Journal of Theoretical and Applied Mechanics

Volume 34, Number 2, 2004

# B. Marinov

Institute of Mechanics, Bulgarian Academy of Sciences,

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

e-mail

ON THE BEHAVIOUR OF POWER TRANSMISSION LINES IN VEHICLES SUBJECTED TO COMPLEX LOADING

Abstract. The power transmission lines are complex mechanical systems. They serve to deliver the working torque from the engine to the working machine. This paper presents analysis of the behaviour of power transmission lines in some vehicles subjected to complex loading. The influence of inertial and elastic characteristics of each unit is studied when applying loads on the one hand, and during the system operation in a resonance regime on the other hand. The author proposes an algorithm for designing of power transmission lines that would operate in a stable regime even under most unfavorable dynamic loads. A numerical example is derived to verify the developed theory.

Key words: engine, machine, resonance, frequency, dynamics, force, torque, axial stress.

R. Blagoeva

Institute of Mechanics, Bulgarian Academy of Sciences,

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

email

ON A NUMERICAL APPROACH TO SOLVING A DIFFUSION WITH RELAXATION PROBLEM

Abstract. A recently proposed numerical approach to solving an initial boundary value problem for diffusion with relaxation in polymers [1] is considered. The correctness of two non-linear numerical schemes is studied and some sufficient conditions for their stability are derived. These conditions representing constraints for the time step in respect to some model parameters assure the solution physical correctness. The present research improves the accuracy and stability of the time difference schemes used in combination with a finite element domain approximation in modelling the penetrant diffusion with relaxation in a polymer matrix.

Kew words: diffusion with relaxation, non-linear numerical scheme, finite element method, time difference method, scheme stability, physical correctness.

Ts. P. Ivanov

Faculty of Mathematics and Informatics,

University of Sofia St. Kl. Ohridski,

5, James Bourchier Blvd., 1126 Sofia, Bulgaria,

e-mail

# R. Savova

Institute of Mechanics, Bulgarian Academy of Sciences,

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

e-mail

### Stability of Relative Equilibria of an Elastic Top

Abstract. Stability of relative equilibria of an elastic top with stress-free surface and a fixed point moving in a gravitational field is considered.  The Koiter's definition for stability with respect to the deformation due to the gravitation and an arbitrary rotation and the usual Lagrange definition for stability of the motion of the deformed top as a rigid one are adopted. Relative equilibrium states are determined and criteria for stability are proved. The obtained results are applied for the case of a sleeping heavy when the top is an elastic circular cylinder.

Key words: stability, elasticity, relative equilibrium, top.

University of Architecture, Civil Engineering and Geodesy,

1, Hr. Smirnenski Str., 1046 Sofia, Bulgaria,

e-mail

### free vibration of skew plates

Abstract. On the basis of the finite strip method a methodology of obtaining the eigenvalues and eigenvectors of skew plates is worked out. A computer program developed by the author was applied in the investigation of the flexural vibration of skew plates for various skew angles and various support conditions. The first six eigen frequencies of skew plates with four different support conditions are obtained. The results presented here are compared with solutions obtained by the other authors.

Key words: plate, free vibration, finite strip method.

# A. Nedev, N. Nikolov

Institute of Mechanics, Bulgarian Academy of Sciences,

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

e-mail: n.nikolov@imbm.bas.bg

# I. Altaparmakov

Technical University of Sofia,

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

e-mail

BLANK DIAMETER ANALYSIS FOR DIFFERENT METALS DURING DEEP DRAWING PROCESS

Abstract. The influence of blank diameter increasing over critical one, calculated theoretically by the permissible coefficient of deep drawing, is investigated. Forming process of cup piece with inside diameter 50 mm and thickness 1.5 mm produced from different materials is the object of this paper. The changes of diameters are within the interval 80 – 117mm. The materials used in simulations of deep drawing processes are Steel 08, Steel 10, Aluminum, Copper and Brass 63. Simulations of the forming processes with an optimal designed forming tool are performed using an Approximate Discrete Method (ADM). The values of punch and die roundness radii and gap between punch and die are taken as optimal ones obtained under a condition characterized by equal weight coefficients of importance of drawing force to punch travel.

The results are presented by diagrams “Drawing force – Punch travel” (“P-S”-diagrams). In some cases numerical results obtained by ADM are compared with the same ones obtained by Finite Element Method (FEM), and for Steel 08 by experiment also. Similar “P-S”-diagrams are obtained for the forming process of steels, aluminum and cooper blank. Different kind of the brass blank forming is obtained, and some reasons for that are given. The influence of flange sizes upon the character and magnitude of “P-S”-diagrams during forming is shown. An increase 2.7% of blank diameter over the theoretically calculated critical one appears for the Aluminum and Copper. The obtained numerical results can be used for next planning of real experiments and enlargement of the investigations in this field.

Key words: deep drawing process, approximate discrete method, finite element method.

###### V. Yanakieva, G. Stefanov

Institute of Metal Science, Bulgarian Academy of Sciences,

67, Shipchenski prohod Str., 1517 Sofia, Bulgaria,

e-mail: veneta@ims.bas.bg

# K. Klyavkov

University of Chemical Technology and Metallurgy,

8, Kliment Ohridski Blvd., 1756 Sofia, Bulgaria,

e-mail:

An improvement of heat treatment conditions of springs made of U8A steel

Abstract. The heat-treatment conditions for springs made of U8A steel with special shape and function have been studied. The investigation concerns the effect of heat treatment on hardness, tensile strength and microstructure. Several methods of heat-treatments have been applied for producing U8A steel made springs with optimal working properties. The optimal properties have been obtained after isothermal hardening.

Key words: spring steel, heat-treatment, microstructure.

# V. Petrov

Institute of Mechanics, Bulgarian Academy of Sciences,

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

e-mail: valko@imbm.bas.bg;

J. Timmer

Centre for Data Analysis and Modelling, University of Freiburg,

1, Eckerstr. 79104 Freiburg, Germany,

e-mail: jeti@fdm.uni-freiburg.de

### Bifurcation analysis of a cell mitosis control model

Abstract. The well-known dynamical model of Novak and Tyson for mitosis (M-phase) control in fertilized Xenopus oocytes is analyzed qualitatively and quantitatively. For this purpose the two nonlinear ordinary differential equations of the model are transformed in a canonical form centered in a steady state solution. On the base of determining Lyapunov value of the steady state solution at the bifurcation point it is concluded that a stable limit cycle emerges or vanishes depending on the direction of parameter variation. This paradigmatic effect in the non-linear dynamics (Hopf bifurcation) is interpreted in terms of the biochemical kinetics of the cell cycle.

Key words: dynamical system, bifurcation, cell mitosis.