Free Vibrations of Cantilever Bars with Linear and Nonlinear Variable Cross-Section

2017 ◽  
Vol 6 (4) ◽  
pp. 489-501 ◽  
Author(s):  
Jacek Jaworski ◽  
Olga Szlachetka
Keyword(s):  
Cross Section ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
Linear And Nonlinear

Free vibrations of circular arches with variable cross-section

1994 ◽  
Vol 2 (4) ◽  
pp. 345-357 ◽  
Author(s):  
James F. Wilson ◽  
Byoung Koo Lee ◽  
Sang Jin Oh
Keyword(s):  
Cross Section ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross

scholarly journals INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS

2019 ◽  
Vol 81 (4) ◽  
pp. 449-460
Author(s):  
V.V. Saurin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Cross Section ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
New Variables ◽  
Differential Relations

Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.

scholarly journals Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

2019 ◽  
Vol 41 (1) ◽  
pp. 1-12
Author(s):  
Józef Szybiński ◽  
Piotr Ruta
Keyword(s):  
Cross Section ◽  
Legendre Polynomials ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Coefficients ◽  
Beam Axis ◽  
Geometrical Parameters ◽  
Beam Model ◽  
Variable Cross ◽  
Thin Walled

AbstractThis article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

scholarly journals INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS

2019 ◽  
Vol 81 (4) ◽  
pp. 449-461
Author(s):  
V.V. Saurin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Cross Section ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
New Variables ◽  
Differential Relations

Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.

scholarly journals Flexural Free Vibrations of Multistep Nonuniform Beams

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Guojin Tan ◽  
Wensheng Wang ◽  
Yubo Jiao
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Cross Section ◽  
Order Differential Equation ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
Exact Approach ◽  
Constant Coefficients ◽  
One Step

This paper presents an exact approach to investigate the flexural free vibrations of multistep nonuniform beams. Firstly, one-step beam with moment of inertia and mass per unit length varying as I(x)=α11+βxr+4 and m(x)=α21+βxr was studied. By using appropriate transformations, the differential equation for flexural free vibration of one-step beam with variable cross section is reduced to a four-order differential equation with constant coefficients. According to different types of roots for the characteristic equation of four-order differential equation with constant coefficients, two kinds of modal shape functions are obtained, and the general solutions for flexural free vibration of one-step beam with variable cross section are presented. An exact approach to solve the natural frequencies and modal shapes of multistep beam with variable cross section is presented by using transfer matrix method, the exact general solutions of one-step beam, and iterative method. Numerical examples reveal that the calculated frequencies and modal shapes are in good agreement with the finite element method (FEM), which demonstrates the solutions of present method are exact ones.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

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