The Ross Linkage drive is a clever way to connect the linear motion of two pistons for either an alpha- or a gamma-type engine with the rotational motion of a crank shaft. A nice animation can be found at http://www.animatedengines.com/ross.html. The mechanism consists of a crank ( center at point 0 , radius 0-1 ) which is connected to the yoke (shaded triangle, vertices 1 4' 4 ). The motion of the yoke due to rotating the crank about point 0 is restricted by the rocking lever ( joints 2 3, length c ) which is free to pivot around the fixed joint 2. The location of joint 2 is described by the lengths a and b which are measured parallel to the axes of the shown x-y coordinate system. The angle β is a measure of the instantaneous angular orientation of the rocking lever. It changes with changes in the crank angle Θ. When the various parameters describing the dimensions of the mechanism are chosen properly the points 4 and 4' move vertically up and down with little side-wise motion and may therefore serve as connecting points to two piston moving in the vertical direction. The kinematics of this drive is quite subtle and no simple equation can be written down which readily describes the motion of the connecting points 4 and 4', in particular their lateral motion. A simplified analysis of this mechanism is discussed by Urieli & Berchovitz¹ and is also available at http://www.ohio.edu/mechanical/stirling/engines/yoke_vol.html. Essential parts of the analysis can also be found in the book by Organ² but no methods have been proposed as to how to choose values for the various parameters in order to minimize the lateral motion of points 4 and 4'. The program below will solve this problem to some extent.

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
r a b c L e d f
such that it is impossible for the crank to turn 360° around. The limiting case occurs when points 2 and 1 are exactly on opposite sides of point 0. In this case the sum of the length c and 1-3 has to be larger than the radius r plus the distance between points 2 and 0. Program Ross will terminate with a corresponding error message. Making length c of the rocking lever arm sufficiently longer will resolve the issue.

Starting program Ross

Into the dialog box below you must enter the eight parameters
r a b c d e f Li. In response program Ross will give a lot of information about the curves the two connecting point, 4 and 4', produce as the crank turns 360 degrees. An animation is provided as well.
The choice of units is yours. The program does not care as long as you choose the same units for all of them.

Option 1 : You start with a geometry Andy Ross³ first used.

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.

 r a b c d e f L Fairly arbitrary values along the lines of above recommendations. The data for this example were obtained with program 4bar using the optimization feature of that program. The input to program 4bar was : desired stroke for point 4 equals 20, the lateral motion of point 4 (x4diff) equals 0.1 and e/a = 0.6. The choice of setting L=e is not prescribed by program 4bar but might be a good choice in general.

Note that you may copy and paste to a text-file the content of above entry box from which you can retreive your data at a later point.

References

¹Urieli 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

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