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MOTION EQUATION AVERAGING IN POTENTIAL AUTONOMOUS SYSTEMS

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Subject of Research. The paper proposes the averaging method of motion equations. In various branches of physics (mechanics, electrodynamics) and while analyzing vibration processes, we may need to average the existing equations of motion over a certain time scale. Most often it is required to consider processes in real time and to exclude high-frequency oscillations. In this case, the averaging procedure leads to the fact that the equations of motion for “slow” time change significantly their form. The usually applied arithmetic mean, i.e. equally all-time values in the given interval, does not solve the problem of determining the explicit form of the new motion equations for a “slow” time scale. Method. For the averaging procedure we propose to use the integral transformation with a smooth normalized kernel. The Gauss function is chosen as this kernel, because it “cuts off” adequately high frequencies and has convenient algebraic properties. The algebra based on these properties gives the possibility to solve efficiently the averaging problem and create a system of equations averaged over a certain scale. Main Results. It is shown that additional terms depending on this scale appear as a result of averaging over a certain small time scale. In contrast to the absence of velocities in the original system of motion equations, additional terms appear in the new averaged system depending not only on the coordinates, but also on the velocities. This fact explains the nature of dissipative forces. Moreover, in the created averaging algebra, the equations remain in their original form. Practical Relevance. The proposed method can be applied to any system of differential equations when it is necessary to obtain smoothed solutions. In particular, deformable solid mechanics and vibration mechanics are the proposed method application areas.

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