Over the years several mechanical drives have been proposed for various reasons, most importantly to reduce sidewise forces onto pistons or sliding parts with the objective to reduce friction and wear and satisfying constraints imposed by the thermodynamics of Stirling engines. Literally, hundreds of different kinematic mechanisms have been invented over the past centruries, a nice collection of these can be found at Cornell University's Model collection. Follow the link to "Models" and click on any of the 3 collections offered to get a nice menu for all documented models. Look for Straight-line Mechanism ( in particular model S35, Peaucellier ) if you are interested in the conversion from straight-line to rotary motion and back.
To first order, two properties of a drive mechanism influence the performance of Stirling engines, the volume amplitude ratio and the phase lag between expansion and compression space.
The volume amplitude ratio, VAR , is defined as the change of the volume of the compression space divided by the change in volume of the expansion space during a complete revolution. VAR = 1 is suggested by many researchers for this quantity.
The volumetric phase lag , VPL , refers to the angular offset between the volume of the compression space as function of crank angle and that of the expansion space. VPL = 90° is an often cited optimal value with the compression space lagging behind the expansions space. For strictly sinusoidal function this definition of phase lag is unique for others one might look at the phase lag of the maxima and the minima of compression and expansion space.
The figure to the left shows schematically those parts of a
beta-type Stirling
engine pertaining to the drive mechanism. Both cranks ,
re and rp,
rotate in counter-clockwise direction in unison as the crank angle &Theta is increased but with a constant angle δ,
referred to as the crank offset angle, in between. During rotation the vertical
distance between the displacer and the endpoint of the crank re
remains constant as does the vertical distance between the power piston
and the endpoint of the crank rp. As a result the displacer
and the power piston move in sinusoidal fashion and the expansion space,
Ve
above the displacer, and the compression space, Vc between
displacer and power
piston, vary in sinusoidal fashion.
Because of the geometric simplicity, only two parameters, the ratio of the crank throws,rp/re, and the crank offset angle δ, are available to achieve particular values for the thermoynamic design design objectives. In particular, if we demand VAR = 1 and VPL = 90° only a single choice : rp/re≅1.4 δ≅45° is available. Click here for further details.
Figure 1 : Schematics of rhombic drivewith power piston connected to the upper and displacer to the lower section. |
In Figure 1 we show the schematics of a rhombic drive with power piston
connected to the upper and displacer to the lower section. The bars
12, 1'2', 13, and 1'3' have identical length, L, and are connected
to the cross-bars
22' and 33' by pin/hole connections. Joints 1 and 1' are pin-hole connections
as well. The crank throws 01 and 0'1' have identical length, r,
and the crank centers 0 and 0' are at equal distance, d+e, from
the piston axis. The size of the cross bars, 22' and 33', d,
has no bearing on the analysis of the variation of the height of
the expansion space, Ve, and the compression space,
Vc, as the angle α changes. In the configuration shown, the left crank will turn clockwise and the right crank counter clockwise in order to achieve proper phase lag between expansion and compression space. They turn at the same angular velocity which can be accomplished by two intermeshing counter-rotating gears. |
This drive has been analyzed many a times before. Click here for relevant equations and numerical results concerning phase lag , VPL , and the volume amplitude ratio, VAR, as function of the dimensions of the drive.
The symmetric Ross drive mechanism consists of a crank ( center at point 0 ,
radius 01 ) which is connected to the solid triangle ( points 1 4' 3 4 ).
The motion of the triangle due to rotating the crank about point 0 is
restricted by the swing arm ( points 2 3 )
which is free to pivot around point 2. 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 equation can be written down which readily describes the motion
of the connecting points 4 and 4'. A simplified form of this mechanism
is discussed by Urieli & Berchovitz¹ . 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 optimize the motion of points 4 and 4'.
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 swing arm 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.
Program 4bar : Analysis of 4-bar linkage
The term symmetric refers only to the solid plate, meaning that the distance 3-4' and 3-4 are equal and that the line 3-1 is perpendicular to the line 4-4'.
The mechanism consists of an upper and lower section which
in its original conception share a common crank ( point 1=1' ) and have
completely identical dimensions, including the position of their respective
pivot points ( 2 and 2' ).
In essence, each section comprises what is known in the kinematics community as a 4-bar linkage and it is in principal sufficent to analyze one of them.
Program 4bar : Analysis of 4-bar linkage
Program symm_bowtie : Analysis of symmetric bowtie drive.
A property of the symmetric bowtie-configuration is that a swept-volume ratio of 1 and a phase-lag of less than 120° can NOT be realized simultaneously. Click here for details.
Program bowtie : Analysis of un-symmetric bowtie drives.
