Natural Frequencies Of Transverse Vibration Research Articles

Уточнение результатов численных расчетов частот поперечных колебаний стержня переменного сечения с упругим закреплением экстраполяционным методом

Thearticle describes a numerical experiment with the results of calculating the natural frequencies of transverse vibrations of a rod of variable cross-section with elastic fixation at one of the ends. The possibility of refining by extrapolation the calculated values of natural frequencies obtained by the method of algebraic polynomials of the fifth degree is analyzed. It is shown that ex-trapolation made it possible to increase the accuracy of calculations by the numerical method by 2-3 orders of magnitude

Seismic response analysis of a hydraulic fill dam

The seismic response of a highly heterogeneous hydraulic fill dam was evaluated by studying the natural frequencies of the first and second modes of vibration and analyzing the crest accelerations of different two-dimensional or 2D sections of the dam when subjected to two different earthquake excitations. The existing methods for determination of the natural frequency of earthen embankment structures can only be used to analyze the structural response at small strain levels. However, during seismic events, the natural frequency of an earthen dam is significantly affected by the nonlinear material behavior exhibited by the geomaterials at high strain levels. Hence, a novel method was devised to evaluate the strain-dependent natural frequency for plane strain 2D dam sections, using a synthesized multi-sine base excitation. The degradation of first and second natural frequencies of transverse vibration for all the 2D sections followed a linear trend when plotted against the respective crest’s root mean square strain on a logarithmic scale. The slope of the degradation curve was found to depend on the constituent material properties prevalent in the individual sections. The observed variations in natural frequencies and crest accelerations of the 2D dam sections were also used to assess the suitability of using two-dimensional plane strain analyses for studying the response of a long earthen dam having variability in material properties. Results indicate that there is a considerable chance of erroneous estimation of the seismic response of such highly heterogeneous earthen dams that are conventionally analyzed using plane strain models. A 2D analysis was found to merely capture the seismic response of the individual sections of the dam as independent entities while ignoring the stiffening or weakening effect of the adjacent neighboring segments that may have different material properties.

A simple formula for predicting the first natural frequency of transverse vibrations of axially loaded helical springs

A new formula that allows the first natural frequency of transverse vibrations of axially loaded steel helical springs to be determined has been presented in the paper. The relationship is easy to use and allows the first natural frequency of spring vibrations to be determined without the necessity of solving analytical or numerical models. According to the Authors’ knowledge, this is the first such formula that allows the first natural transverse vibration frequency to be determined and, consequently, when this frequency becomes zero, it allows the critical axial force or deflection causing the buckling of the spring to be determined. The way of obtaining the described formula is presented in the paper. The results of this formula are compared with those obtained using FEM and experiments. The advantages, drawbacks and limitations of the proposed relationship are also discussed.

Calculation of Natural Frequencies on Transverse Vibration for drill strings in Non-Uniform Temperature Field

It is necessary to consider the effect of temperature which changes the mechanical properties of materials when drill strings work in a non-uniform temperature field. In this paper, the natural frequencies of transverse vibration are studied under axial compression in the nonuniform temperature field. Using an asymptotic solution method, first-order approximation formula of the nonlinear vibration equation is obtained. The effects of temperature changes and axial pressure on the natural frequencies are discussed by an example. The results show that the influence of temperature on the critical load is small; change rate of natural frequency increases with the increase of axial pressure; while the axial pressures become larger, the influence of temperature on the first-order frequency is greater than on the second-order’s.

АВТОМАТИЗАЦИЯ МЕТОДИКИ РАСЧЕТА СОБСТВЕННОЙ ЧАСТОТЫ ПОПЕРЕЧНЫХ КОЛЕБАНИЙ ГРЕБНОГО ВАЛА

Improving the designing quality of engineering systems and accuracy of their calculations is the basis for increasing the technical and economic indicators of modern engineering. Using automated calculations during the design phase raises labour productivity and product quality. There have been discussed the stages of software implementation of engineering calculations of the natural frequency of transverse vibrations of the ship propeller shafts. The problem of determining the first natural frequency of transverse vibrations of the shaft depending on the method of its fixing has been solved. A computational scheme has been build, which presents a uniform cross-section beam based on a point elastic support with a concentrated mass at its end. Differential equations for oscillations have been derived; the method of finding the natural frequencies has been described. For proper calculating natural frequency there has been developed a sequence and its software implementation in Maple environment with application of computer algebra package. The effect of the method of fixing on natural frequency of shaft transverse vibrations has been estimated. The problem of finding natural frequency of transverse vibrations on the extended elastic stern shaft bearing has been considered. The algorithm presented in the paper serves as a basis for formalizing and automating the stage of calculating transverse vibrations and is also applicable for calculating the natural frequencies of various elements, including the shaft and the sliding bearing.

ВЛИЯНИЕ НЕОДНОРОДНОСТИ РАСПРЕДЕЛЕНИЯ КОЭФФИЦИЕНТА ЖЕСТКОСТИ ДЕЙДВУДНЫХ ПОДШИПНИКОВ НА СОБСТВЕННУЮ ЧАСТОТУ ПОПЕРЕЧНЫХ КОЛЕБАНИЙ ВАЛОПРОВОДА СУДНА

One of the most significant factors affecting the natural frequency of transverse vibrations of shaft-slide bearing systems is the stiffness coefficient of the slide bearing. The need to consider the influence of heterogeneity of stiffness coefficient of the bearing on its natural frequency is caused by the fact that when the bearing is worn, the modulus of longitudinal elasticity of the material changes, and since the bearing wears unevenly, the non-uniform distribution of the stiffness coefficient occurs. The problem of determining the natural frequency of transverse vibrations of a ship propeller shaft based on the foundation with a variable stiffness coefficient along the length has been studied. The differential equation of the propeller shaft vibrations written taking into account the stiffness coefficient variable along the shaft length is considered. It has been noted that, in the general case, this equation is a fourth-order partial differential equation and cannot be integrated in quadratures for an arbitrary stiffness distribution function along the shaft length. A numerical-analytical method for determining the natural frequency of a system based on approximation of the stiffness distribution function by a piecewise-linear function is proposed. The method is applied to calculate the natural frequencies of the pipeline section taking into account the functions describing the change in the stiffness coefficient. The proposed method allows to consider the section of the shafting enclosed in the stern bearing, subject to the non-uniform distribution of the stiffness coefficient of the bearing, and is the basis for improving the accuracy of finding the true natural frequency of transverse vibrations of the shafting.

Free and forced nonlinear vibration of a transporting belt with pulley support ends

This paper focuses on the free and forced nonlinear vibration of a viscoelastic transporting belt supported by pulleys for the first time. The transporting belt boundary conditions are usually specified as simply supported or fixed. Therefore, the static equilibrium of the belt is supposed to remain straight. However, the belt wraps around two pulleys in most practical mechanical systems. The round support of the pulleys causes nonhomogeneous terms in the boundary condition. A nonlinear dynamic model is established to describe the transverse vibration of the transporting belt. The static equilibrium configuration caused by the nonhomogeneous boundary conditions is determined by developing a numerical iteration scheme. By means of coordinate transformation, the influence of nonhomogeneous boundary conditions is eliminated. Then, the natural frequencies of transverse vibration of the belt around the equilibrium configuration are obtained by adopting the differential quadrature method. Furthermore, the inertia excitation of the belt system is taken from the periodic motion of the foundation. In dynamic mechanical systems, this kind of fluctuation induced by the engine is very common. By applying the differential quadrature method and the Runge–Kutta time discretization, the time histories of the forced vibration are numerically calculated. Resonance areas of the belt are studied by using the frequency sweep. Moreover, parametric studies are carried out to understand the influences of the system parameters. It is concluded that pulley support ends cause nonhomogeneous boundaries and the static equilibrium configuration, and consequences are exacerbated by transmission speed.

Effects of End Conditions of Cross-Ply Laminated Composite Beams on Their Dimensionless Natural Frequencies

The effects of the end conditions of cross-ply laminated composite beams on their dimensionless natural frequencies of free vibration is investigated. The problem is analyzed and solved by using the energy approach, which is formulated by a finite element model. Various end conditions of beams are used. Each beam has either movable ends or immovable ends. Numerical results are verified by comparisons with other relevant works. It is found that more constrained beams have higher values of natural frequencies of transverse vibration. The values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends because they are generated by longitudinal movements only.

Stability and vibration of a nanoplate under body force using nonlocal elasticity theory

In the present study, the problem of stability and vibration of a square single-layer graphene sheet under body force is studied using Eringen’s theory. The body force is constant and parallel with the plate. The boundary conditions correspond to the dynamical model of a nanoplate clamped at all its sides. Classical plate theory, upgraded with nonlocal elasticity theory, is used to formulate the differential equation of stability and vibration of the nanoplate. Natural frequencies of transverse vibrations, depending on the effects of body load and nonlocality, are obtained using Galerkin’s method. Critical values of the body load parameter, i.e., the values of the body load parameter when the plate loses its stability, are determined for different values of nonlocality parameter. The mode shapes for a square nanoplate under influences of body load and nonlocality are presented as well.

Crack Detection in Cantilever Shaft Beam Using Natural Frequency

Crack is one of the major factors that lead to failure of a machine component. Today, failure of shaft mostly takes place due to material fatigue. Different characteristics of vibration can be used to detect crack in beams. But a question that still remains that which is most sensitive that can be used for damage detection .In present work finite element analysis of beam using Ansys14.5 is done and first three natural frequencies of transverse mode are extracted. Based on which the behaviour of healthy and cracked beam is compared and concluded that natural frequency of transverse vibrations could be used to detect crack in cantilever shaft beam.

Frequencies of transverse vibration of an axially moving viscoelastic beam

One important issue in the investigation of axially moving systems is the viscoelastic constitutive relation. In the present paper, the effects of viscosity on the natural frequency of transverse vibration of an axially moving viscoelastic beam are studied. The viscoelastic material of the moving Euler-Bernoulli beam obeys the Kelvin model. For the first time, the qualitative difference between the natural frequencies with the material time derivative and the partial time derivative in the constitutive relation is investigated. The method of multiple scales with three terms is directly applied to obtain the approximate analytical solutions of the natural frequency. An interesting phenomenon is found in this study. Specifically, for an axially moving viscoelastic beam constituted by the material time derivative, the natural frequencies of transverse vibration may increase with the axial speed. Furthermore, the validity of the analytical results is examined by comparing with two numerical approaches, the differential quadrature methods (DQM) via separating variables and DQM combined with the fast Fourier transforms. There is qualitative difference between the results based on the constitutive relations with the material time derivative and the partial time derivative. Therefore, the results of this work provide an possible approach to determine which kind of the constitutive relation should be adopted to describe the viscoelastic property of axially moving materials.

Optimum design of thermally loaded beam-columns for maximum vibration frequency or buckling temperature

With the cross-sectional area function as the design variable, we seek the optimum design of clamped–clamped, thin, linearly elastic beam-columns under thermal load that maximizes the buckling temperature or the fundamental natural frequency of transverse vibrations. The beam-columns have given length, geometrically similar cross-sections of variable size and a given volume of material. A lower and/or upper bound constraint may be prescribed for the cross-sectional area of the beam-columns. To account for possible bimodality of the optimum designs, both the unimodal and bimodal optimality criterion approaches are applied. For a lower bound smaller than a specific value, bimodal optimum designs are obtained for problems of maximum buckling temperature. When maximizing the thermal fundamental natural frequency, optimum designs may be uni- or bimodal depending on the values of the given thermal load and the lower bound on the cross-sectional area. The geometrically unconstrained optimum design for maximum fundamental natural frequency obtained by Olhoff (1976) without consideration of thermal load, is shown to be at the same time an optimum design that maximizes the fundamental natural frequency at any temperature rise, but it is found to be associated with zero buckling load. Moreover, the optimum design maximizing the critical buckling temperature is very similar to that maximizing the mechanical buckling load obtained by Olhoff and Rasmussen (1977). This interesting analogy is explained by studying optimum design for minimum axial compressive force.

Simulation Analysis of Synchronous Belt Transverse Vibration Frequency

Aiming at the ZA type automotive synchronous belt, natural frequency of transverse vibration of automotive synchronous belt is analyzed theoretically by using model of string vibration, and ZA type automotive synchronous belt with belt length 400mm is analyzed by using ANSYS-WORKBENCH software. Solving natural frequency of belt under preload, the results are compared with the theoretical calculation results, and to verify the theoretical calculation results. It provides a basis to design multi-pulley driving system and reduce transverse vibration of synchronous belt in the process of transmission.

Transverse vibrations of embedded nanowires under axial compression with high-order surface stress effects

Implementing the high-order surface stress model into the Bernoulli–Euler beam theory, the transverse vibration of an axially compressed nanowire embedded in elastic medium is investigated. Closed-form expression is obtained for the natural frequency of a simply supported nanowire. The influences of compressive axial load, high-order surface stress and surrounding elastic medium on the natural frequency are discussed. Additionally, the analytical solution of axial buckling load for the simply supported nanowire is derived, which takes into account the effects of high-order surface stress and surrounding elastic medium. It is concluded from numerical results that the natural frequency of transverse vibration of the nanowire is dependent upon axial load, surrounding elastic medium, and high-order surface stress. Similarly, the dependences of the buckling load on surrounding elastic medium and high-order surface stress are significant.

Parametric excitation of electro-mechanical vibrations of carbon nano tube with distributed surface charge

Electromechanical vibrations of carbon nano tube (CNT), mounted on a metallic base and subjected to a sum of static and radio frequency (RF) electric fields parallel to its length, are investigated. The induced surface charge on the CNT is taken suitably distributed, following Landau and Lifshitz, so that the curved surface remains an equi-potential surface. The natural frequency of transverse vibrations of CNT, ωN depends on the dc electric field, besides the mechanical properties. When the RF frequency ω0 is close to 2ωN, the RF excites the vibrations as three mode parametric instability that saturates via nonlinearity in the restoration force.

Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

Analysis on Mechanics and Reliability of Drill String during Deep Drilling

The deep drill string is subjected to many kinds of loading under the condition of vibration and precession, such as tension, compression, bending and torsion, while the temperature and press are very high. Under this circumstance, the shallow drill string is more likely to failure. Based on the force analysis of the drill string, the drilling process of the deep drill string is simulated using finite element software. The stress distribution and displacement of the drill string is obtained under the interaction of drilling pressure, torque and temperature. At the same time, the effect of transverse vibration, longitudinal vibration and torsional vibration, on the stress and displacement of the drill string, is compared. The change law of the natural frequency of transverse vibration varying with the length of drill string, rotational speed and the hole size is discussed. Connected with concrete examples, the reliability and sensitivity of some influencing factors such as external diameter, wall thickness, the minimum yield strength, temperature, elastic modulus, Poisson’s ratio, axial loading and torque is obtained. The main factors that influence the reliability of the drill string are the yield strength and torque. The reliability of the drill string increases as the increase of yield strength, and decreases as the increase of the torque.

Research on Numerical Computation of Natural Frequency of Transverse Vibration for Simply Supported Beam of Non-Uniform

Recursive and inexplicit differential equation of the second order with variable coefficients is derived from the fourth order linear homogeneous differential equation with variable coefficients of transverse vibration of non-uniform beam, which is about deflection and bending moment according to boundary conditions and order reduction. By finite difference method, numerical computation and accuracy are studied for natural frequency of transverse vibration for simply supported beam of non-uniform. Theoretical analysis and orthogonal computation examples show that numerical computation algorithm is very simple, and accuracy of computation depends on variety rate of gradually changed cross section in vertical direction and numbers of computation step, which is independent of width and length of beam; numerical accuracy of computation is estimable for given length or numbers of computation step; and reasonable length or numbers of computation step is determinable for given accuracy demand.

VIBRATION AND STABILITY OF A HEAVY AND HEATED VERTICAL CIRCULAR PLATE

This paper is concerned with the vibration and stability of a standing, heavy, and circular plate when heated. This investigation also deals with the plate being exposed to inertial forces due to uniform acceleration. The plate edge is clamped. Natural frequencies of transverse vibrations depending on the plate weight and temperature were determined using the Galerkin's method. Mode shapes are given for some frequencies including the influence of weight parameter on changes in mode shapes. It is shown that frequencies split for the asymmetric mode shapes, so that there are two different frequencies in those cases. Critical weight parameter values where plate stability ceases were determined. Critical values of the weight parameter depending on Poisson's ratio are also presented herein.

A Simple Experiment to Demonstrate the Effect of Axial Force on Natural Frequency of Transverse Vibration of a Beam

An axial tension (compression) increases (decreases) the natural frequency of transverse vibration of a beam. The primary objective of the present work is to demonstrate this effect using a universal testing machine, an impact hammer, an accelerometer, an oscilloscope with spectrum facility and a mild steel specimen of circular cross-section. The above equipments are among the essential facilities available in a typical material testing and vibration laboratory. Therefore the proposed experimentation demands no special specific set-up and can be performed at no extra cost.