Chaotic synchronization of vibrations of a coupled mechanical system consisting of a plate and beams

Abstract

In this paper mathematical model of a mechanical system consisting of a plate and either one or two beams is derived. Obtained PDEs are reduced to ODEs, and then studied mainly using the fast Fourier and wavelet transforms. A few examples of the chaotic synchronizations are illustrated and discussed.

Mechanical Design of Innovative Electromagnetic Linear Actuators for Marine Applications

“…Moreover, expensive maintenance and frequent replacements of the components that have been damaged by the local vibrations could be necessary. This kind of vibrations could be classified as chaotic vibrations [24][25][26][27] which are deterministic, but from an analytical point of view, unpredictable. In some real cases the vibrations here defined as "local" and "chaotic" were found.…”

Contact Interaction Plates, Reinforced by Ribs, With Gaps Under the Influence of White Noise

“…ɋ ɷɬɨɣ ɰɟɥɶɸ ɩɨɥɭɱɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜ-ɧɟɧɢɹ ɫɜɨɞɢɥɢɫɶ ɤ ɡɚɞɚɱɟ Ʉɨɲɢ ɦɟɬɨɞɨɦ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɫ ɚɩɩɪɨɤɫɢɦɚɰɢɟɣ 0(h2) ɢ 0(h4) ɢ ɦɟɬɨɞɨɦ Ɏɚɷɞɨ-Ƚɚɥɟɪɤɢɧɚ ɜ ɜɵɫɲɢɯ ɩɪɢɛɥɢɠɟɧɢɹɯ, ɢɫɫɥɟɞɨɜɚɥɚɫɶ ɫɯɨɞɢɦɨɫɬɶ ɦɟɬɨɞɚ ɩɪɢ ɪɚɡɧɨɦ ɱɢɫɥɟ ɱɥɟɧɨɜ ɪɹɞɚ 1,..., 6, N ɩɪɢ ɷɬɨɦ ɬɨɱɧɨɫɬɶ ɭɫɬɚɧɚɜɥɢɜɚɥɚɫɶ ɩɨ ɩɪɚɜɢɥɭ Ɋɭɧɝɟ. ɉɨɥɭɱɟɧɧɵɟ ɨɛɵɤɧɨɜɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɪɟɲɚɥɢɫɶ ɦɟ-ɬɨɞɚɦɢ ɬɢɩɚ Ɋɭɧɝɟ-Ʉɭɬɬɵ 4, 6, 8-ɝɨ ɩɨɪɹɞɤɚ ɬɨɱɧɨɫɬɢ ɩɨ ɜɪɟɦɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ, ɛɵɥ ɩɪɨ-ɜɟɞɟɧ ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɪɟɡɭɥɶɬɚɬɨɜ [1,4]. Ɍɚɤ ɤɚɤ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɧɟɥɢɧɟɣɧɚɹ, ɬɨ ɪɟɲɢɬɶ ɟɟ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɩɭɬɟɦ ɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɵɦ.…”

“…ɇɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɤɨɧɬɚɤɬɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫɥɨɟɜ, ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɤɨɧɫɬɪɭɤɬɢɜɧɨɣ ɧɟɥɢɧɟɣɧɨɫɬɢ [2]. ɇɚɥɢɱɢɟ ɡɚɡɨɪɚ ɦɟɠɞɭ ɷɥɟɦɟɧɬɚɦɢ (ɩɥɚɫɬɢɧɤɚ-ɛɚɥɤɚ, ɩɥɚɫɬɢɧɤɚ-ɩɥɚɫɬɢɧɤɚ, ɛɚɥɤɚ-ɛɚɥɤɚ, ɨɛɨɥɨɱɤɚ-ɛɚɥɤɚ ɢ ɞɪɭɝɢɟ ɫɨɱɟɬɚɧɢɹ), ɚ ɬɚɤɠɟ ɧɟɢɞɟ-ɚɥɶɧɨɫɬɶ ɢɡɝɨɬɨɜɥɟɧɢɹ ɩɪɢɛɨɪɚ ɭɠɟ ɩɪɢ ɦɚɥɵɯ ɩɪɨɝɢɛɚɯ, ɫɨɢɡɦɟɪɢɦɵɯ ɫ ɡɚɡɨɪɨɦ ɦɟɠɞɭ ɷɥɟɦɟɧɬɚɦɢ, ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɢɡɭɱɚɟɦɵɣ ɨɛɴɟɤɬ ɜ ɫɨɫɬɨɹɧɢɟ ɯɚɨɬɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ [3][4]. ɉɨɷɬɨɦɭ ɜɚɠɧɨ ɢɡɭɱɟɧɢɟ ɜɥɢɹɧɢɹ ɭɩɪɚɜɥɹɸɳɢɯ ɩɚɪɚɦɟɬɪɨɜ, ɪɚɡɥɢɱɧɵɯ ɬɢɩɨɜ ɜɧɟɲɧɟɣ ɧɚɝɪɭɡɤɢ ɧɚ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɟ ɫɢɫɬɟɦɵ.…”

Contact Interaction of Plates and Beams Under the Influence of White Noise

“…It is necessary to take into account the contact interaction of layers which is the reason of constructive nonlinearity [2]. A gap between the elements (plate-beam, plate-plate, beam-beam, shell-beam and other combinations), as well as the imperfection of the device, even if there are small deflections comparable with the gap between the elements, can cause the studied object to be in a state of chaotic oscillations [3][4]. Therefore, it is important to study influence of control parameters, and different kinds of external load on performance of the system.…”