Journal of Theoretical and Applied Mechanics

Volume 34, Number 1, 2004

### Contents

Institute of Mechanics, Bulgarian Academy of Sciences,

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

e-mail

# Institute of Biophysics, Bulgarian Academy of Sciences,

Acad. G. Bonchev Str., Bl. 21, Sofia 1113, Bulgaria,

An Approach to Diagonalizing Real 3x3 Symmetric Matrix and Mechanical Applications

Abstract. An easily tractable analytical method of casting a real three-dimensional symmetric matrix into its diagonal form along with explicit formulae for the corresponding eigenvectors is given. This is achieved by two steps relying on a nice geometrical parameterization of the rotational group SO(3) and the fundamental algebraic theorem about solutions of the polynomial equations. Results for the mass inertia matrix in rigid body mechanics and special systems of linear differential equations are examined in some detail. Another aim is the approach to be applied to obtaining a new form of dynamical equations of multi-body mechanical systems, which is quite convenient for the control process.

Key words: eigenvalues, eigenvectors, explicit solutions.

E. V. Zahariev

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

e-mail:

GENERALIZED NEWTON – EULER EQUATIONS IN RIGID AND FLEXIBLE MULTIBODY SYSTEM DYNAMICS

Abstract. In the paper a general approach to deriving the dynamic equations of motion of rigid and flexible multibody systems is presented. An algorithm for differentiation of the Lagrange’s equations is developed. Three-dimensional matrices are applied for working out the configuration space dynamic equations obtained due to differentiation with respect to the generalized co-ordinates, velocities and time. Finite elements are used for modelling of flexible systems. The mass properties are presented by dense 6 × 6 mass-matrices used in finite element discretization and other kinds of mass reduction. Generalized Newton-Euler equations and relations for inertia forces in the nodes of flexible elements are proposed. The application of the equations derived together with the principle of virtual work gives similar results as these obtained directly from the Lagrange’s equations. An example of large flexible deflections of slender beam due to impact over a rigid base displays the effectiveness of the method.

Key words: multibody systems, dynamics. Newton-Euler equations.

# Al. B. Kazakoff

Institute of Mechanics, Bulgarian academy of Sciences,

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

e-mail: alex_kazakoff@yahoo.co.uk

SHAPE FACTOR INFLUENCE ANALYSIS OF VISCOELASTIC MOUNTS  SUBJECTED TO LARGE AMPLITUDE DEFLECTION

Abstract. This paper describes an investigation into the use of the Finite Element (FE) method of predicting large (up to 80% strain) static deformations of “benchmark” elements in the form of viscoelastic solid bodies manufactured in a variety of shapes. For this sake twenty simple “benchmark” models were designed and manufactured, utilizing silastic-j silicon rubber and consisting of ten different shapes with two different lengths for each shape. The cross sectional area of every single sample is the same, while two lengths of the samples for each shape is investigated.

An approach to defining the influence of the shape factor on the static behaviour of the various models is assumed. The objective of this paper is to establish the influence of the shape factor and the “buckling” effect on the non-linear quasi-static response of  “benchmark” rubber mounts utilizing viscoelastic materials and particularly triangular and three point star shape, and subjected to high strain levels of loading. Dynamic transient responses excited by half-sine shock impacts are measured and compared with the optimal mesh prediction and partially validated by modal analysis.

Key words: shape factor analysis, finite element analysis, experimental tests, modal analysis, strain rate, and transient response predictions.

K. Delchev

Institute of Mechanics, Bulgarian Academy of Sciences,

### ROBUST LEARNING CONTROL WITH GIVEN TRACKING ERROR

OF ROBOT MANIPULATORS

Abstract. An algorithm of learning-operator synthesis for considered robust learning control with preliminarily given norm-bound of tracking error of robot manipulators is presented. The algorithm is based on identification of the inertia matrix of robot-arm dynamical equations. A sufficient condition concerning the identification accuracy for robustness and convergence of the self-learning iterative procedure, which leads to feed forward control with desired tracking error, is proven.

Key words: robot manipulator, dynamical equations of motion, learning control, robustness, learning operator, identification and tracking error.

Ch. Tsvetkov

University of Mining and Geology “St. Ivan Rilski”,

#### e-mail: khc@mail.mgu.bg

SPHERICAL HYDROSTATIC BEARING WITHOUT CLEARANCE

Abstract. Hydrostatic bearings guarantee liquid friction of the bearing at heavy duties in a wide range of speeds, including the zero speed. The present paper aims to study the axial spherical hydrostatic bearings without clearance and a circular central chamber. The high loading capacity of that type of bearings makes possible their application for heavy radar antennas, optical and radio telescopes, cone crushers and other up-to-date equipment, working at low speed and under high loads.

The present study performs an analytical investigation of the functional relation between geometrical dimensions, flow rate Q and viscosity m of oil, clearance d, pressure ð and loading capacity Ðó of hydrostatic bearing. The analytical investigations concern the spherical hydrostatic axial bearing without clearance and a central circular chamber. Final formulae are obtained, which allow technological and constructive calculation of the bearing design. Diagrams of maximum pressure p1 and minimum clearance dmin in the bearing, depending on loading and its geometrical parameters are presented. Diagrams and particular values of some individual parameters refer to the spherical bearing of the movable cone of a cone crusher of KUBRIA-210 type of the KRUPP Company.

##### Key words: hydrostatic bearings, pressure, clearance.

J. Turza

Faculty of Materials and Technology in Trnava,

Slovak University of Technology in Bratislava, Slovak Republic,

e-mail: dpmtf@ztsdubnica.sk

PARAMETER PREDICTION trend OF AXIAL PISTON VARIABLE DISPLACEMENT MOTORS

Abstract. A prognosis method of basic parameters of axial piston variable displacement motors is referred to the accession. Parameter option is taken like “Black box” in the sight of motors. These parameters serve like main parameters of basic technical conditions for a new development.

Key words: prediction trend, axial piston hydrostatic variable displacement motors, ground parameters, derivative parameters, comparative parameters.

# A.    Baltov

Institute of Mechanics, Bulgarian Academy of Sciences,

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

e-mail: baltov@eagle.cu.bas.bg

MATERIALS SENSITIVE TO THE TYPE OF THE PROCESS:

A REVIEW

Abstract. Several special effects have been well known for years, concerning sensitivity of materials to:

·       tension or compression; torsion in different directions; different stress combinations, and so on (sensitivity to the type of loading);

·       extension or compression and so on (sensitivity to the type of deformation);

·       creep or relaxation, processes with different strain rates and so on (sensitivity to the type of rheological processes).

There are many successful propositions to modelling these different types of sensitivity for every particular case. This presentation proposes a new approach to generalizing all these events using a unified model.

The model is based on the introduction of extended space of events, including stress and strain subspaces, and time scale (or the well known “internal time scale”). Some different subspaces with “single-type process” ought to be identified. In every “single-type process”, subspace-one material model is preferable. Special attention is paid to the transition from one subspace to another. In different experimentally examined conditions it is possible to introduce some simplified quadratic surfaces of constant stress and strain states (measured by their intensities at fixed times). This approach is applied in different characteristic cases. A discussion is presented on the important problem of the solution of the system of equations for material behaviour in case of complex multi-type processes.

Key words: sensitivity, space of events, numerical procedure.

S. P. Mandjakov, N. G. Gospodinov

Department of Strength of Materials, Technical University of Sofia,

8, Kliment Ohridski Blvd., 1756 Sofia, Bulgaria

A Non-Linear Solution in the Theory of Small Elasto-Plastic Strains

Abstract. The influence of quadratic components of the volumetric strain in solving the classic problem is discussed in the present work. From the analysis of expressions obtained for the case of elastic and perfectly plastic body and their comparison with the conventional expressions it is obvious, that when the quadratic components of the volumetric strain are also taken into consideration, the quantities being searched for vary in accordance with the predictions of the energetic theory in comparison with the theory of maximal tangential stresses.

Key words: volumetric strain, quadratic components of the volumetric strain, elastic and perfectly plastic body.