On oscillations of two joined pendulums with cavities partially filled with an incompressible ideal fluid

Let $G_1$ and $G_2$ be two joined bodies with masses $m_1$ and $m_2$. Each of them has a cavity partially filled with homogeneous incompressible ideal fluids situated in domains $\Omega_1$ и $\Omega_2$ with free boundaries $\Gamma_1(t), \Gamma_2(t)$ and rigid parts $S_1, S_2$. Let $\rho_1, \rho_2$ be densities of fluids. We suppose that the system oscillates (with friction) near the points $O_1, O_2$ which are spherical hinges.

We use the vectors of small angular displacement
\begin{equation*} \vec{\delta}_{k}(t)=\sum\limits_{j=1}\limits^{3}\delta_{k}^{j}(t)\vec{e}_{k}^{\,j}, \quad k=1,2, \end{equation*}
to determine motions of the removable coordinate systems $O_{k}x_{k}^{1}x_{k}^{2}x_{k}^{3}$ (connected with bodies) relative to stable coordinate system $O_{1}x^{1}x^{2}x^{3}$. Then angular velocities $\vec{\omega}_k(t)$ of bodies $G_{k}$ is equal to $d\vec{\delta}_k/dt$.

Let $\vec u_k(x,t)= \vec w_k(x,t) + \nabla \Phi_k (x, t),\ \vec w_k \in \vec J_0(\Omega_k),\nabla \Phi_k \in \vec{G}_{h, \,S_k}(\Omega_k)$ and $p_k(x,t) \in H^1(\Omega_k)$ be fields of fluids velocities and dynamical pressures in $\Omega_k$ (in removable coordinate systems), $\zeta_k (x,t) \in L_{2, \Gamma_k}:= L_2(\Gamma_k) \ominus {\rm sp}\, {1_{\Gamma_k}}$ are functions of normal deviation of $\Gamma_k(t)$ from equilibrium plane surfaces $\Gamma_k(0)=\Gamma_k$. Then we consider initial boundary value problem (2.1), (2.5)–(2.7) with conditions (2.8)–(2.12).

We obtain the law of full energy balance (2.20). Using the method of orthogonal projections with some additional requirements initial problem can be reduced to the Cauchy problem for the system of differential equations
\begin{gather*} \begin{split} C_1 \frac{dz_1}{dt} + A_1 z_1 + gB_{12} z_2 = f_1(t), \quad z_1(0)=z_1^0, \qquad \qquad \ \\ gC_2 \frac{dz_2}{dt} + gB_{21} z_1 = 0, \quad z_2(0)=z_2^0, \qquad \qquad \qquad \qquad \quad \\ z_1 = \Big( \vec w_1; \nabla \Phi_1; \vec\omega_1; \vec w_2; \nabla \Phi_2; \vec\omega_2 \Big)^{\tau} \in \mathcal{H}_1, \quad z_2 = \Big( \zeta_1; P_2 \vec\delta_1; \zeta_2; P_2 \vec\delta_2 \Big)^{\tau} \in \mathcal{H}_2, \end{split} \end{gather*}
in Hilbert spaces
\begin{multline*} \mathcal{H}_1 = (\vec J_0(\Omega_1)\oplus \vec G_{h, S_1}(\Omega_1)\oplus \mathbb{C}^3) \oplus (\vec J_0(\Omega_2)\oplus \vec G_{h, S_2}(\Omega_2)\oplus \mathbb{C}^3), \quad \mathcal{H}_2 = (L_{2, \Gamma_1}\oplus \mathbb{C}^2) \oplus (L_{2, \Gamma_2}\oplus \mathbb{C}^2). \end{multline*}
Here operators of potential energy $C_k$ is bounded, $C_1$ is positive definite, $A_1$ is bounded and nonnegative, $B_{ij}$ is skew self-adjoint operators. Using this properties we prove theorem on existence of unique strong solution for $t\in [0;T]$ if some natural conditions for initial data and given functions $f_1(t)$ are satisfied. As a corollary we obtain theorem on solvability of initial Cauchy problem.

If friction is absent then operator $A_1=0$ and for $z(x,t)=e^{i\lambda t}z(x)$ we obtain spectral operator problem. For the eigenvalues $\mu=\lambda^2/g$ we find new variational principle and prove that spectrum is discrete. It consists of positive eigenvalues with limit point $+\infty$ in stable case, or the positive branch and not more then finite number of negative eigenvalues in unstable case.