New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures with Liquid

Abstract

Nonlinear models of the combined motion of structures carrying liquid with a free surface for translational and rotational motion of the carrying structure were constructed based on the variational approach. Using some specific features this method was applied also for modeling structures with limited volumes of liquid for reservoirs of revolution. The obtained model is free from some restricting assumptions used by other authors and more convenient for numerical implementation. The performed study of the development of resonant processes enables considerable revision of the problem of steady resonant motion of structures with liquid and research of transient modes for translational and rotational motion of the carrying structure. Due to new potential in modeling different processes, we generalized the statement of the problem of parametric oscillations of reservoirs with liquid and found some new features of the system behavior. Some potentials for reduction of the effect of liquid mobility on the motion of the carrying structure by the algorithm of compensation of the liquid response on reservoir walls were studied too.

1. Introduction

The problems of dynamics of structures with a free-surface liquid have significant practical applications and the long period of development of methods of their mathematical modeling. Theoretical research began from linear modeling and about 70 years ago studying this class of problems based on nonlinear approaches started [1, 2, 3, 4, 5]. Until now nonlinear research is mostly focused on resonant processes for the prescribed translational motion of the carrying structure. At the same time, these theoretical approaches used some assumptions, mostly traditional, for the linear theory of oscillations of mechanical systems with one degree of freedom. Therefore, due to the demands of practice and some modern achievements of the theory of oscillations, it is possible to state the following list of insufficient features of modern analytical research of nonlinear problems of structures with a free-surface liquid.

1. Neglecting the combined character of motion of the structure and liquid, which is predetermined by the fact that for the most popular problems, the mass of liquid exceeds the mass of the carrying structure.

2. Neglecting problems with the angular motion of the carrying body although such problems as sea pitching, aircraft control, seismic stability of structures with liquid, and others are mostly connected with the angular motion of the structures.

3. Insufficient level of research for problems with reservoirs of non-cylindrical shape.

4. The use of some hypotheses of the linear theory of oscillations for the mechanical system with one degree of freedom, such as the assumption that oscillations are possible only on resonant frequencies and multiple ones and cannot develop with normal frequencies of non-resonant normal modes, and that steady mode of motion certainly exists [6, 7].

5. Neglecting the potential of a transcendent ratio between normal frequencies considerably changes the character of the oscillation process [6, 7].

6. Considering only the classical Faraday statement of the problem of parametric oscillations of structures with liquid without studying some of these generalizations caused by practice [8].

7. The absence of approaches for studying problems of control for structures with liquid in the nonlinear range of perturbations of liquid amplitudes.

Therefore, we will focus our attention mostly on the ways how it is possible to overcome the mentioned weaknesses of the analytical methods of studying modern problems of structures with liquid.

It is necessary to mention also that despite the great potential of point-wise numerical methods until now they are inferior to the analytical methods constructed based on variational approaches. No suitable solutions for the problems of combined motion of structures with liquid, liquid motion in reservoirs of non-cylindrical shapes with a high level of accuracy of satisfying boundary conditions and others were obtained.

2. Variational approach for the construction of the nonlinear model of the combined motion of structures carrying liquid with a free surface for translational and rotational motion of the carrying structure

Consider a structure with a reservoir, partially filled with liquid with a free surface. The structure is supposed to be a perfectly rigid body, which performs both translational and rotational motion. The liquid is supposed to be homogeneous, incompressible, and ideal and its motion is vortex-free. Therefore, its motion can be represented by the velocity potential. Here we consider that the reservoir has a cylindrical shape with a circular cross-section. The mathematical model of the problem is constructed based on the Hamilton variational principle. As for modeling the system’s nonlinear properties, we restrict ourselves by constructing the model of the third order of smallness of liquid sloshing amplitudes.

We make use the following denotations for mathematical description of the system. $\tau$ is the domain occupied by a liquid (${\tau }_0$ is this domain in the case when liquid surface is not perturbed); S and $S_0$ is a free surface of liquid in disturbed and undisturbed state correspondingly (the surface $S_0$ is bounded by the contour $L_0$); $\mathrm{\Sigma }$ is the moisten surface of liquid (lateral walls and bottom); $\overrightarrow{v}$ and ${\overrightarrow{v}}_r$ are absolute velocities (in a conventionally immovable reference frame) and relative (in the reference frame fixed with the reservoir, where the origin $O$ is in the center of a free surface $S_0$, the axis $\mathit{Oz}$ is directed upward); $\overrightarrow{\epsilon }$ is the vector of displacement of the point $O$ in the immovable reference frame; $\overrightarrow{\omega }$ is the angular velocity of the reservoir motion; $\overrightarrow{r}$ is the radius vector of a point of the domain $\tau$ with respect to the origin $O$; $\rho$ is the liquid density; $M_F$ and $M_T$ are masses of the liquid and the reservoir correspondingly; $J_{\mathit{res}}^{\mathit{ps}}$ is the moment of inertia of the reservoir determined with respect to the reference frame fixed with the reservoir; $h_l$ and $h_r$ are displacements of the mass centers of liquid and reservoir relative to the surface $S_0$; R and H are the reservoir radius and the liquid depth; $z=\xi \left(r,\theta ,t\right)$ is the equation of a liquid free surface with the normal vector $\overrightarrow{n}$ (such representation is possible only for cylindrical domains of liquid), where we call $\xi$ as a free surface excitation. For a description of the rotational motion of the carrying structure we use the so-called airplane angles, $\overrightarrow{\chi }=\left\{{\alpha }_1.{\alpha }_2,{\alpha }_3\right\}$, where ${\alpha }_3$ correspond to rotation about the cylindrical axis; $\sigma$ and ${\theta }_1$ are correspondingly surface tension factor and the capillary angle on the contour $L_0$; ${\overrightarrow{F}}_e$ and ${\overrightarrow{M}}_e$ are the main external force and moment applied to the structure.

We are going to use the Hamilton variational principle for the construction of the system mathematical model. The most fruitful way of this approach foresees the elimination of all kinematic constraints superimposed on the system. If we represent the absolute velocity of liquid as $\overrightarrow{v}=\overrightarrow{\nabla }\left(\phi +\dot{\overrightarrow{\epsilon }}\cdot \overrightarrow{r}+\overrightarrow{\omega }\cdot \overrightarrow{\mathrm{\Omega }}\right),$ were $\phi$ is the velocity potential responsible for liquid motion relative to the reservoir and $\overrightarrow{\mathrm{\Omega }}$ is the Stokes – Zhukovsky potential [1, 5, 9], we can write down the following kinematic constraints

1. $\mathrm{\Delta }\phi =0$ and $\mathrm{\Delta }\overrightarrow{\mathrm{\Omega }}=0$ in the domain $\tau ,$ occupied by liquid, which is the consequence of the continuity equation;

   $$\frac{\partial \phi }{\partial n}=0\text{and}\frac{\partial \overrightarrow{\mathrm{\Omega }}}{\partial n}=\overrightarrow{r}\times \overrightarrow{n}$$

2. on the boundary of liquid contact with rigid walls, which is the nonflowing condition through rigid boundary;

3. $\underset{S_0}{\int }\mathit{\xi ds}=0$, which represents the solvability condition for the Newman problem of the Laplace equation and is equivalent to the mass conservation law;

   $$\frac{\partial \xi }{\partial t}+\overrightarrow{\nabla }\xi \cdot \left[\overrightarrow{\nabla }\phi +\overrightarrow{\nabla }\left(\overrightarrow{\omega }\cdot \overrightarrow{\mathrm{\Omega }}\right)-\dot{\overrightarrow{\epsilon }}-\overrightarrow{\omega }\times \overrightarrow{r}\right]=\frac{\partial \phi }{\partial z}+\overrightarrow{\omega }\cdot \frac{\partial \overrightarrow{\mathrm{\Omega }}}{\partial z}-{\dot{\epsilon }}_z-{\left.\left(\overrightarrow{\omega }\times \overrightarrow{r}\right)\right|}_z$$

4. on the perturbed liquid surface S, which is the requirement of coincidence of normal components of velocities of liquid particles and a free surface of the liquid.

The motion of the system components is described in different ways. Liquid motion is described by the Euler variables in the form of equations in partial derivatives and the motion of the carrying structure is described by the Lagrange variables in the form of ordinary differential equations, which creates difficulties in their mathematical studying. For the transition from this mathematical object of an inhomogeneous structure to a homogeneous one we make use of the method of modal decomposition (or the so-called Kantorovich method). To this end, we represent perturbations of the free surface of the liquid in the form of series with respect to normal modes of oscillations of liquid in a cylindrical reservoir

$$\xi =\sum _n{a}_n\left(t\right){\psi }_n\left(r,\theta \right),$$E1

where ${\psi }_n\left(r,\theta \right)$ and ${\kappa }_n$ are normal modes and eigenvalues, which are determined from the boundary value problem.

$$\mathrm{\Delta }{\psi }_i+{\kappa }_i^2{\psi }_i=0\text{on}{S}_0;\frac{\partial {\psi }_i}{\partial n}=0\text{on}{L}_0,\underset{S}{\int }{\psi }_i\mathit{ds}=0.$$E2

According to the representation of a free surface scalar and vector potentials of the liquid velocity take the following form

$$\phi =\sum _n{b}_n\left(t\right){\psi }_n\left(r,\theta \right)\frac{cosh{\kappa }_n\left(z+H\right)}{{\kappa }_{n}sinh{\kappa }_{n}H};\overrightarrow{\mathrm{\Omega }}={\overrightarrow{\mathrm{\Omega }}}_0+\sum _n{\overrightarrow{q}}_n\left(t\right){\psi }_n\left(r,\theta \right)\frac{cosh{\kappa }_n\left(z+H\right)}{{\kappa }_{n}sinh{\kappa }_{n}H}.$$E3

Here ${\overrightarrow{\mathrm{\Omega }}}_0$ is a linear part of the Stokes – Zhukovsky potential, which is determined from the boundary value problem.

$$\mathrm{\Delta }{\overrightarrow{\mathrm{\Omega }}}_0=0\text{in}{\tau }_0;\frac{\partial {\overrightarrow{\mathrm{\Omega }}}_0}{\partial n}=\overrightarrow{r}\times \overrightarrow{n}\text{on}{S}_0+\mathrm{\Sigma }.$$E4

According to the theorem that the irritational motion of an ideal homogeneous incompressible liquid is completely defined by the motion of its boundaries the amplitude parameters $b_n\left(t\right)$ and ${\overrightarrow{q}}_n\left(t\right)$ should be considered as dependent and the parameters $a_i$ (specify the motion of a free surface of the liquid) and ${\epsilon }_i$, ${\alpha }_i$ (specify the motion of the reservoir walls) should be considered as the independent ones. Moreover, the number of these parameters is equal to the number of degrees of freedom in the considered system, therefore, the model, constructed in these parameters, will be the model with minimum dimension.

Representation of variables $\xi$, $\phi$ and $\overrightarrow{\mathrm{\Omega }}$ (Eqs. (1)–(4)) hold the kinematic constraints a), b), c) exactly, however, the kinematic constraint d) is satisfied only within the framework of the linear approximation. To satisfy this condition within the framework of the nonlinear statement it is necessary to find the dependence of $b_i$ and ${\overrightarrow{q}}_i$ on $a_k$ accurate to terns of the third order of smallness of the values $a_k$. Taking into account that the boundary condition on a free surface of the liquid must hold for an arbitrary motion of the carrying structure, it disintegrates into four independent conditions on a free surface $S$, that is, for $z=\xi$

$$L^{\left(0\right)}\left(\xi ,\phi \right)=\frac{\partial \xi }{\partial t}+\overrightarrow{\nabla }\xi \cdot \nabla \phi -\frac{\partial \phi }{\partial z}=0;$$E5

$$L^{\left(k\right)}\left(\xi ,{\overrightarrow{\mathrm{\Omega }}}_0\right)=\overrightarrow{\nabla }\xi \cdot \overrightarrow{\nabla }{\mathrm{\Omega }}_k-\frac{\partial {\mathrm{\Omega }}_k}{\partial z}+{\overrightarrow{i}}_k\cdot \left({\overrightarrow{i}}_3\times \overrightarrow{r}\right)\left(k=1,2,3\right).$$E6

Then we use the Galerkin method and multiply these differential operators by the system of complete orthogonal functions ${\psi }_i$ and integrate this over the surface $S_0$

$$\underset{S_0}{\int }{\left.L^{\left(i\right)}\left(\xi ,\begin{array}{c}\phi \\ {\mathrm{\Omega }}_k\end{array}\right)\right|}_S{\psi }_k\mathit{dS}=0\begin{array}{c}\left(i=0,1,2,3\right)\\ \left(k=1,2,\dots \right).\end{array}$$E7

The immediate computation of values of the differential operators $L^{\left(i\right)}\left(f,g\right)$ on the in advance unknown free surface $S$ (for $z=\xi$) is done by projecting the mentioned operator on the unperturbed free surface $S_0$ with the use of the decompositions Eqs. (1) and (2). Here this is done by decomposing the hyperbolic functions with respect to the variable $\xi$ in a vicinity of the value $\xi =0$ . This technique enables the determination of the dependence of $a_i,b_i$ and ${\overrightarrow{q}}_i$ accurate to the required order of smallness Eqs. (5)–(7)

$$b_i={\dot{a}}_i+\sum _{n,m}{\dot{a}}_n{a}_m{\gamma }_{\mathit{nmi}}^w+\sum _{n,m,l}{\dot{a}}_n{a}_m{a}_l{\delta }_{\mathit{\text{nmli}}}^w+\sum _{n,m,l,k}{\dot{a}}_n{a}_m{a}_l{a}_k{h}_{\mathit{\text{nmlki}}}^w;$$E8

$${\overrightarrow{q}}_p=\sum _j{a}_j{\overrightarrow{\beta }}_{\mathit{jp}}^u+\sum _{j,k}{a}_j{a}_k{\overrightarrow{\gamma }}_{\mathit{jkp}}^u+\sum _{j,k,l}{a}_j{a}_k{a}_l{\overrightarrow{\delta }}_{\mathit{\text{jklp}}}^u.$$E9

Muultiindex coefficients in these expressions are computed as quadratures from normal modes ${\psi }_i$. The elimination of the nonlinear boundary condition of a free surface of the liquid in an explicit form in quadratures for an arbitrary number of the considered normal modes before solving the variational problem represents the distinctive advantage of the suggested method in comparison with the existing ones [5, 10, 11, 12]. At the same time, this creates convenience for numerical implementation and reduces the number of unknowns.

The Lagrange function for the Hamilton variational principle in the before described parameters takes the form

$$L=\frac{1}{2}\rho \underset{\tau }{\int }{\left(\overrightarrow{\nabla }\phi +\dot{\overrightarrow{\epsilon }}+\overrightarrow{\omega }\cdot \overrightarrow{\mathrm{\Omega }}\right)}^2\mathit{d\tau }+\frac{1}{2}{M}_T{\left(\dot{\overrightarrow{\epsilon }}\right)}^2+\frac{1}{2}{J}_{\mathit{res}}^{\mathit{ij}}{\omega }_i{\omega }_j-\left(M_T+M_F\right)g{\epsilon }_z-$$

$$-\left(M_F+M_T\right)g{\epsilon }_z+\mathit{\rho g}\left(cos{\alpha }_{1}sin{\alpha }_{2}cos{\alpha }_3-sin{\alpha }_{1}sin{\alpha }_2\right)\underset{S_0}{\int }rcos\theta \left(\xi +H\right)\mathit{ds}-$$

$$-\mathit{\rho g}\left(sin{\alpha }_{1}cos{\alpha }_3+cos{\alpha }_{1}sin{\alpha }_{2}sin{\alpha }_3\right)\underset{S_0}{\int }rsin\theta \left(\xi +H\right)\mathit{ds}-\frac{1}{2}\mathit{\rho g}cos{\alpha }_{1}cos{\alpha }_2\underset{S_0}{\int }{\xi }^2\mathit{ds}-$$

$$-\left(M_F{h}_l+M_T{h}_r\right)\left(1-cos{\alpha }_{1}cos{\alpha }_2\right)-\sigma \underset{S_0}{\int }\sqrt{1+{\left(\overrightarrow{\nabla }\xi \right)}^2}\mathit{ds}-\sigma cos{\theta }_1\underset{L_9}{\int }\mathit{\xi dl}+{\overrightarrow{F}}_e\cdot \overrightarrow{\epsilon }+{\overrightarrow{M}}_e\cdot \overrightarrow{\chi }.$$E10

If we substitute the decompositions Eqs. (1) and (3) into the Lagrange function Eq. (10) and make use of the relations Eqs. (8) and (9) we obtain the Lagrange function in parameters $a_i$, ${\epsilon }_i$ and ${\alpha }_i$. It is necessary to note that the determination of the integrals over unknown and variable in the time domain $\tau$ is done according to the two following formulae.

$$\underset{\tau }{\int }\mathit{Ad\tau }=\underset{S_0}{\int }\underset{-H}{\overset{\xi }{\int }}\mathit{\text{Adzds}};\underset{-H}{\overset{\xi }{\int }}\mathit{Bdz}=\underset{-H}{\overset{0}{\int }}\mathit{Bdz}+\mathit{\xi B}\left(0\right)+\frac{{\xi }^2}{2!}{B}^{\prime }\left(0\right)+\dots$$

Here A and B are arbitrary functions, the first relation is obtained due to the cylindrical shape of the domain $\tau$, and the second formula is obtained as the Taylor series for the function

$F\left(\xi \right)=\underset{-H}{\overset{\xi }{\int }}\mathit{Adz}$. The Lagrange equations of the 2nd kind for the obtained Lagrange function can be written as

$$\begin{array}{l}\sum _i{\ddot{a}}_i\left\{{\delta }_{\mathit{ir}}+\sum _j{a}_j{A}_{\mathit{rij}}^3+\sum _{j,k}{a}_j{a}_k{A}_{\mathit{\text{rijk}}}^4\right\}+\ddot{\overrightarrow{\epsilon }}\cdot \frac{1}{{\alpha }_r^v}\left\{{\overrightarrow{B}}_r^1+\sum _i{a}_i{\overrightarrow{B}}_{\mathit{ri}}^2+\sum _{i,j}{a}_i{a}_j{\overrightarrow{B}}_{\mathit{rij}}^3+\sum _{i,j,k}{a}_{+i}{a}_j{a}_k{\overrightarrow{B}}_{\mathit{\text{rijk}}}^4\right\}+\\ +\frac{1}{2{\alpha }_r^v}\sum _{s=1}^3{\ddot{\alpha }}_s\left\{\sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_s}\left[E_{\mathit{pr}}^{1\ast }+\sum _i{a}_i{E}_{\mathit{pri}}^{2\ast }+\sum _{i,j}{a}_i{a}_j{E}_{\mathit{\text{prij}}}^{3\ast }\right]\right\}=\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{C}_{\mathit{ijr}}^3+\sum _{i,j,k}{\dot{a}}_i{\dot{a}}_j{a}_k{C}_{\mathit{\text{ijkr}}}^4+\\ +\frac{1}{2{\alpha }_r^v}\sum _{p=1}^3{\omega }_p\left[\sum _i{\dot{a}}_i\left(E_{\mathit{pir}}^{2\ast }-E_{\mathit{pri}}^{2\ast }\right)+\sum _{i,j}{\dot{a}}_i{a}_j\left(E_{\mathit{\text{pijr}}}^{3\ast }+E_{\mathit{\text{pirj}}}^{3\ast }-E_{\mathit{\text{prij}}}^{3\ast }+E_{\mathit{\text{prji}}}^{3\ast }\right)\right]+\\ +\frac{1}{2{\alpha }_r^v}\sum _{p,s=1}^3{\omega }_p{\omega }_s\left[E_{\mathit{psr}}^2+\sum _i{a}_i\left(E_{\mathit{\text{psir}}}^3+E_{\mathit{\text{psri}}}^3\right)\right]\\ +\frac{1}{2{\alpha }_r^v}\sum _{p=1}^3{\omega }_p^{\left(k\right)}\left[E_{\mathit{pr}}^{1\ast }+\sum _i{a}_i{E}_{\mathit{pri}}^{2\ast }+\sum _{i,j}{a}_i{a}_j{E}_{\mathit{\text{prij}}}^{3\ast }\right]+\\ +\dot{\overrightarrow{\epsilon }}\cdot \left[\sum _i{a}_i{\overrightarrow{D}}_{\mathit{ir}}^2+\sum _i{\dot{a}}_i{a}_j{\overrightarrow{D}}_{\mathit{ijr}}^3+\sum _i{\dot{a}}_i{a}_j{a}_k{\overrightarrow{D}}_{\mathit{\text{ijkr}}}^4\right]+\\ +\frac{1}{2{\alpha }_r^v}\dot{\overrightarrow{\epsilon }}\cdot \sum _{p=1}^3{\omega }_p\left[\begin{array}{c}\\ \end{array}\right.{\overrightarrow{F}}_{\mathit{pr}}^2+\sum _i{a}_i\left({\overrightarrow{F}}_{\mathit{pir}}^3+{\overrightarrow{F}}_{\mathit{pri}}^3\right)+\left.\sum _{i,j}{a}_i{a}_j\left({\overrightarrow{F}}_{\mathit{\text{pijr}}}^4+{\overrightarrow{F}}_{\mathit{\text{pirj}}}^4+{\overrightarrow{F}}_{\mathit{\text{prij}}}^4\right)\right]+\\ +g\frac{N_r}{{\alpha }_r^v}cos{\alpha }_{1}cos{\alpha }_2{a}_r+g\frac{{\alpha }_r^c}{{\alpha }_r^v}\left(cos{\alpha }_{1}sin{\alpha }_{2}cos{\alpha }_3-sin{\alpha }_{1}sin{\alpha }_3\right)+\\ +{\mathit{ga}}_r\frac{N_r}{{\alpha }_r^v}cos{\alpha }_{1}cos{\alpha }_2{a}_r+g\frac{{\alpha }_r^c}{{\alpha }_r^v}\left(cos{\alpha }_{1}sin{\alpha }_{2}cos{\alpha }_3-sin{\alpha }_{1}sin{\alpha }_3\right)+\\ +g\frac{{\alpha }_r^s}{{\alpha }_r^v}\left(sin{\alpha }_{1}cos{\alpha }_3+cos{\alpha }_{1}sin{\alpha }_{2}sin{\alpha }_3\right);\end{array}$$E11

$$\begin{array}{l}\frac{\rho }{M_F+M_T}\sum _i{\ddot{a}}_i\left({\overrightarrow{B}}_i^1+\sum _j{a}_j{\overrightarrow{B}}_{\mathit{ij}}^2+\sum _{j,k}{a}_j{a}_k{\overrightarrow{B}}_{\mathit{ijk}}^3\right)+\ddot{\overrightarrow{\epsilon }}\\ +\frac{\rho }{M_F+M_T}\sum _{s=1}^3{\ddot{\alpha }}_s\left[\sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_s}\left({\overrightarrow{F}}_p^1+\sum _i{a}_i{\overrightarrow{F}}_{\mathit{pi}}^2+\sum _{i,j}{a}_j{a}_k{\overrightarrow{F}}_{\mathit{pij}}^3\right)\right]=\\ =\frac{{\overrightarrow{F}}_e}{M_F+M_T}+\overrightarrow{g}+\frac{\rho }{M_F+M_T}\left(\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{\overrightarrow{B}}_{\mathit{ij}}^2+\sum _{i,j,k}{\dot{a}}_i{\dot{a}}_j{a}_k{\overrightarrow{B}}_{\mathit{ijk}}^3\right)-\end{array}$$E12

$$\begin{array}{l}-\frac{\rho }{M_F+M_T}\sum _{p=1}^3{\omega }_p\left(\sum _i{\dot{a}}_i{\overrightarrow{F}}_{\mathit{pi}}^2+2\sum _{i,j}{\dot{a}}_i{a}_j{\overrightarrow{F}}_{\mathit{pij}}^3+\sum _{i,j,k}{\dot{a}}_i{a}_j{a}_k{\overrightarrow{F}}_{\mathit{\text{pijk}}}^4\right)\\ -\frac{\rho }{M_F+M_T}\sum _{p=1}^3{\omega }_p^{\left(k\right)}\left({\overrightarrow{F}}_p^1+\sum _i{a}_i{\overrightarrow{F}}_{\mathit{pi}}^2\right);\\ \sum _i{\ddot{a}}_i\left\{\sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\left(E_{\mathit{pi}}^{1\ast }+\sum _j{a}_j{E}_{\mathit{pij}}^{2\ast }+\sum _{j,k}{a}_j{a}_k{E}_{\mathit{\text{pijk}}}^{3\ast }\right)\right\}+2\ddot{\overrightarrow{\epsilon }}\cdot \sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\left[{\stackrel{\rightharpoonup }{F}}_p^1+\sum _i{a}_i{\overrightarrow{F}}_{\mathit{pi}}^2+\sum _{i,j}{a}_i{a}_j{\overrightarrow{F}}_{\mathit{pij}}^3\right.+\\ +\left.\sum _i{a}_i{a}_j{a}_k{\overrightarrow{F}}_{\mathit{\text{pijk}}}^4\right]+\sum _{n=1}^3{\ddot{\alpha }}_n\left[2\sum _{p,s=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\frac{\partial {\omega }_s}{\partial {\dot{\alpha }}_n}\left(\frac{1}{\rho }{J}_{\mathit{res}}^{\mathit{ps}}+A_{\mathit{ps}}^2+\sum _i{a}_i{E}_{\mathit{psi}}^2+\sum _{i,j}{a}_i{a}_j{E}_{\mathit{\text{psij}}}^3\right)\right]=\\ =2\sum _{p,s=1}^3\left({\omega }_{p,r}^{\ast }{\omega }_s+{\omega }_p^{\left(k\right)}\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\right)\left(\frac{1}{\rho }{J}_{\mathit{res}}^{\mathit{ps}}+A_{\mathit{ps}}^2+\sum _i{a}_i{E}_{\mathit{psi}}^2+\sum _{i,j}{a}_i{a}_j{E}_{\mathit{\text{psij}}}^3\right)+\\ +\sum _{p=1}^3{\omega }_{p,r}^{\ast }\left(\sum _i{\dot{a}}_i{E}_{\mathit{pi}}^{1\ast }+\sum _{i,j}{\dot{a}}_i{a}_j{E}_{\mathit{pij}}^{2\ast }+\sum _{i,j}{\dot{a}}_i{a}_j{a}_k{E}_{\mathit{\text{pijk}}}^{3\ast }\right)\\ +\dot{\overrightarrow{\epsilon }}\cdot \sum _{p=1}^3{\omega }_{p,r}^{\ast }\left({\overrightarrow{F}}_p^1+\sum _i{a}_i{\overrightarrow{F}}_{\mathit{pi}}^2+\sum _{i,j}{a}_i{a}_j{\overrightarrow{F}}_{\mathit{pij}}^3\right)+\\ +2\sum _{p,s=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}{\omega }_s\left(\sum _i{\dot{a}}_i{E}_{\mathit{psi}}^2+2\sum _{i,j}{\dot{a}}_i{a}_j{E}_{\mathit{\text{psij}}}^3\right)+\sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\left(\sum _i{\dot{a}}_i{\dot{a}}_j{E}_{\mathit{pij}}^{2\ast }+2\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{a}_k{E}_{\mathit{\text{pijk}}}^{3\ast }\right)+\\ +2\dot{\overrightarrow{\epsilon }}\cdot \sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\left(\sum _i{\dot{a}}_i{\overrightarrow{F}}_{\mathit{pi}}^2+2\sum _{i,j}{\dot{a}}_i{a}_j{\overrightarrow{F}}_{\mathit{pij}}^3\right)+\frac{2g}{\rho }\left(M_T{h}_r+M_F{h}_l\right)\frac{\partial }{\partial {\alpha }_r}\left(cos{\alpha }_{1}cos{\alpha }_2\right)+\\ +2g\frac{\partial }{\partial {\alpha }_r}\left[\left(cos{\alpha }_{1}sin{\alpha }_{2}cos{\alpha }_3-sin{\alpha }_{1}sin{\alpha }_3\right)\left(\sum _i{a}_i{\alpha }_i^c+{\mathit{Hl}}^c\right)+\right.\\ +\left.\left(cos{\alpha }_{1}sin{\alpha }_{2}sin{\alpha }_3+sin{\alpha }_{1}cos{\alpha }_3\right)\left(\sum _i{a}_i{\alpha }_i^s+{\mathit{Hl}}^s\right)\right]+\frac{2}{\rho }{M}_e^r\end{array}$$E13

This system of equations Eqs. (11)–(13) is derived for the arbitrary number of normal modes considered in the model. It is reduced to the form when second derivatives of unknowns enter the equation linearly. The generalized form of the system of the following

$$\sum _{n=1}^N{p}_{\mathit{rn}}{\ddot{a}}_n+\sum _{n=N+1}^{N+3}{p}_{\mathit{rn}}{\ddot{\epsilon }}_{n-N}+\sum _{n=N+4}^{N+6}{p}_{\mathit{rn}}{\ddot{\alpha }}_{n-N-3}=q_r,r=\overline{1,N+6},$$E14

where N is the number of the considered normal modes of liquid oscillations, 6 corresponds to the number of degrees of freedom of the carrying structure, ${\delta }_{\mathit{ir}}$ is the Kronecker symbol. Due to this property, the system of equations Eq. (14) can be transformed to the Cauchy form either analytically or numerically and further, it is possible to integrate these equations for the given initial conditions. Here multi index coefficients are computed as quadratures from normal modes ${\psi }_i$ and components of the Stokes – Zhukovsky potential ${\overrightarrow{\mathrm{\Omega }}}_0$.

Finally, we succeeded to construct the model of combined nonlinear dynamics of the structure and a free-surfaced liquid. Within the framework of the usual for this class of problems assumptions, we construct a resolving system of equations for translational and rotational motion of the varying structure for an arbitrary number of the considered normal modes. The algorithm was easily implemented as software.

3. Specific features of construction of the model of nonlinear dynamics of structures with limited volumes of liquid for reservoirs of revolution

In numerous practical cases, reservoirs have non-cylindrical shapes, often they are reservoirs of revolution. Some steps of the previously developed method should be revised in this case. First of all, it is necessary to note that the perturbation of a free surface of the liquid is a significant variable for the construction of the model because namely it is used for the asymptotic decomposition of nonlinear terms. So, it is necessary to conserve it. However, the before-used representation $z=\xi \left(r,\theta ,t\right)$ is valid only for reservoirs of cylindrical shape. Therefore, it is necessary to pass to a curvilinear parametrization of the liquid domain. This step was done before using the tensor calculus [5] but without success in practical applications. A further statement of the material is done for reservoirs of revolution with a simple substitute of variables responsible for the geometry of the liquid domain.

Consider a domain of revolution specified by the generatrix function $r=f\left(z\right)$ given in the cylindrical coordinates. Introduce the non-Cartesian parametrization

$$\alpha =\frac{r}{f\left(z\right)};\beta =\frac{z}{H}$$E15

Here $z=0$ coincides with the unperturbed free surface of the liquid $S_0$. Thus, the parameters $\alpha ,\theta ,\beta$ ($\theta$ is the angular coordinate) vary within $\alpha \in \left[0,1\right];\theta \in \left[0,2\pi \right]$ and for the unperturbed liquid state of the liquid domain $\mathit{\tau }\mathit{\beta }\in \left[-1,0\right]$. For new Eq. (15) and old parametrization, a perturbed free surface of the liquid will have the form correspondingly

$$\beta =\frac{1}{H}\xi \left(\alpha ,\theta ,t\right),\eta \left(r,\theta ,z,t\right)=\frac{z}{H}-\frac{1}{H}\xi \left(\frac{r}{f\left(z\right)},\theta ,t\right)=0$$E16

The new representation of a free surface Eq. (16) enables the successful use of the modal decomposition approach for further construction of the system nonlinear model. Here we restrict ourselves by considering the case only by translational motion of the carrying structure. The kinematic constraints of the system become more complicated.

1. $\mathrm{\Delta }\phi =0$ in the liquid domain$\tau$ (a consequence of the continuity equation);

2. $\frac{\partial \phi }{\partial n}=0$ on moistened walls $\mathrm{\Sigma }$ or in old variables

   $$\frac{1}{\sqrt{1+{f^{\prime }}^2}}\left(\frac{\partial \phi }{\partial r}-f^{\prime }\frac{\partial \phi }{\partial z}\right)=0$$

   (the consequence of the nonflowing condition);

   $$\frac{\partial \xi }{\partial t}+\frac{1}{f^2}\frac{\partial \xi }{\partial \alpha }\frac{\partial {\phi }_0}{\partial \alpha }+\frac{1}{{\alpha }^2{f}^2}\frac{\partial \xi }{\partial \theta }\frac{\partial {\phi }_0}{\partial \theta }-\frac{\alpha f^{\prime }}{f}\frac{\partial \xi }{\partial \alpha }\frac{\partial {\phi }_0}{\partial z}=\frac{\partial {\phi }_0}{\partial z}$$

3. on the perturbed free surface of the liquid $S$;

   $$\underset{\mathrm{\Sigma }}{\int }\frac{\partial \phi }{\partial n}\mathit{ds}+\underset{\mathrm{\Delta \Sigma }}{\int }\frac{\partial \phi }{\partial n}\mathit{ds}+\underset{S}{\int }\frac{\partial \phi }{\partial n}\mathit{ds}=0$$

4. is the revised form of the solvability condition, which will be analyzed in detail.

The first part of this condition is a usual requirement of liquid nonflowing through reservoir walls. However, if for the cylindrical reservoir, it is possible to find the system of functions, which exactly holds this requirement, in the case of the non-cylindrical shape of the reservoir this requirement means that this condition is a part of the solvability condition and its solution must be obtained with high accuracy. The second condition is fundamentally new. Here $\mathrm{\Delta \Sigma }$ is a part of the reservoir wall above the unperturbed free surface of the liquid, where the liquid can reach in perturbed motion. For a cylindrical reservoir, this condition is fulfilled identically, but in the case of a reservoir of revolution, this condition is not usually included in consideration despite its participation in the resolving condition (Figure 1).

Figure 1.

Illustration of denotations.

The reason for this is connected to the problem specificity. The problem statement includes elements only below the free surface of the liquid, and for the usual statement, it does not matter how liquid will rise above the undisturbed free surface. However, it is evident that it must trace the tank wall. Within the usual statement of the problem, this requirement is contradictive. Let us introduce the meridian cross-section of the reservoir with a liquid. The point $A_0$ is at the corner of an undisturbed free surface. The point $A$ is a point, where the liquid can reach in the excited motion. Since the suggested approach uses the reduction of the liquid excitations to the unperturbed domain, to provide fulfillment of the nonflowing condition the following relation must hold

$${\left.\frac{\partial \phi }{\partial n}\right|}_A={\left.\frac{\partial \phi }{\partial n}\right|}_{A_0}+{\left.\xi \frac{{\partial }^2\phi }{\partial n\partial \tau }\right|}_{A_0}+{\left.\frac{1}{2}{\xi }^2\frac{{\partial }^3\phi }{\partial n\partial {\tau }^2}\right|}_{A_0}+\cdots =0.$$E17

Due to the arbitrariness of $\xi$ values, it follows from Eq. (17) that

$${\left.\frac{{\partial }^k\phi }{\partial n\partial {\tau }^{k-1}}\right|}_{A_0}=0\text{for}k=1,2,\dots .$$E18

But the requirement Eq. (18) is contradictive because the problem statement has the main equation with the second order of differential operator, so no conditions of the second and higher order suit for problem statement. Moreover, the profound analysis of the problem of determination of normal modes of oscillations showed that the solution at the point $A_0$ contains singularity, therefore such kinds of derivatives do not exist at all. Unfortunately, the main part of both analytical and numerical studies of this problem neglect fulfillment of this requirement, which result in not reliable results. Usually, a part of the liquid, which penetrated through the surface $\mathrm{\Delta \Sigma }$ later is simply neglected and both the energy and the mass of the system unfoundedly disappear. However, there is a way how it is possible to provide fulfillment of this condition. Zhukovsky proved that if a liquid filled to the level of the point $A$ performs oscillations, then the part of the liquid filled (no “-”) to the level of the point $A_0$ performs the same kinematic motion as the liquid, which fills the reservoir to the level $A_0$. Based on this theorem, we suggested the method of an auxiliary domain [13] for the determination of coordinate functions for nonlinear problems of oscillations of a liquid with a free surface, which hold the nonflowing condition on tank walls not only below a free surface of an unperturbed liquid free surface, but on a certain prolongation of $\mathrm{\Sigma }$ above $S_0$. The idea of the method of auxiliary domain consists of the following. We solve the problem of determination of normal modes of oscillations for the domain, when liquid is filled to the point $A$, further the determined functions are used for the domain $\tau$ as coordinate functions. These functions hold the non-flowing condition on the surface $\mathrm{\Sigma }+\mathrm{\Delta \Sigma }$, and on the free surface of the liquid $S_0$ we take the functions, obtained on the horizontal cross-section on the level, which corresponds to the point $A_0$. This method has approximate character; however, it takes into account the analytical nature of the solution to the problem of determination of normal modes of liquid oscillations and the specificity of its singular properties. The success of the further application of this method is mainly caused by the fact that the contour with singular properties is shifted outside of the liquid. Practical application of this approach for different reservoirs of non-cylindrical shapes (cone, sphere, hyperboloid, ellipsoid, paraboloid [13]) results in reaching the accuracy of satisfying the non-flowing boundary condition on $\mathrm{\Sigma }$ of ${10}^{-5}$ order and ${10}^{-3}$ on $\mathrm{\Delta \Sigma }$, which exceeds the classical approach of approximately 100 times. As a result, this rises the accuracy of satisfying not only the non-flowing boundary condition but the fulfillment of the conservation law of mass, and energy and provides stability in implementation of numerical procedures.

For the representation of unknown variables, we make use of the following decompositions

$$\xi =\overline{\xi }\left(t\right)+\sum _i{a}_i{\overline{\psi }}_i\left(\alpha \right)T_i\left(\theta \right);{\phi }_0=\sum _i{b}_i{\psi }_i\left(\alpha ,\beta \right)T_i\left(\theta \right)$$E19

where ${\left.{\overline{\psi }}_i\left(\alpha \right)=\frac{\partial {\psi }_i}{\partial z}\right|}_{\beta =0}={\left.\left(\frac{1}{H}\frac{\partial {\psi }_i}{\partial \beta }-\frac{\alpha f^{\prime }}{f}\frac{\partial {\psi }_i}{\partial \alpha }\right)\right|}_{\beta =0}$.

These decompositions Eq. (19) hold all kinematic restrictions of the problem, but kinematic constraints c) and d) hold only on the linear level of accuracy. Namely because of this, we use two dependent unknowns $b_i\left(t\right)$ and $\overline{\xi }\left(t\right)$. Here $\overline{\xi }\left(t\right)$ is a function responsible for the correction of the liquid volume, caused by a non-cylindrical shape of the liquid domain in the excited mode of motion; ${\overline{\psi }}_i\left(\alpha \right)$ is a normal mode of a free surface of the liquid determined according to the method of an auxiliary domain, where due to the shape of revolution the angle variable is represented separately $T_i\left(\theta \right)$; ${\psi }_i\left(\alpha ,\beta \right)T_i\left(\theta \right)$ is the potential of the velocity of the liquid, determined according to the method of auxiliary domain with separate angle variable, which corresponds to the $i$-th normal mode; $b_i\left(t\right)$ is an amplitude parameter of excitation of the velocity potential associated with the $i$-th normal mode. According to the theorem that a vortex-free motion of the ideal homogeneous incompressible liquid is completely defined by the motion of its boundaries the variables $a_i\left(t\right)$ (responsible for the motion of a free surface) and $\dot{\overrightarrow{\epsilon }}$ (defines the motion of rigid walls) are the independent variables, while variables $\overline{\xi }\left(t\right)$ and $b_i\left(t\right)$ are depended ones.

The variable $\overline{\xi }\left(t\right)$ is determined from the requirement of the conservation of the liquid volume in its perturbed motion. Using the decomposition of the integral of liquid volume in series relative to $\overline{\xi }\left(t\right)$ in vicinity of $\xi =0$ we obtain accurate to values of the third order the following (here in the latter integral all functions $f$ and their derivatives are taken for $\beta =0$)

$$\begin{array}{l}\mathrm{\Delta }V=\underset{\tau }{\int }\mathit{d\tau }=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{1}{\int }}\left[\underset{0}{\overset{\xi /H}{\int }}{\left[f\left(\mathit{H\beta }\right)\right]}^2\mathit{d\beta }\right]\mathit{\text{αHdαdθ}}=\\ =\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{1}{\int }}\left[f^2\frac{\xi }{H}+f{f}^{\prime }\frac{{\xi }^2}{H^2}+\left({f^{\prime }}^2+f{f}^{″}\right)\frac{{\xi }^3}{3H^3}+\dots \right]\mathit{\text{αHdαdθ}}=0.\end{array}$$E20

If we represent $\overline{\xi }={\overline{\xi }}_1+{\overline{\xi }}_2+{\overline{\xi }}_3+{\overline{\xi }}_4$ as decomposition according to degrees of smallness (the lower index corresponds to the degree of the smallness of a value relative to the value of $\xi$), we obtain Eq. (20)

$${\overline{\xi }}_1=0;{\overline{\xi }}_2=-\frac{e_2}{e_1}\sum _{i,j}{a}_i{a}_j{{\beta }^{\nu }}_{\mathit{ij}};{\overline{\xi }}_3=-\frac{e_3}{e_1}\sum _{i,j,k}{a}_i{a}_j{a}_k{\gamma }_{\mathit{ijk}}^{\nu };{\overline{\xi }}_4=-\frac{e_4}{e_1}\sum _{i,j,k,l}{a}_i{a}_j{a}_k{a}_l{\delta }_{\mathit{\text{ijkl}}}^{\nu }$$E21

In Eq. (21) values with indexes are determined by values of $f$ and its derivatives for $\beta =0$ and by quadratures from coordinate functions ${\psi }_i$. Therefore, the requirement of the liquid volume conservation is provided accurately to the required order of smallness.

Then we substitute decompositions Eq. (19) in the kinematic boundary condition on a free surface of the liquid c), decompose all terms with respect to $\xi$ in vicinity of $\xi =0$. After representation of the coefficient $b_j$ as decomposition according to orders of smallness $b_j=b_j^{\left(1\right)}+b_j^{\left(2\right)}+b_j^{\left(3\right)}+b_j^{\left(4\right)}$ and equating terms with the same order of smallness we get

$$b_p^{\left(1\right)}={\dot{a}}_p;b_p^{\left(2\right)}=\sum _{i,j}{\dot{a}}_i{a}_j{\gamma }_{\mathit{ijp}}^o;b_p^{\left(3\right)}=\sum _{i,j,k}{\dot{a}}_i{a}_j{a}_k{\delta }_{\mathit{\text{ijkp}}}^o;b_p^{\left(4\right)}=\sum _{i,j,k,l}{\dot{a}}_i{a}_j{a}_k{a}_l{h}_{\mathit{\text{ijklp}}}^o$$E22

The relation Eq. (22) contains coefficients determined as quadratures from functions ${\psi }_k$, ${\overline{\psi }}_k$ and $T_k$ over the unperturbed free surface of the liquid. After the determination of the dependents Eqs. (21) and (22) parameters $a_k$ and ${\epsilon }_j$ become the independent system of variables, which completely characterize the motion of the liquid. The studied mechanical system in these parameters represents the mechanical system, for which all kinematical constraints are eliminated, and the number of variables is equal to the number of the system degrees of freedom. Moreover, in these parameters, we pass from the initial mathematical model of inhomogeneous structure (mixture of differential equations in partial derivatives and a system of ordinary differential equations) to the homogeneous mathematical model, i.e., the system of ordinary differential equations in amplitude parameters of motion.

The variational principle of the Hamilton can be constructed based on the following Lagrange function (translational motion of the carrying structure)

$$\begin{array}{l}L=\frac{1}{2}\rho \underset{\tau }{\int }{\left(\overrightarrow{\nabla }\phi +\dot{\overrightarrow{\epsilon }}\right)}^2\mathit{d\tau }+\frac{1}{2}{M}_T{\left(\dot{\overrightarrow{\epsilon }}\right)}^2-\frac{1}{2}\mathit{\rho g}\underset{S_0}{\int }{e}_1{\xi }^2\mathit{\text{αdαdθ}}-\frac{1}{2}\mathit{\rho g}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{1}{\int }}{\xi }^3{e}_2\mathit{\text{αdαdθ}}-\\ -\frac{1}{2}\mathit{\rho g}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{1}{\int }}{\xi }^4{e}_3\mathit{\text{αdαdθ}}-\left(M_T+M_F\right){\epsilon }_{z}g-\sigma \underset{S_0}{\int }\sqrt{1+{\left(\overrightarrow{\nabla }\xi \right)}^2}\mathit{dS}-\sigma \underset{L}{\int }\mathit{\xi dl}cos{\theta }_1+\overrightarrow{F}\overrightarrow{\epsilon },\end{array}$$E23

here for determination of the potential energy, it is necessary to use new terms with coefficients $e_1=f^2\left(0\right);e_2=\frac{4}{3}{f}^{\prime }\left(0\right)f\left(0\right);e_3=\frac{1}{2}\left[f^{″}\left(0\right)f\left(0\right)+{f^{\prime }}^2\left(0\right)\right]$, which reflect the non-cylindrical shape of the liquid domain. After the use of decompositions Eqs. (19)–(22) the Lagrange function Eq. (23) in independent variables $a_k$ and ${\epsilon }_j$ will take the form

$$\begin{array}{l}L=\frac{1}{2}\rho \left[\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{V}_{\mathit{ij}}^1+\sum _{i,j,k}{\dot{a}}_i{\dot{a}}_j{a}_k{V}_{\mathit{ijk}}^2+\sum _{i,j,k,l}{\dot{a}}_i{\dot{a}}_j{a}_k{a}_l{V}_{\mathit{\text{ijkl}}}^3\right]+\\ +\rho \dot{\overrightarrow{\epsilon }}\cdot \left[\sum _i{a}_i{\stackrel{\rightharpoonup }{U}}_i^1+\sum _{i,j}{\dot{a}}_i{a}_j{\stackrel{\rightharpoonup }{U}}_{\mathit{ij}}^2+\sum _{i,j,k}{\dot{a}}_i{a}_j{a}_k{\stackrel{\rightharpoonup }{U}}_{\mathit{ijk}}^3+\sum _{i,j,k,l}{\dot{a}}_i{a}_j{a}_k{a}_l{\stackrel{\rightharpoonup }{U}}_{\mathit{\text{ijkl}}}^4\right]-\\ -\frac{1}{2}\mathit{\rho g}\cdot \left[\sum _i{a}_i{W}_i^1+\sum _{i,j}{\dot{a}}_i{a}_j{W}_{\mathit{ij}}^2+\sum _{i,j,k}{\dot{a}}_i{a}_j{a}_k{W}_{\mathit{ijk}}^3+\sum _{i,j,k,l}{\dot{a}}_i{a}_j{a}_k{a}_l{W}_{\mathit{\text{ijkl}}}^4\right]+\\ +\frac{1}{2}\left(M_{ж}+M_p\right){\left(\dot{\overrightarrow{\epsilon }}\right)}^2-\left(M_{ж}+M_p\right)g{\epsilon }_z+\overrightarrow{F}\cdot \overrightarrow{\epsilon }.\end{array}$$E24

Based on Eq. (24) the Lagrange equations of the 2nd kind can be represented in the form

$$\begin{array}{l}\sum _i{\ddot{a}}_i\left\{V_{\mathit{ir}}^1+\sum _j{a}_j{V}_{\mathit{irj}}^2+\sum _{j,k}{a}_j{a}_k{V}_{\mathit{\text{irjk}}}^3\right\}+\ddot{\overrightarrow{\epsilon }}\cdot \left\{{\stackrel{\rightharpoonup }{U}}_r^1+\sum _i{a}_i{\stackrel{\rightharpoonup }{U}}_{\mathit{ri}}^2+\sum _{i,j}{a}_i{a}_j{\stackrel{\rightharpoonup }{U}}_{\mathit{rij}}^3+\sum _{i,j,k}{a}_i{a}_j{a}_k{\stackrel{\rightharpoonup }{U}}_{\mathit{\text{rijk}}}^4\right\}=\\ =\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{V}_{\mathit{ijr}}^{2\ast }+\sum _{i,j,k}{\dot{a}}_i{\dot{a}}_j{a}_k{V}_{\mathit{\text{ijkr}}}^{3\ast }+\dot{\overrightarrow{\epsilon }}\cdot \left\{\sum _i{\dot{a}}_i{\stackrel{\rightharpoonup }{U}}_{\mathit{ir}}^{2\ast }+\sum _{i,j}{\dot{a}}_i{a}_j{\stackrel{\rightharpoonup }{U}}_{\mathit{ijr}}^{3\ast }+\sum _{i,j,k}{\dot{a}}_i{a}_j{a}_k{\stackrel{\rightharpoonup }{U}}_{\mathit{\text{ijkr}}}^{4\ast }\right\}-\\ -g\left\{\sum _i{a}_i{W}_{\mathit{ir}}^2+\frac{3}{2}\sum _{i,j}{a}_i{a}_j{W}_{\mathit{ijr}}^3+2\sum _{i,j,k}{a}_i{a}_j{a}_k{W}_{\mathit{\text{ijkr}}}^4\right\},r=1,2,\dots N\end{array}$$E25

$$\begin{array}{l}\frac{\rho }{\left(M_r+M_f\right)}\left\{\sum _i{\ddot{a}}_i\left[{\stackrel{\rightharpoonup }{U}}_i^1+\sum _j{a}_j{\stackrel{\rightharpoonup }{U}}_{\mathit{ij}}^2+\sum _{j,k}{a}_j{a}_k{\stackrel{\rightharpoonup }{U}}_{\mathit{ijk}}^3\right]\right\}+\ddot{\overrightarrow{\epsilon }}=\\ =\frac{\overrightarrow{F}}{\left(M_r+M_f\right)}-g{\stackrel{\rightharpoonup }{z}}_0-\frac{\rho }{\left(M_r+M_f\right)}\sum _{i,j}{\dot{a}}_i{\dot{a}}_j\left\{{\stackrel{\rightharpoonup }{U}}_{\mathit{ir}}^2+2\sum _k{a}_k{\stackrel{\rightharpoonup }{U}}_{\mathit{ijk}}^3\right\}.\end{array}$$E26

The general form for these motion equations Eqs. (25) and (26) reflects its linearity with respect to the second derivatives of the unknown variables.

$$\sum _{n=1}^N{p}_{\mathit{rn}}\left(a_k,t\right){\ddot{a}}_n+\sum _{n=N+1}^{N+3}{p}_{\mathit{rn}}\left(a_k,t\right){\ddot{\epsilon }}_{n-N}=q_r\left(a_k,{\dot{a}}_l,{\epsilon }_k,{\dot{\epsilon }}_k,t\right),r=\overline{1,N+3}$$E27

Models for cylindrical and non-cylindrical reservoirs result in practically the same models according to the potential of their further research. The only structural difference is that for cylindrical reservoirs normal modes are orthogonal, while in the case of non-cylindrical reservoir Eq. (27), they are not orthogonal but close to orthogonality. The study of the different properties of these models by the examples enables us to draw the following recommendations for the practical implementation of this algorithm. Both results of experiments [2, 3, 4, 10, 14], some theoretical premises [7], and numerical experiments [15, 16, 17, 18] substantiate the following assumptions.

1. It is possible to restrict the number of the considered normal modes in these models to the level of about 10–12. The increase in this number is mostly connected with the increase of the wave profile steepness, which is important for intense transient processes.

2. It is possible to separate normal modes according to their impact on the formation of the main dynamical processes. Therefore, the first $N_3$ normal modes were studied accurate to the values of the third order of smallness; the first $N_2$ normal modes were studied accurate to the second order of smallness and all normal modes $N_1=N$. This assumption considerably reduces the number of addends in multi index sums and the number of quadratures.

3. Normal modes should be arranged in the acceding order of their frequencies with some exclusions. First of all, it is necessary to consider at least two axisymmetric normal modes, otherwise the property that the height of the wave crest exceeds the depth of the wave foot is violated. Second, in the case of, for example, a rectangular reservoir with a great difference of side lengths it is necessary to take normal modes for both main directions of the reservoir shape despite violation of the ascending order for frequencies.

4. For providing sufficient steepness of the wave profile and due to the combined character of motion of liquid with the reservoir it is necessary to take at least 3 normal modes with the circular number $n=1$.

The study of different examples showed that usually the model with the dimension $N_1=N=12$ and $N_2=6$, $N_3=3$ is the optimal one. Namely, these parameters of the model will be used in examples.

4. Considerable revision of the problem of steady resonant motion of structures with liquid. Transient modes of motion

We find deep contradictions if we try to study the phenomenon of resonant sloshing according to the existing literature. Different authors give different (and in some sense contradictory) representations of the main parameters of sloshing. Studying this problem enables clarification of these contradictions.

1. The main part of the research is based on the idea of the given motion of the carrying structure. Therefore, due to ignoring the combined character of the system motion this studying is done not for resonant frequencies but for below resonance mode.

2. In the mathematical model authors “permit” the system to perform oscillations only with resonant frequencies and multiple to them ones for providing the existence of the so-called steady mode of motion. However, due to the nonlinear properties of the system oscillations with supplementary frequencies can be excited. Moreover, some of these frequencies are transcendent relative to the resonant frequency. It is known that oscillations with transcendent frequencies will never be periodic. Therefore, the hypothesis about the existence of steady motion of the resonance is mistaken. Nobody gave convincing results of experiments that a steady mode of motion exists. At the same time, it is necessary to note that for systems with strong dissipation steady mode of motion manifests.

Here we try to clarify this. Consider the example of the movable spherical reservoir $H=R=1\mathrm{m}$, $M_T=0,2M_F$. The reservoir is moved in the horizontal plane in $\mathit{Ox}$ direction. Initially, the system is at rest. The motion of the system is disturbed by the harmonic force $F=A\cdot \left(M_T+M_F\right)sin{\omega }_{1}t$. Figure 2 shows the variation in time of the liquid elevation on the reservoir wall in the direction of the reservoir motion for the following ratios of the excitation frequencies ${\omega }_1$ and the frequency of normal oscillations ${\omega }_2$, determined within the model of the combined motion of the system 0,5; 0,9; 0,98; 1,0; 1,02; 1.1. The parameter A is determined separately for every variant of the frequency to provide maximum amplitude for free surface elevation close to 0.2 of the radius of a free surface.

Figure 2.

Elevation of liquid on reservoir wall in time.

For the below frequency mode of oscillation, which occurs approximately to the ratio of frequencies 0,5, we obtain the mode of motion, for which drift of mean value and considerable impact of high normal modes of oscillations take place. The effect of modulation of amplitudes is very weak. Since the partial frequency of the resonance is lower than the frequency of oscillation of the system in a combined motion, some authors [10] showed namely this mode of motion as the resonant one.

Close to the resonance the effect of the impact of high normal modes and drift of the amplitude mean practically disappeared and the modulation of oscillations becomes stronger. The closer the disturbance frequency to the resonant one, the longer is the period of modulation. This result is agreed with the theoretical and experimental results of [14]. Some intervals of the mode corresponding to ratios of frequencies 0,98 and 1,0 look like steady modes of motion, however, it is not so. Theoretical prediction, that for frequencies symmetric relative to the resonant one, the system behavior should be similar, occurs weakly, especially for ratios 0,9 and 1,1, because the system near the resonant frequency is two sensitive relative to values of frequency. Theoretical and experimental results close to the ratio of 0,98 were studied in [18]. So, inaccurate determination of the resonant frequency leads to no clear results in studying the resonance of sloshing.

A much more complicated situation takes place for the angular motion of the reservoir with liquid. For the determination of partial and normal frequencies of oscillations using the system of Eqs. (11)–(13) we write down the linear model in the case when we consider oscillations only relative to one normal mode of oscillations.

$${\ddot{a}}_1+\frac{1}{{\alpha }_1^v}{\ddot{\epsilon }}_x{B}_{1x}^1+\frac{1}{2{\alpha }_1^v}{\ddot{\alpha }}_1{E}_{11}^{1\ast }+{\omega }_1^2{a}_1=0;$$

$$\frac{\rho }{M_r+M_F}{\ddot{a}}_1{B}_{1x}^1+{\ddot{\epsilon }}_x+\frac{\rho }{M_r+M_F}{\ddot{\alpha }}_1{F}_2^1=0;$$E28

$${\ddot{a}}_1{E}_{11}^{1\ast }+2{\ddot{\epsilon }}_x{F}_2^1+2{\ddot{\alpha }}_1\left(\frac{1}{\rho }{J}_{\mathit{res}}^{11}+E_{11}^2\right)+{\alpha }_1\frac{2g}{\rho }\left[M_r\left(R_x+H/4\right)+M_l\left(R_x+H/2\right)\right]=0.$$

It is possible to write down partial frequencies for sloshing, translational, and rotational motion of the reservoir from Eq. (28)${\omega }_a^p=\sqrt{\frac{{\mu }_{1}g}{R}tanh\frac{{\mu }_{1}g}{R}}$.${\omega }_a^p=0$. ${\omega }_a^p=\sqrt{\frac{M_r{h}_r+M_l{h}_l}{J_{\mathit{res}}^{11}}}$. However, normal frequencies for the combined motion should be determined from the determinant

$$\left|\begin{array}{ccc}{\lambda }^2-{\omega }_1^2& \frac{1}{{\alpha }_1^v}{B}_{1x}^1{\lambda }^2& \frac{1}{2{\alpha }_1^v}{E}_{11}^{1\ast }{\lambda }^2\\ \frac{\rho }{M_T+M_F}{B}_{1x}^1{\lambda }^2& {\lambda }^2& \frac{\rho }{M_T+M_F}{F}_2^1{\lambda }^2\\ E_{11}^{1\ast }{\lambda }^2& 2F_2^1{\lambda }^2& 2\left(\frac{1}{\rho }{J}_{\mathit{res}}^{11}+E_{11}^2\right){\lambda }^2+\frac{2g}{\rho }\left[M_T\left(R_x+H/4\right)+M_F\left(R_x+H/2\right)\right]\end{array}\right|=0.$$E29

In a particular case of the absence of rotational motion, the normal frequency will be ${\omega }_1=\frac{{\omega }_1^p}{\sqrt{1-\frac{\rho {\left(B_1^{1x}\right)}^2}{{\alpha }_1^v\left(M_T+M_f\right)}}}$. Taking into account that according to the mechanical sense $B_1^{1x}$ is the displacement of the mass center of a liquid for unit excitation of the normal mode and it does not exceed the reservoir radius the denominator of this fraction is less than 1 and greater than 0, so the frequency of the normal mode in a combined motion of the system components is less than the partial frequency. In the case when the liquid mass exceeds the reservoir mass this change of the frequency can be considerable, for example, when liquid mass exceeds the reservoir mass 5 times, the change of frequencies is about 25%.

If we consider the behavior of the reservoir on pendulum suspension, the result of frequencies changing follows from Eq.(29) and it is shown in Figure 3. This result corresponds to the suspension length equal to the reservoir radius. Frequencies with asterisks are partial ones. The partial frequency 2,47 corresponds to pendulum oscillations for the “solidified” liquid, and 4,14 corresponds to the frequency of liquid oscillations in the case of the absence of pendulum oscillations. After passing to the combined mode of motion according to the theorem about changing normal frequencies relative to partial ones, frequencies shift outside the segment [2,47; 4,14] and become correspondingly 2,24 and 6,69. If we arrange frequencies in ascending order then for partial frequencies they were the first and the second ones, but for normal frequencies in the case of the combined motion they become the first and the fourth ones (5,45 and 6,12 are correspondingly frequencies for circular numbers 2 and 0).

Figure 3.

Arrangement of frequencies for pendulum suspension of the reservoir.

The analysis of this arrangement of frequencies enables us to state the following.

1. For the combined motion of the system in the case of the arrangement of frequencies in ascending order location of frequencies corresponding to different circular numbers changes.

2. Only normal frequencies for circular number 1 differ from partial values. This is caused by the property that for excitation of normal modes with circular numbers different from 1 the position of the mass center of the liquid does not change.

3. It is known that the sharpness of manifestation of the resonance depends on the sequence number of its location in the frequency arrangement in the ascending order. Therefore, we can predict that the significance of the resonance relative to the normal mode with the circular number 1 is reduced, resonance on the frequency of pendulum suspension will be the most sensitive, however, resonances for the normal modes with circular numbers 2 and 0 must be analyzed supplementary.

To analyze the effect of the suspension length we consider the following problem statements. The results of the determination of frequencies are shown in Table 1. As before we consider the problem of the reservoir fixing on pendulum suspension with immovable suspension point (column 4 of the table) and for movable suspension point on the horizontal guiding (column 5). Column 1 corresponds to the suspension length, and columns 2 and 3 show partial frequencies of liquid sloshing relative to the normal mode with circular number 1 and pendulum oscillations. It is seen from the table that the mobility of the suspension point considerably increases the normal frequency of the liquid sloshing mode, while the increase of the frequency corresponding to the pendulum mode of oscillations increases insignificantly. Therefore, the sharpness of manifestation of the resonance for the normal mode with the circular number 1 reduces supplementary. Numerical experiments showed that resonance according to this mode of motion was not observed for short suspension lengths. However, for suspension lengths greater than 8R this resonance becomes stronger and stronger.

1	2	3	4	5
$l=1R$	4,14431	2,47842	2,24312	2,26729
			6,69986	16,63085
$l=2R$	4,14431	1,96189	1,85304	1,86112
			6,10821	12,45135
$l=8R$	4,14431	1,07442	1,05895	1,05934
			5,52733	9,75099

Table 1.

Frequencies for different cases of reservoir suspension.

If resonances for normal modes of oscillations corresponding to pendulum oscillations and liquid sloshing relative to the normal mode with the circular number 1 occur for linear systems (we call them primary), the resonance for normal modes with circular numbers 2 and 0 for the linear statement is absent. However, since for combined motion of the system they become the second and the third ones for the nonlinear statement of the problem, they are excited. Figure 4 shows the elevation of the free surface of the liquid on the reservoir wall and the angle of inclination of the reservoir. For the disturbance frequency 5,45 we observe the secondary resonance of sloshing. Despite of the rather small excitation of the normal mode with the circular number 2 (the amplitudes do not exceed 0,05R) amplitude of the inclination angle is considerable and it causes the increase of the normal mode of liquid oscillation with the circle number 1. In some sense, the normal mode with $m=2$ works as a catalyst and within the nonlinear model opens a channel of energy redistribution in the system. A similar study of the resonant for $m=0$ (according to the sequential numbering this resonance is the third) was not successful. The only way to observe the resonance relative to the normal mode $m=0$ was in the case when we adjust the system by changing the suspension length in such a way that frequencies of normal modes for $m=0$ and $m=1$ coincide. Here after rather a durable period of oscillations, both normal modes excite.

Figure 4.

Variation in time of liquid elevation on the walls and the angle of inclination of the reservoir for the secondary resonance.

The general picture of resonance development is the following.

1. Primary resonance corresponding to the pendulum mode of the system motion is always significant.

2. For short suspension lengths, the resonance relative to sloshing mode with the circular number 1 becomes insignificant and its frequency becomes at least the fourth one in the frequency arrangement in ascending order.

3. Secondary (nonlinear) resonances manifest for normal modes with circular numbers $m=2$ and $m=0$ (only for special adjustment of the system).

4. All listed resonances result in considerable angular oscillations of the reservoir and the free surface of the liquid relative to the normal mode with the circle number $m=1$ independently of the frequency of the resonance.

Finally, for the near-resonant oscillations it is significant to consider the problem statement for the combined motion of the system components because in this case, we observe a selective change of frequencies, which can cause even their reordering. The system in the vicinity of resonance is very sensitive relative to variations in the disturbance frequency. The nonlinear statement of the problem causes the excitation of several normal modes of oscillations and this results in the strong manifestation of modulation and excitation of normal modes with a transcendent ratio of frequencies, which lead to the absence of steady mode of oscillations. However, if we increase dissipation 25–30 from the real scale, the system oscillations tend to a steady mode of motion [6].

5. New elements in the problem of parametric oscillations of reservoirs with liquid

The effect of parametric oscillations for the first time was stated and studied by Faraday. The problem used the idealized statement, which enables the parametric mechanism and suppresses all others. The main variants of the problem statement for parametric oscillations are shown in Figure 5. The classical (Faraday) statement of the problem corresponds to the variant Figure 5a. Moreover, here vertical motion of the reservoir is supposed to be given. On the other hand, if we analyze the problem of longitudinal oscillation of a rocket, which is called the POGO problem, we must take into account that for the rocket, which motion is not constrained by other factors, both transversal (Figure 5b) and rotational (Figure 5c) motion for the carrying structure is possible. Moreover, in addition, the longitudinal motion of the rocket should be supposed to be the motion caused by a longitudinal thrust (force) and it is necessary to consider its combined character. Similar situations occur in other engineering systems. Practically all researchers study only the classical problem despite its partial practical significance [3, 19]. In addition, usually, only the circular cylindrical reservoir is considered.

Figure 5.

The main statement of the problem of parametric oscillations.

To show some new elements of the parametric oscillations, we consider two problem statements shown in Figure 5b, c. The following new elements were considered.

1. We consider the problem statement as a combined motion for the supplementary degree of freedom for the motion in the transversal direction (b) and for the rotational motion (c). Moreover, for some situations, we suppose that the law of longitudinal motion of the system is also not given.

2. Combined character of motion results in the necessity to take into account that the resonant frequencies will change.

3. Since we consider the combined character of motion and the free surface of motion performs antisymmetric oscillations it is expedient to wain the manifestation of not only the parametric mode of oscillations but the forced mode of motion too. The frequency of oscillations of the forced mode of motion differs from the frequency of oscillations caused by the parametric mechanism, so the effect of modulation of oscillations will be installed.

Figure 6 shows the difference in the development of oscillations of the conic reservoir, which mass amounts to 13,8% of the mass of liquid. In this case, the resonant frequency of the first antisymmetric normal mode for the classical problem is 3,13 and for the combined statement it is 5,36. We consider the initial excitation of the amplitude of the first antisymmetric normal mode 0,01 and consider the case when the reservoir performs given vertical oscillations with the frequency equal to the double normal frequency of this mode. It is necessary to note that if we compare the results of this problem with the results of the same problem for the cylindrical reservoir, we see that effect of the combined character of motion is manifested stronger. This is caused by the property that the zone of slow-moving liquid in the case of the cylindrical reservoir (near the bottom) considerably exceeds the same zone in the case of the conic reservoir. This is reflected in the mathematical model by the parameter ${\overrightarrow{U}}_r^1$, characterizing the degree of interconnection between the liquid motion relative to the first normal mode and the translational motion of the reservoir.

Figure 6.

Variation of free surface excitations in time.

Similar effects are observed in the case of the angular motion of the carrying reservoir, but in addition to the previous case, two supplementary frequencies appear. The first one corresponds to the pendulum mode motion of the system, and the second one is connected with oscillations at the frequency of the first antisymmetric normal mode.

Manifestation of the forced mechanism results in the following significant property of the system behavior. It is known that the parametric mechanism of oscillations takes place only in the vicinity of the coinciding of the double normal frequency of oscillations relative to the first antisymmetric normal mode and the frequency of the longitudinal excitation. Outside this vicinity oscillations of a free surface of the liquid will be absent. However, in the case of manifestation of the forced mechanism of oscillations, the increase of liquid oscillations will manifest for all frequencies. For different examples studied as computer experiments, we see that this mechanism of the increase of oscillations occurs not so sharply as in the case of the parametric mechanism, but nevertheless, this violates the usual recommendation of the existence of stability zones for liquid in the case of longitudinal excitation of oscillations of the carrying structure.

6. Some potentials in reduction of the effect of liquid mobility on the motion of the carrying structure by the algorithm of compensation of the liquid response of reservoirs wall

Problems of high-precision maneuvering of structures with a fluid are important in modern engineering, including the aerospace and power industries. Modern reviews of the literature show that the greatest success in modeling the nonlinear dynamics of structures with a free-surface fluid was achieved using the variational algorithms, in particular one of them, based on the Hamilton variational principle stated at the beginning of this Chapter. A significant advantage of the Hamilton variational principle consists in the potential of the analytical determination of the forces of interaction between the parts of the system. For the problem of the dynamics of a structure with a liquid, it allows us to determine analytically the main vector of pressure forces of the liquid on the walls of the reservoir (liquid force response). Based on the quantitative characteristic of the force interaction of the structure with the liquid, we can construct a motion control algorithm for the reduction of the effect of liquid mobility on the carrying structure. Its main idea is to compensate by the control of the force response of the liquid. This is extremely important for the high-precision execution of program motions of structures with a liquid [20, 21]. Although this control is not optimal, it enables the study of a control problem for a multidimensional nonlinear mathematical model and provides us with an algorithm ensuring high-precision motion of the carrying structure with a liquid suitable for practical purposes.

If we represent the system of equations in the form of the 2nd Newton law.

$$\ddot{\overrightarrow{\epsilon }}=\frac{\overrightarrow{F}+\overrightarrow{R}}{M_F+M_T}+\overrightarrow{g};\sum _{n=1}^3{\ddot{\alpha }}_n\left[\sum _{p,s=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\frac{\partial {\omega }_s}{\partial {\dot{\alpha }}_n}\left(\frac{1}{\rho }{J}_{\mathit{res}}^{\mathit{ps}}\right)\right]=\frac{1}{\rho }\left(M_R^r+M_{\mathit{ex}}^r\right)\left(r=1,2,3\right),$$E30

where $\overrightarrow{R}$ and $M_R^r$ are the vector of liquid pressure forces and moments correspondingly on the reservoir walls (force and moment response of the liquid), then we can write from Eq. (30)

$$\begin{array}{l}\overrightarrow{R}=\rho \left[\left(\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{\overrightarrow{B}}_{\mathit{ij}}^2+\sum _{i,j,k}{\dot{a}}_i{\dot{a}}_j{a}_k{\overrightarrow{B}}_{\mathit{ijk}}^3\right)-\sum _i{\ddot{a}}_i\left({\overrightarrow{B}}_i^1+\sum _j{a}_j{\overrightarrow{B}}_{\mathit{ij}}^2+\sum _{j,k}{a}_j{a}_k{\overrightarrow{B}}_{\mathit{ijk}}^3\right)\right];\end{array}$$E31

$$\begin{array}{l}{M}_R^r=-\frac{\rho }{2}\sum _i{\ddot{a}}_i\left\{\sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\left(E_{\mathit{pi}}^{1\ast }+\sum _j{a}_j{E}_{\mathit{pij}}^{2\ast }+\sum _{j,k}{a}_j{a}_k{E}_{\mathit{\text{pijk}}}^{3\ast }\right)\right\}-\\ -\rho \sum _{n=1}^3{\ddot{\alpha }}_n\left[\sum _{p,s=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\frac{\partial {\omega }_s}{\partial {\dot{\alpha }}_n}\left(A_{\mathit{ps}}^2+\sum _i{a}_i{E}_{\mathit{psi}}^2+\sum _{i,j}{a}_i{a}_j{E}_{\mathit{\text{psij}}}^3\right)\right]+\\ +\rho \sum _{p,s=1}^3\left({\omega }_{p,r}^{\ast }{\omega }_s+{\omega }_p^{\left(k\right)}\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\right)\left(A_{\mathit{ps}}^2+\sum _i{a}_i{E}_{\mathit{psi}}^2+\sum _{i,j}{a}_i{a}_j{E}_{\mathit{\text{psij}}}^3\right)+\\ +\frac{\rho }{2}\sum _{p=1}^3{\omega }_{p,r}^{\ast }\left(\sum _i{\dot{a}}_i{E}_{\mathit{pi}}^{1\ast }+\sum _{i,j}{\dot{a}}_i{a}_j{E}_{\mathit{pij}}^{2\ast }+\sum _{i,j}{\dot{a}}_i{a}_j{a}_k{E}_{\mathit{\text{pijk}}}^{3\ast }\right)+\\ +\rho \sum _{p,s=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}{\omega }_s\left(\sum _i{\dot{a}}_i{E}_{\mathit{psi}}^2+2\sum _{i,j}{\dot{a}}_i{a}_j{E}_{\mathit{\text{psij}}}^3\right)+\frac{\rho }{2}\sum _{p=1}^3\frac{\partial {\omega }_p}{\partial {\dot{\alpha }}_r}\left(\sum _i{\dot{a}}_i{\dot{a}}_j{E}_{\mathit{pij}}^{2\ast }+2\sum _{i,j}{\dot{a}}_i{\dot{a}}_j{a}_k{E}_{\mathit{\text{pijk}}}^{3\ast }\right).\end{array}$$E32

We consider two examples of using the controlling algorithm based on Eqs. (31) and (32). As was mentioned in Subsection 4 of this Chapter there is certain inaccuracy in the results of a study of the main resonance of liquid in a reservoir, which performs a translation motion. Authors suppose that the reservoir performs the preset harmonic motion neglecting the effect of liquid mobility, however, the graphs of this motion [10] show that this law is partially violated. Therefore, we state a new problem. What control is necessary for providing the reservoir with high-precision harmonic motion? Figures 7 and 8 show variations in time of the controlling force and the velocity of the reservoir for harmonic driving force (solid line) and the dashed line corresponds to the force, determined according to the objective of minimization of the effect of liquid mobility on reservoir walls (using the principle of compensation of the liquid mobility by applying the control equal to the liquid mobility, but with opposite sign). It is seen that for providing harmonic law of motion of the reservoir it is necessary to use a special controlling algorithm. Otherwise in the vicinity of the main resonance, where the system is extremely sensitive to perturbations the incorrect law of motion can distort the general picture of the studied process.

Figure 7.

Variation of the driving force applied to the reservoir in time.

Figure 8.

Variation of reservoir velocity in time.

In practice, the use of the suggested scheme of control will be accompanied by additional perturbations, which are considerable for the accuracy of the algorithm. Therefore, we consider the problem, where only a part of the information characterizing the state of the system with be used. For example, for the construction of the control we use only the information of the excitation of the first normal mode of motion.

Initially, the reservoir with liquid is at rest. Then we apply the force in the form of the rectangular impulse with a duration of 0,25 s (less than a quarter of the period of oscillations of the first normal mode). Parameters of a liquid and the reservoir are selected similarly to the above-considered examples. Figure 9 shows the dependence in time of the translational motion of the reservoir for four cases. First, this corresponds to the case when the liquid is “frozen” (liquid mobility is absent) and is shown by the solid curve 1; second, the solid line 2, which coincides with curve 1 and corresponds to the control with the liquid response compensation obtained based on the whole information about the liquid state; third, corresponds to the case then motion is uncontrollable and it is shown by curve 3 (dash-dotted line); curve 4 (dotted line) corresponds to the case when control is constructed according to not complete information about parameters of a liquid (only including the first normal mode).

Figure 9.

Variation of reservoir velocity in time.

It is seen from Figure 9 that in the case of the “frozen” liquid on the active interval of the action of constant force the velocity of the reservoir increases linearly until the end of the active period (curve 1), further on the stage of free motion (active force is equal to zero) the reservoir velocity is constant. When the controlling action is absent, initially the system moves rapidly and later it performs comparatively large oscillatory motion with the mean velocity peculiar to the velocity of the reservoir with “frozen” liquid on the free stage of motion (curve 3). If we construct the control according to the complete information about the liquid dynamic state (exact compensation) the law of motion (curve 2) coincides with high accuracy with the law in the case of “frozen” liquid, so we eliminate with high accuracy the effect of liquid mobility on the reservoir velocity changing. In the case when we used the simplified (approximate) relation for determination of the liquid response, which is based only on the values of amplitudes of the first normal mode, initially there is a deviation from, the desired law of motion, however, this difference decreases in time and it is practically absent after 9 s. So, this control provides a “convergence” to the desired solution but with a certain delay.

The developed scheme of control is a variant of feedback control relative to accelerations of disturbances of the normal modes of oscillations of a free surface of the liquid with their normalization according to the law of formation of the liquid response. The algorithm is based on some advantages of the Hamilton variational principle, connected with the technique of variation of the functional resulting in the possibility of the analytical determination of forces of interaction between the system components. As a result, this enables the considerable reduction of the effect of liquid mobility on the motion of the carried structure.

7. Conclusions

The Chapter deals with the problem of nonlinear dynamics of reservoirs carrying a free surface liquid. We gave preference to the development of analytical methods suitable for studying nonlinear problems of dynamics of a combined motion of the reservoir and a liquid with a high potential for computer implementation of this method. In aggregate, the following problems were solved.

1. The nonlinear model for studying problems with the combined character of motion of the reservoir and liquid was constructed. This model contains the coefficients, for which there are formulae of their determination for an arbitrary number of the considered normal modes of motion of a free surface of the liquid. At the same time, we consider both the translational and rotational motion of the carrying body.

2. Some specific features of the application of this method to the problems of the non-cylindrical shape of the reservoir were solved. The use of the solvability condition of the corresponding boundary value problem enables the construction of the normal modes, which in advance satisfy nonflowing boundary conditions on the reservoir walls above the free surface of the liquid, where the liquid can reach.

3. The avoiding of some commonly used hypotheses (namely, assumption about the given law of the reservoir motion, omitting motions relative to normal frequencies of all normal modes entrained in the nonlinear model, and the use of low-dimension mathematical models) enables the clarification of the large group of theoretical and experimental results. Mainly this can be stated in five items. The use of the model of combined motion considerably changes resonant frequencies; the system behavior for the below-resonant, near-resonant and above-resonant modes differs considerably; strong manifestation of modulation of oscillations takes place; the presence of frequencies with transcendent ratio results in the absence of the steady mode of resonant oscillations; in the case of rotation motion of the reservoir, we can observe the strong manifestation of the nonlinear secondary resonances and reordering of the location of frequencies, which changes the priority of manifestation of certain resonances.

4. Some generalization of the classical (Faraday) problem statement connected with providing the system either supplementary translational or rotational motion of the carrying body results in the simultaneous manifestation of parametric and active mechanisms of oscillations, which considerably changes the system behavior, because in this case, the increase of liquid oscillations can happen for arbitrary frequency.

5. The problem of control with the objective of minimization of the liquid mobility effect on the motion of the carrying structure was studied by compensating the liquid response on reservoir walls. The success of this approach is grounded on the potential of determination of the liquid response in the analytical form.

The considered examples are evidence of the correctness and high practical efficiency of the stated approach.