The natural frequencies of longitudinal vibrations of bars with random parameters

1967 ◽  
Vol 3 (3) ◽  
pp. 18-19
Author(s):  
I. V. Ignatov ◽  
V. M. Kuz'ma
Keyword(s):  
Natural Frequencies ◽  
Random Parameters ◽  
Longitudinal Vibrations

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.

Effect of compact inclusion on the natural frequencies of vibrations of a pipe string in a well

2021 ◽  
pp. 73-77
Author(s):  
A.M. Svalov ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Natural Frequencies ◽  
Oscillation Period ◽  
Drill String ◽  
Analytical Relation ◽  
Longitudinal Vibrations ◽  
Bottom Hole ◽  
Tubing String ◽  
Analytical Relations

The influence of small-size inclusion of pipes in a well column on the natural frequency of its longitudinal vibrations is investigated. Using the asymptotic expansion in a small parameter, an analytical relation is obtained that describes the change in the period of the column oscillations in the form of some additional small term to the period of the homogeneous column oscillations. Numerical calculations show that the obtained analytical relations almost accurately describe the oscillation period of a column with a massive compact inclusion, while its difference from the oscillation period of a homogeneous column is within ~20%. The results obtained can be useful for preventing resonant phenomena in the drill string when drilling wells, as well as for optimal use of the longitudinal vibrations of the tubing string to influence the bottom-hole zones of producing wells.

Modeling and Free Vibration Analysis of a Complex Cable-Stayed Bridge

2012 ◽  
Author(s):  
D. Q. Cao ◽  
M. T. Song ◽  
W. D. Zhu
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Longitudinal Vibrations ◽  
Simply Supported ◽  
Cable Stayed Bridge ◽  
Linear Partial Differential Equations ◽  
Localized Mode ◽  
Linearized Model

A complex cable-stayed bridge that consists of a simply-supported four-cable-stayed deck beam and two rigid towers is studied. The nonlinear and linear partial differential equations that govern the motions of the cables and segments of the deck beam, respectively, are derived, along with their boundary and matching conditions. The undamped natural frequencies and mode shapes of the linearized model of the cable-stayed bridge, which includes both the transverse and longitudinal vibrations of the cables, are determined. Numerical analysis of the natural frequencies and mode shapes of the cable-stayed bridge is conducted for a symmetrical case with regards to the sizes of the components of the bridge and the initial sags of the cables. The results show that there are very close natural frequencies and localized mode shapes.

scholarly journals Determination of the variable density of the rod from natural frequencies of longitudinal vibrations

2021 ◽  
Vol 2099 (1) ◽  
pp. 012049
Author(s):  
I M Utyashev
Keyword(s):  
Production Process ◽  
Natural Frequencies ◽  
Polynomial Function ◽  
Variable Density ◽  
Density Variation ◽  
Longitudinal Vibrations ◽  
Vibration Diagnostics ◽  
Finite Set ◽  
The Law

Abstract Rods of various configurations are elements of many structures and machines. Therefore, the acoustic and vibration diagnostics of such parts has been widely developed. The paper considers the problem of determining the variable density of the rod from the natural frequencies of longitudinal vibrations. It is assumed that the density changes along the axis and is described by a polynomial function. This approach allows one to determine the law of density variation from a finite set of eigenvalues. The results of the study can find applications for finding hidden defects in steel and composite rods, which arise during the production process or due to corrosion.

Resonance Reliability Sensitivity Analysis for Torsional Vibration of Gear-Rotor Systems With Random Parameters

2017 ◽  
Author(s):  
Jiwei Qiu ◽  
Jianguo Zhang ◽  
Yupeng Ma
Keyword(s):  
Natural Frequencies ◽  
Limit State ◽  
Rayleigh Quotient ◽  
Rotor System ◽  
Random Parameters ◽  
Rotor Systems ◽  
Stiffness Matrices ◽  
Reliability Sensitivity ◽  
Reliability Sensitivity Analysis ◽  
Mc Simulation

In this presented work, a reliability sensitivity analyzing method was proposed for the resonance failures of gear-rotor systems with multiple random parameters. First, eigenvectors corresponding to the natural frequencies of a gear-rotor system governed by deterministic parameters were deduced. Mass and stiffness matrices were then decomposed into sub-matrices in the form of deterministic matrices multiplied by random parameters. Rayleigh quotient formula was utilized to derive the explicit expressions of natural frequencies of the system. Then, limit state functions of resonant failures of the system under an external load with random excited frequency was constructed based on vibration stability criterion. Reliability sensitivity analyzing method was applied to obtain sensitivities of random parameters on the resonant reliability of the gear-rotor system. Finally, a numerical case was given to illustrate the effectiveness and accuracy of the proposed method by comparing with Monte Carlo (MC) simulation.

scholarly journals On the correspondence of theoretical models of longitudinal vibrations of a rod with experimental data

2021 ◽  
Vol 8 (2) ◽  
pp. 270-281
Author(s):  
Alexander L. Popov ◽  
◽  
Sergei A. Sadovsky ◽  
Keyword(s):  
Experimental Data ◽  
Wave Equation ◽  
Theoretical Model ◽  
Natural Frequencies ◽  
Lateral Displacement ◽  
Experimental Spectrum ◽  
Theoretical Models ◽  
Transverse Displacement ◽  
Longitudinal Vibrations ◽  
Longitudinal Waves

A number of theoretical models are known for describing longitudinal vibrations of a rod. The simplest and most common is based on the wave equation. Next comes a model that takes into account lateral displacement (Rayleigh correction). The Bishop model is considered to be more perfect, taking into account both transverse displacement and shear deformation. It would seem that the more perfect the theoretical model, the better it should be consistent with experimental data. Nevertheless, when comparing with a really defined experimental spectrum of longitudinal vibrations of a rod on a large base of natural frequencies, it turns out that this is not quite so. Moreover, in the relative loss is the most complex Bishop model. Comparisons were made for a smooth long cylindrical rod. The questions of refinement with the help of experimentally found frequencies of the velocity of longitudinal waves and the Poisson’s ratio of the rod material are also touched.

Coupled Flexural–Longitudinal Vibrations of Timoshenko Double-Beam Systems Induced by Mass Eccentricities

2016 ◽  
Vol 08 (05) ◽  
pp. 1650067
Author(s):  
Jian Wang ◽  
Zhenguo Zhang ◽  
Hongxing Hua
Keyword(s):  
Natural Frequencies ◽  
Flexural Vibration ◽  
Longitudinal Direction ◽  
Forced Response ◽  
Beam Model ◽  
Vertical Force ◽  
Governing Equations ◽  
Longitudinal Vibrations ◽  
Mass Eccentricity ◽  
Double Beam

In many instances, marine vessel may suffer unexpected flexural–longitudinal vibrations as the consequences of the unevenly distribution of the inner structures and equipment. In order to study the dynamic behaviors of the marine vessel with the involvement of the mass unevenness, the whole vessel is modeled as a discretely connected double-beam system in this paper. In this model, the misalignment between geometric and gravitational center are considered as the simplification of the mass unevenness. The governing equations of coupled flexural–longitudinal vibrations of Timoshenko beam due to mass eccentricity are derived. Extending to the double-beam model, the natural frequencies, modal shapes and forced response of the system are obtained analytically by using the modified transfer matrix method. The comparison between the present method and the conventional finite element method shows a great match. The result indicates that the existence of mass eccentricity can cause significant coupled flexural–longitudinal vibrations, thus, even only vertical force is applied, the vibration in longitudinal direction can be excited and controlled by the natural frequencies of the flexural vibration.

An Analytical Study of Dynamic Characteristics of Multi-Story Timoshenko Planar Frame Structures

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
C. Mei
Keyword(s):  
Vibration Analysis ◽  
Analytical Study ◽  
Natural Frequencies ◽  
Frame Structures ◽  
Longitudinal Vibrations ◽  
Structural Element ◽  
Reflection And Transmission ◽  
Shear Distortion ◽  
Transmission And Reflection ◽  
Good Agreement

This paper concerns in-plane vibration analysis of coupled bending and longitudinal vibrations in multi-story planar frame structures based on the advanced Timoshenko bending theory. It takes into account the effects of both rotary inertia and shear distortion. A wave based vibration analysis approach is proposed. From a wave vibration standpoint, vibrations propagate along a uniform waveguide (or structural element), and are reflected and transmitted at discontinuities (such as joints and boundaries). Reflection matrices at various boundaries, as well as transmission and reflection matrices at joint discontinuities are derived. Natural frequencies of coupled bending and longitudinal in-plane vibrations are obtained by assembling these propagation, reflection, and transmission matrices. Numerical examples are presented along with comparisons to results available in literature. The examples show good agreement with the results presented in the available literature.

Validation of a Modeling Tool for In-Plane Longitudinal Resonators with Zigzag Topology

2019 ◽  
Vol 11 (02) ◽  
pp. 1950013
Author(s):  
Bowei Chen ◽  
Oleg Shiryayev ◽  
Nader Vahdati ◽  
Ameen El-Sinawi
Keyword(s):  
Elastic Waves ◽  
Experimental Validation ◽  
Natural Frequencies ◽  
Frequency Tuning ◽  
Finite Element Models ◽  
Vibration Absorbers ◽  
Longitudinal Vibrations ◽  
Resonant Frequencies ◽  
Modeling Tool ◽  
Proposed Model

Metastructures are viewed as a promising means for suppressing elastic waves and unwanted vibrations. Metastructures comprise of elementary cells with embedded resonators, which act as vibration absorbers. Design and frequency tuning of individual resonators inside the metastructure allows to achieve effective suppression of vibrations over a relatively wide frequency bandwidth, which makes metastructures superior compared to conventional passive vibration absorbers. This paper describes numerical and experimental validation of a modeling tool for design of planar resonators with elastic elements arranged in a zigzag configuration for suppression of longitudinal vibrations. Zigzag topology is advantageous due to its ability to provide higher compliance within a limited space, so as to achieve low resonant frequencies. Natural frequencies predicted by the proposed model agree well with predictions provided by detailed finite element models and experimental measurements.

Coupled Torsional and Longitudinal Vibrations of a Thin Bar

1959 ◽  
Vol 26 (4) ◽  
pp. 510-512
Author(s):  
R. C. Di Prima
Keyword(s):  
Natural Frequencies ◽  
Longitudinal Motion ◽  
Torsional Motion ◽  
Longitudinal Vibrations

Abstract The effect of pretwist on the natural frequencies of coupled torsional-longitudinal oscillations of thin bars is studied. It is found that the natural frequencies of oscillations which consist primarily of torsional motion may be considerably increased depending upon the thinness of the bar, and the amount of pretwist. The natural frequencies of oscillations which consist primarily of longitudinal motion are not significantly altered.

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