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.

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].

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.

scholarly journals The lateral vibrations of bars of variable section

1917 ◽  
Vol 93 (654) ◽  
pp. 506-519 ◽  
Keyword(s):  
Cross Section ◽  
Formal Analysis ◽  
Detailed Examination ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Biological Application ◽  
Lateral Vibrations ◽  
Variable Section ◽  
Set Up ◽  
The Impact

The present investigation, though strictly mathematical in character, arose in connection with a suggestion, put forward by Prof. A. Dendy and the present author in another paper communicated to the Society, that the siliceous deposits found on certain sponge spicules occurred at nodes of the spicules, regarded as vibrating rods. These vibrations, being set up and maintained by the impact of currents of water on the spicules, are necessarily of the lateral type. For the detailed examination of such a suggestion, it is necessary to obtain a comprehensive account of the positions of the funda­mental nodes on a free-free bar, as dependent on the law of variation of its cross-section. The present paper contains, in fact, the formal analysis whose results were quoted without proof in the other paper. This analysis is of considerable generality, as will appear, and the particular examples selected for purposes of illustration, together with the manner in which the variable cross-section is dealt with, have been determined by the requirements of the biological application already mentioned. One general problem is in view throughout the work, and it may be stated as follows

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.

Sign in / Sign up

Export Citation Format

Share Document