Axisymmetric Motion Research Articles

Effect of Thermal Environment and Peripheral Loading on Axisymmetric Vibrations of Non-uniform FG Circular Plates via Generalized Differential Quadrature Method

In this article, axisymmetric vibrations of a non-uniform functionally graded circular plate subjected to uniform in-plane peripheral loading and non-linear temperature rise across the thickness have been analyzed on the basis of classical theory of plates. The thickness of the plate is assumed to vary exponentially along the radial direction. The plate material is graded in thickness direction using a power-law model and its mechanical properties are temperature-dependent. Keeping the uniform thermal environment over the top and bottom surfaces, the equations for thermoelastic equilibrium and axisymmetric motion for such a plate model have been derived by Hamilton’s energy principle. Employing generalized differential quadrature method, the frequency equations for clamped and simply supported plates have been obtained and solved numerically using MATLAB. The lowest three roots have been retained as the frequencies for the first three modes of vibration. The influence of various parameters, such as power-law index, in-plane force parameter and temperature difference with varying values of thickness parameter, has been analyzed on the vibration characteristics of the plate. By allowing the frequency to approach zero, the critical buckling loads with varying values of other parameters have been computed. The benchmark results for linear as well as uniform temperature rise have also been computed. The validity of the present technique is verified by comparing the results with the published work.

An algorithm on dimensional range of machining axisymmetric aspheric surface for machine tool

Based on coordinate transformation method, the mathematical model for machining axisymmetric aspheric surface is set up. Parabolic surface is studied. The functional relationships between workpiece parameter, tool parameter, cutting state parameter and machine tool performance parameter are analyzed. The variation laws of key technical indexes such as the position of tool locating in vertical direction, the maximum vector height and aperture of parabolic surface, swing range of tool, the minimum arbor distance of tool locating tool carrier in lateral are deduced. An algorithm on dimensional range of machining axisymmetric aspheric surface for machine tool is presented. It can be estimated swiftly that whether the machine tool could do it before machining axisymmetric aspheric surface by means of inputting structure parameters of axisymmetric aspheric surface and motion ranges of machine tool. The simulation experiment results show the validity of the algorithm, which facilitates the machining of axisymmetric aspheric surface.

Formation of dual vortex breakdown in a two-fluid confined flow

This experimental and numerical study discusses the formation of double vortex breakdown in a swirling flow of two immiscible fluids where new circulation cells evolve in both fluids. The rotating lid drives the steady axisymmetric motion in a sealed vertical cylindrical container whose other walls are stationary. As the rotation intensifies, multiple topological changes occur in the flow. This study explains how two new circulation cells (vortex breakdown bubbles) almost simultaneously develop near the centers of both upper and lower fluids while the flow remains steady and axisymmetric. Such multi-cell flows can help provide fine, gentle, and nonintrusive mixing in chemical and biological reactors.

Dual vortex breakdown in a two-fluid confined flow

This experimental study reveals a spectacular and important phenomenon—double vortex breakdown—in a swirling flow of two immiscible fluids where vortex breakdown bubbles evolve simultaneously in both fluids. The rotating lid drives the steady axisymmetric motion in a sealed vertical cylindrical container whose other walls are stationary. As the rotation intensifies, topological metamorphoses occur, resulting in a multicellular flow. Two new circulation cells (vortex breakdown bubbles) simultaneously develop near the centers of both fluids while the flow remains steady and axisymmetric. Such a pattern can help provide fine, gentle, and nonintrusive mixing in chemical and biological reactors.

Thermophoresis of a spherical particle straddling the interface of a semi-infinite micropolar fluid

The quasi-steady thermophoretic motion of a spherical particle straddling the flat interface of a semi-infinite micropolar fluid flow is investigated. The Reynolds numbers of the micropolar fluid and also the capillary number are assumed to be small. The case of 90° contact angle with flat surface is considered. The Knudsen number is assumed to be in the slip-flow regime. Various types of slip conditions are considered on the immersed portion of the particle surface: frictional and thermal slip for linear velocity and also microrotation slip. The cases of axisymmetric and asymmetric motions are studied. Expressions for thermophoretic velocities, forces and couples on the half-submerged sphere are obtained and their variations represented graphically as functions of the Fourier thermal conductivities, Knudsen number, micropolarity parameter, frictional slip parameter and the thermal stress slip parameter. The limiting cases of classical viscous fluid with thermophoresis are also discussed.

Flow dynamics and wall-pressure signatures in a high-Reynolds-number overexpanded nozzle with free shock separation

A delayed detached eddy simulation of an overexpanded nozzle flow with shock-induced separation is carried out at a Reynolds number of , based on nozzle throat diameter and stagnation chamber properties. In this flow, self-sustained shock oscillations induce local unsteady loads on the nozzle wall as well as global off-axis forces. Despite several studies in the last few decades, a clear physical understanding of the factors driving this unsteadiness is still lacking. The geometry under investigation is a subscale truncated ideal contour nozzle, which was experimentally tested at the University of Texas at Austin at a nozzle pressure ratio of 30. Under these conditions, the nozzle operates in a highly overexpanded state and comprises a conical separation shock that merges to form a Mach disk at the nozzle centre. The delayed detached eddy simulation model agrees well with the experimental results in terms of mean and fluctuating wall-pressure statistics. Wall-pressure spectra reveal a large bump at low frequencies associated with an axisymmetric (piston-like) motion of the shock system, followed by a broad and high-amplitude peak at higher frequencies associated with the Mach waves produced by turbulent eddies convecting through the detached shear layer. Moreover, a distinct peak at an intermediate frequency ( ) persists in the wall-pressure spectra downstream of the separation shock. A Fourier-based analysis performed in both time and space (azimuthal wavenumber) reveals that this intermediate-frequency peak is associated with the (non-symmetric) pressure mode and is thus related to the generation of aerodynamic side loads. It is then shown how the unsteady Mach disk motion is characterized by an intense vortex shedding activity that, together with the vortical structures of the annular shear layer, contributes to the sustainment of an aeroacoustic feedback loop occurring within the nozzle.

Different types of solitary waves in a thermo-hyperelastic neo-Hookean cylindrical shell

Based on the hyperelastic material models, the propagation of nonlinear travelling waves in cylindrical rods has been studied. Here, different types of nonlinear travelling waves in a thermo-hyperelastic neo-Hookean cylindrical shell are determined. By using the variational principle, a complex differential dynamical system describing the axisymmetric motion of the shell is derived. The effect of the temperature fields is considered. Moreover, the structure is extended from a one-dimensional rod to a three-dimensional cylindrical shell. Then in terms of the bifurcation theory, a detailed qualitative analysis of the system is carried out, and for the bifurcation parameters, their corresponding critical bifurcation values are determined. Combining with the orbits in phase diagrams under different parameters, solitary waves and periodic waves are found in the shell. It is worth pointing out that the solitary waves with the valley form may appear in the radial direction of the shell. The implicit analytical solutions and profiles of these travelling waves are given. There is a promising prospect for the propagation of strongly nonlinear travelling waves to detect structural defects and determine material parameters.

The influence of friction on behavior of a hyperelastic body in a conic channel

The model problem about the axisymmetric quasistatic motion of an elastic sphere considering the interaction with inner surface of a rigid cone die is investigated. The constitutive model is represented by the physically nonlinear generalization of the linear model of Hencky’s elastic material. The tensor of logarithmic strains is used by way of strain measure due to described below special properties. The influence of friction on macro characteristics of a process is studied using proposed numerical model of accounting of mixed boundary conditions on time-changing contact surface.

A Problem on a Viscous Layer Deformation

The axisymmetric motion of a viscous incompressible fluid in a layer bounded by a solid plane and the free surface parallel to it is considered. There are three regimes of motion in the problem: stabilization to rest, collapse in a finite time, and an intermediate self-similar regime in which the viscous layer expands unlimitedly in an infinite time.

Nonlinear traveling wave propagation in a Neo-Hookean cylindrical rod with torsional eigenstrain

This paper investigates the torsional and longitudinal traveling wave propagation through a cylindrical rod with torsional eigenstrain distribution. We derive three dynamic equilibrium equations of axisymmetric motion of a rod which is assumed to be composed of an incompressible, isotropic neo-Hookean solid and then reduce them to one nonlinear differential equation. Employing phase plane analysis, we solve and assess this equation, and accordingly determine all smooth and non-smooth waves. We also examine the effects of torsional eigenstrain on the shapes and velocities of traveling waves. We show that a new type of traveling wave, which is not seen in a normal rod, arises due to the eigentwist. Moreover, we demonstrate that in a cylindrical rod with eigentwist, a torsional traveling wave is always associated with a longitudinal one while it can propagate alone in a normal rod.

Waves and vibrations in a finitely deformed electroelastic circular cylindrical tube

In two recent papers, conditions for which axisymmetric incremental bifurcation could arise for a circular cylindrical tube subject to axial extension and radial inflation in the presence of an axial load, internal pressure and a radial electric field were examined, the latter being effected by a potential difference between compliant electrodes on the inner and outer radial surfaces of the tube. The present paper takes this work further by considering the incremental deformations to be time-dependent. In particular, both the axisymmetric vibration of a tube of finite length with appropriate end conditions and the propagation of axisymmetric waves in a tube are investigated. General equations and boundary conditions governing the axisymmetric incremental motions are obtained and then, for purposes of numerical evaluation, specialized for a Gent electroelastic model. The resulting system of equations is solved numerically and the results highlight the dependence of the frequency of vibration and wave speed on the tube geometry, applied deformation and electrostatic potential. In particular, the bifurcation results obtained previously are recovered as a special case when the frequency vanishes. Specification of an incremental potential difference in the present work ensures that there is no incremental electric field exterior to the tube. Results are also illustrated for a neo-Hookean electroelastic model and compared with those previously obtained for the case in which no incremental potential difference (or charge) is specified and an external field is required.

Converging super-elliptic torsional shear waves in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary.

A theoretical approach was recently introduced [Guidetti and Royston, J. Acoust. Soc. Am. 144, 2312-2323 (2018)] for the radially converging slow shear wave pattern in transverse isotropic materials subjected to axisymmetric excitation normal to the axis of isotropy at the outer boundary of the material. This approach is enabled via transformation to an elliptic coordinate system with isotropic properties. The approach is extended to converging fast shear waves driven by axisymmetric torsional motion polarized in a plane containing the axis of isotropy. The approach involves transformation to a super-elliptic shape with isotropic properties and use of a numerically efficient boundary value approximation.

On the Penetration Capability of Shaped Charges with Conical and Hemispherical Liners

This paper presents a computational comparison of the penetration capability of two shaped charges—with conical and hemispherical (degressive thickness) liners. It is shown that close values of the penetration capability of the charges are provided only in the ideal case of strictly axisymmetric motion of shaped-charge jets.

A Numerical Study of Axisymmetric Wave Propagation in Buried Fluid-Filled Pipes for Optimizing the Vibro-Acoustic Technique When Locating Gas Pipelines

Buried pipeline systems play a vital role in energy storage and transportation, especially for fluid energies like water and gas. The ability to locate buried pipes is of great importance since it is fundamental for leakage detection, pipeline maintenance, and pipeline repair. The vibro-acoustic locating method, as one of the most effective detection technologies, has been studied by many researchers. However, previous studies have mainly focused on vibro-acoustic propagation in buried water pipes. Limited research has been conducted on buried gas pipes. In this paper, the behavior of gas-dominated wave motion will be investigated and compared against water-dominated wave motion by adapting an established analytical model of axisymmetric wave motion in buried fluid-filled pipes. Furthermore, displacement profiles in spatial domain resulting from gas-dominated wave in buried gas pipeline systems will be analyzed, and the effects of pipe material, soil property, as well as mode wave type will be discussed in detail. An effective radiation coefficient (ERC) is proposed to measure the effective radiation ability of gas-dominated wave and water-dominated wave. It is observed that the gas-dominated wave in gas pipes cannot radiate into surrounded soil as effectively as water-dominated wave in water pipes because of the weak coupling between gas and pipe-soil. In this case, gas-dominated wave may not be the best choice as the target wave for locating buried gas pipes. Therefore, the soil displacements result from the shell-dominated wave are also investigated and compared with those from gas-dominated wave. The results show that for buried gas pipes, the soil displacements due to radiation of shell-dominated wave are stronger than gas-dominated wave, which differs from buried water pipe. Hence, an effectively exciting shell-dominated wave is beneficial for generating stronger vibration signals and obtaining the location information. The findings of this study provide theoretical insight for optimizing the current vibro-acoustic method when locating buried gas pipes.

Vibration analysis of FGM circular plates under non-linear temperature variation using generalized differential quadrature rule

In the present paper, free axisymmetric vibrations of functionally graded circular plates subjected to a non-linear temperature distribution along the thickness direction have been studied on the basis of classical plate theory. The top and bottom surfaces of the plate are under uniform thermal environment. The mechanical properties of the plate material are assumed to be temperature-dependent and the plate is graded in thickness direction. The equations for thermo-elastic equilibrium as well as axisymmetric motion for such a plate model have been derived using Hamilton’s principle. Employing generalized differential quadrature rule, the numerical values of thermal displacements from thermo-elastic equilibrium equation and frequencies from the equation of motion for the lowest three modes of vibration have been computed for clamped, simply supported and free plates. The effects of the temperature difference at the surfaces together with material graded index and thickness of the plate on the frequencies have been investigated. The benchmark results for uniform and linear temperature distributions have been computed. For the validity of present considerations and the technique, frequency parameter has been compared for some special cases with published results obtained from other approximate methods.

Vibration analysis of functionally graded circular plates of variable thickness under thermal environment by generalized differential quadrature method

The vibration of functionally graded circular plates of variable thickness under a thermal environment is analyzed when the nodal lines are concentric circles by using the generalized differential quadrature method for the nonlinear temperature distribution in the thickness direction. The parabolic variation in thickness along the radial direction is controlled by a taper constant. The plate material is graded in the transverse direction and its mechanical properties are temperature-dependent. The thermal environment over the top and bottom surfaces of the plate is assumed to be uniform. Hamilton's principle has been used in obtaining the governing differential equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model employing Kirchhoff plate theory. Numerical results for thermal displacements and natural frequencies of clamped and simply supported plates have been obtained using MATLAB. The effect of the taper constant, volume fraction index, and temperature difference on the vibration characteristics has been analyzed for the lowest three modes of vibration. A study in which the plate material has temperature-independent properties has also been performed. The accuracy of the present technique is verified by comparing the results with those available in the literature.

Creeping motion of a solid particle inside a spherical elastic cavity: II. Asymmetric motion.

An analytical method is proposed for computing the low-Reynolds-number hydrodynamic mobility function of a small colloidal particle asymmetrically moving inside a large spherical elastic cavity, the membrane of which is endowed with resistance toward shear and bending. In conjunction with the results obtained in the first part (A. Daddi-Moussa-Ider, H. Löwen, S. Gekle, Eur. Phys. J. E 41, 104 (2018)), in which the axisymmetric motion normal to the surface of an elastic cavity is investigated, the general motion for an arbitrary force direction can now be addressed. The elastohydrodynamic problem is formulated and solved using the classic method of images through expressing the hydrodynamic flow fields as a multipole expansion involving higher-order derivatives of the free-space Green's function. In the quasi-steady limit, we demonstrate that the particle self-mobility function of a particle moving tangent to the surface of the cavity is larger than that predicted inside a rigid stationary cavity of equal size. This difference is justified by the fact that a stationary rigid cavity introduces additional hindrance to the translational motion of the encapsulated particle, resulting in a reduction of its hydrodynamic mobility. Furthermore, the motion of the cavity is investigated, revealing that the translational pair (composite) mobility, which linearly couples the velocity of the elastic cavity to the force exerted on the solid particle, is solely determined by membrane shear properties. Our analytical predictions are favorably compared with fully-resolved computer simulations based on a completed-double-layer boundary integral method.

A Priori Estimates of the Conjugate Problem Describing an Axisymmetric Thermocapillary Motion for Small Marangoni Number

This paper is devoted to the study of equations solution describing the axisymmetric motion of a viscous heat-conducting liquid. The motion is interpreted as a two-layer flow of viscous heat-conducting liquids in a cylinder with a solid wall and a common movable non-deformable interface. From a mathematical point of view, the arising initial-boundary value problem is nonlinear and inverse. Under certain assumptions concerning to apply the problem is replaced by a linear one. As a result, the unimprovable uniform priori estimates for solutions of the problems posed are obtained

Thermal effect on radially symmetric vibrations of temperature-dependent FGM circular plates with nonlinear thickness variation

Thermo-elastic analysis for the free axisymmetric vibration of functionally graded circular plates with quadratic thickness variation along radial direction has been presented on the basis of classical theory of plates. Following a power law model, the plate is assumed to be graded in thickness direction and mechanical properties of the material are temperature-dependent. The plate is subjected to a non-linear temperature distribution in thickness direction while the thermal environment over the top and bottom surfaces is uniform. The equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model have been derived using Hamilton’s principle. Employing differential quadrature method, the numerical values of thermal displacements and frequencies for clamped and simply supported plates have been computed. The influence of thickness parameters, material graded index and temperature difference on the vibration characteristics for the first three modes of vibration has been analyzed. The benchmark results for uniform and linear temperature distributions have been computed. A study with the plate material having temperature-independent mechanical properties has also been performed. Because, functionally graded materials inherently withstand high temperature gradients due to low thermal conductivity, core ductility, low thermal expansion coefficient, etc, the performance of results can help for optimal designing of non-uniform circular plates by controlling one or both the taper parameters. In special cases, the results have been compared with those obtained by other methods.

On radially symmetric vibrations of functionally graded non-uniform circular plate including non-linear temperature rise

In this study, the free axisymmetric vibrations of functionally graded circular plate of linearly varying thickness controlled by a taper parameter in radial direction and non-linear temperature rise along the thickness have been investigated on the basis of classical plate theory. The plate material is graded in transverse direction and its mechanical properties are temperature-dependent. The temperature over the top and bottom surfaces is assumed to be uniformly distributed. Hamilton's principle has been used to derive the governing equations for thermo-elastic equilibrium and axisymmetric motion of such a plate model. The generalized differential quadrature method has been employed to obtain the thermal displacements and characteristic equations, for clamped and simply supported plates. The lowest three roots of these equations have been computed and reported as the values of frequency parameter for the first three modes of vibration. Effect of thickness parameter, volume fraction index and temperature difference has been analyzed on the vibration characteristics of the plate. A study with temperature-independent material properties has also been performed. Results have been compared with the published work.