Vibrations Of Rods Research Articles

Influence of the foundation bed stiffness on the dynamic properties of the building as of a multi-mass cantilever bar

Relevance. The study of the interaction of buildings and structures with the base during an earthquake is one of the most important tasks of the theory of earthquake resistance. The response of the structure to seismic impact depends to a large extent on the ratio of the stiffness characteristics of the soil, foundation, and foundation structure. Moreover, taking into account a rather high degree of statistical variability of the characteristics of the soil foundation, it is possible to ensure the required level of safety of a structure only through the use of probabilistic models and a quantitative assessment of the reliability of the construction-base system as a whole. At present, for the calculation of the construction - base system for seismic loads, deterministic discrete models of the finite element method are mainly used. But these models are poorly adapted for probabilistic calculations and require extensive statistical data, which are currently insufficient. Therefore, in problems of reliability assessment, it is advisable to use simplified analytical models, which make it possible to derive the value of the statistical variability of its reaction with relatively small initial information about the system. The aim of the work - based on the well-known solution for the single-mass model to present an analytical solution in the matrix form of the problem of free horizontal vibrations of a multi-mass cantilever rod on the foundation specified by the elastic half-space model. Methods. A study was made of the effect of the compliance of the soil foundation on the frequencies and forms of horizontal vibrations of the structure. A comparison of the results with the calculation performed by the finite element method is given. Results. The obtained solution is intended to conduct a probabilistic calculation of the construction-base system under seismic loads and evaluate its reliability.

Free vibration analysis of functionally graded cylindrical helices with variable cross-section

The main purpose of this study is to investigate free vibrations of cylindrical helical rods with variable cross-section, composed of functionally graded materials, which may be used to model environmental effects. The material variation is considered to be along the rod axis. Governing differential equations of the free vibration are obtained via Timoshenko's beam theory, where axial and shear deformations and rotary inertia are taken into consideration. The governing equations are solved numerically by using the transfer matrix method. The effects of cross-section variation (βsec), material gradient index (βmat), helix pitch angle (α), and number of active coils (n) on the frequencies are investigated. All results are compared with the ANSYS solutions, in which a very well approximation is achieved.

Determination of frequency characteristics of thin-walled rods with transverse cracks

Vibrations of a thin-walled cantilever rod of a box-shaped profile were studied. The natural vibrations frequencies of a rod with a crack of different depth and located in different places of the rod were determined. The method of conducting an experiment using a vibration analyzer is given. A variant of determining the location of the crack depending on the change in the natural vibrations frequency is proposed. The resulting expression for estimating the location of the crack has an error of no more than 2.7 percent of the length of the rod, which makes it effective in practical application. The puchnost points of each oscillation form of the cantilever rod are determined, taking into account internal friction, which allowed us to specify the locations of sensors for maximum efficiency in evaluating the influence of crack parameters on natural frequencies. We found that when the crack is located in the places where the puchnosty for the corresponding vibration form causes noticeable energy dissipation due to friction in the crack.

Modeling Reactor Noise due to Rod and Thermal Vibrations with Thermal Feedback Using Stochastic Differential Equations

Fluctuations associated with power and detector readings in a nuclear reactor, commonly known as reactor noise, are of great importance in nuclear science and engineering. Two different types of noise are described in the literature: internal noise, which is associated with the inherent stochasticity of fission chains, and external noise, which is governed by physical fluctuations of the macroscopic system. The latter may include temperature fluctuations, vibration of regulation rods, fluent turbulence, bubble formation, and more. It is generally true that in power reactors, where high temperatures and strong hydrodynamic flows are characteristic, the external noise is dominant. The goal of this paper is to propose a stochastic differential equation (SDE) that models the effect of two types of external noise terms: the inlet temperature variations, which affect the power through reactivity feedback, and rod vibrations, which affect the reactivity directly. Although these aspects were studied in the past, they were only treated via nonstochastic equations. It is argued that the SDE approach, previously used only for modeling the effect of internal noise on nuclear reactor dynamics, is also highly suitable for modeling external noise. The main advantage of our approach is the ability to arrive at analytic formulas. The contributions presented in this paper based on the SDE approach are as follows. Under a linear approximation of thermal feedback, the stabilizing effect of thermal feedback is explained and quantified, and a limiting distribution is analyzed in full. An analysis of the detector response on a finite time interval is carried out, leading to a version of the Feynman variance-to-mean-ratio formula in the presence of external noise. Finally, a calculation of the eigenvalues associated with the linearized system alluded to above is performed, showing that in practical cases the rod vibrations and inlet temperature fluctuations correspond to eigenvalues in distinct timescales. The significance of these finding is discussed.

Time-frequency analysis of reactor neutron noise under bubble disturbance and control rod vibration

Time-frequency analysis technique is an effective analysis tool for non-stationary processes. In the field of reactor neutron noise, the time-frequency analysis method has not been thoroughly researched and widely used. This work has studied the time-frequency analysis of the reactor neutron noise experimental signals under bubble disturbance and control rod vibration. First, an experimental platform was established, and it could be employed to reactor neutron noise experiment and data acquisition. Secondly, two types of reactor neutron noise experiments were performed, and valid experimental data was obtained. Finally, time-frequency analysis was conducted on the experimental data, and effective analysis results were obtained in the low-frequency part. Through this work, it can be concluded that the time-frequency analysis technique can effectively investigate the core dynamics behavior and deepen the identification of the unstable core process.

Application of the Crank–Nicolson time integrator to viscoelastic wave equations with boundary feedback damping

Abstract We study the numerical solutions for a boundary feedback mechanism on torsional vibrations of a homogeneous viscoelastic rod. The problem is discretized by the Crank–Nicolson scheme based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step way, a convolution quadrature formula, constructed again from the trapezoidal rule, is used to approximate the integral term. Error estimates of the numerical scheme are derived in the $ l^{\infty }_{t}(0,~\infty ;~L^{2}(0,~1)) $ norm. Results from some numerical experiment are presented.

Vibration of a Segmented Rod

This paper presents the governing equation of motion, boundary conditions and exact vibration frequencies of a segmented rod where the segments are connected by hinges with elastic rotational springs of constant stiffness. The mass of each segment is assumed to be evenly distributed along the length of the rod. Another discrete model called Hencky bar-chain model (short for HBM; which is equivalent to the finite difference model for discretizing continuous rod) assumes the rod mass to be lumped at the ends instead and a different set of boundary conditions are adopted clamped end. The vibration results of a clamped–clamped segment rod are compared with those of the HBM. It is shown that the HBM underestimates the vibration frequencies when compared to the segmented rod model for a finite number of segments while both models furnish vibration solutions that converge to the solutions of Euler beam for infinitely large number of segments.

Analysis of the upflow conversion modification and influence on selected LOCA accidents in a PWR plant

The baffle jetting phenomenon, intensified by pressure difference across baffle plates, may cause excessive fuel rod vibrations and possible damage. The downward flow direction through the baffle-barrel region opposite to the upward flow through the core in pressurized water reactors affects increase of pressure gradient. Gaps between the baffle plates widen with reactor operating time and water jets become more pronounced. During the 2015 outage the NPP Krško has undergone the upflow conversion (UFC) modification in order to minimize baffle jetting and prevent fuel rod failure. The modification consisted in altering reactor vessel internals in such a way that the coolant downflow path in the baffle-barrel region was converted to an upflow path.The impact of the UFC modification on the nuclear power plant behaviour was analyzed in the paper. The double ended guillotine break of one direct vessel injection (DVI) line was selected as a representative scenario because the DVI line break is exhibiting the highest peak cladding temperature among the small break LOCA cases. The thermal-hydraulic analysis was carried out with the RELAP5/Mod3.3 code for core average channel conditions, while additional FRAPTRAN runs were performed to evaluate peak cladding temperatures for high power and high burnup fuel assemblies. Distribution of pressure and velocity fields in the reactor pressure vessel baffle-barrel region and change in bypass flow rates due to modification were determined by the ANSYS-Fluent calculation. The accident analysis has been performed for both the downflow configuration (pre-UFC) and the upflow configuration (post-UFC) to assess the influence of the modification. Additional cases with different reactor coolant pump trip times, higher core decay heat and reduced capacity of safety systems were also analyzed to evaluate the impact on reactor safety.

Longitudinal vibration analysis of sucker rod based on a simplified thermo-solid model

The longitudinal vibration of a sucker rod may change plunger stroke, thus affecting the pumping efficiency. In the study, a simplified thermo-solid coupling model of the longitudinal vibration of a sucker rod is established based on the consideration of the effect of the temperature along the well depth. Then, with the finite element method, the influence of longitudinal vibration of a continuous steel sucker rod on plunger stroke is studied. The calculation model involves two stages. In the first stage, the polished rod position remains stationary, whereas the sucker rod vibrates only under its gravity and liquid column gravity. The final stable state of the sucker rod is the initial state of the second stage. This stage aims to eliminate the effect of the initial tensile vibration. The longitudinal vibration characteristics of the sucker rod in the second stage are calculated based on the consideration of the excitations of both the polished rod displacement and the liquid column load in order to ensure the accuracy of calculation results. The influences of the frequency of stroke, stroke, and the length of sucker rod on plunger displacement are studied. The plunger displacement is slightly lagged behind the excitation on the polished rod due to the joint action of elasticity and load. The frequency of stroke, stroke, and the length of sucker rod affect the overtravel. When the sucker rod length is 2000 m, the overtravel reaches 0.12 m. Other parameters can also be optimized with this model.

BENDING AND FREE OSCILLATIONS OF THE VISCOELASTIC TOROIDAL SHELL, T AL SHELL, TAKING IN AKING INTO ACCOUNT THE RHEOL T THE RHEOLOGICAL PROPERTIES OF THE MATERIAL

The bending of a viscoelastic toroidal shell with diffuse boundary conditions is considered in this work. The relationship between stress and strain is described by the Aviary equations. Based on the principle of calculus of variations, equations of the dynamics of the toroidal shell are obtained. It is assumed that the movement along the normal to the median surface, along the tangents to circular sections and along the shell is not previously imposed any restrictions. As result of the calculation, It was found that shell vibrations occur at higher frequencies than rod vibrations (m)

On contact point motion in the vibration analysis of elastic rods

We present a systematic method for analyzing the vibrations of elastic rods whose effective length is variable, with particular emphasis on rods in unilateral contact with rigid surfaces. Problems of this type abound in engineering applications at all length scales, from the laying of submarine pipelines to the stiction of cantilevers in microelectromechanical systems (MEMS). By a careful treatment of boundary conditions, we elucidate the circumstances under which a rod of variable length can be treated as one of fixed length for the sake of analyzing small-amplitude vibrations. In applying our method to a simple free vibration problem, we encounter an unusual singular limit and observe a close connection between vibration, stability, and existence.

Production of vibration energy harvester with impact-sliding structure using magnetostrictive Fe-Co-V alloy rod

In this work, an axial-type vibration energy harvester with an impact-sliding structure that uses a magnetostrictive Permendur (Fe-Co-V) rod was produced and its power generation properties were measured. It was found that all the peak frequencies analyzed using the FFT method from the experimentally obtained voltage waveforms were consistent with the calculated inherent frequencies of the axial vibration of the rod. Therefore, the power generation by the proposed harvester likely resulted from the inherent axial vibration of the magnetostrictive rod. Furthermore, the maximum output energy obtained in this work was 1592 µJ, which was generated by a harvester with 3000 coil turns and two magnets under a load resistance of 3 kΩ. It was found that the maximum output energy was generated when the load resistance was close to the coil impedance at the inherent frequency of the axial vibration of the rod.

Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods

In this paper, nonlocal free vibration of axial rods embedded in elastic medium is examined by utilizing Love–Bishop rod theory with finite element method for the first time in the literature. Firstly, kinematic relations and dynamic equilibrium of this rod formulation are constituted. Equation of motion and boundary conditions are attained with the help of first variation of total potential of nano rod and solved using separation of variables. The frequency equations of four different nanorod types are obtained. On the other hand, a size–dependent finite element formulation is presented based on Weighted Residual Method. The nondimensional frequency parameters of nanorods are calculated by using analytical and finite element methods for different parameters such as mode number, nonlocal parameter, rod length, length–to–diameter ratio, finite element number and a detailed discussion is performed in the numerical results. In order to understand the characteristics of this rod theory, several comparative results that include nonlocal simple rod formulation are given as well.

Study of the Dynamic Characteristics of A Cone-Shaped Recovery System on Submarines for Recovering Autonomous Underwater Vehicle

National navies equip their submarines with Autonomous Underwater Vehicle (AUV) technology. It has become an important component of submarine development in technologically-advanced countries. Employing advanced and reliable recovery systems directly improves the safety and operational efficiency of submarines equipped with AUVs. In this paper, based on aerial refueling technology, a cone-shaped recovery system with two different guiding covers (closed structure and frame structure) is applied to the submarine. By taking the Suboff model as the research object, STAR-CCM was used to study the influence of the installation position of the recovery system, and the length of the rigid rod, on the Suboff model. It was found that when the recovery system is installed in the middle and rear of the Suboff model at the same velocity and the same length of the rigid rod, the Suboff model has the good stability and less drag. It experiences the largest drag when being installed in the front of the rigid rod. Moreover, when the recovery system is installed in the front and middle of the rigid rod, the drag increases as its length increases, and the lift decreases as its length increases. Compared with the closed structure guiding cover, the Suboff model will have less drag and better stability when the recovery system uses the frame structure guiding cover. Besides, the deflection and vibration of the rigid rod were also analyzed via mathematical theory.

Experimental study of the vibration of vertical rod in liquid flow

Vibration of the vertical tube, induced by an axial flow, were studied experimentally. Experiments were carried out using the sensors based on the electrical impedance. The time chronology of the pass of the tube axis was reconstructed using four sensors. For various flow rates different modes of fluctuations were found. The amplitude, shape, and characteristic frequencies of signals were found to change depending on the liquid flow rate.

Study on Vibration Characteristics of Polishing Rod for Polishing Aeroengine Blade with Abrasive Cloth Wheel

The vibration of grinding tools is very essential to the performance of parts and the safety of machine tools. However, the vibrationis rarely studied in the process of polishing aeroengine blade with abrasive cloth wheel. Polishing force is measured by dynamometer in the process of polishing aeroengine blade with abrasive cloth wheel, and the time-domain expressions of polishing force are obtained. Time-domain and frequency-domain vibration solutions of position at which the abrasive cloth wheel is installed under the coupling of axial force and normal force are obtained by the simulation method. The results show that the total displacement is approximately equal to the displacement in the normal direction. The frequency-domain characteristics of axial force and normal force are similar. The frequency-domain characteristics of vibration in x, y, and z directions are similar. The component frequencies of polishing force are not the same as the component frequencies of rod vibration, which means the vibration is not stable. The range of spindle speed to avoid resonance in polishing process is calculated.

Unsteady pressure and velocity measurements in 5 × 5 rods bundle using grids with and without mixing vanes

The vibration of rods is of prime importance to estimate their wearing due to the frictions with grids’ elements. To predict this wearing, the pressure fluctuations exerted on the rods need to be identified and quantified. Pressure fluctuations are explored using an instrumented rod centred within a 5 × 5 rods bundle. The profiles of mean velocity and velocity fluctuations are explored around the instrumented rod using LDV. Two configurations of spacer-grids are used: with and without mixing vanes respectively noted WMV and NMV. The position of the measurement point is rotated over 360° and its distance from the grids is varied from 0.5Dh to 20Dh. The results are detailed for the Reynolds number 66000. The results show the non-homogeneity of the azimuthal distribution of the pressure fluctuations in the near wake of the spacer-grids for both configurations. The return to a homogeneous azimuthal distribution of the pressure fluctuations is faster without mixing vanes than with mixing vanes. The intensities of the pressure fluctuations are (in average) higher for WMV than for NMV by 13% and 136% at distances 0.5Dh and 20Dh from the grids respectively. The correlation functions and energy spectra highlight significant differences in the spatial and temporal structure of the pressure fluctuations for the configurations with and without mixing vanes. For both configurations, pressure fluctuations spectra reveal a frequency peak in the near wake of the grids. These peaks are around fDh/Vflow≈0.8 (69 Hz). Maxima are found at f∗=0.78 (66.8 Hz) and f∗=0.87 (74.6 Hz) respectively for WMV and NMV configurations. This frequency peaks persist farther from the grid for NMW than for WMV. The integral scale of pressure fluctuations highlight to different behaviour. The pressure integral length scale stays small compared to Dh (ranging from 0.13Dh to 0.2Dh) for the NMV configuration whilst it increases with the distance from the grid for the WMV case. This is coherent with the persistence of frequency peaks for the NMV configurations. The pressure integral length scale is found to be 2–9 times larger for WMV than for NMV with the increase of the distance from the grid between 0.5Dh and 20Dh. A significant anti-correlated domain appears for NMW and not for WMV. In addition, LDV results show oscillations of the velocity profiles and a damping with the distance from the grids. Between 2Dh and 15Dh, the dimensionless ratio of “pressure fluctuations to velocity fluctuations” 〈Prms'〉θ/12ρ〈vrms'2〉Psq is found to increase by about 70% and 100% respectively for the configurations NMV and WMV. This increase further reveals the complexity of the turbulent flow in the wake of spacer-grids.

A fractional nonlocal time-space viscoelasticity theory and its applications in structural dynamics

To overcome the long wavelength and time limits of classical elastic theory, this paper presents a fractional nonlocal time-space viscoelasticity theory to incorporate the non-locality of both time and spatial location. The stress (strain) at a reference point and a specified time is assumed to depend on the past time history and the stress (strain) of all the points in the reference domain through nonlocal kernel operators. Based on an assumption of weak non-locality, the fractional Taylor expansion series is used to derive a fractional nonlocal time-space model. A fractional nonlocal Kevin–Voigt model is considered as the simplest fractional nonlocal time-space model and chosen to be applied for structural dynamics. The correlation between the intrinsic length and time parameters is discussed. The effective viscoelastic modulus is derived and, based on which, the tension and vibration of rods and the bending, buckling and vibration of beams are studied. Furthermore, in the context of Hamilton’s principle, the governing equation and the boundary condition are derived for longitudinal dynamics of the rod in a more rigorous manner. It is found that when the external excitation frequency and the wavenumber interact with the intrinsic microstructures of materials and the intrinsic time parameter, the nonlocal space-time effect will become substantial, and therefore the viscoelastic structures are sensitive to both microstructures and time.

Convergence of Eigenfunction Expansions for a Boundary Value Problem with Spectral Parameter in the Boundary Conditions. I

We consider a spectral problem arising in the description of bending vibrations of a homogeneous rod with a longitudinal force acting in the cross sections, with clamped left end, and with a lumped inertial load at the right end. We give a general characterization of the arrangement of eigenvalues on the real line, study the structure of root subspaces and the oscillation properties of eigenfunctions, and analyze the basis properties of the eigenfunction system of this problem in the space $$L_{p}$$, $$1<p<\infty$$.

Torsional response of pretensioned elastic rods

We examine the torsional response of a pretensioned elastic circular rod. First, we review a procedure for computation of semidiscrete wave modes in the rod. The modes are continuous in the axial and circumferential directions but discrete in the radial direction. In terms of these modes, we synthesize a two-node hyperelement that can represent a rod segment, with discretization only in the radial direction, within the framework of finite element analysis. The effort involved in computations with this hyperelement is independent of the rod segment length. Next, using this efficient formulation, we analyze time-harmonic vibrations of a pretensioned rod subjected to torque at one end and fixed at the other. We show how the rod response is affected by the level of pretension. Assuming that torsional-response measurements are available for a known level of pretension, we show how the coefficient describing the effect of pretension can be estimated. Finally, with this coefficient, we present a simple, yet effective, procedure for estimation of the level of pretension.