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Mathematical modeling in applied mechanics and control

A mathematical model in mechanics is a system of equations allowing to study a mechanical system with required accuracy.

Mathematical modeling of a controlled mechanical systems has a number of specific features:

• The controlled mechanical systems are of great variety. A car, a gyro stabilizer, a walking robot, a simulator of space flight, etc. consist of various rigid constructive elements, devices, gauges, engines. From the point of view of theoretical mechanics all these objects can be considered as systems consisting of a big number of rigid bodies. Attempting to construct a mathematical model of such an object by the means of classical theoretical mechanics, for example in the form of Lagrange equations, usually leads to unimaginably bulky equations with hundreds and thousand of terms. Thus an approach to modeling which describes the necessary phenomena with comprehensible accuracy for a precisely set class of motions.
• For the systems in mechanics of controlled motion strong attenuations of high-frequency components are typical. The approximate modeling of such systems gravitates toward the Poincare decomposition and the Tikhonov and Vasilyeva boundary-layer methods.

The department has significant achievements in mathematical modeling of controlled mechanical systems.

The technique of fractional analysis (I.V. Novozhilov) was developed and introduced to practice. The technique is used for creating approximate mathematical models describing separate components of motion in various time or spatial scales. The fractional analysis of a concrete system is carried out in two stages. On the first stage methods of the dimension theory are applied to normalize the equations and to enter small parameters adequate to the studied class of motion. On the second stage asympthotic methods are applied to create an approximate mathematical model.

Estimates of the errors of asymptotic approximations for regularly and singularly perturbed systems with a small parameter and of the time intervals on which these errors are guaranteed were received (R.P. Kuzmina). These estimates were applied to concrete dynamic systems (A.V. Vlahova).

A technique of drawing up an approximate mathematical model of slow motion components for close to conservative mechanical systems with strongly different frequencies was suggested (A.V. Vlahova, I.V. Novozhilov).

A technique to study dynamic systems with non-continuous right parts was developed. Unlike known approaches, a mathematical model of sliding movements and conditions of their realization are determined with the singular perturbations methods (A.V. Vlahova, I.V. Novozhilov).

For regularly and singularly perturbed systems non-iterative methods of approximation in a small parameter were offered. With this approach the order of the approximate models does not surpass the order of the initial system (A.V. Vlahova, I.V. Novozhilov).

The staff of DAMC is one of leaders in Russia in the gyroscopic systems theory. B.V. Bulgakov's and A.Yu. Ishlinskiy's works became classic in gyroscopy. The research was continued by Ya.N. Rojtenberg, I.V. Novozhilov, E.A. Devjanin, A.I. Kobrin, Y.G. Martynenko, V.I. Borzov, V.V. Tikhomirov, N.P. Stepanenko.

Fractional analysis was used to validate some of the classical mechanics models: precession gyroscopy model, absolutely rigid body, holonomic and non-holonomic constraints (I.V. Novozhilov).

Needs of practice have led A.Ju.Ishlinskiy to formulate a model new to classical mechanics - a rigid body rotating on a string. Unlike a rigid body rotating around a fixed point, here there are no integrable cases, as in the Euler, Lagrange and Kovalevskaya cases. A.Yu. Ishlinskiy together with his pupils investigated bifurcations of stationary movements and their stability.

Applying fractional analysis to the theory of winged aircrafts, V.I. Borzov and I.V. Novozhilov constructed approximate mathematical models of flight dynamics describing fast and slow movements, and estimated the approximation error.

A model of deformable wheel interacting with the road, generalizing Carter's, Rokar's, Fromm's and Keldysh models was suggested. Approximate mathematical models for different classes of motions of a car were constructed (I.V. Novozhilov, P.A. Kruchinin, M.H. Magomedov, I.S. Pavlov). Mathematical models of cross-section " kinematic waggings" for a railway car and a train, taking in mind interaction of the crest of a wheel pair with the head of a rail (I.V. Novozhilov, I.A. Kopylov, A.V. Vlahova, V.N. Phillipov) were proposed.

Among research done on vibrations in the suspension of a car we note results on parametrical excitation of driving wheels, modeling of active suspension dynamics, etc. (S.I. Zlochevsky, A.D. Derbaremdiker, P.A. Kruchinin). Conditions for parasitic fluctuations in anti blocking systems of cars and algorithms for their suppression were suggested (I.V. Novozhilov, P.A. Kruchinin, M.H. Magomedov). These results were used to develop an anti blocking system NPF SAUNO at the Korean Electrotechnical Institute (Pusan, South-Korea).

Mathematical modeling of multi link controlled systems, such as walking robots and multi-DOF dynamic stands was done. The approximate models of such devices describing slow desired modes of motion and fast stabilization modes (E.A. Devjanin, A.M. Formalsky, I.V. Novozhilov, Yu.V. Bolotin, I.V. Bardushkina) were developed.

Power consumption of four-legged and six-legged robots depending on the pace and the kinematic scheme of legs were investigated by I.V. Novozhilov, M.F. Zatsepin, A.V. Panshina. Research of optimal biomechanical power consumption was done (Yu.V. Bolotin). The corresponding optimization problem for a biped device was solved, bifurcation diagrams of a gait type depending on two dimensionless parameters - the Frude number and the relative weights of the legs were obtained. A theory of statically unstable walking robots with dynamically stable gates was suggested (Yu.V. Bolotin).

A new direction of studies at DAMC is connected with mathematical models of bio-mechanical systems. Extensive research on receptors of accelerations of a man was done. The focus is on imitation of the feelings of pilots and cosmonauts in real flight. Research is done in cooperation with the Cosmonaut preparation center (V.V. Alexandrov, T.G. Astakhova, N.V. Kulikovskaja, G.V. But, Yu.O. Mamasueva, N.E. Shulenina).

Together with the Children's Psycho-neurological hospital N 18, DAMC works on mathematical modeling of movements of patients with the children cerebral paralysis diagnosis. Mathematical modeling of complex bending - unbending movements for changing vertical poses was done. These movements were proved to be answered with small changes of two-articulate muscles lengths, i.e. some kind of invariancy of lengths was found. Thus these movements are realized by a rather simple control within the limits of A.G. Feldman's model (I.V. Novozhilov, A.M. Zhuravlyov, P.A. Kruchinin, I.A. Kopylov, P.P. Dyomin, A.A. Grishin).

Significant results were obtained in mathematical modeling of the large-scale patterned whirlwinds, so-called coherent structures. Exampls are the "cloudy streets" in the atmosphere, "circulations of Lengmure " in the seas and lakes, "periodic large whirlwinds" in the jets behind the jet engines, etc. the Constructed models of heat and mass transfer in the turbulent streams use methods of fractioning of components of motion (A.E. Ordanovich, N.S. Blohina, L.A. Mihajlova).

Mathematical models of systems of the type "body - cable " were developed, allowing to calculate configurations, large-scale movements and small fluctuations for common cable devices (cabled flying sensors, towed systems, anchored chisel platforms, etc.). One feature of such systems is that the equations describing motions of a body in a resisting media are complemented with equations in partial derivatives describing the cable. Special attention was given to cable systems where the cable smashing and straightening induces impacts and loops (A.E. Ordanovich, M.V. Los).

A number of problems on stability of stationary motions of a satellite under gravitational, magnetic and aerodynamic moments were solved. Stability of complex mechanical systems consisting of firm and deformable bodies, such as space vehicles and satellites was investigated (V.M. Morozov).

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