Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom
Received February
Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom
0 Shilin Chen Michel Geradin Aerospace Laboratory (LTAS) University
An exact and direct modeling technique is proposed for modeling of rotor-bearing systems with arbitrary selected degrees-of-freedom. This technique is based on the combination of the transfer and dynamic stiffness matrices. The technique differs from the usual combination methods in that the global dynamic stiffness matrix for the system or the subsystem is obtained directly by rearranging the corresponding global tramfer matrix. Therefore, the dimension of the global dynamic stiffness matrix is independent of the number of the elements or the substructures. In order to show the simplicity and efficiency of the method, two numerical examples are given. © 1994 John Wiley & Sons, Inc.
INTRODUCTION
The subject of rotor-bearing dynamics has re
ceived considerable attention over the last few
decades. The reason for this interest in rotor
bearing dynamics is almost certainly the demand
placed on manufacturers to continuously im
prove both the power rating and the reliability of
rotating machinery
(Goodwin, 1992)
. For dy
namic analysis, modeling of the rotor-bearing
systems is the first step. Various methods for
modeling rotor-bearing systems have been devel
oped and widely used during the past few dec
ades. Among these techniques, the transfer ma
trix method (TMM) and the finite element
method (FEM) may be most commonly used for
multi-degree-of-freedom (MDOF) rotor-bearing
systems.
The outstanding advantage of the TMM is that
it requires calculations using matrices of fixed
size, irrespective of the number of DOF in the
problem. This means that the computational
complexity is low even when dealing with sys
tems with hundreds of DOF. However, only the
natural frequencies, mode shapes, and harmonic
response are available so far by this method. The
other important modal parameters such as modal
mass, modal stiffness, and transfer function of
the system cannot be obtained. Furthermore, the
stiffness and mass matrices that are important for
modeling the systems are not available when the
TMM is used.
At the current state of rotor dynamic technol
ogy, the FEM has proved to be powerful and
versatile. It is also the only validated tool avail
able at the present time for nonlinear systems
and for transient dynamic analysis. However, in
this method, the number of DOF is very high for
large rotor-bearing systems. Many unwanted
DOF such as rotational and internal DOF are
used in the model. This makes it difficult for the
FEM model to compare with the experimental
model due to the large difference in the numbers
of coordinates used in both models. If such
comparison is necessary, the FEM model has to be
reduced, which may be uneconomical in some
cases.
Another alternative is the dynamic stiffness
matrix method (DSM). It may be considered as
an improved FEM. Being different from the
FEM, the DSM uses the analytical solutions of
the governing equations as shape functions.
Therefore, the obtained DSM is exact in the
sense ofthe exact governing equation. These ma
trices are in general parametric in terms of the
vibration frequency and the load factor.
One of the most important advantages of the
DSM is that the DOF needed to model a struc
ture is significantly reduced compare to the
FEM. This is due to the fact that a uniform shaft,
for example, can be taken as long as needed in
the DSM. There is a significant number of refer
ences describing the method.
Fergusson and
Pilkey (1991
a,b, 1993a,b) have reviewed the rele
vant literature up to 1992. Most of the literature
up to the end of 1992 may be found in their re
views.
However, there are some drawbacks associ
ated with the DSM. First, the number of DOF is
still high for large structures. Similarly to the
FEM, many unwanted DOF such as rotational
degrees remain in the model. Second, the calcu
lation of modal parameters requires the solution
of a highly nonlinear (trascendental) eigen
problem:
[D(w)]X = o.
To this purpose, some special algorithms should
be used, for example, the algorithms described
by
Williams and Wittrick (1970)
,
Wittrick and
Williams (1971)
, and
Richards and Leung (1977)
.
The former algorithm requires calculation of nat
ural frequencies for each individual beam ele
ment with its end fixed. This increases the com
putational time significantly.
In order to minimize the dynamic DOF with
out any loss of accuracy, a combination method
was introduced by
Dokainish (1972)
for plate vi
bration problems in which the element transfer
matrix was obtained directly from the element
stiffness and mass matrices. In recent years, this
combination method has been improved by other
researchers
(Chiatti and Sestieri, 1979; Ohga
et aI., 1983; Degen et aI., 1985)
for different ap
plications. In this method, the eigenfrequencies
and mode shapes are calculated from the g (...truncated)