Degree of running

The degree of movement F used in transmission technology (also degree of mobility or degree of freedom of the transmission ) determines the number of mutually independent movements that must be initiated in a given coupling mechanism so that it is inevitable.

" Inevitability is the property that every position of any link is clearly assigned the position of the other links".

For a simple kinematic chain with one driven link, F = 1, for two independently driven links, F = 2, and so on. A running degree of zero characterizes a rigid chain.

When determining the degree of running with the help of the Grüblers equation , the respective mobility of all joints occurring in the transmission is used as their respective degree of mobility or degree of freedom . In static systems , the degree of freedom of each of the connections between the components is used to examine whether the system (e.g. a supporting structure ) as a whole is in an unrestrained immobility, whether it is statically determined . ${\displaystyle b}$ ${\displaystyle f}$ ${\displaystyle f}$

Gruger equation

→ Main article : Grübler 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).

Compulsory condition

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

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

Values ​​of the degree of running (gear technology)

• ${\displaystyle F\geq 1}$ : The transmission is operational.
  • ${\displaystyle F=1}$ : The gearbox with one drive is inevitable.
  • ${\displaystyle F>1}$ : The gearbox is inevitable if it has more than one drive. F = 2: two drives; F = 3: three drives; ...
• ${\displaystyle F\leq 0}$ : The gearbox cannot run, is rigid. The exception is the so-called locked gear .

Values ​​of the degree of freedom ( structural engineering )

The symbol for the run degree F .
The symbol for the degree of freedom is f .
For comparison with the above values ​​for F and their respective meanings, the following are the values ​​for f and their respective meanings.

• ${\displaystyle f>0}$. There is a statically undetermined or indeterminate system. The structure (construction) is not stable.
• ${\displaystyle f=0}$. There is a statically determined system. The structure is stable.
• ${\displaystyle f<0}$. There is a statically overdetermined system. There are competing links that hinder the free distribution of forces.

Individual evidence

1. ^ Johannes Volmer : Getriebetechnik - Koppelgetriebe , Verlag Technik, Berlin, 1979, p. 29/30
2. ↑ ^{a b c} Denis Jung: Formula collection gear theory , page 5 ( PDF; 907 KB ( Memento from December 14, 2009 on WebCite ))
3. ↑ Manfred Husty, Adolf Karger, Hans Sachs, Waldemar Steinhilper: kinematics and robotics . Springer-Verlag, Berlin Heidelberg 1997, ISBN 978-3-642-63822-0 , p. 281 .
4. ^ Siegfried Hildebrand : Feinmechanische Bauelemente , Hanser 1968, page 628