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'.

Figure 2 displays a more general form of the Ross linkage with the aim to produce motions of points 4 and 4' which have less lateral movement. In this figure the points 1, 3, 5, and 5' lie on a dashed straight line. Lines 4'-5' (=length e') and 4-5 (length e) are both perpendicular to this line. The distance between points 3 and 5 is called if, which assumes negative values if point 5 lies between points 1 and 3. The same holds true for the points 3 and 5' whose distance is f '. The distance between points 1 and 3 is called d. This setup reduces to the classical (sometimes called symmetric) Ross linkage if f=f '=0 and e=e'.
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.

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 triangular 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 orbit of point 4 might be. 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 including some optimization plus links to images of various orbits of point 4

One hint though before you dive into using this program. It is fairly easy to specify values for the parameters
r a b c d such that it is impossible for the crank to turn 360° around. As a result your values have to satisfy the following three conditions simultaneously :

(1)       c < √ a² + b² - ( r - d )

(2)       c > √ a² + b² + ( r - d )

(3)       d > r

Condition (3) is a consequence of (1) and (2)

#### Starting program Ross

Into the dialog box below you must enter the nine parameters r a b c d e f e' f '. 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.

### Choose r to achieve desired stroke

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 e' but choosing f ' will result in a stroke for point 4' approximately equal to that of point 4 and you can adjust accordingly. The optimizing mode of program 4bar needs as input only the ratios a/r and e/r and determines the remaining parameters.

For either option the choice for the parameters e' and f ' is yours. You might want to start with e'=e and f '=0. This program then will suggest to you a better value for f ', but uses the value you have specified. In general, an increase of e' with optimal f ' will increase the stroke and quality of point 4' and shifts the phase lag between points 4 and 4' to higher values.

Please observe the order in which the nine parameters have to be provided.

 r a b c d e f e' f ' 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 : a/r=2 and e/r = 2 at r=1. Adding to that e'=e=2.0 and f'=0 as my rather arbitrary choice.

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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