Finite-element analysis of the natural vibrations of statically loaded turbomachine blading

The finite-element method of analyzing the natural vibrations of turbomachine blading with allowance for the effect of static loads is presented. Basic stages in the three-dimensional computation are described. The results of computation of test problems are presented. The influence exerted by centrifugal forces and characteristic design features on the vibration characteristics of real blades is investigated.Translated from Problemy Prochnosti, No. 7, pp. 88–94, July, 1990.     

Application of mixed variational formulations based on the finite element method to the solution of problems on natural vibrations of elastic bodies

We consider mixed variational formulations and the application of the mixed approximations of the finite element method to the solution of problems on natural vibrations of elastic bodies. To solve the generalized spectral problem, three forms of the mixed variational formulations are proposed. The correctness and stability of mixed variational formulations for displacements, strains and stresses are investigated. Matrix equations of the mixed method are given whose solution is performed using the modified algorithm of the steepest descent method. The results of calculations for natural frequencies of free vibrations of a straight and a circular beam are presented that are obtained in the solution of the problem in a two-dimensional formulation based on the classical and mixed finite-element method approaches. __________ Translated from Problemy Prochnosti, No. 2, pp. 121–140, March–April, 2008.     

Analysis of nonlinear vibrations of a two-degree-of-freedom mechanical system with damping modelled by a fractional derivative

Free damped vibrations of a mechanical two-degree-of-freedom system are considered under the conditions of one-to-one or two-to-one internal resonance, i.e., when natural frequencies of two modes – a mode of vertical vibrations and a mode of pendulum vibrations – are approximately equal to each other or when one natural frequency is nearly twice as large as another natural frequency. Damping features of the system are defined by the fractional derivatives with fractional parameters (the orders of the fractional derivatives) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. The model put forward allows one to obtain the damping coefficient dependent on the natural frequency of vibrations, so it has been shown that the amplitudes of vertical and pendulum vibrations attenuate by an exponential law with damping ratios which are exponential functions of the natural frequencies. Damped soliton-like solutions have been found analytically.     

Vibration mechanisms in the solid phase of a fluidization bed

The phenomenon of periodic vibrations in a fluidization bed is analyzed in the light of two mechanisms of interaction between solid phase particles. Equations are derived for the frequency, the wave velocity, and the wave vector of periodic-bed vibrations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 1, pp. 76–81, July, 1972.     

Large-amplitude vibrations of nanowires with dissipative surface stress effects

In this study, nonlinear free vibrations of doubly clamped Euler–Bernoulli nanowires (NWs) have been considered. The von Kármán strain–displacement relationships along with the classic Zener model are implemented to derive the nonlinear differential equation of the flexural motion of NW. Nonlinear natural frequencies are calculated using the computer package Mathematica. The effects of size-dependent surface dissipation, mode numbers, and amplitude of vibrations on the nonlinear natural frequencies are investigated. It is shown that the surface dissipation effect on the normalized nonlinear natural frequencies depends on the amplitudes of vibrations. Also, comparisons are made with the results published in previous studies.     

基于谱元法的车辆-轨道结构频域振动特性研究

基于谱元法建立车辆-轨道结构频域振动模型,其中轨道结构模拟为三层铁木辛柯梁,车辆部分考虑为整车模型,运用Lagrange方程实现车辆与轨道结构的耦合,并采用虚拟激励法将轨道不平顺模拟为虚拟荷载,通过求解车辆-轨道整体结构的谱元法方程,得到车辆-轨道结构在频域内的振动响应。结果表明:钢轨、轨道板和底座板的第一、二、四阶振动峰值分别由车体、转向架、车轮自振引起,其他振动峰值由轨道结构系统自振引起;钢轨、轨道板和底座板的振动能量分布在较宽的频率范围;在离开车辆一侧且距离端轮对2.5 m处,1~800 Hz内钢轨振动迅速衰减,当大于800 Hz时,钢轨振动衰减缓慢;在距离端轮对18 m处,25~1171 Hz内钢轨振动衰减基本稳定;在距离端轮对20.5 m处,小于25 Hz时,钢轨振动随着离开端轮对距离的增加迅速衰减,当大于1171 Hz时,钢轨振动则衰减较小。     

Free vibrations of a shallow shell in fluid under geometrically nonlinear deformation

The method of calculating the nonlinear vibrations of shallow shells in fluid is proposed. Normal modes and natural frequencies of linear vibrations in fluid are specified, then the nonlinear vibrations of a shell are expanded in terms of the normal modes of the system. The Shaw–Pierre normal nonlinear modes are determined to get the shell model with the finite number of degrees of freedom.     

Linear dynamical analysis of fractionally damped beams and rods

The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.     

Measurement of cutting tool vibrations during turning

Results are provided for an experimental study of cutting tool vibrations during turning and their effect on the surface roughness of an article being machined. __________ Translated from Izmeritel’naya Tekhnika, No. 7, pp. 22–24, July, 2006.     

Variational Principles for Vibrating Carbon Nanotubes Conveying Fluid,Based on the Nonlocal Beam Model

Variational principles are derived in order to facilitate the investigation of the vibrations and stability of single and double-walled carbon nanotubes conveying a fluid, from a linear time-dependent partial differential equation governing their displacements. The nonlocal elastic theory of Euler-Bernoulli beams takes small-scale effects into account. Hamilton’s principle is obtained for double-walled nano-tubes conveying a fluid. The natural and geometric boundary conditions identified are seen to be coupled and time-dependent due to nonlocal effects.     

Flow induced vibrations of a triangular tube array in various arrangements of orientation and natural frequency

Abstract

Crossflow induced vibrations of a triangular tube array with a pitch ratio 1.33 were investigated experimentally. The streamwise and cross‐stream displacements of a monitored tube in the array were simultaneously measured by two accelerometers to examine the tube response to the cross flow in a water tunnel. The experiment was aimed to study the effects of the array orientation, and the tube's natural frequency on the flow induced vibration of the tube array. It is shown by amplitude diagrams that fluid elastic vibrations exist when the reduced velocity is above a critical value. The critical reduced velocity is found to be sensitive to the orientation of the test array. Based on the measured data of critical reduced velocity, the tube array in a triangular pattern (at a 30‐deg orientation with respect to the flow direction in the experiment) is found to be more stable than when in a rotated triangular pattern (0‐deg orientation). Furthermore, it is illustrated that the discrepancy in natural frequency of the tubes delays the occurrence of the fluid elastic vibrations of the tube array. With all the tubes in the test array having the same natural frequency, the orbits of the tube that exhibits fluid elastic vibrations are an organized, elliptic shape. The corresponding spectra are line‐dominated with peaks at the natural frequency and its harmonics, suggesting that the tube vibration is an organized oscillator. Without the same natural frequency as the surrounding tubes, the monitored tube exhibits fluid elastic vibration at larger reduced velocity, vibrating in a relatively random orbit.     

Errors of electrical methods of determining the natural frequency of electrodynamic seismic detectors

An examination is made of the metrological properties of an electrical method of determining the characteristic parameters of a seismic detector. The method is an alternative to the method of having mechanical vibrations act directly on the detector.Translated from Izmeritel'naya Tekhnika, No. 7, pp. 37–41, July, 1994.     

Stability improvements for an interferometer through study of spectral interference patterns

Conventional Mach-Zehnder interferometer configuration is modified to enhance its stability from the vibrations. To study the effects of vibrations on the Mach-Zehnder interferometer, we used spectral interference fringes from a broadband nanosecond dye laser source. We observed an improvement in the stability of the interferometer by a factor of 3.     

Stability and bifurcations of the steady-state modes in nonlinear vibrations of plates and shells

A method is proposed for numerical investigation of stability and bifurcations of nonlinear vibrations of plates and shells under the action of periodic loads. A system of equations of motion are formed by the methods of finite elements and generalized coordinates by accounting for the geometrical nonlinearity of the objects. The obtained system of differential equations is solved by the parameter extension method. As a result, steady-state curves of the nonlinear vibrations are constructed. Branching solutions at the points of bifurcation are obtained by the method of projections. The proposed method makes it possible to determine the subharmonic solutions and investigate their stability over a wide range of load parameters. Nonlinear vibrations of a series of plates and shells are investigated.Kiev. Translated from Problemy Prochnosti, No. 12, pp. 97–102, December, 1989.     

Test plant for the analysis of deformation vibrations of models of structures by means of a method of contactless excitation

A plant for the analysis of the strain vibrations of a periodic shelf structure by a method of low-frequency, contactless excitation is created and tested. The method involves varying the amplitude of a stepwise force, recording the strain response of the structure at sites where ferroelectric gauges are located, and Fourier analysis of the dynamic strain. The presence of stochastic and high-Q strain vibrations in the model structure is confirmed. It is established that, in general, both natural as well as induced (including stochastic) vibrations may be recorded and distinguished by means of two or three ferroelectric strain gauges.     

Method of acoustic sensor diagnosis

The theoretical dependence of the noise of electrically excited damping vibrations of a piezoelectric sensor on its malfunction is determined. A method is proposed for noise diagnosis for an acoustic piezoelectric sensor for monitoring the plasma source of optical air breakdown. __________ Translated from Izmeritel’naya Tekhnika, No. 7, pp. 65–67, July, 2008.     

平面闸门垂向自激振动机理和稳定性研究

通过分析平面闸门垂向振动机理将平面闸门垂向自激振动分为两类:流量系数及流体惯性引发振动与驰振引发振动,并对两种自激振动的机理与振动稳定性进行研究,重点考察Scruton数对闸门振动稳定性的影响,提出关于Scruton数的稳定性指标,实验结果与稳定性指标相吻合。     

On the procedure of evaluating the damping ability of the materials of specimens subjected to torsional and bending vibrations

We describe a method for the design and numerical analysis of compact mechanical vibrating systems with generators of vibrations on their inertial weights and establish the conditions of damping of unwanted vibrations in these systems aimed at guaranteeing the absence of favorable conditions for the introduction of systematic errors to experimental results. The designed and manufactured vibrating systems are dynamically balanced, and predicted values of the frequencies of their natural vibrations satisfactorily agree with the experimental data. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 145–152, July–August, 1997.     

Dynamic analysis of the micromilling process – influence of tool vibrations on the quality of microstructures

Micromilling is a suitable technique for the manufacturing of microstructured parts. The downscaling of the milling process leads to problems, which are caused by the low stability of microtools and their manufacturing. Tool vibrations and runout occur, resulting in poor surface qualities and undesirable surface location errors. The presented results of the experimental analysis include a stability diagram for the spindle speed range of n = 60,000–120,000 rpm and tool vibrations measured in the xy‐plane. Micrographs illustrate the influence of tool vibrations on the quality of the generated surface structures. In addition, the dynamic runout is characterized and its effects are discussed.     

Free vibration and stability analysis of three-dimensional sandwich beams via hierarchical models

This paper presents a free vibration and a stability analysis of three-dimensional sandwich beams. Several higher-order displacements-based theories as well as classical models (Euler–Bernoulli’s and Timoshenko’s ones) are derived assuming a unified formulation by a priori approximating the displacement field along the cross-section in a compact form. The governing differential equations and the boundary conditions are derived in a nucleal form that corresponds to a generic term in the displacement field approximation. The resulting fundamental nucleo does not depend upon the approximation order N that is a free parameter of the formulation. A Navier-type, closed form solution is used. Simply supported beams are, therefore, investigated. Slender up to very short beams are considered. As far as free vibrations are concerned, the fundamental natural frequency as well as natural frequencies associated to torsional and higher modes such as sheet face bending and twisting (typical of sandwich structures) are investigated. The stability analysis is carried out in terms of critical buckling stress in the framework of a linearised elastic approach. Results are assessed towards three-dimensional FEM solutions. It is shown that upon an appropriate choice of the approximation order, the proposed models are able to match the three-dimensional reference solutions.