We will now proceed to describe how the information from φ0j is used to construct the intrinsic mode shape in sectional forces φ2j in the beam problem. Note that we also require knowledge of the corresponding modes in momentum ψ1j and sectional curvature strains ψ2j. As described previously, the problem is set up as a set of massless beams connecting lumped masses, the sectional mass M are collocated and only non-zero at the nodes themselves (ML,i). However the sectional stiffness C is not well-defined,
which arises from the fact that the Ka matrix can be fully populated as a result of the
Guyan reduction process (and is found to be so). The fully populated stiffness matrix Ka indicates that the force-strain relations are not perfectly local as would have been
expected by a beam-type description, implying the localised displacement of one node while keeping all other nodes fixed will generate internal forces across the entire structure (and not just its adjacent elements), or vice versa. In contrast, under the assumptions of localised stress-strain relationships, which the sectional C matrices describe, a local strain would only generate stresses in its immediate vicinity. The Kamatrix of a model
that uses actual beam elements between nodes will be sparsely populated with the non- zero elements related directly to the connectivity between nodes. In fact, the deviation of the actual Ka matrix from this expected sparse form of a beam problem (i.e. the
magnitude of the entries that should have been zero) would provide a valuable metric on the suitability of using a beam-type description of the structure in question. Due to this problem with the definition of C matrix, in this work we seek to obtain ψ2j, or Cφ2j directly from φ0j without explicit knowledge of C.
We compute ψ2j from φ1j by applying the linearised intrinsic equations. From (3.1b) we can obtain,
ωjψ2j(s) = −φ′1j(s) + E⊤φ1j(s), (5.20)
where subject to the linear interpolation in φ1j and piecewise constant ψ2j, we can also write in discrete form that for each beam element between two nodes,
ψ2j,i+1 2 = ω
−1 j
−(φ1j,i+1− φ1j,i)/(si+1− si) + E⊤(φ1j,i+ φ1j,i+1)/2
, (5.21)
with ψ2j(s) = ψ2j,i+1
2 for si < s < si+1.
Lastly the sectional force modes φ2j will also be computed directly from φ0j, i.e. there is no need for explicit knowledge of the sectional C matrix in this method. Similar
to ψ2j, we can rewrite (3.1a) and obtain
φ′2j(s) + Eφ2j(s) = ωjMφ1j(s). (5.22)
However for a lumped-mass model, this equation encounters issues with geometry and is difficult to apply, as previously described in Section 4.2. In order to clarify the method used in our approach, we seek an alternative but equivalent description to φ2j. We first note that the term Kaxa in (5.8) describes the elastic forces and moments experienced
on each of the 6Na degrees of freedoms on the analysis nodes of the system caused by
the distribution of displacements/rotations x0. Thus at each node i,
! fK,i
mK,i
"
= (Kaxa)|i. (5.23)
We also note that this elastic force can equally be described by an imbalance of sectional (internal) forces, i.e. there is a one-to-one relation between the distribution of x2 and
the distribution of fK and mK(thus xa). In order to obtain the relation, we consider the
linear static problem where the structure experiences an internal stress distribution x2
which arises from the application of fK and mK. We observe that for node i in Figure
8, the relation between the forces described by the Ka matrix, and internal sectional
forces, is fK,i=TL(si+1 2)x2(si+ 1 2)123− TL(si− 1 2)x2(si− 1 2)123, (5.24a) mK,i=TL(si+1 2)x2(si+ 1 2)456+ (˜ri+ 1 2 − ˜ri)TL(si+ 1 2)x2(si+ 1 2)123 − (TL(si−1 2)x2(si− 1 2)456+ (˜ri− 1 2 − ˜ri)TL(si− 1 2)x2(si− 1 2)123). (5.24b)
Due to the piecewise-constant interpolation of x2 within each beam element, it is
natural to evaluate the value of x2 at the midpoint of each element, i.e. si+1
2 and si− 1 2.
The sign in front of each sectional force term is dependent on the direction of increasing s, as reversing this direction reverses the definition of sectional force and thus reverses the sign. We finally note that in the eigenvalue problem we have defined via (3.1) and (5.9) that x2 = cos(ωjt)φ2j and xa= − cos(ωjt)Φ0j, thus we obtain the relation between
Figure 8: Illustration of the beam internal forces TLx2 (red) and the equivalent nodal
applied force fK,i (blue) at node i for the region outlined by dotted line. The negative
sign in one of the internal force terms is due to the definition of the integration direction (in this case equivalent to increasing i).
φ0j, obtained from Φ0j and φ2j at node si as
−(KaΦ0)i,123=TL(si+1 2)φ2j(si+ 1 2)123− TL(si− 1 2)φ2j(si− 1 2)123, (5.25a) −(KaΦ0)i,456=TL(si+1 2)φ2j(si+ 1 2)456+ (˜ri+ 1 2 − ˜ri)TL(si+ 1 2)φ2j(si+ 1 2)123 − (TL(si−1 2)φ2j(si− 1 2)456+ (˜ri− 1 2 − ˜ri)TL(si− 1 2)φ2j(si− 1 2)123). (5.25b) This equation can be easily extended to nodes that contain multiple connectivity between beam elements. The equation can be inverted to obtain φ2j from a knowledge of Φ0j
and ωj, by starting from any end point on the beam assembly and computing φ2j at
each subsequent beam segment from the value of φ2j at the previous segment and KaΦ0
at the node. Note that this method is completely equivalent to that described in Section 4.2. For a single beam without any branching structures, the method can be written as
φ2j(si+1 2) = P k>i (KaΦ0)|k,123 P k>i (KaΦ0)|k,456+ (˜rk− ˜ri+1 2)(KaΦ0)|k,123 . (5.26)
From this complete knowledge of φ1, φ2, ψ1 and ψ2, we can compute the coeffi- cients A and Γ in the nonlinear modal form of the structural equations, as well as the
H coefficients for any aerodynamic forces with the necessary aerodynamic definitions, according to the method described in Section 4.3.1. Thus this method allows as to arrive at a complete modal description of the aeroelastic system by using data from the static condensation on a full 3D FE model to generate the structural model.
To summarise, the condensation method described in this work is carried out as follows,
Step I, the starting point is a 3D FE model of the structure with the requirement that inertia is lumped onto a set of analysis nodes along load paths.
Step II, we carry out the Guyan reduction on the 3D model, arriving at the reduced mass and stiffness matrices describing the linear dynamics of the discrete reduced degrees of freedoms in displacement and rotations xa and a linear system described by (5.8).
Step III, the natural modes described in discrete global displacements and rotations of the analysis nodes Φ0j are obtained from the mass and stiffness matrices.
Step IV, we obtain the equivalent, continuous intrinsic modes φ1, φ2, ψ1 and ψ2 from the discrete natural modes Φ0j.
Step V, the nonlinear intrinsic modal system (3.23) is obtained by computing the coupling coefficients from the continuous intrinsic modes.
Thus, we have developed a method of arriving at a geometrically nonlinear, modal description of a slender structure from a linear 3D FE model of the structure. Chapter 6 will demonstrate the application of this method to an actual FE model.
6
Numerical Results of Structural Model
In this chapter, the implementation of the structural solution method of the proposed simulation framework will be verified against established test cases. The chapter will also demonstrate some numerical aspects of the modal intrinsic formulation in Sections 6.3 and 6.4. First, large, geometrically nonlinear static response of a cantilever beam under external load will be compared against published results. Then the static condensation procedure described in Section 5.2 will be used to create a modal intrinsic system of a cantilever beam modelled in FE using shell elements, which will subsequently be used to validate its dynamic response. Subsequently, a free-flying beam undergoing large rotations as well as elastic deformations will be used to demonstrate various aspects of convergence and momentum conservation of the intrinsic modal formulation. Lastly a similar test will be applied to another model obtained via the condensation process. Combined, all these test cases will provide confidence in the proposed method and its implementation, and also a good understanding of the numerical issues.