scholarly journals Design of a beam of variable cross-section on the elastic base by the quasi-analytical method considering boundary conditions

2021 ◽  
pp. 73-76
Author(s):  
Р. Shtanko ◽  
S. Ryagin ◽  
І. Geletiy ◽  
А. Kononenko
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Nonlinear Differential Equation ◽  
Cross Section ◽  
Analytical Method ◽  
Variable Cross Section ◽  
Elastic Base ◽  
Variable Cross ◽  
Equation Solution ◽  
Linear Algebraic Equations

Purpose. Improvement of the quasi-analytical method of nonlinear differential equation solution and its approbation with reference to beams of variable cross-section on the elastic base with two base factors. Research methods. Boundary conditions in the form of required number of correspondently transformed equations are added to the system of the linear algebraic equations which results from substitution of approximating function with constant factors (for example – power function) in the nonlinear differential equation and fixation of a set of variable values. The total number of the equations have to correspond to quantity of constant factors if the further solution will be carried out by an analytical method. Results. Deflection diagram of a trapezoid concrete beam with rectangular cross-section of variable height on the elastic base with two base factors has been calculated during approbation. Average solution error was equal to 0.06%. Distributions of the bending moments and normal stresses along the beam have been researched. Scientific novelty. The authors did not meet in literature such method of nonlinear differential equation solution. Practical value. The quasi-analytical method with realised consideration of boundary conditions that has been offered can be used for solution of differential equations of any order with various types of nonlinearity, including calculations of beams of variable cross-section on the elastic base.

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.

scholarly journals Closed-form solutions for vibrations of a magneto-electro-elastic beam with variable cross section by means of Green’s functions

2018 ◽  
Vol 30 (1) ◽  
pp. 82-99 ◽  
Author(s):  
Xuan Ling Zhang ◽  
Xiao Chao Chen ◽  
Echuan Yang ◽  
Hai Feng Li ◽  
Jian Bo Liu ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Electric Potential ◽  
Green's Functions ◽  
Green’S Functions ◽  
Elastic Beam ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Cross Sectional ◽  
Closed Form Solutions

In this article, closed-form solutions are obtained for vibrations of a magneto-electro-elastic beam with variable cross section. Based on Timoshenko beam assumptions, governing equation for the non-uniform beam with exponentially varying width is obtained. Laplace transform approach applied to the governing equation results in the corresponding Green’s functions for the beams with various boundary conditions. The equations, which are solved to obtain Green’s functions, are degenerated for the analyses of the characters of free vibration. For free vibrations of the beams under different mechanical boundary conditions, the effects of the non-uniformly cross-sectional parameters and magneto-electric boundary conditions on the dynamic characters are studied. In addition, the magneto-electric potential modal variables’ distributions through the thickness are presented. In the discussions of forced vibration, two points in the beam are selected to investigate frequency responses in terms of displacement and magneto-electric potential. Moreover, the influences of excitation frequency and cross-sectional parameter on through-the-thickness distributions of electric potentials are investigated.

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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