Rigid Disk Research Articles

Model based identification of crack and bearing dynamic parameters in flexible rotor systems supported with an auxiliary active magnetic bearing

In this paper, possibility of identification of a crack appearing in flexible rotor systems supported with active magnetic bearing has been investigated. A cracked rotor with active magnetic bearing (AMB) support in auxiliary bearing configuration has been analysed for identification of crack and end support bearing stiffness. Presence of AMB in an auxiliary bearing configuration provides additional damping to the rotor system, thus attenuates the vibrations induced by rotor flaws. Identification strategies based on vibration responses of such a rotor may not produce correct results. However, loss of diagnostic information due to attenuation of the vibration signal could be compensated for, by supplementing the vibration signal with the AMB control current history. With inclusion of the control current history, the crack force and other parameters could be identified. A cracked rotor with multiple discs has been modelled by standard finite elements method, with consideration of gyroscopic effect due to rigid discs. Breathing behaviour of the crack has been modelled with a switching crack excitation function, containing multiple harmonics of both forward and backward nature. Full spectrum of the frequency domain of the response is utilised to develop the identification algorithms. This algorithm identifies the crack force in form of additive crack stiffness and simultaneously estimates the disc unbalances, end support bearing stiffness and active magnetic bearing dynamic parameters as well. The algorithm has been tested in a simple rotor system for the measurement noise and bias errors in system parameters, and found robust.

Disc–halo interactions in ΛCDM

We present a new method for embedding a stellar disc in a cosmological dark matter halo and provide a worked example from a {\Lambda}CDM zoom-in simulation. The disc is inserted into the halo at a redshift z = 3 as a zero-mass rigid body. Its mass and size are then increased adiabatically while its position, velocity, and orientation are determined from rigid-body dynamics. At z = 1, the rigid disc is replaced by an N-body disc whose particles sample a three-integral distribution function (DF). The simulation then proceeds to z = 0 with live disc and halo particles. By comparison, other methods assume one or more of the following: the centre of the rigid disc during the growth phase is pinned to the minimum of the halo potential, the orientation of the rigid disc is fixed, or the live N-body disc is constructed from a two rather than three-integral DF. In general, the presence of a disc makes the halo rounder, more centrally concentrated, and smoother, especially in the innermost regions. We find that methods in which the disc is pinned to the minimum of the halo potential tend to overestimate the amount of adiabatic contraction. Additionally, the effect of the disc on the subhalo distribution appears to be rather insensitive to the disc insertion method. The live disc in our simulation develops a bar that is consistent with the bars seen in late-type spiral galaxies. In addition, particles from the disc are launched or "kicked up" to high galactic latitudes.

Numerical analysis of MHD Casson Navier’s slip nanofluid flow yield by rigid rotating disk

Abstract An exertion is perform to report analysis on Casson liquid equipped above the rigid disk for z ¯ > 0 as a semi-infinite region. The flow of Casson liquid is achieve through rotation of rigid disk with constant angular frequency Ω ¯ . Magnetic interaction is consider by applying uniform magnetic field normal to the axial direction. The nanosized particles are suspended in the Casson liquid and rotation of disk is manifested with Navier’s slip condition, heat generation/absorption and chemical reaction effects. The obtain flow narrating differential equations subject to MHD Casson nanofluid are transformed into ordinary differential system. For this purpose the Von Karman way of scheme is executed. To achieve accurate trends a computational algorithm is develop rather than to go on with usual build-in scheme. The effects logs of involved parameters, namely magnetic field parameter, Casson fluid parameter, slip parameter, thermophoresis and Brownian motion parameters on radial, tangential velocities, temperature, nanoparticles concentration, Nusselt and Sherwood numbers are provided by means of graphical and tabular structures. It is observed that both tangential and radial velocities are decreasing function of Casson fluid parameter.

Flow Analysis of Viscous Nanofluid Due to Rotating Rigid Disk with Navier's Slip: A Numerical Study

Flow Analysis of Viscous Nanofluid Due to Rotating Rigid Disk with Navier's Slip: A Numerical Study

Potentialities of diffusion weighted MRI in the assessment of the degree of adjacent intervertebral disc degeneration: rigid lumbosacral stabilization and total intervertebral disc arthroplasty

Purpose: to evaluate the condition of adjacent intervertebral discs (IVD) after single level rigid lumbosacral stabilization and total arthroplasty by calculating IVD height index and apparent diffusion coefficient (ADC). Patients and methods. The study included 117 patients (64 women and 53 men) after rigid lumbosacral stabilization or total arthroplasty of the degenerative IVD at L5-S1 level. Values of ADC and height of the adjacent IVD were assessed prior to surgery, at discharge and in 6, 12, 24 and 36 months after surgical intervention. Results. The value of the height of the adjacent IVD in patients after rigid stabilization in the early postoperative period averaged 0.58±0.046, in 6 months - 0.58±0.044 and 0.52±0.037 in 36 months after surgery. In patients after total arthroplasty it made up 0.59±0.041, 0.60±0.038 and 0.56±0.02, respectively. Comparison of the adjacent IVD height indices showed significant difference starting from the 12th observation months (p

Drag coefficient of a liquid domain with distinct viscosity in a fluid membrane

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

Viscoelastic analysis of a spring-connected dielectric elastomer actuator undergoing large inhomogeneous deformation

A viscoelastic model is developed for a conical dielectric elastomer actuator (CDEA) connected by a spring. The CDEA is composed of a dissipative dielectric elastomer membrane, a rigid disk and a spring. The CDEA undergoes large out-of-plane inhomogeneous deformation by applying the electro-mechanical loadings. The nonlinear spring-dashpot model is employed to model the viscoelastic behavior of the membrane. A differential-algebraic system of equations is derived and the solving technique is developed. Both the creep behaviors under the constant instantaneous electro-mechanical loadings and the cyclic performance under the cyclic mechanical and electrical loadings are presented. The time-dependent behaviors of the CDEA can be tuned by changing the stiffness of the spring, the pre-stretch of the membrane as well as the loading rate and the loading type.

Response to Discussion of “Rocking vibration of a rigid disc embedded in any depth of a coupled seawater-visco-poro-elastic seabed half-space”

Response to Discussion of “Rocking vibration of a rigid disc embedded in any depth of a coupled seawater-visco-poro-elastic seabed half-space”

Buckling and vibration of composite lattice elliptical cylindrical shells

An approach to the finite element study of the buckling and dynamic behaviour of composite lattice cylindrical shells with elliptical cross sections is presented in this paper. The lattice shells are modelled as three-dimensional frame structures composed of curvilinear ribs using beam finite elements. A specialised algorithm is developed to generate the finite element model of the lattice shells based on multiple use of the repeating unit cell of the composite lattice structure. Using this model, the buckling behaviour of the shells subjected to axial loading and transverse bending are investigated. Fundamental frequencies of axial and transverse vibrations of the shells with a massive rigid disk attached to their ends are determined based on the modelling approach proposed in this work. The effects of parameters of the lattice structure on the values of critical buckling loads, buckling and vibration mode shapes, and the fundamental frequencies are examined using parametric analyses. Based on the computations, the angles of orientation of helical ribs delivering maximum critical loads and fundamental frequencies are identified. The results of this study can be applied to the design of the composite tubular bodies of spacecraft made in the form of cylindrical lattice shells with elliptical cross sections.

Bonded contact of a rigid disk inclusion with a penny-shaped crack in a transversely isotropic solid

An analytical treatment is presented for bonded contact of a rigid disk inclusion embedded in a penny-shaped crack in a transversely isotropic full-space. Theoretical analysis is carried out using a complete potential function method, and with the aid of Hankel transforms. Boundary conditions propel the problem toward a set of triple integral equations, which are solved analytically and then reduced to a pair of Fredholm integral equations of the second kind. Furthermore, the results are evaluated and presented graphically using numerical schemes, and comparison is made with well-known classical solutions in transversely isotropic and isotropic media. This can be obtained as special cases for the problem to reveal the efficacy of the proposed method. Eventually, it can be seen that not only the presence of the crack around the disk inclusion decreases the axial stiffness, but also the extension of its length also reduces the fracture parameter, stress intensity factor, for different degrees of material anisotropy.

Axial translation of a rigid disc inclusion embedded in a penny-shaped crack in a transversely isotropic solid

In this paper, an analytical solution for the axisymmetric interaction of a rigid disc inclusion embedded in bondedcontact with the surfaces of a penny-shaped crack and a transversely isotropic medium is investigated. By using amethod of potential functions and treating dual and triple integral equations, the mixed boundary value problem iswritten in the form of two coupled integral equations, which are amenable to numerical treatments. The axial sti nessof the inclusion and the shearing stress intensity factor at the tip of the penny-shaped crack for di erent degrees ofmaterial anisotropy are illustrated graphically. Useful limiting cases such as a rigid disc inclusion in an uncrackedmedium and in a completely cracked solid are recovered

Evaluating and improving the performance of thin film force sensors within body and device interfaces

Thin film force sensors are commonly used within biomechanical systems, and at the interface of the human body and medical and non-medical devices. However, limited information is available about their performance in such applications. The aims of this study were to evaluate and determine ways to improve the performance of thin film (FlexiForce) sensors at the body/device interface. Using a custom apparatus designed to load the sensors under simulated body/device conditions, two aspects were explored relating to sensor calibration and application. The findings revealed accuracy errors of 23.3±17.6% for force measurements at the body/device interface with conventional techniques of sensor calibration and application. Applying a thin rigid disc between the sensor and human body and calibrating the sensor using compliant surfaces was found to substantially reduce measurement errors to 2.9±2.0%. The use of alternative calibration and application procedures is recommended to gain acceptable measurement performance from thin film force sensors in body/device applications.

Dynamic Characteristics of Rotor-Bearing System with a Labyrinth Seal

In this paper, the dynamic characteristics of rotor-bearing system with a labyrinth seal are investigated. The dynamic coefficients of labyrinth seal and the whirl speeds of rotor-bearing system are analyzed. The simplified rotor-bearing model consists of a rigid disk and a flexible shaft. The rotor-bearing system supported by two ball bearings, which are modeled as the spring-damper systems. A labyrinth seal supports one of the rigid disks. The perturbation approach has been applied to the governing equations of the bulk-flow model to calculate the dynamic coefficients of labyrinth seal. The Finite Element Method (FEM) is utilized to model the flexible-rotor system. The numerical results show the labyrinth seal effect on the dynamic characteristics of rotor-bearing system, which depend on a great variety of parameters such as geometry of the labyrinth seal.

Effect of Crack Patterns on the Stress Distribution of Hard Chromium Coatings Under Sliding Contact: Stochastic Modeling Approach

In this work, the influence of different crack arrangements in the stress distribution of hard chromium (HC) coatings was determined. Three parameters for position and length of the cracks for two different types of coatings were probabilistically modeled based on measured scanning electron microscopy (SEM) images. Probability density functions (PDF) for those parameters were obtained to characterize each kind of coating. A two-dimensional finite element (FE) model of the coating in contact with a rigid disk was developed, modeling cracks with elliptical shapes. A Monte Carlo method was used to simulate different crack distributions for each kind of coating, and values of stress and strains in the domain were obtained. Both the J-integral and the stress intensity factors (SIFs) were taken as comparative parameters of the results. Coatings which statistically present larger quantities of shorter cracks have lower values of J-integral and SIFs, and, therefore, distribute stresses better than those with low density of longer cracks.

A simple and rapid method for preparing a diversity of powdered materials for analysis by laser ablation inductively coupled plasma mass spectrometry

Laser ablation inductively coupled plasma mass spectrometry is increasingly being accepted as a leading method for elemental and isotopic microanalyses of a large number of solid matrices. The lack of suitable standards (i.e., concentration-, abundance-, or matrix-matched), however, is a significant challenge preventing the application of this technique to a broader range of solid samples. In this paper, we report a simple method for preparing samples and standards from powdered materials using resin impregnation under vacuum. High quality, rigid disks were prepared and verified as calibration standards through analysis of a number of well-characterized NIST standard reference materials (NIST 610 and NIST 612) and USGS reference materials (BIR-1G, BHVO-2G and BCR-2G). Analysis of NIST 610 and 612 glasses for 32 elements using BHVO-1 as an external calibration standard, which was prepared from the powdered reference material, showed good agreement (less than 10% difference for most elements) with the published values. Precision of better than 4% relative standard deviation (RSD) was obtained for most investigated elements. Similar results were obtained for basalt powders (BIR-1, BHVO-2, and BCR-2) when analyzed using NIST 610 glass as an external standard. Improved accuracy was obtained when the external calibration standard and the sample were matrix matched (i.e., both were prepared from powders).

Amorphization model of nanostructured composite solid electrolytes

In the last decades, extensive research has been undertaken to find solid electrolytes that might increase the power and safety of promising electrochemical devices such as lithium batteries, supercapacitors, and solid oxide fuel cells. It is mainly based on the screening of advanced functional suprastructures and developing special synthesis procedures to obtain nanosized materials, allowing one to increase ionic conductivity. That’s why special attention is paid to composite solid electrolytes. In this work we use the accumulated knowledge on the physical chemistry of metals and alloys to describe the amorphization effects. It is known that amorphization at the interphase boundaries (interfaces) and grain boundaries noticeably increases (sometimes by orders of magnitude) the ionic transfer rate and, therefore, affects the functional properties of nanostructured composite solid electrolytes. In the theoretical model proposed, we have attempted to elucidate the reasons, inducing the amorphization effects which are observed upon the crystallization of inorganic eutectics and composite formation. We have especially used the approximation of rigid discs, considered, unit-cell volumes. In the context of the theory, describing the amorphization as an excess molar volume arising upon the crystallization of metals and alloys, we have established that the degree of amorphization depends not only on the synthesis conditions, but also on the incommensurability of crystal unit cells in the components. The findings can be useful in the elaboration of novel inorganic materials for various applications.

Buckling of composite cylindrical shells with rigid end disks under hydrostatic pressure

An analytical solution of the buckling problem formulated for a composite cylindrical shell with its ends closed by rigid disks and subjected to hydrostatic pressure is presented in the paper. The problem is solved using Fourier decomposition and the Galerkin method. The boundary conditions are assigned in the form accounting for axial displacements of the end disks caused by an axial contraction of the deformed shell. The hoop displacement and deflection of the shell are approximated by the beam function corresponding to the first mode shape of vibration of a clamped-clamped beam. The axial displacement is approximated by the third derivative of the beam function. Based on this solution, a number of analytical formulas enabling calculations of critical hydrostatic pressure for composite orthotropic cylindrical shells are derived. Using these formulas, the critical loads are calculated for the shells with various elastic and geometric properties. The calculations are verified by comparisons with the results of finite-element analyses. The efficiency of analytical solutions in the search for fibre reinforcement arrangements providing maximum resistance to buckling is demonstrated by several examples.

Characterization of interface stresses and lubrication of rough elastic surfaces under ball-on-disc rolling

This paper presents a statistical method for characterizing the interface stresses between an elastic rough ball and a smooth rigid disc under lubrication by considering the percolation effect of lubricant flow. The statistical analysis was carried out to integrate the random, microscopic asperity deformation with the asperity–lubricant interaction. An efficient solution process with the aid of the fast Fourier transform was developed for the multi-scale analysis. To avoid numerical instability, the Poiseuille term of the average Reynolds equation was turned off when the average lubricant film thickness was below the percolation threshold and the average hydrodynamic pressure was solved by using the Couette term. This successfully overcomes the limitation in the conventional statistical treatment of mixed lubrication. Moreover, it enables a smooth transition from mixed to boundary lubrication and allows a large range of dimensionless rolling speed, which cannot be achieved by conventional approaches. Also, an empirical formula was developed to evaluate the average central and minimum film thicknesses between lubricated rough surfaces in contact. It was found that the method developed can precisely predict the effect of surface roughness on contact stresses, can effectively estimate the contribution of the direct asperity–disc contacts, and can be used to describe the lubricant performance at a rolling–sliding contact interface.

A hybrid element method for dynamics of piles and pile groups in transversely isotropic media

A hybrid analytical-numerical method is proposed for the dynamic analysis of single piles and pile groups embedded in semi-infinite transversely isotropic media. In the method proposed, the soil-pile system is modeled using finite elements combined with massless rigid radiation discs representing pile-soil-pile interaction. The elasto-dynamic response of the radiation discs buried at different depths in a transversely isotropic half-space is analytically derived in a transform domain using a set of complete potential functions. A Boussinesq-type loading distribution is introduced to act on the disc region to achieve the proper mode of deformation at the cross sections of piles. Numerical results and comparisons with known analytical/numerical solutions are presented, demonstrating the application of the method.

Dynamics of low speed geared shaft systems mounted on rigid bearings

Analytical model for the linear vibration analysis of a pair of coupled spur geared shaft system is developed. The model is a hybrid discrete-continuous one in which the gears are modeled as rigid disks mounted on the elastic shafts having transverse as well as torsional flexibilities and supported by rigid bearings. The non-dimensional governing equations along with the natural boundary conditions are developed using the Hamilton’s principle. An extended operator formulation is employed to simplify the system to a compact analytical form and demonstrate that the system is self-adjoint. The natural frequencies and vibration modes are determined by discretizing the system using the assumed modes method, where the critical feature is the use of Lagrange multipliers to enforce the matching boundary conditions at the disk-shaft interfaces. The sensitivities of the natural frequencies to various system parameters are examined and correlated with the modal energy distribution. Response due to the loaded static transmission error at the gear mesh is presented.