scholarly journals TRANSVERSE OSCILLATIONS OF THE BEAM ON AN ELASTIC BASE IF THE BOUNDARY CONDITIONS CHANGE

2020 ◽  
Vol 16 (3) ◽  
pp. 35-46
Author(s):  
Yevgeny Leontiev
Keyword(s):  
Boundary Conditions ◽  
Dynamic Load ◽  
Arbitrary Point ◽  
Elastic Base ◽  
Base Change ◽  
Attached Mass ◽  
Dynamic Action ◽  
Transverse Oscillations ◽  
Free Edges

The article deals with the proper transverse oscillations of a beam with free edges while the conditions of support on an elastic base change, taking into account its own weight and the influenceof the attached mass m1. The problem of determining the forces in the beam is being solved taking into account the dynamic load F(t) applied at an arbitrary point d while the conditions for the support of a part of the beam on an elastic base change.The conditions that must be taken into account while analyzing the dynamic action of the structure under the influenceof variable loads in the case of changes in the conditions of support on an elastic base are formulated.

Analysis of vibration in rotating pretwisted functionally graded graphene platelets reinforced nanocomposite laminated blades with an attached point mass

2021 ◽  
pp. 095440622110084
Author(s):  
Amin Ghorbani Shenas ◽  
Parviz Malekzadeh ◽  
Sima Ziaee
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Polymer Matrix ◽  
Natural Frequencies ◽  
Weight Fraction ◽  
Functionally Graded ◽  
Point Mass ◽  
Attached Mass ◽  
Graphene Platelets ◽  
Twisted Beam

This work presents an investigation on the free vibration behavior of rotating pre-twisted functionally graded graphene platelets reinforced composite (FG-GPLRC) laminated blades/beams with an attached point mass. The considered beams are constituted of [Formula: see text] layers which are bonded perfectly and made of a mixture of isotropic polymer matrix and graphene platelets (GPLs). The weight fraction of GPLs changes in a layer-wise manner. The effective material properties of FG-GPLRC layers are computed by using the modified Halpin-Tsai model together with rule of mixture. The free vibration eigenvalue equations are developed based on the Reddy’s third-order shear deformation theory (TSDT) using the Chebyshev–Ritz method under different boundary conditions. After validating the approach, the influences of the GPLs distribution pattern, GPLs weight fraction, angular velocity, the variation of the angle of twist along the beam axis, the ratio of attached mass to the beam mass, boundary conditions, position of attached mass, and geometry on the vibration behavior are investigated. The findings demonstrate that the natural frequencies of the rotating pre-twisted FG-GPLRC laminated beams significantly increases by adding a very small amount of GPLs into polymer matrix. It is shown that placing more GPLs near the top and bottom surfaces of the pre-twisted beam is an effective way to strengthen the pre-twisted beam stiffness and increase the natural frequencies.

Vibration of Stiffened Rectangular Plates

1963 ◽  
Vol 67 (629) ◽  
pp. 305-307 ◽  
Author(s):  
S. Mahalingam
Keyword(s):  
Boundary Conditions ◽  
Present Note ◽  
Ritz Method ◽  
Arbitrary Parameter ◽  
Rectangular Plates ◽  
Various Boundary Conditions ◽  
Beam Function ◽  
Rayleigh Method ◽  
The Given ◽  
Free Edges

The free flexural vibrations of rectangular plates with various boundary conditions have been considered by Warburton. The natural frequencies were calculated by the Rayleigh method, the mode assumed being the product of the characteristic beam functions for the given boundary conditions. Comparison with experimental results shows that the method gives reasonably good approximations. The present note describes a method of obtaining the approximately equivalent characteristic beam functions to enable Warburton's method to be extended to plates having one or more stiffeners parallel to an edge. As a numerical example expressions for the frequencies are derived for a plate, simply supported along two opposite edges, and having a central stiffener parallel to the other two free edges. The results are compared with those given in a recent note by Kirk, who solved the same problem by the Rayleigh-Ritz method, using a mode with one arbitrary parameter. In the case of the fundamental frequency of the unstiffened plate, the characteristic beam function in a direction perpendicular to the free edges is simply a constant, and the solution is less accurate than that given by the Rayleigh-Ritz method. However, numerical analysis of a square plate shows that above a certain stiffener depth the characteristic beam function method is more accurate than the Rayleigh-Ritz method. The two methods are also compared for the 2/2 mode.

A Novel and Accurate Ritz Formulation for Free Vibration of Rectangular and Skew Plates

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
S. A. Eftekhari ◽  
A. A. Jafari
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Rectangular Plates ◽  
Natural Boundary ◽  
Boundary Points ◽  
Natural Boundary Conditions ◽  
Free Edges

One of the major limitations of the conventional Ritz method is its difficulty in implementation to the differential equations with natural boundary conditions at the boundary points/lines. Plates involving free edges/corners and irregularly shaped plates are two historical and classical examples which show that their solutions cannot be accurately approximated by the conventional Ritz method. To solve this difficulty, a simple, novel, and accurate Ritz formulation is introduced in this paper. It is revealed that the proposed methodology can produce much better accuracy than the conventional Ritz method for rectangular plates involving free edges/corners and skew plates.

scholarly journals Vibration analysis of carbon nanotubes-based zeptogram masses sensors and taking into account their rotatory inertia

2018 ◽  
Vol 149 ◽  
pp. 02087 ◽  
Author(s):  
A. Azrar ◽  
L. Azrar ◽  
A. A. Aljinaidi
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Methodological Approach ◽  
Nonlocal Parameter ◽  
Research Work ◽  
Mode Shapes ◽  
Attached Mass ◽  
Rotatory Inertia ◽  
Mass Sensors ◽  
Walled Carbon Nanotubes

In this research work, the transverse vibration behaviour of single-walled carbon nanotubes (SCNT) based mass sensors is studied using the Timoshenko beam and nonlocal elasticity theories. The nonlocal constitutive equations are used in the formulations and the CNT with different lengths, attached mass (viruses and bacteria) and the general boundary conditions are considered. The dimensionless frequencies and associated modes are obtained for one and two attached masses and different boundary conditions. The effects of transverse shear deformation and rotatory inertia, nonlocal parameter, length of the carbon nanotubes, and attached mass and its location are investigated in detail for each considered problem. The relationship between the frequencies and mode shapes of the sensor and the attached zeptogramme masses are obtained. The sensing devices for biological objects including viruses and bacteria can be elaborated based on the developed sensitivity and frequency shift methodological approach.

scholarly journals QUASIHARMONIC BENDING WAVE, DISTRIBUTING IN THE BALK OF TIMOSHENKO, LYING ON A NONLINEAR ELASTIC BASE

2021 ◽  
Vol 83 (1) ◽  
pp. 61-75
Author(s):  
V.I. Erofeev ◽  
A.V. Leontieva
Keyword(s):  
Cross Sections ◽  
Modulation Instability ◽  
Good Description ◽  
Elastic Base ◽  
Base Change ◽  
Flexural Wave ◽  
Bending Vibrations ◽  
Technical Theory ◽  
Basic Equations ◽  
Nonlinear Elastic

In this paper, we consider the modulation instability of a quasiharmonic flexural wave propagating in a homogeneous beam fixed on a nonlinear elastic foundation. The dynamic behavior of the beam is determined by Timoshenko's theory. Timoshenko's model, refining the technical theory of rod bending, assumes that the crosssections remain flat, but not perpendicular to the deformable midline of the rod; normal stresses on sites parallel to the axis are zero; the inertial components associated with the rotation of the cross sections are taken into account. The uniqueness of the model lies in the fact that, allowing a good description of many processes occurring in real structures, it remains quite simple, accessible for analytical research. The system of equations describing the bending vibrations of the beam is reduced to one nonlinear fourthorder equation for the transverse displacements of the beam particles. The nonlinear Schrödinger equation, one of the basic equations of nonlinear wave dynamics, is obtained by the method of many scales. Regions of modulation instability are determined according to the Lighthill criterion. It is shown hot the boundaries of these areas shift when the parameters characterizing the elastic properties of the beam material and the nonlinearity of the base change. Nonlinear stationary envelope waves are considered. An equation that generalizes the Duffing equation, which contains two additional terms in negative powers (first and third), is obtained and qualitatively analyzed. Solutions of the Schr?dinger equation in the form of envelope solitons are found and the dependences of their main parameters (amplitude, width) on the parameters of the system are analyzed. The dynamics of the points of intersection of the amplitudes and widths of "light" solitons in the case of soft nonlinearity of the base is shown within the region of modulation instability.

scholarly journals ASYMPTOTICS OF CRITICAL LOADS OF A COMPRESSED NARROW ELASTIC PLATE WITH INTERNAL STRESSES

2021 ◽  
Vol 83 (2) ◽  
pp. 227-234
Author(s):  
I.M. Peshkhoev
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Small Parameter ◽  
Critical Load ◽  
Boundary Value Problems ◽  
Stress Function ◽  
Boundary Value ◽  
Elastic Base ◽  
Internal Stresses ◽  
Short Side

The problem of the asymptotic solution of a modified system of nonlinear Karman equilibrium equations for a longitudinally compressed elongated elastic rectangular plate with internal stresses lying on an elastic base is considered. Internal stresses can be caused by continuously distributed edge dislocations and wedge disclinations, or other sources. The compressive pressure is applied parallel to the long sides of the plate to the two short edges. The boundary conditions are considered: the long edges of the plate are free from loads, and the short edges are freely pinched or movably hinged. A small parameter is introduced, equal to the ratio of the short side of the plate to the long side. The solution of the system – the compressive load, the deflection function, and the stress function – is sought in the form of series expansions over a small parameter. The system of Karman equations with dimensionless variables is reduced to an infinite system of boundary value problems for ordinary differential equations with respect to the coefficients of asymptotic expansions for the critical load, deflection, and stress function. In this case, to meet the boundary conditions, the boundary layer functions are additionally introduced, which are concentrated near the fixed edges and disappear when moving away from them. Boundary value problems for determining the functions of the boundary layer are constructed. It is shown that the main terms of the small parameter expansions for the critical load and deflection are determined from the equilibrium equation of a compressed beam on an elastic base with the boundary conditions of free pinching or movable hinge support of the ends. In this case, the main term of the expansion into a series of the stress function has a fourth order of smallness in the parameter of the relative width of the plate.

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