The Dynamics of Rotating Machines with Cracks ... - Michael I Friswell

312
Modern Practice in Stress and Vibration Analysis
2 Title of Publication (to be inserted by the publisher)
2 The Mayes model for an opening and closing crack
This model is defined, in rotating coordinates, by
and
k
k
kM
kDC1
(1a)
kM
kDC1
(1b)
where k
1
M k0
k
, k
1
D
k0
k
, etc., and C
2
2
1 cos . When
closed and kk k0
. When cos 1
k k .
the crack is fully open and
cos 1 the crack is
k
k ,
The stiffnesses in fixed coordinates can be determined by transforming from rotating coordinates so
that
T
K F T K RT
(2)
where
C1
S1
T . (3)
S1
C1
In (3), S1 sin
. Carrying out the matrix multiplications of (2) and expanding the trigonometric
expressions in multiple angles gives
1
K
1
k
k S
k
k S
F 12 2 M M
2 4 D
D
1 S3
or
K 1
k k S S S
1
F
12 4 2
2 1 3
(4)
where S
and
3 sin 3 , etc. Similarly
1 1
1
F D D D D 3
D D D D
K k 11 0 k k k k C
2 2 2 k k C
4
1 k k C3
(5)
1 1
1
F D D D D 3
D D D D
K k 22 0 k k k k C
2 2 2 k k C
4
1 k k C3
. (6)
Now, if t , where is the rotor angular velocity, then in fixed coordinates the Mayes model
generates a constant term plus 1X, 2X and 3X rotor angular velocity components in the diagonal
stiffness terms and 1X, 2X and 3X rotor angular velocity components in the off-diagonal stiffness
terms.
3. Approximating the Equations of motion
The analysis may be performed in fixed or rotating coordinates. If the bearings and foundations are
axi-symmetric then the stator dynamic stiffness will appear constant in the rotating frame, and there
is some benefit in analysing the machine in rotating coordinates. Typically foundations will be