Entry
Physics: Dynamics: 3D: Rotation: Euler: Equation: What are the Euler rotation equations?
Feb 2nd, 2006 13:13
Knud van Eeden,
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--- Knud van Eeden --- 30 January 2021 - 09:07 pm --------------------
Physics: Dynamics: 3D: Rotation: Euler: Equation: What are the Euler
rotation equations?
---
Ixx . d/dtime( xAngularVelocity ) +
( Izz - Iyy ) . zAngularVelocity . yAngularVelocity =
SUM( xPositionVectorForce X xForce )
Iyy . d/dtime( yAngularVelocity ) +
( Ixx - Izz ) . xAngularVelocity . zAngularVelocity
= SUM( yPositionVectorForce X yForce )
Izz . d/dtime( zAngularVelocity ) +
( Iyy - Ixx ) . yAngularVelocity . xAngularVelocity
= SUM( zPositionVectorForce X zForce )
===
1. To recognize this structure, in case that the rotation takes place
around 1 axis only, say the x-axis, and no rotations around the
other axis, then all the angular velocities, but the one around the
x-axis are zero. In that case this formula reduces to
Ixx . d/dtime( xAngularVelocity ) =
-------------------- ------
SUM( xPositionVectorForce X xForce )
To recognize it better, letting Ixx equal to I, and assuming for
simplicity that the force is perpendicular to the position vector, you
get the familiar formula
I . d/dtime( xAngularVelocity ) = r . F
or thus
.
I . d/dtime( angle ) = r . F
or thus
..
I . angle = r . F
or thus
..
angle = r . F
-----
I
you can then calculate the angle by integrating twice
.
angle = Integral( r . F . dt )
-----
I
integrating again
angle = Integral( Integral( r . F . dt ) . dt )
-----
I
---
To further simplify, assuming that r, F and I are constants this
becomes thus respectively
.
angle = r . F . time + constant 1
-----
I
and
2
angle = 1 . r . F . time + constant1 . time + constant2
- -----
2 I
This is by the way the equation of motion for a circular disk on which
a constant force is working. You see there the rotated angle as
function of time.
.<--------Force
. ^ .
. | .
| r
. | .
x
. .
. .
. circular disc with moment of inertia I
This is thus the rotational equivalent of a constant linear
acceleration (e.g. a stone falling under influence of gravity), which
gives you a linearly growing velocity, and a quadratically growing
distance.
===
Symmetry:
Notice further that the above simple example should be similar for each
axis, thus the above simple example should be similar when rotating
only around the x axis, versus only around the y-axis, versus only
around the z-axis.
This is shown in the first part of the formulas, which are clearly very
similar in structure.
Ixx . d/dtime( xAngularVelocity )
Iyy . d/dtime( yAngularVelocity )
Izz . d/dtime( zAngularVelocity )
===
Influence of rotation and moment of inertia of other axes:
The other terms show that the rotation around the other axes
has also an influence
e.g. the term
( Izz - Iyy ) . zAngularVelocity . yAngularVelocity
which includes moments of inertia and rotations around the
other axis, thus influences the rotation around the x-axis.
This other axes influence is a symmetric influence
(the contribution of both axes is equivalent, which
is expressed in equalness of the terms in the formula)
===
The mathematical structure of the Euler equation is a non linear system
of coupled, second order, ordinary differential equations in 3
variables.
The structure looks similar like:
.. . .
constant1 . x + constant2 . y . z = constant3
.. . .
constant4 . y + constant5 . x . z = constant6
.. . .
constant7 . z + constant8 . x . y = constant9
A specific example of this looks similar like:
.. . .
3 . x + 6 . y . z = 15
.. . .
4 . y + 2 . x . z = 9
.. . .
8 . z + 11 . x . y = 10
You could then solve this in the usual way, e.g. using Runge Kutta or
Euler methods of numerically solving systems of differential equations.
Given are the initial
. . .
x, y and z
The solution gives you then the dependent variables x, y and z as
function of the independent variable (e.g. t). Or thus translated back
to the Euler rotation equation, the solution gives you the angles
rotated around each axis, as function of the time.
===
The Euler rotatation equations and rotation around the principal axes:
[book: source: Joos, Georg / Freeman, Ira M. - Theoretical Physics Ira
M Freeman - publisher=Courier Dover Publications - p. 844]
Given this 3 Euler equations:
.. . .
I1 . x + (I3 - I2) . z . y = Mx
. . .
I2 . y + (I1 - I3) . x . z = My
. . .
I3 . z + (I2 - I1) . y . x = Mz
---
If
-
M equals zero
then this equations are for example satisfied if
-- or --
.
x = constant, in other words the rotation velocity
around the x-axis remains constant
.
y = 0, in other words no rotation around
the y-axis
.
z = 0, in other words no rotation around
the z-axis
-- or --
.
x = 0, in other words no rotation around
the x-axis
.
y = constant, in other words the rotation velocity
around the y-axis remains constant
.
z = 0 in other words no rotation around
the z-axis
-- or --
.
x = 0, in other words no rotation around
the x-axis
.
y = 0 in other words no rotation around
the y-axis
.
z = constant, in other words the rotation velocity
around the z-axis remains constant
---
That is, in the absence of any torque, a rigid body
can continue to rotate about one of the principal
axes.
---
Stability considerations show that rotation about
either the axis of greatest or least inertia will
be stable, while that about the middle one will
be unstable.
===
Internet: see also:
---
Physics: Dynamics: 3D: Link: Overview: Can you give an overview of
links?
http://www.faqts.com/knowledge_base/view.phtml/aid/39259/fid/1857
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