2.2 Model of a 2-DOF Cracked Rotor with AMB Support
2.2.3 Modeling of the Crack
A schematic representation of a transverse crack and the associated loads along coordinate directions is shown in Figure 2-3.
Figure 2-3 A rotor element showing various loads at the crack section
It is known from Dimarogonas (1995) that the form of crack flexibility matrix based on the fracture mechanics approach is widely used as the crack model. Elements of the flexibility matrix, in general, includes the coupling of lateral, longitudinal, and torsional vibration effects. As
enunciated in Papadopoulos and Dimarogonas (1988) and Darpe (2007), the crack flexibility matrix for a transverse crack is defined as follows
11 14 15
22 26
33 36
FULL
41 44 45
51 54 55
62 63 66
0 0 0
0 0 0 0
0 0 0 0
0 0 0
0 0 0
0 0 0
h h h
h h
h h
h h h
h h h
h h h
∆ ∆ ∆
∆ ∆
∆ ∆
=
∆ ∆ ∆
∆ ∆ ∆
∆ ∆ ∆
∆H (2.7)
In the case of slant crack, the flexibility matrix is more populated (Darpe, 2007). In Eqn. (2.7), ∆h terms denote the change in flexibility (or compliance) due to the crack effect. Flexibility is defined as the deflection in ith direction due to the force in jth direction. As depicted in (2.7), subscript 1 corresponds to the axial force (z-axis direction), subscripts 2 and 3 correspond to shearing forces, subscripts 4 and 5 correspond to the bending moment about two lateral directions, and subscript 6 corresponds to the torque. A combination of subscripts corresponds to cross-coupled terms.
The crack flexibility matrix in Eqn. (2.7) corresponds to a 6-DOF rotor system. In the case of a 4- DOF model, crack flexibilities and loads in the axial direction are not considered thus the flexibility matrix is obtained considering the deflections and loads in the two lateral directions alone. The crack flexibility matrix for a 4-DOF model thus can be obtained by reducing terms in 1st and 6th rows and respective columns in Eqn. (2.7). The crack flexibility matrix so obtained for 4- DOF analysis is
22 33 4-DOF
44 45
54 55
0 0 0
0 0 0
0 0
0 0
h h
h h
h h
∆
∆
= ∆ ∆
∆ ∆
∆H (2.8)
The crack flexibility matrix for a 2 DOF analysis based on linear response in two directions is similarly obtained by retaining the rows and columns corresponding to shearing forces along two orthogonal directions. The 3rd and 4th rows and columns of the matrix in Eqn. (2.8) thus get eliminated to give the flexibility matrix for the 2-DOF analysis, as
22 2-DOF
33
0 0 h
h
∆
=
∆
∆H (2.9)
Except under the situation of slow roll or very deep cracks, the cross flexibility, ∆h33, introduced by the presence of a crack are many times smaller than the main flexibility,∆h22. So, for the sake of simplicity this has been neglected in the further derivations of this chapter. Deflection at the position of the disc with an open crack adjacent to it, in the rotating coordinate system ξ– η, as given by Gasch (2008) is
0 22
0
0
0 0
u h h f
h f u
ξ ξ
η η
+ ∆
=
+
(2.10)
where h0 is the flexibility of the round intact shaft, ∆h2 2, is the additional flexibility due to the crack, and fξ and fη are excitation forces along the ξ and η coordinate axis directions, respectively. With rotation of the shaft, as the crack enters the compressive zone, the crack will close and ∆h2 2 becomes zero. Now the shaft behaves as if the crack is not present. Since h0 is
constant and ∆h2 2 has a periodicity of 2π, the flexibility matrix in equation (2.10) can be factored into matrices containing h0 and ∆h2 2. The behaviour of the crack opening and closure in the period of 2π rad is controlled by many factors, the most prominent being the depth of the crack.
The crack opening and closing is generally modelled by a suitable crack excitation function (CEF).
Irrespective of the type of crack excitation function made use of, a value of 1 of this function represents an open crack and a value of 0 represents the closure of the crack. The rotor cracks in early stage of propagation are likely to have a sudden opening/closing as compared to the advance stage of crack propagation when the opening and closing may be smoother. A hinge model simulates the opening/closing profile of a rotor crack in early stage of propagation and is given by a bi-linear square-wave function as follows (Gasch, 1993)
( )
1s t = when the crack is in the tensile portion of the shaft, i.e. the crack is open
( )
0s t = when the crack is in the compressed portion of the shaft, i.e. the crack is closed Pictorial representation of this crack excitation function is presented in Figure 2-4.
Figure 2-4 Variation of a switching crack excitation function over a shaft rotation
It may be seen in the figure that the crack suddenly closes at 90° and opens at 270°. Such cracks are also called as switching cracks (Mayes and Davies, 1984; Gasch, 1993) and the crack excitation function accordingly named as switching crack excitation function (SCEF). This bi- linear square wave SCEF may be approximated as a Fourier series as
1 2 2 2 2 2
( ) cos cos(3 ) cos(5 ) cos(7 ) cos(9 )
2 3 5 7 9
s t ωt ωt ωt ωt ωt
π π π π π
= + − + − + − ⋅⋅⋅⋅ (2.11)
In contrast to the switching crack, a breathing crack has more gradual opening and closing profile.
Any suitable non-linear function such as Mayes’ modified function (Gasch, 1993) simulates its opening/closing profile. Mayes’ function for the breathing crack is given as
( )
12(
1 cos)
s t = + ωt (2.12)
0 90 180 270 360
0 1
0.5
θ (deg)
s(t)
In the present work, the switching crack function represented by Eqn. (2.11) is used for numerical simulations. Thus, a crack switching/breathing function s(t) accounts for time variation into the crack flexibility matrix and it is defined as,
0 22
0
0 0
0 ( ) 0 0
u h h f
u h s t f
ξ ξ
η η
∆
= +
(2.13)
After the introduction of the steering function s(t), the restoring force vector in the rotating coordinate system may be obtained by inverting the flexibility matrix as
22 22
22
0 0
0 ( ) 0 0
f k k u
f k s t u
ξ ξ
η η
∆
= −
(2.14)
Here, (1 /h0)=k22 and 1 / (h0+ ∆h22)=k22− ∆k22 and k22 is the stiffness of the intact shaft and k22
∆ is the loss in stiffness due to opening of the crack. Eqn. (2.14) can be written in terms of the intact shaft stiffness and the additive stiffness due to crack as
(
rot ( ) rot)
f u
f s t u
ξ ξ
η η
= +
K ∆K (2.15)
with
22 22
22
0 0
and
0 0 0
rot rot
k k
k
∆
= = −
K ∆K (2.16)
Eqn. (2.15) represents a mathematical model of a crack in 2 DOF. Here subscript rot denotes matrices defined in the rotating coordinate system. The negative sign of the additive crack stiffness implies reduction in the stiffness due to appearance of the crack.