A Tabular Collocation Method for Beam Vibration

1961 ◽  
Vol 83 (4) ◽  
pp. 373-376 ◽  
Author(s):  
R. Chicurel ◽  
E. Suppiger
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Integral Equation Method ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Lateral Vibration ◽  
Static Deflection ◽  
Preliminary Step ◽  
Stepped Beam

This paper presents a procedure, based on the integral equation method, for the calculation of the natural frequencies of lateral vibration of beams with variable cross section. The approximate solution is obtained by collocation [1, 2]. A preliminary step in the analysis is the determination of static deflection curves; this is carried out in a convenient tabular form. An example of a stepped beam is given and the results are compared to those obtained by Myklestad’s method [3].

Determination of the Natural Frequencies of the Bending Vibrations of Beams

1947 ◽  
Vol 14 (1) ◽  
pp. A1-A6
Author(s):  
A. I. Bellin
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Moment Equation ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Bending Vibrations ◽  
Elastic Beams ◽  
Lateral Vibrations

Abstract This paper presents a method for determining the natural frequencies of lateral vibrations for elastic beams. The beams may be of variable cross section and may have any number of spans. The five-moment equation is developed and is then applied to beams supported in various ways. The author reduces the necessary calculations to a simple tabular scheme. Several illustrative examples are included to demonstrate the method of computation.

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.

Geometrical Model for Permeable Material’s Porous Structure

2019 ◽  
Vol 7 (2) ◽  
pp. 42-54 ◽  
Author(s):  
А. Синцов ◽  
A. Sintsov ◽  
Владимир Девисилов ◽  
Vladimir Devisilov
Keyword(s):  
Porous Material ◽  
Porous Structure ◽  
Cross Section ◽  
Structural Characteristics ◽  
Variable Cross Section ◽  
Geometrical Model ◽  
Variable Cross ◽  
Experimental Stand ◽  
Experimental Technology

In this paper have been presented a new model of the porous structure, as well as an analysis of possibilities of a new method for experimental investigation of porous permeable materials and determination of their structural characteristics. An analysis for the majority of used in analytical calculations geometric models for a porous medium has been presented, and a model for a porous material in the form of porous matrix’s elementary cells has been proposed. Each of the cells contains a capillary channel with a variable cross-section. Volumetric structural characteristics, as well as dependencies of surface structural characteristics over the porous matrix’s thickness, are identical to these parameters, which have been obtained during the experimental study of a porous material. As a result of use of an original experimental technology offered by authors, and of experiment processing the porous matrix’s structure can be completely defined. The problem of creating an experimental setup, allowing determine the porous matrix’s characteristics, has been formulated. One of possible options for the experimental stand has been considered.

Vibration analysis of sandwich beams with variable cross section on variable Winkler elastic foundation

2013 ◽  
Vol 20 (4) ◽  
pp. 359-370 ◽  
Author(s):  
Ersin Demir ◽  
Hasan Çallioğlu ◽  
Metin Sayer
Keyword(s):  
Elastic Foundation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Slenderness Ratio ◽  
Sandwich Beam ◽  
Variable Cross Section ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Variable Cross ◽  
Winkler Elastic Foundation

AbstractIn this study, free vibration behavior of a multilayered symmetric sandwich beam made of functionally graded materials (FGMs) with variable cross section resting on variable Winkler elastic foundation are investigated. The elasticity and density of the functionally graded (FG) sandwich beam vary through the thickness according to the power law. This law is related to mixture rules and laminate theory. In order to provide this, a 50-layered beam is considered. Each layer is isotropic and homogeneous, although the volume fractions of the constituents of each layer are different. Furthermore, the width of the beam varies exponentially along the length of the beam, and also the beam is resting on an elastic foundation whose coefficient is variable along the length of the beam. The natural frequencies are computed for conventional boundary conditions of the FG sandwich beam using a theoretical procedure. The effects of material, geometric, elastic foundation indexes and slenderness ratio on natural frequencies and mode shapes of the beam are also computed and discussed. Finally, the results obtained are compared with a finite-element-based commercial program, ANSYS®, and found to be consistent with each other.

Dynamical Stress Distribution Analysis of a Non-Uniform Cross-Section Beam Under Moving Mass

2006 ◽  
Author(s):  
M. T. Ahmadian ◽  
E. Esmailzadeh ◽  
M. Asgari
Keyword(s):  
Stress Distribution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Moving Load ◽  
Variable Cross Section ◽  
Moving Mass ◽  
Variable Cross ◽  
Distribution Analysis ◽  
Concentrated Force ◽  
Direct Integration Method

One of the engineers concern in designing bridges and structures under moving load is the uniformity of stress distribution. In this paper the analysis of a variable cross-section beam subjected to a moving concentrated force and mass is investigated. Finite element method with cubic Hermitian interpolation functions is used to model the structure based on Euler-Bernoulli beam and Wilson-Θ direct integration method is implemented to solve time dependent equations. Effects of cross-section area variation, boundary conditions, and moving mass inertia on the deflection, natural frequencies and longitudinal stresses of beam are investigated. Results indicates using a beam of parabolically varying thickness with constant mass can decrease maximum deflection and stresses along the beam while increasing natural frequencies of the beam. The effect of moving mass inertia of moving load is found to be significant at high velocity.

scholarly journals The Vibration of Tubular Beam Conveying Fluid with Variable Cross Section

2020 ◽  
Vol 65 (1) ◽  
pp. 56-62
Author(s):  
Mohamed Gaith
Keyword(s):  
Flow Velocity ◽  
Cross Section ◽  
Natural Frequencies ◽  
Variable Cross Section ◽  
System Stability ◽  
Variable Cross ◽  
Cross Sectional ◽  
Critical Flow Velocity ◽  
Constant Flow Rate ◽  
The Impact

The dynamics and stability of flow induced vibration of flow conveying in pipes particularly in case of high velocity flow may lead to severe damage. Predicting the circular natural frequencies and critical fluid velocities is an important tool in design and prevent system failures. In this study transverse dynamic response of simply supported pipe with variable tubular cross sectional area carrying fluid with a constant flow rate is investigated. Euler Bernoulli's beam theory is used to model the pipe. Hamilton's principle will be used to produce the governing equation of motion for the system. The resulting partial differential equation is solved using Galerkin's technique. The impact of the flow velocity and non-uniform variable cross section on the natural frequencies of the system, critical flow velocity and system stability is presented.

scholarly journals DETERMINATION OF DIMENSIONAL PARAMETERS FOR ANNULAR CONCENTRATOR OF ULTRASONIC SYSTEM

2018 ◽  
Vol 17 (1) ◽  
pp. 51-55
Author(s):  
V. P. Lugovoi ◽  
I. V. Lugovoi
Keyword(s):  
Cross Section ◽  
Acoustic Waves ◽  
Visual Analysis ◽  
Variable Cross Section ◽  
Vibrational Amplitude ◽  
External Diameter ◽  
Variable Cross ◽  
Thin Sections ◽  
Dimensional Parameters

Conventional ultrasonic systems contain concentrators of longitudinal type to amplify and transfer vibrations to a tool. Along with them, elastic rings with variable cross section thickness can be used effectively as concentrators for ultrasonic vibrations of acoustic systems. Their practical application requires a scientifically substantiated methodology for determination of geometric parameters. The paper provides substantiation of the method used to determine dimensions of annular concentrators with a variable cross section which can enhance effectiveness of the ultrasound equipment while performing various technological tasks. Visual analysis of acoustic waves radiated by annular concentrators has shown that the most intensive vibrations are produced in the most thin sections. Computer simulation of oscillations in rings with an external diameter of 50 mm and a variable cross section has demonstrated that the largest increase of vibration amplitude is achieved at a certain ratio of ring thicknesses and diameters. An analysis of numerical values for amplication factor of vibrational amplitude Kд has revealed that there is a limit boundary for the ratio of ring wall thicknesses which, in its turn, depends on the ratio of ring outer and inner diameters at certain values of hole axis eccentricity. The ratio of diameters is expressed quantitatively by the coefficient Kд. An analysis of the results concerning numerical calculations of amplitude amplification factor performed for the specified model of the ring having 50 mm diameter have illustrated that this ratio should lie between 1.3 > Kд > 1.15. The obtained results can be used in ultrasonic devices with annular concentrators in order to perform various technological tasks.

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