1 Introduction

The variational principles of solid mechanics and fluid mechanics have attracted much attention,since they are the theoretical foundation of the finite element method,and have important applications^{[1, 2, 3, 4, 5, 6]}.

Starting from the basic equations of hydrodynamics,Chien^[5] established the maximum power-type variational principle of the hydrodynamics of viscous fluids in 1984. Through long-term research,it shows that,with the power-type variational principles, some transformations in the time domain space may be omitted in the process of establishing the variational principles,and it is convenient to numerically model the rigid-elastic-liquid coupling dynamics. Therefore,it is convenient to establish the power-type variational principle.

The spaceflight liquid-filled system is a nonlinear multi-field coupling complex system with an unsteady infinite degree of freedom, and it is hard to be studied^{[7, 8, 9]}. In order to apply the power-type variational principle to aerospace dynamics,the power-type variational principle is extended from hydrodynamics to rigid-body dynamics,elasto-dynamics,and rigid-elastic-liquid coupling dynamics. Liang and Shi deduced the power-type variational principles of hydrodynamics by use of the variational integral method^[10],and established the power-type variational principles for the initial value problem^[11]. Liang et al.^[12] and Liang and Song^[13] studied the rigid-elastic coupling dynamics by variational principles,which laid a foundation for the research of rigid-elastic-liquid coupling dynamics.

In this paper,the power-type variational principle is extended to rigid-body dynamics, elasto-dynamics,and rigid-elastic-liquid coupling dynamics. The governing equations of the rigid-elastic-liquid coupling dynamics in the liquid-filled system are obtained by deriving the stationary value conditions.

2 Basic theory 2.1 Variational principle of hydrodynamics of viscous fluids

In order to establish the power-type variational principle in the spacecraft liquid-filled system,we employ the power-type variational principle of hydrodynamics in the liquid-filled system, and extend it to rigid-body dynamics and elasto-dynamics. It has been shown that it is convenient to achieve the extension from the Jourdian principle.

The Jourdian principle is one of the most fundamental mechanics principles. For a system composed of arbitrary particles ,the general dynamics theorem of the Jourdian principle can be expressed as follows:

where $v_i $ is the velocity vector of the particle $m_i $, $f_i $ is the active force applied on the particle $m_i $, and $\delta v_i$ is the Jourdian velocity vector variation of the particle $m_i $. Equation (1) can be interpreted in the following manner: for any arbitrary particle in the system satisfying the constraint condition prescribed by the Jourdain velocity vector variation, the total virtual work change rate of the active force and the inertial force is equal to zero. According to the Jourdian principle, the virtual power principle and the power-type variational principle of the hydrodynamics of viscous fluids can be derived.

2.2 Power-type variational principle of rigid-body dynamics

The governing equations of rigid-body dynamics are

where ${{v}^{\text{c}}}$ is the velocity vector of the center of the mass,$t$ is the time,$m$ is the mass,$\omega $ is the angular velocity of rotation,$J$ is the moment of the inertia tensor,$J\omega $ is the moment of the momentum vector to the center of the mass,$F$ is the principal vector,and $M$ is the principal couple vector.

Using the Jourdian principle yields

When the principal vector $F$ and the principal couple vector $M$ are all non-conservative generalized forces,according to the method in Ref.^[5],the above formula can be translated into the functional stationary value problem,and the functional $\pi _{\rm r} $ can be written as follows: where Equation (5) is the power-type variational principle of rigid-body dynamics. $\pi_{\rm r0}$ is the total kinetic energy of the rigid-body per unit time. It is also called the kinetic energy change rate,whose dimension is the same as that of the power.

2.3 Power-type variational principle of elasto-dynamics

The governing equations of elasto-dynamics are

where $v^{\rm e}$ is the velocity vector of the elastic body, $\sigma $ is the stress tensor of the elastic body,$n^{\rm e}$ is the unit normal vector of the elastic boundary,$\varepsilon $ is the strain tensor of the elastic body,$D$ is the stiffness coefficient tensor,$\nabla $ is the gradient operator (the Hamilton operator),$f^{\rm e}$ is the body force vector per unit volume of the elastic body,$T^{\rm e}$ is the surface force vector of the elastic body,$S_{\rm u} $ is the displacement boundary surface,and $S_\sigma $ is the stress boundary surface.

Since it is a continuum and the summation signs become integral signs in the Jourdian principle, using the Jourdian principle yields

Using the Green theorem yields Substituting Eq.(13) into Eq.(12) yields With Eqs.(9) and (10),we can rewrite Eq.(14) as follows: Substituting Eq.(11) into Eq.(15) yields The above formula can be translated into the functional stationary value problem. Then,we can obtain the functional $\pi _{\rm e} $ as follows: where The pre-conditions are Eqs.(9) and (10).

Equation (17) is the power-type variational principle of elasto-dynamics. $\pi _{\rm e0}$ is the total kinetic energy of the elastic body per unit time. It is also called the kinetic energy change rate,whose dimension is the same as that of the power.

3 Power-type variational principle of hydrodynamics of viscous fluids in liquid-filled system

Starting from the basic equations of hydrodynamics,Chien^[5] established the maximum power-type variational principle of the hydrodynamics of viscous fluids in 1984. In this paper,the power-type variational principle of the hydrodynamics of viscous fluids is used to study the liquid-filled system.

We think that the fluids in the liquid-filled system can be regarded as incompressible viscous fluids. For the liquid-filled system,the governing equations of the hydrodynamics of steady incompressible viscous fluids can be expressed as follows.

The dynamics equation is

The continuity equation is

The constitutive relation is

The boundary condition is $S=S_{\rm w} +S_{\rm f}.$

On the free boundary surface $S_{\rm f} $,

On the solid boundary surface $S_{\rm w} $,

In the above equations,$\rho $ is the mass density (scalar), $v^{\rm q}$ is the velocity vector of the fluids,$f^{\rm q}$ is the body force vector per unit volume fluid,$\tau $ is the stress tensor,$\mu $ is the viscosity coefficient (scalar),$I$ is the second-order unit tensor,$p$ is the hydro-static pressure (scalar), $n^{\rm q}$ is the unit normal vector of the fluid boundary,$p_{\rm u} $ is the gas pressure at the stress boundary,$\nabla $ is the gradient operator (the Hamilton operator),$S_{\rm w} $ is the solid boundary surface,and $S_{\rm f} $ is the free boundary surface.

Here,some remarks regarding the $(\rho \eta g+p_{\rm u} )$ term are made,where $\eta $ is the difference between the height of the actual surface and the height of the average surface^[14],and $g$ is the acceleration of gravity. In Ref.^[15],the effective range of the gravity on spacecraft was discussed. ``A common misconception is that the satellites orbiting at high altitudes are free from gravity. Nothing could be further from the truth. The force of gravity on a satellite 200 kilometers above the Earth's surface is nearly as strong as at the surface. The high altitude is to put the satellite beyond the Earth's atmosphere,where the air drag is almost totally absent,but not beyond Earth's gravity.'' The discussion makes us realize that we must seriously consider the effect of the gravity on spacecraft. Through the above discussion, the meaning of the $(\rho \eta g+p_{\rm u} )$ term is clarified. With the development of aerospace science and technology,how to deal with the boundary conditions of mechanics must be improved.

Then,using the Jourdian principle yields

Using the Green theorem yields

Substituting Eq.(21) into Eq.(26) yields

Substituting Eq.(21) into Eq.(26) yields

With Eqs.(20) and (23),we can rewrite Eq.(27) as follows:

Considering the symmetry of the $\nabla v^{\rm q}+v^{\rm q}\nabla $ term,the above formula can be translated into the functional stationary value problem. Then,the functional $\pi _{\rm q} $ can be expressed by

The pre-conditions are Eqs.(20) and (23).

Equation (29) is the power-type variational principle of the hydrodynamics of incompressible viscous fluids. According to the method in Ref.^[5],the variation of $\pi _{\rm q0} $ can be expressed as follows:

where $\pi _{\rm q0} $ is the total kinetic energy of the fluids per unit time. It is also called the kinetic energy change rate,whose dimension is the same as that of the power. Judging from the last term in the volume integral,we can see that $\mathop{\iiint}\limits_{\kern-5.5pt V} {f^{\rm q} v^{\rm q}{\rm d}V} $ is the power of the body force. The dimensions of the rest terms are the same as that of the body force.

4 Power-type variational principle of rigid-elastic-liquid coupling dynamics in liquid-filled system

The power-type variational principles of rigid-body dynamics, elasto-dynamics,and hydrodynamics of incompressible viscous fluids in the liquid-filled system have been established in the above section,respectively. Establishing the power-type variational principles of the rigid-elastic-liquid coupling dynamics in the spacecraft liquid-filled system is the focus of this section.

The power-type variational principle of rigid-elastic-liquid coupling dynamics is not a simple sum of the power-type variational principles of rigid-body dynamics,elasto-dynamics,and hydrodynamics,but coupling problems. The reference coordinate system is shown in Fig.1.

There are three aspects for the coupling problems.

(i) The rigid-elastic coupling effect can be studied in the manner of Refs.^[12] and ^[13]. Assume that the Coriolis effect caused by the elastic deformation velocity will not be considered. Then,the rigid-elastic coupling effect can be expressed by

(ii) The rigid-liquid coupling effect can be studied in the manner of Refs.^[12] and ^[13]. For simplicity,assume that the Coriolis effect caused by the liquid velocity will not be considered. Then, the rigid-liquid coupling effect can be expressed by

(iii) The elastic-liquid coupling effect is mainly embodied in the velocity coordination relationship in the interface between the elastic body and the liquid^[6],i.e.,

It should be emphasized that all the solid boundary surfaces are assumed to be the interfaces between the elastic body and the liquid. In the interfaces between the elastic body and the liquid, the contribution of the elastic-liquid coupling effect for the function of the variational principle can be expressed as follows:

The contribution will be implicit in the function of the power-type variational principle of the coupling dynamics in the liquid-filled system.

With the above results and considering

we can obtain the function of the power-type variational principles of rigid-elastic-liquid coupling dynamics as follows: where

Equation (35) is the power-type variational principle of the coupling dynamics in the liquid-filled system. Using the direct variational method,such as the Ritz method,we can carry out approximate calculations for the coupling dynamics in the liquid-filled system. The calculation models of the finite element method of the coupling dynamics in the liquid-filled system can be established after further discretization. Moreover,the approximate numerical solution of the coupling dynamics in the liquid-filled system can be obtained by the calculation models. It shows that the analytical analysis of the coupling dynamics in the liquid-filled system in this paper and the numerical analysis of the coupling dynamics in the liquid-filled system presented by other scholars are complementary.

5 Governing equations of rigid-elastic-liquid coupling dynamics The governing equations of the rigid-elastic-liquid coupling dynamics in the liquid-filled system can be obtained by deriving the stationary value conditions of the power-type variational principle of the rigid-elastic-liquid coupling dynamics in the liquid-filled system. Taking variation and setting $\delta \pi _{\rm req} =0$ yield The variation of Eq.(9) can be expressed by Equation (37) can be further transformed into Using the Green theorem yields

Substituting Eqs.(40)--(42) into Eq.(39) and considering the following velocity boundary conditions:

Consider the following velocity compatibility condition on $S_{\rm w} $: $ \delta v^{\rm e}=\delta v^{\rm q}.$ Then,Eq.(43) can be simplified as follows:

Because of the arbitrariness of $\delta v^{\rm c}$,$\delta \omega$, $\delta v^{\rm e}$,and $\delta v^{\rm q}$,the stationary value conditions can be derived from Eq.(44) as follows:

The governing equations of the coupling dynamics in the liquid-filled system are obtained by deriving the stationary value conditions of the power-type variational principle of the coupling dynamics in the liquid-filled system. The analytic solution of the coupling dynamics in the liquid-filled system can be obtained by the governing equations. Through the computer simulation,the approximate solution of the coupling dynamics in the liquid-filled system can also be obtained by the governing equations. Due to the requirement of the calculation precision,the method is rarely used now. The coupling dynamics problem in the liquid-filled system can be numerically derived by the finite difference method. It indicates once again that the analysis of the coupling dynamics in the liquid-filled system proposed in this paper and the numerical analysis of the coupling dynamics in the liquid-filled system proposed by other scholars are complementary.

6 Conclusions

It is clarified that the power-type variational principle coincides with the Jourdian principle,which is one of the common principles of analytical mechanics. In the paper,the power-type variational principle is extended to rigid-body dynamics,elasto-dynamics,and rigid-elastic-liquid coupling dynamics. The governing equations of the rigid-elastic-liquid coupling dynamics in the liquid-filled system are obtained by deriving the stationary value conditions. The rigid-elastic coupling effect,the rigid-liquid coupling effect,the elastic-liquid coupling effect,and the power-type variational principle of the rigid-elastic-liquid coupling dynamics in the liquid-filled system are considered. It shows that,with the power-type variational principles being studied directly in the state space,some transformations in the time domain space may be omitted in the process of establishing the variational principles, and the rigid-elastic-liquid coupling dynamics can be conveniently modeled by numerical methods.

The analysis of the coupling dynamics in the liquid-filled system proposed in this paper and the numerical analysis of the coupling dynamics in the liquid-filled system offered by other scholars are complementary.