Subharmonic Regions Research Articles

Nonlinear vibration, stability, and bifurcation analysis of axially moving and spinning cylindrical shells

The nonlinear vibration characteristics of the rotating axially moving circular cylindrical shells in subharmonic regions are investigated in the present paper. The motion equations are carried out based on the Hamilton principle in cylindrical coordinates utilizing Donnell’s nonlinear shell theory. By introducing the suitable airy stress function, three equilibrium equations in the cylindrical coordinates are simplified into two nonlinear coupled nonhomogeneous PDEs, including a compatibility equation and the transverse motion equation. The compatibility equation solution is obtained employing the seven degrees of freedom for the flexural mode shape of the system. By implementation of the Galerkin method, the motion equation would be projected into seven nonlinear coupled nonhomogeneous ODEs. This set of equations is solved using a direct normal form method validated by the numerical method and available data. The effect of angular velocity and axial speed is investigated employing frequency and force response curves, bifurcation diagrams, time history, and the system’s phase portraits. Always axially moving and rotation speed of the system intensifies the nonlinear behavior of the frequency responses.

Nonlinear vibration analysis of an axially moving thin-walled conical shell

The purpose of the current study is the nonlinear vibration analysis of an axially moving truncated conical shell. Based on the classical linear theory of elasticity, Donnell’s nonlinear theory assumptions with Von Kármán nonlinear terms, Hamilton principle, and Galerkin method, the nonlinear motion equations of axially moving truncated conical shells are derived. Then, a set of nonlinear motion equations has solved using the normal form method in the sub-harmonic region. Also, to validate the accuracy of the mentioned method, the fourth-order Runge–Kutta technique and arclength continuation has applied as a numerical solution. Therefore, the frequency response curves are investigated using the semi-analytical method and validated using the numerical method. The results show that by increasing the velocity up to a limited value, the system’s maximum amplitude increases, and then increasing the velocity decreases the maximum amplitude. Also, the softening or hardening of nonlinear behaviour of the system’s frequency responses due to change of cone angle has investigated.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.

Lock-in regions of laminar flows over a streamwise oscillating circular cylinder

In this paper, flow over a streamwise oscillating circular cylinder is numerically simulated to examine the effects of the driving amplitude and frequency on the distribution of the lock-in regions in laminar flows. At $Re=100$, lock-in is categorized according to the spectral features of the lift coefficient as two different lock-in phenomena: harmonic and subharmonic lock-in. These lock-in phenomena are represented as maps on the driving amplitude–frequency plane, which have subharmonic lock-in regions and two harmonic lock-in regions. The frequency range of the subharmonic region is shifted to lower frequencies with increasing amplitude, and the lower boundary of this subharmonic region is successfully predicted. A symmetric harmonic region with a symmetric vortex pattern is observed in a certain velocity range for a moving cylinder. Aerodynamic features induced by different flow patterns in each region are presented on the driving amplitude–frequency plane. The lock-in region and aerodynamic features at $Re=200$ and $40$ are compared with the results for $Re=100$. A subharmonic region and two harmonic regions are observed at $Re=200$, and these show the same features as for $Re=100$ at a low driving amplitude. Lock-in at $Re=40$ also shows one subharmonic region and two harmonic regions. However, compared with the $Re=100$ case, the symmetric harmonic lock-in is dominant. The features of aerodynamic force at $Re=200$ and $40$ are represented on a force map, which shows similar characteristics in corresponding regions for the $Re=100$ case.

Study of a periodically forced magnetohydrodynamic system using Floquet analysis and nonlinear Galerkin modelling

We present a detailed study of Rayleigh–Benard magnetoconvection with periodic gravity modulation and uniform vertical magnetic field. Linear stability analysis is carried out using Floquet theory to construct the stability boundaries in order to estimate the magnitude of forcing amplitude $$\epsilon $$ required for having convection in the system for a fixed Rayleigh number Ra, wave number k and modulating frequency $$\Omega $$ . The effects of varying Prandtl number Pr and Chandrasekhar number Q on the threshold of convection are also investigated. A higher Pr value reduces the value of the threshold, whereas a higher Q value increases it. Bicritical states are also observed at which the minimum forcing amplitude needed for convection to begin occurs at two different k values in harmonic and sub-harmonic regions, respectively. We also construct a nonlinear Galerkin model and compare the results with those obtained from linear stability analysis. Two-dimensional (2D) oscillatory convection is observed at the onset, while quasiperiodic and chaotic behaviours are found at higher Ra values. 2D as well as nonlinear convective flow patterns are observed for primary and higher-order instabilities, respectively. Bifurcation diagrams with respect to different parameters such as $$\epsilon $$ , Ra and Q are provided for thorough understanding of the forced nonlinear system.

A computational model for the dynamic stabilization of Rayleigh-Bénard convection in a cubic cavity.

The dynamic stability of Rayleigh-Bénard convection with vertical vibration in a cubic container is computationally modeled. Two parametric drives are considered (sinusoidal and rectangular), as well as two thermal boundary conditions on the sidewalls (insulating and conducting). The linearized equations are solved using a spectral Galerkin method and Floquet analysis. Both the synchronous and the subharmonic regions of instability are recovered. The conditions necessary for dynamic stability are reported for a range of Rayleigh numbers from critical to 10(7) and for Prandtl numbers in the range of 0.1-7. The linear model is compared to the data set available in the literature where the performance of an inverted pulse tube cryocooler is measured.

Chaos and beyond in a water filled ultrasonic resonance system

Finite amplitude ultrasonic wave resonances in a one dimensional liquid-filled cavity are reported. The resonances are observed to include not only the expected harmonic and subharmonic signals but chaotic signals as well. The nonlinear features of this system were recently investigated and are the focus of this presentation. An ultrasonic interferometer having optical precision was constructed. The transducers having the frequency range 1-10 MHz, driven by a high power amplifier. Both an optical diffraction system and a receiving transducer were used to assess the generated resonance response in the cavity. Five regions of excitation are identified: 1. Linear region: at low intensity of the ultrasonic wave the diffraction pattern of a light beam is symmetric. 2. Nonlinear region: with increased sound amplitude the diffraction pattern becomes asymmetrical. 3. Subharmonic region: further increase of the amplitude above a threshold value leads to the generation of subharmonics. 4. Chaos: increasing the drive amplitude to a second threshold level the diffraction pattern is smeared out indicating a time-chaotic region. 5. Beyond chaos: further increase of the amplitude results again a stable diffraction pattern. A first-principle-based explanation is presented. This work is supported by the Aircraft Aging Program, at NASA Langley Research Center.

Dynamic stabilization of the Rayleigh-Bénard instability in a cubic cavity

The dynamic stability of the Rayleigh-Bénard instability with vertical vibration in a cubic container is computationally modeled. Two periodic parametric drives are considered (sinusoidal and rectangular), as well as two thermal boundary conditions on the sidewalls (insulating and conducting). The linearized equations are solved using a spectral Galerkin method and Floquet analysis. Floquet analysis recovers both the synchronous and the subharmonic regions of instability. The conditions necessary for dynamic stability are reported for Rayleigh numbers from critical to 107 and for Prandtl numbers in the range of 0.1-7.0, and the approach produces maps over a wide range of Rayleigh number and vibration parameters for stability. The linear model is compared to data set available in the literature [G. W. Swift and S. Backhaus J. Acoust. Soc. Am. 126, 2273 (2009)] where the performance of system simulating an inverted pulse tube cryocooler is measured. The relevant instability for this case is the synchronous instability. Over this limited data set, the model appears to bound the empirically observed conditions for stability, but in some cases the model would seem to predict significantly higher required periodic acceleration amplitudes that appear to have been observed by Swift. Comparison with another data set is on-going. [Research supported by the Office of Naval Research and ARL Exploratory and Foundational Research Program.]

Dynamic Stability Analysis of Periodically Time-Varying Rotor System with a Transverse Crack

Fatigue cracks may appear in horizontal rotating machinery due to periodic stresses imposed to its shaft. The investigation of stability behavior of cracked rotors can lead to proper diagnosis of machinery and to prevent possible accidents caused by the rotor failure. In this study, the dynamic stability of a rotor with a transverse crack is investigated. Models of both open and breathing cracks are developed and then used in the model of a cracked Jeffcot (de Laval) rotor. The stability of rotor motion equations represented by differential equations with periodic coefficients is investigated using Floquet theory. While both crack models show instability regions around the first un-damped frequency, sub-harmonic regions are predicted by the breathing crack models. Compared to perturbation methods frequently used to determine the stability regions, the transition matrix approach used in this study can be applied to complex models of rotors and consequently may help in the identification of cracks in rotating machinery.

Repetitive impact response of a beam structure subjected to harmonic base excitation

This paper investigates the forced response dynamics of a clamped–clamped beam to which a rigid body is attached, and in the presence of periodic or non-periodic impacts between the body and a comparatively compliant base structure. The assembly is subjected to base excitation at specified frequency and acceleration, and the potentially complex responses that occur are examined analytically. The two sets of natural frequencies and vibration modes of the beam-rigid body structure (in its in-contact state, and in its not-in-contact state), are used to treat the forced response problem through a series of algebraic mappings among those states. A modal analysis based on extended operators for the (continuous) beam and (discrete) rigid body establishes a piecewise linear state-to-state mapping for transition between the in-contact and not-in-contact conditions. The contact force, impulse, and displacement each exhibit complex response characteristics as a function of the excitation frequency. Periodic responses occurring at the excitation frequency, period-doubling bifurcations, grazing impacts, sub-harmonic regions, fractional harmonic resonances, and apparently chaotic responses each occur at various combinations of damping, excitation frequency, and contact stiffness. The results of parameter studies in structural asymmetry and the eccentricity of the contact point's location are discussed.

COMPLEXITY IN FARADAY EXPERIMENT WITH VISCOELASTIC FLUID

A systematic experimental study of the Faraday instability in viscoelastic fluid is presented. We have used a shear thinning polymer solution in which the elastic effects are predominant within our work range. We have analyzed the dependence of the threshold instability as a function of the depth in layer. Depending on the fluid layer depth and the driving frequency, harmonic or subharmonic regimes are developed. We have focused our work on the subharmonic region and temporal and spatial behaviors were analyzed. In addition, we have used the onset acceleration to estimate the rheological properties of the fluid. These predictions are supported by experimental measurements.

Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

The linear stability theory is used to investigate analytically the effects of gravity modulation on solutal convection in the mushy layer of solidifying binary alloys. The gravitational field consists of a constant part and a sinusoidally varying part, which is synonymous to a vertically oscillating mushy layer subjected to constant gravity. The linear stability results are presented for both the synchronous and subharmonic solutions. It is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. Beyond the transition point, further increases in the frequency tend to destabilize the solutal convection, but gradually. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (η 0 ) is to destabilize the solutal convection.

Subharmonic resonances and criteria for escape and chaos in a driven oscillator

Escape and chaos phenomena are investigated in the subharmonic region of frequencies of the twin-well potential Duffing system. Approximate periodic solutions are found for low order subharmonic resonances by using a near linear perturbation method. Then first and higher order instabilities of the solutions are examined. The theoretical results are applied to estimate escape regions in the parameter space of the system and compared with numerical simulations. Three scenarios of transitions between chaotic and regular motion are observed: period doubling cascade, type 3 intermittency and boundary crisis.

BIFURCATION STRUCTURE OF THE DRIVEN VAN DER POL OSCILLATOR

The bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurcation curves. The two regions in which the driving frequency ω is greater or less than the limit cycle frequency ω0 of the nondriven oscillator are considered separately. For the case ω > ω0, the subharmonic region, the extent and location of the largest Arnol'd tongues are shown, as well as the period-doubling cascades and chaotic attractors that appear within most of them. Special attention is paid to the pattern of the bifurcation curves in the transitional region between low and large dampings that is difficult to approach analytically. In the case ω < ω0, the ultraharmonic region, a recurrent pattern of the bifurcation curves is found for small values of the damping d. At medium damping the structure of the bifurcation curves becomes involved. Period-doubling sequences and chaotic attractors occur.

Nonlinear electrohydrodynamic Rayleigh–Taylor instability. Part 1. A perpendicular field in the absence of surface charges

Nonlinear electrohydrodynamic Rayleigh-Taylor instability is investigated. A charge-free surface separating two semi-infinite dielectric fluids influenced by a normal electric field is subjected to nonlinear deformations. We use the method of multiple-scale perturbations in order to obtain uniformly valid expansions near the cutoff wavenumber separating stable from unstable flows. We obtain two nonlinear Schrödinger equations by means of which we can deduce the cutoff wavenumber and analyse the stability of the system. It is found that if a finite-amplitude wave exists then its small modulation is stable. We also obtain the surface elevation for such waves. The electric field plays a dual role in the stability criterion and the dielectric constant plays a distinctive role in this analysis. If the dielectric constant of the upper fluid is smaller than that of the lower fluid the field has a destabilizing effect for large wavenumbers. For relatively smaller wavenumbers the electric field stabilizes considerable parts of the first and second subharmonic regions in the stability diagrams; a result which is in contrast with the linear theory. If the dielectric constant of the upper fluid is larger than that of the lower fluid, then the field is stabilizing for larger values of the wavenumber K′ when ρ is small (ρ is the density ratio) and destabilizing for smaller values of K′.