Moment of inertia

The moment of inertia is a physical quantity that does not conform to the SI , which was previously often used instead of the moment of inertia in rotating machines .

From the definition

${\displaystyle SM=G\cdot D^{2}=G\cdot 4\cdot i^{2}}$

With

• Weight  G
• "Inertia diameter" (i: radius of gyration ) ${\displaystyle D=2\cdot i}$

follows as the unit of measurement : kilopond times meter squared (kp · m ² ).

Relationship with the moment of inertia

The moment of inertia of an arbitrarily shaped rigid body with respect to an arbitrary axis of rotation can be expressed purely formally with the moment of inertia of a point mass of equal mass : ${\displaystyle J}$ ${\displaystyle J_{p}}$ ${\displaystyle m}$

${\displaystyle J=J_{p}=m\cdot i^{2}=m\cdot \left({\frac {D}{2}}\right)^{2}}$

To do this, the distance between the point mass and the axis of rotation must be selected so that the moment of inertia of the point mass corresponds to the moment of inertia of the rigid body under consideration. This distance is called the radius of gyration, twice the value corresponding to the diameter of gyration D. ${\displaystyle i}$

The mass is given by its weight on the earth's surface: ${\displaystyle G}$

${\displaystyle m={\frac {G}{g}}}$

with the acceleration of gravity . ${\displaystyle g}$

This is listing

${\displaystyle {\begin{aligned}\Rightarrow J&={\frac {G\cdot D^{2}}{4\cdot g}}\\\Leftrightarrow G\cdot D^{2}&=4\cdot g\cdot J\end{aligned}}}$

Accordingly, the moment of inertia is identical to the moment of inertia except for the factor . ${\displaystyle GD^{2}}$ ${\displaystyle 4g}$

Individual evidence

1. ^ R. Fischer: Electrical machines , 2009, Carl Hanser Verlag Munich, ISBN 978-3-446-41754-0
2. ^ A. Böge: Mechanics and strength theory , 1971, Vieweg Verlag, ISBN 978-3-5281-4010-6
3. ↑ solution Note 13.3 in machine elements of exercises , Roloff / Matek, 14th edition 2007, Vieweg Verlag, ISBN 978-3-8348-0340-5