scholarly journals CALCULATION OF THE STABILITY OF A PLANE BENDING OF BEAMS WITH A VARIABLE CROSS-SECTIONAL WIDTH

2021 ◽  
Vol 9 (2) ◽  
pp. 41-45
Author(s):  
Serdar Yazyev
Keyword(s):  
Eigenvalue Problem ◽  
Twist Angle ◽  
Variable Cross ◽  
Rectangular Section ◽  
Concentrated Force ◽  
Cross Sectional ◽  
Generalized Eigenvalue ◽  
Angle Function ◽  
The Stability ◽  
Bending Of Beams

The article proposes a technique for calculating the side buckling of beams of variable rectangular section based on the energy method. This method is considered on the example of a two-section cantilever beam of variable width under the action of a concentrated force. The twist angle function was set in the form of a trigonometric series. As a result, the problem is reduced to a generalized eigenvalue problem.

scholarly journals Flat bending shape stability of the beams with variable section width

2020 ◽  
Vol 164 ◽  
pp. 02016
Author(s):  
Anastasia Lapina ◽  
Serdar Yazyev ◽  
Anton Chepurnenko ◽  
Irina Dubovitskaya
Keyword(s):  
Trigonometric Series ◽  
Cross Section ◽  
Secular Equation ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Shape Stability ◽  
Lateral Buckling ◽  
Concentrated Force ◽  
Angle Function ◽  
Variable Section

The paper proposes a methodology for calculating lateral buckling of beams of variable rectangular cross section based on the energy approach. The technique is considered on the example of a cantilever beam of variable width with two sections under the action of a concentrated force. The twist angle function was set in the form of a trigonometric series. As a result, the problem is reduced to a generalized secular equation.

scholarly journals CALCULATION OF THE FLAT BENDING SHAPE STABILITY OF RECTANGULAR CROSS SECTION BEAMS WITH REGARD TO CREEP

2019 ◽  
Vol 46 (1) ◽  
pp. 169-176
Author(s):  
I. M. Zotov ◽  
A. S. Chepurnenko ◽  
S. B. Yazyev
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Euler Method ◽  
Shape Stability ◽  
Concentrated Force ◽  
Flat Shape ◽  
The Stability

Objectives. The article presents the conclusion of the resolving equation for calculating the stability of the flat form of deformation of prismatic beams, taking into account the rheological properties of the material.Method. The problem is reduced to a second-order differential equation for the twist angle, which is solved numerically by the finite difference method in combination with the Euler method.Result. The obtained differential equation allows one to take into account the presence of initial imperfections in the form of the initial deflection of the beam, the initial angle of twist, and also the eccentricity of the applied load. The solution of the test problem for a cantilever polymer beam under the action of a concentrated force is presented. The non-linear Maxwell-Gurevich equation is used as the creep law. The value of the long-term critical load is introduced and it is shown that with a load less than the long-term critical creep is limited. It has been established that, as with the squeezed rods, with a load less than the long-term critical, the growth rate of the displacements with time decays. When F = F_dl, the displacements grow at a constant speed, and when F> F_dl, the growth rate of displacements increases with time. The results obtained confirm the validity of the chosen method.Conclusion. A universal resolving equation is obtained for calculating the stability of a flat shape of bending of rectangular beams, suitable for arbitrary creep laws.

A Numerical Approach to the Stability of Rotor-Bearing Systems

1982 ◽  
Vol 104 (2) ◽  
pp. 356-363 ◽  
Author(s):  
K. Athre ◽  
J. Kurian ◽  
K. N. Gupta ◽  
R. D. Garg
Keyword(s):  
Eigenvalue Problem ◽  
Characteristic Polynomial ◽  
Eigenvalue Problems ◽  
Numerical Technique ◽  
Numerical Approach ◽  
Generalized Eigenvalue ◽  
Generalized Eigenvalue Problems ◽  
Rotor Bearing ◽  
The Stability ◽  
Bearing System

The stability characteristics of a rotor-bearing system which indicate the threshold of instability are generally obtained by applying the Routh-Hurwitz criterion to the characteristic polynomial. Usually the characteristic polynomial is obtained analytically from the characteristic determinant. In the case of the generalized eigenvalue problems, this is practically impossible. To study the stability characteristics of a floating bush bearing, the characteristic polynomial is constructed from the generalized eigenvalue problem using a recently developed numerical technique. Results obtained through this computer package are compared with those already available in the literature.

Research on Calculation Method of the Stability Factor of Variable Cross-Sectional Concrete Filled Steel Tubular Laced Columns

2014 ◽  
Vol 587-589 ◽  
pp. 1420-1423
Author(s):  
Zhi Jing Ou ◽  
Lei Huang ◽  
Qiao Ling Yan
Keyword(s):  
Slenderness Ratio ◽  
Stability Factor ◽  
Computational Formula ◽  
Variable Cross ◽  
Analysis Method ◽  
Element Analysis ◽  
Cross Sectional ◽  
Finite Element Analysis Method ◽  
The Finite Element Analysis ◽  
The Stability

The finite element analysis method of variable cross-sectional Concrete-Filled Steel Tubular (CFST) laced columns is put forward. The influence of longitudinal elements slope and slenderness ratio on stability factor are analyzed, and the computational formula of slenderness ratio of variable cross-sectional CFST laced columns is presented. On the basis of the analytical results, a rational methodology for calculating the stability factor of four-element variable cross-sectional CFST laced columns is proposed.

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.

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.

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

Stability and Natural Characteristics of a Supported Beam

2011 ◽  
Vol 338 ◽  
pp. 467-472 ◽  
Author(s):  
Ji Duo Jin ◽  
Xiao Dong Yang ◽  
Yu Fei Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Axial Force ◽  
Critical Values ◽  
Rotational Spring ◽  
Analytical Expression ◽  
The Stability ◽  
Spring Constants ◽  
Critical Axial Force ◽  
Natural Characteristics

The stability, natural characteristics and critical axial force of a supported beam are analyzed. The both ends of the beam are held by the pinned supports with rotational spring constraints. The eigenvalue problem of the beam with these boundary conditions is investigated firstly, and then, the stability of the beam is analyzed using the derived eigenfuntions. According to the analytical expression obtained, the effect of the spring constants on the critical values of the axial force is discussed.

Stability of two-layer stratified flow in inclined channels: applications to air entrainment in coating systems

1996 ◽  
Vol 312 ◽  
pp. 173-200 ◽  
Author(s):  
Yuan C. Severtson ◽  
Cyrus K. Aidun
Keyword(s):  
Spectral Decomposition ◽  
Air Entrainment ◽  
Density Variation ◽  
Generalized Eigenvalue ◽  
Long Wave ◽  
Inclined Channels ◽  
The Stability ◽  
Air Liquid Interface ◽  
Sommerfeld Equation ◽  
Interfacial Mode

To understand the physics of air entrainment in thin-film liquid coating and other applications, the stability characteristics of general stratified two-layer Poiseuille-Couette flow are examined in inclined channels. Only one mode of instability, the interfacial mode, is obtained in the long-wave asymptotic limit. The generalized eigenvalue problem, formed by spectral decomposition and solution of the general two-layer Orr-Sommerfeld equation, is solved to obtain all of the critical modes. Analysis of the air/liquid interface corresponding to experiments reveals that because of the large density variation between the two layers, the interfacial mode is the only mode of instability in air entrainment. Results from the stability analysis of the flow near the contact line where air entrainment occurs are consistent with previous experimental observations.

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