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The nonlinear equations of flap-lag coupled with oscillation of a torsionally rigid blade are derived. The solution is established by a simplified asymptotic expansion procedure of multiple time scales. The region of stability and limit cycle oscillation and its comparison with numerical results are presented.
The pointing stability of the low-cost large space telescope (LST) system was investigated. The low-cost LST is characterized by the use of reaction wheels for the generation of control torques. Because of the critical requirement on the pointing accuracy of the LST, the nonlinear frictional characteristics of the bearings of the reaction wheels were studied which can cause limit cycles in a closed-loop system. Another possible source of pointing error in the LST is due to the effect of quantization and sensor noise. Since the LST is a digital system, digital to analog and analog to digital converters and sensors for positional and rate feedbacks were used. Sensor noise and amplitude quantization causes pointing error in the LST and, in addition, quantization is a nonlinear phenomenon which can cause self-sustained oscillations in the closed-loop system. The dynamic modeling of the single-axis LST is described, and several methods of evaluating the attitude error of the digital LST due to quantization and noise inputs are given.
An approximate theoretical method is presented which determined the limit cycle behavior of a helicopter model which has one or two nonlinear dampers. The relationship during unstable ground resonance oscillations between lagging motion of the blades and fuselage motion is discussed. An experiment was carried out on using a helicopter scale model. The experimental results agree with those of the theoretical analysis.
For steady multidimesional convection, the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme has several attractive properties. However, for highly convective simulation of step profiles, QUICK produces unphysical overshoots and a few oscillations, and this may cause serious problems in nonlinear flows. Fortunately, it is possible to modify the convective flux by writing the normalized convected control-volume face value as a function of the normalized adjacent upstream node value, developing criteria for monotonic resolution without sacrificing formal accuracy. This results in a nonlinear functional relationship between the normalized variables, whereas standard methods are all linear in this sense. The resulting Simple High Accuracy Resolution Program (SHARP) can be applied to steady multidimensional flows containing thin shear or mixing layers, shock waves, and other frontal phenomena. This represents a significant advance in modeling highly convective flows of engineering and geophysical importance. SHARP is based on an explicit, conservative, control-volume flux formation, equally applicable to one, two, or three dimensional elliptic, parabolic, hyperbolic, or mixed-flow regimes. Results are given for the bench-mark purely convective first-order results and the nonmonotonic predictions of second- and third-order upwinding.
For steady multidimensional convection, the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme has several attractive properties. However, for highly convective simulation of step profiles, QUICK produces unphysical overshoots and a few oscillations, and this may cause serious problems in nonlinear flows. Fortunately, it is possible to modify the convective flux by writing the normalized convected control-volume face value as a function of the normalized adjacent upstream node value, developing criteria for monotonic resolution without sacrificing formal accuracy. This results in a nonlinear functional relationship between the normalized variables, whereas standard methods are all linear in this sense. The resulting Simple High Accuracy Resolution Program (SHARP) can be applied to steady multidimensional flows containing thin shear or mixing layers, shock waves, and other frontal phenomena. This represents a significant advance in modeling highly convective flows of engineering and geophysical importance. SHARP is based on an explicit, conservative, control-volume flux formation, equally applicable to one, two, or three dimensional elliptic, parabolic, hyperbolic, or mixed-flow regimes. Results are given for the bench-mark purely convective first-order results and the nonmonotonic predictions of second- and third-order upwinding.
The present use of a numerical model developed for the prediction of high-frequency combustion stabilities in liquid propellant rocket engines focuses on (1) the overall behavior of nonlinear combustion instabilities (2) the effects of acoustic oscillations on the fuel-droplet vaporization and combustion process in stable and unstable engine operating conditions, oscillating flowfields, and liquid-fuel trajectories during combustion instability, and (3) the effects of such design parameters as inlet boundary conditions, initial spray conditions, and baffle length. The numerical model has yielded predictions of the tangential-mode combustion instability; baffle length and droplet size variations are noted to have significant effects on engine stability.
Plasma loops in solar active regions have been observed in EUV and soft X-rays for decades. Their formation mechanism and properties, however, are still not fully understood. Predictions by early models, based on 1D hydrostatic equilibria with uniform plasma heating, are not consistent with high-resolution measurements. In this Letter, we demonstrate, via 3D simulations, that a class of heating models can lead to the dynamic formation of plasma loops provided the plasma is heated sufficiently to match SXT soft X-ray measurements. We show that individual flux tubes in a 3D magnetic structure tend to stand out against their neighbors. The loops have large aspect ratios and nearly uniform cross sections in the corona, similar to those observed by EIT and TRACE. The coronal EUV emission from these thermally unstable solutions is roughly consistent with EIT measurements. The solution oscillates in time through a large-amplitude, nonlinear cycle, leading to repeated brightening and fading of the loops.
We analyze the effect of a highly dispersive element placed inside a modulated optical cavity on the frequency and amplitude of the output modulation to determine the conditions for enhanced gyroscopic sensitivities. The element is treated as both a phase and amplitude filter, and the time-dependence of the cavity field is considered. Both atomic gases (two-level and multi-level) and optical resonators (single and coupled) are considered and compared as dispersive elements. We find that it is possible to simultaneously enhance the gyro scale factor sensitivity and suppress the dead band by using an element with anomalous dispersion that has greater loss at the carrier frequency than at the side-band frequencies, i.e., an element that simultaneously pushes and intensifies the perturbed cavity modes, e.g. a two-level absorber or an under-coupled optical resonator. The sensitivity enhancement is inversely proportional to the effective group index, becoming infinite at a group index of zero. However, the number of round trips required to reach a steady-state also becomes infinite when the group index is zero (or two). For even larger dispersions a steady-state cannot be achieved, and nonlinear dynamic effects such as bistability and periodic oscillations are predicted in the gyro response.
Conditions of self-sustained oscillations in a two-axis model of the nonlinear LST system are studied. The describing function of the CMG frictional nonlinearity of the LST system is used for the analysis, as well as continuous-data and discrete-data models of the simplified LST control system. A numerical-iterative method is described for the analysis of the two-axis system. Approximation methods and the direct plotting of the stability equation are implemented in the study. It is shown that although the dynamics of the two axes are identical, the amplitudes of self-sustained oscillations in the two axes may in principle be different. Analysis shows that the LST systems are of equal amplitudes but with 180-degree phase shift.
Ulysses has shown that fast solar wind is extremely smooth, with a variance of less than 5%, in contrast to slow wind with a variance of approximately 30%. Now UVCS has produced the surprising result that the flow speed within coronal holes, the source of fast wind, is not at all smooth. Specifically, Giordano et al. (ApJ, v531, L79-L82, 2000) report that at 1.7 R(sub SUN) the interplume flow speed is typically more than twice the plume flow speed. Other less direct evidence supports this same result, with speeds from less than 300 to over 1000 km/s reported at approximately 5 R(sub SUN). This presents the paradox of how strongly differing plume and interplume flow speeds can exist near the Sun and be absent far from the Sun. The only answer is that plume and interplume material or momentum must be strongly mixed and that the mixing must occur mainly inside 0.3 AU to be consistent with Helios observations of smooth fast wind. Pressure balance structures (PBSs) and He abundance anomalies (Reisenfeld, et al., GRL, v26(13), 1805-1808, 1999) have been identified as interplanetary remnants of plumes, implying momentum mixing is the dominant coronal process. One possible source for plume/interplume momentum mixing is MHD Kelvin-Helmholtz (KH) shear instabilities occurring on the velocity shear interfaces. The velocity shear is a source of free energy and KH fluctuations could, through nonlinear cascade, provide the forcing required for the plasma oscillations (Cranmer, ApJ, v532, 1197-1208, 2000) reported to exist in coronal holes. The physical properties in coronal holes are now sufficiently well known that we can show plume/interplume shear interfaces become unstable to the KH instability at 5 - 10 R(sub SUN). The KH dispersion relation can be used to analyze marginal stability, the most unstable wavelengths, and linear growth rates. Numerical simulations can be used to verify results from the linear analysis and study the nonlinear development of KH modes. Here we will describe the marginal stability criterion for the KH instability, how and where this condition is met in the corona, and the possible character of the resultant fluctuations.
A simple physical interpretation is advanced by means of an analysis of the shock cone in the accretion flows past a compact object and with an examination of the accretion-line stability analyses. The stability of the conical shock is examined against small angular deflections with attention given to several simplifying assumptions. A line instability is identified in the Bondi-Hoyle accretion flows that leads to the formation of a large opening-angle shock. When the opening angle becomes large the instability becomes irregular oscillation. The analytical methodology is compared to previous numerical configurations that demonstrate different shock morphologies. The Bondi-Hoyle accretion onto a compact object is concluded to generate a range of nonlinear instabilities in both homogeneous and inhomogeneous cases with a quasiperiodic oscillation in the linear regime.
A test technique and data analysis method has been developed for determining the longitudinal aerodynamic characteristics from free-flight tests of rocket-propelled models. The technique makes use of accelerometers and an angle-of-attack indicator to permit instantaneous measurements of lift, drag, and pitching moments. The data, obtained during transient oscillations resulting from control-surface disturbances, are analyzed by essentially nonlinear direct methods (such as cross plots of the variation of lift coefficient with angle of attack) and by linear indirect methods by using the equations of motion for a transient oscillation. The analysis procedure has been set forth in some detail and the feasibility of the method has been demonstrated by data measured through the transonic speed range on several airplane configurations. It was shown that the flight conditions and dynamic similitude factors for the tests described were reasonably close to typical full-scale airplane conditions.
Abstract Extreme El Niño events have outsized global impacts and control the El Niño Southern Oscillation (ENSO) warm/cold phases asymmetries. Yet, a consensus regarding the relative contributions of atmospheric and oceanic nonlinearities to their genesis remains elusive. Here, we isolate the contribution of oceanic nonlinearities by conducting paired experiments forced with opposite wind stress anomalies in an oceanic general circulation model, which realistically simulates extreme El Niño events and oceanic nonlinearities thought to contribute to ENSO skewness (Tropical Instability Waves (TIWs), Nonlinear Dynamical Heating (NDH)). Our findings indicate a weak contribution of oceanic nonlinearities to extreme El Niño events in the eastern Pacific, owing to compensatory effects between lateral (NDH and TIWs) and vertical processes. These results hold across different vertical mixing schemes and modifications of the upper‐ocean heat budget mixed layer criterion. Our study reinforces previous research underscoring the pivotal role of atmospheric nonlinearities in shaping extreme El Niño events.
A two-dimensional nonlinear integrable system was experimentally demonstrated at the Fermilab Integrable Optics Test Accelerator. The system was implemented by inserting a special nonlinear magnet in a conventional accelerator lattice. We characterized the system by measuring lifetimes, transverse profiles and transverse oscillation frequencies of the 150-MeV electron beam as a function of the strength of the nonlinear insert. The measured shift of the working point and the amplitude-dependent detuning were consistent with theoretical predictions. We also observed the predicted bifurcation of the stable closed orbit. A striking consequence of the system's implementation was the possibility to operate the storage ring with integer tunes without lifetime degradation. This research opens up novel ways to design particle accelerators and to stabilize particle beams.
For Courant numbers larger than one and cell Reynolds numbers larger than two, oscillations and in some cases instabilities are typically found with implicit numerical solutions of the fluid dynamics equations. This behavior has sometimes been associated with the loss of diagonal dominance of the coefficient matrix. It is shown that these problems can be related to the choice of the spatial differences, with the resulting instability related to aliasing or nonlinear interaction. Appropriate filtering can reduce the intensity of these oscillations and possibly eliminate the instability. These filtering procedures are equivalent to a weighted average of conservation and nonconservation differencing. The entire spectrum of filtered equations retains a three point character as well as second order spatial accuracy. Burgers equation was considered as a model.
For Courant numbers larger than one and cell Reynolds numbers larger than two, oscillations and in some cases instabilities are typically found with implicit numerical solutions of the fluid dynamics equations. This behavior has sometimes been associated with the loss of diagonal dominance of the coefficient matrix. It is shown here that these problems can in fact be related to the choice of the spatial differences, with the resulting instability related to aliasing or nonlinear interaction. Appropriate 'filtering' can reduce the intensity of these oscillations and in some cases possibly eliminate the instability. These filtering procedures are equivalent to a weighted average of conservation and non-conservation differencing. The entire spectrum of filtered equations retains a three-point character as well as second-order spatial accuracy. Burgers equation has been considered as a model. Several filters are examined in detail, and smooth solutions have been obtained for extremely large cell Reynolds numbers.
To facilitate analysis, ALS systems are often assumed to be linear and time invariant, but they usually have important nonlinear and dynamic aspects. Nonlinear dynamic behavior can be caused by time varying inputs, changes in system parameters, nonlinear system functions, closed loop feedback delays, and limits on buffer storage or processing rates. Dynamic models are usually cataloged according to the number of state variables. The simplest dynamic models are linear, using only integration, multiplication, addition, and subtraction of the state variables. A general linear model with only two state variables can produce all the possible dynamic behavior of linear systems with many state variables, including stability, oscillation, or exponential growth and decay. Linear systems can be described using mathematical analysis. Nonlinear dynamics can be fully explored only by computer simulations of models. Unexpected behavior is produced by simple models having only two or three state variables with simple mathematical relations between them. Closed loop feedback delays are a major source of system instability. Exceeding limits on buffer storage or processing rates forces systems to change operating mode. Different equilibrium points may be reached from different initial conditions. Instead of one stable equilibrium point, the system may have several equilibrium points, oscillate at different frequencies, or even behave chaotically, depending on the system inputs and initial conditions. The frequency spectrum of an output oscillation may contain harmonics and the sums and differences of input frequencies, but it may also contain a stable limit cycle oscillation not related to input frequencies. We must investigate the nonlinear dynamic aspects of advanced life support systems to understand and counter undesirable behavior.