Rigid Circular Disc Research Articles

On the sound field of an oscillating elliptical disk in free space with an open and closed back.

An oscillating elliptical disk in free space has a uniform surface velocity distribution and is therefore a rigid sound radiator. The radiated sound also represents that scattered from a stationary disk in the presence of a normal plane wave. Using the Fourier (Hankel) Green's function in cylindrical coordinates, together with the monopole Kirchhoff-Helmholtz boundary integral, rigorous analytical formulas are derived for calculating the surface pressure distribution, radiation impedance, on-axis pressure, and, directivity pattern of an elliptical open-back disk in free space, and these are plotted over a range of normalized frequencies. The directivity is compared with that of a rigid circular disk in free space, and asymptotic low-frequency approximations are derived for the radiation impedance and on-axis pressure. Using the Gutin concept, the radiated sound is combined with that from a rigid elliptical disk in an infinite baffle to obtain the radiation impedance, on-axis pressure, and, directivity pattern of a closed-back elliptical disk in free space. The latter has a uniform velocity distribution on the front surface and zero velocity over the entire back surface.

Stress State of a Piezoelectric Medium with a Noncentrally Loaded Rigid Circular Disk

Stress State of a Piezoelectric Medium with a Noncentrally Loaded Rigid Circular Disk

Magnetohydrodynamic flow of Copper-Water nanofluid over a rotating rigid disk with Ohmic heating and Hall Effects

In this article, a steady flow model is developed and analyzed for a water-based nanofluid that is generated by a rigid circular disk undergoing uniform rotation. Copper nanoparticles have been used for the analysis. Region over the plate is considered porous possessing uniform porosity whereas a uniformly impinged magnetic flux is assumed in the axial direction. Considerations of Hall currents and Ohmic heating are undertaken. To simplify the problem at hand, similarity transformations are utilized to transform it into a set of ordinary differential equations (ODEs), which can be more easily solved. Additionally, no-slip conditions are enforced at the boundary of the rigid disk. Numerical solutions for resulting nonlinear system are obtained via shooting method and are presented graphically for the detailed analysis. Results indicate that for magnetohydrodynamic flow with Hall effects through a porous medium, there is a direct proportion between axial velocity and volume fraction of copper-nanoparticles. Also, as the concentration of nanoparticles in the nanofluid increases, there is a corresponding increase in the temperature of the fluid. Additionally, at higher values of the Hall parameter, the impact of the Hartman number on flow variables is reduced.

Coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic and layered poroelastic half‐space

The coupled horizontal and rocking vibrations of a rigid circular disc resting on a transversely isotropic (TI) and layered poroelastic half-space is solved by a newly developed semi-analytical method for the first time. The poroelastic medium is modelled via Biot's poroelastodynamic theory. The Green's function corresponding to the horizontal uniform and vertical linearly changing patch loads on the surface of the layered half-space are first derived by virtue of the cylindrical system of vector functions and dual variable and position method. To solve the dynamic interaction, the influence function by the ring load is then introduced and calculated via the method of superposition. Together with the mixed boundary-value conditions, the dynamic horizontal, rocking, and coupling compliances are finally solved by a novel discretization scheme and an integral least-square approach. The developed fundamental solutions are verified by comparing with existing solutions in the purely elastic and poroelastic media. Numerical examples are presented to investigate the influence of poroelastic properties on the coupled vibration characteristics, which could be useful to the dynamic design of foundation on the saturated soil. It should be emphasized that the effect of the coupling compliance must be considered since its contribution can be as large as 26%, compared to the diagonal compliances.

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.

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.

The Reissner-Sagoci problem for an interfacial crack in an elastic bilayer medium under torsion of an embedded rigid circular disc

This paper studies an axisymmetric problem of a penny-shaped crack at the interface of a bi-material under a static torsion by an embedded circular rigid disc. By using the Hankel integral transformation method, the angular displacements and the shear stresses are formulated. The faces of the crack are supposed stress free, and the displacement is continuous outside in the crack plane while the rigid circular disc inclusion rotates about the axis passing through their centers and the stress is continuous outside in the rigid disc plane. Considering these mixed conditions associated with the embedded rigid disc and the interfacial crack, the mixed boundary values are taken into account, that are transformed, to a system of dual integral equations. By appropriate transform, the dual integral equations are converted into a regular system of Fredholm integral equations of the second kind with two unknown functions which is then solved by quadrature rule. Numerical results for displacements and stresses in the interaction zone, the stress intensity factor at the edges of the crack and the rigid disc and the applied torque are obtained and discussed according to certain relevant parameters. The torsional effects of the disc on the elastic bilayer are evaluated by the analysis of the Mode III stress intensity factor in the crack vicinity depending on the degree of nonhomogeneity at the interfacial plane, crack size and the depth of embedment of the rigid disc. The efficiency of the method and of the mathematical formulation are checked by comparison with the results available for a relevant analysis in homogeneous solids.

Vertical vibration of a rigid circular disc embedded in a transversely isotropic and layered poroelastic half-space

Fundamental solution for the time-harmonic vertical vibration of a rigid circular disc embedded in a transversely isotropic and layered poroelastic half-space is developed using a semi-analytical method. First, based on the Green's function of the time-harmonic vertical circular load embedded in the layered half-space, function of the corresponding annular ring load is determined via the method of superposition. Then the disc-medium contact area is discretized into annular ring elements with unknown densities which are determined via an integral least-square approach. Finally, the dynamic vertical compliance is derived by virtue of the equilibrium between the applied load on the disc and the resultant contact stress. Based on the derived fundamental solution, selected numerical examples on the dynamic vertical compliance are presented to investigate the influence of the surface hydraulic condition, fluid saturation, embedment depth of the disc, material anisotropy, material layering, and input frequency.

Coupled horizontal and rocking vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces

A new semi-analytical method is developed for solving the coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic multilayered half-space with imperfect interfaces. By virtue of the dual variable and position method, the recursive relation among different layers is established so that the final expansion coefficients of the displacement and traction vectors corresponding to the surface loading can be expressed in terms of the cylindrical system of vector functions. The physical-domain response by the surface loading (i.e., the forward solution) is obtained by carrying out the semi-infinite line integral involved. To solve for the corresponding coupled interaction problem between the surface rigid disc and layered elastic medium, the loading disc area is discretized into annular load-rings containing basic horizontal constant and vertical linear loads with unknown weights. These weights are then determined by applying an integral least-square approach with a Lagrange multiplier. Finally, the coupled time-harmonic compliances are derived by virtue of the equilibrium of the applied load on the disc and the reaction force on the half-space side. Selected numerical examples on the horizontal, moment and coupling compliances are presented to study the effect of material layering, anisotropy, interface behavior as well as input frequency. The coupling compliance is investigated in detail for the first time as compared to the horizontal and moment ones. The fully coupled, semi-coupled and decoupled schemes are also discussed with numerical examples so that one can easily access the relative accuracy among them when an actual layered structure is under both horizontal and rocking vibrations. This is important since the coupling compliance could significantly contribute to the vibrational behavior of the layered system at certain frequency.

Analytical investigation of a two-layered elastic half-space containing a cylindrical elastic inhomogeneity under forced torsional rotation

This paper presents a rigorous investigation for a two-layered linear elastic half-space containing a cylindrical elastic inhomogeneity with length equal to the thickness of the top layer undergoing forced torsional rotation applied on a rigid circular disc with the same radius as the cylinder, which is welded on the top of the cylinder. To investigate this complicated system analytically, a combination of Fourier sine and cosine integral transforms for depth, respectively in the domain of inhomogeneity and its matrix at the top layer, and Hankel integral transforms for radial distance in the lower half-space as the remaining part of the matrix are used, which translate the related boundary value problem to a system of coupled singular integral equations for the non-dimensional shear stresses resulted from the continuity and boundary conditions. The system of coupled singular integral equations is solved for some collocation points with a smoothed variable of distance, which is adapted with the use of a free parameter. The validities of both the analytical procedure and numerical evaluations are verified with its degenerated case for the homogeneous case, which is the Reissner–Sagoci problem. In addition, the degree of singularities of the shear stresses in between different regions are estimated in terms of material properties of the layer, inhomogeneity, and the half-space. Moreover, some new graphical results are presented for more understanding of related engineering behavior.

Vertical and torsional vibrations of an embedded rigid circular disc in a transversely isotropic multilayered half-space

In this paper, we present a fundamental solution to the mixed boundary-value problem of vertical and torsional vibrations of a rigid circular disc embedded in a transversely isotropic multilayered half-space. We first introduce the cylindrical system of vector functions to express displacements and stresses of the layered half-space induced by the time-harmonic vertical and torsional loads acting within the half-space. We then derive the recursive relation of the expansion coefficients of the displacement and traction vectors from one layer to any other layer via the dual variable and position (DVP) method. Finally, by virtue of the boundary/interface and internal loading conditions, the expansion coefficients of the displacements and tractions are determined, which are integrated back to the physical domains. To satisfy the mixed boundary conditions at the embedded rigid-disc level, we divide the load disc area into many annular load-rings with the corresponding solutions being superposed together via an integral least-square approach. Based on the derived fundamental solutions, various numerical examples on the vertical and torsional dynamic compliances are presented to demonstrate the influence of the embedment depth of the disc, material layering, anisotropy, and input frequency. Furthermore, the derived fundamental solution is applied to a homogeneous half-space with a thin coating to investigate the suitability of two effective boundary conditions on the half-space.

Vertical and torsional vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces

Rigid disc vibration over a layered half-space is essential to understand soil-structure interaction (SSI) between a foundation and the structure over it. In this article, we present a novel semi-analytical and very efficient method to calculate the SSI coefficients on the layered foundation due to both vertical and torsional vibrations. The material properties in each layer are transversely isotropic to take care of the possible different Young's moduli in different directions (vertical vs. horizontal). Furthermore, the interface between the layers is assumed to be general so that possible loose bonding between the layers (where the displacements and/or tractions are discontinuous across the interface) can be considered. The semi-analytical solution is based on the recently developed forward solution of the layered structure with imperfect interface under time-harmonic loadings within the circle on the surface. Since the present SSI problem is a mixed boundary-value problem, the method of superposition in terms of the powerful cylindrical system of vector functions combining with the integral least-square approach is proposed. To take care of multiple layers, the dual variable and position (DVP) is further introduced. The cylindrical system of vector functions has the advantage of separating the torsional vibration from the vertical vibration and yet expressing the two types of solutions uniformly. After validating the proposed semi-analytical method, numerical examples are presented to demonstrate the effect of material layering, interface imperfection, elastic anisotropy, and input frequency on the SSI coefficients, and on the surface displacement and stress variation. These new results should be also good benchmarks for future numerical methods.

A transversely isotropic medium containing a penny-shaped crack subjected to a non-uniform axisymmetric loading via an anchored smooth rigid disk

A smooth rigid circular anchor disk encapsulated by a penny-shaped crack is embedded in and unbounded transversely isotropic medium. The lamellar rigid disk exerts a nonuniform axisymmetric loading to the upper face of the crack. With the aid of an appropriate stress function and Hankel transform, the governing equations are converted to a set of triple integral equations which in turn are reduced to a Fredholm integral equation of the second kind. For some transversely isotropic materials the normalized stiffness of the system falls well outside of the envelope pertinent to isotropic media. It is shown that mode I stress intensity factor is independent of the material properties and solely depends on the ratio of the radius of the rigid disk to that of the crack; moreover, for the cases where this ratio is less than about 0.9 a simple explicit approximate expression for the mode I stress intensity factor is derived. In contrast, the normalized mode II stress intensity factor is independent of the mentioned geometrical parameters but depends on the elastic properties of the material; depending on the material properties, the normalized mode II stress intensity factor can vary between 0 to ∞ for transversely isotropic materials and between 0 to π/4 for isotropic materials.

Forced Horizontal Vibration of Rigid Disc in Transversely Isotropic Trimaterial Full-Space

AbstractA theoretical investigation is presented for the dynamic interaction of a rigid circular disc and its surrounding space, which is a trimaterial transversely isotropic full-space, and the disc is undergoing a prescribed horizontal harmonic vibration. Because the equations of motion form a system of coupled partial differential equations in a cylindrical coordinate system, a system of two scalar potential functions are used to decouple the equations of motion. This approach results in a set of two separated partial differential equations, which may be transformed to some ordinary differential equations by using Fourier series along the angular coordinate and the Hankel integral transformations with respect to the radial coordinate in a cylindrical coordinate system. After solving the ordinary differential equations, the unknown functions are determined by satisfying relaxed boundary conditions, which themselves are transformed to a set of four coupled integral equations. These coupled integral equat...

Dynamic analysis of a vertically loaded rigid disc in a transversely isotropic multilayered half-space

This paper focuses on the investigation of the response of a vertically loaded rigid circular disc embedded in a transversely isotropic multilayered half-space. The solutions for this dynamic interaction problem are obtained by utilizing the boundary element method, which takes the analytical layer-element solution of the transversely isotropic multilayered half-space subjected to an interior ring time-harmonic loading as the fundamental solution. The accuracy of present solutions is verified by the comparisons with existing solutions. In addition, selected numerical results are presented to illustrate the influence of embedded depth, frequency of excitation, material anisotropy and layering of the embedding medium on vertical impedance and resultant contact stresses.

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.

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.

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.

Vibration Analysis of a Rigid Circular Disk Embedded in a Transversely Isotropic Solid

AbstractThe analytical treatment of normal, rocking, and torsional forced time-harmonic vibrations of a rigid circular disk in a transversely isotropic full-space is revisited. A complete discussion on frequently used contact assumptions of adhesive and smooth models is given, and the effects of different contact models on the results of each vibration mode are discussed. With the aid of appropriate dynamic Green’s functions, the in-plane mode of vibration of the disk is treated analytically for the first time, and the results are expressed in terms of the solution of a Fredholm integral equation. For all four vibration modes, the relations for the contact stress, the resultant force acting on the disk, and the dimensionless compliance factor are given. The available closed-form results in the literature corresponding to the static loading are exactly recovered as the limiting cases of the current study. Furthermore, the results are verified with the special case of an isotropic full-space. By virtue of t...

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.