The system (3.55) describes a modal aeroelastic system whose coupling coefficients, de- scribed in (3.18) and (3.40), are computed from an airframe definition that is continuous in space. This is by nature infinite-dimensional and, thus, in order to arrive at a practical implementation of such a system, it is necessary to develop a suitable spatial discreti- sation to the airframe structure which we will use to compute the coefficients. For simplicity, in the current work we seek a low-order interpolation scheme and, if required, increase the number of nodes to reduce the errors associated with a low-order scheme.
As this will eventually be required by Guyan reduction as part of the condensation process from full 3D FE description to a beam-type model used in this work (a method that will be described later in Chapter 5), the discretised structural problem will be approximated by lumping of the structural inertia onto a series of discrete nodes that are joined via massless, flexible beams. This lumping of masses onto nodes is also commonly used in finite-element descriptions [49] but will require special treatment in the current method of projection onto global modes. The distributed external loads
fA will also be integrated onto the nodes. Thus, we define an equivalent problem of
a structure consisting a series of point masses on the Na nodes, linked by massless,
flexible beams with external loads applied on the nodes only. We will now find a suitable spatial interpolation of the primary intrinsic variables (local velocities and sectional force resultants) on this modified structural problem.
We start by examining the governing intrinsic equations for which the interpolation schemes will be used. If, on the massless beam segment connecting the nodes, we assume that the intrinsic variables are small in magnitude, (i.e. remaining in the linear regime) and setting M to zero in (2.79a), obtaining
−x′2− Ex2 = 0 (4.28a)
as fAis also zero on the beam segment. Expanding x2 into f and m gives
−f′= 0, (4.29a)
−m′− ˜ef = 0. (4.29b)
As can be seen in the first equation, the sectional force f in a straight, massless beam segment between two nodes must be constant in the linear case as there is no external forces applied on the segment other than at the end points. The sectional moment m should also vary linearly with s, resulting from integrating the constant force along the length of the beam element in the second equation. Between adjacent elements however the sectional load x2 will not be continuous as our problem regards external load as
lumped, concentrated loads applied on each node and also due to M not being zero at the nodes. If we further assume a constant but not necessarily diagonal C throughout the beam segment between two nodes, then Cx2 also varies linearly with s.
In contrast, the velocity field x1 must be continuous as can be shown via enforcing
a finite x2 on the compatibility equation (2.79b). Again by assuming small magnitudes
of variables in (2.79a), we obtain
C ˙x2− x′1+ ETx1= 0, (4.30a)
or, expanding in individual components,
˙γ − v′− ˜ev = 0, (4.31a)
It can be seen that, since components of Cx2, γ and κ vary linearly with s, the
angular velocity ω component in x1 must vary quadratically with s whereas the linear
velocity component varies with s in a cubic manner. Note that such interpolation is very similar to that used in beam elements in commercial FE packages, e.g. CBEAM elements in NASTRAN [52] or B31 elements in ABAQUS [137]. As we have lumped the masses onto the nodes and made the beam segment massless, the momentum Mx1 is not
continuous and is only non-zero at the nodes. We will refer to the lumped mass at node i as ML,i to differentiate this variable with the distributed mass (inertia)-per-unit-length
M(s).
It will be shown in Section 5.2.3 that the Guyan reduction technique can result in slightly different values of C in each natural mode for the same beam segment. This means that similarly we could not compute the quadratic and cubic interpolation to x1to
a high precision using data from Guyan reduction. The use of a linear analysis with the assumption of a constant C matrix throughout further limits the accuracy advantage of the cubic interpolation over lower-order schemes to small deformations of each element, where geometric nonlinearities are insignificant, and material properties that do not vary along the length of the element. Particularly, the latter requirement is not likely to hold in a real-life situation, however the condensation method does not allow us to have a more detailed knowledge of the sub-element variations in C. Therefore rather than attempting to use a physically accurate interpolation scheme based on a constant C assumption, we will instead simplify and reduce the order of the interpolation. If required, the node count can be increased to reduce the size of each beam segment, so that the error caused by using a low-order interpolation and geometrically nonlinear effects become far less significant.
In the interpolation used in this work we will therefore regard the sectional forces and moments x2, together with sectional force/moment strains Cx2, as piecewise constant,
i.e. we discard the linear variation of the sectional moment with s and assume it to be the midpoint value in the element, shown in the right image in Figure 6. As the velocity variables need to be continuous, the velocities and angular velocities x1 will be a linear
interpolation between the nodal values at the end of each element as shown in the left image in Figure 6, again discarding higher-order variations.
We will now provide a mathematical description of the interpolation used in this work. Note that this is not a discussion of the basis functions used in the formulation but rather on the interpolation that the states, and the basis functions use. For simplicity and clarity of presentation, the following argument will be made with a beam with Na
Figure 6: Interpolation scheme of velocity x1 and sectional force x2, with various
contributions from interpolation functions coloured, used throughout the current work.
multi-beam problem in a practical implementation by taking account of the topology of the structure. The definition of s in this case will be more complex but would not affect the arguments in this section.
If we define x1,i as the discretised value of x1defined at node i, that is, x1,i= x1(si),
linear interpolation results in the definition of the continuous x1 corresponding to the
discretised values being
x1 ∼=
X
i
ϕ1,ix1,i, (4.32)
with the shape function
ϕ1,i(s) = s−si−1 si−si−1 si−1 < s < si s−si+1 si−si+1 si < s < si+1 0 otherwise . (4.33)
Similarly, the discretised x2, being piecewise constant between the nodes, can be
represented as x2 ∼= X i ϕ2,i+1 2x2,i+ 1 2 (4.34)
with the variable being x2,i+1
2 and the shape function
ϕ2,i+1 2(s) =
(
1 si < s < si+1
0 otherwise (4.35)
This thus defines the interpolated version of the intrinsic states x1 and x2 at every
point s on the structure as a function of the collection of discretised intrinsic variables x1,iand x2,i+1
2. These definitions form the basis on which we compute our modal system
and the eventual aeroelastic model we will use in subsequent analysis and simulations. Similarly, the intrinsic modes φ1 and φ2 defined at every point s will also follow this interpolation scheme and will be defined as
φ1j ∼=X i ϕ1,iφ1j,i, (4.36) φ2j ∼=X i ϕ2,i+1 2φ2j,i+ 1 2, (4.37)
where φ1j,i and φ2j,i+1
2 are the discretised values defined at the nodes and mid-points
respectively. In the next section we will make use of these interpolations in computing the nonlinear coupling coefficients in the modal system.