Putting aside for the moment the option of connecting a piston to point 4', we concentrate on the motion of point 4 which is facilitated by the rocking lever 2-3, the crank 0-1, plus only the right side of the trianglular plate described by the points 1-3-4. This reduced drive is now essentially what is known in the kinematics community as a 4-bar linkage and it is helpful to analyze that part first and get a first feeling on how varied the curve point 4 outlines for each complete rotation of the crank shaft. These orbits can take on the shape of a simple oval all the way up to a double-eight figure.
Program 4bar : Analysis of 4-bar linkage with links to images of various orbits of point 4
At times the term "Ross Linkage" is preceeded by the term "symmetric" which means that the distance 1-4' and 1-4 are equal and that joint 3 is located on the line 4-4' half-way inbetween its end points. Of course these conditions do not render the mechanism completely symmetric because of the rocking lever 2-3.
We first specify the solid triangle with its vertices 1, 4, 4'. Point 5 is an auxiliary point located such that the dashed line 1-5 is perpendicular to the line 4-4'. The connecting point 3 is located exactly on this line with f being the distance between point 3 and 5 and d the distance from point 1 to 3. Negative values for f are allowed, Fig. 3 shows f with a positive value. The distance 5-4 is measured as e and the distance 5-4' as L. The angle γ signifies the instantaneous angular orientation of the triangle with respect to the same x-y coordinate system as shown in Figure 1. Of course γ changes as the crank 0-1 turns.
The derivation of the relevant equations is
described here. One hint though before you dive into using the program.
It is fairly easy to specify values for the eight parameters
Starting program Ross
Into the dialog box below you must enter the eight parameters
To produce a useful geometry for your Ross linkage you can of course follow your own ideas or follow one of the following two options :
Option 1 : You start with a geometry Andy Ross³ first used.
f = 0 ; L = e = b = d = half of desired horizontal distance between point 4 and 4'
a = c > b
Choose r to achieve desired stroke
Obey : r ≤ c + (d+f) - sqrt(a²+b²)
Option 2: Use the optimization feature of program Program 4bar to provide you with values for the first seven parameters, r a b c d e f, with very limited input. Connecting point 4 will move almost exactly on a vertical straight line (you specify the maximum lateral motion) and will have the specified stroke. Program 4bar will not provide any information about the length L but choosing L=e will result in a stroke for point 4' approximately equal to that of point 4 and you can adjust accordingly. Program 4bar needs the ratio e/a as input. As this ratio increases from 0.4 to 0.7 for example the the phase angle between the motion of point 4 and 4' moves closer to 90°.
Please observe the order in which the eight parameters have to be provided.
References¹Urieli I and Berchovitz D M , Stirling Engine Cycle and Analysis, Adam Higler, Bristol, 1984
² Organ Allan J. , Thermodynamics and Gas Dynamics of the Stirling Cycle Machine, Cambridge, 1992
³ Andy Ross, Making Stirling Engines, 1993