Keywords

1 Introduction

A detailed study on the dynamic behavior of various elements of engineering structures is widely possible using modern numerical methods. It has been used in areas such as geological surveys, sound reduction, crack detection or even in earthquake propagation studies. Although 3D finite elements are universal and can be used to solve dynamic problems for structures of various complex geometry, they need huge computing capacity. The aim of this paper is to design an algorithm by combining several well-established mathematical methods with some new approaches for the efficient solution of dynamic problems for axisymmetric structures in a way involving less computational cost. Different structures of cylindrical shape elements are very often used as parts of important engineering constructions. For this reason, dynamic behavior of such structures has been extensively investigated in recent years. FEM in the form of the semidiscrete Galerkin method is so far the currently dominating one for dynamic analysis of such objects. The use of the eigenfunction expansion method for time integration together with semi-analytical FEM has become a good basis for the algorithms of parallelization [1].

2 Literature Review

Currently, there is renewed interest in this area due to advances in the development of ultrasonic and microsonic based devices for trapping of biological cells and micro particles [2]. Two types of dynamic problems are usually considered: dynamic response of a structure sustaining arbitrary loading [3, 4] and free vibration [5, 6]. As noted in [7], despite the significant growth in the breadth of its applicability, two areas have received relatively less attention in p/hp-FEM for solid mechanics: dynamic problems and parallelism [8]. A Chebyshev-Ritz numerical procedure based on the 3D elasticity theory is employed in [5] to extract the full vibration spectrum of natural frequencies along with selected 3D deformed mode shapes. To study a number of problems of harmonic and impulse oscillations, an approach of fundamental solutions for layers was developed on the basis of which amplitude characteristics, as well as stress diagrams for layers with one or two cavities of different types, were obtained [9]. Paper [3] presents an explicit smoothed finite element method (SFEM) for elastic dynamic problems. The central difference method for time integration is used in presented formulations. A spectral method for elastic wave calculations which is based on a Chebychev expansion in the vertical direction is presented in [4]. The results indicate that the method presents an improvement over the ordinary Fourier method in handling the free surface boundary condition. An algorithm that is able to solve dynamic contact problems with complex local geometries was proposed in [10]. The authors combined domain decomposition with mortar coupling, contact modeling via semismooth Newton methods and energy-consistent time integration. Numerical examples confirm the optimality of the approach and its stabilization effect applied to dynamic contact problems.

3 Research Methodology

3.1 Governing Equations

Let us consider the deformation of a three-dimensional object that occupies the axi-symmetric domain of an anisotropic, linearly elastic material. The domain is referred to a right-hand system of orthogonal, cylindrical coordinates r, \( \theta {\kern 1pt} \) and z which represent the radial, angular and axial coordinates, respectively.

The dynamic problem of elasticity is investigated in a 3D statement. The independent displacement components that describe the motion of the body can be written as

$$ u_{r} \equiv u_{r} (r,\theta ,z,t),\quad u_{\theta } \equiv u_{\theta } (r,\theta ,z,t),\quad u_{z} \equiv u_{z} (r,\theta ,z,t),\quad {\mathbf{u}} \equiv (u_{r} ,u_{\theta } ,u_{z} )^{T} , $$
(1)

The equations of motion corresponding to the displacement field (1) can be expressed as [6]:

$$ \begin{aligned} & \frac{{\partial \sigma_{rr} }}{\partial \, r} + \frac{1}{r}\frac{{\partial \sigma_{r\theta } }}{\partial \theta } + \frac{{\partial \sigma_{rz} }}{\partial \, z} + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r} = \rho \frac{{\partial^{2} u_{r} }}{{\partial t^{2} }}, \\ & \frac{{\partial \sigma_{r\theta } }}{\partial \, r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \theta } }}{\partial \theta } + \frac{{\partial \sigma_{\theta \, z} }}{\partial \, z} + \frac{{2\sigma_{r\theta } }}{r} = \rho \frac{{\partial^{2} u_{\theta } }}{{\partial t^{2} }}, \\ & \frac{{\partial \sigma_{rz} }}{\partial \, r} + \frac{1}{r}\frac{{\partial \sigma_{\theta \, z} }}{\partial \theta } + \frac{{\partial \sigma_{zz} }}{\partial \, z} + \frac{{\sigma_{rz} }}{r} = \rho \frac{{\partial^{2} u_{z} }}{{\partial t^{2} }} \\ \end{aligned} $$
(2)

Here \( \rho {\kern 1pt} \) is the density of the material. We also assume that the relations for strain components and constitutive relations are given in matrix form as

$$ {\varvec{\upvarepsilon}} \equiv \left\{ {\begin{array}{*{20}l} {\varepsilon_{rr} } \hfill \\ {\varepsilon_{\theta \theta } } \hfill \\ {\varepsilon_{zz} } \hfill \\ {\varepsilon_{rz} } \hfill \\ {\varepsilon_{r\theta } } \hfill \\ {\varepsilon_{z\theta } } \hfill \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial \, r}} & 0 & 0 \\ {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 \\ 0 & 0 & {\frac{\partial }{\partial \, z}} \\ {\frac{1}{2}\frac{\partial }{\partial \, z}} & 0 & {\frac{1}{2}\frac{\partial }{\partial \, r}} \\ {\frac{1}{2r}\frac{\partial }{\partial \theta }} & {\frac{\partial }{\partial \, r} - \frac{1}{r}} & 0 \\ 0 & {\frac{1}{2}\frac{\partial }{\partial \, z}} & {\frac{1}{2r}\frac{\partial }{\partial \theta }} \\ \end{array} } \right] \cdot \left\{ {\begin{array}{*{20}l} {u_{r} } \hfill \\ {u_{\theta } } \hfill \\ {u_{z} } \hfill \\ \end{array} } \right\} , $$
(3)

and

$$ {\varvec{\upsigma}} \equiv \left\{ {\begin{array}{*{20}l} {\sigma_{rr} } \hfill \\ {\sigma_{\theta \theta } } \hfill \\ {\sigma_{zz} } \hfill \\ {\sigma_{rz} } \hfill \\ {\sigma_{r\theta } } \hfill \\ {\sigma_{z\theta } } \hfill \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {d_{11} } & {d_{12} } & {d_{13} } & 0 & 0 & 0 \\ {d_{12} } & {d_{22} } & {d_{23} } & 0 & 0 & 0 \\ {d_{31} } & {d_{32} } & {d_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {d_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {d_{55} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {d_{66} } \\ \end{array} } \right] \cdot \left\{ {\begin{array}{*{20}l} {\varepsilon_{rr} } \hfill \\ {\varepsilon_{\theta \theta } } \hfill \\ {\varepsilon_{zz} } \hfill \\ {\varepsilon_{rz} } \hfill \\ {\varepsilon_{r\theta } } \hfill \\ {\varepsilon_{z\theta } } \hfill \\ \end{array} } \right\} , $$
(4)

respectively. The constitutive relations (4) contain nine independent elastic constants and involve the most widely used kinds of anisotropic media: transversely isotropic and orthotropic.

There are external loads applied to the surface of the object

$$ \sum\limits_{i,j}^{3} {\sigma_{ij} n_{j} } = \zeta_{i} ,\quad {\text{i}},{\text{j}} = {\text{r}},\theta ,{\text{z}},\quad {\text{on}}\quad\Gamma _{N} . $$
(5)

here \( \Gamma _{N} \) is the part of the surface of the cylinder, \( n_{r} ,n_{\theta } ,n_{z} \) are the components of the outward unit normal vector to this surface in radial, angular and axial directions respectively, and \( \zeta_{i} { (}i = r ,\theta ,{\kern 1pt} z ) \)—are prescribed loads. Moreover, we also have the kinematic boundary conditions

$$ {\mathbf{u}} = {\mathbf{g}},\quad {\text{on}}\;\Gamma _{D} ,\quad\Gamma \equiv \partial\Omega =\Gamma _{N} \cup\Gamma _{D} ,\quad\Gamma _{N} \cap\Gamma _{D} = {\varnothing}, $$
(6)

where \( \varvec{g} = \left( {g_{\text{r}} ,g_{\theta } ,g_{z} } \right)^{T} \) is the given vector of displacements and initial conditions:

$$ {\mathbf{u}}|_{t = 0} = {\mathbf{0}},\quad \frac{{\partial \, {\mathbf{u}}}}{\partial t}|_{t = 0} = {\mathbf{0}} ,\quad x \in\Omega . $$
(7)

\( {\varvec{\upvarepsilon}} \equiv \left( {\varepsilon_{rr} ,\varepsilon_{zz} ,\varepsilon_{\theta \theta } ,\varepsilon_{rz,} \varepsilon_{r\theta } ,\varepsilon_{\theta \, z} } \right)^{T} \)—is the vector of the deformations component in (3), and \( {\varvec{\upsigma}} \equiv \left( {\sigma_{rr} ,\sigma_{zz} ,\sigma_{\theta \theta } ,\sigma_{rz,} \sigma_{r\theta } ,\sigma_{\theta \, z} } \right)^{T} \)—is the vector of the stresses component in (4).

3.2 Weak Formulation

Let us introduce the space of kinematically admissible vectors of displacement, which is analogous to [6]

The weak variational form of the initial-boundary problem (2), (5)–(7) is formulated as:

Find \( {\mathbf{u}} \in H \) such that

$$ \begin{aligned} & m\left( {{\mathbf{u}}^{\prime\prime } (t),{\mathbf{v}}} \right) + a\left( {{\mathbf{u}}(t),{\mathbf{v}}} \right) = \left\langle {l(t),{\mathbf{v}}} \right\rangle ,\quad \forall {\mathbf{v}} \in V,\quad \forall t \in (0;T], \\ & {\mathbf{u}}(0) = {\mathbf{0}},\quad {\mathbf{u}}^{\prime } (0) = {\mathbf{0}}. \\ \end{aligned} $$
(8)

where bilinear forms:

$$ m({\mathbf{u}},{\mathbf{v}}) = \int\limits_{\Omega } {\rho \left( {u_{r} v_{r} + u_{\theta } v_{\theta } + u_{z} v_{z} } \right)d\Omega } , $$

and

$$ \begin{aligned} a({\mathbf{u}},{\mathbf{v}}) & = \int\limits_{\Omega } {\left\{ {d_{11} \frac{{\partial \, u_{r} }}{\partial \, r}\frac{{\partial \, v_{r} }}{\partial \, r} + d_{12} \left( {\frac{{\partial \, u_{z} }}{\partial \, z}\frac{{\partial \, v_{r} }}{\partial \, r} + \frac{{\partial \, u_{r} }}{\partial \, r}\frac{{\partial \, v_{z} }}{\partial \, z}} \right)} \right.} \\ & \quad + \frac{{d_{13} }}{r}\left( {u_{r} \frac{{\partial \, v_{r} }}{\partial \, r} + v_{r} \frac{{\partial \, u_{r} }}{\partial \, r} + \frac{{\partial \, u_{\theta } }}{\partial \, \theta }\frac{{\partial \, v_{r} }}{\partial \, r} + \frac{{\partial \, u_{r} }}{\partial \, r}\frac{{\partial \, v_{\theta } }}{\partial \, \theta }} \right) + d_{22} \frac{{\partial \, u_{z} }}{\partial \, z}\frac{{\partial \, v_{z} }}{\partial \, z} \\ & \quad + \frac{{d_{23} }}{r}\left( {\frac{{\partial \, u_{z} }}{\partial \, z}v_{r} + \frac{{\partial \, v_{z} }}{\partial \, z}u_{r} + \frac{{\partial \, u_{z} }}{\partial \, z}\frac{{\partial \, v_{\theta } }}{\partial \, \theta } + \frac{{\partial \, v_{z} }}{\partial \, z}\frac{{\partial \, u_{\theta } }}{\partial \, \theta }} \right) + \frac{{d_{33} }}{{r^{2} }}\left( {u_{r} v_{r} } \right. \\ & \quad \left. { + \frac{{\partial \, u_{\theta } }}{\partial \, \theta }\frac{{\partial \, v_{\theta } }}{\partial \, \theta } + \frac{{\partial \, u_{\theta } }}{\partial \, \theta }v_{r} + \frac{{\partial \, v_{\theta } }}{\partial \, \theta }u_{r} } \right) + d_{44} \left( {\frac{{\partial \, u_{r} }}{\partial \, z}\frac{{\partial \, v_{r} }}{\partial \, z} + \frac{{\partial \, u_{z} }}{\partial \, r}\frac{{\partial \, v_{z} }}{\partial \, r}} \right. \\ & \quad \left. { + \frac{{\partial \, u_{z} }}{\partial \, r}\frac{{\partial \, v_{r} }}{\partial \, z} + \frac{{\partial \, u_{r} }}{\partial \, z}\frac{{\partial \, v_{z} }}{\partial \, r}} \right) + d_{55} \left( {\frac{1}{r}\frac{{\partial \, v_{r} }}{\partial \, \theta }\left( {\frac{{\partial \, u_{\theta } }}{\partial \, r} - \frac{{u_{\theta } }}{r}} \right)} \right. \\ & \quad \left. { + \frac{1}{r}\frac{{\partial \, u_{r} }}{\partial \, \theta }\left( {\frac{{\partial \, v_{\theta } }}{\partial \, r} - \frac{{v_{\theta } }}{r}} \right) + \left( {\frac{{\partial \, u_{\theta } }}{\partial \, r} - \frac{{u_{\theta } }}{r}} \right)\left( {\frac{{\partial \, v_{\theta } }}{\partial \, r} - \frac{{v_{\theta } }}{r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial \, u_{r} }}{\partial \, \theta }\frac{{\partial \, v_{r} }}{\partial \, \theta }} \right) \\ & \quad \left. { + d_{66} \left( {\frac{1}{r}\frac{{\partial \, u_{\theta } }}{\partial \, z}\frac{{\partial \, v_{z} }}{\partial \, \theta } + \frac{1}{r}\frac{{\partial \, v_{\theta } }}{\partial \, z}\frac{{\partial \, u_{z} }}{\partial \, \theta } + \frac{1}{{r^{2} }}\frac{{\partial \, u_{z} }}{\partial \, \theta }\frac{{\partial \, v_{z} }}{\partial \, \theta } + \frac{{\partial \, u_{\theta } }}{\partial \, z}\frac{{\partial \, v_{\theta } }}{\partial \, z}} \right)} \right\}d\Omega , \\ \end{aligned} $$

and linear functional

$$ \left\langle {l,{\mathbf{v}}} \right\rangle = \int\limits_{{\Gamma _{N} }} {\left( {\xi_{r} v_{r} + \zeta_{z} v_{z} + \zeta_{\theta } v_{\theta } } \right)d\Gamma } . $$

Since axisymmetric objects are considered, we will present the components of the vectors of the surface load in the form of an infinite Fourier series for an angular coordinate

$$ \zeta_{r} = \sum\limits_{i = 0}^{\infty } {p_{r}^{(i)} \phi_{i} (\theta )} ,\quad \zeta_{z} = \sum\limits_{i = 0}^{\infty } {p_{z}^{(i)} \phi_{i} (\theta )} ,\quad \zeta_{\theta } = \sum\limits_{i = 0}^{\infty } {p_{\theta }^{(i)} \tilde{\phi }_{i} (\theta ).} $$
(9)

Then the solution of the problem (8) will be found in the form

$$ {\mathbf{u}} = \left( {\begin{array}{*{20}l} {u_{r} } \hfill \\ {u_{z} } \hfill \\ {u_{\theta } } \hfill \\ \end{array} } \right) = \sum\limits_{i = 0}^{\infty } {\left( {u_{r}^{(i)} (r,z,t),u_{z}^{(i)} (r,z,t),u_{\theta }^{(i)} (r,z,t),} \right)\left( {\begin{array}{*{20}c} {\phi_{i} (\theta )} & 0 & 0 \\ 0 & {\phi_{i} (\theta )} & 0 \\ 0 & 0 & {\tilde{\phi }_{i} (\theta )} \\ \end{array} } \right)} , $$
(10)

where the following notation is introduced

$$ \begin{aligned} \phi_{i} (\theta ) & = \left\{ {1,\cos \theta ,\sin \theta ,\cos 2\theta ,\sin 2\theta , \ldots ,\cos n\theta ,\sin n\theta , \ldots } \right\}, \\ \tilde{\phi }_{i} (\theta ) & = \left\{ {1,\sin \theta ,\cos \theta ,\sin 2\theta ,\cos 2\theta , \ldots ,\sin n\theta ,\cos n\theta , \ldots } \right\} \\ \end{aligned} $$

  • complete orthogonal systems of trigonometric functions on the interval \( \left[ {0;2\pi } \right] \).

Partial sum

$$ {\mathbf{u}}_{n} = \sum\limits_{i = 0}^{n} {{\mathbf{u}}^{(i)} (r,z,t)\Phi _{i} (\theta )} $$
(11)

is named as the approximation of the weak solution of the problem (8).

Let us select in the subspace of the approximations \( V_{h} \) a basis of piecewise continuous test functions constructed on a regular partition of serendipity quadrilaterals:

$$ V_{h} = span\left\{ {N_{1} ,N_{2} ,N_{3} , \ldots ,N_{L} } \right\}. $$

Then Galerkin’s semidiscretization \( {\mathbf{u}}_{h}^{(i)} \left( t \right) \in V_{h} \) of the weak solution (11) is presented in the form

$$ {\mathbf{u}}_{h}^{(i)} \left( {r,z,t} \right) = \sum\limits_{k = 1}^{L} {u_{kh}^{(i)} \left( t \right)N_{k} \left( {r,z} \right)} $$
(12)

with unknown coefficients \( u_{kh}^{(i)} \left( t \right), \) which are functions of time. We find these coefficients for one harmonic from the Cauchy problem for a system of ordinary differential equations, which can be presented in matrix notation as

$$ {\mathbf{M}}^{(i)} {\mathbf{\ddot{U}}}^{(i)} \left( t \right) + {\mathbf{K}}^{(i)} {\mathbf{U}}^{(i)} \left( t \right) = {\mathbf{R}}^{(i)} \left( t \right),\quad {\mathbf{U}}^{(i)} \left( 0 \right) = {\mathbf{0}},\quad {\dot{\mathbf{U}}}^{(i)} \left( 0 \right) = {\mathbf{0}}. $$
(13)

The matrices of mass \( {\mathbf{M}}{\kern 1pt}^{(i)} \) and stiffness \( {\mathbf{K}}{\kern 1pt}^{(i)} \) have the structure

$$ {\mathbf{M}}^{(i)} = \left( {\begin{array}{*{20}c} {C_{1}^{(i)} M_{kj}^{1} + C_{6}^{(i)} M_{kj}^{2} } & 0 & 0 \\ 0 & {C_{1}^{(i)} M_{kj}^{1} + C_{6}^{(i)} M_{kj}^{2} } & 0 \\ 0 & 0 & {C_{1}^{(i)} M_{kj}^{1} + C_{6}^{(i)} M_{kj}^{2} } \\ \end{array} } \right), $$
$$ {\mathbf{K}}^{(i)} = \left( {\begin{array}{*{20}c} {C_{1}^{(i)} K_{kj}^{11} + C_{2}^{(i)} \tilde{K}_{kj}^{11} } & {C_{1}^{(i)} K_{kj}^{12} } & {C_{3}^{(i)} K_{kj}^{13} + C_{4}^{(i)} \tilde{K}_{kj}^{13} } \\ {C_{1}^{(i)} A_{kj}^{21} } & {C_{1}^{(i)} \tilde{K}_{kj}^{22} + C_{2}^{(i)} \tilde{K}_{kj}^{22} } & {C_{3}^{(i)} K_{kj}^{23} + C_{4}^{(i)} \tilde{K}_{kj}^{23} } \\ {C_{3}^{(i)} K_{kj}^{31} + C_{4}^{(i)} \tilde{K}_{kj}^{31} } & {C_{3}^{(i)} K_{kj}^{32} + C_{4}^{(i)} \tilde{K}_{kj}^{32} } & {C_{5}^{(i)} K_{kj}^{33} + C_{6}^{(i)} \tilde{K}_{kj}^{33} } \\ \end{array} } \right),\quad k,j = 1,2, \ldots ,L, $$

and vector of load \( {\mathbf{R}}{\kern 1pt}^{(i)} \left( t \right) \)

$$ {\mathbf{R}}^{(i)} \left( t \right) = \left( {\begin{array}{*{20}c} {C_{1}^{(i)} R_{k}^{1} \left( t \right)} \\ {C_{1}^{(i)} R_{k}^{2} \left( t \right)} \\ {C_{6}^{(i)} R_{k}^{3} \left( t \right)} \\ \end{array} } \right),\quad \quad k = 1,2, \ldots ,L, $$

with the coefficients

$$ R_{k}^{1} \left( t \right) = \int\limits_{{\Gamma _{\sigma }^{ \, 1} }} {\zeta_{r}^{(i)} \left( t \right)N_{k} d\Gamma _{\sigma }^{1} } ,\quad R_{k}^{2} \left( t \right) = \int\limits_{{\Gamma _{\sigma }^{ \, 1} }} {\zeta_{z}^{(i)} \left( t \right)N_{k} d\Gamma _{\sigma }^{1} } ,\quad R_{k}^{3} \left( t \right) = \int\limits_{{\Gamma _{\sigma }^{ \, 1} }} {\zeta_{\theta }^{(i)} \left( t \right)N_{k} d\Gamma _{\sigma }^{1} } . $$

Here the elements \( C_{l}^{(i)} , \, l = 1, \ldots ,6, \) are the integrals of trigonometric functions.

3.3 Finding a Non-stationary Solution

The solution of problem (13) can be obtained using standard procedures for the solution of the Cauchy problem for ordinary second-order differential equations with constant coefficients. In direct integration, the solution of the system (13) is obtained by a numerical stepwise procedure. The number of operations in this case is directly proportional to the number of steps per time. For integration in time we use the method of expansion according to eigenfunctions.

The nodal values \( {\mathbf{U}}^{(i)} \left( t \right) \) of displacement were written as

$$ {\mathbf{U}}^{(i)} \left( t \right) = {\varvec{\Psi}}_{i} {\mathbf{X}}_{i} \left( t \right), $$
(14)

where \( {\mathbf{X}}_{i} \left( t \right) \)—an unknown vector whose components are generalized displacements, and \( {\varvec{\Psi}}_{i} \)—the matrix whose columns are eigenvectors obtained as solutions of a generalized algebraic eigenproblem

$$ {\mathbf{K}}^{(i)} {\varvec{\uppsi}} = \omega^{2} {\mathbf{M}}^{(i)} {\varvec{\uppsi}}, $$
(15)

Let us consider that in the time coordinate the solution of variational Eq. (8), can be presented as

$$ {\mathbf{u}} = {\mathbf{U}}\exp (i\omega t). $$
(16)

If we denote \( {\mathbf{u}} = \left( {u_{r} ,u_{\theta } ,u_{z} } \right)^{T} \), the amplitude of displacements \( {\mathbf{u}} = {\mathbf{U}} \), then we obtain the variational equation for \( l(t) \equiv 0 \)

$$ a\left( {{\mathbf{u}},{\mathbf{v}}} \right) - \omega^{2} m\left( {{\mathbf{u}},{\mathbf{v}}} \right) = 0,\quad \forall {\mathbf{v}} \in V. $$

Here the bilinear forms have the form \( a({\mathbf{u}},{\mathbf{v}}),\;m({\mathbf{u}},{\mathbf{v}}) \) as above.

Let us formulate the following variational problem [6]:

Find a pair \( \left( {{\mathbf{u}},\omega } \right) \in V \times {\mathbf{R}} \) that

$$ a\left( {{\mathbf{u}},{\mathbf{v}}} \right) = \omega^{2} m\left( {{\mathbf{u}},{\mathbf{v}}} \right) ,\quad \forall {\mathbf{v}} \in V. $$
(17)

Problem (17) is a variational problem of finding eigenvalues \( \nu = \sqrt {\omega /2\pi } \) and their corresponding eigenfunctions. Applying the same approximation as for problem (8), the solution of (17) is reduced to the solution of (15).

Substituting (14) into equations and initial conditions (13) and multiplying to the left at \( {\varvec{\Psi}}_{i}^{T} \), we obtain a system of equilibrium equations for generalized displacements:

$$ {\mathbf{\ddot{X}}}_{i} (t) +\Omega _{i}^{2} {\mathbf{X}}_{{\mathbf{i}}} (t) = {\varvec{\Psi}}_{{\mathbf{i}}}^{{\mathbf{T}}} {\mathbf{R}}^{(i)} (t), $$
(18)

with initial conditions \( {\mathbf{X}}_{{\mathbf{i}}} {\mathbf{(0)}} = {\mathbf{0}},\quad {\dot{\mathbf{X}}}_{{\mathbf{i}}} {\mathbf{(0)}} = {\mathbf{0}}. \)

The system (18) splits into \( \tilde{k} \) separate equations

$$ \ddot{x}_{i}^{j} (\tau ) + \left( {\omega_{i}^{j} } \right)^{2} x_{i}^{j} (\tau ) = ({\varvec{\uppsi}}_{i}^{j} )^{T} {\mathbf{R}}^{(i)} (\tau ),\quad j = 1, \ldots ,\tilde{k}. $$
(19)

The solution of each Eq. (19) is represented in the form of the Duhamel integral:

$$ x_{i}^{j} (t) = \frac{1}{{\omega_{i}^{j} }}\int\limits_{0}^{t} {({\varvec{\uppsi}}_{i}^{j} )^{T} {\mathbf{R}}^{(i)} (\xi )} \sin \omega_{i}^{j} (t - \xi )d\xi + \alpha_{i}^{j} \sin \omega_{i}^{j} t + \beta_{i}^{j} \cos \omega_{i}^{j} t. $$
(20)

To obtain a complete system response, it is necessary to find a solution of all \( \tilde{k} \) Eq. (19). We find the displacements of node points for i—harmonic as a superposition of the reactions of the system in all its eigenvectors.

$$ {\mathbf{U}}^{(i)} (t) = \sum\limits_{k = 1}^{{\tilde{k}}} {{\varvec{\Psi}}_{i}^{k} } {\mathbf{x}}_{i}^{k} (t). $$

Finally, the solution is obtained as a linear superposition according to (11) for each Fourier component present in the load.

4 Results

The approbation of the proposed approach with the parallelization was carried out on the cluster of Ivan Franko National University of Lviv, which consists of 14 computing nodes and a server under the Scientific Linux 6.2 (core 3.6.6) operating system. All computing nodes for sharing data between parallel processes are united by 1Gbit/s Ethernet. The problem of determining the dynamic reaction of a hollow cylinder, with internal radii \( R_{1} = 0.8\;{\text{m}} \) and external radii \( R_{2} = 1.2\;{\text{m}} \) and height \( L = 10\;{\text{m}} \), with a rigidly pinched lower end was solved. On the outside of the cylinder a non-axisymmetric, non-stationary, normal load is given

$$ \tilde{\Psi}(\theta ,t) = \left\{ {\begin{array}{*{20}l} {\Psi (t)\cos \theta ,} \hfill & { - \pi /2 \le \theta \le \pi /2} \hfill \\ {0,} \hfill & {\left| \theta \right| > \pi /2} \hfill \\ \end{array} } \right. ,\quad\Psi (t) = 1,\quad {\text{for}}\quad 0 \le t < \infty . $$

Since the applied load is symmetric with respect to the cross section \( \theta = 0 \), then only the coefficients with cosines are not nonzero in the expansion in a series of trigonometric functions. They are given by formulas

$$ a_{0} = 1/\pi ,\quad {\text{a}}_{1} = 0.5,\quad {\text{a}}_{n} = - \frac{{2( - 1)^{n/2} }}{{\pi (n^{2} - 1)}} ,\quad n = 2,4,6, \ldots $$

The physical characteristics of the cylinder were chosen as follows: the density \( \rho = 2.7 \times 10^{3} \;{\text{kg/m}}^{3} \), Poisson’s coefficient \( \nu = 0.17 \) and Young’s modulus \( E = 0.146 \times 10^{11} \;{\text{N/m}}^{2} \).

The first step for the solution of the dynamic elasticity problem is to find eigenvalues and eigenvectors of free oscillations of the object. Table 1 shows the values of the first five cyclic frequencies of free cylinder oscillations for the first five harmonics, including the zero harmonic. Here i—the column corresponds to i—the harmonic, and j—the row is j—free frequency. It is seen in Table 1 that the smallest free frequency of a cylinder corresponds to the first harmonic. This fact demonstrates that in the study of the dynamic characteristics of cylindrical objects, it is not enough to take into account only symmetric frequencies and their corresponding forms of free vibration.

Table 1. The five cyclic frequencies of free oscillations of the cylinder.
Full size table

There are graphs of the component of the displacement vector by the angular coordinate at the time (t = 0.17 s) on Fig. 1, that corresponds to the largest value of the amplitude of the oscillations. The graphs are marked with: solid line—displacement \( u_{r} \), dashed line—displacement \( u_{z} \) and dotted-dotted—displacement \( u_{\theta } \). In the section \( \theta = 0 \) the components \( u_{r} ,u_{z} \) of displacement, and in the section \( \theta = \pi /2 \) the displacement \( u_{\theta } \) take the maximum value.

Fig. 1.
figure 1

The values of displacements \( u_{r} ,u_{z} ,u_{\theta } \) by angular coordinates \( \theta \) (rad).

Full size image

Figure 2 presents graphs of stress changes \( \sigma_{\theta \theta } ,\sigma_{r\theta } ,\sigma_{zz} \) by angular coordinates at the moments of time when the stresses exceed the maximum of the amplitude. The solid, dashed and dotted-dashed lines represent the stresses \( \sigma_{\theta \theta } ,\sigma_{r\theta } ,\sigma_{zz} \), respectively.

Fig. 2.
figure 2

The values of stresses \( \sigma_{\theta \theta } ,\sigma_{r\theta } ,\sigma_{zz} \) by angular coordinates \( \theta \) (rad).

Full size image

5 Conclusion

The numerical investigation of dynamic problems for anisotropic axisymmetric objects within the framework of 3D linear elasticity theory has been developed. Two kinds of dynamic problems have been considered: dynamic response of a structure under an arbitrary load and free vibrations. Semidiscrete FEM with trigonometric series for an angular coordinate is used. The possibility of obtaining asymmetric oscillation frequencies in this scheme is very important. The results of numerical solutions of engineering problems confirm the effectiveness of the proposed approach. The algorithm of parallel implementation provided the opportunity to significantly reduce the time of obtaining the solution.