Forced Longitudinal Vibrations Research Articles

Comparative Non-Destructive Investigations of the Effects of Structural Changes After Heat Treatment of Carbon Steel Samples by Magnetic Noise and Vibration Methods

The microstructure of some components that work for a long time at high temperatures changes, which leads to a reduction in their life cycle. Traditionally, the study of the resulting structures is carried out by mechanical tests and metallographic analysis. The application of non-destructive methods increases the possibilities to monitor the quality of the elements and to prevent destruction. The aim of the study is to detect changes in the microstructure after different heat treatment and ageing heating of materials via magnetic noise and vibration methods. To realize the goal, samples of high-quality medium carbon steel were prepared through targeted heat treatments, quenching and tempering at different temperatures, as well as ageing heating at temperatures of 700 o C for up to 16 hours by vibration and magnetic noise method. The natural frequencies and damping properties of forced longitudinal vibrations of various heat treated steel specimens were obtained and analyzed. Using magnetic noise method, the following parameters were obtained and analyzed: root mean square voltage (RMS), energy and envelope parameters of magnetic noise signals. The comparative application of several non-destructive methods increases the knowledge of methods and materials and makes it possible to establish their sensitivity in studying microstructural characteristics.

Asymptotic Behavior of Solutions of a Complete Second-Order Integro-Differential Equation

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin-Voigt friction.

Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

Analysis of Forced Longitudinal Vibrations of Columns Taking into Account Internal Resistance in Resonance Zones

In analytical form, formulas are obtained for the amplitude of forced harmonic longitudinal vibrations of reinforced concrete and fiber-reinforced concrete columns with fixed edges. In order to verify the proposed approach, columns were simulated in the ANSYS program and calculated by the finite element method. Analysis of the calculations shows that a significant raise in the amplitude of the forced vibrations is observed only in the region of the first resonant frequency. It has been established that the value of the maximum amplitude of the vibrations of the fiber reinforced concrete column is 16% less than that for a reinforced concrete column.

Longitudinal vibration of double nanorod systemsusing doublet mechanics theory

This paper investigates the free and forced longitudinal vibration of a double nanorod system using doublet mechanics theory. The doublet mechanics theory is a multiscale theory spanning between lattice dynamics and continuum mechanics. Equations of motion and boundary conditions for the double nanorod system are obtained using Hamilton\'s principle. Clamped-clamped and clamped-free boundary conditions are considered. Frequencies and dynamic displacements are determined to demonstrate the effects of length scale parameter of considered material and geometry of the nanorods. It is shown that frequencies obtained by the doublet mechanics theory are bounded from above (van Hove singularity) and unlike classical elasticity theory doublet mechanics theory predicts finite number of modes depending on the length of the nanotube. The present doublet mechanics results have been compared to molecular dynamics, experimental and nonlocal theory results and good agreement is observed between the present and other mentioned results. The difference between wave frequencies of graphite is less than 10% between doublet mechanics and experimental results near to the end of the first Brillouin zone.

Dynamic magneto-mechanical properties of magneto-sensitive elastomers determined using a new experimental test involving forced longitudinal vibration

A new experimental method involving forced longitudinal vibration is presented for experimentally determining the dynamic magnetomechanical properties of a magnetosensitive elastomer in a magnetic field. A cylindrical sample is attached to a vibration platform and placed in a longitudinal magnetic field generated by a solenoid electromagnet, and a resonant technique is used to obtain the dynamic magnetomechanical properties of the tested sample. The results indicate that the resonant frequency (i) increases with the intensity of the applied magnetic field and the content of the magnetic-particle filler in the matrix but (ii) decreases with the sample length. The dynamic properties of the storage and loss moduli depend significantly on the excitation frequency and the magnetic-particle content. The testing process shows that it is simple and easy to evaluate the dynamic properties using forced longitudinal vibration, with the additional advantage of it being a nondestructive technique. This method could be extended to characterize the coupled magnetomechanical behavior of magnetosensitive functional elastomers such as Terfenol-D/epoxy composites.

Application of the Kantorovich-Galerkin Method for Solving Boundary Value Problems with Conditions on Moving Borders

The approximate Kantorovich-Galerkin method is considered for solving problems describing the vibrations of viscoelastic objects with conditions on moving boundaries and analyzing the resonance properties of these objects. The method makes it possible to take into account the effect of forces of environmental resistance on the system, flexural rigidity, and also boundary conditions with weak nonstationarity. The mathematical formulation of the problem involves a partial differential equation with respect to the desired displacement function and inhomogeneous boundary conditions. The Kantorovich-Galerkin method makes it possible to take into account the initial conditions, but they do not affect the resonance properties of linear systems, so in this case they are not taken into account. By introducing a new function into the problem, the boundary conditions are reduced to homogeneous ones. The solution is carried out in dimensionless variables to within a second order of smallness with respect to small parameters characterizing the velocity of the boundary motion and viscoelasticity. Using the Kantorovich-Galerkin method, an approximate solution of high accuracy of the problem of forced longitudinal vibrations of a viscoelastic rope of variable length, one end of which is wound on a drum, and the second is rigidly fixed, is found. The results obtained for the amplitude of oscillations corresponding to the nth dynamical mode are presented. The phenomenon of steady resonance and passage through resonance is investigated using numerical methods. A graphical dependence of the maximum amplitude of the rope oscillations as it passes through the resonance, depending on the coefficient characterizing the viscoelasticity of the object based on the Voigtmodel, is presented. The accuracy of the Kantorovich-Galerkin method is estimated.

Phantom partials in the piano sound and the role of the soundboard

The presence of phantom partials in the spectra of the piano sound are well known, and for decades they have been attributed to forced longitudinal vibrations in the string. These vibrations are generally assumed to originate with the stretching of the string, which produces frequency components at the sums, differences and second harmonics of the resonant frequencies of the transverse string motion. Results of recent experiments indicate that phantom partials can be produced in other parts of the piano as well as the string, and in some cases these contributions to the sound can be significant. Measurements of the power in multiple phantom partials produced simultaneously by steady-state excitation are providing insight into the process that creates them.

Способ определения модуля Юнга поверхностно-упрочнённых материалов

The issue of strength analysis as well as foreknowledge of component and structure life is of interest at this time for new composite materials. New composite materials and alloys elastic response is often unknown, and standard methods for their determining can not be applied to, e.g. materials featuring surface gradient properties. 
 Elasticity modulus method of examination is based on stepwise removal of part of the material from the rod sample surface or application of a coating onto the sample surface. Each step is accompanied by measurement of the sample lateral dimensions, and density. To determine Young´s modulus, resonance frequencies of the sample subjected to forced longitudinal vibrations are measured using an ultrasonic device. The device principle of operation is based on excitation and tracking of the sample forced mechanical oscillations close to its own frequencies, measurement of resonance peak parameters followed by calculations of the Young´s modulus. To induce and track the forced oscillations in the sample, use is made of piezoelectric transducers attached to damped acoustic transmission rods, while the sample is fixed between pointed rod ends. This arrangement makes it possible to examine also Young´s modulus temperature dependences.
 Young´s modulus measurement results for case-hardened hard alloy are provided. It is shown that Young´s modulus of surface layers is lower than that of the base material, which is explained by formation of new compounds in the surface layers owing to chemical and diffusion reactions between an alloying element and alloy components. The doped layer extends to 0.45 mm. The results obtained can be used for strength calculations of case-hardened materials, estimation of cohesive and adhesive strength of individual layers of coatings, structure and other physical and mechanical properties of materials.

Longitudinal vibrations of arbitrary non-uniform rods

Free and steady-state forced longitudinal vibrations of non-uniform rods are investigated by an iteration method, which results in a series solution. The series obtained are convergent and linearly independent. Its convergence is verified by convergence tests, its linear independence confirmed by the nonzero value of the corresponding Wronski determinant. Then, the solution obtained is an exact one reducible to a classical solution for the case of uniform rods. In order to verify the method, two examples are presented as an application of the proposed method. The results obtained are equivalent to the method in literature. In contrast to the proposed method capable of dealing with arbitrary non-uniform rods in principle, the method in literature is confined to work on special cases.

The generalization of the results of research of dynamic bearing capacity of hydrodynamic thrust bearings

The analysis of traditional and developed calculation methods of dynamic bearing capacity of hydrodynamic thrust bearings of various technical systems was conducted. Due to generalization of existing data and the results of the research, it was found that each of the operational factors, affecting the TB reduces its dynamic load capacity. The combined impact of these factors can deplete the dynamic load capacity of the thrust assembly and lead to the destruction of the lubricating layer, which will cause the breakdown of the TB and the failure of the technical system. This problem is particularly acute nowadays because of an increase in service life of ship machinery and a decrease in the number of planned preventive maintenance. The mathematical model was developed and calculation method of the dynamic bearing capacity of the thrust assemblies of fluid friction, which allows to determine the allowable amplitude of forced longitudinal vibrations of the shaft, face motion variation of thrust collar and allowable misalignment of the bearing housing was improved. At the design stage, improved calculation method allows to perform vibration calculations to assess the dynamic bearing capacity of the TB of various technical systems.

Nonlinear forced vibrations of a hysteretic bar: Revisited

An earlier analytical model describing forced longitudinal vibrations of a bar with quadratic hysteretic nonlinearity is revised. To this end, the nonlinear constitutive there introduced is corrected and the origin of the errors is explained. The new resulting equation of motion, which includes also linear dissipative forces, is solved by means of a perturbation approach which makes use of the modulus defect of the material as a perturbation parameter. The experimentally observed linear dependence on the excitation amplitude of both resonance frequency shift and nonlinear attenuation is recovered. Similarly, the model predicts the generation of only odd harmonics with amplitude proportional to the square of the source strength. The solution includes the effects of the material hysteretic nonlinearity both in the time and in the spatial dependence of the vibration patterns, the latter feature being neglected by all other models. The relevance of this improvement resides in the potential the present model offers to deal with damage confined in narrow regions containing stress concentrators responsible for crack initiation. Furthermore, it is argued that such an approach may allow modeling of recent advanced imaging methods of nonlinear crack-like defects, which exploit the break-down of reciprocity in material systems with damage.

Microwave absorption by an array of carbon nanotubes: A phenomenological model

A simple model to explain microwave-induced heating of carbon nanotubes (CNTs) through transformation of electromagnetic energy into mechanical vibrations is proposed and analyzed. The model provides a way to understand recent observations of heating of CNTs exposed to microwaves in the range of $2--20\phantom{\rule{0.3em}{0ex}}\mathrm{GHz}$. It is shown that transverse vibrations of CNTs during microwave irradiation can be associated with parametric resonance, as occurs in the analysis of acoustic experiments on forced longitudinal vibrations of a stretched elastic string. For carbon nanotubes [single wall nanotube (SWNT), double wall nanotube (DWNT), multiwall nanotube (MWNT), ropes, and strands] the resonant parameters are shown to be located in a region of instability of the Mathieu's equation. Wave equations with cubic nonlinearity were used to qualitatively describe the effects of phonon-phonon interactions and energy transfer from microwaves to CNTs at a rate much exceeding the traditional Joule heating via electron-phonon interaction.

NONLINEAR VIBRATIONS AND VIBROHEATING OF A VISCOELASTIC ROD WITH CUBIC ELASTICITY

The model problem on the forced longitudinal vibrations of a physically nonlinear viscoelastic rod is solved. An assessment is made of how well the single-frequency mode is maintained during the resonance vibrations of the rod.

Analiza gibanja ventilske opruge

The paper considers the surge of a spring, retained at oneend and controlled by a cam at the other end. The periodic liftfunction of the cam was approximately replaced by the Fourierpoly nom. The forced displacement amplitudes were determinedfor each harmonic of this poly nom. The theory of forced longitudinal vibrations of a bar with continuously distributed mass,elasticity and dumping was applied to simulate the motion ofspring coils. It was found out that two or more modes may consistconcurrently in the spring. This was obtained by analyzingmathematical formulas and by comparing computed and publishedmeasured results. The line spectrum of the Fourier serieswas introduced as a useful tool for estimating the influence ofthe high order harmonics on the spring surge.

Vibration analysis of a rod with complex boundary conditions

This paper deals with the forced longitudinal vibration of a rod carrying a concentrated mass and supported by a spring at one end. The vibration of the rod is excited by the motion of the support point at the other end. Since the boundary conditions of the problem are complex and it is necessary to consider the damping, we determine only the steady state periodic solution. First the linear system is analysed; then the material nonlinearity is considered and the approximate analytic solution of nonlinear partial differential equation with nonlinear boundary conditions is obtained by the perturbation method.

Longitudinal Vibrations of Embankment Dams

This paper presents an analytical method to study free and forced longitudinal vibrations of embankment dams. Both shear and dilatational deformations are taken into account, and the dam is modeled as a linear homogeneous prism with a wedge-shape cross section, bounded by two vertical (abutment) and one horizontal (riverbed) planes. Numerical results demonstrate the effect of the length and height of the dam on its natural frequencies and its modal displacement and strain shapes. It is shown that shear deformations are more important in relatively long dams, whereas the opposite is true with dams built in narrow canyons. The method is evaluted through two case histories involving an earthfill and a rockfill dam. Close agreement exists between predicted and observed natural frequencies, but the distribution of peak accelerations within the dam is badly predicted.

Forced longitudinal vibration of an elastic circular rod on an elastic half-space

A solution is given for the problem of the forced longitudinal vibration of an elastic circular rod, one end of which is attached to the surface of an elastic half-space and to another end of which a periodic disturbing force is applied. Resonance curves for the motions of rods are derived using specific boundary conditions. Attention is directed to the effective damping of the motion of the rod due to dissipation of waves to infinity. Subject Classification: [43]40.20; [43]20.22.

On the Vibrational Properties of High-Elastic Gels

A new method is presented for determining dynamic mechanical properties of high-elastic gels by measuring the amplitude and phase of forced longitudinal vibrations excited in get blocks themselves. It is shown that the Young's modulus and viscosity can be determined by analysing these quantities. Theoretical analyses are also made for the vibrations excited in the vertical cylindrical samples with no viscosity by horizontal oscillations of the bottom ends. The calculated results are illustrated by strains (displacement, inclination and shearing angle) for various values of cylinder diameters and frequencies used. It is shown that an approximate value of rigidity can be obtained from the frequency dependence of the strains. On an basis of this method the dependence of dynamic mechanical constants of agar-agar gels on cylinder diameters, concentrations and frequencies is studied, and reasonable values are obtained.

Pressure Distribution and Bubble Formation Induced by Longitudinal Vibration of a Flexible Liquid-Filled Cylinder

Analytical and experimental results are presented for a flexible liquid-filled cylinder subjected to forced longitudinal vibration. Analytical predictions are correlated with experimental measurements of pressure amplitudes and threshold acceleration levels required to produce bubble formation, as well as visual observations of the regions in which formations occur.