Longitudinal Vibration of Variable Cross-Sectional Nanorods

2020 ◽  
Vol 64 ◽  
pp. 49-60
Author(s):  
Mustafa Arda ◽  
Metin Aydogdu
Keyword(s):  
Differential Equation ◽  
Longitudinal Vibration ◽  
Mode Shapes ◽  
Variable Cross ◽  
Cross Sectional ◽  
Axial Coordinate ◽  
Atomic Force ◽  
Governing Differential Equation ◽  
Area Variation ◽  
Linear Differential

Vibration problem of variable cross-sectional nanorods have been investigated. Analytical solutions have been determined for the variable cross-sectional nanorods for a family of cross-sectional variation. Cross-sectional area variation has been assumed as power function of the axial coordinate. Nonlocal governing equation of motion has been obtained as a second order linear differential equation. Bessel functions have been used in analytical solution of the governing differential equation. Effect of nonlocal and area variation power parameters on dynamics of nanorods have been analyzed. Mode shapes of nanorod have been depicted in various cases and boundary conditions. Present results could be useful at design of atomic force microscope’s probe tip selection.

scholarly journals Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections

2011 ◽  
Vol 18 (4) ◽  
pp. 555-562 ◽  
Author(s):  
Bulent Yardimoglu ◽  
Levent Aydin
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Cross Sections ◽  
Longitudinal Vibration ◽  
Transformation Method ◽  
Variable Cross ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
Sinusoidal Functions

Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions ofax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.

A New Approach to Compute Natural Frequencies and Mode Shapes of Non-Uniform Continuous Bar, Circular Shaft and Beam Vibration

2019 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Mode Shapes ◽  
Engineering Structures ◽  
Cross Sectional ◽  
Closed Form Solutions ◽  
Continuous Structure ◽  
Circular Shaft ◽  
Spatial Parameters

Abstract The wave equation governing longitudinal vibration of a bar and torsional vibration of a circular shaft, and the Euler-Bernoulli equation governing transverse vibration of a beam were developed in the eighteenth century. Natural frequencies and mode shapes are easily obtained for uniform or constant spatial parameters (cross sectional area, material property and mass distribution). But, real engineering structures seldom have constant parameters. For non-uniform continuous structure, a large number of papers have been written for more than 100 years since the publication of Kirchhoff’s memoir in 1882. There are analytical solutions only in few cases, and there are approximate numerical methods to deal with other (almost all) cases, most notably Stodola, Holzer and Myklestad methods in addition to Rayleigh-Ritz and finite element methods. This paper presents a novel approach to compute natural frequencies and mode shapes for arbitrary variations of spatial parameters on the basis of linear time-varying system theory. The advantage of this approach is that now it can be claimed that “almost” closed-form solutions are available to find natural frequencies and mode shapes of any non-uniform, linear and one-dimensional continuous structure.

On Application of a Quasi-Static Variational Principle to a System With Damping

1954 ◽  
Vol 21 (1) ◽  
pp. 8-10
Author(s):  
Morris Morduchow
Keyword(s):  
Differential Equation ◽  
Mode Shapes ◽  
Stationary Condition ◽  
Important Class ◽  
Damping Force ◽  
Principal Mode ◽  
Rayleigh Method ◽  
Modes Of Vibration ◽  
Linear Differential ◽  
Bending Modes

Abstract The principal bending modes of vibration of a beam with a damping force proportional to the velocity are considered. It is shown that, in an important class of cases, the damping has exactly no effect on the mode shapes. It is further shown that the linear differential equation for the vibrating beam with damping can be transformed mathematically into a stationary condition after eliminating the time as a variable. Application of the Rayleigh method to this condition then leads to general approximate results for the logarithmic decrement, and for the effect of damping on the natural frequency, not only in the fundamental mode, but also in any higher principal mode.

A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures With Arbitrary Nonuniformities

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Numerical Approach ◽  
Mode Shapes ◽  
Cross Sectional ◽  
One Dimensional ◽  
Dirac Delta ◽  
Mass Distributions ◽  
Jump Discontinuities

Abstract One-dimensional continuous structures include longitudinal vibration of bars, torsional vibration of circular shafts, and transverse vibration of beams. Using the linear time-varying system theory, algorithms are developed in this paper to compute natural frequencies and mode shapes of these structures with nonuniform spatial parameters (mass distributions, material properties and cross-sectional areas) which can have jump discontinuities. A general numerical approach has been presented to include Dirac-delta functions and their spatial derivatives due to jump discontinuities. Numerical results are presented to illustrate the application of these techniques to the solution of different types of spatial variations of parameters and boundary conditions.

Exact Solutions for Longitudinal Vibration of Multi-Step Bars with Varying Cross-Section

1999 ◽  
Vol 122 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Q. S. Li
Keyword(s):  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Transmission Tower ◽  
Power Functions ◽  
Closed Form Solutions ◽  
High Rise ◽  
One Step ◽  
Area Variation

Using appropriate transformations, the equation of motion for free longitudinal vibration of a nonuniform one-step bar is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the area variation. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step nonuniform bar are derived and used to obtain the frequency equation of multi-step bars. The new exact approach is presented which combines the transfer matrix method and closed form solutions of one step bars. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a television transmission tower are in good agreement with the corresponding experimental data, and the selected expressions are suitable for describing the area variation of typical high-rise structures. [S0739-3717(00)00302-0]

Acoustic Field in Ducts With Sinusoidal Area Variation

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
N. S. Vikramaditya ◽  
R. B. Kaligatla
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Acoustic Field ◽  
Second Order ◽  
Ordinary Linear Differential Equation ◽  
Order Partial Differential Equation ◽  
Mean Flow ◽  
Area Variation ◽  
The Mean ◽  
Linear Differential

The purpose of this article is to provide an analytical solution for the acoustic field in a duct with sinusoidal area variation along the length. The equation describing the acoustic field in a variable area duct is a second-order partial differential equation. It is converted into a second-order ordinary linear differential equation, whose solution is dependent on the choice of area variation. The solution for the differential equation is obtained in terms of the area and is obtained neglecting the mean flow. Therefore, it is applicable in the absence of mean flow or in cases where the effects of mean flow are insignificant.

scholarly journals An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections

2019 ◽  
Vol 38 (2) ◽  
pp. 403-414 ◽  
Author(s):  
Deshui Xu ◽  
Jingtao Du ◽  
Zhigang Liu
Keyword(s):  
Boundary Condition ◽  
Fourier Series ◽  
Boundary Conditions ◽  
Cross Section ◽  
Longitudinal Vibration ◽  
Variable Cross ◽  
Arbitrary Boundary ◽  
Cross Section Area ◽  
Section Area ◽  
Area Variation

Longitudinal vibration of non-uniform rod has been of great significance in various engineering occasions. The existing works are usually limited to the certain area variation and/or classical boundary condition. Motivated by this limitation, an efficient accurate solution is developed for the longitudinal vibration of a general variable cross-section rod with arbitrary boundary condition. Displacement function is invariantly expressed as the summation of standard Fourier series and supplementary polynomials, with an aim to make the calculation of all derivatives more straightforwardly in the whole solving region [0, L]. Energy principle is employed for system dynamics formulation, with the elastic boundaries considered as potential energy stored in the restraining spring. Arbitrary cross-section area variation is uniformly expanded into Fourier series. Numerical examples are presented for the natural frequency and mode shapes of non-uniform rod of free and clamped boundary conditions and compared with literature data. Results show good agreement with the previous analytical solutions. Influence of cross section area variation on vibration characteristics of non-uniform rods is then studied and discussed. One of the most advantages of the proposed model is that there is no need to reformulate the problem or rewrite the codes when the cross-section area distribution and/or boundary conditions change arbitrarily.

Transverse Vibration of Non-Uniform Euler-Bernoulli Beam, Using Differential Transform Method (DTM)

2011 ◽  
Vol 110-116 ◽  
pp. 2400-2405
Author(s):  
K. Torabi ◽  
H. Afshari ◽  
E. Zafari
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Method ◽  
Transverse Vibration ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Transform Method ◽  
Special Cases ◽  
Linear Differential ◽  
Euler Bernoulli Beam

Analysis of transverse vibration of beams is presented in this paper. Unfortunately, complexities which appear in solving differential equation of transverse vibration of non-uniform beams, limit analytical solution to some special cases, so that the numerical method is presented. DTM is a numerical method for solving linear and some non-linear, ordinary and partial differential equations. In this paper, this technique has been applied for solving differential equation of transverse vibration of conical Euler-Bernoulli beam. Natural circular frequencies and mode shapes have been calculated. Comparing results with the cases which exact solution have been presented, shows that DTM is a strong method especially for solving quasi-linear differential equations.

Adiabatic One-Dimensional Flow of a Perfect Gas through a Rotating Tube of Uniform Cross Section

1954 ◽  
Vol 4 (4) ◽  
pp. 373-399 ◽  
Author(s):  
K. Kestin ◽  
S. K. Zaremba
Keyword(s):  
Differential Equation ◽  
Gas Turbines ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Velocity Variation ◽  
Perfect Gas ◽  
Cross Sectional ◽  
One Dimensional ◽  
Linear Differential ◽  
Entrance Velocity

SummaryThe paper contains an analysis of the flow of a perfect gas with constant specific heats through a rotating channel of constant cross-sectional area, as used in certain helicopter propulsion systems and wind-driven gas turbines. The analysis is restricted to the adiabatic one-dimensional treatment, the Coriolis accelerations acting across a section being disregarded.The equations of motion and energy are deduced and, together with the equation of continuity, reduced to an ordinary non-linear differential equation of the first order, involving a dimensionless form of velocity and distance.The patterns of the integral curves of the differential equation are discussed and sketched by examining their asymptotic behaviour as well as that in the neighbourhood of singular points. It is shown that there exists a critical value of the angular velocity below which the flow remains subsonic in the pipe, if the entrance velocity is subsonic; it may, however, become sonic at the exit of the pipe. For supercritical angular velocities the flow may become sonic or supersonic in the pipe if the entrance velocity attains a given “ correct” but subsonic value. A method of examining for the possibility of shock formation is indicated.The initial conditions for the differential equation are deduced for the design and for the performance problem, two new flow functions being introduced and tabulated to facilitate practical calculations. Formulae are also deduced for the calculation of the pressure and Mach number variation from the previously calculated velocity variation along the pipe.Finally an approximate solution in closed terms is given for the case of small entrance velocities.

DETERMINATION OF THE FUNCTION OF THE ROD’S CROSS-SECTIONAL AREA USING VIBRATION EIGENFREQUENCIES

2020 ◽  
Vol 0 (4) ◽  
pp. 19-24
Author(s):  
I.M. UTYASHEV ◽  
◽  
A.A. AITBAEVA ◽  
A.A. YULMUKHAMETOV ◽  
◽  
...  
Keyword(s):  
Cross Sectional Area ◽  
Variable Cross ◽  
Sectional Area ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Method Error ◽  
Various Boundary Conditions ◽  
Elastic Fixation ◽  
Crosssectional Area

The paper presents solutions to the direct and inverse problems on longitudinal vibrations of a rod with a variable cross-sectional area. The law of variation of the cross-sectional area is modeled as an exponential function of a polynomial of degree n . The method for reconstructing this function is based on representing the fundamental system of solutions of the direct problem in the form of a Maclaurin series in the variables x and λ. Examples of solutions for various section functions and various boundary conditions are given. It is shown that to recover n unknown coefficients of a polynomial, n eigenvalues are required, and the solution is dual. An unambiguous solution was obtained only for the case of elastic fixation at one of the rod’s ends. The numerical estimation of the method error was made using input data noise. It is shown that the error in finding the variable crosssectional area is less than 1% with the error in the eigenvalues of longitudinal vibrations not exceeding 0.0001.

Sign in / Sign up

Export Citation Format

Share Document