scholarly journals Собственные частоты изгибных колебаний углеродных нанотрубок

2021 ◽  
Vol 91 (11) ◽  
pp. 1732
Author(s):  
С.В. Дмитриев ◽  
И.Р. Сунагатова ◽  
М.А. Ильгамов ◽  
И.С. Павлов
Keyword(s):  
Carbon Nanotubes ◽  
Molecular Dynamics ◽  
Degrees Of Freedom ◽  
Electromechanical Coupling ◽  
Natural Frequencies ◽  
High Accuracy ◽  
Dynamics Model ◽  
Natural Vibrations ◽  
Bending Vibrations ◽  
Low Amplitude

Using a molecular dynamics model with a reduced number of degrees of freedom, the natural frequencies of bending vibrations of carbon nanotubes (CNTs) of various diameters are calculated under plane strain conditions. It is shown that the theory of thin cylindrical shells provides high accuracy in estimating the frequencies of low-amplitude natural vibrations even for relatively small CNT diameters. It is shown that with an increase in the amplitude, the frequency of natural vibrations decreases, which is consistent with the data available in the literature. The results obtained are necessary for the design of terahertz resonators based on CNTs and high-precision mass and force nanosensors based on the effect of electromechanical coupling that CNTs exhibit.

scholarly journals QUALITATIVE PROPERTIES OF VIBRATIONS OF ELASTICALLY SUPPORTED RIGID BODY

2020 ◽  
Vol 2 (2) ◽  
pp. 85-94
Author(s):  
S Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Natural Vibrations ◽  
Qualitative Properties ◽  
Engineering Structures ◽  
Natural Oscillations ◽  
Geometric Characteristics ◽  
Operating Modes

The paper investigates free vibrations of an absolutely rigid body, supported by a set of linearly elastic springs and performing a plane-parallel motion. The proposed system has two degrees of freedom, which makes it elementary to determine the frequencies and modes of its natural oscillations by using exact analytical expressions. However, these expressions are rather cumbersome, which makes it difficult to study the behavior of frequencies and modes when the characteristics of the model change. Therefore, the aim of the work was to find out the qualitative properties of the modes of free vibrations depending on the elastic, inertial and geometric characteristics of the system, as well as to study the effect of changing the position of elastic supports on its natural frequencies. The main qualitative characteristic of the mode of natural vibrations of the system in consideration is the position of its node – a point that remains stationary during natural vibrations. For the practically important case of a system with two supports, it has been established in the work that, in the general case, of two modes corresponding to two different natural frequencies, one has a node located inside the gap between the supports, and the other – outside this gap. Analytical conditions are found that must be satisfied by the inertial and geometric characteristics of the system, which make it possible to determine which of the two modes corresponds to the internal position of the node. It is noted that these conditions do not depend on the stiffness of the supports. Analytical results were also obtained, allowing to determine a more accurate qualitative localization of the node. To clarify the behavior of natural frequencies when the position of the supports changes, an explicit expression is obtained for the derivative of the square of the natural frequency of the system with respect to the coordinate defining the position of the support. This expression can be used to solve a variety of problems related to the control and optimization of the operating modes of engineering structures subjected to dynamic, in particular periodic, effects. The results of the work were obtained using qualitative methods of the mathematical theory of oscillations. In particular, the theorem on the effect of imposing constraints on the natural frequencies of an elastic system is systematically used.

scholarly journals AIMED CONTROL OF THE FREQUENCY SPECTRUM OF EIGENVIBRATIONS OF ELASTIC PLATES WITH A FINITE NUMBER OF DEGREES OF FREEDOM OF MASSES BY SUPERIMPOSING ADDITIONAL CONSTRAINTS

2021 ◽  
Vol 17 (2) ◽  
pp. 76-82
Author(s):  
Leonid Lyakhovich ◽  
Pavel Akimov
Keyword(s):  
Finite Number ◽  
Frequency Spectrum ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Elastic Plates ◽  
Natural Vibrations ◽  
Design Constraint ◽  
Elastic Systems ◽  
The Masses ◽  
Additional Constraints

As is known, for some elastic systems with a finite number of degrees of freedom of masses, for which thedirections of motion of the masses are parallel and lie in the same plane, methods have been developed for creatingadditional constraints that purposefully change the spectrum of natural frequencies. In particular, theory and algorithm forthe formation of aimed additional constraints have been developed for the rods, the introduction of each of which doesnot change any of the modes of natural vibrations, but only increases the value of only one frequency, without changingthe values of the remaining frequencies. The distinctive paper is devoted to the method of forming a matrix of additionalstiffness coefficients corresponding to such aimed constraint in the problem of natural vibrations of rods. This method canalso be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which thedirections of motion of the masses are parallel, but not lie in the same plane. In particular, such systems include plates.However, the algorithms for the formation of aimed additional constraints, developed for rods and based on the propertiesof rope polygons, cannot be used without significant changes in a similar problem for plates. The method for the formationof design constraint schemes that purposefully change the spectrum of frequencies of natural vibrations of elastic plateswith a finite number of degrees of freedom of masses, will be considered in the next work.

scholarly journals QUALITATIVE PROPERTIES OF VIBRATIONS OF ELASTICALLY SUPPORTED RIGID BODY

2020 ◽  
Vol 2 (2) ◽  
pp. 85-94
Author(s):  
S Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Natural Vibrations ◽  
Qualitative Properties ◽  
Engineering Structures ◽  
Natural Oscillations ◽  
Geometric Characteristics ◽  
Operating Modes

The paper investigates free vibrations of an absolutely rigid body, supported by a set of linearly elastic springs and performing a plane-parallel motion. The proposed system has two degrees of freedom, which makes it elementary to determine the frequencies and modes of its natural oscillations by using exact analytical expressions. However, these expressions are rather cumbersome, which makes it difficult to study the behavior of frequencies and modes when the characteristics of the model change. Therefore, the aim of the work was to find out the qualitative properties of the modes of free vibrations depending on the elastic, inertial and geometric characteristics of the system, as well as to study the effect of changing the position of elastic supports on its natural frequencies. The main qualitative characteristic of the mode of natural vibrations of the system in consideration is the position of its node – a point that remains stationary during natural vibrations. For the practically important case of a system with two supports, it has been established in the work that, in the general case, of two modes corresponding to two different natural frequencies, one has a node located inside the gap between the supports, and the other – outside this gap. Analytical conditions are found that must be satisfied by the inertial and geometric characteristics of the system, which make it possible to determine which of the two modes corresponds to the internal position of the node. It is noted that these conditions do not depend on the stiffness of the supports. Analytical results were also obtained, allowing to determine a more accurate qualitative localization of the node. To clarify the behavior of natural frequencies when the position of the supports changes, an explicit expression is obtained for the derivative of the square of the natural frequency of the system with respect to the coordinate defining the position of the support. This expression can be used to solve a variety of problems related to the control and optimization of the operating modes of engineering structures subjected to dynamic, in particular periodic, effects. The results of the work were obtained using qualitative methods of the mathematical theory of oscillations. In particular, the theorem on the effect of imposing constraints on the natural frequencies of an elastic system is systematically used.

Molecular Dynamics Simulation of Double-Walled Carbon Nanotube Vibrations: Comparison With Continuum Elastic Theories

2009 ◽  
Vol 25 (4) ◽  
pp. 337-343 ◽  
Author(s):  
S. Shayan-Amin ◽  
H. Dalir ◽  
A. Farshidianfar
Keyword(s):  
Carbon Nanotubes ◽  
Molecular Dynamics ◽  
Molecular Dynamics Simulation ◽  
Degrees Of Freedom ◽  
Dynamics Simulation ◽  
Matrix Stiffness ◽  
Resonant Frequencies ◽  
Radial Displacements ◽  
Surrounding Matrix ◽  
The Matrix

ABSTRACTDouble-walled carbon nanotubes (DWNTs) are expected to be useful as elements in improving conventional polymer-based fibers and films. An extensive molecular dynamics simulation and continuum analyses are carried out to estimate the influence of matrix stiffness and the intertube radial displacements on free vibration of an individual DWNT. The effects of nanotube length and chirality are also taken into account. The continuum analyses are based on both Euler-Bernoulli and Timoshenko beam theories which considers shear deformation and rotary inertia and for both concentric and non-concentric assumptions considering intertube radial displacements and the related internal degrees of freedom. New intertube resonant frequencies are calculated. Detailed results are demonstrated for the dependence of resonant frequencies on the matrix stiffness. The results indicate that internal radial displacement and surrounding matrix stiffness could substantially affect resonant frequencies especially for longer doublewalled carbon nanotubes of larger innermost radius at higher resonant frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies.

Vibration of Cantilevered Double-Walled Carbon Nanotubes Predicted by Timoshenko Beam Model and Molecular Dynamics

2015 ◽  
Vol 12 (04) ◽  
pp. 1540017 ◽  
Author(s):  
Rumeng Liu ◽  
Lifeng Wang
Keyword(s):  
Carbon Nanotubes ◽  
Molecular Dynamics ◽  
Natural Frequencies ◽  
Md Simulations ◽  
Beam Model ◽  
Timoshenko Beams ◽  
Fundamental Frequencies ◽  
The Difference ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

Vibration of double-walled carbon nanotubes (DWCNTs) with one end fixed and the other end free is studied by using different beam models of continuum mechanics and the molecular dynamics (MD) simulations. The models of the double-Euler beams (DEB) and the double-Timoshenko beams (DTB) of cantilevered case, with the van der Waals interaction between layers taken into consideration, are applied to predict the natural frequencies of DWCNTs. An analytical solution is first obtained for the DTB model with cantilevered boundary condition. The fundamental frequencies obtained by the DEB model and the DTB model are very close, for the relatively long DWCNTs. The MD simulations show that these two models can predict the natural frequencies well. However, the difference between the DEB model and the DTB model becomes obvious, for the vibration of the relatively short DWCNTs. The DTB model can offer a much better prediction than the DEB model when the DWCNT is very short especially for high-order frequencies.

The Natural Vibration of Fluid-Structure Interaction Systems Subject to the Sommerfeld Radiation Condition

2006 ◽  
Author(s):  
Jing T. Xing
Keyword(s):  
Eigenvalue Problem ◽  
Degrees Of Freedom ◽  
Natural Vibration ◽  
Natural Frequencies ◽  
Fluid Structure Interaction ◽  
Complex Conjugate ◽  
Natural Vibrations ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Natural Modes

A fluid-structure interaction system subject to a Sommerfeld condition is defined as a Sommerfeld system in this paper. It is well known that the natural vibration of a dynamic system is defined by the eigenvalue problem of the corresponding idealized system with no material damping assumed and external forces. From the defined eigenvalue problem, the real natural frequencies and the corresponding natural modes of the system can be derived. What are the characteristics of natural vibrations of a Sommerfeld system? This paper intends to address this problem by investigating three selected fluid-structure interaction systems. The systems chosen involve the solid structures with one, two and infinite degrees of freedom coupling to an infinite fluid domain subject to a Sommerfeld condition, respectively. The governing equations describing these coupled systems are presented using the theory of continuum mechanics. The theoretical solution for each problem is derived and discussed. The analysis demonstrates that a Sommerfeld system undergoing a natural vibration behaves energy dissipative characteristics although there is no material damping in solid and fluid of the system. The natural vibrations of a Sommerfeld system are governed by a complex eigenvalue problem which has only pairs of complex conjugate natural frequencies. The number of the complex conjugate natural frequencies and corresponding natural modes of this Sommerfeld system equals to the number of the degrees of freedom of the dry solid structure in the system and it is independent of the infinite fluid domain. The natural vibration forms of the solid structure in natural vibrations do not satisfy the orthogonal relationship. The findings in this research reveal some common dynamic characteristics of Sommerfeld systems. An approach for the dynamic response analysis of a Sommerfeld system is proposed based on the orthogonal natural modes of the dry structure in the system which is more efficient for engineering analysis.

scholarly journals Oversampled Modal Approach of a Turbocharger Rotor from the Experimental Lateral Vibrations

2019 ◽  
Vol 24 (4) ◽  
pp. 774-783
Author(s):  
Luis Alvaro Montoya ◽  
Eloy Edmundo Rodríguez ◽  
Helen Janeth Zúñiga ◽  
Israel Mejía
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shaft Speed ◽  
Natural Vibrations ◽  
Transient Conditions ◽  
Modal Approach ◽  
Lateral Vibrations ◽  
First Approximation ◽  
Proposed Model

The elements of rotors have inherent characteristics as geometry and material composition, which causes natural vibrations at frequencies that, due to the rotor unbalance, may coincide with the harmonics of the shaft speed, increasing stress and the probability of fractures even in transient conditions. Therefore, in this work, a theoretical-experimental hybrid method for calculating the natural frequencies and the mode shapes, at rest and non-supported conditions, of a turbocharger rotor is proposed. Firstly, a discrete model of low number of degrees of freedom is considered, and from an oversampled modal approach (OSMA) based on the axial oversampling, sectioning and coupling of the rotor, it is possible to use the oversampled mode shapes to increase the degrees of freedom of the system without major complications in the model. This spatial oversampling criterion is based on the Nyquist-Shannon theorem, and it is used to reduce the error in the estimates of the natural frequencies and to get a first approximation of the mode shapes. The natural frequencies were estimated by the transfer matrix method (TMM) and finite element method (FEM) in order to compare the proposed model results with well-founded numerical methods.

Algorithm for diagnosing fastenings of pipe with liquid

2007 ◽  
Vol 5 ◽  
pp. 96-100
Author(s):  
A.M. Akhtyamov ◽  
F.F. Safina
Keyword(s):  
Frequency Spectrum ◽  
Natural Frequencies ◽  
Algebraic Equations ◽  
Bending Vibrations ◽  
Narrow Tube

An algorithm is considered for diagnosing fastening of a narrow tube filled with a fluid by a spectrum of natural frequencies of its bending vibrations. The constructed algorithm, based on the solution of systems of algebraic equations, allows one to determine any pipe fastenings by 9 values from the frequency spectrum of its vibrations when the liquid is flowing through the pipe.

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