Gruger equation

The Grübler's equations were set up in 1917 and 1918 almost simultaneously and independently of each other by both Martin Fürchtegott Grübler (1851-1935) and Maurice d'Ocagne . They are used in technology to determine the mobility of gears and their degree of running . The mobility of the joints connecting the transmission parts is considered in relation to the number of links. A distinction must be made as to whether these movements take place in the plane (plane gear) , on a curved ( spherical ) surface (spherical gear) or arbitrarily in space (spatial gear) .

the equation

The general form of Grübler's equation is:

${\displaystyle F=\ T\cdot (n-1-g)+\sum _{i=1}^{g}b_{i}}$

it stands

• Q: Degree of running
• T: type of gear (T = 6 for three-dimensional, T = 3 for spherical or plane gear)
• n: number of transmission links (including the frame)
• g: number of joints
• b _i : mobility of a single joint i (b _i = 1, 2, ...).

Spatial transmission

T = 6

${\displaystyle F=\ 6\cdot (n-1-g)+\sum _{i=1}^{g}b_{i}}$

Flat and spherical gear

T = 3

${\displaystyle {\begin{aligned}F&=\ 3\cdot (n-1-g)+\sum _{i=1}^{g}b_{i}\\&=\ 3\cdot (n-1-g)+c+2\cdot d\end{aligned}}}$

It stands

• c: number of joints with mobility b _i = 1
• d: Number of joints with mobility b _i = 2 (e.g. rolling and sliding at the contact points of spur gear flanks or combined thrust-pivot bearing).

Result

The degree of running F of the gear must be ≥1 so that the gear can move:

• if F = 1, the gear is called inevitable . This corresponds to the usual use in which one link is moved (driven) and the other links inevitably follow it.
• with F = 2, the gearbox requires two drives that act on two different links in order to move it in a defined manner.

The inevitability of a gear (F = 1) is checked with the so-called Grübler's forced running condition - a corresponding change in the Grübler's equation. For planar gears, which only have joints with b _i = 1 (d = 0 and c = g), the condition is:

${\displaystyle 2\cdot g-3\cdot n+4=0}$

Individual evidence

1. ↑ Friedrich melting, Erich Aucktor: joints and drive shafts: Analysis, Design, Applications , 1988, Springer-Verlag , ISBN 3540417591
2. ↑ Martin Fürchtegott Grübler : Gear theory. A theory of the forced running and the plane mechanisms. VDM Verlag, 2007, ISBN 3836404265
3. ↑ ^{a b c d} Denis Jung: Formula collection gear theory , page 5 ( PDF; 907 kB ( Memento from December 14, 2009 on WebCite ))
4. ↑ Manfred Husty, Adolf Karger, Hans Sachs, Waldemar Steinhilper: kinematics and robotics . Springer-Verlag, Berlin Heidelberg 1997, ISBN 978-3-642-63822-0 , p. 281 .
5. ^ Siegfried Hildebrand : Feinmechanische Bauelemente , Hanser 1968, page 628