Rotor on overhung shaft
The equations of motion for a rotor mounted on the end of a rotating cantilever are derived in analytically and complex variables are introduced. Physical interpretations of the different terms of the equations are given.
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Rotor on overhung shaft
2. Rotor on overhung shaft
2. 1. Equations of motion. The equations of motion for a rotor mounted on the end of a rotating cantilever are derived in analytically and complex variables are introduced. Physical interpretations of the different terms of the equation.s are given. Fig. 1 shows the notations
n
and coordinate systems. The origin 0 of the fixed 1 2 3 coordinate system coincides initially with the center W of the rotor. 1 -2 -3 coThe body-fixed = ordinate system, with its
origin at the center of mass S
of the rotor, is directed
3-axis is paralso that the = lel to the axis of symmetry
-
of the rotor, that the center
w
Fig. 1 ity is
ei. =
[- e ) 0 J 0
lies on the 1-axis and
that the vector of eccentric-
J
T.
Only eccentricity of the center of
mass is taken into account. Let the rotor itself be symmetricG. Schweitzer, Critical Speeds of Gyroscopes © Springer-Verlag Wien 1972
8
Rotor on overhung shaft
al, its tensor of inertia is then
J ..~~ =
A
0
0
0
A
0
0
0
c
The transformation between the two coordinate systems is expressed by the cardanic angles mation matrix
a, f.>, y with the transfor-
of Fig. 2. c .. ~~
Fig. 2
I q. = c ~I< c
=
coso: sin
kt
- cosfJ cosy
cos]l cos"'{ C·· ~A
•t =I
1+
COS0 is valid, i.e.
if
the support of the rotor is statically stable.
Then the gyroscopic matrix
G cannot remove stability no mat-
ter how fast the rotor turns or if the rotor has an elongated form ( C < A) or a flat one
( C > A) •
The motion will in one form or another be subject to damping. Let,for example,the disk vibrate in a damping fluid, supposing that the damping force is proportional to the rate of deflectio~ then (2. 1 0) has to be completed by a damping matrix
As
0 is positive definite the aforesaid stability criter.i
on, that was only sufficient, now becomes necessary as well. A stable solution of (2. 10) will now be asymptotically stable. This special kind of damping is named external damping, as the damping forces react on the externally fixed parts of the sup-
16
Rotor on overhung shaft
port. By way of cont:rast an internal damping, resulting from internal friction in the material of the shaft, cannot improve the system 1s behavior but can even cause instability as shown in chapter 6. 2. Now the eigenvalues of the system {2. 12) are determined. The function
w
.we At
satisfies the homogeneous part of (2. 12) if the characteristic equation of A hol~s
ciet [
MA.l- ~GA. A. •
In detail, with
+
~ 00 ,
F] -
0 •
it follows
(2. 15) This equation has four real roots that are plotted in Fig. 3 against rotor velocity as dimensionless variables. It is seen that there are two positive natural frequencies and two negative ones zero rotation where
W1
(A) 1
,
w3
(1)2. , (J) 4
that are all different except for
=- oo 2 and
0> 3
=- (1) 4 • The curves are
symmetrical which means that for + .0. and -
.n
the same
values occur, in other words, the four natural frequencies do not care whether the disk rotates clockwise or counterc
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