Analysis Of Free Oscillations Research Articles

Free oscillation analysis of rectangular plate of porous FGM material placed on a Winkler elastic base by analytical method

In this paper, classical plate theory is used to analyse free oscillation of rectangular plates made of porous FGM material, edge joints on Winkler elastic base. Three different types of pore distribution: uniform distribution, symmetric irregular distribution and asymmetrical irregular distribution are investigated. The reliability of the analytical solution as well as the calculation program written on Matlab are verified with some published results and with the results calculated by the SAP2000 structural calculation software. The influence of the material parameters, the geometrical dimensions of the plate, as well as the background coefficient on the free oscillation frequency of the plate are evaluated.

Vibration analysis of multi-scale hybrid nanocomposite shells by considering nanofillers’ aggregation

Free oscillation analysis of shells consisted of multi-scale hybrid nanocomposites is carried out with regard to the destroying effect of the nanofillers’ agglomeration on the system’s dynamics. The equivalent material properties of the hybrid nanocomposite are obtained in the framework of a bi-level micromechanical procedure. The influence of existence of agglomerated carbon nanotubes (CNTs) on the stiffness of the nanocomposite is included with the aid of the Eshelby-Mori-Tanaka method. Thereafter, the first-order shear deformation theory (FSDT) of the shells will be combined with the dynamic form of the principle of virtual work to reach the Euler-Lagrange equations of the problem. Next, the governing equations will be extracted from the Euler-Lagrange ones by using the constitutive equations of the nanocomposite. To obtain the natural frequencies of the system for shells with clamped and simply supported ends, the Galerkin’s method is hired. The validity check reveals that the utilized methodology is powerful enough to predict the frequency behaviors of nanocomposite shells. According to the results, it can be inferred that the hybrid nanocomposite shells may be involved in resonance phenomenon in low-range frequencies if the impact of the CNTs’ aggregation is neglected. Obviously, the volume fraction of the CNTs inside the inclusions plays a magnificent role in the determination of the system’s dynamic behavior.

Analysis of free oscillations of round thin plates of variable thickness with a point support

This paper reports the derived general analytical solution to the IV-order differential equation with variable coefficients for the problem on free axisymmetric oscillations of a circular plate of variable thickness. The plate thickness changes along the radius ρ in line with the parabolic law h=H0(1–µρ)2. When building a solution, the synthesis of the factorization method and the symmetry method was used. The factorization method has enabled us to represent the solution to the original IV-order equation as the sum of the solutions to the two respectively constructed II-order equations. The method of symmetry has produced precise solutions to these two equations.The problem on a point fixation of the plate has been considered as a boundary case of the problem on the rigid fixation of the inner contour of a circular plate whose ρ→0. To this end, the general solution has been transformed into the form that pre-meets the conditions on a rigid point support. The result of such a transformation is a simpler solution, with only two permanent integration variables instead of four. As a result, the frequency equation for a plate under any conditions on the outer contour is significantly simplified because it is derived from the second-order determinant. The frequency equation for a plate with a point support and with a free edge at µ=1.39127, which corresponds to the ratio of the limit thicknesses equal to 10.8, yielded the first five eigenvalues λi (i=1÷5). The oscillation shapes have been constructed as a graphic illustration for λi (i=1÷3). The numerical values of amplitude ratios have been given, as well as coordinates (relative radii) of the oscillation antinodes and nodal circles for each of the five oscillation shapes (i=1÷5). The derived numerical values of the oscillation parameters could in practice be used to initially identify the type of an oscillatory system and its possible characteristics for the case when a plate is fixed inside the inner contour of the small diameter. The criterial ratio of the fastening contour diameter to the plate diameter could serve the same purpose. If this ratio is equal to or less than 0.2, then it is permissible to assume that it is a point-type fastening. In this case, it is possible to calculate the circular plate oscillations with its internal contour fixed using the algorithm set out for a plate with a point support

Analysis of free oscillations of circular plates with variable thickness based on the symmetry method

Based on symmetry and factorization methods, a general analytical solution to the fourth-order differential equation has been derived for a problem on the free axisymmetric oscillations of a circular plate of variable thickness. The law of the thickness change is the concave parabola h=H0(1–μρ)2, where µ is a constant coefficient that determines the degree of plate concaveness. The solution has been given by the Bessel functions of zero and the first order of the actual and imaginary argument. A circular ring plate has been considered whose inner contour is rigidly fixed and whose outer edge is free, for three values of the µ coefficient. We have determined the first three natural values for the problem (frequency numbers) and their natural functions (oscillation shapes). It has been shown that the natural frequencies of the first three shapes of oscillations decrease, with the increase in concaveness (increase in µ), to varying degrees, determined by the number of the frequency number λi (i=1, 2, 3). At µ=1.21417 and µ=1.39127, the frequencies decrease, compared to the case of µ=0.5985, by (1;1.3) %, (17.6;24) %, (22.85;30.35) %, respectively, for λ1, λ2, λ3. One can see a significant drop in frequency on the higher shapes of oscillations (λ2, λ3) and a slight drop in the basic shape (λ1). We have established the values and coordinates of extreme deflections (antinodes of oscillations) and the indicative coordinates of the nodal cross-sections. The reported numerical parameters, along with the frequency indicators, are a means of identifying the oscillational properties of a plate when it is studied in practice. We have built the graphic dependences for radial σr and tangential σθ cyclical stresses at the basic shape for each of the three variants of the concaveness of a parabolic plate. It has been established that the increase in the ratio of edge thickness, that is, concaveness, leads to an increase in σr in the cross-sections outside the end constraint. These stresses, which operate far from the free edge, for example at the end constraint or the area of the maximum σθ, are greater than σθ in varying degrees. Because of this, these stresses pose a major threat in terms of the cyclical strength of the plate when σr reaches destructive values. We have pointed to the possibility to provide, by increasing the concaveness of the parabolic plate, the optimal ratio between the value of σr at the end constraint and σr operating away from the fastening. This ratio, approximately equal to 1, is ensured at µ=1.39127 considered in the current work.

Hydraulic Oscillation and Instability of A Hydraulic System with Two Different Pump-Turbines in Turbine Operation

Hydraulic oscillation mainly reveals the undesirable pressure fluctuations which can cause catastrophic failure of any hydraulic system. The behavior of a hydraulic system equipped with two different pump-turbines was investigated through hydraulic oscillation analysis to demonstrate severe consequences induced in turbine operation, including S-shaped characteristics. The impedance of a pump-turbine has an essential role in the determination of the instability of the hydraulic system. The conventional way to determine the instability solely using the slope of a characteristic curve was improved, including the effect of guide vane opening in pump-turbine impedance, which consequently modified the instability expression. With this pump-turbine impedance, hydraulic oscillation analysis, including free oscillation analysis and frequency response analysis, was carried out. The free oscillation analysis entails the computation of complex natural frequencies and corresponding mode shapes of the system. These computations provided necessary information about the vulnerable position of vital hydraulic components and the scenario for self-excited oscillation. Further, the analysis illustrates the significant role of guide vane opening to prevent the system from becoming unstable. Lastly, frequency response analysis was performed for the system with an oscillating guide vane to obtain the frequency response spectrum, which revealed that the resonating frequencies are consistent with natural frequencies, and it supported free oscillation results.

Radar analysis of free oscillations of rail for diagnostics defects

One of the tasks of developing and implementing defectoscopy devices is the minimal influence of the human factor in their exploitation. At present, rail inspection systems do not have sufficient depth of rail research, and ultrasonic diagnostics systems need to contact the sensor with the surface being studied, which leads to low productivity. The article gives a comparative analysis of existing noncontact methods of flaw detection, offers a contactless method of diagnostics by excitation of acoustic waves and extraction of information about defects from the frequency of free rail oscillations using the radar method.

Analysis of free oscillations of thin-walled cylindrical shells containing a compressible liquid

Free oscillations of a thin-walled cylindrical cell containing a compressible liquid were investigated. At some values of the parameters of the system, its eigenfrequencies were determined, and the influence of the geomet- ric and physical parameters of the cylindrical shell-liquid system on the free oscillation of cylinder was in- vestigated. Introduction. Shells as elements of machines and constructions are widely used in aircraft industry, shipbuild- ing, and so on. Therefore, in recent years the problems associated with the dynamic behavior of thin-walled structures that are in contact with environment under operating conditions attract considerable interest of researchers. The frequencies and forms of free oscillations of spherical and cylindrical shells coming in contact with elas- tic and liquid media were investigated in (1, 2). Approximate simple formulas were obtained by asymptotic methods for calculating the frequency and determining the form of oscillations of the system considered; this limits the use of the results obtained, because in a number of cases the possibility of carrying out a qualitative analysis of investigated processes is eliminated. Statement of the Problem. In this work we consider free asymmetric oscillations of an infinite thin-walled cylindrical shell containing a compressible liquid. Since finding the eigenfrequencies of the cylindrical shell-liquid sys- tem is associated with the solution of transcendental equations, the frequency of oscillations of a shell without a liquid is expressed in terms of the frequency of oscillations of the system in an explicit form; this allows one to investigate the spectra of the system's frequencies both analytically and graphically. The equations of radials oscillations of cylindrical shells have the form (3) ∂ 2 u

Analysis of free oscillations of construction elements

Substantiated is the choice of a system of equations for culculation of energy dissipation characteristics in test effect on constuction elements at the moment of control

Measuring complex spectra of long‐period surface waves for earthquake source analysis

Long‐period surface waves from large earthquakes can provide reliable estimates of the source depth, rupture duration, seismic moment, and fault orientation. This requires accurate surface wave phase and amplitude spectra and precise propagation corrections. Standard procedures for estimating spectra of traveling waves in source analyses are inaccurate for periods exceeding 300 s, so the longer periods are usually analyzed by free oscillation analysis if at all. We apply a multiple filter analysis to extract complex spectra for periods up to 500 s suitable for traveling wave spectral inversions. This will allow improved constraints on the rupture process of large earthquakes to be obtained by inversion methods using a single formalism.

NONLINEAR-COUPLED FREE OSCILLATION ANALYSIS OF SUSPENSION BRIDGES

NONLINEAR-COUPLED FREE OSCILLATION ANALYSIS OF SUSPENSION BRIDGES

Discovery of the Earth’s core

In 1896 when Emil Wiechert proposed his model of the Earth with an iron core and stony shell, scientists generally believed that the entire earth was a solid as rigid as steel. R. D. Oldham’s identification of P and S waves in seismological records allowed him to detect a discontinuity corresponding to a boundary between core and shell (mantle) in 1906, and Beno Gutenberg established the depth of this boundary as 2900 km. But failure to detect propagation of S waves through the core was not sufficient evidence to persuade seismologists that it is fluid (contrary to modern textbook statements). Not until 1926 did Harold Jeffreys refute the arguments for solidity and establish that the core is liquid. In 1936 Inge Lehmann discovered the small inner core. K. E. Bullen argued, on the basis of plausible assumptions about compressibility and density, that the inner core is solid. Attempts to find seismic signals that have passed through the inner core as S waves have so far failed (with one possible exception), but analysis of free oscillations provided fairly convincing evidence for its solidity.