# An efficient modeling of a flexible airships

Selima Bennaceur, Naoufel Azouz, Djaber Boukraa

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Selima Bennaceur, Naoufel Azouz, Djaber Boukraa. An efficient modeling of a flexible airships. ASME 8th Biennial Conference on Engineering Systems Design and Analysis, Jul 2006, Turin, Italy. pp.573-582, 10.1115/ESDA2006-95741. http://hal-00342772

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
AN EFFICIENT MODELLING OF FLEXIBLE AIRSHIPS

Selima BENNACEUR, Naoufel AZOUZ, Djaber BOUKRAA

Laboratoire des Systèmes Complexes, CNRS FRE 2494

Université d'Evry Val d'Essonne

40 Rue du Pelvoux, 91020 Evry cedex, France

E-mail: Selima.Bennaceur, Azouz,boukraa@iup.univ-evry.fr

**Abstract**

This paper presents an efficient modelling of airships with small deformations moving in an ideal fluid. The formalism is based on the Updated Lagrangian Method (U.L.M.). This formalism proposes to take into account the coupling between the rigid body motion and the deformation as well as the interaction with the surrounding fluid. The resolution of the equations of motion is incremental. The behaviour of the airship is defined relatively to a virtual non-deformed reference configuration moving with the body. The flexibility is represented by a deformation modes issued from a Finite Elements Method analysis. The increment of rigid body motion is represented similarly by rigid modes. A modal synthesis is used to solve the general system equations of motion. Time constant matrices appears (i.e. mass and structural stiffness matrices), and we show a convenient technique to actualise the time dependant matrices.

**Keywords:** flexible airships, small deformations, incremental scheme, modal synthesis.

**1. Introduction**

The interest for the modelling and control of airships increases significantly in the last years. The complexity and capability of airships are expanding rapidly and the range of missions they designed to support is growing. However in order for airships to reach this potential, significant technical issue must be overcome. One main point of this challenge is the modelling of the structural flexibility. It is important to note that several kinds of airships, usually called blimps, are mainly constituted of a balloon filled of gas. The only solid parts are the careen and the tail fins. The integration of the structural flexibility in the

dynamic analysis is then useful, but it is now in an embryonic state and is only just emerging. Several researches were done using the assumption of rigid body behaviour for airships [1, 2]. The flexibility effects are sometimes modelled as a perturbation. However in other flying objects, such as light aircrafts, the introduction of the flexibility in the dynamic model becomes essential [3, 4]. We try by this study to contribute to the analysis of the deformation of the airships by introducing the effect of flexibility as non controlled supplementary degrees of freedom. The deformation of the blimp is not considered as a perturbation but rather acting on the motion of the airship.

The influence of structural flexibility on dynamics of mechanical systems has become increasingly important in classical robotics [5, 6] and recently in flying robots (i.e. Airships, UAVs...).

Several approaches, to study the problems of flexible bodies, have been proposed in the literature. These approaches can be classified into two groups. The first group uses the Newton-Euler description [7] which is an interesting method in regard of the time computation. However it is not easy to use it in the case of the flexible body and it is sometimes sensitive to the numerical simulation. In the other hand the total Lagrangian method [8], this consists to define the motion relatively to a fixed reference frame, leads also to complex relations when describing stresses and strains in the flexible body.

An Updated Lagrangian Method (U.L.M.) was proposed by Bathe & al. [9] and developed for deformable bodies that undergo large translational and rotational displacements. The resolution of the dynamic problem is incremental. The configuration and the motion of the body are identified using a
moving reference configuration representing the position of the deformable body in the preceding step. Azouz & al. [10] propose as reference configuration a rigid body configuration which follows the motion of the body without coinciding with it. This approach is convenient for a body with small deformations. Using this approach, we develop an efficient formalism to describe the behaviour of airships with small deformations. The motion is given by coupled sets of rigid and elastic variables. The nonlinear equations are formulated in terms of a set of time invariant matrices expressed in a reference configuration (i.e. mass and stiffness matrices). Time-variant quantities appear in the nonlinear terms that represent the dynamic coupling between the rigid body modes and the elastic deformation. We show in this study a suitable technique to actualise these terms using matrix partitioning and canonical decomposition. The dynamic system is reduced and solved through a modal synthesis. The elastic deformation is represented by a set of generalised variables and natural modes issued from a Finite Element method (FEM) discretization.

Airships are also governed by the aerodynamic forces that have to be modelled. The basis to analyse the motion of a rigid body in a perfect fluid has been established in the 19th century and has been described by Lamb [11]. In his work, Lamb considered the case of simple displacement in a big infinite mass of fluid and where the movement of this last is entirely due to the motion of a solid, and it is irrotational and acyclic. He proved that the kinetic energy of the fluid can be expressed as a quadratic shape of the six velocities of translation and rotation of the vehicle. The derivations given by Lamb will be used in the description of the airship, in a stationary uniform atmosphere. The terms depending on the acceleration or the added masses come from the fact that the fluid considered perfect is accelerated. When an ellipsoid body moves in an incompressible and infinite inviscid fluid so that the external flow is everywhere irrotational and continuous, the kinetic energy of the fluid produces an effect equivalent to an important increase of the mass and of the moments of inertia of the body [12, 13]. In our model, the fluid effects have been introduced into the dynamical system through a modal synthesis to build the global dynamic system of the flexible airship.

## 2. Dynamic model of the airship

The dynamics of the airship with small deformation is highly non-linear. The non linearities are essentially due to the large rotations of the body and the interaction between rigid body motion and the deformation.

<figure><img src="image_2.png" /><figcaption>Figure 1. Presentation of the different frames</figcaption></figure>

In our formulation, we use an Update Lagrangian Method. The description of the airship behaviour is made relatively to a reference configuration which changes in the time accordingly to the airship configuration. This method is chosen because it takes into account the buckling between the rigid body motion and the deformation, and could be coupled with results of classical structural dynamics. In fact, many dynamic parameters will be kept unchanged during the simulation. In the case of a small deformation, the configuration at the time t ($C_t$) is not so different from the one obtained by the transformation of a rigid body in the initial configuration. We choose as a reference configuration, the configuration of a virtual rigid body following the motion. Hence, we can linearize the airship displacement around this configuration. We consider two frames in the derivation of the motion equations. These frames are: the fixed frame related to the earth $R_f$ and the moving frame $R_m$ related to the reference configuration ($C_{ref}$). The position and the orientation of the vehicle should be described relatively to the reference frame. The origin $O'$ of $R_m$ coincides with the inertia centre of the undeformed vehicle in the reference frame. Its axes are the principal axes of symmetry.

### 2.1 Description of the reference configuration

<figure><img src="image_3.png" /><figcaption>Figure 2. Incremental scheme</figcaption></figure>

In this model we use an incremental scheme. The displacement of each point L is given by:
$$U(L) = U_{\text{ref}}(L) + \underbrace{\Delta U_{\text{ref}}(L) + U_d(L)}_{\tilde{Q}} \quad (1)$$

Where $U_d$ is the small displacement related to the whole deformation, $\Delta U_{\text{ref}}$ is the increment of the rigid body motion. Let us note $H$ the rotation matrix between the initial frame and the local reference $R_m$ such as $H^T H = H \cdot H^T = I_d$, and let us note $\tilde{Q} = H^T Q$; from the equation (1) we can obtain the following expression:

$$\ddot{U} = \ddot{U}_{\text{ref}} + (H \ddot{Q} + 2\dot{H} \ddot{Q} + \ddot{H} \ddot{Q}) \quad (2)$$

$\tilde{Q}$, and $\ddot{Q}$ represent the time derivative of the displacement $\tilde{Q}$.

## 2.2 Description of the strains

To describe the strains we use the symmetric Green-Lagrange tensor $\varepsilon$ which is related to the reference configuration. We note $J$ the gradient tensor relative at the reference configuration co-ordinates $\bar{J} = \nabla \tilde{Q}$ and the Green tensor $\varepsilon$ can be expressed as: $\bar{\varepsilon} = \frac{1}{2} [ (\bar{J}^t + \bar{J}) + \overline{\bar{J}^t \cdot J} ]$.

In this case we have small deformations, consequently $\bar{J} \ll 1$; we can then neglect the non linear terms, hence the strain tensor becomes:

$$\bar{\varepsilon} = \frac{1}{2} [ \bar{J}^t + \bar{J} ] .$$

When using the vector notation such as: $\bar{\varepsilon} = [\varepsilon_{11} \ \varepsilon_{22} \ \varepsilon_{33} \ \varepsilon_{12} \ \varepsilon_{13} \ \varepsilon_{23}]^T$, we can obtain this relation:

$$\varepsilon = D\tilde{Q} \quad (3)$$

Where $D$ is a differential operator, $\varepsilon_{ij}$ are the components of the strain tensor.

## 2.3 Description of the stresses

For the description of the stresses applied on the airship, we use the second tensor of Piola-Kirchoff $\overline{\overline{P}}_k$ defined on the reference configuration $\overline{\overline{P}}_k = \frac{\rho^{\text{ref}}}{\rho} J^t \cdot \overline{\sigma} \cdot J^{-t}$, with $\overline{\sigma}$ is the stress tensor of Cauchy, $J = I_d + \bar{J}$, $I_d$ is the 3x3 identity matrix, and $\rho^{\text{ref}}$ is the mass density

of the reference configuration equal to its initial value $\rho^0$ defined on the initial configuration (C$^0$). The material of the careen is assumed elastic, homogeneous and isotropic such as we have a linear behaviour law between stresses and strains:

$$\underline{\underline{\sigma}} = E\underline{\underline{\varepsilon}} \quad (4)$$

Here $\underline{\underline{\sigma}}$ is the vector of the stress components and $E$ the symmetric tensor of elastic properties of the material.

## 2.4 Dynamic equilibrium

Let's consider the dynamic equilibrium of an airship. According to the classical mechanics laws, the equilibrium is represented by the two following relations [14]:

$$\text{div}\underline{\underline{\sigma}} + f_v = \rho \ddot{U} \quad \text{on } (C^t) \quad (5)$$

$$\underline{\underline{\sigma}} \cdot \bar{\bar{n}} = t_n \quad \text{on board } (\partial C^t) \quad (6)$$

$f_v$ are the volumic forces acting on the elements of volume such as gravity. $t_n$ is the boundary stress vector, it represents the boundary surfacic forces such as the air pressure.

Now, we consider the following assumption: $V \approx V_{\text{ref}}$ and $S \approx S_{\text{ref}}$ the volume and surface of the actual configuration (C$^t$) are close to those of the reference configuration. This permits to confuse the Piola stress tensor $\overline{\overline{P}}_k$ with the Cauchy stress tensor $\underline{\underline{\sigma}}$.

If we use the principle of the virtual works and consider a small displacement $\delta\tilde{Q}$, the equation of equilibrium becomes:

$$\int_{V_{\text{ref}}} \rho \underline{\underline{\sigma}} \cdot \ddot{\underline{\underline{Q}}} d\underline{\underline{\sigma}} V_{\text{ref}} + \int_{V_{\text{ref}}} \underline{\underline{\sigma}} \cdot \partial \underline{\underline{\varepsilon}} d\underline{\underline{\varepsilon}} V_{\text{ref}} = \int_{V_{\text{ref}}} f_v^T \cdot \delta\tilde{Q} d\underline{\underline{V}}_{\text{ref}} + \int_{S_{\text{ref}}} t_n^T \cdot \delta\tilde{Q} dS_{\text{ref}} \quad (7)$$

Consequently, the equation (7) is similar to the dynamic equation of the deformable bodies in structural mechanics.

## 3. Resolution of the dynamic equation

To define the exact configuration of the deformable airship, we should use an infinite number of co-ordinates in order to calculate the
location of every point of the body. It has been shown that the motion of the deformable airship is governed by a set of partial differential equations that are space and time dependent. The discretization consists to subdivide the airship in a number of sub-domains with simple shapes that could be studied easily. We will use approximate techniques namely Rayleigh-Ritz method coupled with Finite Elements discretization.

## 3.1 Discretization of the airship

We indicate that the discretization consists to subdivide the airship in a number of sub-domains with simple shape. The displacement of the airship is then described through some interpolations functions and nodal displacement. We must note that the number of elements required for an acceptable approximation of the displacement should be high. However when using modal synthesis, this number of elements play a part only in a preliminary computation. Just a few numbers of "useful" nodes are kept.

We should note that the nacelle is represented by three points in the bottom of the airship. In the central point we apply its actual heavy, in the other point we model the effect of the two propellers.

<figure><img src="image_4.png" /><figcaption>Figure 3. Discretization of the airship</figcaption></figure>

Let us now consider a displacement field $U^i$ of an element i of the body. It can be expressed in function of the column matrix $\bar{u}$ representing the displacement, relatively to the local reference frame, of all nodes as follows:

$U^i = N^i \bar{u}$ where $N^i$ is an interpolation space function [15]. This relation is valid for any given displacement, especially for:

$$ \tilde{Q} = N \cdot \bar{q} \qquad (8) $$

We note $M^i$ and $K_L^i$ the mass and stiffness matrices of the element i defined as follows:

$$ M^i = \int_{V_i} \rho_{ref} \cdot N^{iT} \cdot N^i \cdot dV^i_{ref} \qquad (9) $$

$$ K_L^i = \int_{V_i} N^{iT} \cdot D^T \cdot E^i \cdot D^i \cdot N^i \cdot dV^i_{ref} \qquad (10) $$

$V^i$ and $E^i$ are respectively the volume and the tensor of the elastic properties of the element i, $D^i$ is the differential operator of an element i.

The mass and stiffness matrices of the whole airship are built by an adequate assembly of elementary matrices of mass and stiffness:

$$ M = \sum_{i=1}^{ne} M^i \qquad (11) $$

$$ K_L = \sum_{i=1}^{ne} K_L^i \qquad (12) $$

ne is the number of elements of the airship.

## 3.2 Equation of motion of the discretized airship

Lets us now analyse the different terms of the equilibrium equation (7).

According to the equation (2), the first left hand term could be expressed as:

$$ \int_{V_{ref}} \rho_{ref} \cdot \ddot{\bar{Q}} \cdot \delta \bar{Q} dV_{ref} = \int_{V_{ref}} \rho_{ref} \cdot \ddot{U}_{ref}^T \cdot \delta \bar{Q} dV_{ref} + \int_{V_{ref}} \rho_{ref} \cdot (H \ddot{\bar{Q}} + 2 \dot{H} \dot{\bar{Q}} + \ddot{H} \bar{Q})^T \cdot \delta \bar{Q} dV_{ref} \qquad (13) $$

We just add here that, in the incremental scheme the reference acceleration $\ddot{U}_{ref}$ related to the reference configuration is known.

Using the equations (3) and (7), we can write the second left hand term as:

$$ \int_{V_{ref}} \sigma^T \cdot \partial \epsilon \cdot dV_{ref} = \int_{V_{ref}} \bar{q}^T \cdot N^T \cdot D^T \cdot E(DN) \partial \bar{q} \cdot dV_{ref} \qquad (14) $$

Relations (3; 8-13) give :

$$ K \cdot \bar{q} + B \cdot \bar{q} + M \cdot \bar{q} = \tilde{F}_e + \tilde{F}_f - \tilde{F}_r \qquad (15) $$

We define here the diagonal bloc matrix of rotation $\bar{H}$ which concerns the rotation of all the nodes:
$$[\bar{H}] = \begin{bmatrix} [H] & & & \\ & [H] & & \\ & & \ddots & \\ & & & [H] \end{bmatrix} \quad (16)$$

The stiffness matrix K is constituted by:

$K_L$ the structural stiffness, and $K_D = M\bar{H}^T\bar{H}$ the dynamic stiffness due to the Coriolis effect. $B = C_s + C_g$ with $C_g = 2M\bar{H}^T\bar{H}$ the anti-symmetric matrix of gyroscopic terms; $C_s$ is a matrix of structural damping. We suppose in the following that we can neglect this matrix is front of $C_g$. $K_D$ and $B$ are consistent matrices which take into account the inertial coupling between the overall motion and the deformation. $\tilde{F}_e = \bar{H}^T F_e$, $\tilde{F}_f = \bar{H}^T F_f$ are respectively the external (volumic and boundary forces) and fluid forces; $\tilde{F}_r = M\bar{H}^T\bar{u}_{ref}$ is the column matrix of residual inertia terms due to the dynamics of the reference configuration.

The structural stiffness and mass matrices are defined relatively to a moving reference frame $\mathbf{R}_m$ in the “rigid” reference configuration. They are then constant. The dynamic equation (15) has a huge size and the number of degree of freedom (d.o.f) is tremendous, especially if we choose to model the airship with small elements; this leads to a very high computational cost and many problems to establish control laws. For this reason, we tried to solve this dynamic equation using a modal synthesis.

### 3.3 Modal synthesis

We decompose the increment $\tilde{Q}$ into a rigid contribution $\tilde{Q}_r$ and a deformable part $\tilde{Q}_d$ such as in its discretized form we have:

$$\bar{q}_r = \sum_{i=1}^{6} Y_{ri} X_{ri} \quad (17)$$

$Y_{ri}$ being the modal amplitude of the rigid mode i. Hence for a 3D motion, we have six variables $(Y_{trax}, Y_{tray}, Y_{traz}, Y_{rotx}, Y_{roty}, Y_{rotz})$.

$$\mathbf{X}_{\text{rig}} = \begin{bmatrix} [H] & & & \\ & [H] & & \\ & & \ddots & \\ & & & [H] \end{bmatrix} \quad (18)$$

$\tilde{\mathbf{X}}_{\text{tra}} \quad \tilde{\mathbf{X}}_{\text{rot}}$

$X_{\text{tra}}$ : are three modes of the translation motion.

$X_{\text{rot}}$ : are the three modes of rotation.

The displacement due to the deformation of the airship is:

$$\bar{q}_d = \sum_{i=1}^{nd} Y_{di} X_{di} \quad (19)$$

nd: is the number of significant deformable modes kept for the study, $Y_{di}$ is a time dependent variable of the deformable mode i, $X_{di}$ is a free-free natural mode of rank i. The choice of these boundary conditions is in harmony with the real configuration of the airship. O' is the origin of $\mathbf{R}_m$, L is a given nodal point, $\widehat{OL}$ is the skew matrix associated with the vector $\overline{OL}$.

The deformable airship could be considered as low frequency structure. We can model its vibration accurately with the first modes of lower frequencies.

The projection of dynamical equation of motion (15) in the modal basis composed of rigid and flexible modes permits to reduce the number of d.o.f. The dynamic equation can then be written as:

$$\mathbf{M} \cdot \ddot{\mathbf{Y}} + \mathbf{B} \cdot \dot{\mathbf{Y}} + \mathbf{K}_D \mathbf{Y} + \mathbf{K}_L \mathbf{Y} = \bar{\mathbf{F}}_e + \bar{\mathbf{F}}_f - \bar{\mathbf{F}}_R \quad (20)$$

$\mathbf{M}$ and $\mathbf{K}_L$ are the constant mass and stiffness matrices projected in the modal basis ($\mathbf{M} = \mathbf{X}^T \cdot \mathbf{M} \cdot \mathbf{X}$), $\mathbf{X}$ is the matrix of the modal basis issued from the collection of all rigid and flexible modes. The other bold terms are terms projected in the modal basis:

$$\mathbf{M} = \begin{bmatrix} M_{TT} & M_{TR} & M_{Td} \\ & M_{RR} & M_{Rd} \\ \text{Sym} & & M_{dd} \end{bmatrix} \quad \text{and}$$
Denoting by **v** the velocity field in the fluid domain Ω<sub>air</sub>, the incompressibility and irrotational assumptions leads to:

$$
\nabla \cdot \mathbf{v} = 0 \quad ; \quad \nabla \wedge \mathbf{v} = 0 \tag{23}
$$

$\wedge$ is the vectorial product of two vectors. $\nabla$ is the gradient symbol, and the flow field may be described in terms of a potential F such that:

$$
\mathbf{v} = \nabla \Phi \tag{24}
$$

From the incompressibility constraint, it is easy to
show that the potential obeys to the homogeneous
Laplace equation:

$$
\nabla^2 \Phi = 0 \quad \text{in } \Omega_{\text{air}} \qquad (25)
$$

with Newman boundary conditions:

$$
\nabla \Phi \cdot \mathbf{n} = - \dot{\mathbf{q}} \cdot \mathbf{n} = -(\mathbf{X}] \dot{\mathbf{Y}}) \cdot \mathbf{n} \quad \text{on} \quad (\partial \mathbf{C}^t) \quad (26)
$$

[X] is the matrix of all the modes (rigid and
deformable modes), Y is the column matrix of all
the time dependent variables defined in §3.4, n is a
unit vector, normal to (∂C¹).

Thus, one of the important characteristic of this
representation is that **v** only depends on the current
boundary conditions, and not on the history of the
flow: the model is quasi-steady. To solve the
potential equation, we use the boundary integral
representation of the Laplace equation, together
with standard boundary element method. It
consists in the determination of a piecewise
constant distribution of singularities over (∂C¹)
(see [16] for details on the numerical treatment).

4.2 Fluid forces

For this assumption, the pressure at any point
in the fluid domain (including (∂C¹)) is given by
Bernoulli theorem:

$$
P + \rho_{\text{air}} \left[ \frac{1}{2} v \cdot v + \frac{\partial \Phi}{\partial t} \right] = P_{\infty} + \rho_{\text{air}} \cdot \frac{1}{2} v_{\infty} \cdot v_{\infty} \quad (27)
$$

The subscript ∞ denotes the undisturbed
conditions far from the airship. This pressure
distribution over the airship surface can be
integrated to compute the resulting forces and
torques which in turn are projected on the modal
basis. At the end, and with the linear property of
the Laplace equation, the generalised fluid forces
vector can be rewritten as:

$$
F_f = -M_{ad} \ddot{Y} - B_f \dot{Y} \quad (28)
$$

where M_ad is the matrix of the added masses
(virtual masses), and Y is the column matrix of all
Yᵢ, B_f is a modal damping due to the flexibility of
the hull.

4.3 Modal projection

Taking into account the previous
developments, the dynamic equation (20) becomes:

$$
\mathbf{M}' \cdot \ddot{\mathbf{Y}} + \mathbf{B}' \cdot \dot{\mathbf{Y}} + \mathbf{K}_d \mathbf{Y} + \mathbf{K}_e \mathbf{Y} = \bar{\mathbf{F}}_e - \bar{\mathbf{F}}_R \quad (29)
$$

we note **M**'=**M**+**M**<sub>ad</sub>, **B**'=**B**+**B**<sub>f</sub>. The effect of the
fluid on the structure is then represented mainly by
the adjunction of the added masses matrix **M**<sub>ad</sub> to
the mass matrix of the structure.

Where

$$
M_{\text{ad}} \approx \begin{pmatrix}
-k_1 m & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -k_2 m & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -k_2 m & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -k_3 I_y & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -k_3 I_y & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}
$$

m is the mass of the body; k₁, k₂ and k₃ are
constants depending of the shape of the airship,
Iₓ, Iᵧ and I_z are moments of the inertia's body.

For a quasi-ellipsoid airship the extra-diagonal
terms of **M**<sub>ad</sub> can be neglected (for more details
about the constitutive terms of **M**<sub>ad</sub>, the reader can
see [11, 1]).

5. Simulation results

To illustrate this incremental formalism we
study the blimp belonging to the L.S.C having the
following characteristics:

-The envelope:

Length: 6.25 m.

Diameter: 1.52 m.

Volume: 7.48 m³.

-Mass of the airship: 5.8 Kg.

- Payload: 1.58 Kg.

The blimp is thrusted by two contrarotating
propellers.
<figure><img src="image_5.png" /><figcaption>Figure 4. The LSC airship</figcaption></figure>

To illustrate our incremental formalism, two kinds of motions are tested for this airship.

In the first step we apply to the blimp two opposite forces in the propellers to generate a yaw motion (Fig 7). The flexible airship should rotate about 90° around the z-axis. A P.I.D controller will impose this task, and we study the behaviour of the airship during this manoeuvre. The number of deformable modes kept is nd = 2. This number seems to give an acceptable approximation of the flexible behaviour. We were guided for that by the modal masses of these two modes which represent roughly 65% of the total mass of the airship.

In this simulation one applied around the yaw angle a torque consigns with:

$$f_{eg} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ k_d (\psi_d - \psi) + k_v (\dot{\psi}_d - \dot{\psi}) \end{pmatrix}$$

$K_d = 800$, $K_v = 50$, $\psi_d$: desired angular position and $\psi$: angular position of the airship.

One notices in figure 5 that the amplitude of deviation of the yaw angle decreases significantly in few seconds, it stabilizes oneself while merging with the instruction.

We show the position of the tip of the airship in figure 6.

<figure><img src="image_6.png" /><figcaption>Figure 5. The superposition of trajectories (yaw angle and desired).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6. Position of the tip of the airship</figcaption></figure>

In figure 7 one superimposes total displacement along the X-axis of the rigid airship and the flexible device. It is noticed that the flexible device continues to oscillate what proves the impact of flexibility on displacement. For the rigid behaviour we eliminate the two deformable modes and kept only the six rigid modes.

<figure><img src="image_8.png" /><figcaption>Figure 7. Superposition of the motions of the flexible and rigid airship</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8. The deformation's displacement</figcaption></figure>

The deformations (in figure 8) are about 0.15m
what is more or less significant considering one
has small deformations. In this simulation one
visualized the tip of the airship, and one sees well
the oscillation of the latter with non-negligible
amplitude. One classical motion of an airship is the
helicoidally motion. This usually represents the
optimal trim trajectories.

<figure><img src="image_10.png" /><figcaption>Figure 9. Rise of the airship.</figcaption></figure>

This represents a complete aerial motion of the
airship. The propellers were oriented adequately to
assure a combined motion along the moving x-axis
and z-axis and the tails were oriented to give a
homogeneous yaw motion.

One can see (in figure 9) the influence of the
deformation on the overall motion of the airship.
This proves that in our model, the deformation is
not a sample perturbation around the main rigid
motion, but it acts on this motion.
Experimental data will be available in few months
to validate the results of our incremental scheme.

Finally we show the influence of the added masses
on the total displacement of the airship.

<figure><img src="image_11.png" /><figcaption>Figure 10. Superposition of the displacement of the airship with and without air environing</figcaption></figure>

It is noticed (in figure 10) that the immersed
airship has a delay compared to the body alone and
this is because of the addition of the virtual masses
representing the influence of the displacement of
the mass of air around the airship when this last
accelerates.

**6. Conclusion**

The study of the flexible structures in the space
has different performance requirements which may
lead to troublesome calculation and inextricable
problems of controllability.

We introduced in this paper the flexibility of
the airship as an extension of the classical
structural dynamics taking into account the
coupling between the rigid body motion and the
deformation.

The Updated Lagrangian Method (U.L.M) applied on rigid reference configuration and the uses of an updated modal synthesis permit to diminish the dynamic equations of the flexible airship to a reduced set of degrees of freedom.

Simulation results prove that the integration of
the flexibility in the dynamic system of the airship
is very important and could not be neglected. The
influence of surrounding air is also taken into
account in our global model by the mean of added
masses and through the same modal algorithm.

The effect of surrounding air with the
flexibility of the hull will be treated in future
works.

**References**

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