Moment of deviation

The deviation moment (often called centrifugal or by- moment of inertia referred to) is a physical size as a measure of the tendency of a rotating body ( gyro ), its axis of rotation to change, can be considered. ${\displaystyle J_{xy}}$

It always occurs when a body does not rotate about one of its main axes of inertia . Deviation and inertia moments are combined to form the inertia tensor , the deviation moments are its secondary diagonals .

From a mathematical point of view, the moment of deviation is an expression of form

${\displaystyle J_{xy}=\int x\cdot y\cdot \mathrm {d} m}$

with the crowd . ${\displaystyle m}$

Derivation

The centrifugal force , from which the term centrifugal moment is derived, is:

${\displaystyle F_{\mathrm {Z} }=m\cdot \omega ^{2}\cdot r}$

With

• the angular velocity ${\displaystyle \omega }$
• the radius . ${\displaystyle r}$

Here you can see the ground through an infinitesimally replace dm little added mass and the mass integrate :

${\displaystyle F_{\mathrm {Z} }=\omega ^{2}\cdot \int r\cdot \mathrm {d} m}$

By introducing an xy coordinate system, is . ${\displaystyle r=x}$

If there is now a lever arm  for the centrifugal force in the system, a torque acts on the system , of which the deviation moment is a part: ${\displaystyle y}$ ${\displaystyle M=F_{\mathrm {Z} }\cdot y}$

${\displaystyle M=\omega ^{2}\cdot \int x\cdot y\cdot \mathrm {d} m}$

The lever arm can only occur if the body does not rotate around its main axes of inertia, otherwise any lever arms for all infinitesimally small mass components would add up to zero. For this reason, the moment of deviation is also a measure of the asymmetry of a body.