Multidisciplinary Journal of Science and Technology

THE PROBLEM OF THE STABILITY OF VIBRATIONS OF A DISCRETE MECHANICAL SYSTEM PROTECTED FROM VIBRATIONS

Bakhtiyor Ashurov

Samarkand State Architectural and Civil Engineering University, 140147, Samarkand, Uzbekistan

Keywords: mechanical system, dynamic damper, hysteresis, dissipative, random parametric excitation, stochastic process, root mean square value, priority conditions.

---

Abstract

In this work, the issue of analytically verifying the priority of vibrations of a mechanical system with elastic dissipative characteristics of the hysteresis type under the influence of random parametric excitations together with a dynamic damper is considered, depending on the design parameters of the system. In this case, solutions are sought in the form of exponents, a system of equations in normal form is formed, and the system of motion differential equations is averaged using the averaging method. The system of motion differential equations of the vibration-protected system is brought to the system of Ito differential equations. Based on the solutions obtained in the form of exponents, characteristic equations and their roots are determined. According to the main theorem of priority theory, analytical expressions of priority conditions are obtained based on the roots of characteristic equations. The boundaries of the priority and unstable regions are determined and analyzed in an analytical form.

---

References

[1] Zhang H., Wu Zh., Xi Y., Exponential stability of stochastic systems with hysteresis switching Journal of Automatica. Volume 50, 2014, p.599-606.

[2] Jia Y. Review of nonlinear vibration energy harvesting: Duffing, bistability, parametric, stochastic and others. Journal of Intelligent Material Systems and Structures. 2020; 31(7):921-944. doi:10.1177/1045389X20905989

[3] Proppe C., Stochastic linearization of dynamical systems under parametric Poisson white noise excitation. International Journal of Non-Linear Mechanics. (2003). 38. 543-555. 10.1016/S0020-7462(01)00083-X.

[4] Pavlovskii, M.A., Ryzhkov, L.M. Random parametric oscillations of elastic systems with hysteresis energy dissipation. Soviet Applied Mechanics 26, 890–895 (1990). https://doi.org/10.1007/BF00888776

[5] Kuz'ma V.M. Dynamic instability of the random vibrations of a rod. Soviet Applied Mechanics 2, 71–74 (1966). https://doi.org/10.1007/BF00887762

[6] Bobryk R.V., Chrzeszczyk A., Numerical studies of dynamic stability under small random parametric excitations, journal of Computer Assisted Mechanics and Engineering Sciences, Volume 17, 2010, pp.89–97.

[7] Plaut R.H., Limam W.,Oscillations of weakly non-linear, self-excited systems under multi-frequency parametric excitation Journal of Sound and Vibration. Volume 144, Issue 2, 22 january 1991. https://doi.org/10.1016/0022-460X(91)90744-5.

[8] Sveshnikov, A. A., Applied Methods of the Theory of Random Functions. Moscow, 1968.

[9] Alifov A. A.,Oscillations under a nonlinear parametric action and combinations of delays, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. 17 (1) (2024) 47–55. DOI: https://doi.org/10.18721/JPM.17105

[10] Malara. G., Pomaro B., Spanos P., Nonlinear stochastic vibration of a variable cross-section rod with a fractional derivative element. International Journal of Non-Linear Mechanics. (2021). 135. 103770. 10.1016/j.ijnonlinmec.2021.103770.

[11] M. M. Mirsaidov, О. М. Dusmatov, М. U. Khodjabekov; Dynamics of the rod protected from vibration under kinematic excitations. AIP Conf. Proc. 15 March 2023; 2612 (1): 030005. https://doi.org/10.1063/5.0113225

[12] Mirsaidov, M.M., Dusmatov, O.M., Khodjabekov, M.U. (2023). Mode Shapes of Hysteresis Type Elastic Dissipative Characteristic Plate Protected from Vibrations. Lecture Notes in Civil Engineering, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-031-10853-2_12

[13] M. M. Mirsaidov, O. M. Dusmatov, M. U. Khodjabekov; Mathematical modeling of hysteresis type elastic dissipative characteristic plate protected from vibration. AIP Conf. Proc. 20 October 2022; 2637 (1): 060009. https://doi.org/10.1063/5.0118289

[14] M. U. Khodjabekov, Kh. M. Buranov, A. E. Qudratov; Modal mass and stiffness of hysteresis type elastic dissipative characteristic plate. AIP Conf. Proc. 20 October 2022; 2637 (1): 050004. https://doi.org/10.1063/5.0118292