The theory behind analytical algorithms for computer-algebra systems is discussed. A computer system is described which is capable of optimizing input data, constructing equations of equilibrium and motion, formulating and solving the basic mechanics problems for a broad class of holonomic systems with elastic and dissipative constraints on the basis of the Lagrange-D'Alembert principle Keywords: computer-algebra system, equation of motion, holonomic system, Lagrange-D'Alembert principle Problem Formulation. The advent of computer-algebra systems (CASs) stimulated the development of problem-oriented systems (POSs) for PCs with the purpose of research automation and solution of the basic mechanics problems. The major results obtained in this filed are reported in [2][3][4][5][6][7][8][9][10][11] and pertain to discrete mechanical systems modeled by rigid bodies with elastic, dissipative, kinematic, and geometrical constraints. However, no unified approach to the problem has been worked out yet. Developments in the field are based on different principles of analytical mechanics [5]. Because of the specific nature of analytic transformations on PCs, methods having proved themselves well for manual use (Lagrange equations of the second kind, for one) may appear poorly suited for computer use. The classical and original methods may produce better results [3,6].The present paper discusses algorithms for computer-aided analytic representation and analysis of holonomic rigid-body systems with elastic and dissipative constraints based on the D'Alembert-Lagrange principle, proposes a theory and analytic algorithms for CASs and POSs, and describes a computer system that can optimize input data, construct analytic equations of motion and equilibrium, and formulate and solve the basic mechanics problems.1. Analytic Description of a Mechanical System. Let us consider a holonomic mechanical system consisting of n bodies, each having mass m i and principal central moments of inertia J ix , J iy , and J iz . The velocities of the centers of mass are defined in projections onto the axes of an absolute coordinate system (CS); and the angular velocities of the bodies, in projections, ω ix , ω iy , and ω iz , onto the principal central axes of the body-fixed CSs. The kinetic energy of the system is

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We present results of the investigations on flexural-flexural-torsional nonlinear vibrations of twisted rotating beams described by the system of three nonlinear integro-differential equations expressed in partial derivatives. Cross-sectional deplanation of beams is adjusted to the equations and it is assumed that centre of gravity and shear centre are located at different points. Vibration systems are expressed as a row by free forms of the linear problem. Free vibrations are studied with the help of the Show-Pierre nonlinear normal mode.Keywords: flexural-flexural-torsional vibrations, nonlinear normal mode method, backbone curves. Introduction.Rotating beams are the components of the helicopter blades, manipulators, gas and steam turbine blades and propeller blades. In service, such beam structures often make oscillations and cause fatigue damage. Such vibrations are difficult for studying because their cross sections are asymmetric. As the cross section is asymmetric, the centre of gravity and shear centre of a beam cross-section are not assumed to coincide.The attempts to study nonlinear vibrations of beams with asymmetric cross section have been already undertaken. Timoshenko [1] has derived the equations of linear flexural-torsional vibrations of direct untwisted beams with asymmetric cross section. Vorob'ev and Shorr [2, 3] obtained the equations of linear flexural-flexuraltorsional-longitudinal vibrations of twisted rotating beams with cross-sectional deplanation for displacements and under torsion. System of the equations expressed in partial derivatives which describes geometrically nonlinear flexural-flexural-torsional-longitudinal vibrations of rotating beams is presented in [4]. In these equations, an assumption is made that centre of gravity of the cross section and shear centre are located at the same point. Crespo da Silva [5, 6] obtained the equations which describe flexural-flexural-torsional vibrations of beams with allowance for inextensibility of the median line, assuming that centre of gravity and shear centre are located at the same point. The theory of flexible beams is described in [7]. We introduce the system of three nonlinear integro-differential equations expressed in partial derivatives which describe flexural-flexural-torsional vibrations of rotating beam with allowance for cross-sectional deplanation. In deriving the equation, it was assumed that center of gravity and shear centre do not coincide. Discrete nonlinear dynamic system which approximates the indicated vibrations has been obtained. Motion of system is studied with the help of a nonlinear normal mode method. The effect of cross-sectional deplanation on vibrations is analyzed.Vibration Equations. A rotating beam with constant angular velocity W is considered (Fig.

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