Vibration Of Disk Research Articles

A Study on the Transverse Vibration of the High Speed Rotating Circular Saw and the Countermeasure

In this paper, we considered a new tensioning method by the centrifugal stress on the base metal of rotating circular saw. First, the residual stress distribution on the base metal generated by roll tensioning and the centrifugal stress distribution was compared by calculation using a disk model (Φ305mm). As the results, it was confirmed that the tangential stress σθR by centrifugal force at outer of disk became larger than the residual stress σθP by roll tensioning when the rotational speed is higher than 7300 rpm. This rotational speed is larger than 1.4 times of the critical speed (resonance rotational speed) of disk. And it is concerned that the unstable vibration may be generated. Therefore, we investigated the behavior of the unstable vibration and established the countermeasure by the rotating experiment. As the results of the rotating experiment, we considered that the unstable vibration is the self-excited vibration as the negative damping caused by the effects of the high-speed airflow around the rotating disk. According to this consideration, we investigated the relationship of the damping coefficient of disk and the behavior of the unstable vibration. As the result, it is confirmed that the rotational speed of the unstable vibration occurred becomes higher with the increase of the damping coefficient of disk. It is confirmed that it is possible to suppress the unstable vibration of disk and provide the tensioning effect by centrifugal force of disk.

Rocking vibration of a rigid circular disc in a transversely isotropic full‐space

AbstractIn this paper, forced rocking vibration of a rigid circular disc placed in a transversely isotropic full‐space, where the axis of material symmetry of the full‐space is normal to the surface of the plate, is analytically investigated. Because of using the Fourier series and Hankel integral transforms, the mixed boundary‐value problem is transformed into two separate pairs of integral equations called dual integral equations. The dual integral equations involved in this paper are reduced to Fredholm integral equations of the second kind. With the aid of contour integration, the governing integral equation is numerically evaluated in the general dynamic case. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure and the static impedance/compliance function are explicitly determined, and it is shown that the pressure in between the plate and the full‐space and the compliance function reduced for isotropic half‐space are identical to the previously published solutions. The dynamic contact pressure in between the disc and the space and also the related impedance function are numerically evaluated in general dynamic case and illustrated. It is shown that the singularity exists in the contact pressure at the edge of the disc is the same as the static case. To show the effect of material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared. Copyright © 2010 John Wiley & Sons, Ltd.

DETECTION OF STICK-SLIP OCCURRENCES THROUGH NEAR FIELD NOISE MEASUREMENTS

The present study is concerned with evaluating the pin-on-disk test rig usage and the measurements conducted to recognize the occurrence of the stick-slip phenomenon. The test rig consists of two separate blocks to ensure that the vibration of the elastic disk and emitted noise are completely adverted to friction between the moving pin and stationary steel disk. The occurrence of stick slip is clearly picked up using near field noise measurements. It can be distinguished from the vibration signal as high consecutive spikes, but the existence of such spikes is clearer and more definite in the noise signature.

Vibration Response of Elastic Disks in Surrounding Fluid: Viscous Versus Acoustic Effects

The vibrations of thin, elastic, circular disks such as musical cymbals, hard disk drives, and microscale resonators are significantly influenced by the presence of a surrounding fluid. The energy of disk vibrations is known to dissipate into viscous losses and to radiate away as sound. However, the relative importance of these mechanisms is not well understood. In this paper, we present three-dimensional computations of the fluidic impedance of thin, elastic disks vibrating with small amplitudes under ambient conditions. These computations encompass both macroscale and microscale disks, a wide range of operating frequencies, and different fluidic environments. Viscous fluidic impedances are computed using a finite element model, whereas acoustic fluidic impedances are computed using a boundary element method. For a disk with a given clamping ratio vibrating in a specific mode, the nondimensional viscous impedance depends on the unsteady Reynolds number, while the nondimensional acoustic impedance depends on the ratio of structural to acoustic wavelengths. It is shown that viscous losses dominate the fluid damping of disks in data storage and circular saw applications and of conventional disk microresonators. However, for ultrahigh frequency resonators, acoustic radiation must be taken into account to correctly estimate the overall fluid damping. The computed fluidic impedances are expected to be an important aid in the design of a wide range of disk resonators up to the megahertz regime.

Transverse vibration of spinning disk with attached distributed patch and discrete point masses using finite element analysis

Free and forced transverse vibration characteristics of a thin spinning disc attached to a rigid core have been investigated by finite element analysis using ANSYS software. The effect of discrete point masses and patches of distributed masses attached at the periphery of the plate on free and forced vibration behavior of a spinning disc has been investigated. Discrete point masses and distributed patch masses have been placed at higher strain regions to look into its influence on the natural frequencies, mode shapes and response of the plate to external excitation. Results for eight and sixteen point masses and patches on the vibration characteristics have been compared. It has been observed that discrete patches and point masses have significant influence on the modal frequencies and these can also act as dynamic vibration absorbers in reducing vibration of a spinning disc. It has also been shown that discrete patches of piezoelectric patches can also be used to actively control vibration of the spinning disc. Keywords: Free and forced vibration, discrete mass and patch, vibration control.

A Comprehensive Procedure to Estimate the Probability of Extreme Vibration Levels Due to Mistuning

A new procedure is developed to find the probabilities of extremely high amplification factors in mistuned bladed disk vibration levels, typical of events which occur rarely. While a rough estimate can be made by curve-fitting the distribution function generated in a Monte Carlo simulation, the procedure presented here can determine a much more accurate upper bound and the probabilities of amplification factors near to that bound. The procedure comprises an optimization analysis based on the conjugate gradient method and a stochastic simulation using the importance sampling method. Two examples are provided to illustrate the efficiency of the procedure, which can be 2 or 3 orders of magnitude more efficient than Monte Carlo simulations.

Finite Element Analysis of Thermoelastic Damping in Contour-Mode Vibrations of Micro- and Nanoscale Ring, Disk, and Elliptical Plate Resonators

Thermoelastic damping in contour-mode in-plane vibrations of rings, disks, and elliptical plates is investigated on various size scales, using a reduced finite element formulation. The Fourier scheme is applied to the axisymmetric geometries including circular rings and disks, and is found to be remarkably efficient in searching solutions. The numerical accuracy is further improved by the implementation of quadratic interpolation functions. The computational results are validated by comparing with the commercial software packages as well as the existing analytical solutions in literature. For resonators of elliptical shapes, the dominant frequency has a weak dependence on the geometric aspect ratio γ, whereas the effect of γ on the quality factor (Q value) is much stronger and the peak Q value of the leading mode consistently occurs in the vicinity of γ=1.42.

Applications of continuous tracking SLDV measurement methods to axially symmetric rotating structures using different excitation methods

This paper presents the latest developments of the continuous-scanning laser Doppler vibrometry (C-SLDV) measurement methods applied to rotating axially symmetric structures such as bladed discs. Measurement of vibrations of rotating structures are still difficult to perform. The main reason is due to the limitations of conventional transducers used for capturing the vibrations, these are often unable to measure and/or transfer a good quality signal remotely. The LDV became an ideal candidate to replace/aid such transducers, but only the introduction of two scanning mirrors in front of a laser head produced a step forward in measurements of vibrations of rotating targets; SLDV systems are now widely used both in industry and academic communities. This work shows how a commercial SLDV system can be used for measurement of vibrations of bladed discs under rotating conditions. The features of the scanning head, which was not modified for this study, were exploited up to the limit to achieve the synchronization between the scanner and a rotating target. The simplest tracking technique is performed when a circle-line scan pattern is synchronized with the rotating bladed disc so as to produce the point tracking measurement method. The extension of such a discrete measurement method to a full-field one is made when the laser beam is capable of tracing continuously either a line (LineScan tracking) or an area (AreaScan tracking) over the surface of the rotating structure. The development of tracking C-SLDV measurement techniques is achieved by the use of the traditional excitation methods and the development of a new excitation system, which will bring the excitation and the measurement to be in the same rotating frame of reference. Several experimental results are provided to illustrate the use of such techniques in turbomachinery industries.

Transverse nonlinear vibrations of a circular spinning disk with a varying rotating speed

We analyze the transverse nonlinear vibrations of a rotating flexible disk subjected to a rotating point force with a periodically varying rotating speed. Based on Hamilton’s principle, the nonlinear governing equations of motion (coupled equations among the radial, tangential and transverse displacements) are derived for the rotating flexible disk. When the in-plane inertia is ignored and a stress function is introduced, the three nonlinearly coupled partial differential equations are reduced to two nonlinearly coupled partial differential equations. According to Galerkin’s approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes is derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including the periodic, period-n and multi-pulse type chaotic motions for the spinning disk with a varying rotating speed. It is also found that among all parameters, the damping and excitation have great influence on the nonlinear responses of the spinning disk with a varying rotating speed.

Sound Field Characteristics in Air Gaps of Noncontact Ultrasonic Motor Driven by Two Flexural Standing Wave Vibration Disks

The sound field characteristics in the air gaps of the noncontact ultrasonic motor driven by two flexural standing wave vibration disks were analyzed by finite element method (FEM). Standing wave sound fields with an opposite phase were generated in the two air gaps by a single driving stator. Traveling wave sound fields in the two air gaps are formed as the superposition of the standing wave sound fields generated by two stators whose temporal phases and spatial positions are different from each other. The traveling direction of the calculated sound fields coincided with the rotating direction of the rotor observed in the experiments. The intensity of the sound field generated in the air gap by the single driving stator was measured by detecting the voltage induced in another stator. A gap distance wider than 0.5 mm is required for a high revolution speed because the sound field generated in the gap on the opposite side of the rotor is weak in an air gap narrower than 0.3 mm.

Euler's disk: examples used in engineering and applied mathematics teaching

Euler's disk is a toy described at http://www.eulersdisk.com. Aspects of its motion are modelled as an ideal disk rolling on a horizontal plane. In the final stages of Euler disk motions, the disk is nearly flat to the plane. Asymptotic approximations to the frequency of finite amplitude oscillations on steady (non-dissipative) rolling motions of the Euler disk are described. There are two different approximations which are appropriate in different limits. When the parameters are such that both apply, the formulae for the frequency agree: this appears to be new and simple. The material has been used in teaching; the teaching, and related, materials are available via the web [Keady, Math2200 Lecture Handouts, UWA]. References M. Batista. The nearly horizontally rolling of a thick disk on a rough plane. of a thick rigid disk Regular and Chaotic Dynamics 13 (4) (2008), 344--354. doi:10.1134/S1560354708040084 H. Caps, S. Dorbolo, S. Ponte, H. Croisier and N. Vandewalle. Rolling and slipping motion of Euler's disk. Physical Review E 69 (6) (2004), 056610. doi:10.1103/PhysRevE.69.056610 R. H. Cushman and J. J. Duistermaat. Nearly flat falling motions of the rolling disk. Regular and Chaotic Dynamics 11 (1) (2006) 31--60. doi:10.1070/RD2006v011n01ABEH000333 G. Keady. MATH2200 Lecture Handouts. The rolling of a disk on a horizontal plane. Part I, General; and Part II, Euler disk motions. http://school.maths.uwa.edu.au/ keady/papers.html H. K. Moffatt. Euler's disk and its finite-time singularity. Nature 404 (2000) 833. doi:10.1038/35009017 A. H. Nayfeh and D. T. Mook. Nonlinear Oscillations (Wiley: 1979). O. M. O'Reilly. The dynamics of rolling disks and sliding disks. Nonlinear Dynamics 10 (1996) 287--305. M. Srinivasan and A. Ruina. Rocking and rolling: a can that appears to rock might actually roll. Phys. Rev. E 78 (6) (2008) 066609. doi:10.1103/PhysRevE.78.066609 A. A. Stanislavsky and K. Weron. Nonlinear oscillations in the rolling motion of Euler's disk. Physica D 156 (2001) 247--259. doi:10.1016/S0167-2789(01)00281-0 J. L. Synge and B. A. Griffith. Principles of Mechanics. (McGraw-Hill, 3rd ed, 1959). W. T. Thomson. Introduction to Space Dynamics. (Dover: 1986 reprinting of Wiley 1963). R. Villanueva and M. Epstein. Vibrations of Euler's disk. Phys. Rev. E 71 (7) (2005) 066609. doi:10.1103/PhysRevE.71.066609 E. T. Whittaker. A treatise on the analytical dynamics of particles and rigid bodies. (CUP, 2nd ed, 1917).

Experimental Research on the Vibration Characteristics of Flexural Vibration Disk

The vibration characteristics of flexural vibration disk in ultrasonic vibration system are studied through experiments. The results show that the resonant frequency of flexural vibration disk reduces with the reducing of its thickness and the resonant frequency increases with the reducing of its diameter. When designing the ultrasonic vibration system, because the transformer horn and the flexural vibration disk are designedly integrated, the defects caused by the way of threaded connection can be avoided. It is found that flexural vibration is dominating in the disk with some reflected ultrasonic through experiments, at the same time, there exist amplifying characteristics in the vibration of the disk and the maximal amplitude can reach 18 μm. The corresponding results provide certain reference value for the structure design of ultrasonic vibration system and the application in ultrasonic vibration deep-hole honing.

Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half space

This paper is concerned with the investigation of the vertical vibration of a rigid circular disc buried at an arbitrary depth in a transversely isotropic half space in such a way the axis of material symmetry of the half space is normal to the surface of it and parallel to the vibration direction. By using the Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations called dual integral equations, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration, the governing integral equation is numerically solved in the general dynamic case. Two degenerated cases (i) the disc is buried in a transversely isotropic full space, and (ii) rigid circular disc is attached on the surface of the half space are discussed. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure and the static impedance/compliance function are explicitly found. It is shown that the vertical pressure and the compliance function reduced for isotropic half space are identical to the previous solutions reported in the literature. The dynamic contact pressure under the disc and the impedance function are numerically evaluated in general dynamic case and graphically shown that the singularity exists in the contact pressure at the edge of the disc is the same as the static case. In addition, the impedance functions evaluated here for the isotropic domain are collapsed on the solution given by Luco and Mita. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.

Synthesis of arbitrarily discrete, diffractionless beams using dual-wavelength diffractive objective lens for optical pickup head

Sensitivity to disk vibration and large component number, which complicates assembly and optical alignment, are the drawbacks of traditional optical pickup systems. Here, a numerical method of designing a dual-wavelength diffractive objective lens with high numerical aperture for generating arbitrarily discrete, diffractionless beams with extended depth of focus is presented. Simulation and experimental results show that the optimized design provides better resolution, longer depth of focus and higher diffractive efficiency. The proposed design is promising for next-generation optical pickup systems that are more robust to disk vibration and easier to assemble.

Frequency Equations for the In‐Plane Vibration of Circular Annular Disks

This paper deals with the in‐plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in‐plane free vibration of an elastic and isotropic disk is studied on the basis of the two‐dimensional linear plane stress theory of elasticity. The exact solution of the in‐plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson′s ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods.

Analysis of flexural vibration thick disk

The flexural vibration of thick disk is studied in this paper. The general solutions of radial displacement and transverse displacement for flexural vibration thick disk under the Mindlin theory are derived. The resonance frequency equations are derived when the boundary conditions are free, simply supported and clamped, and The results obtained from the analyticae method are in good agreement with those from the finite element method and also with the experimental results. These results can serve as reference for the design of flexural vibration thick disk radiator.

Methods of modulation of light wave propagated in optical fiber using ultrasonic wave

Glass optical waveguides are a promising medium for mixed laser-ultrasonic applications. Initial analysis of the possibilities of compensating for the flaws associated with laser and ultrasonic technologies, e.g. in the process of cutting biological tissues, showed that such a mixed technology could be more effective. The measurements of ultrasonic wave transmission ware performed for three representative types of optical fibers: a multimode step-index fiber, a single mode step-index fiber and multimode gradient-index fiber. The material the optical fiber is made of enables simultaneous transmission of light and ultrasonic wave. This study presents amplitude characteristics of the measured output voltages of the capacitive sensor for the types of optical fibers in question. The work also presents methods of modulation of light wave propagated in optical fiber by means of ultrasonic wave, which is performed with the use of radial vibrations of piezoelectric disk and longitudinal vibrations generated by a sandwich ultrasonic transducer with a wave velocity transformer. Experiments were performed using a Mach-Zehnder interferometer.

Bessel-like response in transducer based on homogeneously poled piezoelectric disks: Modeling and experimental analysis

The study of ultrasonic Bessel transducers is motivated by their ability to produce limited diffraction ultrasonic beams, providing improved lateral resolution and large penetration depth in the related ultrasonic field. There are three principal methods to construct a Bessel transducer. On the other hand, it is well known that radial modes in homogeneously poled piezoelectric disks (HPPD) are governed by a typical Bessel differential equation. In this paper, the possibility to generate Bessel radiation using a thickness HPPD is proposed and experimentally evaluated, by analyzing the field effects of the disk vibrations when it is driven in its radial modes PACS.

충만 디스크의 면내 진동 해석을 위한 1차원 환상 평판 요소

We present a one-dimensional annular plate element with which the in-plane vibration of full disks can be analyzed efficiently and accurately by using the FEM. Its elementary mass matrix and stiffness matrix are derived, respectively, from the virtual work by effective forces and the virtual strain energy. The static deformation modes obtained from an integration of the differential equilibrium equations of the annular plate are used as interpolation functions of the one-dimensional annular plate element. The in-plane natural vibration characteristics of a 2-step full disk and a uniform full disk are analysed. Its results are compared with the results obtained by utilizing two-dimensional 8-node quadrilateral plane elements and cyclic symmetry of the disk. And also, by comparing with the theoretical results of previous researchers, the efficiency and accuracy of the presented element are verified.

Vibration and critical speeds of composite-ring disks for data storage

A new concept for data storage disks is proposed to increase operating speed with minimum changes in the geometry and design of conventional data storage disks. The disk—named a composite-ring disk—is composed of a storage material inside and a thin composite ring outside. Stress distributions are found for the rotating disk composed of two annular disks, of which the inside is made of isotropic material and the outside is made of orthotropic material. The dynamic equation for a composite-ring disk in rotation is formulated to calculate its natural frequencies and critical speeds. For the solution of transverse vibration, a rotational symmetry condition is applied in the circumferential direction and a finite element interpolation with Hermite polynomials is performed in the radial direction. The results show that reinforcing a disk at the rim increases critical speeds drastically, and can cause buckling in mode (0,0) which occurs above the lowest critical speed.