Motion Of Shell Research Articles

1/3 Subharmonic resonance of a nonlinear composite laminated cylindrical shell in subsonic air flow

The 1/3 subharmonic resonance of a composite laminated circular cylindrical shell with clamped boundary conditions at both ends in subsonic air flow under radial harmonic excitation is investigated. The equation of motion of the composite cylindrical shell is derived from the Donnell’s nonlinear shell theory and the Galerkin’s method is adopted to transform the equation of motion of the shell into a nonlinear ordinary differential equation. The 1/3 subharmonic resonance of the shell is analyzed by using the method of multiple scales and the sufficient and necessary conditions for the stability of the steady-states of the 1/3 subharmonic resonance is obtained by solving the eigenvalue problem of the linearized equations. The influences of the subsonic air flow on the 1/3 subharmonic resonance of the shell are discussed for different modes and the effects of the ply angles are analyzed for the fundamental mode. The numerical results of the threshold curves and the amplitude–frequency relations of the 1/3 subharmonic resonance are illustrated. From the results it can be seen that the existence region for the 1/3 subharmonic resonance is reduced and the amplitude of the subharmonic resonance is dropped with the flow velocity increasing. The composite shell with 45° ply angle exhibits better dynamic characteristic than that with other ply angles. The interval of the frequency ratio for the instability of the steady-state of the subharmonic resonance is broader when the ply angle is chosen as 45° than that of other ply angles for given amplitude of the external excitation.

Violation of the Third Law of Black Hole Thermodynamics inHigher Curvature Gravity

We examine the weak version of the third law of black hole thermodynamics inthe n-dimensional Einstein–Gauss–Bonnet system with a negative cosmological constant.To see whether the extreme black hole solution with zero temperature is formed, weinvestigate the motion of the thin shell that has the equal mass to the extreme black holein the background described by the non-GR branch solutions. The interior of the shell isempty. Our analysis using the generalized Israel's junction condition shows that the shellcan contract beyond a certain radius and the degenerate horizon is formed for same range ofparameters. Hence, this model can be a counterexample of the third law.

Free Vibration Analysis on Combined Cylindrical-Spherical Shell

Based upon the Reissner-Naghdi-Berry’s shell theory, a domain decomposition method (DDM) is utilized to investigate the vibration characteristics of the combined cylindrical-spherical shell with different boundary conditions. The combined shell was first apart from prescribed-displacement boundary and then divided into some cylindrical and spherical shell subdomains, respectively. The boundary equations were introduced into the energy functional of the combined shell as well as the constraint equations derived from interface continuity conditions between two adjacent shell subdomains. Fourier series and Chebyshev orthogonal polynomials were employed as the admissible displacement functions for each shell subdomain in the circumferential direction and axial direction in order to obtain the discretization equations of motion of the combined shell. Exact free vibration solutions of the combined shell has been performed via the DDM and were compared with those obtained by the finite element software ANSYS to confirm the reliability and accuracy.

부분 탄성지지된 유체 저장 원통셸의 자유진동

The free vibration characteristics of fluid-filled cylindrical shells on partial elastic foundations are investigated by an analytical method. The cylindrical shell is fully or partially surrounded by the elastic foundations, these are represented by the Winkler or Pasternak model. The motion of shell is represented by the first order shear deformation theory to account for rotary inertia and transverse shear strains. The steady flow of fluid is described by the classical potential flow theory. The fluid-structure interaction is considered in the analysis. The effect of internal fluid can be considered by imposing a relation between the fluid pressure and the radial displacement of the structure at the interface. To validate the present method, the numerical example is presented and compared with the available existing results.

Advanced formulation for laminated composite shells: 3D stress analysis and rigid-body motions

The application of the sampling surfaces (SaS) method to laminated composite plates is presented in a companion paper (Kulikov GM, Plotnikova SV. Exact 3D stress analysis of laminated composite plates by sampling surfaces method. Compos Struct 2012;94:3654–3663). In this paper, we extend the SaS method to shells to solve the 3D elasticity problems for cylindrical and spherical laminated composite shells. For this purpose, we introduce inside the nth layer In not equally spaced SaS parallel to the middle surface of the shell and choose displacements of these surfaces as basic kinematic variables. Such choice of displacements allows the derivation of strain–displacement equations, which are invariant under all rigid-body shell motions in any convected curvilinear coordinate system. This gives in turn the opportunity to find the 3D elasticity solutions for laminated composite shells with a prescribed accuracy utilizing a sufficiently large number of SaS, which are located at Chebyshev polynomial nodes and layer interfaces as well.

Expanding and collapsing scalar field thin shell

This paper deals with the dynamics of scalar field thin shell in the Reissner-Nordstr$\ddot{o}$m geometry. The Israel junction conditions between Reissner-Nordstr$\ddot{o}$m spacetimes are derived, which lead to the equation of motion of scalar field shell and Klien-Gordon equation. These equations are solved numerically by taking scalar field model with the quadratic scalar potential. It is found that solution represents the expanding and collapsing scalar field shell. For the better understanding of this problem, we investigate the case of massless scalar field (by taking the scalar field potential zero). Also, we evaluate the scalar field potential when $p$ is an explicit function of $R$. We conclude that both massless as well as massive scalar field shell can expand to infinity at constant rate or collapse to zero size forming a curvature singularity or bounce under suitable conditions.

Vibration and stability of ring-stiffened thin-walled cylindrical shells conveying fluid

Based on the Flügge shell theory, equations of motion of ring-stiffened thin-walled cylindrical shells conveying fluid are developed with the aid of the Hamilton's principle. Analysis is carried out on the vibration and stability of the ring-stiffened shells conveying fluid, and the effects of fluid velocity, the Young modulus, the size, and the number of the ring stiffeners on the natural frequency and the instability characteristics are examined. It is found that stiffeners can reduce the number of circumferential waves for the fundamental mode, and increase the shell's natural frequency, and thus the critical fluid velocity. For the number of longitudinal half waves being equal to one, the natural frequency and the corresponding critical fluid velocity are the largest for the internal-ring stiffened shell and are the smallest for the symmetrical-ring stiffened shell. The natural frequencies and the corresponding critical fluid velocity predicted by the established model increase with the increase in the Young modulus, the size, or the number of the stiffeners.

On some generalization of the area theorem with applications to the problem of rolling balls

This publication contributes to the series of RCD translations of Sergey Alexeevich Chaplygin’s scientific heritage. Earlier we published three of his papers on non-holonomic dynamics (vol. 7, no. 2; vol. 13, no. 4) and two papers on hydrodynamics (vol. 12, nos. 1, 2). The present paper deals with mechanical systems that consist of several spheres and discusses generalized conditions for the existence of integrals of motion (linear in velocities) in such systems. First published in 1897 and awarded by the Gold Medal of Russian Academy of Sciences, this work has not lost its scientific significance and relevance. (In particular, its principal ideas are further developed and extended in the recent article “Two Non-holonomic Integrable Problems Tracing Back to Chaplygin”, published in this issue, see p. 191). Note that non-holonomic models for rolling motion of spherical shells, including the case where the shells contain intricate mechanisms inside, are currently of particular interest in the context of their application in the design of ball-shaped mobile robots. We hope that this classical work will be estimated at its true worth by the English-speaking world.

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.

The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium

We consider a rectilinear quasi-stationary motion of a two-mass system in a viscous medium. The motion of the system as a whole occurs due to periodic movements of the internal mass relatively to the shell. The problem is to describe the law of motion of the internal mass that provides the minimum energy consumption with a specified average velocity of the shell. We propose an algorithm for solving the problem with any law of the resistance of the medium. We obtain the energy-optimal law of motion of a spherical shell in a viscous liquid.

Acoustic scattering from a finite periodically bulkheads in cylindrical shell

Acoustic scattering from a submerged finite cylindrical shell with periodically bulkheads, insonified by an incident plane wave, is studied. The motion of shell is described by Donnell equations, while the motion of bulkheads is described by thin plate motion equations and plane stress equations with considering axial force circumferential force radial force and bending moment simultaneously. Through angle-frequency spectrum besides the surface elastic waves, Bragg waves and Bloch-Floquet waves for the periodically bulkheads, the interactions between bulkheads resonance can be seen. Finally the theory is verified by experiment. Bragg waves, partial Bloch-Floquet waves and resonance of bulkheads highlight are all accordant with theory.

Apparent motion of a spherical shell collapsing onto a black hole

We model the collapse of a spherically symmetric, constant density, pressureless shell onto a preexisting black hole. Contrary to a recent claim of Liu and Zhang (2009), we show that an observer at infinity never sees any part of the shell cross the event horizon. The entire shell appears to “freeze” outside the black hole. We find that during intermediate stages of the collapse, the change in apparent size of the inner surface is non-monotonic. However, at late times, the apparent sizes of the inner and outer surfaces of the shell both approach 33GM/c2 asymptotically, where M is the combined mass of the initial black hole and the shell.

Active vibration control of conical shells using piezoelectric materials

In this paper, the active vibration control of conical shells is studied using velocity feedback and linear quadratic regulator methods. Up to now, many researches on the active vibration control of beams, plates and cylindrical shells have been published, however, to our knowledge, few people have studied the active vibration control of conical shells. Normally, in the equation of motion of the conical shells, some coefficients are variables, which makes the equation of motion of the conical shells very complicated and difficult to solve analytically. In order to solve this problem, Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the complex electromechanical coupling system. This equation of motion for the conical shell and piezoelectric patch system can be easily solved and effectively used for the structural active vibration control. Based on the traditional theory of structural dynamics, this method is easy to understand and is verified by numerical simulations. The forced vibration responses of the conical shells with two piezoelectric patches are computed to study the active vibration control. The optimal design for the locations of the piezoelectric patches is also developed by the genetic algorithm. From the results it can be seen that the control gain has a significant effect on the vibration control of the conical shell, but the effect of the size of the piezoelectric patches on controlling the vibration amplitudes is not so obvious. The overall vibration of the conical shell can be effectively reduced by the velocity feedback control method. With the increase of the control gain, the active damping characteristics of the conical shell are improved. Moreover, the optimal placement scheme of the piezoelectric patches obtained by the genetic algorithm can significantly reduce the vibration amplitudes of the conical shell.

An adjustable law of motion for relativistic spherical shells

Abstract A classical and a relativistic law of motion for an advancing shell are deduced applying the thin layer approximation. A new parameter connected with the quantity of absorbed matter in the expansion is introduced; this allows the matching of theory with observation.

Theoretical and Experimental Investigations of Piezoelectrically Excited Travelling Waves in Cylindrical Tubes

AbstractFlügge's equations of motion for cylindrical shells are used to analyze the generation of travelling waves in a tube of an ultrasonic travelling wave transport system. The vibration‐shape of a semi‐infinite acrylic tube with structural damping excited by a piezoelectric ring at the end is calculated and the influence of the driving frequency is shown.Starting with an exponential ansatz the frequency dependent wavenumbers can be determined. Three frequency bands with different combinations of imaginary, real and complex wavenumbers can be found. Supposing a semi‐infinite tube with given boundary conditions at the end of excitation the vibrations of the tube can be calculated depending on the frequency. Simulation agrees well with experimental results. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Free vibration of symmetric angle-ply laminated cylindrical shells of variable thickness including shear deformation theory

Free vibrational study of symmetric angle-ply laminated cylindrical shells of variable thickness including first order shear deformation theory using spline function approximation is studied. The equations of motion for the cylindrical shells are derived using first order shear deformation theory. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms of displacement and rotational functions, and these functions are approximated by Bickley-type splines of order three. The vibrations of three and five layered shells, made up of two different types of order of the layers of materials and two types of boundary conditions are considered. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. Parametric studies are made for the frequency parameters with respect to the coefficients of thickness variations, length-to-radius ratio, length-to-thickness ratio and ply angles under different boundary conditions. In the present work, the results are expected to be more accurate and more suitable for immediate application in the areas of missiles, aviation, shipping, surface transport and a large number of industries related to the cement and chemicals.

On high-altitude projectile motion

In this study, the formulas for projectile velocity components and coordinates in a vacuum were derived with the altitude decrease in gravitational acceleration factored in. A model of cannon shell motion in the air is presented that accounts for the altitude dependence of gravitational acceleration, air density, the speed of sound up to an altitude of 84 km, and the speed dependence of the drag coefficient at trans- and supersonic speeds. The drag coefficient function is obtained by fitting to experimental data taken for the US M101 155 mm shell. The model gives flight parameters that agree with the published ones. The motion of the Paris Gun projectile is then modeled. The model shows that a range of 120 km is possible if the projectile mass is about 150 kg. A flat Earth approximation was used in the computations. Changing the launch angle to 90°, super high-altitude vertical ascent and free fall are modeled.

Levitation characteristics of a squeeze-film air journal bearing at its normal modes

A tubular squeeze-film journal bearing was designed such that it flexed its shell at its normal modes producing a triangular modal shape. The shell motion was created by a single-layer piezoelectric actuator powered at 75 V AC with a 75 V DC offset and the driving frequency coincided with the modal frequency of the bearing. The paper provided a theory that shows the existence of a positive pressure in a squeeze film responsible for the levitation phenomenon. The various modes of vibration of the tubular bearing, made from AL2024-T3, were obtained from a finite element model implemented in ANSYS. Two normal modes, the 13th and 23rd, at the respective theoretical frequencies of 16.37 and 25.64 kHz, were identified for further investigation by experiments with respect to the squeeze-film thickness and its load-carrying capacity. While the bearing at both modes could cause levitation, the 13th mode has a greater load-carrying capacity because its modal shape produced a much lower end leakage.

DESIGN PARAMETERS AFFECTING TUMBLING MILL NATURAL FREQUENCIES

Tumbling mills describe a class of mechanical systems defined by a cylindrical chamber filled with balls and/or rock that rotates around its own longitudinal axis. This class of mechanical systems is used to grind to a desired quality different material in at least three industries: mining, cement and metal powders industries (ore, clinker and metal powder). These tumbling mills range in size from small 1 ft diameter lab mills to a 40 ft diameter semi-autogenous industrial mill and are composed of three main interactive and interdependent elements: the mill shell, liners/lifters and charge. These three elements work together to impact energy to the mill charge through the rotational motion of the mill shell. As with any mechanical system, the natural frequency of that system is essentially dictated by its mass and stiffness. In the case of a tumbling mill, a rotating system, the natural frequency of a given mill, as seen by a stationary observer, is also a function of the rotation speed. The aim of this paper The consequence for a rotating mill, and for that matter any rotating system, is that if the natural frequency of the mill should happen to match the operating rotating speed of the mill, resonance of the mill will occur. The objective of this paper is to explore the effect of mill diameter, length on mill natural frequencies as well as outline further work.

Large-scale outflows in galaxies

Abstract We discuss massive outflows in galaxy bulges, particularly the ones driven by accretion episodes where the central supermassive black hole reaches the Eddington limit. We show that the quasar radiation field Compton-cools the wind shock until this reaches distances ∼1 kpc from the black hole, but becomes too dilute to do this at larger radii. Radiative processes cannot cool the shocked gas within the flow time at any radius. Outflows are therefore momentum driven at small radii (as required to explain the M—σ relation). At large radii, they are energy driven, contrary to recent claims. We solve analytically the motion of an energy-driven shell after the central source has turned off. This shows that the thermal energy in the shocked wind can drive further expansion for a time ∼10 times longer than the active time of the central source. Outflows observed at large radii with no active central source probably result from an earlier short (few Myr) active phase of this source.