Free Vibration Frequencies Research Articles

Parametric vibrations of the underground oil pipeline

This paper is devoted to obtaining a solution to the problem of parametric vibrations and dynamic stability of thin-walled underground oil pipelines, taking into account the effect of a damper. An equation has been obtained that allows estimating the dynamic stability of underground oil pipelines, which consists in constructing regions on the plane of the parameters ωmn, ω0 and the longitudinal force parameter P, as well as the internal pressure p0, where ωmn is the frequency of free vibrations, ω0 is the excitation frequency. The assessment of the dynamic stability of the oil pipeline section is made for the known values of ωmn, ω0, p0, P, k by finding a point. If the point does not fall into the outlined area, it means that dynamic stability is ensured, otherwise there is a danger of parametric resonance.

Modeling and simulation of Vibro-isolator Rapid Prototyping using additive manufacturing

The emerging concept of mechanical meta-materials has gained increasing attention in recent years, partly due to advances in additive manufacturing techniques (additive manufacturing, 3D printing) that have allowed the fabrication of materials with arbitrarily complex micro / nanostructures. The designed geometry of these structures gives rise to unprecedented or rare mechanical properties that can used to create advanced elements with new, unprecedented functions. This paper presents the method of prototyping a vibro-isolator made of a material that has the properties of storing and disposing of mechanical vibration energy and that can freely be shaped by modern manufacturing techniques. An example of a vibration isolator of the proprietary concept analyzed, using the concept of stepped and adding elements. The method is based on the use of two main tools. Mechanical properties verified by analyzing the frequency and mode of free vibrations of the prototype using the finite element method. The same CAD model of the vibro-isolator prototype produced by 3D printing. The paper presents one of the production techniques, which is 3D printing in SLS (Selective Laser Sintering) technology, and the SolidWorks and Comsol Multhiphysics software used in the CAD / CAE prototyping method.

СВОБОДНЫЕ КОЛЕБАНИЯ ТОНКОСТЕННОГО ДВУХСЛОЙНОГО ТРУБОПРОВОДА С УЧЕТОМ ВЛИЯНИЯ ПРОДОЛЬНОЙ СЖИМАЮЩЕЙ СИЛЫ ПРИ ПОЛУПОДЗЕМНОЙ ПРОКЛАДКЕ

The article considers a cylindrical two-layer shell subjected to a longitudinal compressive force. An inhomogeneous cylindrical two-layer shell of finite length, consisting of a steel pipe and a protective reinforced concrete layer, was adopted as the calculation scheme. Based on the use of the assumptions of the theory of shells by Vasiliy Z. Vlasov, the equation of motion of the shell in efforts and displacements was obtained. A system of differential equations has been formed and solved taking into account the assumptions of this theory of cylindrical shells, the solution for determining the frequency of free vibrations taking into account the effect of internal working pressure, inhomogeneous ground pressure on the pipe wall and a longitudinal compressive force has been obtained.

Сравнительный анализ определения частот собственных колебаний прямоугольных панелей с защемленно-свободными краями

An iterative method for assessing the frequency spectrum of free vibrations of rectangular panels with different support conditions in combination with the parameter enumeration method is proposed. Two hyperbolo-trigonometric series on two coordinates with indeterminate coefficients containing the desired frequency of free vibrations as a parameter are taken as a basis. The obtained natural frequencies and waveforms are compared with the results provided by other authors, and are presented very fully in tabular form, the accuracy of calculations and methods are analyzed.

Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application

A new type of thin plate model and the related nonclassical boundary value problems were established within the framework of strain gradient and velocity gradient elasticity. The closed-form solutions of deflections and free vibrational frequencies of a simply supported plate resting on an elastic foundation were obtained. The results of the present model agree well with those predicted by the molecular dynamics. Numerical results show that, the elastic foundation and the strain gradient parameter have a stiffness-hardening effect, while the velocity gradient parameter has a stiffness-softening effect. The proposed boundary value problems are of great significance to the study of the mechanical behaviors of plates under complex boundary conditions and external loadings. Furthermore, it will be useful for developing effective numerical methods such as the finite element method, the finite difference method and the Garlerkin method.

Vibration and dynamic analysis of a cantilever sandwich microbeam integrated with piezoelectric layers based on strain gradient theory and surface effects

In this paper, size-dependent vibration and dynamic analysis of an advanced sandwich composite microbeam including piezoelectric layers based on higher-order strain gradient and surface effects theories are investigated. To evaluate and analyze the effect of surface energy, length scale parameters and dynamic response on the transverse vibration and stability of sandwich smart composite microbeam with piezoelectric layers, the partial differential equation of motion is obtained using Hamilton's principle. Subsequently, this equation is transformed into a set of ordinary differential equations according to the generalized differential quadrature method. The influences of various material length scale associated with strain gradient theory, geometry and surface parameters such as thickness to material length scale parameter, surface residual stress, Young's modulus of surface layer, surface mass density and surface piezoelectric constant on structural dynamic deflection behavior and natural frequency have been investigated. In addition, the vibration and dynamic responses of the model were compared with those micro sandwich beams based on modified couple stress and classical theories. The obtained results are compared with the results available in the literature to validate the correctness of the present solution method. The results indicated that strain gradient theory with considering the surface effects increases the model stiffness and shifts the amplitudes to lower magnitude faster in comparison to other theories. Moreover, the influences of material length related to dilatation gradient on vibrational frequency are significant among the other parameters. Also, the free vibrational frequency increases with decreasing the length to thickness ratio.

Nonlinear impulsive and vibration analysis of nonlocal FG-CNT reinforced sandwich plate by considering agglomerations

This paper is focused on impulsive and subsequent vibration response of a nano-scale composite sandwich plate considering both nonlocal effect and nanoparticle agglomerations. In particular, the core is homogeneous polymethyl methacrylate (PMMA) and the face sheet is PMMA matrix with functionally graded carbon nanotubes (short for FG-CNT). The material properties are given by Eshelby-Mori-Tanaka's approach taking into account of CNT agglomerations. The equations of motion for the FG-CNT reinforced nanoplate are derived based on nonlocal strain gradient theory, Reddy's higher order shear deformation theory and von-Karman nonlinear deformation theory. The impulsive and vibration problems for simply supported FG-CNT plates are solved using Galerkin method and Newton iteration method where the solutions have been verified with the literature. By considering impulsion of step load, exponential load and sinusoidal load and, it is found that weaker size effect, fewer CNT agglomerations, higher CNT content, smaller functional gradient, thinner core, bigger aspect ratio and larger length thickness ratio can enhance the stiffness of the nanoplate and thus will induce smaller center displacement during the load process and higher free vibration frequency after unloading, regardless of load type. However, the subsequent vibration amplitude is significantly influenced by the impulsive load type. Among the material parameters, minimizing size effect and agglomerations could greatly improve the structural stiffness whereas the improvement by varying functional gradient is very limited. It is the first time that a FG-CNT reinforced sandwich plate, especially considering nonlocal effect and nanoparticle agglomerations, is treated in the context of impulsive and subsequent vibration problems. The solutions presented in this paper could provide a benchmark for the design and manufacture of such a composite nanoplate.

Transverse vibration responses of the in-plane-wise functionally graded piezoelectric composite plates

Aided by the scaled boundary finite element method (SBFEM), analytical precise integration solutions to the transverse free vibration responses of functionally graded piezoelectric composite plates are provided for the first time. Distributions of the functionally graded piezoelectric materials follow arbitrary mathematical formulae as concerns the in-plane unidirectional or bidirectional coordinates. The vibration behaviors of piezoelectric composite plates with various geometrical shapes, multifarious boundary constraints and any number of laminae can be explored by the introduced technique. Additionally, the shear correction factors and assumptions on the changing patterns of mechanical and electrical variables are not needed to be offered in the employed methodology. The proposed approach regards only four quantities containing three elastic displacement components and the electric potential as the primary unknowns, which can be denoted as an analytical matrix exponent in terms of the thickness coordinate. Only two-dimensional high-order spectral elements are adopted to discretize the in-plane surface of the plate with less nodes and degree of freedoms, which promotes to lowering the calculative expense and advancing the computational efficiency. The introduced scaled boundary coordinate system and the dual vector technology are helpful to simplify the basic partial differential equations of piezoelectric materials into the first order ordinary differential SBFEM governing equation, from which the highly accurate global stiffness matrix are constructed by dine of the precise integration technique (PIT). With the aid of the kinetic energy scheme and compatibility conditions between neighboring layers, the whole mass matrix for the laminated piezoelectric plates can be built. According to solving the eigenvalue equation, the precise integration solutions to free vibration frequencies can be acquired. Finally, numerical experiments are afforded to elucidate the high precision and fast convergence of the present method. Moreover, the effect of different boundary constraints, gradation parameters and aspect ratios on the transverse vibration responses of in-plane-wise functionally graded piezoelectric composite plates is revealed.

Compensatory bellows oscillations as a corrugated shell with liquid inside

The paper deals with a relevant problem of shipbuilding, i.e. calculation of free and forced vibrations of pipeline compensatory bellows. These devices are used to reduce the vibration load caused by ship power machines. When analyzing the vibrations of the compensatory bellows, it is necessary to take into account the liquid contained in the bellows. In this work, the design model of the bellows is represented by a corrugated elastic shell as a material surface with five degrees of freedom. A variant of the classical theory of shells, built on the basis of Lagrangian mechanics, is used. The influence of the liquid is taken into account by two models. First, the liquid is considered to be ideal and incompressible and is considered through the attached mass to the shell. The shell is replaced by a cylindrical surface with a radius in the middle line of the corrugation. To account for the influence of the frequency of bellows oscillations on the attached inertia of the liquid in the calculation we also used the acoustic approximation; and derived a formula for a generalized attached mass of the ideal compressible liquid. The equations of the bellows oscillations under the periodic loading are obtained. The problem has been solved by the finite difference method. The values of natural frequencies of free vibrations are obtained for the compensatory bellows from the corrosion-resistant heat-resistant steel. It is shown that by taking account of the liquid, we significantly change the natural frequencies of the bellows. With high-frequency vibrations it is necessary to take into account the compressibility of the liquid. The problem of the forced vibrations of the bellows caused by a displacement of its end face by the harmonic law is solved. The internal forces and moments are determined, as well as occurring stresses by Mises criterion in the bellows. We found the critical value of the end face displacement at a frequency of 50 Hz, at which the bellows goes into a plastic state.

Research on Hovercraft – Fatigue Cracks in the Engine Frame

Abstract Rescue patrol hovercrafts must meet the basic condition – high reliability of use in extreme conditions. The introduction to the work shows damage to the propulsion system and the fan tunnel structure resulting from a fatigue fracture of the attachment wound to the propulsion unit hull. In this paper, the author describes some ways of improving the engine frame structure. In the first phase of the exploitation crack testing of the hovercraft frame, the probable causes of damage were determined. The necessary output data for analysis of the load course were obtained from the operating documentation. The approximate number of variable load cycles acting on the frame truss rod was determined. Using the comparative testing methods, the service life of the frame was estimated. Probable resonance frequencies of the vibrating bars in the truss were determined. Vibration tests of the power transmission assemblies were carried out, which allowed to determine the amplitudes and frequencies of free vibrations. Finally, a modification of the frame shifting the resonant frequency range was proposed. In conclusions, changes to the design and a schedule of inspections were proposed. The newly designed engine frame should have an extended service life.

Free vibrations of a thin-walled curvilinear section of an oil-trunk pipeline during ground laying

In this article authors obtained solutions to determine the ground curved sections free vibrations of thin-walled large-diameter pipelines with a liquid flow, based on a geometrically nonlinear version of the semi-momentless toroidal shells medium bending theory by V.Z. Vlasov and V.V. Novozhilov. The pipeline is a toroidal shell, the design scheme is presented in the form of a half torus. Angle β = 1800. The shell is laid on an elastic foundation and makes contact with the ground along a narrow strip. The problem of the soil pressure influence on the shell along a narrow strip is solved as a contact problem, using Fourier's series and an impulse function. The shell is exposed to the cooperative effect of the internal operating pressure, the liquid flowing pressure, the elastic soil foundation, and changes in the geometric characteristics. Motion equations of the middle shell surface are obtained taking into account the geometric and mechanical characteristics, and all the components of the shell material inertial forces. The hinged fastening of the shell ends is taken as the limiting condition. Using the semi-momentless shell theory assumptions, displacements in the longitudinal and circumferential directions are obtained. The solution to the problem of determining the free vibrations comes down to solving the problem of determining AB matrix values. The solutions obtained make it possible to determine the free vibrations frequencies at various wavenumbers values in the longitudinal and circumferential directions, and also make it possible to determine the internal operating pressure contribution, the soil bed coefficient, and geometric characteristics to the pipeline free vibrations frequencies.

Analysis of Changes in Parameters of Free Vibration of Single and Multi-span Girders with Semi-rigid Joints

This study presents the analysis carried out for changes in parameters of free vibrations of single-span corrugated web girders with a semi-rigid joint at midspan and multi-span girders with spans connected by semi-rigid joints. Based on the experimental tests and the theoretical analysis, the behavior of six simply supported girders with the semi-rigid joint at midspan was analyzed. They were straight and double-slope girders with a span of 6.02 m, made of corrugated web sections WTA 500/300x15 with different types of semi-rigid end plate joints. It was demonstrated that the variable rotational stiffness Sj of the joint affected the equivalent concentrated mass of the girder mz, the frequency of damped free vibrations α, damping ρ, and the frequency of free vibrations ω. The theoretical analysis was conducted for a change in the equivalent concentrated mass of the single-span girders fixed at both ends and of the multi-span girders. It was described how the change in the support stiffness in the single-span girders fixed at both ends and the change in joint stiffness of spans affected the equivalent concentrated mass mz as a function of non-dimensional stiffness k. The equivalent concentrated mass mz of the girder was found to affect the values of maximum vibrations in the structure. A continuous change in the rotational stiffness of joints was taken into account from the pinned to the rigid joint.

Dynamic stability of CNTs-reinforced non-uniform composite beams under axial excitation loading

This paper presents an analytical study on the dynamic stability of carbon nanotubes (CNTs) reinforced functionally graded composite beams when subjected to axial excitation loading. The composite beams are functionally graded using a I-type CNTs reinforcement. The section properties of the non-uniform functionally graded composite beams are assessed using Halpin–Tsai model. The analysis of dynamic stability of the composite beams is performed by using Bolotin’s method. Analytical expressions of determining free vibration frequency, critical buckling load, and excitation frequency of the non-uniform functionally graded composite beams are derived, in which both the shear deformation and rotary inertia effects are considered. The study demonstrates that the dynamic stability of the beam can be improved significantly when it is functionally graded using the I-type CNTs reinforcement.

Free vibration analysis of coupled structures of laminated composite conical, cylindrical and spherical shells based on the spectral-Tchebychev technique

In this article, a numerical spectral-Tchebyshev (ST) technique is applied to solve the free vibration strong solution of the coupled structures of laminated composite conical, cylindrical and spherical shells under various boundary conditions. The artificial spring technique is introduced to realize the coupling connection of substructures and simulate the arbitrary boundary conditions. Considering the first-order shear deformation shell theory (FSDST), and using Hamilton variational analysis under its framework to derive the governing equations of motions of three coupled structures: coupled conical-cylindrical shells, coupled spherical-cylindrical shells and coupled spherical-cylindrical-conical shells. The solution of the governing equations of motions is obtained using the spectral-Tchebyshev technique. The variables along the generatrix direction are discretized based on the Gauss-Lobatto nodes, while the Tchebyshev polynomials are used for spectral approximation. The variables along the circumferential direction are approximated by Fourier series. The calculated free vibration characteristics of the coupled structures are compared with the numerical results based on previous references and the finite element method, which completely reflects the excellent convergence and calculation accuracy of the spectral-Tchebyshev technique. On this basis, the effect of the material properties and geometric dimensions of the substructures (conical, cylindrical and spherical shells) on the free vibration frequencies of the coupled structure is further studied.

Influence of crack on the damping properties of recycled concrete

This paper investigates the damping property of a recycled concrete beam with prefabricated cracks through the lens of free vibration attenuation. The experiment is designed to evaluate the effects of various recycled aggregate replacement percentages and fracture depths. The frequency and damping characteristics of recycled concrete and cracking, as well as their evolution through time, were explored, as well as the influence of a variety of factors and their interactions. For various recycled aggregate replacement ratios, the testing results indicate that the frequency of free vibration is between 3000 and 4000 Hz. The damping ratio of unconstrained vibration varies between 0 and 0.04 and is typically about 0.015. The analysis demonstrates that as the replacement ratio of recycled aggregate increases, the average frequency decreases but the damping ratio increases overall; vibration frequency has a parabolic relationship with the replacement ratio of recycled aggregate but a linear relationship with the crack depth; and damping ratio has a parabolic relationship with the replacement ratio of recycled aggregate. The average frequency decreases as the recycled aggregate replacement ratio increases. The average frequency decreases as the recycled aggregate replacement ratio increases. The frequency of vibration is proportional to the recycled aggregate replacement ratio and is linear in relation to the crack depth. The damping ratio and the recycled aggregate replacement ratio are related to the crack depth in a parabolic fashion. In the test, no statistically significant relationship exists between the vibration amplitude and the damping ratio.

Static stability and vibration behavior of graphene platelets reinforced porous sandwich cylindrical panel under non-uniform edge loads using semi-analytical approach

Buckling and free vibration characteristics of sandwich cylindrical panel with porous functionally graded graphene platelets (FG-GPL) core are investigated using semi-analytical approach. The effective mechanical properties are obtained by using properties of open cell foams and Halpin–Tsai micro mechanical model. The governing equations are obtained using Hamilton’s principle, considering a higher order theory to account the transverse shear and solved by Galerkin’s method. Effects of nature of in-plane edge load, distribution of porosity and GPL, porosity coefficient, GPL loading, core to total thickness ratio are analyzed in detail. It is shown that for a FG-GPL core sandwich cylindrical panel with high core thickness, even at higher amount of porosity the buckling resistance and free vibration frequency can be improved by properly tailoring both the GPL and porosity distribution. Moreover, a much variation in buckling and free vibration response with the type of in plane loading is observed and evident mode shape changes are observed with increase in aspect ratio. The cylindrical sandwich panel having a core with D-PD porosity variation and I-GPL-P pattern of GPL distribution has the maximum buckling resistance and free vibration frequency value.

Decoupling the effects of material thickness and size scale on the transverse free vibration of BNNTs based on beam models

Here we establish a new model to decouple the material thickness and elastic property of single-walled boron nitride nanotubes, and thus we can evaluate natural frequencies of transverse free vibration by directly using the Timoshenko beam and Euler-Bernoulli beam models. Full atomic simulations regarding on the vibration behaviors are also presented for comparison. The determination procedure of thickness and the physical meaning of scale parameter for one-atomic thick material are clearly defined. In order to comprehensively unfold the influence of the selection of tubular thickness on natural frequencies, a series of assumed tubular thickness is also adopted to analyze the sources of error and its trend, and finally confirms the accuracy of our proposed result. The accuracy of the beam models is also found to be dependent on the type of boundary constraints. The precision of the beam theory improves for weaker boundary constraints due to the release of shear stresses. The paper heals the rift between atomic simulations and continuum theories.

Accurate Results for Free Vibration of Doubly Curved Shallow Shells of Rectangular Planform (Part.1)

A method is presented for determining the free vibration frequencies of doubly curved, isotropic shallow shells under general edge conditions and is used to obtain accurate natural frequencies for wide range of geometric parameters. Based on the shallow shell theory applicable to thin thickness shells, a method of Ritz is extended to derive a frequency equation wherein the displacement functions are modified to accommodate arbitrary sets of edge conditions for both in-plane and out-of-plane motions. In numerical computation, convergence is tested against series terms and comparison study is made with existing results by other authors. Twenty one sets of frequency parameters are tabulated for a wide range of shell shape and curvature ratio to serve as data for future comparison and practical design purpose.

Pilot Studies of Vibrations Induced in Perambulators When Moving on Different Surfaces

The ergonomics of transport is a topic widely described in the literature. One of the fields of ergonomics that researchers are engaged in is vibrometry (both laser and accelerometry) of travel and its translation into NVH (Noise, Vibration and Harshness). However, so far, the influence of baby carriage movement on the generated vibrations has not been described in more detail. The topic seems to be particularly important considering occurrence of vibrations with significant amplitudes, whose frequency range can have a direct bearing on the resonance frequencies of the child’s internal organs. The article presents the results of research consisting in the measurement of vibrations to which an infant, lying in two different types of prams, may be exposed when being transported on different surfaces. The author’s measurement system, based on accelerometry, was used for the research. The obtained weighted RMS acceleration values not only exceeded human comfort level in all cases (according to ISO standard) but several times were in the range of the highest discomfort (>2 m/s2). Furthermore, the observed vibration frequency range (≈0 ÷ 32 Hz) coincided with the frequencies of free vibration of organs and parts of the child’s body.

Evaluation of the adequacy of a spring‐mass model in analyses of liquid sloshing in anchored storage tanks

AbstractThis research addresses the influence of the load characteristics, that is, frequency content and maximum acceleration, on the wall stresses of an anchored water storage tank. A low‐density polyethylene tank with a range of six different aspect ratios (water height to tank radius) was tested using a shake table. Eight sine excitations that cover the lowest free vibration frequency of the tank‐water system were applied. Additionally, two sets of five Ricker wavelet excitations were utilized. Each set represents potential earthquakes with a bandwidth between a low and a high dominant frequency. The experimentally determined maximum stresses and those obtained from calculations using a common spring‐mass model employed for seismic analysis of tanks were compared. The results reveal that the relationship between the excitation frequency and the wall stresses strongly depends on the sloshing behavior, especially when the frequency of loading is in the vicinity of the lowest free–vibration frequency of the tank–water system. When the frequencies are dissimilar, there is a proportional relationship between stress and the maximum acceleration of the excitation. The spring‐mass model was found to underestimate both the maximum hoop stress (for aspect ratios greater than two) and axial stress (for aspect ratios equal to 0.5). This occurs because is the spring‐mass model cannot capture, in all cases, the contribution of chaotic sloshing to wall stress.