Flexural vibrations of conical shells with free edges
Resonant frequencies and mode shapes of truncated conical shells with free edges in transverse vibration
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The Stability of Radiatively Cooling Jets. 2: Nonlinear Evolution
We use two-dimensional time-dependent hydrodynamical simulations to follow the growth of the Kelvin-Helmholtz (K-H) instability in cooling jets into the nonlinear regime. We focus primarily on asymmetric modes that give rise to transverse displacements of the jet beam. A variety of Mach numbers and two different cooling curves are studied. The growth rates of waves in the linear regime measured from the numerical simulations are in excellent agreement with the predictions of the linear stability analysis presented in the first paper in this series. In the nonlinear regime, the simulations show that asymmetric modes of the K-H instability can affect the structure and evolution of cooling jets in a number of ways. We find that jets in which the growth rate of the sinusoidal surface wave has a maximum at a so-called resonant frequency can be dominated by large-amplitude sinusoidal oscillations near this frequency. Eventually, growth of this wave can disrupt the jet. On the other hand, nonlinear body waves tend to produce low-amplitude wiggles in the shape of the jet but can result in strong shocks in the jet beam. In cooling jets, these shocks can produce dense knots and filaments of cooling gas within the jet. Ripples in the surface of the jet beam caused by both surface and body waves generate oblique shock "spurs" driven into the ambient gas. Our simulations show these shock "spurs" can accelerate ambient gas at large distances from the jet beam to low velocities, which represents a new mechanism by which low-velocity bipolar outflows may be driven by high-velocity jets. Rapid entrainment and acceleration of ambient gas may also occur if the jet is disrupted. For parameters typical of protostellar jets, the frequency at which K-H growth is a maximum (or highest frequency to which the entire jet can respond dynamically) will be associated with perturbations with a period of - 200 yr. Higher frequency (shorter period) perturbations excite waves associated with body modes that produce internal shocks and only small-amplitude wiggles within the jet. The fact that most observed systems show no evidence for large-amplitude sinusoidal oscillation leading to disruption is indicative that the perturbation frequencies are generally large, consistent with the suggestion that pro- tostellar jets arise from the inner regions (r less than 1 AU) of accretion disks.
Stone, James M.↗
Pick-Up Ion Instabilities at Planetary Magnetospheres
This effort involved the analysis of low frequency waves as observed by the Galileo spacecraft near the Galilean moon, Io. Io is a significant source of material, especially SO2, and various products of dissociation, and further these atoms and molecules are readily ionized. The initial velocity of the ions is essentially that of the neutral species, i.e., the Keplerian velocity. The plasma, on the other hand is co-rotating, and there is a differential flow of the order 57 km/s between the plasma and the neutral particles. Thus pick-up ion instabilities are Rely to occur within the Jovian magnetosphere. Indeed, magnetometer observations from the Galileo spacecraft clearly show ion cyclotron waves that have been identified with a large variety of plasma species, such as O+, S++ (which has the same gyro-frequency as O+), S+, and SO2+. Typically, however, the dominant frequency is near the SO2+ gyro-frequency. The research effort was originally planned to be a team effort between Robert J. Strangeway as the Principal Investigator, and Debbie Huddleston, who was an Assistant Research Geophysicist at UCLA. Unfortunately, Dr. Huddleston took a position within Industry. The effort was therefore descoped, and Dr. Strangeway instead pursued a collaboration with Dr. Xochitl Blanco-Cano, of the Instituto de Geofisica, Universidad Nacional Autonoma de Mexico. This has proved to be a productive collaboration, with several papers and publications arising out of the effort. The magnetic field oscillations near lo generally fall into two types: ion cyclotron waves, with frequencies near an ion gyro-frequency, and lower frequency mirror-mode waves. The ion cyclotron waves are mainly transverse, and frequently propagate along the ambient magnetic field. The mirror-mode waves are compressional waves, and they have essentially zero frequency in the plasma rest frame. One of the purposes of our investigation is to understand what controls the types of wave modes that occur, since both wave modes can be drive unstable by the pressure anisotropy associated with the pick up ions. The pick ion velocity is perpendicular to the ambient magnetic field, and is generally much larger than the thermal velocity, at least initially. At its simplest, we found that the ion cyclotron waves are controlled by the parameters of the species in gyro-resonance with the wave. Thus, while the growth rates for the lower mass (higher gyro-frequency) pick-up ions are generally larger, we found that the heavier SO2+ ion cyclotron waves are generally preferred. This is because one of the effects of the wave instability is to diffuse the ions in pitch angle and energy. The lower mass ions therefore consist of both a ring of recently created pick-up ions, and a thermal background. This thermal background quenches the ion cyclotron instability. SO2+ is different, however. Being a molecule, the species can also dissociate. Our analysis suggests that the dissociation acts on a time scale comparable to or faster than the velocity space diffusion time scale. There are consequently no thermal SO2+ ions to quench the instability. We have also investigated the mirror-mode. This mode can at times grow more rapidly than the individual ion cyclotron waves. This is mainly because the mirror-mode can grow off the pressure anisotropy of the individual species with which the waves are in resonance. Lastly, as part of this effort we have begun to investigate the instability for obliquely propagating modes. Galileo observations show that at times the ion cyclotron waves are significantly elliptically polarized, and further the wave vector is at a large angle to the field, significant growth can occur for oblique propagation. Depending on the group velocity of the waves, it is possible that obliquely propagating modes have higher advective growth rate, but this has yet to be determined.
Strangeway, Robert J.↗
Application of the Galerkin Method in the Design of Stable Liquid Rocket Motors
A brief description of results obtained in the continuing investigations is presented. The following areas are covered: (1) the development of the third-order, multimode theory to study the behavior of large amplitude transverse instabilities, (2) nonlinear axial mode instability, and (3) the influence of the functional form of the unsteady combustion response function upon the stability characteristics of rocket motors. Also a study was begun to improve the second order potential theory in order to provide a better approximation to the nozzle boundary condition.
Zinn, B. T.↗
Applications of optical holography to applied mechanics.
This paper provides a brief summary of applications of optical holography and holographic interferometry to applied solid mechanics. Basic equations commonly used in fringe interpretation are described and used to reduce the data contained on holographic interferograms. A comparison of data obtained holographically with analytical prediction is given wherever possible. Applications contained herein include front surface physics, study of bomb breakup, transverse wave propagation, study of mode shapes of panels at elevated temperatures, nondestructive testing, and vibration analysis.
Aprahamian, R.↗
Electrostatic ion instabilities in the presence of parallel currents and transverse electric fields
The electrostatic ion instabilities are studied for oblique propagation in the presence of magnetic field-aligned currents and transverse localized electric fields in a weakly collisional plasma. The presence of transverse electric fields result in mode excitation for magnetic field aligned current values that are otherwise stable. The electron collisions enhance the growth while ion collisions have a damping effect. These results are discussed in the context of observations of low frequency ion modes in the auroral ionosphere by radar and rocket experiments.
Ganguli, G.↗
Relationship between mechanical-property and energy-absorption trends for composite tubes
U.S. Army helicopters are designed to dissipate prescribed levels of crash impact kinetic energy without compromising the integrity of the fuselage. Because of the complexity of the energy-absorption process it is imperative for designers of energy-absorbing structures to develop an in-depth understanding of how and why composite structures absorb energy. A description of the crushing modes and mechanisms of energy absorption for composite tubes and beams is presented. Three primary crushing modes of composite structures including transverse shearing, lamina bending, and local buckling are described. The experimental data presented show that fiber and matrix mechanical properties and laminate stiffness and strength mechanical properties cannot reliably predict the energy-absorption response of composite tubes.
Farley, Gary L.↗
On Dirac equations for linear magnetoacoustic waves propagating in an isothermal atmosphere
A new analytical approach to study linear magnetoacoustic waves propagating in an isothermal, stratified, and uniformly magnetized atmosphere is presented. The approach is based on Dirac equations, and the theory of Sturm-Liouville operators is used to investigate spectral properties of the obtained Dirac Hamiltonians. Two cases are considered: (1) the background magnetic field is vertical, and the waves are separated into purely magnetic (transverse) and purely acoustic (longitudinal) modes; and (2) the field is tilted with respect to the vertical direction and the magnetic and acoustic modes become coupled giving magnetoacoustic waves. For the first case, the Dirac Hamiltonian possesses either a discrete spectrum, which corresponds to standing magnetic waves, or a continuous spectrum, which can be clearly identified with freely propagating acoustic waves. For the second case, the quantum mechanical perturbation calculus is used to study coupling and energy exchange between the magnetic and acoustic components of magnetoacoustic waves. It is shown that this coupling may efficiently prevent trapping of magnetoacoustic waves instellar atmospheres.
Alicki, R.↗
Rectangular Waveguides with Two Conventional and Two Superconducting Walls
The propagation properties of transverse electric TE(sup pm) modes and their dispersion relations in rectangular waveguides with two conventional and two superconducting walls, derived by using the Meissner boundary conditions on the superconducting walls, are presented. In addition to recovering some previously known results, some novel results have been obtained: the cut-off wavelength of the dominant TE(sup 10) mode is greater than that of the conventional TE(sub 10) mode, and the tangential electric field and normal magnetic field for the dominant mode TE(sup 10) exist on the superconducting surfaces. Expressions for electromagnetic components, surface currents, attenuation coefficient, maximum transmitted power, dispersion and wave impedance are also presented.
Yalamanchili, Raj↗
The Impact Response of Carbon/Epoxy Laminates (Center Director's Discretionary Fund, Project No. 94-13)
Low velocity dropweight impact tests were conducted on carbon/epoxy laminates under various boundary conditions. The composite plates were 8-ply (+45,0,-45,90)s laminates supported in a clamped-clamped/free-free configuration with varying amounts of in-plane load, N(sub x), applied. Specimens were impacted at energies of 3.4, 4.5, and 6 Joules (2.5, 3.3, and 4.4 ft-lb). The amount of damage induced into the specimen was evaluated using instrumented impact techniques, x-ray inspection, and cross-sectional photomicroscopy. Some static identation tests were performed to examine if the impact events utilized in this study were of a quasi-static nature and also to gain insight into the shape of the deflected surface at various impact load combinations. Load-displacement curves from these tests were compared to those of the impact tests, as was damage determined from x-ray inspection. The finite element technique was used to model the impact event and determine the stress field within the laminae. Results showed that for a given impact energy level, more damage was induced into the specimen as the external in-plane load, N(sub x), was increased. The majority of damage observed consisted of back face splitting of the matrix parallel to the fibers in that ply, associated with delaminations emanating from these splits. The analysis showed qualitatively the results of impact conditions on maximum load of impact, maximum transverse deflection, and first failure mode and location.
Nettles, A. T.↗
Analytic and Computational Perspectives of Multi-Scale Theory for Homogeneous, Laminated Composite, and Sandwich Beams and Plates
This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C0-continuous finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite elements provide a unified and reliable computational platform for the analysis and design of high-performance load-bearing aerospace structures.
Tessler, Alexander↗
Analytic and Computational Perspectives of Multi-Scale Theory for Homogeneous, Laminated Composite, and Sandwich Beams and Plates
This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C(sup 0)-continuous finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite element approximations thus provide a unified and reliable computational platform for the analysis and design of high-performance load-bearing aerospace structures.
Tessler, Alexander↗
Propagation and scattering of elastic waves in fiber-reinforced plates
The objectives of this program are as follows: modelling of guided waves in fiber-reinforced plates in terms of different modes; and analysis of scattering by transverse cracks using modal representation. A hybrid numerical method combining the finite element representation of a region around the crack with the modal representation in the exterior region will be used in this program. Modes will be obtained using the through-the-thickness discretization of the displacement field.
Bratton, Robert L.↗
Test and Analysis of Sub-Components of Aluminum-Lithium Alloy Cylinders
Integrally machined blade-stiffened panels subjected to an axial compressive load were tested and analyzed to observe the buckling, crippling, and postcrippling response of the panels. The panels were fabricated from aluminum-lithium alloys 2195 and 2050, and both alloys have reduced material properties in the short transverse material direction. The tests were designed to capture a failure mode characterized by the stiffener separating from the panel in the postbuckling range. This failure mode is attributed to the reduced properties in the short transverse direction. Full-field measurements of displacements and strains using three-dimensional digital image correlation systems and local measurements using strain gages were used to capture the deformation of the panel leading up to the failure of the panel for specimens fabricated from 2195. High-speed cameras were used to capture the initiation of the failure. Finite element models were developed using an isotropic strain-hardening material model. Good agreement was observed between the measured and predicted responses for both alloys.
Haynie, Waddy T.↗
Effects of transverse shear on large deflection random response of symmetric composite laminates with mixed boundary conditions
Nonlinear equations of motion of symmetrically laminated anisotropic plates are derived considering von Karman strains and transverse shear effects. Using a single-mode Galerkin procedure, the nonlinear modal equation is obtained. The direct equivalent linearization method is employed for solution of this equation. The effects of transverse shear on large deflection vibration of laminated plates with mixed boundary support conditions under random excitation are studied. Mean-square deflections and mean-square inplane strains are obtained for symmetric graphite-epoxy laminates. Using equilibrium equations and the continuity requirements, the mean-square transverse shear stresses are calculated. The results obtained will be useful in the sonic fatigue design of composite aircraft panels. The analysis is presented in detail for plates with two opposing edges simply supported and the remaining edges clamped.
Prasad, C. B.↗
Finite-amplitude pressure waves in the radial mode of a cylinder
A numerical study of finite-strength, isentropic pressure waves transverse to the axis of a circular cylinder was made for the radial resonant mode. The waves occur in a gas otherwise at rest, filling the cylinder. A method of characteristics was used for the numerical solution. For small but finite amplitudes, calculations indicate the existence of waves of permanent potential form. For larger amplitudes, a shock is indicated to occur. The critical value of the initial amplitude parameter in the power series is found to be 0.06 to 0.08, under various types of initial conditions.
Kubo, I.↗
Effect of adhesive interleaving and discontinuous plies on failure of composite laminates subject to transverse normal loads
Results of a series of tests to determine the effects of adhesive interleaving and discontinuous plies (plies with end-to-end gaps) on the displacements, failure loads and failure modes of graphite-epoxy laminates subjected to transverse normal loads are presented. Adhesive interleaving can be used to contain local damage within a group of plies, i.e., to arrest crack propagation on the interlaminate level, and it can increase the amount of normal displacement the laminate can withstand before failure. However, the addition of adhesive interleaving to a laminate does not significantly increase its load carrying capability. A few discontinuous plies in a laminate can reduce the normal displacement and load at failure by 10 to 40 percent compared to a laminate with no discontinuous plies, but the presence of the ply discontinuities does not generally change the failure location or the failure mode of the laminate.
Jegley, Dawn C.↗
Modes in light wave propagating in semiconductor laser
The study of semiconductor laser based on an analogy of the Schrodinger equation and an equation describing light wave propagation in nonhomogeneous medium is developed. The active region of semiconductor laser is considered as optical waveguide confining the electromagnetic field in the cross-section (x,y) and allowing waveguide propagation along the laser resonator (z). The mode structure is investigated taking into account the transversal and what is the important part of the suggested consideration longitudinal nonhomogeneity of the optical waveguide. It is shown that the Gaussian modes in the case correspond to spatial squeezing and correlation. Spatially squeezed two-mode structure of nonhomogeneous optical waveguide is given explicitly. Distribution of light among the laser discrete modes is presented. Properties of the spatially squeezed two-mode field are described. The analog of Franck-Condon principle for finding the maxima of the distribution function and the analog of Ramsauer effect for control of spatial distribution of laser emission are discussed.
Manko, Margarita A.↗