over their straight counterparts in modeling curved cables by yielding higher accuracy using coarse meshes.
However, the formulation of curved beam elements is not a simple extension of the straight beam elements
because of the so-called locking problem. For instance, the earliest curved beam elements exhibited excessive
bending stiffness when using the independent C
0
-continuous axial and C
1
-continuous transverse displacement
functions (Cantin and Clough, 1968). Stolarski and Belytschko (1981) and Prathap and Bhashyam (1982)
pointed out that the locking in the curved beam element was due to the independently interpolated axial
and bending displacement functions unable to recover correct constraints from the zero membrane strain
of pure bending state. Later, Prathap and Bhashyam (1986) developed a concept of coupled consistency dis-
placement fields where the axial displacement is required to be one order higher than the transverse displace-
ment. It can predict a priori any poor convergence due to the locking and yield high accuracy. Recently, more
works have been done to improve its computational efficiency. Detailed reviews of the locking-free curved
beam elements can be found in the works of Raveendranath et al. (2001) and Kulikov and Plotnikova
(2004). In addition to the locking, the problem is further complicated by the fact that the aerial refueling hose
will experience very large rigid body motion and small elastic deformation simultaneously. To authors’ knowl-
edge, no attempt has been made to extend the method of coupled consis tency displacement fields to the geo-
metric nonlinear analysis. The incremental update Lagrangian (UL) method has the advantages in handling
the large displacement and rotation because (1) it reduces the non-commutativity of large spatial rotations
into the additive incremental rotations, and (2) it simplifies the state update procedures. Although the incre-
mental UL formulation requires smal l time steps in order to keep the incremental rotations small and additive,
the real time step is actually dictated by the stability requirement of the numerical integration, which effec-
tively satisfies the requirement of small incremental rotations. Based on the above rationale, Zhu and Meguid
(2006a,b) recently proposed a new three-noded locking-free curved beam element using the lower order, cou-
pled and co nsistent axial and transverse polynomial interpolations to achieve higher accuracy, faster conver-
gence rate and better computational efficiency. In this paper, the new curved beam element was used to study
the dynamic characteristics of the aerial refueling hose-and- drogue system by examining its response to tow
point disturbances and vortex wake excitation.
3. Finite element formulation of generalized aerial cable towed model
The generalized model of aerial cable towed system is analyzed by the finite element method where the hose
is modeled by the curved beam elements and the towed body is simplified as a lumped mass element. The
geometry of the curved element is characterized by its length L and natural curvature j while the position
and the orientation of the curved element in space is described by its nodal coo rdinates X
i
(i = 1, 2,3) in a glo-
bal Cartesian coordinate system as shown in Fig. 2 . The incremental translational and rotational displace-
ments of the neutral axis
t
u
1
,
t
u
2
,
t u3
,
t
h
1
,
t
h
2
,
t
h
3
with respect to time t configuration are shown in the
curvilinear coordinate system in their positive directions.
The incremental Green–Lagrange strain vector in the curved beam can be expressed in terms of membrane
and torsional shear strains by
De ¼fDE
11
; DE
13
; DE
23
g¼fDe x
1
ðDx
2
2jDeÞþx
2
Dx
1
; x
2
Dg=2; x
1
Dg=2gð1Þ
1
2
3
X
1
X
2
3
x
1
,
Δ
u
1
,
Δθ
1
L
κ
=
1
/
R
x
3
Δ
u
3
,
Δθ
3
x
2
Δ
u
2
,
Δθ
2
Fig. 2. Three-dimensional curved beam element in curvilinear coordinates.
8060 Z.H. Zhu, S.A. Meguid / International Journal of Solids and Structures 44 (2007) 8057–8073