Dynamics Of Thin Shell Research Articles

Geometric control of topological dynamics in a singing saw

The common handsaw can be converted into a bowed musical instrument capable of producing exquisitely sustained notes when its blade is appropriately bent. Acoustic modes localized at an inflection point are known to underlie the saw’s sonorous quality, yet the origin of localization has remained mysterious. Here we uncover a topological basis for the existence of localized modes that relies on and is protected by spatial curvature. By combining experimental demonstrations, theory, and computation, we show how spatial variations in blade curvature control the localization of these trapped states, allowing the saw to function as a geometrically tunable high-quality oscillator. Our work establishes an unexpected connection between the dynamics of thin shells and topological insulators and offers a robust principle to design high-quality resonators across scales, from macroscopic instruments to nanoscale devices, simply through geometry.

Scalar field thin shell collapse in the Brane-world geometry

In this paper, we studied the dynamics of thin shell in the perfect fluid composed of scalar field. To formulate the equation of motion of the shell, we used the Israel thin-shell formalism for the Brane-world black hole in the two surrounding vacuum regions (interior and exterior). In this study, we considered the potential function as a quadratic function of scalar field. The resulting dynamical equations have been analyzed numerically for both the cases, massless and massive scalar field through the effective potential and radius of the shell by considering different settings of the parameters involved. We found that there are three possibilities in this geometry, thin shell in the scalar field can expand, collapse or attain equilibrium for a while, however, in most of the cases for large value of radius, thin shell collapses to zero size. The effects of the parameters [Formula: see text] and [Formula: see text] (involved due to the Brane-world geometry) on the expansion and collapsing rates have been analyzed and the obtained results compared with the Schwarzschild case ([Formula: see text], [Formula: see text]).

Dynamics of scalar shell for rotating and charged rotating BTZ black holes

This paper studies the dynamics of thin-shell for (2 + 1)-dimensional rotating and charged rotating Bañados–Teitelboim–Zanelli black holes by using Israel thin-shell formalism. We consider the matter distribution located at thin-shell associated with a scalar field and analyze its effects on the dynamics of thin-shell through equations of motion and effective potential. The corresponding dynamical equations are numerically studied for both massless as well as massive scalar fields. For rotating case, the rate of expansion and collapse increases with massless scalar shell but decreases for massive case. For charged rotating, the rate of expansion and collapse decreases by increasing angular momentum for both massless as well as massive case. We conclude that the rate of expansion and collapse of the rotating case is greater than charged rotating black hole.

Nonlinear Dynamical Responses of Rotary Cylindrical Shells with Internal Resonance

The nonlinear forced vibration response of a thin, elastic, rotary cylindrical shell to a harmonic excitation is investigated in this study. Nonlinearities due to the large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structural damping. Different from the conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing the Airy stress function is utilized to study the nonlinear dynamics of thin shells. The system is discretized using the Galerkin method, while a model involving two degrees of freedom and allowing for the traveling wave response of the shell is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the two-degree-of-freedom system. In addition, the stability of steady-state solutions is analyzed in detail. Finally, results are given for exploring the effects of different parameters on the nonlinear dynamic response with internal resonance.

Transition to light-like trajectories in thin shell dynamics

It was recently shown that a massive thin shell that is sandwiched between a flat interior and an exterior geometry given by the outgoing Vaidya metric becomes null in a finite proper time. We investigate this transition for a general spherically-symmetric metric outside the shell and find that it occurs generically. Once the shell is null its persistence on a null trajectory can be ensured by several mechanisms that we describe. Using the outgoing Vaidya metric as an example we show that if a dust shell acquires surface pressure on its transition to a null trajectory it can evade the Schwarzschild radius through its collapse. Alternatively, the pressureless collapse may continue if the exterior geometry acquires a more general form.

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.

On the dynamics of thin shells of counter-rotating particles

In this paper, we study the dynamics of self-gravitating spherically symmetric thin shells of counter-rotating particles. We consider all possible velocity distributions for the particles, and show that the equations of motion by themselves do not constrain this distribution. We therefore consider the dynamical stability of the resulting configurations under several possible processes. This includes the stability of static configurations as a whole, where we find a lower bound for the compactness of the shells. We analyze also the stability of the single-particle orbits, and find conditions for ‘single-particle evaporation’. Finally, in the case of a shell with particles whose angular momenta are restricted to two values, we consider the conditions for stability under splitting into two separate shells. This analysis leads to the conclusion that under certain conditions that are given explicitly, an evolving shell may split into two or more separate shells. We provide explicit examples to illustrate this phenomenon. We also include a derivation of the thick-to-thin shell limit for an Einstein shell that shows that the limiting distribution of angular momenta is unique, covering continuously a finite range of values.

Thin shell dynamics and equations of state

A relation between stress-energy and motion is derived for accelerated Israel layers. The relation, for layers between two Schwarzschild manifolds, generalizes the equation of state for geodesic collapse. A set of linked layers is discussed.

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.

LQR Control of Thin Shell Dynamics: Formulation and Numerical Implementation

A PDE-based feedback control method for thin cylindrical shells with surface-mounted piezoceramic actuators is presented. Donnell-Mushtari equations modified to incorporate both passive and active piezoceramic patch contributions are used to model the system dynamics. The well-posedness of this model and the associated LQR problem with an unbounded input operator are established through analytic semigroup theory. The model is discretized using a Galerkin expansion and basis functions constructed from Fourier polynomials tensored with cubic splines, and convergence criteria for the associated approximate LQR problem are established. The effectiveness of the method for attenuating the coupled longitudinal, circumferential and transverse shell displacements is illustrated through a set of numerical examples.

Spline approximation of thin shell dynamics

A spline-based method for approximating thin shell dynamics is presented here. While the method is developed in the context of the Donnell–Mushtari thin shell equations, it can be easily extended to the Byrne–Flügge–Lur'ye equations or other models for shells of revolution as warranted by applications. The primary requirements for the method include accuracy, flexibility and efficiency in smart material applications. To accomplish this, the method was designed to be flexible with regard to boundary conditions, material non-homogeneities due to sensors and actuators, and inputs from smart material actuators such as piezoceramic patches. The accuracy of the method was also of primary concern, both to guarantee full resolution of structural dynamics and to facilitate the development of PDE-based controllers which ultimately require real-time implementation. Several numerical examples provide initial evidence demonstrating the efficacy of the method. © 1997 John Wiley & Sons, Ltd.

Spline approximation of thin shell dynamics

Spline approximation of thin shell dynamics

Bragg-modulated hybrid ray-mode algorithm for sound scattering from a periodically ribbed submerged elastic cylindrical shell: Theory and asymptotics

A wave-based hybrid ray-mode algorithm has recently been developed to parametrize mid- and high-frequency sound scattering from submerged empty and simply loaded cylindrical shells [Felsen et al., J. Acoust. Soc. Am. 87, 554–569 (1990)]. The present study extends the hybrid ray-mode scheme to acoustic scattering from a submerged infinite cylindrical elastic shell with annular ribs attached internally at periodic intervals. Assuming thin shell dynamics, the problem is solved by adapting to the nonperiodic azimuthally unbounded domain (−∞<φ<∞) the azimuthally periodic harmonic series procedure used by Burroughs [J. Acoust. Soc. Am. 75, 715–722 (1984)] for this problem; the unbounded azimuthal (φ) domain with its continuous wave-number spectrum is the essential starting point for the wave-based ray-acoustic analysis. The solution in the spectral domain is separated into the previously derived contributions for the nonloaded shell and into contributions from the infinite array of periodic ring forcings on the shell surface that account for the internal loads. Spectral synthesis furnishes the formal integral representation that is then manipulated by integration path deformations into a form amenable to reduction by saddle point asymptotics and residue calculus so as to yield the desired hybrid ray-mode combination. The final algorithm for the scattered field contains (a) specular and Bragg-modulated diffracted ray fields; (b) new unmodulated and Bragg-modulated helically traveling shell-guided trapped, creeping and leaky modes, which are phase-matched to the fluid, in all wave species (compressional, flexural, shear) that the loaded thin shell geometry can support; (c) helically resonant shell-guided modes, formed by phase coherent circumnavigations of the loaded cylinder by the traveling modes in (b). The dispersion equations for the various Bragg-modulated wave types and species are determined from the internal impedance functions for the periodically distributed loadings.

The dynamics of thin shells with variable viscosity and the origin of toroidal flow in the mantle

SUMMARY The observed equipartition of kinetic energy between the poloidal (convergence/divergence) and toroidal (strike-slip) parts of the Earth’s surface velocity field is evidence that large lateral viscosity variations exist in the mantle. The largest such variations probably occur within the lithosphere, in the form of relatively weak boundaries between nearly rigid plates. To understand better the dynamical effects of such viscosity variations, we study an Earth model consisting of a lithospheric shell of outer radius a with laterally variable thickness h(8, q) and viscosity ti(@, q) overlying a mantle whose viscosity fj(r) is a function of depth only. Flow in the model is driven by lateral density variations beneath the shell. An asymptotic (long wave) analysis reduces the dynamics of the shell to a pair of boundary conditions, and shows that the effect of the shell is governed by a single dimensionless parameter, the ‘stiffness’ f = ~(y - l), where E = h/a and y = fj/fj(u). A perturbation analysis for the case of small lateral stiffness variations shows that the poloidal flow generated by an internal load interacts with the stiffness variations to generate additional poloidal and toroidal fields of various wavelengths according to definite selection rules. A simple model in which the shell consists of stiff (f >> 1) ‘plates’ separated by narrow weak (f<O.5) zones shows that (1) large lateral stiffness variations give rise to surface motion with a substantial toroidal component; (2) the observed surface velocity field is extremely sensitive to lateral stiffness variations, and thus provides poor constraints on the mantle viscosity; (3) lateral stiffness variations have a large (-50 per cent) effect on geoid anomalies if the mantle viscosity increases with depth; and (4) geoid anomalies predicted by models that ignore the order-unity relative change of long-wavelength normal stress across the shell are likely to be in error by 30-50 per cent. These results suggest that previous models may have overestimated the viscosity increase in the lower mantle that is required to fit the geoid data.

Ray acoustic formulation of sound wave scattering from submerged cylindrical thin elastic shells with endcaps.

Cylindrical shells with endcaps of various types are useful prototypes for a class of submerged elastic structures of current interest period. In the mid- and high-frequency ranges (ka≳3, where ‘‘a’’ is the radius of the cylinder), ray acoustics offers a physical-based methodology for computing the scattered signature. Restricted to thin shell dynamics, hemispherically capped cylinders are investigated by combining results from our previous studies of infinite cylindrical and full spherical shells [Felsen etal., J. Acoust. Soc. Am. 87, 554–569 (1990); J. M. Ho and L. B. Felsen, J. Acoust. Soc. Am. 88, 2389–2414 (1990)]. Rays from arbitrarily incident plane waves are traced to the cylinder and give rise there to geometrically reflected and phase matched shell guided supersonic wavefields, each with its own distinct trajectories. The surface guided fields follow helical paths on the cylindrical portion, which are smoothly matched to great circle paths on the hemispheres, and are turned back there toward the cylinder at a pitch-dependent caustic. The total field is synthesized by the sum of all possible ray paths, traversing the structure azimuthally and longitudinally, which reach the observer. Quantitative expressions are obtained by including ray field launching, detachment, and propagation coefficients. Attention is also given to conical endcaps and to conical sections with smoothly joined spherical caps. [Work supported by ONR.]

Synthesis of Optimal Control for an Infinite Dimensional Periodic Problem

We prove an existence and uniqueness result on periodic solutions of an infinite dimensional Riccati equation.

Elasto-plasticity of the club sandwich

The kinematics and dynamics of thin shells are well established. The constitutive equations of Hookean shells are linear in 6 strains and 6 stresses, but the equations of elastoplastic shells are incremental and require additional internal variables, notably stresses. In typical computations, the shell is divided into N layers: With the usual hypotheses, 6 strains and 3N stresses are required. The storage of 3N stresses poses practical limitations. The ideal sandwich of 2 layers is useful for limit analyses, but inadequate for general purposes. The club sandwich of 4 layers offers a practical alternative: Relative strengths, stiffnesses and positions of the layers are selected to fit the conditions of yielding under actions of membrane and bending stresses. Here, the mechanics of the club sandwich are presented and the behavior is compared with that of the multi-layered shell.