Finite-amplitude Oscillations Research Articles

Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation

This article introduces a new, physically-based Lyapunov function definition that can be used as acoustic energy for finite-amplitude pressure oscillations in reacting systems, especially in combustion chambers. Reacting flow is seen as an open, non-equilibrium (parameters are distributed unevenly locally) thermodynamic system. This Lyapunov function is defined as the maximum mechanical work that could be extracted by a heat engine from the studied system if this system were disconnected from the inlet and outlet and from any other surrounding environment and the engine could transfer to the surrounding environment only mechanical work.

Non-linear finite-amplitude oscillations of the large beam arrays oscillating in viscous fluids

Over the past decade, several studies have been conducted on a single and multiple oscillating thin cantilever beams in an unbounded viscous fluid. With an increase in the applications of large array oscillators in a fluid environment for fields like medicine, biology, and energy harvesting devices, it is crucial to understand the nature of the surrounding fluid dynamics. In this present study, we perform a two-dimensional computational fluid dynamics (CFD) analysis of an array of beams oscillating in an unbounded viscous fluid. The two-dimensional Navier Stokes and continuity equations are solved to investigate the hydrodynamic forces exerted on the array members from interaction with the fluid environment. A complex hydrodynamic function is proposed here to represent the distributed hydrodynamic loading experienced by the oscillating beams. Results suggest that there is an increase in viscous damping with an increase in the size of the array. In addition, the nonlinearities become dominant when an array of beams is subjected to large amplitude oscillations. The number of beams in an array determines the overall hydrodynamics and the array effect. CFD analysis can predict the non-linearities unlike boundary integral method (BIM) approach, which is limited for low amplitudes. The results from the full Navier–Stokes simulations compared favorably with results using the BIM for the time-harmonic linearized Stokes equations.

Causation of Supersonic Limit Cycle Oscillations in Atmospheric Entry Vehicles

A successful atmospheric entry vehicle (AEV) design demonstrates manageable aerodynamic heating, bearable structural load, smooth deceleration, and intended trajectory during a descent into the atmosphere. Interestingly, the supersonic regime of the AEV manifests limit cycle oscillations (LCO) that restrict the maneuver potential and deployment of the drag chute. In this paper, efforts are made numerically and analytically to link the causation of LCO with external geometric variables of AEV. Computational fluid dynamics is used to calculate damping derivatives. The numerical results are validated with the Orion Crew Exploration Vehicle. The multiple time scales method, which belongs to the class of perturbation methods, is explored to develop an approximate closed-form solution of the nonlinear dynamic behavior of AEV. The analytical solution identifies that higher-order nonlinearities associated with pitch damping and static lift govern the onset of LCO. Finally, a parametric interaction study is carried out to determine the effect of two design variables, apex angle and length, on the vehicle’s dynamic stability. The mean data values from the main geometric effects plot show the condition of finite-amplitude oscillations. The results indicate that variation in these two parameters significantly impacts the magnitude of identified higher-order nonlinearities.

Computational fluid-structure interaction of a restrained ogive-cylindrical body with a blunt elliptical base at a high incidence

The fluid-structure interaction of an elastically restrained inclined tangent ogive-cylindrical body with a blunt elliptical base is investigated numerically. The flow is three-dimensional, compressible, and laminar, and the slender body is allowed to yaw at high incidence. The resulting response exhibits an intricate bifurcation structure that includes bistable periodic (finite amplitude) and nonstationary (small amplitude) limit cycles for moderate angles of attack, and nonstationary (finite amplitude) oscillations for high angles of attack.

On the Finite Amplitude Oscillations in a Viscous Two-Dimensional Fluid

On the Finite Amplitude Oscillations in a Viscous Two-Dimensional Fluid

Suppression of liquid surface instability induced by finite-amplitude oscillation in liquid piston Stirling engine

This study addresses a problem introduced by finite-amplitude oscillations in a liquid piston thermoacoustic Stirling engine and proposes methods to overcome it. After the engine was operated by external heat, the liquid piston oscillated, keeping a flat surface when the amplitude was sufficiently low. However, it became unstable to cause ejection of droplets and formation of air bubbles because the acceleration amplitude exceeded gravitational acceleration. Two methods were proposed to recover the stable liquid surface: one method was the enlargement of the cross-sectional area of the tube around the liquid surface and the other method was the use of submerged polyethylene floats. These methods successfully suppressed the instability of the liquid piston Stirling engine and increased the pressure amplitude by almost twice of the original engine. In addition, adjusting the liquid filling level resulted in the ratio of pressure amplitude over the mean pressure reaching 18% when the acceleration amplitude was elevated to 36m/s2 by the installation of floats. The high-pressure amplitude is an advantage of a liquid piston thermoacoustic Stirling engine over thermoacoustic engines that use only gas as the working fluid.

Analysis of degenerate mechanisms triggering finite-amplitude thermo-acoustic oscillations in annular combustors

A simplified model is introduced to study finite-amplitude thermo-acoustic oscillations in $N$-periodic annular combustion devices. Such oscillations yield undesirable effects and can be triggered by a positive feedback between heat-release and pressure fluctuations. The proposed model, comprising the governing equations linearized in the acoustic limit, and with each burner modelled as a one-dimensional system with acoustic damping and a compact heat source, is used to study the instability caused by cross-sector coupling. The coupling between the sectors is included by solving the one-dimensional acoustic jump conditions at the locations where the burners are coupled to the annular chambers of the combustion device. The analysis takes advantage of the block-circulant structure of the underlying stability equations to develop an efficient methodology to describe the onset of azimuthally synchronized motion. A modal analysis reveals the dominance of global instabilities (encompassing the large-scale dynamics of the entire system), while a non-modal analysis reveals a strong response to harmonic excitation at forcing frequencies far from the eigenfrequencies, when the overall system is linearly stable. In all presented cases, large-scale, azimuthally synchronized (coupled) motion is observed. The relevance of the non-modal response is further emphasized by demonstrating the subcritical nature of the system’s Hopf point via an asymptotic expansion of a nonlinear model representing the compact heat source within each burner.

Controlling self-excited vibration using acceleration feedback with time-delay

This paper investigates the control of vibration of a self-excited system, namely Rayleigh Oscillator, by using Acceleration Feedback Control method. In this control scheme, acceleration of the vibrating system is fed back to a second-order compensator and the control force is produced by amplifying the signal obtained from the compensator. Linear and non-linear stability analyses are performed. Linear stability analysis is used to obtain the stability regions and the optimal system parameters. Non-linear analysis is performed using Describing Function method. Acceleration feedback control is found to be effective in controlling the self-excited vibration. The presence of time-delay is also studied in this paper. It is observed that the presence of uncertain time-delay in the feedback loop can be detrimental. The optimal system (optimized for no delay case) can result in instability of the static equilibrium leading to finite amplitude oscillation. However, one can improve the situation by increasing the loop-gain. In order to circumvent this problem, it is proposed that a pre-determined time-delay may be introduced in the feedback circuit and the control parameters are re-optimized considering this time-delay. As a result, the system equilibrium can be stabilized even in the presence of time-delay. The results of theoretical analysis are validated with the simulation results performed in MATLAB Simulink.

Theory of Weakly Exothermic Oblique Detonations

A simplified formulation, based on treating the ratio of the heat release to the postshock thermal enthalpy as a small parameter, accommodating arbitrary chemistry descriptions, is shown to reproduce computationally the same variety of phenomena as more complex formulations for oblique detonations with supersonic postshock flow. The resulting small relative variations of velocity and thermodynamic properties across the reaction region are described by linearized Euler equations written in characteristic form, supplemented by the linearized Rankine–Hugoniot jump conditions across the leading shock. The simplified formulation is used to analyze the interaction of the oblique detonation with a weak vortex sheet, for an Arrhenius irreversible reaction with an activation energy large compared with the postshock thermal enthalpy. The analysis reveals that, as , the product of the activation energy and the heat release divided by the square of the postshock thermal enthalpy, increases through values of order unity, decaying spatial oscillations, found for small values, are replaced by persistent nonlinear oscillations of finite amplitude for larger values. Beyond a critical value of the growth of the oscillation amplitude leads to the development of a singularity at the shock, an explosion, consistent with the formation of a triple point. Many related problems can be clarified with this formulation.

Bubble shape oscillations of finite amplitude

Shape oscillations arising from the spherical instability of an oscillating bubble can be sustained in a stationary acoustic field. Describing such a steady state requires that nonlinear saturation effects are accounted for to counteract the natural exponential growth of the instability. In this paper, we analyse the establishment of finite-amplitude bubble shape oscillations as a consequence of nonlinear interactions between spherical and non-spherical modes. The set of coupled dynamical equations describing the volume pulsation and the shape oscillations is solved using a perturbation technique based on the Krylov–Bogoliubov method of averaging. A set of first-order differential equations governing the slowly varying amplitudes and phases of the different modes allows us to reproduce the exponential growth and subsequent nonlinear saturation of the most unstable, parametrically excited, shape mode. Solving these equations for steady-state conditions leads to analytical expressions of the modal amplitudes and derivations of the conditionally stable and absolutely stable thresholds for shape oscillations. The analysis of the solutions reveals the existence of a hysteretic behaviour, indicating that bubble shape oscillations could be sustained for acoustic pressures below the classical parametric threshold.

Dynamics of nonspherical microbubble oscillations above instability threshold.

Time-resolved dynamics of nonspherical oscillations of micrometer-sized bubbles are captured and analyzed using high-speed imaging. The axisymmetry of the bubble shape is ensured with certainty for the first time from the recordings of two synchronous high-speed cameras located at 90^{∘}. The temporal dynamics of finite-amplitude nonspherical oscillations are then analyzed for various acoustic pressures above the instability threshold. The experimental results are compared with recent theories accounting for nonlinearities and mode coupling, highlighting particular effects inherent to these mechanisms (saturation of the instability, triggering of nonparametric shape modes). Finally, the amplitude of the nonspherical oscillations is given as function of the driving pressure both for quadrupolar and octupolar bubbles.

Sliding Mode Control with Nonlinear Disturbance Observer based on Genetic Algorithm for Inverted Pendulum with Mismatched Disturbances

Objective: This study proposes the application of optimized Sliding Mode Control (SMC) with Nonlinear Disturbance Observer (NDO) for the Inverted pendulum, which is subjected to mismatched disturbances. Methods/Statistical Analysis: Sliding Mode Control (SMC) with Nonlinear Disturbance Observer (NDO) solves the main problem of chattering present in traditional sliding mode controllers while retaining the nominal performance even with mismatched disturbances. Chattering defines the presence of finite frequency and finite amplitude oscillations in the control input. Genetic Algorithm is used for searching the optimal controller parameters, which includes switching function coefficient, switching gain and observer gain. Findings: Simulation results shows the advantages of NDO based SMC and proves that NDOSMC can observe the mismatched disturbances very effectively. The conventional SMC method is also used in the control design the Inverted pendulum for comparison studies. The two main issues to be tackled in conventional Sliding Mode Control (SMC) are convergence time and chattering. Chattering and convergence time can be reduced in SMC by getting the optimal values of controller parameters using intelligent optimization techniques such as Genetic Algorithm (GA). In the present study convergence time and chattering of the controlled system are considered as the performance index of objective function. Application/Improvements: This study can further be extended for Multi-Input Multi-Output (MIMO) systems subjected to mismatched disturbances and parametric uncertainties.

The effects of finite amplitude drop shape oscillations on the inference of material properties

Acoustic levitation of drops provides a non-contact means of isolation, stable positioning and static and dynamic manipulation. Despite a long history of the use of drop shape oscillations to infer both surface and bulk material properties, and an equally long history of observing behaviors that result in incorrect inference of material properties, researchers continue to employ finite amplitude oscillations for their experimental observations. In this study we use the inference of surface tension and viscosity of known samples of glycerin-water mixtures to illustrate the fact that finite-amplitude oscillations are the dominant mechanism for many reports of ill-behaved drop dynamics, including modal peak-splitting, vorticity, and strong field-coupling. Since this “bad” behavior leads to incorrect inferences of surface tension and viscosity, we show that small amplitude oscillations yields recovery of correct inferences for these properties for known samples. Finally, we show results from experiments with bovine and human blood. [Work supported by NSF grant # 1438569.]

An Investigation of Bend Connection Effect on the Finite Amplitude Oscillation in Circular Ducts

An Investigation of Bend Connection Effect on the Finite Amplitude Oscillation in Circular Ducts

An Investigation of Bend Connection Effect on the Finite Amplitude Oscillation in Circular Ducts

This paper presents an experimental and theoretical study of finite amplitude oscillations in circular ducts ended with/without bend with connections. The acoustic oscillations are created by reciprocating piston at the closed end of the circular duct while the other end is left free end or it is fitted with constant cross section area bend with different turning angles. Three different turning angles for bends of 450, 900, and 1350 degrees were used for experimental study. The pressure time histories along the duct were recorded at three different axial locations using a capacitance pressure transducer connected to a data acquisition system and DAQ VIEW software. The results showed that, the bend angle has a significant effect on wave deformation inside the duct. Also, it is observed that, close to the piston, the free end duct (duct without bending) has a larger acoustic wave amplitude than the same duct fitted with the bend and the vice versa for the duct close to its exit exactly before the attached bend entrance. Also, the amplitude of the wave in the duct close to the piston increases with rising the bend angles from 450 to 1350, while it decreases close to the duct exit exactly before the bend entrance with increasing the bend angle. That is due to the wave reflections from the bend at the duct exit and wave interaction between different wave phases at the duct exit. Two different piston reciprocating speeds have been used and the results are compared for the different bend turning angles. The numerical simulation and analytical solution are compared with the experimental results for qualitative validation. It is noticed that the experimental results were in agreement with the numerical and analytical results within an accepted error. Also, fast Fourier transform is performed for the experimental results and higher harmonics are noticed for duct fitted with bends at its exit. These higher harmonics are significantly noticed for duct fitted with 1350 bend.

Modelling nonlinear electrostatic oscillations in plasmas

The nonlinear 1-D plasma electrostatic oscillation is formulated in an analytic framework that allows closed-form analytic solutions along the characteristics, and solved numerically in configuration space. Additionally, a novel iterative analytical form for the finite-amplitude oscillation solution is derived, which compares favourably with the other two techniques. A fresh insight into the evolution of the oscillation is gained, including defining the least achievable density in the nonlinear oscillation as half of the equilibrium value, and relating the associated maximum density achievable in terms of that minimum.

Finite amplitude oscillations of flanged laminas in viscous flows: Vortex–structure interactions for hydrodynamic damping control

In this paper, we study the problem of harmonic oscillations of a flanged lamina in a quiescent Newtonian incompressible viscous fluid. We conduct a comprehensive fluid–structure interaction investigation with the goal of assessing the effect of the presence of the flanges on the added mass and hydrodynamic damping experienced by the oscillating solid. We determine the complex nonlinear hydrodynamic function incorporating these effects via its real and imaginary parts, respectively, and its dependence on three nondimensional parameters that govern the flow evolution. We further investigate in detail the flow physics and the effects of nonlinearities on vortex shedding, convection, and diffusion in the vicinity of the oscillating structure. We find that the added mass effect is relatively independent of the oscillation amplitude and increases with the flange size. On the other hand, the hydrodynamic damping effect is remarkably affected by the interplay of geometry and dynamic parameters resulting into a peculiar non-monotonic behavior. We show the existence of a minimum in the hydrodynamic damping which can be attained via specific control of vortex–structure interaction dynamics and discuss its properties and significance from a physical perspective through analysis of the relevant flow fields. This novel finding has potential application for damping reduction in elastic systems where reduction of energy losses and increase of oscillation quality factor are desired.

The rupture of viscoelastic shell bubble under high intensity ultrasound drive

A model for encapsulated bubble (EB) used in ultrasonic therapy is presented discussing EB's finite amplitude oscillation and rupture under high intensity ultrasound drive. The shell of EB, which is the key to EB dynamics, is treated as a layer of viscoelastic material and its large deformation and stress are expressed analytically. By introducing the critical tolerable stress of the shell material, the threshold drive pressure for the rupture of an EB can be calculated in the framework of the model. The oscillation of an EB on the verge of its rupture is simulated and the results are in agreement with the reported experimental data. The relationship between the drive frequency and EB's rupture is also investigated.

The influence of joints on friction induced vibration in brake squeal

The effect of joints on flutter type friction induced oscillation in friction brakes is investigated. The work is based on a small characteristic system with elementary joint models. Linear stability analysis is conducted and a harmonic balance method is applied to determine limit cycle amplitudes and bifurcation diagrams. It turns out that in finite amplitude oscillation the resulting dynamic state of the joints is dominated largely by the deflection shape of the underlying linear instability. Both local and global bifurcations do exist and large limit cycle amplitudes may result for quite unexpected parameters, depending on the kind of joints involved and related to their dynamical state.

Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part II: POD and DMD applied to reduced domains

Three-dimensional direct numerical simulations (DNS) have been performed on a finite-size hemisphere-cylinder model at angle of attack AoA = 20° and Reynolds numbers Re = 350 and 1000. Under these conditions, massive separation exists on the nose and lee-side of the cylinder, and at both Reynolds numbers the flow is found to be unsteady. Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are employed in order to study the primary instability that triggers unsteadiness at Re = 350. The dominant coherent flow structures identified at the lower Reynolds number are also found to exist at Re = 1000; the question is then posed whether the flow oscillations and structures found at the two Reynolds numbers are related. POD and DMD computations are performed using different subdomains of the DNS computational domain. Besides reducing the computational cost of the analyses, this also permits to isolate spatially localized oscillatory structures from other, more energetic structures present in the flow. It is found that POD and DMD are in general sensitive to domain truncation and non-educated choices of the subdomain may lead to inconsistent results. Analyses at Re = 350 show that the primary instability is related to the counter-rotating vortex pair conforming the three-dimensional afterbody wake, and characterized by the frequency St ≈ 0.11, in line with results in the literature. At Re = 1000, vortex-shedding is present in the wake with an associated broadband spectrum centered around the same frequency. The horn/leeward vortices at the cylinder lee-side, upstream of the cylinder base, also present finite amplitude oscillations at the higher Reynolds number. The spatial structure of these oscillations, described by the POD modes, is easily differentiated from that of the wake oscillations. Additionally, the frequency spectra associated with the lee-side vortices presents well-defined peaks, corresponding to St ≈ 0.11 and its few harmonics, as opposed to the broadband spectrum found at the wake.