Rigid Disk Research Articles

The development of a mature collagen network in cartilage from human bone marrow stem cells in Transwell culture

Damaged hyaline cartilage shows a limited capacity for innate repair. Potential sources of cells to augment the clinical repair of cartilage defects include autologous chondrocytes and mesenchymal stem cells. We have reported that culture of human bone marrow mesenchymal stem cells with specific growth and differentiation factors as shallow multilayers on Transwell permeable membranes provided ideal conditions for chondrogenesis. Rigid translucent cartilaginous disks formed and expressed cartilage-specific structural proteins aggrecan and type II collagen. We report here the analysis of the collagen network assembled in these cartilage constructs and identify key features of the network as it became mature during 28days of culture. The type II collagen was co-polymerized with types XI and IX collagens in a fibrillar network stabilized by hydroxylysyl pyridinoline cross-links as in epiphyseal and hyaline cartilages. Tandem ion-trap mass-spectrometry identified 3-hydroxylation of Proline 986 and Proline 944 of the α1(II) chains, a post-translational feature of human epiphyseal cartilage type II collagen. The formation of a type II collagen based hydroxy-lysyl pyridinoline cross-linked network typical of cartilage in 28days shows that the Transwell system not only produces, secretes and assembles cartilage collagens, but also provides all the extracellular mechanisms to modify and generate covalent cross-links that determine a robust collagen network. This organized assembly explains the stiff, flexible nature of the cartilage constructs developed from hMSCs in this culture system.

Nonlinear forced vibrations analysis of overhung rotors with unbalanced disk

In this paper, primary resonances of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) are investigated. Unbalance forces due to eccentricity and disk skew are simultaneously considered, and the shaft has initially static deflection due to the rotor’s weight which causes asymmetry in the equations of motion. The rotor has large amplitude vibrations, which lead to nonlinearities in curvature and inertia. In the model, rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The method of multiple scales is applied to the discretize differential equations of motion. It is shown that the static deflection creates second-order nonlinear terms and near the primary resonances only the forward modes are excited. With considering the gravity, the inertial nonlinearities become stronger near the primary resonances. So, gravity decreases the hardening effect, and the nonlinear system tends to a linear system. It is concluded that the gravity effect has a softening effect. By using this property of gravity, a relation between weight and external forces is derived, in which by applying this relation the jumping phenomenon is eliminated. Numerical examples are presented, and the result is verified by numerical simulations.

Self-balancing system of the disk on an elastic shaft

This paper presents an analysis of the automatic balancing of a rigid disk mounted on an elastic shaft. The balancing system consists of two drums at a variable distance from the disk and free balls (or rollers) inside the disk. The balls are able to change positions with respect to the rotor and compensate for rotor unbalance. This paper presents the equations of motion for the disk as well as for the balls during balancing. It is shown that the balls can compensate a part or all of the rotor unbalance depending on the positioning of the drums. There are vibratory forces that push the balls to new positions; these are responsible for the behavior of the balls and the final results. The vibratory forces are defined as a function of the system׳s parameters and they determine the position of equilibrium of the balls. The stability and efficiency of the method is analyzed in this paper.

A new dynamic model of rotor–blade systems

A new dynamic model of rotor–blade systems is developed in this paper considering the lateral and torsional deformations of the shaft, gyroscopic effects of the rotor which consists of shaft and disk, and the centrifugal stiffening, spin softening and Coriolis force of the blades. In this model, the rotating flexible blades are represented by Timoshenko beams. The shaft and rigid disk are described by multiple lumped mass points (LMPs), and these points are connected by massless springs which have both lateral and torsional stiffness. LMPs are represented by the corresponding masses and mass moments of inertia in lateral and rotational directions, where each point has five degrees of freedom (dofs) excluding axial dof. Equations of motion of the rotor–blade system are derived using Hamilton׳s principle in conjunction with the assumed modes method to describe blade deformation. The proposed model is compared with both finite element (FE) model and real experiments. The proposed model is first validated by comparing the model natural frequencies and vibration responses with those obtained from an FE model. A further verification of the model is then performed by comparing the model natural frequencies at zero rotational speed with those obtained from experimental studies. The results shown a good agreement between the model predicted system characteristics and those obtained from the FE model and experimental tests. Moreover, the following interesting phenomena have been revealed from the new model based analysis: The torsional natural frequency of the system decreases with the increase of rotational speed, and the frequency veering phenomenon has been observed at high rotational speed; The complicated coupling modes, such as the blade–blade coupling mode (BB), the coupling mode between the rotor lateral vibration and blade bending (RBL), and the coupling mode between the rotor torsional vibration and blade bending (RBT), have also been observed when the number of blades increases.

Uncertainty analysis of a flexible rotor supported by fluid film bearings

THIS PAPER IS DEDICATED TO THE ANALYSES OF THE EFFECT OF UNCERTAIN PARAMETERS ON THE DYNAMIC BEHAVIOR OF A FLEXIBLE ROTOR CONTAINING TWO RIGID DISCS AND SUPPORTED BY TWO FLUID FILM BEARINGS. A STOCHASTIC METHOD HAS BEEN EXTENSIVELY USED TO MODEL UNCERTAIN PARAMETERS, I.E., THE SO-CALLED MONTE CARLO SIMULATION. HOWEVER, IN THE PRESENT CONTRIBUTION, THE INHERENT UNCERTAINTIES OF THE BEARINGS' PARAMETERS (I.E. THE OIL VISCOSITY AS A FUNCTION OF THE OIL TEMPERATURE, AND THE RADIAL CLEARANCE) ARE MODELED BY USING A FUZZY DYNAMIC ANALYSIS. THIS ALTERNATIVE METHODOLOGY SEEMS TO BE MORE APPROPRIATED WHEN THE STOCHASTIC PROCESS THAT MODELS THE UNCERTAINTIES IS UNKNOWN. THE ANALYSIS PROCEDURE IS CONFINED TO THE TIME DOMAIN, BEING GENERATED BY THE ENVELOPES OF THE ROTOR ORBITS AND THE UNBALANCE RESPONSES OBTAINED FROM A RUN-DOWN OPERATING CONDITION. THE HYDRODYNAMIC SUPPORTING FORCES ARE DETERMINED BY CONSIDERING A NONLINEAR MODEL, WHICH IS BASED ON THE SOLUTION OF THE DIMENSIONLESS REYNOLDS' EQUATION FOR CYLINDRICAL AND SHORT JOURNAL BEARINGS. THIS NUMERICAL STUDY ILLUSTRATES THE VERSATILITY AND CONVENIENCE OF THE MENTIONED FUZZY APPROACH FOR UNCERTAINTY ANALYSIS. THE RESULTS FROM THE STOCHASTIC ANALYSIS ARE ALSO PRESENTED FOR COMPARISON PURPOSES.

Fundamental frequency of a cantilever composite filament-wound anisogrid lattice cylindrical shell

Free vibrations of a cantilever composite lattice cylindrical shell with the rigid disk attached to its free end are considered in the paper. The filament-wound lattice cylinder is modelled as a continuous shell characterised by the effective stiffness parameters. The governing equations of motion are derived using the semi-membrane theory of orthotropic cylindrical shells. The free vibration problem formulated based on these equations is reduced to a transcendental equation. The solution of this equation yields the fundamental frequency of the composite lattice cylinder under consideration. The results of calculations are verified by comparison with a finite-element solution. The effect of the angle of orientation of unidirectional helical ribs on the magnitude of fundamental frequency is studied using a parametric analysis.

A novel method to evaluate Young׳s modulus of ceramics at high temperature up to 2100 °C

Although it is important to understand Young׳s modulus and creep behavior of ceramics at ultra-high temperature from 1500°C to 2300°C, measuring these properties accurately at high temperature above 1600°C is not very easy due to the difficulty of the deformation measurement or invalid testing instruments at such high temperature. This work presented a solution to this problem via a combination of the split ring test and relative method (viz. relative split ring method). The real deformation of the split ring at ultra-high temperature could be easily determined from the difference of cross-beam displacements between the split ring and a rigid disk under the same loads. Experiments of graphite and C/SiC/ZrB2 composite at ultra-high temperature demonstrated the validity and convenience of the relative split ring method, which would provide a simple and efficient approach to evaluate Young׳s modulus of ceramic materials at ultra-high temperature.

Analysis of Three-dimensional Earthquake Responses of a Floating Offshores Structure with an Axisymmetric Floating Structure

A seismic response analysis method for three-dimensional floating offshore structures due to seaquakes is developed. The hydrodynamic pressure exerted on the structure is calculated taking into account the compressibility of the sea water, the fluid-structure interaction, the energy absorption by the seabed, and the energy radiation into infinity. To validate developed method, the hydrodynamic pressure induced by the vibration of a floating massless rigid circular disk is calculated and compared with an exact analytical solution. The developed method is applied to seismic analysis of a support structure for a floating offshore wind turbine subjected to the hydrodynamic pressures induced from a seaquake. Analysis results show that earthquake response of a floating offshore structure can be greatly influenced by the compressibility of fluid, the depth (natural frequencies) of the fluid domain, and the energy absorption capacity of the seabed.

Adhesion-induced instabilities and pattern formation in thin films of elastomers and gels

A hydrostatically stressed soft elastic film circumvents the imposed constraint by undergoing a morphological instability, the wavelength of which is dictated by the minimization of the surface and the elastic strain energies of the film. While for a single film, the wavelength is entirely dependent on its thickness, a co-operative energy minimization dictates that the wavelength depends on both the elastic moduli and thicknesses of two contacting films. The wavelength can also depend on the material properties of a film if its surface tension has a pronounced effect in comparison to its elasticity. When such a confined film is subjected to a continually increasing normal displacement, the morphological patterns evolve into cracks, which, in turn, govern the adhesive fracture behavior of the interface. While, in general, the thickness provides the relevant length scale underlying the well-known Griffith-Kendall criterion of debonding of a rigid disc from a confined film, it is modified non-trivially by the elasto-capillary number for an ultra-soft film. Depending upon the degree of confinement and the spatial distribution of external stress, various analogs of the canonical instability patterns in liquid systems can also be reproduced with thin confined elastic films.

Model updating of a rotating machine using the self-adaptive differential evolution algorithm

Despite the good accuracy of finite element (FE) models to represent the dynamic behaviour of mechanical systems, practical applications show significant discrepancies between analytical predictions and experimental results, which are mostly due to uncertainties on the geometry configuration, imprecise material parameters and vague boundary conditions. Thereby, different approaches have been proposed to solve the inverse problems associated with the updating of FE models. Among them, the techniques based on minimization processes have shown to be some of the most promising ones. In this paper, a self-adaptive heuristic optimization method, namely the self-adaptive differential evolution (SADE), is evaluated. Differently from the canonical differential evolution (DE) algorithm, the SADE strategy is able to update dynamically the required parameters such as population size, crossover parameter and perturbation rate. This is done by considering a defined convergence rate on the evolution process of the algorithm in order to reduce the number of evaluations of the objective function. For illustration purposes, the SADE strategy is applied to the solution of typical mathematical functions. Additionally, the strategy is equally used to update the FE model of a rotating machine composed by a horizontal flexible shaft, two rigid discs and two unsymmetrical bearings. For comparison purposes, the canonical DE is also used. The results indicate that the SADE algorithm is a recommended technique for dealing with this kind of inverse problem.

An incompressible smoothed particle hydrodynamics method for the motion of rigid bodies in fluids

A two-dimensional incompressible smoothed particle hydrodynamics scheme is presented for simulation of rigid bodies moving through Newtonian fluids. The scheme relies on combined usage of the rigidity constraints and the viscous penalty method to simulate rigid body motion. Different viscosity ratios and interpolation schemes are tested by simulating a rigid disc descending in quiescent medium. A viscosity ratio of 100 coupled with weighted harmonic averaging scheme has been found to provide satisfactory results. The performance of the resulting scheme is systematically tested for cases with linear motion, rotational motion and their combination. The test cases include sedimentation of a single and a pair of circular discs, sedimentation of an elliptic disc and migration and rotation of a circular disc in linear shear flow. Comparison with previous results at various Reynolds numbers indicates that the proposed method captures the motion of rigid bodies driven by flow or external body forces accurately.

Dynamic response analysis of a floating offshore structure subjected to the hydrodynamic pressures induced from seaquakes

A seismic analysis method for a floating offshore structure subjected to the hydrodynamic pressures induced from seaquakes is developed. The hydrodynamic pressure exerted on the structure by the ocean water, modeled as compressible non-viscous ideal fluid, is calculated taking into account the fluid–structure interaction, the energy absorption by the seabed, and the energy radiation into infinity. From the hydrodynamic pressure, the added mass and the effective load, resulting from the hydrodynamic pressure, are obtained. They are combined with the finite element model of the structure, resulting in a numerical model for the entire coupled system. For validation of the numerical model, the hydrodynamic pressures induced by the vertical motion of the seabed at free field, and due to the vertical vibration of a floating massless rigid disk, are calculated and compared with exact analytical solutions. The developed method is applied to seismic analysis of a simplified support structure for floating offshore wind turbines subjected to the hydrodynamic pressures induced from seaquakes. Analysis results show that dynamic response of a floating offshore structure induced by the vertical seismic motion of the ocean bed can be greatly influenced by the compressibility of sea water and the energy absorption capacity of the seabed.

Deformation of a cantilever composite anisogrid lattice cylindrical shell loaded by transverse inertia forces

Bending of a cantilever composite anisogrid lattice cylindrical shell with the rigid disk attached to its free end is considered in the paper. The structure is subjected to transverse inertia loading. An analytical model has been developed based on the membrane theory of orthotropic cylindrical shells. The lattice structure is modeled as a continuous shell characterized by the effective membrane stiffnesses. The analytical formula was derived to calculate displacements of the disk and predict the bending stiffness of the structure under the transverse inertia load. The solution has been verified by comparison with a finite-element analysis. Applications of the formula to the design analysis and selection of the structural parameters of a spacecraft body lattice structure are demonstrated considering constraints imposed on the displacements of the disk.

Structural Dynamics Optimization of Rotor Systems for a Small-Size Turboprop Engine

This paper presents the results of the rotordynamic analysis and design optimization performed for a turboprop engine-rotor system consisting of a free-power turbine and a gas generator. Predictions of static stresses, natural frequencies, and amplitudes are obtained by the in-house finite-element code. A detailed analysis of the finite-element method model (various beam formulations, three-dimensional axisymmetric model, point masses, and one-dimensional rigid disk elements) is provided. A sizing optimization problem of total shaft mass minimization is considered. Design variables are inner radii and wall thicknesses of shaft sections. Constraints are imposed on static stresses, natural frequencies, and amplitudes of the unbalance response. Changes in design variables are constrained to avoid contacts between the shafts of the twin-spool configuration. An in-house implementation of the sequential quadratic programming method is coupled with hybrid, analytical and numerical, sensitivity analysis. Results of the optimization show that a significant mass reduction in comparison with the baseline configuration can be achieved with all constraints being simultaneously satisfied. The optimal designs, however, must be finalized to meet other requirements not considered during the optimization. The Campbell diagram of the free-power turbine rotor system is significantly affected by the optimization.

Stabilization of memory type for a rotating disk–beam system

In this article, a rotating disk–beam system is considered. Specifically, the system consists of a flexible beam and a rigid disk which rotates with a time-varying angular velocity. The beam, free at one end and clamped at the other one to the center of the disk, is supposed to rotate with the disk in another plane perpendicular to that of the disk. To stabilize the system, we propose a feedback law which consists of a control torque applied on the disk, while either a boundary or distributed internal control with memory is exerted on the beam. Then, it is shown, in both cases, that the closed-loop system is stabilized under suitable conditions on the angular velocity and the memory terms.

Diffraction of a Plane Acoustic Wave from a Finite Soft (Rigid) Cone in Axial Irradiation

The problem of diffraction of a plane acoustic wave by a finite soft (rigid) cone is investigated. This one is formulated as a mixed boundary value problem for the three-dimensional Helmholtz equation with Dirichlet (Neumann) boundary condition on the cone surface. The diffracted field is sought as expansion of unknown velocity potential in series of eigenfunctions for each region of the existence of sound pressure. The solution of the problem then is reduced to the infinite set of linear algebraic equations (ISLAE) of the first kind by means of mode matching technique and orthogonality properties of the Legendre functions. The main part of asymptotic of ISLAE matrix element determined for large indexes identifies the convolution type operator amenable to explicit inversion. This analytical treatment allows one to transform the initial diffraction problem into the ISLAE of the second kind that can be readily solved by the reduction method with desired accuracy depending on a number of truncation. All these determine the analytical regularization method for solution of wave diffraction problems for conical scatterers. The boundary transition to soft (rigid) disc is considered. The directivity factors, scattering cross sections, and far-field diffraction patterns are investigated in both soft and rigid cases whereas the main attention in the near-field is focused on the rigid case. The numerically obtained results are compared with those known for the disc.

Horizontal Vibration of an Underwater Foundation on a Poroelastic Stratum

This paper studies the horizontal vibration of a rigid disc on a poroelastic soil layer in contact with a fluid half-space using the poroelastic theory of potentials. The solution of this problem is expressed in terms of dual integral equations, and then converted into Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the horizontal dynamic impedance coefficients are given. The results of this study are helpful for designing an underwater foundation on a soil stratum due to dynamic horizontal forces.

A Skyrme model approach to the spin-orbit force

The spin-orbit force is a vital tool in describing finite nuclei and nucleon interactions; however its microscopic origin is not fully understood. In this paper we study a model inspired by Skyrmions which provides a classical explanation of the force. To simplify the calculations the Skyrmions are approximated as two-dimensional rigid discs which behave like quantum cogwheels.

First-order doubly-asymptotic formulation of the direct stiffness method for elastodynamic problems

A first-order formulation to analyze the dynamic response of layered soil profiles is presented as an alternative to the widely used second-order thin-layer method by the direct stiffness approach, including an efficient simulation of the underlaying elastic half-space. In contrast to the thin-layer method where response is expressed through a combination of second-order propagation modes, the proposed procedure uses first-order modal parameters that have the capacity to provide a good approximation in the complete wave number domain k, including the exact stiffness values for k=0 and k→∞, thus justifying its designation of doubly-asymptotic. This feature allows obtaining the exact soil profile response for static loads, while the proposed treatment of the elastic half-space reproduces naturally the radiation condition without a need of artificial damping. The capacity of the proposed formulation to solve elastodynamic problems is assessed by comparing its results with those of exact solutions available in the literature, and numerical solutions of rigid disks supported on the surface of different soil profiles.

Dynamic Reissner–Sagoci problem for a transversely isotropic half-space containing finite length cylindrical cavity

A transversely isotropic linear elastic half-space containing a circular cylindrical cavity of finite length with a depth-wise axis of material symmetry is considered to be under the effect of a mono-harmonic torsional motion applied on a rigid circular disc with the same radius of the cavity and welded at the bottom of the cavity. With the aid of Fourier sine and cosine integral transforms, the mixed boundary value problem is reduced to a generalized Cauchy singular integral equation for the unknown shear stress. The Cauchy integral equation involved in this paper is analytically investigated such that the solution is written in the form of a known singular function multiplied by an unknown regular function. The regular part of the shear stress is numerically determined with the use of Gauss–Jacobi integration formula. Series representation of the stress and displacement are obtained, and it is shown that their degenerated form to the static problem of isotropic material is coincide with the existing solutions in the literature. To investigate the effects of material anisotropy and the length of cavity, the tangential displacement and the shear stress in between the rigid disc and the bottom of cavity are numerically evaluated and illustrated, where some differences are distinguished. With the differences illustrated in this paper for different length of cavity, it is recognized that the effect of length of cavity cannot be neglected in analysis and design. Different results for different degrees of anisotropy shows that the anisotropy of the material is a normal behavior, which should be considered in this kind of medium.