THE NON-CLASSICAL PROBLEM OF AN ORTHOTROPIC BEAM OF VARIABLE THICKNESS WITH THE SIMULTANEOUS ACTION OF ITS OWN WEIGHT AND COMPRESSIVE AXIAL FORCES

2019 ◽  
Vol 53 (3 (250)) ◽  
pp. 183-190
Author(s):  
R.M. Kirakosyan ◽  
S.P. Stepanyan
Keyword(s):  
Collocation Method ◽  
Variable Thickness ◽  
Transverse Shear ◽  
Specific Problem ◽  
Classical Problem ◽  
Compressive Force ◽  
Simultaneous Action ◽  
Axial Forces ◽  
The Stability

thickness, the equations of the problem of bending of an elastically clamped beam in the case of simultaneous action of its own weight and axial compressive forces are obtained. The effects of transverse shear and the effect of reducing the compressive force of the support are taken into account. Turning to dimensionless quantities, the specific problem for a beam of linearly variable thickness is solved by the collocation method. The question of the stability of the beam is discussed.

ON THE NUMERICAL SOLUTION TO A NON-CLASSICAL PROBLEM OF BENDING AND STABILITY FOR AN ORTHOTROPIC BEAM OF VARIABLE THICKNESS

2021 ◽  
pp. 111-120
Author(s):  
S.P. Stepanyan ◽  
Keyword(s):  
Variable Thickness ◽  
Classical Problem ◽  
Compressive Force ◽  
Variable Coefficients ◽  
Simultaneous Action ◽  
Refined Theory ◽  
Orthotropic Plates ◽  
Axial Compressive Force ◽  
Axial Forces ◽  
The Stability

The mathematical model of the problem of bending of an elastically clamped beam is constructed on the basis of the refined theory of orthotropic plates of variable thickness. To solve the problem in the case of simultaneous action of its own weight and compressive axial forces, a system of differential equations with variable coefficients is obtained. The effects of transverse shear and the effect of reducing compressive force of the support are also taken into account. Passing on to dimensionless quantities, the specific problem for a beam of linearly varying thickness is solved by the collocation method. The stability of the beam is discussed. The critical values of forces are obtained by varying the axial compressive force. Results are presented in both tabular and graphical styles. Based on the results obtained, appropriate conclusions are drawn.

THE NON-CLASSICAL PROBLEM OF AN ELASTICALLY CLAMPED ORTHOTROPIC BEAM OF VARIABLE THICKNESS UNDER THE COMBINED ACTION OF COMPRESSIVE FORCES AND TRANSVERSE LOAD

2018 ◽  
Vol 52 (2 (246)) ◽  
pp. 101-108
Author(s):  
R.M. Kirakosyan ◽  
S.P. Stepanyan
Keyword(s):  
Variable Thickness ◽  
Classical Problem ◽  
Compressive Force ◽  
Transverse Load ◽  
Simultaneous Action ◽  
Refined Theory ◽  
Orthotropic Plates ◽  
Combined Action ◽  
Beam Bending ◽  
The Stability

On the basis of the refined theory of orthotropic plates of variable thickness, the equations of the beam bending problem are obtained with the simultaneous action of compressive forces and transverse load. It is accepted that the edges of the beam have an elastically clamped support and the reduction of the compressive force by the support due to friction is taking into account. Passing to dimensionless quantities, a certain problem is solved. The stability of a beam is discussed. Based on the results obtained, conclusions are drawn.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

scholarly journals Determination of Transverse Shear Stiffness of Sandwich Panels with a Corrugated Core by Numerical Homogenization

Materials ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 1976
Author(s):  
Tomasz Garbowski ◽  
Tomasz Gajewski
Keyword(s):  
Transverse Shear ◽  
Shear Stiffness ◽  
The Finite Element Method ◽  
Homogenization Procedure ◽  
Plate Structures ◽  
Corrugated Board ◽  
Energy Equivalence ◽  
The Stability ◽  
Global Stiffness Matrix

Knowing the material properties of individual layers of the corrugated plate structures and the geometry of its cross-section, the effective material parameters of the equivalent plate can be calculated. This can be problematic, especially if the transverse shear stiffness is also necessary for the correct description of the equivalent plate performance. In this work, the method proposed by Biancolini is extended to include the possibility of determining, apart from the tensile and flexural stiffnesses, also the transverse shear stiffness of the homogenized corrugated board. The method is based on the strain energy equivalence between the full numerical 3D model of the corrugated board and its Reissner-Mindlin flat plate representation. Shell finite elements were used in this study to accurately reflect the geometry of the corrugated board. In the method presented here, the finite element method is only used to compose the initial global stiffness matrix, which is then condensed and directly used in the homogenization procedure. The stability of the proposed method was tested for different variants of the selected representative volume elements. The obtained results are consistent with other technique already presented in the literature.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

scholarly journals The Krylov problem in the case of a beam on Vlasov inertial foundation

2018 ◽  
Vol 19 (6) ◽  
pp. 728-736
Author(s):  
Wacław Szcześniak ◽  
Magdalena Ataman
Keyword(s):  
Closed Form ◽  
Compressive Force ◽  
Elastic Beam ◽  
Forced Vibrations ◽  
Critical Force ◽  
Winkler Foundation ◽  
Static Deflection ◽  
Numerical Examples ◽  
Moving Force ◽  
Axial Forces

The paper deals with vibrations of the elastic beam caused by the moving force traveling with uniform speed. The function defining the pure forced vibrations (aperiodic vibrations) is presented in a closed form. Dynamic deflection of the beam caused by moving force is compared with the static deflection of the beam subjected to the force , and compressed by axial forces . Comparing equations (9) and (13), it can be concluded that the effect on the deflection of the speed of the moving force is the same as that of an additional compressive force . Selected problems of stability of the beam on the Winkler foundation and on the Vlasov inertial foundation are discussed. One can see that the critical force of the beam on Vlasov foundation is greater than in the case of Winkler's foundation. Numerical examples are presented in the paper

Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach

2021 ◽  
Vol 24 (3) ◽  
pp. 895-922
Author(s):  
Platon G. Surkov
Keyword(s):  
Control Theory ◽  
Fractional Derivative ◽  
Approximate Calculation ◽  
Classical Problem ◽  
Tikhonov Regularization Method ◽  
Inaccurate Data ◽  
Local Modification ◽  
Ill Posed ◽  
Computational Errors ◽  
The Stability

Abstract A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

scholarly journals Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983445
Author(s):  
Ma’en S Sari ◽  
Wael G Al-Kouz ◽  
Rafat Al-Waked
Keyword(s):  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Engineering Structures ◽  
Axial Forces ◽  
Double Beam ◽  
Tensile Forces ◽  
Torsional Coupling ◽  
The Stability

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

Stability Prediction in Turning of Flexible Components

2010 ◽  
Vol 112 ◽  
pp. 149-157 ◽  
Author(s):  
Gorka Urbicain ◽  
David Olvera ◽  
Luis Norberto López de Lacalle ◽  
Francisco Javier Campa
Keyword(s):  
Machine Tool ◽  
Surface Quality ◽  
Classical Problem ◽  
Cutting Conditions ◽  
Stability Prediction ◽  
Nose Radius ◽  
The Stability ◽  
Tool Nose Radius ◽  
Flexible Components

Chatter is the most classical problem in machining. It is prone to occur in low rigidity structures generating poor surface quality and harmful vibrations which could damage any part of the machine-tool system. In finishing operations, the effect of the tool nose radius should be taken into account in order to obtain safe and reliable cutting conditions. The present paper uses a simple SDOF model to study the stability during finishing operations.

Sign in / Sign up

Export Citation Format

Share Document