DYNAMIC OF THE CONNECTED PENDULUMS IN A MAGNETIC FIELD

YURI MIKHLIN, YULIIA SURHANOVA

NATIONAL TECHNICAL UNIVERSITY «KHARKIV POLYTECHNIC INSTITUTE»

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DSMSI 2023- DYNAMICAL SYSTEM MODELING AND STABILITY INVESTIGATION

THE CONFERENCE IS DEDICATED TO THE 77TH ANNIVERSARY OF THE OUTSTANDING UKRAINIAN SCIENTIST PROFESSOR DENYS KHUSAINOV

CONTENTS

2

THE EQUATIONS OF MOTION OF A SYSTEM AND THE MODEL OF MAGNETIC INTERACTION OF PARTICLES

3

(1)

(2)

Fig. 1. The pendulum system under consideration

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Fig. 2. Experimental data on the magnetic moment in comparison with the numerically consistent model (2)

MULTI-SCALE ANALYSIS OF THE COUPLED VIBRATION MODE

4

(3)

(4)

(5)

A solution of the system (1) is presented by the following decomposition:

Represent the time scaling:

MULTI-SCALE ANALYSIS OF THE COUPLED VIBRATION MODE

5

(6)

The solution of (4) is

It corresponds to the in-phase (or coupled) mode of vibrations.

(7)

(8)

(9)

INFLUENCE OF SYSTEM PARAMETERS AND INITIAL VALUES ON THE COUPLED VIBRATION MODE UNDER MAGNETIC INFLUENCE

6

INFLUENCE OF SYSTEM PARAMETERS AND INITIAL VALUES ON THE COUPLED VIBRATION MODE UNDER MAGNETIC INFLUENCE

7

Fig. 4. Phase portrait and curves in configuration space at different mass ratios of two pendulums

INFLUENCE OF THE SYSTEM PARAMETERS AND INITIAL VALUES ON THE COUPLED VIBRATION MODE UNDER MAGNETIC INFLUENCE

8

Fig. 5. Phase portraits and curves in the configuration space at different values of the masses ratio of two pendulums and the distance between the center of gravity and the axis of rotation of the pendulums

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MULTI-SCALE ANALYSIS OF THE COUPLED VIBRATION MODE

9

Fig. 6. Phase portraits and curves in configuration space at different values of initial conditions, masses ratio, and the distance between the center of gravity and the axis of rotation of the pendulums, and coupling coefficient

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10

ANALYSIS OF THE STABILITY OF THE COUPLED OSCILLATION MODE DEPENDING ON THE PARAMETERS 𝝋(𝟎), 𝝁

11

Fig. 8. Phase portraits, curves in configuration space, and time representations and spectrum for unstable (a) and stable (b) coupled mode oscillations

Instability

Stability

12

ANALYSIS OF THE STABILITY OF THE COUPLED OSCILLATION MODE DEPENDING ON THE PARAMETERS 𝝋(𝟎), 𝝁, 𝒔

13

Fig. 11. Phase portraits, curves in configuration space (a), and time representations and spectrum (b, c) for unstable coupled mode oscillations

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11a

11b 11c

14

15

MULTI-SCALE ANALYSIS OF THE LOCALIZED VIBRATION MODE

16

(10)

Time transformation:

(11)

(12)

MULTI-SCALE ANALYSIS OF THE LOCALIZED VIBRATION MODE

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(13)

(14)

In zero approximation one has the following presentation of the localized mode:

The magnetic moment is represented by the shortened Fourier series in the form (7).

One has from the modulation equations of the first approximation:

INFLUENCE OF THE SYSTEM PARAMETERS AND INITIAL VALUES ON THE LOCALIZED VIBRATION MODE

18

Fig. 16. Phase portraits and trajectories in configuration space under different initial angle values and the pendulum masses ratio

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INFLUENCE OF THE SYSTEM PARAMETERS AND INITIAL VALUES ON THE LOCALIZED VIBRATION MODE

19

Fig. 17. Phase portraits and trajectories in configuration space at different values of initial angles, masses ration and distance between the center of gravity and the axis of rotation of pendulums

INFLUENCE OF THE SYSTEM PARAMETERS AND INITIAL VALUES ON THE LOCALIZED VIBRATION MODE

20

Fig. 18. Phase portraits and modal lines in configuration space at different values of initial angles, masses ratio of the pendulums, and the distance between the center of gravity and the axis of rotation of the pendulums, and coupling coefficients

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CONCLUSIONS

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The stable coupled mode of oscillation does not exist over the entire range of initial conditions. It is unstable at small initial values of pendulum deflection angles if pendulum masses differ significantly. Increasing the proportionality factor of the pendulum masses leads to a decrease in the wandering of trajectories near the mode. The coupled mode is more pronounced with a greater distance between the center of mass and the axis of rotation, since then the effect of magnetic moment is less. This mode is observed when both the distance and the mass of the smaller pendulum increase. In most of the cases considered, with significant values of the distance from the axis of rotation to the center of mass of the pendulum and the mass ratio, an increase in the value of the coupling coefficient leads to stabilization of the coupled mode and a decrease in the walks of trajectories near such a mode. Increasing dissipation does not always tighten trajectories to the mode of oscillation.

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Like the coupled mode, localized does not exist over the entire range of initial pendulum deviations. As the magnitude of the pendulum mass proportionality factor increases, the walks near the mode decrease and the mode becomes more defined. It turned out, as for the coupled mode, that localization is manifested by increasing the connection and distance between the center of mass and the axis of rotation of the pendulums or by increasing both the connection and the proportionality coefficient of the masses of the pendulums. The considerable mass proportionality coefficients, together with the significant coupling and distance values as the friction coefficients increase, reduce the walk of trajectories near the localized mode, or contract trajectories to this mode.

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Thank You

For Your Attention

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