Analysis of the Small Oscillations of a Pendulum Partially Filled by a Heavy Capillary Liquid

The authors study the small oscillations of a compound pendulum partially filled by a heavy liquid, in presence of surface tensions. Using the variational formulation of the problem, they prove that, under a very simple geometrical condition, it is a classical vibration problem. On the other hand, introducing the operatorial equations of motion, they prove that the eigenvalues equation can be obtained by equaling to zero an absolutely convergent infinite determinant.


Introduction
The problem of the small oscillations of a heavy system formed by a container partially filled by an inviscid liquid has been the subject of a number of works (Moiseyev and Rumiantsev 1968).The same problem has been studied more recently in the case of a liquid submitted to surface tensions and oscillating in a fixed container (Morand et Ohayon 1992, Kopachevsky and Krein, Vol 1, 2001).

H. Essaouini et al.
In this work, the authors consider the problem of a moving container partially filled by a capillary liquid, restricting themselves to the simple case of the compound pendulum.Using the variational formulation of the problem and suitable Hilbert spaces, they prove that, under a simple geometrical condition, the problem is a classical vibration problem.On the other hand, introducing the operatorial equations of motion, they obtain the eigenvaluesequation by equaling to zero an absolutely convergent infinite determinant.

Position of the Problem
The container is a compound pendulum oscillating about a fixed point O .We use the axes Ox , Oy , Oy vertical upwards (unit vectors x , y ).In the equilibrium position, the container is symmetrical with respect to Oy  We suppose that the pressure above the free line is equal to zero and we take into account the constant surface tension τ .
We are going to study the possible small oscillations of the system pendulum-liquid about its equilibrium position, in linear theory.

Let us consider the liquid
If P is the pressure, we introduce the dynamic pressure p by ( ) u x y t is the small displacement of a particle from its equilibrium position, we have (1) grad u p Integrating from the equilibrium position to the instant t , we have The kinematic condition can be written

Let us write the equations deduced from the laws of the capillarity a)
Denoting by α the angle n Γ , n Σ at the points A and B , the conditions of contact angle are (Morand et Ohayon 1992) where R is the radius of curvature of Σ at the points A and B .
b) The Laplace law on the free line Γ in the equilibrium position is verified, because the pressure on the line y h = − is equal to zero.

H. Essaouini et al.
We where m ℓ is the mass of the liquid and G ℓ its center of mass in the equilibrium position, and ( ) On the other hand, we have, using the Euler's equation (1) and the Green formula Finally, using the condition (6), we obtain the equation K being positive if the pendulum is preponderant.

Variational formulation of the problem
Using the condition (6), we obtain Integrating by parts the last two integrals and taking into account the condition (5), we obtain the variational equation so that the problem is self adjoint.We are going to study the equation (8).

4.2) Since
We are going to prove that: Introducing the multipliers λ and µ , we write or, after integration by parts ( ) So, we obtain for calculating λ the Steklov eigenvalues problem: We find easily two values for λ : 2 σ and 6 σ , so that 2 λ σ = .
4.3) Now, we consider the hermitiansesquilinear form, continuous in ( ) ( ) Using the precedent result, we have ( ) In the following, we will suppose and ( ) It is easy to see that de condition (9) express that the center of curvature of Σ in A must be of the right of the axis Oy.

4.4)
Let use prove that Indeed, if c does not exist, there is a sequence { } ( ) , we can deduce a sequence, denoted still by { } n v , that is strongly convergent in the space Therefore, the sequence { } n v is strongly convergent in ( )

4.5)
In the variational equation (8), we consider the hermitian sesquilinear form ( ) ( ) By means of the precedent result and little long calculation, it is possible to prove that, under the condition (9), ( ) a norm that is equivalent to the classical norm of this space.

4.6)
Since, by the Lagrange's theorem, u is a gradient, we introduce the space ( ) equipped with the hilbertian norm defined by To find ( ) Using a method and calculations that can be found in the book (Sanchez Hubert, Sanchez Palencia 1989, pp 66-68)-so that we omit the proofwe see that the problem (10) is a classical vibration problem.Therefore, if the pendulum is preponderant and under the simple condition Using the equations ( 2), ( 3), ( 4), we see that ϕ verifies ( ) We are going to seek ϕ in the form is a Joukowski's potential, depending on the form of Ω .
On the other hand, if , it is well-known that the first problem has a generalized solution and only one and that K , self-adjoint, positive definite and compact.From the Euler's equation (1), we deduce ( ) Using the condition (6), we obtain the equation ( ) Integratingon Γ , we calculate ( )  c t and we obtain finally a first equation between ( )  t θ and ( ) 2) Now, we transform the left-hand side of the equation of the moment of momentum.We have ( )   In order to obtain the equations with bounded operators, we set ( ) Applying 1/2 0 A − to the equation (13), we obtain ( ) and the equation ( 12) can be written

The eigenvalues equation
We set N is obviously compact from ( ) it is self-adjoint and positive definite.Therefore, N has a countable infinity of positive eigenvalues It is a Von Koch's determinant ( Riesz, 1913) and it is absolutely convergent because the series Truncating this determinant, it is possible to calculate the first eigenvalues.

,
. It contains an incompressible inviscid liquid (density ρ ) that occupies a domain Ω bounded by the wetted part Σ of the wall of the container and the horizontal free line Γ or AB ( ) mass of the pendulum is denoted by m , its center of mass by 0 is the acceleration of the gravity.At the instant t , the liquid occupies the domain t Ω , bounded by the wetted part t Σ of the wall of the container and the free line t Γ or A'B' .If G is the position of 0 G at the instant t , θ and its derivatives being small.Introducing the axes OX , OY , Ox,OX θ = , we denote by ( ) the equation of t Γ , ζ and its derivatives being small.

n
Σ and n Γ being the unit vectors normal to Σ and Γ .

4. 1 )
Let use introduce θ ɶ arbitrary in R, a smooth function u ɶ defined in Ω and verifyingdiv 0 u = ɶ , in Ω , χ , completion of V for the norm defined by ɶ , we obtain the precise variational formulation of the problem:

5 . Operatorial equations of motion 5 . 1 )
basis in χ and an orthogonal basis in V equipped with the scalar product ( ) , a U U ɶ .We introduce the displacement potential ( )

5 . 3 )
We transform the equation (11) by introducing the unbounded operator η being constants and ω being real, we obtain the system of a countable infinity of linear equations for 0 θ and the 0n η .