Kutzbach plan
The Kutzbach plan is a graphic process with which the speeds and directions of rotation of all wheels in a gear transmission can be determined. If the method is reversed, the dimensions of the wheels can be determined for a given gear ratio .
This process was developed by Karl Kutzbach , who has been researching and further developing gear drives as a professor at the TH Dresden since 1913 . It is used to advantage in planetary gears , in which the conditions are less easy to see in contrast to stationary gears.
Procedure
Their circumferential speeds are plotted in the radial direction of the wheels ( r axis) . They are straight lines (so-called rope rays), because the peripheral speed u on a wheel increases linearly with the radius r . The speeds of the wheels are proportional to the inclination of their cable beams with respect to the r axis. Opposite direction of inclination means opposite direction of rotation.
- The cable beams from wheels mounted in the frame intersect the r axis at the points of their axis of rotation ( u = 0).
- The rope rays of two wheels intersect where they mesh with each other. r values and peripheral speeds are the same. The directions of rotation are opposite.
This part of the Kutzbach plan is the so-called speed plan .
Investigation of a given transmission
In case 1 in the figure above, the bar (yellow) is fixed, i.e. added to the frame. An ordinary stationary transmission was created. The rope beam of the web coincides with the r axis. The sun gear (blue) is driven. Its speed of rotation is shown with the blue rope beam of any chosen inclination. The planet gear (red) meshes with the sun gear and consequently cuts its cable beam at the corresponding point on the r axis. Its axis of rotation is fixed together with the bridge, and its rope beam has a zero point there, which is the second point that determines it. Its extension to the rolling radius of the ring gear (green) with which it meshes results in a point for its straight rope. The connection from there to the intersection of the axis of rotation of the ring gear with the r axis is its cable line. If the ring gear or the planetary gear is driven, you first draw its straight line and the result is the straight line of the two other gears.
The speed and direction of rotation relationships are shown on a perpendicular to the r- axis (called speed line in the figure). This is cut by the rope rays, which all have to start from one point. A parallel line is drawn for the planet gear through the otherwise common point of intersection ( theorem of rays ). This part of the Kutzbach plan is the so-called speed plan .
The particular advantage of the Kutzbach plan becomes evident with a rotating web and rotating planet gears of a planetary gear. Sun or ring gear are held. In case 2 in the figure above, the ring gear is fixed. The sun gear or web are driven, and the web or sun gear is the driven part.
The relationships can also be represented when all parts rotate. There is only forced running if two of the three parts that can rotate in the frame are driven, the third is the driven part. In case 3 in the figure above, for example, the sun gear and the web are driven, and the rotation of the ring gear results from this. The plan looks basically the same if the sun gear and ring gear or web and ring gear are driven.
Dimensioning a gear
The pitch circle diameters of the wheels (and the center distances that depend on them) are to be determined at the specified gear ratio (or at specified speeds of the two wheels in the ratio).
The speeds are plotted on the speed line as vectors and the speed plan is created with them. The speed plan is then drawn with the help of the rope beams created in it. Some of the geometrical quantities are to be specified, all others are thus determined, that is, they result from the speed plan. If the resulting gearbox is not considered appropriate, the specifications must be changed and a new plan drawn up.
In the example of case 1 in the figure, a certain ratio is to be achieved between the speed of the sun gear and that of the ring gear. If, for example, the pitch circle diameter of the sun gear and the axial position of the planet gear on the web are specified, the pitch circle diameter of the ring gear and the planet gear are obtained. The rope beam of the planet gear is drawn.
Swamp's scheme
The Kutzbach plan is not sufficient for the exact determination of the revolutions and speeds due to the limited drawing accuracy. The geometric relationships represented by it are unambiguous and can therefore be precisely determined by calculation. One can use a summarizing mathematical scheme from Swamp for this purpose .
Individual evidence
- ^ A b Siegfried Hildebrand : Feinmechanische Bauelemente , Hanser 1968, page 543
- ↑ Hildebrand, p. 546
- ^ TU Dresden, Institute for Machine Elements: Drive Elements , Formula Collection