Environmental Engineering Reference
In-Depth Information
The blade is modeled with NASTRAN tapered beam (CBEAM) elements (developed
by the MSC Software Corp.) that have six degrees of freedom at either end. These elements
also have provisions for shear deformation and warping of the cross section, although these
features were not used in this analysis. Additional terms are added to the various matrices
to provide for rotating coordinate system effects. These are the Coriolis and centrifugal
softening terms, C C (W) and K cs (W), respectively, that will be discussed in detail later. The
aeroelastic matrices, M a (W), C a (w,W) and K a (w,W), all depend on W, since the free-stream
velocity at any blade radius is governed by it (it is assumed that the rotor is turning in still air).
Linear shape functions are used in the development of these aeroelastic matrices.
As a demonstration of the method used to generate the aeroelastic matrices, a single
beam element's contribution to K a is presented in Equation (11-61). The element has a
length, l , and is bounded by nodes 1 and 2. The stiffness contribution is due to the a terms
of Equations (11-59) and can be developed with the principle of virtual work using only a
virtual displacements. Thus, for a varying linearly along the element (i.e. a (x) = a 1 (1-x/l) +
a 2 x / l ) the contribution of this single element is as follows:
2
x
l
1- l
x
l
1 - l
é
ê ê ê ê ë
æ ç ç
í
L 1
L 2
M 1
M 2
2
l
l 1 - l
d 1 ( x ) 1 - l
x
a 1
a 2
ç
ò
a 0 ( x ) U 2 ( x ) b ( x ) C ( k ( x ))
= r
dx
2
d 1 ( x ) l 1 - l
d 1 ( x ) l
ç ç è
0
2
d 1 ( x ) l
1 - l
(11-61)
Here the quantity 2p of Equations (11-59), which represents the lift curve slope for a flat
plate, is replaced by a 0 (x) representing the lift curve slope of an airfoil. This slope varies
along the length of the element as does the free stream velocity, U , the semichord, b , the
Theodorsen function C , and the distance from the elastic axis to the quarter-chord, d 1 . The
integral is evaluated numerically. Elemental contributions to M a and C a are developed in an
entirely similar manner.
Commensurate with the use of the NASTRAN CBEAM element above, the NASTRAN
commercial finite element software is used for this aeroelastic stability investigation. The
contributions to the stiffness, mass, and damping matrices discussed above are supplied by
means of a NASTRAN input option. NASTRAN can accommodate non-symmetric, com-
plex-valued matrices as are required in this effort, and it provides a number of complex
eigenvalue solvers for the stability analysis.
In addition to W, the aeroelastic matrices, C a (w,W) and K a (w,W), also depend on w, the
natural frequency of the mode shape of interest, which occurs in the argument of the The-
odorsen function. Since this frequency is unknown at the onset of the computations an itera-
tive process is required for obtaining accurate results. The iterative procedure developed for
the aeroelastic stability analysis of the rotor blade is composed of the following steps:
1. Select a value for W .
2. In a quasi-static NASTRAN run, create K ( u 0 ,W) for subsequent eigenvalue
analysis.
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