Issue 1
JTAM, Sofia, vol. 4 Issue 1 (1973)

On an Algebraic Condition for Periodic Trajectories in Autonomous Systems of Differential Equations with Polynomial Non-linearities of a Definite Class

G. Bradistilov1, Sp. Manolov2
1Sofia, United Centre of Mathematics and Mechanics, Geo Milev district, Bl. VIII
2Higher Institute of Mechanical and Electrical Engineering, Durvenitsa


Autonomous systems of differential equations with small parameter, the non-linear parts of which comprise polinomials of a definite class, are investigated. The motions of a number of mechanical configurations are described by means of such systems. The cases of internal resonances are considered when the initial conditions depend in a definite way on a parameter N. A condition sufficient for the existence and the construction of corresponding periodic trajectories is found. This condition represents an algebraic equation towards the parameter N and its degree x(k, q) is a function of the resonance k and of the degree q of the polynomials used. The basic problem for determining the x(k, q) function of the resonance k and of q as well as the type of the algebraic condition for periodicity, is set. In this connection some concrete cases concerning q have been studied.

JTAM, Sofia, vol. 4 Issue 1 pp. 009-020 (1973)

Constitutive Visco-plastic Equations of Metal Single Crystals

Zh. Zarka
Chargé de Recherches an C.N.R.S., Laboratoire de Mécanique de l'Ecole Polytechnique, Paris 5

Constitutive equations describing visco-plastic properties of metal single crystals are proposed. The elastic and plastic components of deformations in single crystals are determined on the basis of microstructural analysis. A suitable geometric description of the deformation process is given. An equation determining the speed of the plastic deformation is proposed and a visco-plastic potential is introduced.

JTAM, Sofia, vol. 4 Issue 1 pp. 021-030 (1973)
Oscillations and Stability of Cylindric Membranes, Subjected to Finite Deformations

M. Kozarov, Al. Rachev
Sofia, Institute of Technical Mechanics, Bulgarian Academy of Sciences, Geo Milev distr., Bl. IV

Oscillations of a circular cylindric membrane with small amplitudes are considered. The membrane has been preliminary subjected to finite deformation. The material is considered elastic, isotropic and non-linear of most general type. The oscillations are examined as an adiabatic reversal process, imposed on the initial isotermic deformation. The equation of the eigen frequencies is obtained, when the edges of the basis of the membrane are freely supported. The equation of stability is deduced on the dynamic criterion, the total geometric non-linearity of the next-to-critical state being taken into account.

JTAM, Sofia, vol. 4 Issue 1 pp. 031-044 (1973)
Parametric Stability of the Main Shaft of Chip-removing Machines

D. Mangeron1, M. Oguztiorelli2, S. Chiriakesku3
1Technische Hochschule zu Jassy, Iasi Rumanien
2Dept. Math, University of Alberta, Canada
3Universitat Brasov, Rumanien

The problem for stability of oscillations in metal cutting machines during their operation is considered, and a general formulation of the task is given. It is accepted that the motion of the considered mechanical system is described by means of independent linear non-homogeneous equations with periodic coefficients. The stability of the periodic solutions of the considered equations is examined.

JTAM, Sofia, vol. 4 Issue 1 pp. 045-056 (1973)
On the Solution of the Plane Problem of the Elasticity Theory for an Uniform Anisotropic Rectangular Region

U. Stojchev
Russe, "9 Septemvri" st. 70, Entr. B

The Solution of the first boundary problem of the elasticity theory is sought for an uniform anisotropic rectangular region at arbitrary boundary conditions expressed with strains, the mass forces being taken into account. The strains are presented in double trigonometric series.

JTAM, Sofia, vol. 4 Issue 1 pp. 057-062 (1973)
A System of Equations for the Investigation of Spatial Pending Constructions at Static and Dynamic Loading and Temperature Variation

A. Popov
Sofia, Higher Institute of Civil Engineering, Blvd. "Chr. Smirnenski", 1

System of equations for the investigation of a spatial net of filaments subjected to arbitrary static or dynamic loadings and temperature variation is deduced on the basis of most general premises. The total translations of the knots are taken into account and the slopes of the filaments are not limited. Thus the system is valid in the case of not-bent nets as well. In comparison with equations [4] and [5] used at similar premises the proposed ones are more precise, which is due to the lack of additional simplifications during the deduction and the uniform treatment of all small magnitudes from the same order. The higher accuracy does not principally affect the possibilities for and the complexity of the solution.

JTAM, Sofia, vol. 4 Issue 1 pp. 063-072 (1973)
Model Investigation of Corrugated Shell Constructions on Circular Base

K. Yamboliev, M. Kmetov
Sofia, Institute of Engineering Cybernetics, Bulgarian Academy of Sciences, Geo Milev distr., Bl. IV

Some experimental investigations of a model type corrugated reinforced glass-plastic shell construction on circular base are surveyed and the experimental methods and the results obtained are given. Stresses in different points of the hip and valley are obtained as well as the deflections of the shell in its most typical points. For the existing geometric dimensions, loading and supporting conditions, the shell is calculated at a premise that practically it works in meridional direction. On this basis the shell is calculated as a three-hinged arch. The obtained experimental results confirm the appropriateness of the accepted premise. The similarity theory is used for the transfer of the model results on the big shell. Conclusions for the stress and strain state of this type of corrugated shell on circular base, made of reinforced glass-plastic, are drawn.

JTAM, Sofia, vol. 4 Issue 1 pp. 073-084 (1973)
Equilibrium Orientations of Symmetric Gyrostat Satellites with a Specified Internal Angular Momentum

A. Anchev
Mathematical Institute with Calculating Center, Bulgarian Academy of Sciences, Geo Milev distr., Bl. VIII

All possible eguilibrium orientations are found for a gyrostat satellite with dynamic symmetry in case of an internal angular momentum arbitrarily specified. The satellite is orbiting in a circular orbit and only gravitational torques are present. The elements of the matrix which determines the orientations of the satellite in the orbital reference frame are expressed by the elements of the inertia matrix and by the projections of the internal angular mo¬mentum vector on the principal axes. The sufficient conditions for stability of the equilibrium orientations considered are found by means of Liapunov's methods.

JTAM, Sofia, vol. 4 Issue 1 pp. 085-093 (1973), [Full Article]