title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ²Ð¾Ð¹ÑÑÐ² Ð¸Ð·Ð³Ð¸Ð±Ð½ÑÑ Ð²Ð¾Ð»Ð½ Ð² Ð¼Ð°ÑÐµÑÐ¸Ð°Ð»Ð°Ñ Ð¼Ð°ÑÑÐ°-Ð²-Ð¼Ð°ÑÑÐµ: Ð´Ð¸ÑÐ¿ÐµÑÑÐ¸Ð¾Ð½Ð½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð· Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Reaserch on the properties of bending waves in mass-in-mass metamaterials: dispersion analysis and numerical simulation
Creators: name: Ð§Ð¶Ð°Ð¾ Ð®Ð¹ÑÐ¸Ð½
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2024
Subjects: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÐÐ¾Ð»Ð½Ñ
Ð°ÐºÑÑÑÐ¸ÑÐµÑÐºÐ¸Ð¹ Ð¼ÐµÑÐ°Ð¼Ð°ÑÐµÑÐ¸Ð°Ð»
Ð´Ð¸ÑÐ¿ÐµÑÑÐ¸Ñ
acoustic metamaterial
dispersion

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description: Bending wave, as a special wave form, widely exists in nature and engineering applications, such as sound wave propagation inÂ  pipes, seismic wave propagation inÂ  layers of the earths crust, etc. Therefore, a deep understanding of its properties is important for both theoretical research and practical applications. In this work, the properties of bending waves are deeply discussed. By means of dispersion analysis and numerical simulation, the propagation characteristics of bending waves in media are comprehensively studied. The theoretical part introduces the concepts, mathematical principles, and applications of the continuum long-wavelength approximation, the Lagrange equation, and the dispersion equation, providing theoretical guidance for subsequent dispersion analysis. In terms of dispersion analysis, this work presents in detail the process of deriving discrete differential-difference equations of motion from the variational principle for potential strain energy and kinetic energy for the basic model of a mass-in-mass chain during bending. Using dispersion analysis of coupled differential equations, the presence of a band gap for both frequency and phase velocity was established. The higher order continuum approximation can provide a more accurate prediction of dispersion characteristics than the basic order continuum approximation. In the numerical simulation part, coupled partial differential equations of the basicÂ  order continuum approximation are studied, and physical constant parameters are specified in such a way that the results can be easily observed in graphs. Initial and boundary conditions corresponding to the excitation of periodic and localized waves are imposed. Numerical experiments on the boundary excitation of a harmonic bending wave are carried out. Numerical results show that there is a band gap for the values of the boundary frequencies when no wave propagation happens. This is completely consistent with the results of dispersion analysis. Studies of the evolution of localized flexural waves show that strong dispersion prevents their stable propagation within a linear problem. The results of numerical simulations not only confirm the theoretical predictions of dispersion analysis, but also provide an intuitive visualization method for furtherÂ  understandingÂ  the mechanism of bending wave propagation.