Journal of Theoretical and Applied Mechanics

Volume 31, Number 3, 2001

 

Contents

Cl. Mladenova

 

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, Sofia 1113, Bulgaria,

e-mail: clem@bgcict.acad.bg

 

Modelling and Control of Multibody Systems on a Configurational Space with a Lie Group Structure

Abstract. This paper is a review of our research activity during the last ten years concerning the problems of modelling and control of multibody mechanical systems. Because the treatment of the above topics is quite sensitive with respect to the different parameterizations of the rotation group in three dimensional space SO (3) and because the properties of the parametrization more or less influence the efficiency of the dynamic model, here the so called vector-parameter is used for parallel considerations.

The consideration of the mechanical system on the configurational space of pure vector-parameters with a group structure opens the possibilities for the Lie group theory to be applied in the problems of the dynamics and control.

 

Key words: multibody system, manipulator, kinematics, dynamics, Lie group, Lie algebra.

 

 

Al. Kazakoff, V. Abadjiev

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria,

e-mail: abadjiev@imech.imbm.bas.bg

 

INVESTIGATION ON SPATIAL VIBRATION PROCESSES IN A HELICON GEAR SET

Abstract. In this paper are studied monofrequency and polyfrequency spatial vibrations of a double mass vibration dynamic system with a suitable adjustment. The physical object is a Helicon gear set. The mechanical model is created on the basis of a double mass spatial dynamic model with twelve degrees of freedom - six for each mass. The influence of the elastic links between the two masses and also the energy mutual links between the co-ordinates of the two bodies are investigated. Natural frequencies and amplitudes of motions are predicted in the three dimensional solution. The obtained results can be used for optimisation of the constructive parameters of the Helicon gear set and also for improving of harmful resonance effects.

 

 

Key words: mathematical modelling, Helicon gear set, vibration dynamic excitation, natural frequencies, amplitudes of motion.

 

 

R. Blagoeva

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, Sofia 1113, Bulgaria,

e-mail: rumi@imech.imbm.bas.bg

 

NUMERICAL SOLUTION OF A CLASS OF REACTION-DIFFUSION PROBLEMS FOR COMPOSITE MATERIALS

Abstract. The numerical solution of the initial boundary-value problems arising in modelling the diffusion of a reactant in composite materials with a finite sorption capacity is considered. A numerical scheme, developed on the base of the Galerkin semidiscrete method and an appropriate difference method, is proposed. Unlike the methods used till now, it enables the numerical simulation of the diffusion of the reactant in two- and three-dimensional domains and under nonhomogenuous initial conditions, as well as time dependent and nonhomogenuous boundary conditions. The derived sufficient conditions for the stability of the scheme are included in the criterion for the time step choice. The numerical procedure is verified in the one-dimensional case under available numerical results for the penetrant sorption in epoxies. Corresponding numerical results are obtained in the two-dimensional case.

 

 

Key words: reactant-diffusion problem, finite sorption capacity, finite element method, finite difference method, numerical scheme stability.

 

 

Ia. V. Hristov

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria,

e-mail: iavhri@fmi.uni-sofia.bg

 

FINITE DEFORMATIONS OF AN ECCENTRIC COMPOUND DROP SUBJECTED TO AN ELECTRIC FIELD

 

Abstract. The finite deformations of an eccentric compound drop in an electric field are studied. The drop consists of two homogenous, incompressible and Newtonian fluids of different properties. The Reynolds number is assumed small enough and the hydrodynamic problem is solved in quasi-steady Stokes approximation. The initial forms of the drops are spherical ones.

The electric and hydrodynamic problems are separated and the electric field has influence on the hydrodynamic one by the Maxwell’s stress tensor appearing in the boundary conditions at the fluid surfaces. The Maxwell equations are reduced to Laplace equations, which are solved together with Stokes equations by means of semianalytical-seminumerical method, based on boundary elements. Using the kinematics condition, the forms of the particles are obtained at each time step.

The results indicate that interaction between different fluid phases in an electric field leads to the shape deformations of the inner and the outer drop. The influence of some fluid parameters and the electric field intensity on the deformation of the eccentric drop is represented in figures.

 

 

Key words: compound eccentric drop, electric field, boundary element method, drop deformations.

 

 

T. Penchev, I. Altaparmakov

Technical University of Sofia,

8, St. Kl. Ohridski Blvd., 1756 Sofia, Bulgaria,

e-mail: tpenchev@vmei.acad.bg

 

MODELLING OF GRAIN SHAPE IN COLD PLASTIC DEFORMATION THROUGH THE FINITE ELEMENT METHOD.

PART I. THEORETICAL MODEL.

Abstract. In the paper, a mathematical model for description of the grain rotation and grain shape changes during plastic deformation is presented. This model is based on the theory of plasticity. The data concerning the strains in the volume of the deformed body is obtained by Finite Element Method (FEM) modelling of the deformation process. Using strains instead of stresses allows obtaining more realistic data for shape changes and rotation of the grains during the process of cold plastic deformation.

Key words: plastic deformation, modelling, grain shape

 

V. Al. Dzhupanov

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria,

e-mail: dzhupan@imbm.bas.bg

 

TWelve METHODICAL NOTES ON THE Paradoxical results

in a class of dynamical problems

(Part 1)

Abstract. The dynamics of a complicated mechanical system consisting of three components, namely: (i) cantilevered pipe, (ii) high speed flowing fluid conveyed, and (iii) - linearly elastic resisting foundation is under methodical analytical investigation. The nature of the just accounted components is considered, and the need to introduce a structural resisting base is deduced. Samples of structural supports and their responses are shown. The end effects of three pairs of supported beam-like pipes are demonstrated by their continuous diagram (fixed + free; hinged + free; and free + free). The general follower character of the PDE modeling the case under consideration is underlined. Two theorems based on a purposeful use of the Symmetry Principle of infinite canonical structures motivate the Winklers concept on the resisting base. The concept of “critical combination of subcritical parameters” is formulated and discussed. Using the known (from the Hydroelasticity) concept of the added liquid mass, the concept of the joined rigidity (added stiffness) due to the dynamic structurefoundation interaction is introduced. By using the latter, as well as the concept on the local and global stability of the pipe, the dependence of the critical follower force on the stiffness of the resisting medium is proved - in contrast to some paradoxical results for the dynamic stability of the Dzhanelidzes column lying on elastic foundation (considered in the basic text). Because of its methodical character, the entire text had to be exposed into two parts (see the Foreword).

 

 

Key words: cantilevered pipe; fluid conveyed, resisting medium (foundation), added mass; added stiffness; local stability; global stability, critical velocities.

 

 

P. A. Djondjorov

Institute of Mechanics, Bulgarian Academy of Sciences,

 

Acad. G. Bonchev Str., Bl.4, 1113 Sofia, Bulgaria,

e-mail: padjon@imbm.bas.bg

 

Dynamic Stability of pipes partly resting on winkler foundation

Abstract. The present study concerns with the dynamic stability of straight elastic cantilevered pipes conveying inviscid fluid. A part of the pipe span is assumed to rest on Winkler foundation. The objective is to examine the influence of the foundation length, position and rigidity on the critical velocities of such pipes. Numerical solution of the associated boundary value problem is suggested, applying Galerkin method. It is established that for small ratios of the masses of the fluid and the pipe, the maximal stabilizing effect of a Winkler foundation of certain length is achieved if it supports the free end of the pipe. For higher values of the mass ratio a foundation applied at the free end can either stabilize or destabilize the pipe depending on the foundation length. An elastic foundation applied at the midpoint is found to have a stabilizing effect on the pipe.

 

Key words: fluid conveying pipes, Winkler foundation, dynamic stability, critical flow velocity, Galerkin method.