Mode Coalescence Research Articles

Comment on invalid JWKB unstable solutions of combined inertial-Rossby waves

In three recent papers (McKenzie, 2009, 2011a, b) on the combined theory of gravity inertial-Rossby waves on a β-plane it was shown that “local” JWKB solutions exhibit complex roots for the wave frequency if the speed of the low-frequency, long-wavelength westward propagating Rossby wave exceeded that of the higher frequency (KelvinPoincare) inertial wave speed. If the ratio of the former (Rossby) speed to the latter (inertial) speed is denoted by m (a function of β, f the Coriolis parameter and V the inertial speed) the condition for complex roots m> 1 translates into a region in the parameter space of the latitude of the β-plane and the rotational Mach/Froude number of the planet. Therefore it is tempting to associate these complex roots in this parameter space with instability of the system arising from the coalescence of modes (Akhiezer et al., 1967), or from the interaction between positive and negative energy waves (Cairns, 1971). However a note of caution was sounded in the latest paper of the series (McKenzie, 2011b) to the effect that JWKB solutions whose latitudinal wave number remained unconstrained, may not, in fact, provide an acceptable solution to the “global” problem which involves the imposition of latitudinal boundary conditions. Indeed Paldor (2010) showed that if mid-latitude JWKB type solutions are subject to, for example, rigid wall conditions in one hemisphere, the corresponding eigenvalues precluded the fulfillment of the “instability condition” m> 1, and therefore that the system was globally stable. Furthermore Paldor emphasized that in the well known case of an equatorial β-plane, the eigenvalues and eigenfunctions correspond to stable (real) values for the wave frequency. Finally, a global energy invariant for the system (Pedlosky, 1987; Paldor, 2010) demonstrates that, in the absence of shear flow or negative buoyancy, the system is stable. In summary the latitudinal dependence in the equations arising from the variation of the Coriolis force, which is an essential feature of the Rossby wave, together with appropriate boundary conditions gives rise to an eigenvalue problem instead of a purely local solution with its associated dispersion equations. This eigenvalue problem (whether the βplane is centred on mid-latitudes or at the equator) has no unstable solutions. Therefore it is clear that the claim that the system is unstable, based solely upon local JWBK solutions, made by me is invalid.

Parametric study on supersonic flutter of angle-ply laminated plates using shear deformable finite element method

The influence of fiber orientation, flow yaw angle and length-to-thickness ratio on flutter characteristics of angle-ply laminated plates in supersonic flow is studied by finite element approach. The structural model is established using the Reissner-Mindlin theory in which the transverse shear deformation is considered. The aerodynamic pressure is evaluated by the quasi-steady first-order piston theory. The equations of motion are formulated based on the principle of virtual work. With the harmonic motion assumption, the flutter boundary is determined by solving a series of complex eigenvalue problems. Numerical study shows that (1) The flutter dynamic pressure and the coalescence of flutter modes depend on fiber orientation, flow yaw angle and length-to-thickness ratio; (2) The laminated plate with all fibers aligned with the flow direction gives the highest flutter dynamic pressure, but a slight yawing of the flow from the fiber orientation results in a sharp decrease of the flutter dynamic pressure; (3) The angle-ply laminated plate with fiber orientation angle equal to flow yaw angle gives high flutter dynamic pressure, but not the maximum flutter dynamic pressure; (4) With the decrease of length-to-thickness ratio, an adverse effect due to mode transition on the flutter dynamic pressure is found.

Enhanced meta-modelling technique for analysis of mode crossing, mode veering and mode coalescence in structural dynamics

In this paper, an enhanced meta-modelling procedure for the approximation of structural eigenfrequencies and eigenvectors is introduced. The procedure allows for correct prediction of the modal parameters in case of mode crossing, veering, and coalescence phenomena that can be observed when variations of the structural parameters occur. The procedure overcomes the erroneous approximation of these phenomena which results from a direct approximation of the modal parameters. The methodology is based on the response surface approximation of the structural matrices and on the concept of modal reduction. A comparison with a direct response surface approximation of the eigenfrequencies shows a considerable improvement in the accuracy as it is presented for a finite element frame structure.

Aeroelastic Stability of Axially Moving Webs Coupled to Incompressible Flows

The aeroelastic flutter of thin flexible webs severely limits their transport speeds and consequently the machine throughputs in a variety of paper, plastics, textiles, and sheet metal industries. The aeroelastic stability of such high-speed webs is investigated using an assumed mode discretization of an axially moving, uniaxially tensioned Kirchhoff plate coupled with cross and machine direction flows of a surrounding incompressible fluid. The corresponding aerodynamic potentials are computed using finite element solutions of certain mixed boundary value problems that arise in the fluid domain. In the absence of air coupling, the cross-span mode frequencies tightly cluster together, and the web flutters via mode coalescence at supercritical transport speed. Web coupling to an initially quiescent incompressible potential flow significantly reduces the web frequencies, substantially modifies the mode shapes, and separates the frequency clusters, while only marginally affecting the flutter speed and frequency. The inclusion of machine direction base flows significantly modifies the web stability and mode shapes. Cross machine direction flows lead to the flutter with vanishing frequency of very high cross-span nodal number modes, and the unstable vibration naturally localizes at the leading free edge. These results corroborate several previous experimental results in literature and are expected to guide ongoing experiments and the design of reduced flutter web handling systems.

밀폐된 용기 안에서 동시에 회전하는 디스크의 진동과 안정성에 관한 연구

The vibration and stability of thin, flexible corotating disks in an enclosed compressible fluid is investigated analytically and compared with the results of a single rotating disk. The discretized dynamical system of the corotating disks is derived in the compact form of a classical gyroscopic system similar with a single disk. For the undamped system, coupled structure-acoustic traveling waves destabilize through mode coalescence leading to flutter instability. However, it is found that the flutter regions of the corotating disks are wider than those of a single disk. A detailed investigation of the effects of dissipation arising from acoustic or disk damping is also performed. Finally, in the presence of both acoustic and disk dampings, the instability regions are found and compared with those of a single disk. Although this study does not allow a radial clearance between the disk and the enclosure, the computational frame work of the problem can be expanded to the system having the radial clearance in an enclosure.

Coalescence of Tollmien–Schlichting and interfacial modes of instability in two-fluid flows

An extensive parametric study is performed to bring out the details of the mode coalescence phenomenon in a two-fluid boundary-layer stability problem. The solution consists of numerically solving the Orr–Sommerfeld equation. The mode coalescence occurs between the Tollmien–Schlichting (TS) mode also found in the classical hydrodynamic stability theory and the interfacial mode that exists in the presence of a viscosity stratification in two-fluid flows. It was previously shown that the coalescence occurs in the unstable part of the domain when the sheared fluid thickness is sufficiently large. It was also shown that if the sheared fluid thickness is further increased, a composite mode dominates the stability problem, which carries the characteristics of the TS mode at moderate wave numbers and the interfacial mode at low and high wave numbers. In this study, the influence of sheared fluid thickness, viscosity ratio, surface tension, density stratification, and gravity on the composite mode is further elaborated. It is demonstrated that for certain parameter combinations, the mode coalescence occurs not only in the unstable part of the domain but also in the stable part. This feature had been previously reported by other researchers by using a different solution method and is now demonstrated in the Orr–Sommerfeld context as well. Moreover, mode coalescence occurs in the presence of density stratification and gravity also.

Vibrations and stability of a flexible disk rotating in a gas-filled enclosure—Part 2: Experimental study

The vibrations and acoustic oscillations of a rotating disk coupled to surrounding fluids in an air-filled enclosure are investigated experimentally. Aeroelastically excited disk transverse vibrations are measured in a specially designed enclosure on a high-speed test stand using different disk materials. In addition, acoustic modes are monitored as a function of disk rotating speed using microphones which are installed at the top and bottom covers of the enclosure. It is shown from the acoustic pressure measurements that acoustic modes can be classified as in-plane modes that decouple from disk vibration or out-of-phase modes that couple to disk vibration. Further, it is also shown that with disk rotation both these types of acoustic modes split into forward and backward traveling waves, offering direct evidence of a rotating core in the fluid flow in the enclosure. It is confirmed from measurements of the disk vibration that disk flutter does not occur by mode coalescence but by a damping-induced instability as described in the accompanying theoretical paper. Beyond the flutter speed, it is shown that acoustic pressures oscillate severely at the frequency of disk instability. These phenomena are, for the most part, predicted by accompanying theory. Other phenomena are observed that are not predicted by current theory including moderate amplitude 1/2 rotation speed subharmonics in the disk vibration spectrum. Put together these experiments confirm several theoretical predictions, but also reveal new phenomena that can occur in enclosed rotating disk systems such as in CD/DVD ROMs, hard disk drives, and micro-gas turbines.

Vibrations and stability of a flexible disk rotating in a gas-filled enclosure—Part 1: Theoretical study

The vibrations and linear stability of a flexible disk rotating at sub- and supercritical speeds, and coupled to the acoustic oscillations of the surrounding fluid are investigated theoretically. The surrounding fluid is contained in a cylindrical enclosure. The coupled gyroscopic system equations are formulated using a Kirchhoff plate model for the disk, and the wave equation for the compressible fluid. The formulation includes systematically the effects of geometric perturbations such as radial clearances and asymmetric disk positioning, as well as bulk rotating fluid flows driven by fluid viscosity and disk rotation. A rigorous spatial discretization of this coupled gyroscopic system leads to a singular generalized non-Hermitian eigenvalue problem for which a special computational treatment is presented. The underlying physics of acoustic–structure coupling in the presence of these effects is complex—acoustic oscillations of the fluid above and below the disk couple through disk vibrations and through the radial clearance, while the bulk fluid rotation splits the acoustic modes into forward and backward traveling waves. Flutter instabilities arising from acoustic–structure mode coalescence and various damping mechanisms are discussed. The predictions reveal significant influences of radial clearance, asymmetric disk positioning, and bulk fluid rotation on the vibration and acoustic characteristics of the system that are likely to be observed in experiments, and in practical applications such as in CD/DVD drives and hard disk drives.

Supersonic flutter suppression of piezolaminated cylindrical panels based on multifield layerwise theory

For the aerothermoelastic analysis of cylindrical piezolaminated shells, geometrically nonlinear finite elements based on the multifield layerwise theory have been developed. Present multifield layerwise theory describes zigzag displacement, thermal and electric fields providing a more realistic multiphysical description of fully and partially piezolaminated panels. By applying a Hans Krumhaar's supersonic piston theory, supersonic flutter analyses are performed for the cylindrical piezolaminted shells subject to thermal and piezoelectric loads. The possibility to increase flutter boundary and reduce thermoelastic deformations of piezolaminated panels is examined using piezoelectric actuation. Results show that active piezoelectric actuation can effectively increase the critical aerodynamic pressure by retarding the coalescence of flutter modes and compensating thermal stresses.

Aeroelastic Flutter Mechanisms of a Flexible Disk Rotating in an Enclosed Compressible Fluid

The aeroelastic stability of a thin, flexible disk rotating in an enclosed compressible fluid is investigated analytically through a discretization of the field equations of a rotating Kirchhoff plate coupled to the acoustic oscillations of the surrounding fluid. The discretization procedure exploits Green’s theorem and exposes two different gyroscopic effects underpinning the coupled system dynamics: One describes the gyroscopic coupling between the disk and acoustic oscillations, and another arises from the disk rotation. The discretized dynamical system is cast in the compact form of a classical gyroscopic system and acoustic and disk mode coupling rules are derived. For the undamped system, coupled structure-acoustic traveling waves can destabilize through mode coalescence leading to flutter instability. A detailed investigation of the effects of dissipation arising from acoustic and disk damping predicts previously unknown instability mechanisms for this system. The results are expected to be relevant for the design of high speed, low vibration, low-noise hard disk drives, and optical data storage systems.

Flow Angle, Temperature, and Aerodynamic Damping on Supersonic Panel Flutter Stability Boundary

The effects of flow yaw angle, temperature, and aerodynamic damping on supersonic flutter of plates are investigated. Quasisteady, first-order piston theory is employed for formulation of aerodynamic forces. The von Karman large-deflection plate theory is adapted for the aerothermal deflection. Two types of thermal effects are considered: 1) plate expansion by uniform temperature and 2) thermal moment induced by temperature gradient across the plate thickness. A finite element modal formulation and a two-step procedure are presented for the predictions of stability boundaries and nonlinear aerothermal deflection and shown to be efficient in solution. Results have shown that flow angle has lesser effect on stability boundaries as compared with temperature for isotropic square plates. However, both flow angle and temperature have a large influence on stability boundaries for rectangular isotropic and laminated composite plates. The presence of the ripple characteristics of stability boundaries for composite plates caused by the frequency coalescence of higher modes and the smaller effect of aerodynamic damping is investigated. The stabilization effects on panel motions induced by variations of flow angle, temperature, and aerodynamic damping are discussed.

Floquet Modal Analysis of a Teetered-Rotor Wind Turbine

This paper examines the operating modes of a two-bladed wind turbine structural model. Because of the gyroscopic asymmetry of its rotor, this turbine’s dynamics can be quite distinct from that of a turbine with three or more blades. This asymmetry leads to system equations with periodic coefficients that must be solved by the Floquet approach to extract the correct modal parameters. A discussion of results is presented for a series of simple models with increasing complexity. We begin with a single-degree-of-freedom system and progress to a model with seven degrees-of-freedom: tower fore-aft bending, tower lateral bending, tower twist, nacelle yaw, hub teeter, and flapwise bending of each blade. Results illustrate how the turbine modes become more dominated by the centrifugal and gyroscopic effects as the rotor speed increases. Parametric studies are performed by varying precone angle, teeter stiffness, yaw stiffness, teeter damping, and yaw damping properties. Under certain levels of yaw stiffness or damping, the gyroscopic coupling may cause yaw and teeter mode coalescence, resulting in self-excited dynamic instabilities. Teeter damping is the only parameter found to strictly stabilize the turbine model.

Active control and a simple estimation method for the onset of Rayleigh-Bénard convection in temperature-dependent-viscosity liquids

Active feedback control for the onset of Rayleigh-Benard Convection in temperature-dependent-viscosity liquids is investigated. In this paper, three major problems are addressed: (1) The results of Tang-Bau control are improved by considering the effects of temperature-dependent viscosity; (2) A more efficient two-plate control strategy is presented. A phenomenon of coalescence of the unstable modes is observed as the controller gain is large enough; (3) A simple way to estimate the critical Rayleigh number under the effects of temperature-dependent viscosity is described. Numerical results show that the effects of temperature-dependent viscosity on the critical Rayleigh number should be taken into account in some cases and the onset of Rayleigh-Benard convection can be effectively delayed or advanced by the active feedback control strategies studied here.

Damage detection in composite materials using frequency response methods

Cost-effective and reliable damage detection is critical for the utilization of composite materials. This paper presents part of an experimental and analytical survey of candidate methods for the in situ detection of damage in composite materials. The experimental results are presented for the application of modal analysis techniques applied to graphite/epoxy specimens containing representative damage modes. Changes in natural frequencies and modes were found using a laser vibrometer, and 2-D finite element models were created for comparison with the experimental results. The models accurately predicted the response of the specimens at low frequencies, but coalescence of higher frequency modes makes mode-dependant damage detection difficult for structural applications. The frequency response method was found to be reliable for detecting even small amounts of damage in a simple composite structure, however the potentially important information about damage type, size, location and orientation were lost using this method since several combinations of these variables can yield identical response signatures.

On the frontal condition for finite amplitude α2Ω-dynamo wave trains in stellar shells

In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443–1481, 1999) (BKS) investigated finite amplitude αΩ-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from α-quenching. There asymptotic solutions were developed based upon the small aspect ratio ε of the shell. Specifically, as a consequence of a prescribed latitudinally dependent α-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude θF. Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point θf; this is essentially the Dee–Langer criterion, which determines both the wave frequency and front location. Recently, Griffiths et al. (Geophys. Astrophys. Fluid Dynam. 94, 85–133, 2001) (GBSK) obtained solutions to the α2Ω-extension of the model by application of the Dee—Langer criterion. Its justification depends on the linear solution in a narrow layer ahead of the front on the short O(θf—θF) length scale; here conventional WKB-theory, used to describe the solution elsewhere, is inadequate because of mode coalescence. This becomes a highly sensitive issue, when considering the transition from the linear solution, which occurs when the dynamo number D takes its critical value D c corresponding to the onset of kinematic dynamo action, to the fully nonlinear solutions, for which the Dee—Langer criterion pertains. In this paper we investigate the nature of the narrow layer for α2Ω-dynamos in the limit of relatively small but finite α-effect Reynolds numbers R α, explicitly ε½ ≪ R 2 α ≪ 1. Though there is a multiplicity of solutions, our results show that the space occupied by the corresponding wave train is generally maximised by a solution with θf—θF small; such solutions are preferred as evinced by numerical simulations. This feature justifies the application by GBSK of the Dee—Langer criterion for all D down to the minimum D min that the condition admits. Significantly, the frontal solutions are subcritical in the sense that |D min| ≤ |D c|; equality occurs as the α-effect Reynolds number tends to zero. We demonstrate that the critical linear solution is not connected by any parameter track to the preferred nonlinear solution associated with D min. By implication, a complicated bifurcation sequence is required to make the connection between the linear and nonlinear states. This feature is in stark contrast to the corresponding results for αΩ-dynamos obtained by BKS valid in the limit R 2 α ≪ ε½, which, though exhibiting a weak subcriticality, showed that the connection follows a clearly identifiable nonbifurcating track.

Mode coalescence in a two-fluid boundary-layer stability problem

The triple-deck analysis of a two-fluid boundary layer stability at high Reynolds numbers in Timoshin [J. Fluid Mech. 353, 163 (1997)] is extended to include broader parameter variations, most notably variations in the scaled film thickness. The two underlying instabilities, the Tollmien–Schlichting and interfacial waves, are shown to have points of mode coalescence located in the stable or unstable part of the spectrum. An interpretation of the instability mechanisms operational in two-fluid triple-deck flows is proposed in terms of the disturbance vorticity equation.

Kelvin-Helmholtz driven modes of the magnetosphere

Ultralow frequency (ULF) waves in the magnetosphere are thought to be driven by disturbances of the magnetopause caused by the flow in the magnetosheath. In this paper a model showing how the trapping and excitation of these modes depends upon the shear flow and propagation angle is presented. The ideal magnetohydrodynamics (MHD) equations are used and the perturbations are assumed to be linear. A bounded, uniform magnetospheric cavity, with a finite plasma beta, separated by a vortex sheet from a semi-infinite, field-free, flowing magnetosheath is considered. It is shown that the bounded model allows the trapping and excitation of both fast and slow cavity modes, and that unstable surface modes may also exist. Slow surface modes are unstable only for a small interval of flow speed, becoming fast surface modes for higher flows. Slow cavity modes have small growth rates and are unlikely to be significant observationally. It is shown that fast modes propagating quasiparallel to the flow may be excited for realistic flow speeds, but that for nonparallel modes, much higher flows are required. Finally, an exact method for predicting the onset of instability for fast modes is derived and is shown to occur at the coalescence of modes of opposite energy.

Flutter Control of Wing Boxes Using Piezoelectric Actuators

Abstract : This paper examines the use of piezoelectric actuators to control supersonic flutter of wing boxes. Aluminum built-up wing boxes are used as examples to analyze the free vibration, aeroelastic, and control concepts associated with flutter control. Finite elements are used to calculate deflections due to input forces, the member stresses and strains, natural frequencies, and mode shapes. Linear strip theory with steady aerodynamics are applied to find the frequency coalescence of modes indicating flutter. The variables of interest are the skin, web, and rib thicknesses associated with torsional rigidity, and the spar cap and vertical post areas associated with bending rigidity. Piezoelectric actuators are implemented in a configuration which generates torsional control of the wing box. Pole assignment concepts are applied to change the free vibration frequencies. A parametric study changing the free vibration frequencies using piezoelectric actuators is conducted to determine which thicknesses of skins, webs, and ribs will meet a specified flutter requirement. The addition of piezoelectric actuators will allow the flutter requirements to be met at smaller thicknesses of skins, webs, and ribs so that the overall weight of the wing box, including actuators, is decreased.

Crack coalescence in a rock-like material containing two cracks

Abstract This paper investigates the pattern of crack coalescence and strength of a sandstone-like material containing two parallel inclined frictional cracks under uniaxial compression, with changing values of inclination of preexisting cracks α , bridge angle β (inclination between the inner tips of the two preexisting cracks), and the frictional coefficient μ on the surfaces of the preexisting cracks. Three main modes of crack coalescence are observed: the shear (S) mode (shear cracking between the two preexisting cracks); the mixed shear/tensile (M) mode (propagation of both wing and shear cracks within the bridge area); and the wing tensile (W) mode (coalescence of wing cracks from the tips of the preexisting cracks). The M-mode and W-mode of crack coalescence can further be divided into two and six types, respectively. Simple regime classifications of coalescence in the α – β space are proposed for different values of μ (=0.6, 0.7 and 0.9). In general, the S-mode mainly occurs when α=β or when β β *( α , μ )=a−b α , with both a and b depending on μ ; the M-mode dominates when β L > β > β *( α , μ ) (where β L ≈82.5°); and the W-mode is only observed when β > β L . However, more experiments are still required to refine the classification. The observed peak strength, in general, increases with μ . Our results show that the peak strength predicted by the Ashby and Hallam (1986) model basically agrees with experiments. A minimum occurs at about α =65° when the peak strength is plotted against α . For α >45°, the peak strength is essentially independent of the bridge angle β .

Nonlinear analysis of instability produced by linear mode coupling

We consider a system of two first-order complex partial differential equations with cubic nonlinear terms, which is a generic asymptotic model for wave envelopes in the situation when a linear instability is produced by mode coalescence (intersection of the dispersion curves) in the presence of a weak linear coupling between the two modes. Analytical considerations produce a simple condition for onset of collapse in this system, in the form of an inequality on the coefficients of the nonlinear terms. Exactly the same inequality turns out to determine the modulational stability of the continuous wave solutions with the wave number which corresponds to the center of the linear instability band. When collapse is absent, the continuous wave solutions are subject only to a long-wave modulational instability which can be eliminated by appropriate periodic boundary conditions, but in the opposite case modulational instability extends to all wave numbers. Numerical simulations demonstrate that there are two basic scenarios of principal interest. Either collapse occurs (exactly when the collapse condition is satisfied), or otherwise an initial periodic wave breaks down due to successive sideband instabilities into an apparently chaotic state. The predicted behaviors may be expected as a result of an interface instability in a number of hydrodynamic systems.