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NASA NTRS1992

Pulsatile instability in rapid directional solidification - Strongly-nonlinear analysis

In the rapid directional solidification of a dilute binary alloy, analysis reveals that, in addition to the cellular mode of Mullins and Sekerka (1964), there is an oscillatory instability. For the model analyzed by Merchant and Davis (1990), the preferred wavenumber is zero; the mode is one of pulsation. Two strongly nonlinear analyses are performed that describe this pulsatile mode. In the first case, nonequilibrium effects that alter solute rejection at the interface are taken asymptotically small. A nonlinear oscillator equation governs the position of the solid-liquid interface at leading order, and amplitude and phase evolution equations are derived for the uniformly pulsating interface. The analysis provides a uniform description of both subcritical and supercritical bifurcation and the transition between the two. In the second case, nonequilibrium effects that alter solute rejection are taken asymptotically large, and a different nonlinear oscillator equation governs the location of the interface to leading order. A similar analysis allows for the derivation of an amplitude evolution equation for the uniformly pulsating interface. In this case, the bifurcation is always supercritical. The results are used to make predictions about the characteristics of solute bands that would be frozen into the solid.

Merchant, G. J.↗
NASA NTRS2018

Airspace Technology Demonstration 2 (ATD-2): Integrated Arrival/Departure/Surface (IADS) System Demonstration, Technology Transfer Document Summary (ReadMe)

Airspace Technology Demonstration - 2 (ATD-2) is part of NASA's Airspace Technology Demonstrations (ATD) Project under its Airspace Operations and Safety Program (AOSP). ATD-2 is a multi-year research and development effort to improve the predictability and the operational efficiency of the air traffic system in metroplex environments, through the enhancement, development, and integration of the nation's most advanced and sophisticated arrival, departure, and surface prediction, scheduling, and management systems. In order to ensure that the products of this knowledge and technology transfer are relevant and useful, NASA has created strong partnerships with the FAA and key industry stakeholders. This summary document and accompanying artifacts satisfy the first of three Research Transition Products (RTPs) defined in the IADS Research Transition Team (RTT) Plan. This transfer consists of NASA's ATD-2 Phase 1 Baseline IADS System capabilities.

IADS↗
NASA NTRS2004

Modified Coulomb-Dipole Theory for 2e Photoionization

In the light of recent experiment on 2e photoionization of Li near threshold, we have considered a modification of the Coulomb-dipole theory, retaining the basic assumption that the threshold is dominated by asymmetric events in phase space [implies r(sub 1), k(sub 1)) greater than or equal to 2(r(sub 2), k(sub )]. In this region [in a collinear model, 2/r(sub 12) approached + 2/(r(sub 1)+r(sub 2)] the interaction reduces to V(rIsub 1) is greater than or equal to 2r(sub 2) is identically equal to [(-Z/r(sub 2)-(A-1)/r(sub 1)] + [(-2r(sub 2)/r(sub 1 exp 2)] is identically equal to V(sub c)+[V(sub d)]. For two electron emission Z = 2, thus both electrons see a Coulomb potential (V(sub c)) asymptotically, albeit each seeing a different charge. The residual potential (V(sub d)) is dipole in character. Writing the total psi = psi (sub c) + psi(sub d) = delta psi, and noting that. (T+V(sub c)-E)psy(sub c) = 0 and (T+V(sub c))psi(sub d) = 0 can be solved exactly, we find, substituting psi into the complete Schrod. Eq., that delta psi = -(H-E)(exp -1)(V(sub d) psi(sub 0)+V(sub c psi (sub 1). Using the fact that the absolute value of V(sub c) is much more than the absolute value of V(sub d) in almost all of configuration space, we can replace H by H(sub 0) in 9H-E)(exp -1) to obtain an improved approximation psi (improved) = psi(sub c) + psi(sub d) -(H(sub 0)-E)(exp -1) (V(sub c) psi (sub 0) + V(sub c) psi(sub 1). Here's the Green's function (H(sub 0)-E)(exp -1), can be exhibited explicitly, but the last term in psi (improved) is small, compared to the first two terms. Inserting them into the transition matrix element, which one handles in the usual way, we obtain in the limit E approaches 0, the threshold law: Q(E) alpha E + M(E)E(exp 5/4) + higher order (Eq. 1a). The modulation function, M(E), is a well-defined (but very non-trivial integral, but it is expected to be well approximated by a sinusoidal function containing a dipole phase term (M(E) = c sin[alpha log (E) + micron] (Eq. 1b). Experimental results show definite modulations, and are well fitted by Eqs (1).

Source record↗
NASA NTRS1984

High pressure cosmochemistry applied to major planetary interiors: Experimental studies

Progress is reported on a project to determine the properties and boundaries of high pressure phases of the H2-He-H2O-NH3-CH4 system that are needed to constrain theoretical models of the interiors of the major planets. This project is one of the first attempts to measure phase equilibria in binary fluid-solid systems in diamond anvil cells. Vibrational spectroscopy, direct visual observations, and X-ray diffraction crystallography of materials confined in externally heated cells are the primary experimental probes. Adiabats of these materials are also measured in order to constrain models of heat flow in these bodies and to detect phase transitions by thermal anomalies. Initial efforts involve the NH3-H2O binary. This system is especially relevant to models for surface reconstruction of the icy satellites of Jupiter and Saturn. Thermal analysis experiments were completed for the P-X space, p4GPa:0 or = 0.50, near room temperature. The cryostat, sample handling equipment, and optics needed to extend the optical P-T-X work below room temperature was completed.

Nicol, M. F.↗
NASA NTRS1997

Nitrogen sulfide in giant molecular clouds

We report a survey for nitrogen sulfide (NS) toward regions of massive star formation. NS was observed by means of its 2 pi 1/2, J = 3/2 --> 1/2, J = 5/2 --> 3/2, and J = 7/2 --> 5/2 transitions at 69, 115, and 161 GHz, respectively, and was detected toward 12 of 14 giant molecular clouds (GMCs) observed. Analysis of the hyperfine component relative line strengths suggests that NS emission is optically thin toward these sources, with the possible exception of Sgr B2(M). The fractional abundance of NS relative to molecular hydrogen is best defined for the Orion molecular cloud, where it is typically (1-4) x 10(-10), which is about an order of magnitude larger than found by some recent gas-phase chemistry models developed for quiescent clouds. Toward OMC-1, the NS integrated intensity is strongly peaked toward KL, but also extends all along the Orion ridge, resembling the distribution of SO and CH3OH. We have identified a spectral feature seen toward several sources as the ortho-NKK = 4(04) --> 3(13) J = 3 --> 2, fine-structure component of methylene (CH2; cf. Hollis, Jewell, & Lovas). We also report the first detection of the SO+ 2 pi 1/2, J = 3/2 --> 1/2, parity-e transition toward W51(MS) and L134N.

NASA Discipline Exobiology↗
NASA NTRS2000

Hybrid Active/Passive Control of Sound Radiation from Panels with Constrained Layer Damping and Model Predictive Feedback Control

There has been considerable interest over the past several years in applying feedback control methods to problems of structural acoustics. One problem of particular interest is the control of sound radiation from aircraft panels excited on one side by a turbulent boundary layer (TBL). TBL excitation appears as many uncorrelated sources acting on the panel, which makes it difficult to find a single reference signal that is coherent with the excitation. Feedback methods have no need for a reference signal, and are thus suited to this problem. Some important considerations for the structural acoustics problem include the fact that the required controller bandwidth can easily extend to several hundred Hertz, so a digital controller would have to operate at a few kilohertz. In addition, aircraft panel structures have a reasonably high modal density over this frequency range. A model based controller must therefore handle the modally dense system, or have some way to reduce the bandwidth of the problem. Further complicating the problem is the fact that the stiffness and dynamic properties of an aircraft panel can vary considerably during flight due to altitude changes resulting in significant resonant frequency shifts. These considerations make the tradeoff between robustness to changes in the system being controlled and controller performance especially important. Recent papers concerning the design and implementation of robust controllers for structural acoustic problems highlight the need to consider both performance and robustness when designing the controller. While robust control methods such as H1 can be used to balance performance and robustness, their implementation is not easy and requires assumptions about the types of uncertainties in the plant being controlled. Achieving a useful controller design may require many tradeoff studies of different types of parametric uncertainties in the system. Another approach to achieving robustness to plant changes is to make the controller adaptive. For example, a mathematical model of the plant could be periodically updated as the plant changes, and the feedback gains recomputed from the updated model. To be practical, this approach requires a simple plant model that can be updated quickly with reasonable computational requirements. A recent paper by the authors discussed one way to simplify a feedback controller, by reducing the number of actuators and sensors needed for good performance. The work was done on a tensioned aircraft-style panel excited on one side by TBL flow in a low speed wind tunnel. Actuation was provided by a piezoelectric (PZT) actuator mounted on the center of the panel. For sensing, the responses of four accelerometers, positioned to approximate the response of the first radiation mode of the panel, were summed and fed back through the controller. This single input-single output topology was found to have nearly the same noise reduction performance as a controller with fifteen accelerometers and three PZT patches. This paper extends the previous results by looking at how constrained layer damping (CLD) on a panel can be used to enhance the performance of the feedback controller thus providing a more robust and efficient hybrid active/passive system. The eventual goal is to use the CLD to reduce sound radiation at high frequencies, then implement a very simple, reduced order, low sample rate adaptive controller to attenuate sound radiation at low frequencies. Additionally this added damping smoothes phase transitions over the bandwidth which promotes robustness to natural frequency shifts. Experiments were conducted in a transmission loss facility on a clamped-clamped aluminum panel driven on one side by a loudspeaker. A generalized predictive control (GPC) algorithm, which is suited to online adaptation of its parameters, was used in single input-single output and multiple input-single output configurations. Because this was a preliminary look at the potential constrained layer damping for adaptive control, static feedback control with no online adaptation was used. Two configurations of CLD in addition to a bare panel configuration were studied. For each CLD configuration, two sensor arrangements for the feedback controller were compared. The first arrangement used fifteen accelerometers on the panel to estimate the responses of the first six radiation modes of the panel. The second sensor arrangement was simpler, using the summed responses of only four accelerometers to approximate the response of the first radiation mode of the panel. In all cases a PZT patch was mounted at the center of the panel for control input. The performance of the controller was quantified using the responses of the fifteen accelerometers on the panel to estimate radiated sound power. The paper begins with a brief discussion of the GPC algorithm and the experimental setup. The experimental results are discussed next, comparing the CLD and sensor configurations, followed by discussion and conclusions.

Cabell, Randolph H.↗
NASA NTRS2013

Advanced Colloids Experiment (ACE) Science Overview

The Advanced Colloids Experiment is being conducted on the International Space Station (ISS) using the Light Microscopy Module (LMM) in the Fluids Integrated Rack (FIR). Work to date will be discussed and future plans and opportunities will be highlighted. The LMM is a microscope facility designed to allow scientists to process, manipulate, and characterize colloidal samples in micro-gravity where the absence of gravitational settling and particle jamming enables scientists to study such things as:a.The role that disordered and ordered-packing of spheres play in the phase diagram and equation of state of hard sphere systems,b.crystal nucleation and growth, growth instabilities, and the glass transition, c.gelation and phase separation of colloid polymer mixtures,d.crystallization of colloidal binary alloys,e.competition between crystallization and phase separation,f.effects of anisotropy and specific interactions on packing, aggregation, frustration and crystallization,g.effects of specific reversible and irreversible interactions mediated in the first case by hybridization of complementary DNA strands attached to separate colloidal particles,h.Lock and key interactions between colloids with dimples and spheres which match the size and shape of the dimples,i.finding the phase diagrams of isotropic and interacting particles,j.new techniques for complex self-assembly including scenarios for self-replication, k.critical Casimir forces,l.biology (real and model systems) in microgravity,m.etc. By adding additional microscopy capabilities to the existing LMM, NASA will increase the tools available for scientists that fly experiments on the ISS enabling scientists to observe directly what is happening at the particle level. Presently, theories are needed to bridge the gap between what is being observed (at a macroscopic level when photographing samples) with what is happening at a particle (or microscopic) level. What is happening at a microscopic level will be directly accessible with the availability of the Light Microscopy Module (LMM) on ISS. To meet these goals, the ACE experiment is being built-up in stages, with the availability of confocal microscopy being the ultimate objective. Supported by NASAs Physical Sciences Research Program, ESAESTEC, and the authors respective governments.

Microgravity Biology Research↗
NASA NTRS1997

Peptide Folding and Translocation Across the Water-Membrane Interface

The ability of small peptides to organize at aqueous interfaces was examined by performing a series of large-scale, molecular dynamics computer simulations of several peptides composed of two amino acids, nonpolar leucine (L) and polar glutamine (Q). The peptides differed in size and sequence of the amino acids. Studies on dipeptides LL, LQ, QL and QQ were extended to two heptamers, LQQLLQL and LQLQLQL, designed to maximize interfacial stability of an alpha-helix and a beta-strand, respectively, by exposing polar side chains to water and nonpolar side chains to a nonpolar phase. Finally, a transition of an undecamer, composed entirely of leucine residues, from a disordered structure in water to an alpha-helix in a nonpolar phase representing the interior of the membrane was investigated. Complete folding of a peptide in solution was accomplished for the first time in computer simulations. The simulations revealed several basic principles governing the sequence-dependent organization of peptides at interfaces. Short peptides tend to accumulate at interfaces and acquire ordered structures, providing that they have a proper sequence of polar and nonpolar amino acids. The dominant factor determining the interfacial structure of peptides is the hydrophobic effect, which is manifested at aqueous interfaces as a tendency for polar and nonpolar groups of the solute to segregate into the aqueous and nonpolar phases, respectively. If peptides consist of nonpolar residue's only, they become inserted into the nonpolar phase. As demonstrated by the example of the leucine undecamer, such peptides fold into an alpha-helix as they partition into the nonpolar medium. The folding proceeds through an intermediate, called 3-10-helix, which remains in equilibrium with the alpha-helix. Once in the nonpolar environment, the peptides can readily change their orientation with respect to the interface from parallel to perpendicular, especially in response to local electric fields. The ability of nonpolar peptides to modify both the structure and orientation with respect to the interface from parallel to perpendicular, especially in response to local electric fields. The ability of nonpolar peptides to modify both the structure and orientation with changing external conditions may have provided a simple mechanism of transmitting signals from the environment to the interior of a cell.

Pohorille, Andrew↗
NASA NTRS1990

Self-organized criticality in sandpiles - Nature of the critical phenomenon

The scaling behavior of sandpile models is investigated analytically. First, it is shown that sandpile models contain a set of domain walls, referred to as troughs, which bound regions that can experience avalanches. It is further shown that the dynamics of the troughs is governed by a simple set of rules involving birth, death, and coalescence events. A simple trough model is then introduced, and it is proved that the model has a phase transition with the density of the troughs as an order parameter and that, in the thermodynamic limit, the trough density goes to zero at the transition point. Finally, it is shown that the observed scaling behavior is a consequence of finite-size effects.

Carlson, J. M.↗
NASA NTRS2002

Wholly Aromatic Ether-imides. Potential Materials for n-Type Semiconductors

We report on the synthesis and characterization of a novel series low-molar-mass ether-imide rod-shaped model compounds. All ether-imides were obtained by terminating the appropriate rigid core dianhydride, i.e. pyromellitic dianhydride (PMDA), 1,4,5,8-naphthalenetetracarboxylic dianhydride (NDA), 3,3',4,4'-biphenyltetracarboxylic dianhydride (BPDA), and 3,3,4,4'-oxydiphthalic dianhydride (ODPA) with three flexible aryl-ether tails of different chain length. The mono-functional aryl-ether amines, i.e. 4-(3-phenoxy-phenoxy)-phenylamine (2) and 4-(3-phenoxy-3-phenoxy-phenoxy)-phenylamine (4), were synthesized using standard fluoro-displacement and Ullmann condensation techniques. The corresponding ether-imide model compounds were obtained in high yields using a one-step solution imidization procedure. Increasing the number of meta-substituted aryl-ether units reduces the melt transition temperatures and at the same time it increases the solubility of the model compounds. Most model compounds are crystalline solids and form isotropic melts upon heating. 2,7-Bis-(-4-phenoxy-phenyl)-benzo[Imn][3,8]phenanthroline1,3,6,8-tetraone (NDA-n0), however, displays a smectic A (SA) when cooled from the isotropic phase, followed by what appears to be either a highly ordered smectic phase or a, columnar phase. This is the first example, known to date, in which a mesophase is detected in a wholly aromatic ether-imide compound. For all compounds we present spectroscopic data and X-ray diffraction data. Cyclic voltammetry was used to determine the redox behavior and pertinent energy levels of the model compounds.

Dingemans, Theo J.↗
NASA NTRS2013

A Suzaku X-ray Observation of One Orbit of the Supergiant Fast X-ray Transient IGR J16479-4514

We report on a 250 ks long X-ray observation of the supergiant fast X-ray transient (SFXT) IGR J16479-4514 performed with Suzaku in 2012 February. During this observation, about 80% of the short orbital period (P(sub orb) approximates 3.32 days) was covered as continuously as possible for the first time. The source light curve displays variability of more than two orders of magnitude, starting with a very low emission state (10(exp −13) erg / sq cm/s; 1-10 keV) lasting the first 46 ks, consistent with being due to the X-ray eclipse by the supergiant companion. The transition to the uneclipsed X-ray emission is energy dependent. Outside the eclipse, the source spends most of the time at a level of 6-7X10)(exp−12) erg/sq. cm/s) punctuated by two structured faint flares with a duration of about 10 and 15 ks, respectively, reaching a peak flux of 3-4X10(exp −11) erg/sq. cm./S, separated by about 0.2 in orbital phase. Remarkably, the first faint flare occurs at a similar orbital phase of the bright flares previously observed in the system. This indicates the presence of a phase-locked large scale structure in the supergiant wind, driving a higher accretion rate onto the compact object. The average X-ray spectrum is hard and highly absorbed, with a column density, NH, of 10*exp 23)/sq cm, clearly in excess of the interstellar absorption. There is no evidence for variability of the absorbing column density, except that during the eclipse, where a less absorbed X-ray spectrum is observed. A narrow Fe K-alpha emission line at 6.4 keV is viewed along the whole orbit, with an intensity which correlates with the continuum emission above 7 keV. The scattered component visible during the X-ray eclipse allowed us to directly probe the wind density at the orbital separation, resulting in rho(sub w)=7X10(exp −14) g/cubic cm. Assuming a spherical geometry for the supergiant wind, the derived wind density translates into a ratio M(sub w)/v(sub infinity) = 7X10(exp −17) Solar M/km which, assuming terminal velocities in a large range 500-3000 km/s, implies an accretion luminosity two orders of magnitude higher than that observed. As a consequence, a mechanism should be at work reducing the mass accretion rate. Different possibilities are discussed.

J164794514↗
NASA NTRS1980

Melting relations and elemental distribution of portion of the system Fe-S-Si-O to 32 KB with planetary application

The melting relations and distribution of K and Cs in portions of the system was determined at high pressures. Ferrosilite is stable as a primary phase at high pressures because of the incongruent melting of ferrosilite to quartz plus liquid and the boundary between the one and two liquid fields on the joint Fe(1-x) O-FeS-SiO2 shifts away from silica with increasing pressures. Potassium K was found to have limited solubility in metal sulfide liquids at pressures up to 45 kb. The speculation that K may dissolve significantly in metal-metal sulfide liquids after undergoing first order isomorphic transition was tested by determining the distribution of Cs between sulfide and silicate liquids as an analogy to K. At 45 kb, 1400 C and 27 kb, 1300 C only limited amounts of Cs were detected in quench sulfide liquids even at pressures beyond the isomorphic transition of Cs.

Huang, W. L.↗
NASA NTRS2010

The Evolution of Gas in Protoplanetary Systems: The Herschel GASPS Open Time Key Programme

The Gas in Protoplanetary Systems (GASPS) Open Time Key Programme for the Herschel Space Observatory will be the first extensive, systematic survey of gas in circumstellar disks over the critical transition from gas-rich protoplanetary through to gas-poor debris. The brightest spectral lines from disks lie in the far-infrared and arise from radii spanning roughly 10 to 100 AU, where giant planets are expected to form. Herschel is uniquely able to observe this wavelength regime with the sensitivity to allow a large scale survey. We will execute a 2-phase study using the PACS instrument. Phase I is a spectroscopic survey about 250 young stars for fine structure emission lines of [CII] (at 157 microns) and [OI] (at 63 microns). In Phase II, the brightest sources will be followed up with additional PACS spectroscopy ([OI] at 145 microns and some rotational lines of water). We expect that the gas mass sensitivity will be more than an order of magnitude lower than that achieved by ISO and Spitzer or expected for SOFIA. We will also measure the dust continuum to an equivalent mass sensitivity. We will observe several nearby clusters with ages from 1 to 30 Myr, encompassing a wide range of disk masses and stellar luminosities. The sample covers disk evolution from protoplanetary disks through to young debris disks, i.e. the main epoch of planet formation. With this extensive dataset, the GASPS project will: 1) trace gas and dust in the planet formation region across a large multivariate parameter space, 2) provide the first definitive measurement of the gas dissipation timescale in disks, 3) elucidate the evolutionary link between protoplanetary and debris disks, 4) investigate water abundances in the planetforming regions of disks, and 5) provide a huge database of disk observations and models with long-lasting legacy value for follow-up studies.

Roberge, A.↗
NASA NTRS2010

AIRS Water Vapor and Cloud Products Validate and Explain Recent Short Term Decreases in Global and Tropical OLR as Observed by CERES

A strong equatorial SST cooling occurred from 160E westward to 120W during the period of September 2002 through August 2010, surrounded by a weaker warming ring to the west. This is the result of a transition from a strong El Nino in late 2002 to a strong La Nina in 2008. Late 2009 is characterized by the beginning of another El Nino. Average rates of change (ARC's) in 500mb specific humidity and cloud cover are in phase with those in the Sea surface temperature (SST). In the El Nino and surrounding region causing outgoing longwave radiation (OLR), to decrease significantly near the dateline and increase in the vicinity of Indonesia. Tropical OLR ARC's in these two areas cancel each other to first order. The negative zonal mean tropical OLR ARC from a drop in equatorial OLR in region 1 from 140W to 40E. This results from increasing water vapor and cloud cover in this area during La Nina with the reverse holding during El Nino.

Susskind, Joel↗
NASA NTRS1997

Hydrogenation of interstellar molecules: a survey for methylenimine (CH2NH)

Methylenimine (CH2NH) has been convincingly detected for the first time outside the Galactic center as part of a study of the hydrogenation of interstellar molecules. We have observed transitions from energy levels up to about 100 K above the ground state in the giant molecular clouds W51, Orion KL and G34.3 + 0.15. In addition, CH2NH was found at the " radical-ion peak" on the quiescent ridge of material in the Orion molecular cloud. The abundance ratio CH2NH/HCN at the radical-ion peak agrees with the predictions of recent gas-phase chemical models. This ratio is an order of magnitude higher in the warmer cloud cores, suggesting additional production pathways for CH2NH, probably on interstellar grains.

Non-NASA Center↗
NASA NTRS1988

Modified non-linear Burgers' equations and cosmic ray shocks

A reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.

Zank, G. P.↗
NASA NTRS1986

Will cosmic strings be discovered using the Space Telescope?

Cosmic strings are topologically stable defects in the vacuum of space which may be produced by a phase transition in the early universe. Here, it is suggested that observations of very distant galaxies are a more useful means of discovering strings than quasar observations. It is argued that if there is only one string out to redshift z - about 1 the probability that it crosses a random image obtained using the Wide Field Camera (WFC) of the Space Telescope is about 0.0001. In order to discover a cosmic string the Space Telescope WFC will be required to operate almost continuously in primary and serendipity modes, and a cosmic string, if it exists, may be discovered within the first few years of operation.

Pacynski, B.↗
NASA NTRS1991

Nonlinear response of infinitely long circular cylindrical shells to subharmonic radial loads

The nonlinear response of infinitely long circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode is analyzed with the multiple-scale method. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting models. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcation. As excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddle-node bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.

Nayfeh, Ali H.↗