Rigid Disk Research Articles

Dynamic Stiffness Matrix Approach to Free Vibration Analysis of Functionally Graded Rotor Bearing System Subjected to Thermal Gradients.

The dynamic stiffness matrix (DSM) method, an analytical method that provides exact solutions, has been used for the first time for the free vibration analysis of a functionally graded (FG) rotor bearing system subjected to temperature gradients and to investigate its application to FG rotors. The material gradation occurs based on the power law between the inner metal core and the outer ceramic rich layer of the FG rotor. The temperature gradation follows the Fourier law of heat conduction which leads to non-linear temperature distribution (NLTD) in the radial direction of the FG rotor. The development of the DSM formulations for Timoshenko FG rotor elements using the governing equations derived from translational and rotational equilibrium conditions is the novelty of the present work. The DSM of the FG rotor elements, rigid disk and linear isotropic bearings are assembled to obtain the global dynamic stiffness matrix of the FG rotor bearing system. The natural whirl frequencies are computed from the global DSM using the Wittrick–William algorithm as a root searching technique. The natural and whirl frequencies are validated with the results available in the literature and the exactness of the DSM method has been exemplified.

Buckling of a Composite Cylindrical Shell with Cantilever-like Boundary Conditions under Hydrostatic Pressure

In this study, an analytical solution for the buckling of a composite cylindrical shell subjected to hydrostatic pressure is proposed. The boundary conditions of the composite cylindrical shell are cantilever-like, with one end fixed and the other end connected to a rigid disk. The differential equations are solved using the Galerkin method. The axial displacement of the shell is approximated by the first mode shape of the transverse vibration of the clamped sliding beam. The circumferential displacement and deflection are approximated by the first derivation of the beam function. Based on this solution, an analytical formula enabling prediction of the critical buckling pressure and buckling mode of composite orthotropic cylindrical shells is derived. A finite element analysis and external hydrostatic pressure test are conducted to verify the proposed approach. The efficiency and accuracy of the analytical solution in predicting the critical buckling pressure and buckling mode are demonstrated.

Numerical investigation for non-axisymmetric Homann stagnation point flow of a SWCNT/MWCNT-water nanofluid over a disk

In this article, the non-axisymmetric MHD Homann stagnation point flow over an impermeable rigid cylindrical disk is discussed. Thermal convection is also studied in the Homann flow of water by adding the multi-wall carbon nanotube/ single-wall carbon nanotube (MWCNT/SWCNT) as nanoparticles. The analysis of energy transport in nanofluids is done by adopting the dynamic and static models namely the Buongiorno and Xue models for nanofluids, respectively. Furthermore, the effect of non-linear thermal radiations and viscous dissipation on thermal energy transport is also examined. The appropriate transformations are used in order to obtain the similar differential equations. The resulting non-linear transformed coupled equations are solved by the bvp4c method in MATLAB. It is noted that for MWCNT-Water nanofluid there is significant enhancement in the heat conduction rate as compared to SWCNT-Water nanofluid.

МАТЕМАТИЧНЕ ТА КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ НАПРУЖЕНОГО СТАНУ ПРИ ПЕРЕМІЩЕННІ ЖОРСТКОГО КРУГОВОГО ДИСКУ У П’ЄЗОЕЛЕКТРИЧНОМУ ПРОСТОРІ

Using the mathematical model that takes into account the coupling of force and electric fields in piezoelectric bodies, the stress distribution that occurs when a rigid thin circle disk moves along the axis of symmetry of an electroelastic transversely isotropic material is studied. The exact solution of the space problem of electroelasticity is obtained on the basis of the solution representing of the static equations of electroelasticity for a transversely isotropic body through harmonic functions and using a harmonic potential of a special type. As special cases, from the found analytical solution follow the expressions for evaluating the stress state in an elastic transversely isotropic material for the displacement of a circle hard disk. Computer simulation have been carried out, the influence of the coupling of the force and electric fields on the distribution of stresses under a rigid circle disk. 233 KEY WORDS: MATHEMATICAL MODEL, COMPUTER SIMULATION, PIEZOELECTRIC SPACE, HARD CIRCLE DISK, ELECTRICAL AND STRESS STATE, STRESS DISTRIBUTION, FIELD COUPLING EFFECT.

Factorization method for inverse time-harmonic elastic scattering with a single plane wave

<p style='text-indent:20px;'>This paper is concerned with the factorization method with a single far-field pattern to recover an arbitrary convex polygonal scatterer/source in linear elasticity. The approach also applies to the compressional (resp. shear) part of the far-field pattern excited by a single compressional (resp. shear) plane wave. The one-wave factorization is based on the scattering data for a priori given testing scatterers. It can be regarded as a domain-defined sampling method and does not require forward solvers. We derive the spectral system of the far-field operator for rigid disks and show that, using testing disks, the one-wave factorization method can be justified independently of the classical factorization method.</p>

Viscoacoustic squeeze-film force on a rigid disk undergoing small axial oscillations

This paper investigates the air flow induced by a rigid circular disk or piston vibrating harmonically along its axis of symmetry in the immediate vicinity of a parallel surface. Previous attempts to characterize these so-called ‘squeeze-film’ systems largely relied on simplifications afforded by neglecting either fluid acceleration or viscous forces inside the thin enclosed gas layer. The present viscoacoustic analysis employs the asymptotic limit of small vibration amplitudes to investigate the flow by systematic reduction of the Navier–Stokes equations in two distinct flow regions, namely, the inner gaseous film where streamlines are nearly parallel to the confining walls and the near-edge region of non-slender flow that features gas exchange with the surrounding stagnant atmosphere. The flow in the gaseous film depends on the relevant Stokes number, defined as the ratio of the characteristic viscous time across the film to the characteristic oscillation time, and on a compressibility parameter, defined as the square of the ratio of the acoustic time for radial pressure equilibration to the oscillation time. A Strouhal number based on the local residence time emerges as an additional governing parameter for the near-edge region, which is incompressible at leading order. The method of matched asymptotic expansions is used to describe the solution in both regions, across which the time-averaged pressure exhibits comparable variations that give opposing contributions to the resulting time-averaged force experienced by the disk or piston. A diagram structured with the Stokes number and compressibility parameter as coordinates reveals that this steady squeeze-film force, typically repulsive for small values of the Stokes number, alternates to attraction across a critical separation contour in the parametric domain that exists for all Strouhal numbers. This analysis provides, for the first time, a unifying viscoacoustic theory of axisymmetric squeeze films, which yields a reduced parametric description for the time-averaged repulsion/attraction force that is potentially useful in applications including non-contact fluid bearings and robot locomotion.

Fast Water Waves in Stationary Surface Disk Arrays.

Traditionally, the phase and group velocities of water waves can be increased by increasing water depth but possess upper bounds, which are related to the gravitational acceleration and difficult to exceed. Here, we theoretically propose and experimentally demonstrate that when water is covered with a periodic array of stationary rigid disks, both the gravitational acceleration and reduced water depth can be effectively increased in the lowest frequency band. As a result, fast water waves can occur in the system, with both the phase and group velocities exceeding those in water without disks. Unusual effects, such as total reflection at oblique incidence and unidirectional transmission of water waves, are further realized.

On the compression of a rigid disc by finitely deformed elastic halfspaces

The paper examines the problem of a flat rigid circular disc that is compressed between two finitely deformed incompressible elastic halfspace regions with smooth surfaces. This problem yields a unilateral contact problem where the zone of separation needs to be determined. The analysis of the unilateral contact problem is reduced to the solution of a set of triple integral equations associated with the internal indentation of a penny-shaped crack and the internal tensile pressurization of an annular crack. The solutions to these problems can be obtained in an approximate series form in terms of the non-dimensional parameter involving the radius of the rigid disc to the radius of the separation zone. The extent of the separation zone is determined from the vanishing of the contact stress at the point of separation. Specific solutions are developed for the case where the initial finite deformation is for halfspace regions with a strain energy function of the Mooney-Rivlin form.

Second-virial theory for shape-persistent living polymers templated by disks.

Living polymers composed of noncovalently bonded building blocks with weak backbone flexibility may self-assemble into thermoresponsive lyotropic liquid crystals. We demonstrate that the reversible polymer assembly and phase behavior can be controlled by the addition of (nonadsorbing) rigid colloidal disks which act as an entropic reorienting "template" onto the supramolecular polymers. Using a particle-based second-virial theory that correlates the various entropies associated with the polymers and disks, we demonstrate that small fractions of discotic additives promote the formation of a polymer nematic phase. At larger disk concentrations, however, the phase is disrupted by collective disk alignment in favor of a discotic nematic fluid in which the polymers are dispersed antinematically. We show that the antinematic arrangement of the polymers generates a nonexponential molecular-weight distribution and stimulates the formation of oligomeric species. At sufficient concentrations the disks facilitate a liquid-liquid phase separation which can be brought into simultaneously coexistence with the two fractionated nematic phases, providing evidence for a four-fluid coexistence in reversible shape-dissimilar hard-core mixtures without cohesive interparticle forces. We stipulate the conditions under which such a phenomenon could be found in experiment.

Modeling of joint work of steel beam constructions with reinforced concrete ribbed floor slabs

Traditional methods of calculation of beam constructions of floors and coverings of industrial buildings assume their consideration when calculating separately from the frame structures, in particular, reinforced concrete slabs, without taking into account their joint work, which leads to a significant margin of safety. Today in Ukraine there is a significant number of industrial buildings and structures that need strengthening and reconstruction.
 In this regard, of particular importance are studies of the actual load-bearing capacity of the frames of single-storey and multi-storey industrial buildings, and both in the reconstruction and in new construction, the results of which will significantly reduce costs and more rationally design structures.
 At the same time, one of the most relevant areas is the study of the joint work of metal load-bearing structures with prefabricated reinforced concrete structures of rigid disks of coatings and floors in their calculation.
 
 
 
 
 
 Moreover, in the national building codes, as well as in the educational and methodological literature, the calculation methods of taking into account the joint work of such constructions are not fully covered.
 The purpose of this work is to estimate the reduction of mass of the metal beam structure in its calculation in bending, taking into account the joint work with the rigid disk of the floor consist of precast concrete.
 As part of the study, the calculation of the floor beam according to the traditional calculation scheme - without taking into account the joint work with the floor slab, the calculation of its cross-section taking into account the joint work with floor slabs and experimental numerical study of the floor by the finite element method. Modeling of the floor fragment was performed in the software packages "SCAD Office" and "LIRA CAD 2019".
 Numerical research is aimed at verifying the feasibility of using the calculation methodology of DBN B.2.6-98-2009 to determine the effective width of the shelf when calculating the T-sections for prefabricated reinforced concrete slabs, which are included in the joint work with the floor beams.
 A comparative analysis of the obtained cross-section of the beam with the beam which was previously calculated by the traditional method of calculation in stresses in the most dangerous cross section and the total mass of the beams.
 According to the results of the analysis, the correctness of the application of the above normative method for determining the effective width of the shelf of T-bending reinforced concrete elements was confirmed.

Dynamic impedances of ring disks buried in arbitrary depths

This paper addresses the vertical, horizontal and rocking vibrations of a rigid ring disk, with especially small radial wall-thicknesses, embedded in any depth of a fully saturated poroelastic half-space. The mixed boundary value problem is expressed in terms of dual integral equations that are solved semi-analytically with all boundary conditions satisfied. Numerical results of quasi-static stiffness, dynamic impedances (including dynamic stiffness and radiation damping ratio) are obtained. The influence of Poisson’s ratio, soil permeability, disk embedded depth and excitation frequency on the stiffness and damping ratio of the ring disk is illustrated. It is found that for a ring disk, although its thickness-to-radius ratio is only about 2%-4%, its vertical, horizontal and rocking impedances are around 70% of the corresponding values of a circular disk. Besides, the vertical, horizontal and rocking stiffness of a deeply embedded ring disk is all about 2.0 times that of the surface ring disk. Simplified equations for the initial base stiffness of pipe piles/caissons are obtained. The results can be combined with Winkler-based models to address thin-walled pile/caisson-soil interaction problems in the field of offshore wind turbines or bridges.

Enhanced 3D solid finite element formulation for rotor dynamics simulation

In this paper, an enhanced solid finite element formulation for rotor dynamics analysis is presented. The element kinematics is first defined with a Total Lagrangian formulation in order to account for possible centrifugal stiffening effect and its associated pre-stress deformation. The novelty of the three-dimensional element proposed in this study is then to correctly consider the effect of inertia due to small rotational deformation occurring around the pre-stressed equilibrium position. The rotational motion is characterized by interpolation of the translational degrees of freedom of the element. The accuracy and convergence of the proposed finite element are tested on several numerical examples consisting of a rotating rigid disk and a rotating Timoshenko beam with different slenderness ratios. All results agree well with analytical solutions excerpted from literature and demonstrate the usefulness and applicability of the proposed finite element formulation. • Initial stress effect from centrifugal force is included. • A total-Lagrangian formulation is used. • The change of direction of the spin velocity induced by deformation is accounted for. • Results show good accuracy and convergence rate at high rotating speed.

Significance of nanoparticles' shape and thermo-hydrodynamic slip constraints on MHD alumina-water nanoliquid flows over a rotating heated disk: The passive control approach

This work aims at the numerical exploration of the hydromagnetic alumina –water nanoliquid flow due to a rotating rigid disk. The nanoliquid flow considering different nanoparticle shapes (namely sphere, platelet, cylinder, and brick) and the thermo-hydrodynamic slip constraints have been modeled utilizing the two-phase modified Buongiorno model (MBM). MBM modifies the conventional two-phase Buongiorno model by additionally considering the volume fraction-dependent nanoparticle properties. The present study is relevant in both engineering and industrial fields like aerodynamics, spin coating, extrusion process, etc. Von Kármán's similarity transformations are exercised in the transmutation of the mathematically modeled equations into a system of first-order ODEs and treated numerically using the generalized differential quadrature method. The consequence of pertinent parameters on the physical quantities and the flow profiles are explained with the aid of tables and graphs. It is observed that the nanoliquid temperature increases with the hydrodynamic slip parameter and decreases with the thermal slip parameter. Further, it is noted that the highest drag is exhibited by platelet-shaped alumina nanoparticles.

Response and stability analysis of a two-spool aero-engine rotor system undergoing multi-disk rub-impact

The main aim of this paper is to propose a numerical procedure for capturing the nonlinear dynamic characteristics of a two-spool aero-engine rotor system undergoing multi-disk rub-impact. In aero-engines, the possibility for the multi-disk rub-impact is high during the fan blade-out (FBO) event and subsequent windmilling action. It intensifies the nonlinear effects on the rotor vibrations and leads to certain undesired circumstances in the engine. The dual-rotor model consists of multi-stage compressors and single-stage turbines that undergo rubbing whenever their deflection exceeds the clearance. The dynamic model of the dual-rotor system is constructed using the tapered Timoshenko beam elements, rigid disks and rolling contact bearings. A proper model reduction technique based on component mode synthesis coupled with the Craig-Bampton substructuring is utilized to reduce the size of the finite element model. A semi-analytic technique called the approximate time variational method is employed to investigate the steady-state response of the system under multi-disk rub impact. Based on the proposed method, the response characteristics of the model are obtained and are verified with the numerical integration results. Compared to the single-disk rub-impact, the nonlinearities are intensified and significant variations are observed in the response characteristics and stability of the system. Period-5, quasi-periodic, and dry friction backward whirl motions are observed in the response for different values of the system parameters. During quasi-periodic motion, some unknown fractional components such as 0.716ω1, 0.766ω1, 0.916ω1 and 0.964ω1 are appeared in the response. Moreover, the dry friction backward whirl happened with a very large amplitude and it contains a superharmonic frequency component in the response.

Active Compliant PID Learning Control for Grinding Robots

This article proposes a Proportional, Integral, and Derivative (PID) learning controller for rigid robotic disk grinding mechanism. It has been observed that the stiffness of the robotic arm for a grinder has a direct correlation with the sensitivity of the grinding forces. It is also drastically influenced by the end-effector path tracking error resulting in limited accuracy of the robot. The error in robot’s accuracy is also increased by external interferences, such as surface imperfections and voids in the subject material. These errors can be mitigated via efficient feedback. In the proposed methodology, the controller gain is tuned by implementing a learning-based methodology to PID controllers. The learning control for the robotic grinding system helps by progressively decreasing error between the actual grinded paths and required trace. Experimental results demonstrate that as the grinder machines the required path iteratively, its grinding accuracy improves due to the learning algorithm.

Alignment of slender fibers and thin disks induced by coherent structures of wall turbulence

The angular dynamics of small rigid fibers and disks in turbulent channel flow has been investigated numerically, focusing on interactions between particles and near-wall coherent vortices. Three distinctly different alignment patterns of fibers and disks around ensemble-averaged vortices were identified. From the wall to the channel center we observed a shear-dominant, a structure-dominant, and an isotropic region, each with its unique alignment pattern. These regions are different from those based on the conventional Reynolds-averaging view. The structure-dominant region, for instance, extends from about 3 to 60 wall units from the channel wall. Unlike the prevailing view that preferential alignment of fibers and disks are more common in the viscous sublayer than in the buffer layer, where the turbulence intensity reaches its maximum, the conditional ensemble-averaged approach revealed that fibers and disks align preferentially in the structure-dominant region. The particles moreover align differently in sweep and ejection events. The physical alignment mechanism we proposed may also be carried over to polymer- and fiber-induced drag reduction in wall-turbulence.

An Exact Solution for Whirling Speeds and Mode Shapes of Multi-span Rotating Shafts with Each Span Carrying a Number of Rigid Disks

An Exact Solution for Whirling Speeds and Mode Shapes of Multi-span Rotating Shafts with Each Span Carrying a Number of Rigid Disks

A RESEARCH PAPER ON SHAPE OPTIMIZATION OF CLUTCH DISC OF A PASSANGER CAR USING FEA AND IMPACT HAMMER TEST

This work represents the statement of computing stresses and the deformations and the vibration characteristics of a rigid clutch disc. Newer models for this rigid clutch disc has been recommended the material will remain the same however the shape of disc has been varying. The outcome of the newly recommended models has been recorded and compared with the reference model. 3D-CAD modeling of this clutch disc is created using CATIA-V5R20. The software used to determine total deformation andequivalent stress is ANSYS Version 19.0 in clutch disc. The experimental enquiry has been done by using the strain gauge process and the UTM. Relative analysis of FEA & Data-based results are carried out for the validation of work.

Rigid multibody dynamic modeling for a semi-submersible wind turbine

This paper presents a rigid multibody dynamic model for a Floating Offshore Wind Turbine, and its dynamic responses are simulated. This system consists of four important mechanical rigid parts, referred to as platform, tower, nacelle, and rotor. The platform is constrained by three cables connected to the seabed, known as the mooring system. Buoyancy and hydrodynamic forces are also applied to the platform. The semi-submersible platform is investigated as one of the latest proposed designs for the platform, which plays an important role in the stability and minimization of applied forces to the whole structure. Turbine’s rotor is modeled as a rigid disk rotating about its shaft axis and under the influence of aerodynamic force and torque. Therefore, the platform with six, and rotor with one Degree of Freedom in combination with the tower and nacelle, form a seven Degree of Freedom holonomic system. Euler–Lagrange equations are utilized to formulate the system’s equations of motion. In numerical simulation section, dynamic responses under four load cases are presented and validated against “Fatigue, Aerodynamics, Structures and Turbulence” module outputs under the same conditions. The effects of considering pontoons and cross braces on dynamic responses are also investigated. Moreover, the effect of the rotor’s gyroscopic moment is examined.

Steady azimuthal flow field induced by a rotating sphere near a rigid disk or inside a gap between two coaxially positioned rigid disks

Geometric confinements play an important role in many physical and biological processes and significantly affect the rheology and behavior of colloidal suspensions at low Reynolds numbers. On the basis of the linear Stokes equations, we investigate theoretically and computationally the viscous azimuthal flow induced by the slow rotation of a small spherical particle located in the vicinity of a rigid no-slip disk or inside a gap between two coaxially positioned rigid no-slip disks of the same radius. We formulate the solution of the hydrodynamic problem as a mixed-boundary-value problem in the whole fluid domain, which we subsequently transform into a system of dual integral equations. Near a stationary disk, we show that the resulting integral equation can be reduced into an elementary Abel integral equation that admits a unique analytical solution. Between two coaxially positioned stationary disks, we demonstrate that the flow problem can be transformed into a system of two Fredholm integral equations of the first kind. The latter are solved by means of numerical approaches. Using our solution, we further investigate the effect of the disks on the slow rotational motion of a colloidal particle and provide expressions of the hydrodynamic mobility as a function of the system geometry. We compare our results with corresponding finite-element simulations and observe very good agreement.