On Travelling Wave Modes of Axially Moving String and Beam

Lü, Lei; Yang, Xiao‐Dong; Zhang, Wei; Lai, Shih-Kung

doi:10.1155/2019/9496180

Cited by 18 publications

(3 citation statements)

References 39 publications

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“…For the first time, Shen et al 46 introduced nonlocal strain gradient theory into the axially moving functionally graded nanoplates to study the dynamic stability, but lacked the systematic analysis on free vibration. Considering that the axially moving structure is often subjected to initial axial tensions in practical engineering, 47 this paper studies the transverse free vibration of axially moving functionally graded nanoplates subjected to the initial tensile force.…”

Section: Introductionmentioning

confidence: 99%

Free Vibration of Axially Moving Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory

Li¹,

Wang²,

Luo³

et al. 2020

The International Journal of Acoustics and Vibration

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Using the nonlocal strain gradient theory, we explore vibration behaviours of initially bidirectional tensioned functionally graded nanoplates with axial speed. The governing equation of motion can be obtained based on the differential type of nonlocal strain gradient constitutive relation, which characters the dynamics of nanostructures containing kinematic relation. The simply supported boundary constraints on four sides are considered and subsequently the numerical results are determined. It shows that natural frequencies of axially moving nanoplates decrease when increasing the axial speed and the nonlocal parameter. Hence the nonlocal and kinematic factors cause the natural frequencies to decrease or, weaken the equivalent bending rigidity. On the other hand, natural frequencies increase with an increase in the axial tension and material characteristic scale. Hence the strain gradient and tensile stress factors cause the natural frequencies to increase or, strengthen the equivalent bending rigidity. In addition, the natural frequencies get higher with a larger aspect ratio of the functionally graded nanoplate. The larger one between the nonlocal parameter and the material characteristic scale plays a dominant role in the softening and stiffening mechanisms of the nonlocal strain gradient effect. In case of the same magnitude of the nonlocal parameter and the material characteristic scale, the softening and hardening phenomena disappear. The equivalent bending rigidity neither increases nor decreases in such a situation, and its value degenerates to the classical one.

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Section: Introductionmentioning

confidence: 99%

Free Vibration of Axially Moving Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory

Li¹,

Wang²,

Luo³

et al. 2020

The International Journal of Acoustics and Vibration

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“…В работе [1] для задачи (1)-( 3) строятся пробные решения в виде тригонометрических многочленов, состоящие из конечного числа слагаемых, и устанавливаются механические свойства колебаний, которые возникают в движущемся полотне. Работа [2] содержит прием из вариационного исчисления для вывода уравнения колебаний движущейся струны.…”

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“…. , λ N , то для каждого значения имеем функцию, которая определяется по формуле (22) и является решением уравнения (1). Тогда линейная комбинация этих решений так же является решением и определяется следующей формулой:…”

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On solutions of the equation of small transverse vibrations of a moving canvas

Romanenkov¹

2022

Vestnik SPbSU. Mathematics. Mechanics. Astronomy

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The paper considers a model problem of one-dimensional small transverse vibrations of a canvas moving at a constant speed, which is fixed in a hinged manner. The oscillatory process is described by a linear differential equation of the 4th order with constant coefficients. In the model under consideration, the Coriolis force is taken into account, which leads to the appearance of a term with a mixed derivative in the differential equation. This effect makes it impossible to use the classical method of separating variables. However, families of exact solutions of the oscillation equation in the form of a traveling wave have been constructed. For the initial-boundary value problem, it was established that the solution can be constructed in the form of a Fourier series according to the system of eigenfunctions of the auxiliary problem on beam vibrations. For the considered oscillatory process, the law of conservation of energy is established and the uniqueness of the solution to the initialboundary value problem is proved.

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