# An Improvement of Port-Hamiltonian Model of Fluid Sloshing Coupled by Structure Motion

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

_{v}is the mass of the empty vessel; X(t) represents the horizontal position of the vessel (with respect to an inertial frame); x is the position along the vessel (measured in local coordinates from the middle of the vessel); h(x,t) is the height profile of fluid (−a/2 < x < a/2); and u(x,t) is the fluid speed (measured relative to the vessel).

- The fluid and the tank are viewed as an open system.
- The fluid governing equations are obtained from Euler–Lagrange formulation.
- There is no flow through the tank walls.
- The effect of rigid-body inertias of the tank and the mass of the tank is considered.
- The structure is considered a beam with two independent bending and torsion motions.
- Because of symmetrical installation of the piezoelectric patches along the torsion axis, their effect on torsional motion is neglected.
- The piezoelectric patches actuation affects only the bending motion of the beam.
- Here, the linear Euler–Bernoulli beam with distributed piezoelectric actuators is used for the bending.
- The electric field is constant through the piezoelectric patch thickness.
- The voltage is uniform along the patch.
- The beam is considered as a long beam under small displacements.
- The fixed end for the torsion and for the bending is considered at the clamped point.
- A weak formulation is used to overwhelm the derivation of the non-smoothness function.

_{b}, the discretized bending equations are

_{t}, the discretized torsion equations are

_{f}, the discretized shallow water equations are

_{e}= ρπ·L

_{out}·(D

_{out}

^{2}− D

_{in}

^{2})/4), the tip mass (m

_{tip}), the mass of two fixation rings (m

_{ring}= 2ρ

^{ring}·π·t

^{ring}·((D

_{out}

^{ring})

^{2}− (D

_{in}

^{ring})

^{2})/4), and reduction of plate hole mass (m

_{h}= ρ

_{b}π·t

_{b}·D

_{out}

^{2}/4). The tip mass (m

_{tip}), located at the vessel tip, consists of plastic disc, glue, and a plug. Although it is not a simple geometry, here, the effects on the system mass and inertia are approximated as a point mass (m

_{R}= m

_{e}+ m

_{tip}+ m

_{ring}− m

_{h}).

_{r}is total vessel mass moment of inertia around the z-direction, since vessel and fixation rings are symmetric around the z-axis. The total vessel mass moment of inertia is the sum of the empty vessel inertia (I

_{e}= m

_{e}(3/4(D

_{out}

^{2}− D

_{in}

^{2}) + L

_{out}

^{2})/12), the tip mass inertia (I

_{tip}= m

_{tip}L

_{tip}

^{2}), the mass of two fixation rings (I

_{ring}= m

_{ring}/12(3/4 ((D

_{out}

^{ring})

^{2}+ (D

_{in}

^{ring})

^{2}) + (t

^{ring})

^{2})), and reduction of plate hole mass (I

_{h}= m

_{h}·D

_{out}

^{2}/16).

_{R}is total mass moment of inertia around the y-direction (J

_{R}= I

_{r}+ I

_{f}) Total vessel mass moment of inertia around the y-direction is the sum of vessel moment of inertia around the y-direction, which is similar to I

_{r}·(J

_{R}= I

_{e}+ I

_{tip}+ I

_{ring}− I

_{h}) and the frozen fluid mass moment of inertia $\left({I}_{f}=\text{}{m}_{f}{L}_{eq}{}^{2}/12\right)$

_{in}), then the total volume of the fluid is

_{eq}= V

_{f}/W

_{v}).

## 3. Numerical Results

_{i}for finite element method with the property of being one at the point i and zero at the other points, the simple linear functions are used. As shown in Figure 4, the simplest choice of course linear functions (blue lines) caused the values of basis functions equal to the maximum value (here is one) at the desired point and zero other grid nodes (plus symbols). During the semi-discretization of the equations, a weak formulation is used to overcome the problem of discontinuous derivate which made some difficulties in existence and uniqueness results for such systems. Finite-element method discretization of the second order derivate in Equation (50) using spline approximation leads to the matrix equation as

^{−4}s. Figure 5 illustrates the tip displacement versus time. Initial condition considered here is the stationary case, with four Newtons tip force. The figure shows the maximum downward deflection of the beam is 3 mm. Figure 5 also shows that in steady conditions beam tip deflection started from 3 mm with two small oscillations in each period. The classical laminated plate theory gives the relationships among the plate bending moments, torsional moment and curvatures. The dimensionless coupling term $\psi =K/\sqrt{EIGJ}$ here is small, where EI, GJ and K are bending stiffness about the y-axis, torsion stiffness and bending–torsion coupling stiffness, respectively. Different modes of operation lead to a wide range of frequencies, which are partially visible for example in the fluid forces acting on the structure.

## 4. Conclusions

- A new model for rotating–translating shallow water equations in moving containers is offered within the port-Hamiltonian equations.
- The suggested model was used to solve a fluid–structure interaction problem, which includes a piezoelectric actuator on a flexible beam (bending element and torsional element), and a rigid vessel. Each subsystem was considered within the port-Hamiltonian equations and coupled using the interaction ports.

## Funding

## Conflicts of Interest

## References

- Molin, B.; Remy, F. Experimental and numerical study of the sloshing motion in a rectangular tank with a perforated screen. J. Fluids Struct.
**2013**, 43, 463–480. [Google Scholar] [CrossRef] - Nicolici, S.; Bilegan, R.M. Fluid structure interaction modeling of liquid sloshing phenomena in flexible tanks. Nucl. Eng. Des.
**2013**, 258, 51–56. [Google Scholar] [CrossRef] - Shadloo, M.S.; Oger, G.; le Touzé, D. Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges. Comput. Fluids
**2016**, 136, 11–34. [Google Scholar] [CrossRef] - Zhang, Z.; Qiang, H.; Gao, W. Coupling of smoothed particle hydrodynamics and finite element method for impact dynamics simulation. Eng. Struct.
**2011**, 33, 255–264. [Google Scholar] [CrossRef] - Chen, B.F.; Nokes, R. Time-independent finite difference analysis of fully non-linear and viscous fluid sloshing in a rectangular tank. J. Comput. Phys.
**2005**, 209, 47–81. [Google Scholar] [CrossRef] - Koutandos, E.; Prinos, P.; Gironella, X. Floating breakwaters under regular and irregular wave forcing: Reflection and transmission characteristics. J. Hydraul. Res.
**2005**, 43, 174–188. [Google Scholar] [CrossRef] - Wei, Z.; Faltinsen, O.M.; Lugni, C.; Yue, Q. Sloshing-induced slamming in screen-equipped rectangular tanks in shallow-water conditions. Phys. Fluids
**2015**, 27, 032104. [Google Scholar] [CrossRef] - Monaghan, J.J. Smoothed particle hydrodynamics. Ann. Rev. Astron. Astrophys.
**1992**, 30, 543–574. [Google Scholar] [CrossRef] - Fatehi, R.; Shadloo, M.S.; Manzari, M.T. Numerical investigation of two-phase secondary kelvin-helmholtz instability. Proc. Inst. Mech. Eng. Part C
**2013**. [Google Scholar] [CrossRef] - Benson, D.J. Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Methods Appl. Mech. Engrg.
**1992**, 99, 235–394. [Google Scholar] [CrossRef] - Rahmat, A.; Tofighi, N.; Shadloo, M.S.; Yildiz, M. Numerical simulation of wall bounded and electrically excited Rayleigh–Taylor instability using incompressible smoothed particle hydrodynamics. Colloids Surfaces A Physicochem. Eng. Asp.
**2014**, 460, 60–70. [Google Scholar] [CrossRef] - Monaghan, J.J. Smoothed particle hydrodynamics. Rep. Prog. Phys.
**2005**, 68, 1703–1759. [Google Scholar] [CrossRef] - Libersky, L.D.; Petschek, A.G. Smooth Particle Hydrodynamics with Strength of Materials. In Proceedings of the Next Free-Lagrange Conference, Moran, WY, USA, 3–7 June 1990; pp. 248–257. [Google Scholar]
- Benz, W.; Asphaug, E. Simulations of brittle solids using smoothed particle hydrodynamics. Comput. Phys. Commum.
**1995**, 87, 253–265. [Google Scholar] [CrossRef] - Monaghan, J.J. SPH without a tensile instability. J. Comput. Phys.
**2000**, 159, 290–311. [Google Scholar] [CrossRef] - Libersky, L.D.; Randles, P.W.; Carney, T.C.; Dickinson, D.L. Recent improvements in SPH modeling of hypervelocity impact. Int. J. Impact Eng.
**1997**, 20, 525–532. [Google Scholar] [CrossRef] - Johnson, G.R.; Stryk, R.A.; Beissel, S.R. SPH for high velocity impact computations. Comput. Methods Appl. Mech. Engrg.
**1996**, 139, 347–373. [Google Scholar] [CrossRef] - Johnson, G.R. Linking of Lagrangian particle methods to standard finite element methods for high velocity impact computations. Nucl. Eng. Des.
**1994**, 150, 265–274. [Google Scholar] [CrossRef] - Attaway, S.W. Coupling of smoothed particle hydrodynamics with the finite element method. Nucl. Eng. Des.
**1994**, 150, 199–205. [Google Scholar] [CrossRef] - Fernández-Méndez, S.; Bonet, J.; Huerta, A. Continuous blending of SPH with finite elements. Comput. Struct.
**2005**, 83, 1448–1458. [Google Scholar] [CrossRef] [Green Version] - Yu, Y.; Ma, N.; Fan, S.; Gu, X. Experimental and numerical studies on sloshing in a membrane-type LNG tank with two floating plates. Ocean Eng.
**2017**, 129, 217–227. [Google Scholar] [CrossRef] - Kotyczka, P.; Maschke, B. Discrete port-Hamiltonian formulation and numerical approximation for systems of two conservation laws. Automatisierungstechnik
**2017**, 65, 308–322. [Google Scholar] [CrossRef] - Cardoso-Ribeiro, F.L.; Matignon, D.; Pommier-Budinger, V. A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system. J. Fluids Struct.
**2017**, 69, 402–427. [Google Scholar] [CrossRef] - Ibrahim, R.A. Liquid Sloshing Dynamics; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Lasiecka, I.; Tuffaha, A. Boundary feedback control in fluid-structure interactions. In Proceedings of the 47th IEEE Conference on Decision Control, Cancun, Mexico, 9–11 December 2008; pp. 203–208. [Google Scholar]
- Petit, N.; Rouchon, P. Dynamics and solutions to some control problems for water-tank systems. IEEE Trans. Autom. Control
**2002**, 47, 594–609. [Google Scholar] [CrossRef] [Green Version] - Pommier-Budinger, V.; Richelot, J.; Bordeneuve-Guib, J. Active control of a structure with sloshing phenomena. In Proceedings of the IFAC Conference on Mechatronic Systems, Berkeley, CA, USA, 9–11 December 2006. [Google Scholar]
- Coron, J.-M. Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations. ESAIM Control Optim. Calc. Var.
**2002**, 8, 513–554. [Google Scholar] [CrossRef] [Green Version] - Prieur, C.; de Halleux, J. Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations. Syst. Control Lett.
**2004**, 52, 167–178. [Google Scholar] [CrossRef] - Robu, B. Active Vibration Control of a Fluid/Plate System. Ph.D. Thesis, University of Toulouse, Toulouse, France, December 2010. [Google Scholar]
- van der Schaft, A.J.; Jeltsema, D. Port-Hamiltonian Systems Theory: An introductory Overview; Trends Controls: Horsham, UK, 2014; pp. 173–378. [Google Scholar]
- Robu, B.; Baudouin, L.; Prieur, C.; Arzelier, D. Simultaneous H∞ Vibration Control of Fluid/Plate System via Reduced-Order Controller. IEEE Trans. Control Syst. Technol.
**2012**, 20, 700–711. [Google Scholar] [CrossRef] - Cardoso-Ribeiro, F.L.; Pommier-Budinger, V.; Schotte, J.-S.; Arzelier, D. Modeling of a coupled fluid-structure system excited by piezoelectric actuators. In Proceedings of the 2014 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Besançon, France, 8–11 July 2014; pp. 216–221. [Google Scholar]
- Cardoso-Ribeiro, F.L.; Matignon, D.; Pommier-Budinger, V. Control design for a coupled fluid-structure system with piezoelectric actuators. In Proceedings of the 3rd CEAS EuroGNC, Toulouse, France, 13–15 April 2015. [Google Scholar]
- Ardakani, H.A.; Bridges, T.J. Dynamic coupling between shallow-water sloshing and horizontal vehicle motion. Eur. J. Appl. Math.
**2010**, 21, 479–517. [Google Scholar] [CrossRef] - Ardakani, H.A.; Bridges, T.J. Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions. J. Fluid Mech.
**2011**, 667, 474–519. [Google Scholar] [CrossRef] - Hamroun, B.; Lefevre, L.; Mendes, E. Port-based modelling and geometric reduction for open channel irrigation systems. In Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, 12–14 December 2007; pp. 1578–1583. [Google Scholar]
- Hamroun, B.; Dimofte, A.; Lefèvre, L.; Mendes, E. Control by Interconnection and Energy-Shaping Methods of Port Hamiltonian Models. Application to the Shallow Water Equations. Eur. J. Control
**2010**, 16, 545–563. [Google Scholar] [CrossRef] [Green Version]

Parameter | Value | Unit | Description |
---|---|---|---|

L_{b} | 1.36 | m | Plate length |

w_{b} | 0.16 | m | Plate width |

t_{b} | 0.005 | m | Plate thickness |

ρ_{b} | 2970 | kg·m^{−3} | Plate density |

E_{b} | 75 | GPa | Plate Young modulus |

ν_{b} | 0.33 | - | Plate Poisson coefficient |

L_{pa} | 0.14 | m | Actuator length |

w_{pa} | 0.075 | m | Actuator width |

t_{pa} | 0.5 | mm | Actuator thickness |

ρ_{pa} | 7800 | kg·m^{−3} | Actuator density |

E_{pa} | 67 | GPa | Actuator Young modulus |

ν_{pa} | 0.3 | - | Actuator Poisson coefficient |

d_{31} | −210 | pm·V^{−1} | Actuator piezoelectric strain coefficient |

L_{ps} | 0.015 | m | Sensor length |

w_{ps} | 0.025 | m | Sensor width |

t_{ps} | 0.5 | mm | Sensor thickness |

ρ_{ps} | 7800 | kg·m^{−3} | Sensor density |

E_{ps} | 67 | GPa | Sensor Young modulus |

ν_{ps} | 0.3 | - | Sensor Poisson coefficient |

c_{ps} | −9.6 | Nm^{−1}V^{−1} | Sensor piezoelectric coefficient |

x_{r} | 1.28 | m | rigid body distance from clamp along z axis |

D_{out} | 0.11 | m | Vessel exterior diameter |

D_{in} | 0.105 | m | Vessel interior diameter |

L_{out} | 0.5 | m | Vessel outside length |

L_{in} | 0.47 | m | Vessel inside length |

ρ | 1180 | kg·m^{−3} | Vessel density |

E | 4.5 | GPa | Vessel Young modulus |

D_{out}^{ring} | 0.1445 | m | Fixation rings exterior diameter |

D_{in}^{ring} | 0.11 | m | Fixation rings interior diameter |

t^{ring} | 28.76 | mm | Fixation rings thickness |

d^{ring} | 16.9 | mm | Fixation rings distance from plate mean axis |

ρ^{ring} | 2970 | kg·m^{−3} | Fixation rings density |

E^{ring} | 75 | GPa | Fixation rings Young modulus |

m^{ring} | 588.6 | g | Total fixation ring masses |

m_{tip} | 313 | g | Tip masses |

L_{tip} | 0.24 | m | Distance of tip masses from plate mean axis |

Sub-system. | Type | Formula |
---|---|---|

Beam bending | Potential | $\frac{1}{2}{\displaystyle {\int}_{z=0}^{z=L}\left({E}_{s}{I}_{s}+{E}_{p}{I}_{p}{\Pi}_{ab}(z)\right){\left(\frac{{\partial}^{2}w}{\partial {z}^{2}}\right)}^{2}dz}$ |

Kinetic | $\frac{1}{2}{\displaystyle {\int}_{z=0}^{z=L}\left({\rho}_{s}{t}_{s}{w}_{s}+{\rho}_{p}{t}_{p}{w}_{p}{\Pi}_{ab}(z)\right){\left(\frac{\partial w}{\partial t}\right)}^{2}dz}$ | |

Beam Torsion | Potential | $\frac{1}{2}{\displaystyle {\int}_{z=0}^{z=L}{G}_{s}{J}_{s}{\left(\frac{\partial \varphi}{\partial z}\right)}^{2}dz}$ |

Kinetic | $\frac{1}{2}{\displaystyle {\int}_{z=0}^{z=L}{I}_{p}{\left(\frac{\partial \varphi}{\partial t}\right)}^{2}dz}$ | |

Rotating–translating shallow water | Potential | ${\int}_{x=-a/2}^{x=a/2}{\rho}_{f}bg\left(\frac{{h}^{2}}{2}\mathrm{cos}\theta +hz\mathrm{sin}\theta \right)dx$ |

Kinetic | $\frac{1}{2}{m}_{T}{\dot{D}}^{2}+\frac{1}{2}{\displaystyle {\int}_{x=-a/2}^{x=a/2}{\rho}_{f}bh\left[{\left(u+\dot{D}\mathrm{cos}\theta \right)}^{2}+{\left(z\dot{\theta}-\dot{D}\mathrm{sin}\theta \right)}^{2}\right]dx}$ |

Sub- | Formula |
---|---|

Beam bending | $\begin{array}{l}\left(\rho S+{\rho}_{p}{S}_{p}{\Pi}_{ab}\right)\ddot{w}=\\ -\frac{{\partial}^{2}}{\partial {z}^{2}}\left(\left(EI+{\Pi}_{ab}{E}_{p}{I}_{p}\right)\frac{{\partial}^{2}}{\partial {z}^{2}}w\right)\\ +\frac{{\partial}^{2}}{\partial {z}^{2}}\left({\Pi}_{ab}\frac{\gamma {I}_{p,1}}{{t}_{p}}V\right)\end{array}$ |

Beam Torsion | ${I}_{p}\frac{{\partial}^{2}\theta}{\partial {t}^{2}}=\frac{\partial}{\partial z}\left(GJ\frac{\partial \theta}{\partial z}\right)$ |

Rotating–translating shallow water | $\begin{array}{l}\frac{\partial \eta}{\partial t}+{h}_{0}\sigma \frac{\partial}{\partial x}\left[\varphi u(y=\eta )\right]=0\\ \frac{\partial u(\eta )}{\partial t}+(1-{T}_{H}^{2})u(\eta )\frac{\partial u(\eta )}{\partial x}+\\ \left(1+h\sigma \phi \frac{{\partial}^{2}\eta}{\partial {x}^{2}}\right){a}_{y}\frac{\partial \eta}{\partial x}=\lambda u(\eta )+{a}_{x},\end{array}$ |

Sub- | Formula |
---|---|

Beam bending | $\begin{array}{l}w(0,t)=\frac{\partial w}{\partial z}(0,t)=0\\ EI\frac{{\partial}^{2}w}{\partial {z}^{2}}(L,t)=M\\ EI\frac{{\partial}^{3}w}{\partial {z}^{3}}(L,t)={F}_{y}\end{array}$ |

Beam Torsion | $\begin{array}{l}\theta (0,t)=0\\ GJ\frac{\partial \theta}{\partial z}=T\end{array}$ |

Rotating–translating shallow water | $u(0,t)=u(L,t)={\dot{X}}_{0}=\dot{w}$ |

Filling Ratio | Previous [23] | Current | Experiment |
---|---|---|---|

0.25 | 0.43 | 0.44 | 0.47 |

0.5 | 0.59 | 0.57 | 0.55 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdollahzadeh Jamalabadi, M.Y.
An Improvement of Port-Hamiltonian Model of Fluid Sloshing Coupled by Structure Motion. *Water* **2018**, *10*, 1721.
https://doi.org/10.3390/w10121721

**AMA Style**

Abdollahzadeh Jamalabadi MY.
An Improvement of Port-Hamiltonian Model of Fluid Sloshing Coupled by Structure Motion. *Water*. 2018; 10(12):1721.
https://doi.org/10.3390/w10121721

**Chicago/Turabian Style**

Abdollahzadeh Jamalabadi, Mohammad Yaghoub.
2018. "An Improvement of Port-Hamiltonian Model of Fluid Sloshing Coupled by Structure Motion" *Water* 10, no. 12: 1721.
https://doi.org/10.3390/w10121721