1 semester


Name of discipline

Multibody dynamics


Course, specialty

Undergraduates, specialty “Mechanics”


Training semester



Quantity of the credits



Full name lecturer

PhD in Physical and Mathematical Sciences N.A. Dokukova


Purposes of studying of discipline

–    assimilation of the basic principles of modeling of movements of multielement mechanical systems, technical objects;

–    high-quality estimation of types of mathematical models of dynamic multielement systems;

–    ability of use of knowledge of linearization of nonlinear oscillatory systems;

–    finding of solutions of the received mathematical models;

–    a research of models of dynamic systems by means of phase charts;

–    development of abilities in a research on stability of oscillating motions of multielement mechanical systems.

As a result of studying the student has to be able:

–   to apply the gained knowledge to assessment of efficiency of the chosen models, methods of calculation and optimum parameters of multielement dynamics system of mechanical objects;

–  to solve problems of all sections of a course.



Theoretical mechanics, resistance of materials, mechanics of a deformable solid body, mechanics of the continuous environment


Content of discipline

The basic concepts and definitions in the theory of fluctuations of mechanical systems

Fluctuations and the sizes characterizing them. Modeling of oscillatory systems. Settlement schemes and equations of the movement.

Fluctuation of systems with final number of degrees of freedom. Own frequencies and forms of fluctuations.

Images of fluctuations on the plane.

Types of fluctuations and their signs. Own, compelled, parametrical, connected, self-oscillations. Subharmonics, superharmoniscs and main oscillations.

Transitional functions.

The vector image of fluctuations in the complex plane. Addition of two harmonic oscillations with identical frequencies. Addition of two fluctuations with any frequencies and amplitudes.

Phase trajectories and phase portrait. General features of phase trajectories. Balance position, special points.

Characteristics of external and internal forces

Characteristics of friction forces. Amonton’s hypothesis, Pendant formula. Pure friction, dry, boundary, liquid. Friction forces in kinematic couples. Characteristics of pulse and striking powers.

Characteristics of elastic forces. Linear, rigid, soft elastic forces. Nonlinear elastic force like “gap”.

Characteristics of driving forces and forces of resistance. Inelastic resistance.

Constructional hysteresis. Losses on internal friction in material.

Methods of solutions of the nonlinear equations of the movement of multielement mechanisms

Methods of solutions of the nonlinear equations of the movement of mechanisms. Linearization of nonlinear characteristics of forces.

Linearization on small sites of the characteristic and on big sites.

Application of mean square approach, linearization on a discrete set of broken lines.

The theory of the best approach across Chebyshev.

Method of small parameter. Quasilinear equations of the movement of mechanisms. The generating equation. Use of small parameter for the quasilinear equations, which right part, is obvious function of time.

Use of small parameter and method of consecutive approximations for the quasilinear equations.

Use of small parameter for autonomous systems.

Method of harmonious balance.

Galerkin’s method.

Method of slowly changing parameters Wang-der-Polya. Fluctuations of a circular saw of friction.

Chebyshev’s linearization. Fluctuations in the mechanism with the nonlinear elastic coupling. Nonlinear AFC. Unstable operating modes of the mechanism, failure of amplitude.

Methods of the analysis of oscillatory mechanical systems

Matrix methods of the analysis of oscillatory mechanical systems.

Application of a method of integrated transformations of Laplace for the solution of the linear equations of the movement of the mechanism. Decision, phase portrait, amplitude-frequency characteristic.

Main definitions and concepts of the theory of an impedance

Full mechanical resistance, mobility, dynamic firmness, dynamic susceptibility, coefficients of transfer and vibration insulation. Mechanical impedance of rigidity of a spring, viscous damper, weight.

Equivalent systems. Physical elements. The given elements.

Vibration insulation bases

Vibration insulation calculation bases. Dynamic clearing of fluctuations. AFC. Dynamic quencher with friction. Transfer function.

Calculation of limiters of scope of vibrations fluctuations at casual overloads.

Criteria of stability of the oscillatory modes

Criterion of stability of the oscillatory regime of Mikhaylov.

Criterion of stability of the oscillatory regime of Nyquist.

Algebraic criteria of stability of Raus and Gurvits.

Algebraic criteria of stability of V.S. Voronov.


The recommended literature

1.     Andronov A.A., Vit A.A., Khaykin S. E. Theory of fluctuations of M.: Science, 1981.

2.     Biderman V.L. Thorium of mechanical oscillations. M.: Science, 1980.

3.     Bogolyubov N.N., Mitropolsky Yu.A. Asymptotic methods in the theory of nonlinear fluctuations. M.: Science, 1974.

4.     Vibrations in the equipment. Reference book. M.: Science, T. 2-6, 1981.

5.     Levitical N.I. Fluctuations in mechanisms. M.: Science, 1988.

6.     Magnus K. Fluctuations. M.: Science, 1982.

7.     Cyril M. Harris, Charles I. Crete. Reference book on shock loadings. L.: Shipbuilding, 1980.


Teaching methods

Lectures, laboratory works with attraction of software, practical training with the solution of tasks


Training language



Conditions (requirement), current control

– laboratory works;

– performance of tasks on a practical training.

At examination, the acquired skills taking into account rating are estimated: 35% – laboratory works; 5% – the performed tasks on a practical training; 60% – the oral answer.


Form of the current certification