NUMERICAL STUDY OF FREE VIBRATIONS OF AN ELASTIC BODY OF REVOLUTION

ABSTRACT

We consider the three-dimensional problem of calculating the free oscillations of the first boundary value problem of elasticity theory in the body rotation. On a grid of 900 nodes available for calculations, natural frequencies were found that coincide with the one-dimensional test with 3–7 decimal places.

Keywords: three-dimensional theory of elasticity, free vibrations, numerical algorithm without saturation.

1. Introduction

Consider the vector equation of free vibrations of the theory of elasticity for a homogeneous isotropic medium:

                                                    (1)

where  and  Poisson's constant, density,  vibration frequency, μ - shear modulus. The problem is posed by studying the spectrum of the operator on the left side of equation (1) under the boundary conditions of the first problem:

                                                                               (2)

Here Ω is a body of revolution around the axis , G - its meridional section.

For all finite values of ω, except for ω = −1, the operator = Δ + ω grad div is elliptic. Consequently, the solutions of the first boundary value problem of the theory of elasticity are sufficiently smooth. To use this smoothness, below we construct a method for discretizing the first boundary value problem of the theory of elasticity, which does not have saturation [1, 2]. The most widespread method for solving problems in the mechanics of deformable solids is the finite element method. Its disadvantages are well known: approximating the displacement by a piecewise linear function, we find that the stresses are discontinuous. At the same time, it should be noted that the majority of problems in the mechanics of a deformable solid are described by equations of elliptic type, which have smooth solutions. It seems relevant to develop algorithms that would take into account this smoothness. The idea of such algorithms belongs to K.I. Babenko [2]. This idea was expressed by him in the early 70s of the last century. The long-term application of this technique in elliptic eigenvalue problems by the author of this work has proved their high efficiency. For example, the eigenvalue problem for the zero Bessel equation was considered; on a grid of 23 nodes, the first eigenvalue of this problem is determined with 28 decimal places. In contrast to the classical difference methods and the finite element method, where the dependence of the convergence rate on the number of grid nodes is power-law, here we have an exponential decrease in the error.

2. Discretization.

We introduce a system of curvilinear coordinates (, θ, ϕ) related to the Cartesian coordinates by the relations

                                            (3)

Let G be the domain obtained by the meridional section of the body Ω. Let us choose the functions u and v as follows. Let  be a conformal map of the circle  to the interior of area G. It is convenient to consider (, θ, ϕ) to be spherical coordinates; then relations (3) define a map of the ball of unit radius onto the interior of the body Ω. The surface of the ball of unit radius is mapped under map (3) into the surface of the body Ω. Then the boundary conditions on ∂Ω are transferred to the surface of the ball.

Usually, when using curvilinear coordinates, the equations for vector quantities are written in projections on the axes of their basis, the coordinate vectors of which are directed tangentially to the coordinate lines. This basis depends on the coordinates of a point in space. In this case, such an approach is inconvenient, since the mapping (3) loses its uniqueness on the  axis (if  = 0, then ϕ is arbitrary). This causes the appearance of peculiarities in the solution, which are caused not by the essence of the case, but by the "bad" coordinate system. Note that the spherical coordinate system has a similar "disadvantage". The way out of this situation is as follows: we leave as the required functions the projections of the velocity vector ( = 1, 2, 3) on the axis of the Cartesian coordinate system, and replace the independent variables by substitution (1) by r, θ, ϕ. Then the partial derivatives for the Cartesian coordinates  ( = 1, 2, 3) are expressed in terms of the derivatives for r, θ, and ϕ: U () = U (, )

where

Since the Cauchy – Riemann conditions are satisfied:  then the coordinate system (r, θ, ϕ) is orthogonal, and in this coordinate system the Laplacian of the scalar function has the form

 , .                          (4)

Then, instead of problem (1) - (2), we have an inner problem in a ball of unit radius. Moreover, zero boundary conditions are set on its border. In what follows, we will assume that the conformal mapping of the circle of unit radius onto the interior of the domain G is known. Note that there are reliable algorithms for the numerical construction of a conformal mapping [3].

To discretize the Laplacian (4) with a homogeneous boundary condition, we apply the technique described in [1].

Thus, we obtain a discrete Laplacian in the form of an h-matrix:

Here the prime means that the term at k = 0 is taken with a coefficient of 1/2; the  sign ⊗ is the Kronecker product of matrices; h is an L × L matrix with elements

 is the matrix of the discrete operator corresponding to the differential operator

                                       (5)

with boundary condition

                                                                     (6)

To discretize the differential operator (5), (6), we choose a grid for θ, consisting of n nodes:          

and also apply the interpolation formula

                                       (7)

The first and second derivatives for θ included in relations (5) are obtained by differentiating the interpolation formula (7).

For r, choose a grid consisting of m nodes:

,

and also apply the interpolation formula

                                  (8)

The first and second derivatives for r included in expression (5) can be found by differentiating the interpolation formula (8).

The original system will take the form:

                           (9)

3. Results of numerical calculations

The calculations were carried out for a ball and a region close to a ball with an epitrochoid in the meridional section (for problems 2 and 3, the conformal mapping is given by the formula ψ (z) = z (1 + ), where  = 4, ε = 1/6 for problem 2 and  = 12, ε = 1/16 for problem 3), on a grid of 900 = 10 × 10 × 9 nodes. Thus, the size of the spectral problem being solved is 2700 × 2700. The calculation results are presented in the table. The first column of the table shows the number of the real eigenvalue for the ball, the second column shows the simple eigenvalues for the ball, the number of signs is kept, which coincides with the one-dimensional test [4], the 3rd and 4th columns show the simple eigenvalues for the perturbed ball, σ = 0.25.

Calculation results of prime frequencies  for the sphere and regions close to the sphere, σ = 0.25

4. Conclusion

Thus, on a grid of 900 nodes, it is possible to determine the first natural frequencies of the sphere and the region close to the sphere with 3-7 decimal places. This effect was achieved by using the method without saturation to solve the spectral problem.

References:

1. Algazin S.D. Numerical algorithms of classical mathematical physics. Moscow: Dialogue-MEPhI, 2010.240 p.
2. Babenko K.I. Fundamentals of Numerical Analysis. Moscow: Nauka, 1986.744 p.; Second edition, revised and enlarged, edited by A.D. Bruno. Moscow-Izhevsk, RKhD, 2002.847 p.
3. Kazanjian E.P. On one numerical method for conformal mapping of simply connected domains. Moscow: Institute of Applied Mathematics, Academy of Sciences of the USSR, 1977. Prepr. No. 82. 60 p.
4. Novitsky V. Theory of elasticity. Moscow: Mir, 1975.872 p.