Aggregation of identical mechanical systems with oscillations

We consider identical mechanical systems described by Lagrange’s equations of the second kind and subject to the action of positional forces. It is assumed that a separate system allows for single-frequency oscillation. The problem of aggregating the set of systems into a coupled system with an attractive cycle close to the oscillation of uncoupled systems is solved. For this purpose, weak universal coupling controls are found.


Introduction
Aggregation of individual systems is performed to obtain a coupled system with a given property. This procedure should be accompanied by the study of the coupled system.
Usually, aggregation is performed after decomposition. Methods for aggregating complex systems by the Lyapunov method are given in [1]. It is proposed to study the set of systems in the framework of a model containing weakly coupled subsystems [2]. The idea of [2] was implemented for autonomous and periodic coupled systems described by general differential equations in [3][4][5][6]. In this paper, we consider identical mechanical systems.

Problem statement
Consider k n -degree-of-freedom identical mechanical systems described by Lagrange's equation of the second kind under the action of positional forces. Each system is assumed to allow for a singlefrequency oscillation.
The phase portrait of an individual mechanical system is symmetrical with respect to the fixed space { , : 0} SPMs cannot be stabilized in the framework of the considered mechanical model. In order to stabilize an SPM, it is necessary to apply a force that violates the symmetry of the phase portrait. If the applied force is  -small, then the oscillation of the controlled mechanical system, which is stabilized, is  -close to the SPM. The appropriate controlled mechanical system that possesses an orbitally asymptotically stable cycle is constructed in [12,13]. A small force becomes a weak coupling control for a set of mechanical systems.
Stated is the problem of aggregation of considered identical mechanical systems such that the resulting system allows for an attracting cycle, which is close to the oscillations of uncoupled systems. At that, universal coupling controls (i.e. suitable for any mechanical systems) are found.

Aggregation technique
The proposed technique interprets a set of uncoupled identical systems as a single mechanical system. This system, according to the problem statement, allows for a family ( ) ( ) l hh    of nondegenerate SPMs. A controlled mechanical system is constructed, where an  -small control is For k identical mechanical systems with n degrees of freedom, the dimension of system (1) is ,1 00 In [13] it is proved that an orbitally asymptotically stable cycle always occurs in system (1),(2) at the proper choice of matrix sj r . In (2), multiplier is applied. Therefore, the problem of stabilization of k mechanical systems can be solved in principle, provided that appropriate function b and matrix sj r in (2) are found.

Cycle of the coupled mechanical system
Necessary and sufficient conditions for a  -periodic SPM to exist are expressed as follows  [14].
Since (1) is satisfied on  , the following lemma holds.

Lemma. A family of nondegenerate SPMs is described by a one-degree-of-freedom conservative system.
Proof. Equation (3) implies that the following linear equation hold.
The above equations become identity on  .
For  , the existence of the linear transform mentioned above means that there exist coordinates 1   w c w  Therefore, the dynamics on  is described by a one-degree-of-freedom conservative system, q.e.d.
According to Lemma, a one-degree-of-freedom conservative system is a key subsystem of the multi-degree-of freedom system that implement the family () h  . The whole system can be described in the neighborhood of  by the generalized coordinates  

 
Kh is the characteristics of the individual system. As a result, we obtain the following theorem that gives the solution to the problem of aggregation.