机械膜和生物膜振荡的模型问题

IF 2.1 Q2 ENGINEERING, MULTIDISCIPLINARY Inventions Pub Date : 2023-10-30 DOI:10.3390/inventions8060139
Yury Kostikov, Aleksandr Romanenkov
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引用次数: 0

摘要

本文讨论了机械系统弹性理论、脊椎动物内耳生物力学和电路理论中出现的膜振荡的各种模型。所考虑的振荡具有不同的性质,但它们的数学模型是用具有非平凡边界条件的二阶双曲方程的类似初始边值问题来描述的。这些问题的微分方程是一样的。因此,例如,电报线路中的电压分布模型出现了一维振荡方程。对于二维和三维算符,分别建立了圆形均匀固体膜、带孔膜的振荡模型,以及球面和球面区域的气体振荡模型,但考虑了振荡的径向对称性。膜振荡模型问题可以看作是耳鼓膜振荡问题。提出了将相应问题统一化为零边界条件下的初边值问题的方法。描述了利用相应的Sturm-Liouville问题的特征函数将其表述为傅里叶级数形式的方法。
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Model Problems on Oscillations of Mechanical and Biological Membranes
Various models of membrane oscillations emerging in the theory of elasticity of mechanical systems, biomechanics of the internal ear of vertebrata, and in the theory of electrical circuits are discussed in the article. The considered oscillations have different natures, but their mathematical models are described using similar initial boundary value problems for the second-order hyperbolic equation with the nontrivial boundary condition. The differential equations in these problems are the same. Thus, for example, the model of voltage distribution in the telegraph line emerges for the one-dimensional equation of oscillations. The model of oscillations of a circular homogeneous solid membrane, a membrane with a hole, and the model of gas oscillations in a sphere and spherical region emerge for the two-dimensional and three-dimensional operators, but take into account the radial symmetry of oscillations. The model problem on membrane oscillation can be considered as the problem on ear drum membrane oscillations. The unified approach to reducing the corresponding problems to the initial boundary value problem with zero boundary conditions is suggested. The technique of formulating the solution in the form of a Fourier series using eigenfunctions of the corresponding Sturm–Liouville problem is described.
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来源期刊
Inventions
Inventions Engineering-Engineering (all)
CiteScore
4.80
自引率
11.80%
发文量
91
审稿时长
12 weeks
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