacturing accuracy, maybe out of reach for many of us. Information in form of contour plots should help you in making a sound decision. To give you a feeling for the range of curves (orbits) point 4 moves on as the crank ( center at point 0 ) rotates once around click on the following links : oval , ovals with cusps , ovals with "ears" , leaning double-8 , double-8 with vanishing waist , and perfect double-8 orbit.

Figure 1 : Principle layout of 4-bar linkage
4connecting point, to be moved along a vertical line as accurately as possible
01crank, length r, rotating about fixed point 0 = origin of x-y coordinate system
23swing bar, length c, pivoting about fixed point 2. The position of point 2 with respect to point 0 is given by the lengths a and b
1354 bellcrank = rigid body, pin/hole connection at points 1, 3 and 4 with distances :
d from 1 to 3; f from 3 to 5 (positive as shown, negative when 5 is closer to point 1 than 3); e from 5 to 4.
Line 1-5 is perpendicular to line 5-4

For detailed information about the inner workings of the 4-bar linkage feel free to browse the mathematics behind the 4-bar drive page. Related to that is the scaling property of the 4-bar linkage which means that if you multiply all of your seven parameters ( a .... f ) by the same factor then the stroke and lateral motion of point 4 will change by the same factor. The ratio of stroke to lateral motion of point 4 ( called quality ) of course remains unchanged. For that reason I often first find a suitable configuration with r=1 and then scale to the desired stroke. Another consequence of this scaling property is that the units in which you express the various length is inconsequential as long as you use the same units for all parameters.

2. What Information can I expect from the program

The program is written in javascript. As a consequence the entire code is downloaded from my server but runs solely on your computer using the javascript capabilities of your browser. I am using Firefox with Ubuntu ( a Linux version ) as operating system with excellent success. Old versions of Firefox and IE will probably fail because such browsers are not HTML-5 compatible. Assuming that your browser works perfectly you have two options to utilize the program. You either supply your own values for the seven parameters mentioned above or you provide just two input variables, a/r and e/r, and let the program suggest the remaining parameters by finding a semi-optimal configuration ( = set of parameters). In either case you will be provided a generous amount of information about your configuration with ample information in graphical form.
In a nut shell : you will see quite a bit of numerical output like the obtained stroke and information about the lateral motion of point 4. This is followed by an animation which you can start and stop or advance step wise as you see fit. This animation reflects precisely your chosen drive. Subsequently you will see an image of the orbit of point 4 with the lateral (x) coordinate scaled up for easy viewing. Documentation about your drive is concluded with a plot tracing the vertical motion of point 4 as function of crank angle Θ together with a matching sine-wave.

After that, the program provides information in terms of several contour plots. Each shows you how the so-called quality ( = stroke divided by extent of lateral motion of point 4) changes as function of two of the seven parameters. Often these contour plots clearly show how quickly the quality can change in response to small changes of one or the other parameters. Due to that, the employed plotting method at times is not capable of resolving the sharp changes in quality with the default zoom factor of 1. Choosing a zoom factor greater that 1 is helpful in such cases.

In order to generate all of this information, particularly the graphical part, the program needs a certain amount of computation time. Depending on your browser setting and your computer's processing speed your browser may issue a message asking you to either stop or continue to run a javascript program. This is normal, so please click on the continue-button.

3. Investigate your own configuration

Before you bother to enter the values of your own configuration you might perform a simple check on the values of your parameters. Your values have to pass the following tests :

(1)       c < √ a² + b² - ( r - d )

(2)       c > √ a² + b² + ( r - d )

Both of these conditions have to be satified simultaneously to insure that the crank can turn all the way (360°) around, in particular when points 0, 1, and 2 are on a straight line (either in order 0, 1 , 2 or in order 1, 0, 2).

An immediate consequence is the requirement that :

(3)       d > r

Step 1 : Populate the box below with your data manually or use one of the buttons to do so

Step 2 : Choose zoom factor
>1 will zoom in
<1 will zoom out
Step 3 : Activate computations and display of results

4. Find a semi-optimal configuration

The search for configurations with high quality ( ratio of vertical to lateral extent of the point 4 in Figure 1 ) can be quite frustrating. The method employed in this section is based on two observations. First, the quality of configurations always improves if the ratios a/r and/or e/r become larger. This can be readily seen from inspection of Figure 1. An increase in the length a immediately demands an increase of the distance c. With that the angle β in Figure 1 becomes smaller and smaller and point 3 will move up and down with little deflection in the x-direction. In similar fashion one can argue about the influence of an increasing size of length e. Second, it seems to me that an orbit of point 4 which resembles that of a completely balanced double-eight as shown here might be particularly advantageous. This was confirmed by various contour plots in which we display the influence of pairs of parameters (like quality versus lengths b and c, etc.). The contour plots always show that varying any of the parameters b,c,d, at fixed a/r and e/r either individually or in pairs leads to a decrease in the quality.

We cannot prove though the configurations produced by our program are indeed optimal and therefore use the phrase semi-optimal.

Figure 2a : Quality and stroke/r as function of a/r ≤ 8 and e/r ≤ 8
The"radial" lines represent constant stroke lines with values
of stroke/r as indicated by the numbers next to the lines.
The arced lines are lines of constant quality with values ranging
from 20 to 400


Figure 2b : Quality and stroke/r as function of a/r ≤ 3 and e/r ≤ 3


In Figure 2a and 2b (subset of Figure 2a) shows the quality and stroke/r as function of the variables e/r and a/r for these semi-optimal configurations. As an example take the case of a/r=2 and e/r=3. The corresponding configuration has a quality of about 40 (purple arc) and a ratio of stroke to radius, r, (torquois "straight" line) of the crank of about 3.2.

The program, which you can access below, requires as input values for the quantities a/r and e/r and provides you with the entire configuration and displayes various information about it.

Enter value for a/r : ( 1.1 .... 8.0 )
Enter value for e/r : ( 0.8 .... 8.0 )
Enter zoom factor : ( > 1 zoom in ; < 1 zoom out )

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Zig Herzog © 2014
Last revised: 11/06/15