# Three-Dimensionnal Hydroelastic Water Entry: Preliminary Results

Bundy Donguy, Bernard Peseux, Laurent Gornet, Emmanuel Fontaine

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Bundy Donguy, Bernard Peseux, Laurent Gornet, Emmanuel Fontaine. Three-Dimensionnal Hydroelastic Water Entry: Preliminary Results. 11th International Offshore and Polar Engineering Conference, ISOPE, Jun 2001, Stavanger, Norway. hal-01009051

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# Three-Dimensional Hydroelastic Water Entry: Preliminary Results

B. Donguy, B. Peseux and L. Gornet

Ecole Centrale de Nantes
Nantes, France

E. Fontaine

Institut Français du Pétrole
Rueil-Malmaison, France

## ABSTRACT

Slamming loads during the hydrodynamic impact of a three-dimensional body are studied numerically. First, the finite element method is used to solve the fluid part. For the water entry problem of a rigid body, the numerical results are successfully compared with analytical solutions for wedge, cone and the three-dimensional case of an elliptic paraboloid shaped bodies. In the second part of the paper, the coupled formulation and the coupling matrix between fluid and structure part which will be used to treat the hydroelasticity impact problem is presented and validated by solving for sloshing in a tank with an elastic wall problem.

**KEY WORDS:** slamming, hydroelasticity, asymptotic, three-dimensional.

## INTRODUCTION

In severe sea conditions, impact loads with high pressure occur when the hull of a ship strikes the water surface. These impulsive loads may generate plastic deformations of the local hull structure. Fractures have also been observed as the result of severe slamming events. In extreme cases, the integrity of the overall structure may be threatened due to a large increase of the global bending stresses. The ability to better predict the structural response of the ship hull to slamming loads, both locally and globally, appears therefore necessary.

Reviews on the subject of slamming have been proposed by Korobkin & Pukhnachov (1988), Mizoguchi & Tanizawa (1996) and more recently by Faltinsen (2000) who focus attention on the influence of hydroelasticity effects. From a theoretical point of view, slamming loads have been mostly studied within the framework of potential flow theory, assuming blunt and rigid body, and planar flow. Under these assumptions, first order asymptotic solutions were found for the case of a wedge with small deadrise angles (Wagner, 1932), a cylinder (ointe, 1987), and more generally, for arbitrary two-dimensional blunt body shapes (Cointe, 1989, Howison, Ockendon & Wilson, 1991). The latter also consider arbitrary axi-symmetric bodies for which the solution is very similar. These classical two-dimensional asymptotic results were successfully validated against experiments in the cases of wedges (Chuang, 1967, Fontaine & Cointe, 1997),

cylinders (Cointe & Armand, 1987), and realistic ship cross-sections (Zhao et al, 1996, Magee & Fontaine, 1998). Good agreement is generally reported although the asymptotic solution tends to over predict the maximum of the peak pressure. Comparisons require sometimes to take into account three-dimensional side effects. The first order asymptotic solution was also compared with fully non-linear simulations using Boundary Element Method together with a cutting algorithm for the jet (Zhao & Faltinsen, 1992, Fontaine & Cointe, 1997, Iafrati & Korobkin, 2000). For a wedge, the comparisons show that the pressure distribution is correctly predicted as the deadrise angle goes to zero. For larger values of the deadrise angle, the agreement remains very good if second order effects are accounted for in the asymptotic pressure distribution. For constant vertical impact velocity, most of the second order effects are accounted for when the matching technique proposed by Cointe (1987) is applied.

From a practical point of view, the two-dimensional asymptotic solutions for the flow around rigid bodies are classically used to represent the three-dimensional flow within the framework of a strip theory. Nevertheless, the use of a strip theory assumes the ship hull to be slender near the impact point, which is not always the case, thus the need for three-dimensional solutions. Some problems also arise when strip theory is applied to study locally the hydro-elastic effects associated with severe impact events. Indeed, the structural deformations of the hull near the impact point exhibit a strong three-dimensional character (see e.g. Donguy, Peseux, Fontaine, 2000). Unlike the two-dimensional problem for which analytical solutions can be derived for arbitrary blunt body shapes, few results are available in the three-dimensional case. Existing rigid body solutions remains limited to elliptic paraboloid body like shapes (Korobkin, 1988, Scolan & Korobkin, 2001). One of the next improvements in the modeling of the hydrodynamic impact problem is therefore to take into account the three-dimensionality of the flow. If analytic or quasi-analytic solutions can be obtained in the two-dimensional case, a purely numerical approach has to be developed to treat the general three-dimensional situation. The aim of this study is to develop such a numerical tool in order to evaluate impact loads on rigid or deformable, three-dimensional blunt bodies.
In the present paper, the asymptotic equations describing the fluid flow during the first time of impact are first presented together with the physical assumptions sustaining the modelling. Within the framework of this analysis, the problem is solved once the velocity potential on the body and the contact line between the body and the free surface are known. The numerical procedure used to solve the problem is then described. The velocity and displacement potentials are computed based on a variational formulation of the problem together with the Finite Element Method. The displacement potential is introduced for iterative resolution of the contact line. Once the wetted area is known, pressure distribution is then computed using velocity potential. The accuracy of the numerical scheme is tested through systematic comparisons between numerical and analytical results. Classical rigid body solutions are recovered for the two-dimensional case of a wedge, the axi-symmetric case of a cone, and the three-dimensional case of an elliptic paraboloid, therefore validating the approach for the rigid body case. The second part of the paper is devoted to hydroelasticity. The equations describing the structural deformations are included in the analysis and a coupled approach is proposed to solve the resulting fluid structure interaction problem. Numerical simulations for the impact of deformable bodies impact will be presented in subsequent publications. In the present study, attention is focussed on the validation of the coupling terms and of the resolution procedure. More specifically, the numerical scheme is successfully applied to determine the high frequency resonant frequencies of a sloshing tank with an elastic wall. For this test case problem, the numerical results by Peseux, Cartraud and Argouar'ch (1999) are successfully recovered.

### THREE DIMENSIONAL RIGID BODY WATER ENTRY

#### Fluid flow modelling

The water entry problem is classically formulated within the framework of potential flow theory. The fluid is assumed to be ideal and incompressible, and the flow to be irrotational. Under these assumptions, the velocity field can be deduced from a velocity potential $\phi = \phi(x,y,z,t)$, according to $\vec{v} = gr\bar{a}d\phi$.

<figure><img src="image_2.png" /><figcaption>Figure 1. Geometrical definitions</figcaption></figure>

The velocity potential must satisfies the following boundary value problem:

$$ \Delta\phi = 0 \qquad \text{in the fluid} \qquad (1) $$

$$ \frac{\partial\phi}{\partial n} = \dot{u}_s \cdot \bar{n} \qquad \text{on the body} \qquad (2) $$

$$ \frac{\partial\phi}{\partial t} + \frac{1}{2}(gr\bar{a}d\phi)^2 + gz = 0 \qquad \text{on the free surface} \qquad (3) $$

$$ \frac{dh}{dt} = \frac{\partial\phi}{\partial t} \qquad \text{on the free surface} \qquad (4) $$

where the displacement of the body is given by its $\bar{u}_s(x,y,t)$ and the free surface elevation is written as $z = h(x,y,t)$. The body boundary condition, Eq. 2, expresses the continuity of the normal velocity on the body surface. The kinematic and dynamic free surface conditions, Eq. 3 and Eq. 4 respectively, state that the free surface is a material surface along which the pressure is constant. Moreover, the fluid is assumed to be initially at rest:

$$ h(x,y,0) = 0 \qquad \phi(x,y,z,0) = 0 \qquad (5) $$

and the flow to be unperturbed far from the body:

$$ |gr\bar{a}d\phi| \to 0 \quad \text{when } (x^2+y^2+z^2)^{1/2} \to \infty \qquad (6) $$

Once the velocity potential is known, the pressure on the body is calculated by Bernoulli's equation:

$$ \frac{p}{\rho} = -\frac{\partial\phi}{\partial t} - \frac{1}{2}(gr\bar{a}d\phi)^2 - gz \qquad (7) $$

#### Asymptotic formulation

The above mentioned boundary value problem is not easily tractable, even numerically, due to the violent deformations of the free surface. The latters are experienced as jets developing along the side of the body (see e.g. Greenhow, 1987). Classically, the hydrodynamic impact problem (1)-(4) is simplified into:

$$ \Delta\phi = 0 \qquad \text{in } \Omega_f \qquad (8) $$

$$ \frac{\partial\phi}{\partial n} = \dot{u}_s \cdot \bar{n} \qquad \text{on } \Gamma_B \qquad (9) $$

$$ \phi = 0 \qquad \text{on } \Gamma_L \qquad (10) $$

$$ \frac{\partial h}{\partial t} = \frac{\partial \phi}{\partial z} \qquad \text{on } \Gamma_L \qquad (11) $$

where $\Gamma_B$ is the projection on $z = 0$ of the wetted body surface and $\Gamma_L$ the position of the unperturbed free surface. $\Omega_f$ represent the linearized fluid domain (see fig. 1). The simplified equations can be formally derived using perturbation technique (see Cointe, 1989, Wilson, 1989). The perturbation procedure relies strongly on the blunt body assumption since the small parameter used in the asymptotic expansion is the ratio between the immersion and characteristic length scale of the body wetted width. Within a far field point of view, the body boundary condition, Eq. 2, can be written on $z = 0$ without introducing significant error, therefore justifying Eq. 9. The exact dynamic condition, Eq. 3, has been replaced by a Dirichlet condition for the potential on the undisturbed position of the free surface, Eq. 10. Physically the acceleration in the fluid is assumed to be large compared to gravity which can be neglected during the first instants. Finally, the simplified kinematical free surface condition, Eq. 11, states that the vertical displacement of the free surface is equal to the fluid vertical motion, evaluated on the undisturbed position of the free surface. The quadratic terms have to remain small compared to the linear ones for this approximation to be valid. The resulting problem, Eqs. 8-11 is often referred as the generalised Wagner (1932) problem. In the present formulation, three-dimensional effects are retained through the Laplace's equation, and structural deformations of the body surface are taken into account in the body boundary condition. In this section the body is assumed to be rigid. Structural equations driving the displacement $\bar{u}_s$ and hydroelastic effects will be presented in the second part of the paper.

It is important to keep in mind that the asymptotic equations are only valid in the far-field since the solution is singular at the intersection between the body and the free surface. Nevertheless, the
far-field solution is matched near the intersection to a local solution describing the formation of a jet following Wagner (1932), leading to a composite solution which is regular by construction. Since the jet solution is completely defined by the far-field solution, attention is focused on the latter.

## Determination of the wetted surface

Due to the deformations of the free surface, the wetted surface $Γ_B$ is part of the unknown. An additional equation is therefore needed to close the problem. Physically, the contact line is determined by imposing the existence of an intersection point between the outer expansion of the free surface elevation and the body:

$$h(x,y,t) = \bar{u}_r \bar{z} \qquad (12)$$

This condition, intuitively introduced by Wagner (1932), also states volume conservation to the leading order (Wilson, 1989, Fontaine & Cointe, 1997). In order to satisfy this condition, the asymptotic problem is formulated for the displacement potential:

$$\psi(x, y, z; t) = \int_0^t \phi(x, y, z; s) \, ds \qquad (13)$$

as suggested by Korobkin (1982). The main advantage of introducing this transformation is that solving for $\psi$ does not require the knowledge of the temporal evolution of the free surface elevation. The boundary value problem satisfied by $\psi$ is similar to the one satisfied by $\phi$ except for the body boundary condition which reads:

$$\frac{\partial \psi}{\partial n} = \bar{u}_s \cdot \bar{n} \quad \text{on the body} \qquad (14)$$

The problem for $\psi$ is solved through an iterative procedure until Eq. 12 is satisfied. A precise description of the method is given in Donguy et al. (2000).

## Numerical resolution

The resolution procedure for the velocity and displacement potentials is based on the Finite Element Method. The weighted residue method is applied considering the weighting functions $\phi$ which verify the boundary condition $\phi = 0$ on $Γ_L$. The problem comes to minimize the integral quantity:

$$W(\phi) = \int_{\Omega_f} \phi R(\phi) dD \qquad (15)$$

where the residue $R(\phi)$ is set equal to $\Delta\phi$. Then, applying Green's identity and taking into account of boundary conditions, Eq. 16, leads to the weak formulation:

$$\int_{\Omega_f} gr\bar{a}d\phi \cdot gr\bar{a}d\phi \cdot dD = \int_{\Gamma_s} \phi(\dot{\bar{u}}_s \cdot \bar{n}) \cdot dS \qquad (16)$$

The potential $\phi$ and weight functions $\phi$ are approximated using Galerkin's method leading to:

$$\phi = \langle N_f \rangle (\phi)^e \qquad \phi = \langle N_f \rangle (\phi)^e \qquad (17)$$

where the shape functions $N_f$ depend on the types of elements, and $(\phi)^e$ denotes the nodal potential vector of the finite element (e). Finally, the following linear system is deduced from the discretization of the bilinear forms, Eq. 16:

$$[H]\phi = \{G_\phi\} \qquad (19)$$

$$H_{ij}^e = \int_{\Omega_e} N_{i,j} \cdot N_{j,i} dD \qquad (20)$$

$$(G_{\phi})^e = \int_{\Gamma_B^e} (N_f)(\dot{\bar{u}}_s \cdot \bar{n})dS \qquad (21)$$

and for the displacement potential:

$$[H]\psi = \{G_{\psi}\} \qquad (14)$$

$$(G_{\psi})^e = \int_{\Gamma_B^e} (N_f)(\dot{\bar{u}}_s \cdot \bar{n})dS \qquad (15)$$

## Validation

The numerical method previously described is applied to solve several test case problems of rigid bodies penetrating with constant velocity a free surface initially at rest.

The two-dimensional test case of a wedge is presented in Fig. 2 and 3 where the free surface elevation and the pressure distribution are plotted, respectively. Despite a relatively coarse mesh based on triangular and quadrilateral elements is used, the agreement between the numerical and analytical solution remains excellent, even close to the intersection point were the solution is singular since the pressure and the free surface slope tends to infinity. The singularity of the pressure distribution is nevertheless correctly captured as can be seen in Fig. 3 where only the far-field solution has been represented. One should keep in mind that the singularity is to be removed by matching the far-field solution to a jet solution near the intersection.

The axisymmetric case of a cone is presented in Fig. 4 and 5. Although the body shape is axis-symmetric, the computation is performed on a three-dimensional mesh based on hexahedron types of elements. This allows to check independently the three-dimensional resolution of the Laplace's equation and the iterative procedure for solving the wetted area. Good agreement between numerical and analytical results is obtained again. All parameters being similar to the previously described two-dimensional computation, the relative error remains of the same order, despite the mesh is this time really three-dimensional.

Finally, the fully three-dimensional case of an elliptic paraboloid is presented in Fig. 6 to 9. This test case allows to validate the resolution procedure in the general case. Fig. 6 shows that the free surface elevation is correctly predicted along angular directions corresponding to the two main axis of the body. Fig. 7 shows that good agreement is also obtained for the pressure distribution too. In order to capture correctly the singularity near the intersection point, the mesh has to be refined locally, see Fig. 8. Finally, Fig. 9 shows that good agreement is obtained for the evolution of the wetted width along the two main axis. Being able to predict the wetted width as a function of the angular direction is essential to match the three-dimensional outer solution to the two-dimensional inner solution describing the jet. In particular, the jet thickness and the pressure distribution in the jet region are directly proportional to the time derivative of the wetted width along the angular direction.
<figure><img src="image_3.png" /><figcaption>Figure 2: Wedge shaped body. Comparison between numerical and analytical solutions for the outer free surface elevation.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 4: Cone shaped body. Comparison between numerical and analytical solutions for the outer free surface elevation.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 6: Elliptic paraboloid shaped body. Comparison between numerical and analytical solutions for the outer free surface elevation.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 3: Wedge shaped body. Comparison between numerical and analytical for the outer pressure distribution. The error distribution is plotted on the right.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Cone shaped body. Comparison between numerical and analytical for the outer pressure distribution. The error distribution is plotted on the right.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Elliptic paraboloid shaped body. Comparison between numerical and analytical solutions for the outer pressure distribution. The error distribution is plotted on the right.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: Elliptic paraboloid shaped body.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9: Elliptic paraboloid shaped body. Comparison between numerical and analytical solutions for the wetted length along the two main axis. The blue line represents the penetration depth.</figcaption></figure>

## HYDROELASTICITY

In this section, the previous analysis is extended to treat the case of a deformable body. The equations governing the structural deformations are first recalled for the sake of completeness, together with the boundary conditions. A coupled method to solve the fluid-structure interaction problem is then proposed and validated on an example.

### Governing structural equations

The usual assumptions of small perturbations is made. The equilibrium equation reads:

$$ \rho \ddot{u}_s = d \dot{v}_d \Sigma + \bar{f} \quad \text{in } \Omega_s \quad (22) $$

where $\Sigma$ denotes the stress tensor, $\bar{u}_s$ corresponds to the structural displacement, and $f$ corresponds to volume force associated with the gravity field. If gravity effects do not contribute to the local deformations here, they nevertheless influence the global motion of the body before impact. Boundary condition on the structure is:

$$ \bar{u} = \bar{\Sigma}_f \bar{n} = \bar{\Sigma}_s \bar{n} \quad \text{on } \Gamma_B \quad (23) $$

where the subscript $f$ and $s$ refer to the fluid and the structure, respectively.

The structural modeling is performed within the framework of linear elasticity. Hooke's law is used to express the relation between the

stresses and the strains which are given by the linearised Green-Lagrange tensor $\varepsilon$:

$$ \varepsilon_{i,j} = \frac{1}{2} (u_{i,j} + u_{j,i}) \quad (24) $$

### Coupled problem

The structural problem is also solved using the Finite Element Method. The principle of virtual works is applied to Eq. 22 together with boundary condition, Eq. 23, leading classically to the discretised form:

$$ [M_s]\ddot{u}_s + [K]\{U\} = \{F\} \quad (25) $$

$$ \{F\} = \int_{\Gamma_B} [N_s \Gamma \{n\} p dS \quad (26) $$

where the mass and stiffness matrices, respectively $[M_s]$ and $[K]$, together with the generalized load vector $\{F\}$, are obtained by assembly of the elementary terms calculated on each finite element (e). The nodal displacements $\{U\}$ is linked to the interpolation of the displacement field through:

$$ \{u_s\} = [N_s]\{U\} \quad (27) $$

By definition, hydroelastic effects arise as the result of a strong interaction between the fluid flow and the structural response. From Eq. 26, it can be seen that the pressure imposed by the fluid will drive the structural deformations. On the other hand, the structural deformations influence the pressure field through the body boundary condition as shown by Eq. 2. In order to obtain a robust method to solve the coupled fluid-structure interaction problem, all the evolution equations are integrated simultaneously. The structural deformations are included in the fluid flow discrete formulation, while the pressure is expressed as the time derivative of the velocity potential in the discrete structural analysis. The following problem is then obtained:

$$ [H]\phi = [FS]\dot{u} \quad (28) $$

$$ [M_s]\ddot{u}_s + [K]\{U\} = -\rho_f [FS]\dot{\phi} \quad (29) $$

$$ [FS]^T = \int_{\Gamma_B} [N_f \Gamma \{n\} p dS \quad (30) $$

where the coupling terms $[FS]$ representing the fluid-structure interaction appear explicitly. Eq. 30 shows that the coupling term involve only the values of the structural displacement $\bar{u}_s$ and the velocity potential $\phi$ on the wetted surface of the body, $\Gamma_B$. For the numerical resolution, Eqs. 28 and 29 are rewritten using with a new nodal vector $\{W\}^T = \{\phi, U\}^T$, leading to:

$$ [M]\ddot{w} + [B]\dot{w} + [K]\{W\} = 0 \quad (31) $$

where:

$$ [M] = \begin{bmatrix} [0] & [0] \\ [0] & [M_s] \end{bmatrix} \quad [B] = \begin{bmatrix} [0] & -[FS] \\ \rho_f [FS] & [0] \end{bmatrix} \quad [K] = \begin{bmatrix} [H] & [0] \\ [0] & [K_s] \end{bmatrix} $$

The fluid-structure interaction problem, Eq. 31, is then integrated starting from the initial conditions:

$$ W(t=0) = W_0 \quad (32) $$

$$ \dot{W}(t=0) = \dot{W}_0 \quad (33) $$
## Sloshing in a tank with an elastic wall

The numerical resolution of Eq. 31 is performed using commercial Finite Element Method numerical code Castem. An external procedure is implemented for the evaluation of the coupling matrix [FS]. The correct implementation of the approach is checked for the test case problem of sloshing in a rectangular tank including an elastic beam in its center, as shown in Fig. 10. The mechanical excitation consists in a harmonic load applied to the beam extremity. A frequency scanning is performed, see Fig. 12, to evaluate the eigenfrequencies which correspond to the peaks of the beam displacement. The free surface elevation for the first four modes of the coupled problem is shown in Fig. 11, together with the modal shape of the elastic beam. The numerical value of the eigenfrequencies are presented in table 1, column C1. They are compared to the results by Peseux et al (1999), column entitled "Ref". A toolbox that allows the direct computation of the resonant frequencies was also applied, see column C2. All the results are relatively closed to each other, especially, the two last columns. The computation by Peseux et al (1999) were performed using another f.e.m. code, which may explains the small differences in the results. The agreement between the three method is very satisfactory, which give confidence into the external numerical procedure that has been developed in the present study. Developing this procedure, and controlling precisely the coupling terms resulting from the fluid-structure interaction is necessary to treat the water impact problem of an elastic body.

## CONCLUSIONS

In the present paper, a numerical method is proposed to solve the three-dimensional water entry problem of a blunt and rigid body. A variational formulation together with a finite element method are used to solve the so called Wagner problem. The wetting correction is obtained through an iterative procedure. The numerical resolution is validated by simulating simple problems, such as the water impact problem of a wedge with small deadrise in 2D, or a cone for the axisymmetric case, and, an elliptic paraboloid in the three-dimensional case. Good agreement is obtained between numerical and analytical solutions. The approach is then extended to treat the case of a deformable body, for which a strong fluid-structure interaction is expected. Structural elastic deformations are computed simultaneously in the f.e.m. analysis. The fluid-structure interaction arise through a coupling matrix. The correct evaluation of the coupling term has checked for the sloshing problem of a tank with an elastic wall. Earlier numerical results are recovered, therefore validating the methodology of the coupled approach.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support of the French Navy, Délégation Générale de l'Armement and Bassin d'Essais des Carènes.

## REFERENCES

* Castem2000. Département de Mécanique et Technologie du Commissariat à l'Energie Atomique.

* Cointe, R (1987). "L'impact hydrodynamique en deux dimensions," J'ère Journées de l'Hydrodynamique, Nantes, France.

* Cointe, R, Armand, JL (1987). "Hydrodynamic impact analysis of a cylinder," J. Offshore Mechanics and Artic Engineering, Vol 9, pp 237-243.

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* Chuang, SL (1967). "Experiments on slamming of wedge-shaped bodies," Journal of Ship Research, pp 190-198.

<figure><img src="image_11.png" /></figure>

**Mechanical characteristic**

$$E = 2,1 \times 10^{11} \text{ Pa}$$

$$\rho_f = 7800 \text{ kg.m}^{-3}$$

$$e = 6.10^{-4} \text{ m}$$

$$H = 0,266 \text{ m}$$

**Fluid characteristic**

$$\rho_f = 1000 \text{ kg.m}^{-3}$$

$$L = 0,231 \text{ m}$$

$$h_f = 0,210 \text{ m}$$

<table><caption>Figure 10: elastic beam in a heavy fluid problem</caption><thead><tr><th>Hertz</th><th>Ref.</th><th>C1</th><th>C2</th></tr></thead><tbody><tr><td>f<sub>1</sub></td><td>1.49</td><td>1.60</td><td>1.61</td></tr><tr><td>f<sub>2</sub></td><td>1.86</td><td>1.84</td><td>1.83</td></tr><tr><td>f<sub>3</sub></td><td>2.39</td><td>2.46</td><td>2.44</td></tr><tr><td>f<sub>4</sub></td><td>2.66</td><td>2.63</td><td>2.61</td></tr><tr><td>f<sub>5</sub></td><td>2.97</td><td>3.06</td><td>3.01</td></tr></tbody></table>

<table><caption>Table 1. Eigenfrequencies in Hz</caption><thead><tr><th>Hertz</th><th>Ref.</th><th>C1</th><th>C2</th></tr></thead><tbody><tr><td>f<sub>6</sub></td><td>3.33</td><td>3.29</td><td>3.21</td></tr><tr><td>f<sub>7</sub></td><td>3.46</td><td>3.51</td><td>3.42</td></tr><tr><td>f<sub>8</sub></td><td>3.95</td><td>3.91</td><td>3.73</td></tr><tr><td>f<sub>9</sub></td><td>4.0</td><td>3.99</td><td>3.81</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Figure 11: Eigenmodes shapes.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12: Eigenfrequencies search using scanning.</figcaption></figure>
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