Self-sustained Vibrations Research Articles

Frequency stabilization of self-sustained oscillations in a sideband-driven electromechanical resonator

We present a method to stabilize the frequency of self-sustained vibrations in micromechanical and nanomechanical resonators. The method refers to a two-mode system with the vibrations at significantly different frequencies. The signal from one mode is used to control the other mode. In the experiment, self-sustained oscillations of micromechanical modes are excited by pumping at the blue-detuned sideband of the higher-frequency mode. Phase fluctuations of the two modes show near-perfect anticorrelation. They can be compensated in either of the modes by a stepwise change of the pump phase. The phase change of the controlled mode is proportional to the pump phase change, with the proportionality constant independent of the pump amplitude and frequency. This finding allows us to stabilize the phase of one mode against phase diffusion using the measured phase of the other mode. We demonstrate that phase fluctuations of either the high-frequency mode or the low-frequency mode can be significantly reduced. The results open new opportunities in generating stable vibrations in a broad frequency range via parametric down-conversion in nonlinear resonators. Published by the American Physical Society 2024

Fluid and Structure Coupling Analysis on Frequency Locking of an Elastic Hydrofoil

Vortex-induced vibration (VIV) has always been a significant research topic in relation to marine structures, with structural resonance and locking vibrations liable to occur at particular Reynolds numbers. To investigate this problem, fluid–structure coupling simulations of an elastic hydrofoil within a viscous flow are conducted. The correlation between the structural response and vortex shedding was explored. By analyzing the frequency eigenvalues, the structural response and vortex shedding at different frequencies can be decoupled. The findings suggest that the self-sustainability of natural vibration plays a significant role in frequency lock-in once structural resonance has been triggered. The phase difference between the lift pulsation and vortex shedding is determined, and is confirmed to be a significant factor influencing the resonant vibration of elastic foils. This phase difference can be used to determine whether the lock-in phenomenon has occurred. It is also found that veering phase, the resonance appears as forced vibration, while at locking phase, the resonance appears as self-sustained vibration at a natural frequency.

Effect of vortex-induced vibrations on mass transfer enhancement in a channel with two cylinders in tandem arrangement

The vortex-shedding phenomenon leads to a self-sustained vibration named Vortex-Induced Vibration (VIV), an important mass transfer enhancement method. This work studies the VIV effects of two circular cylinders inside a channel in a tandem arrangement on the mixing performance. A comprehensive study is conducted on different modes of cylinder vibrations, i.e., both cylinders vibrate, one cylinder vibrates, another cylinder is stationary, and both cylinders are stationary. The finite volume method is applied to solve the fluid flow equations, and the cylinders' vibration is modeled as a mass-spring-damping system. At low ranges of the reduced velocity, the vibration's amplitude of the cylinder is significant, and the mixing index reaches its highest values. For the case in which the distance between two cylinders is four times their diameter, the best mixing performance is related to two vibrating cylinders. In this mode, the mixing index increased up to 35% and 78% with respect to two stationary cylinders and a single stationary cylinder, respectively. Also, the investigation shows that increasing the distance between the cylinders improves the mixing process. For cylinders' distances of 4.5 and 5, the mixing index reaches 98% and up at the channel outlet.

Flutter analysis of compressor blades under travelling wave modes using an efficient fluid–structure interaction method

Flutter is a self-sustained vibration which could create serious damage to compressor blades. Improving the efficiency and accuracy of Fluid-Structure Interaction (FSI) method is crucial to flutter analysis. An efficient FSI method which combines a fast mesh deformation technology and Double-Passage Shape Correction (DPSC) method is proposed to predict blades flutter under traveling wave modes. Firstly, regarding the fluid domain as a pseudo elastic solid, the flow mesh deformation and blade vibration response can be quickly obtained by solving the governing equations of the holistic system composed of blade and pseudo elastic solid. Then, by storing and updating the Fourier coefficients on the circumferential boundary, the phase-lagged boundary condition is introduced into the computational domain. Finally, the aerodynamic stability for the blades of an axial compressor under various Inter-Blade Phase Angle (IBPA) is analyzed. The results show that the proposed method can effectively predict the characteristics of aerodynamic damping, aerodynamic force and blade displacement. And a conceptual model is proposed to describe the motion behavior of the shock wave. Compared with the multi-passage method, the proposed method obtains almost the same unstable IBPA interval and the blade displacement error is less than 3.4%. But the calculation time is significantly shortened especially in small IBPA cases.

A Study on Self-Sustained Vibrations of a Tire Operating above Peak Friction

ABSTRACT When a tire operates at a side slip level above peak friction, large vibrations in the produced forces and moments are observed. In the lateral direction, these vibrations are typically centered at frequencies close to 50 Hz and significantly affect the force that is produced by the tire. As an example, high dynamics vehicle maneuvers with an electronic stability program control system in the loop can be affected by this behavior. To accurately reproduce the tire response in these operating conditions, it is important to employ a model that can capture this phenomenon. Based on analysis of nonlinear, unstable systems and limit cycle phenomenon, a theory is presented that provides a physical background for the source of vibrations. The theory also gives insight in how the frequency and amplitude of the vibrations are influenced by the tire physical characteristics and the operating conditions. It is found that the most dominant characteristics are the shape of the steady-state force response of the tire around peak friction, the contact patch mass, the carcass damping and stiffness. Simulations with physical models of different complexity levels, as well as with the commercial Simcenter Tire MF-Tyre/MF-Swift tire software model, validate the theory and demonstrate how the vibrations can be reproduced in a simulation environment. The simulation results are compared with tire measurements in different operating conditions, validating the exposed theory and employed models.

DYNAMIC PROPERTIES OF NANOCOMPOSITE AND THREE-LAYER THIN-WALLED AEROSPACE ELEMENTS MANUFACTURED BY ADDITIVE TECHNOLOGIES

Nanocomposite and sandwich plates with a honeycomb core are characterized by a high strength-to-mass ratio. Thus, such a solution is very promising for the aerospace and aircraft industry. This paper represents a mathematical model for a nanocomposite functionally gradient cylindrical shell interacting with a supersonic gas flow. To obtain such a model, the predetermined form method is used. An ordinary nonlinear differential equations system is obtained to describe the self-sustained vibrations of the shell. The structure model is developed using nonlinear strain-displacement relationships to analyze self-sustained vibrations. A model describing self-sustained vibrations of a sandwich conical shell interacting with a supersonic gas flow is obtained. The core layer of the shell is an FDM-manufactured honeycomb. The stress state of the structure is analyzed using the highorder shear deformations theory. Each layer’s stress state is described by five coordinates which are the three displacements of the midsurface and two angles of rotation of the normal to the midsurface. At the layers’ junctions, the border conditions of displacements’ continuity are used. To analyze self-sustained vibrations, the nonlinear strain-displacement relationships are utilized. Using the normal modes technique allows us to obtain a nonlinear autonomous dynamic system. Results of numerical simulations of self-sustained vibrations are provided. They are obtained by solving a nonlinear boundary value problem for the ordinary differential equations system using shooting and continuation techniques. Experimental investigation of sandwich plates’ fatigue with honeycomb core is considered. A method of fatigue testing of sandwich plates is described. The testing results are presented using S-N diagrams.

Modelling of wheel/rail squeal noise in curves from mono-harmonic vibratory limit cycles

Most of the works in the literature agree to attribute the generation of wheel/rail squeal noise in curves to the important lateral slip imposed in the curve and the resulting instabilities. In models, the occurrence of the phenomenon is thus generally studied through a stability analysis based on the linearization of the contact forces. Despite its undeniable interest, the stability analysis does not allow the prediction of the amplitudes of the nonlinear self-sustained vibrations resulting from the instabilities. These nonlinear vibrations are most often calculated using a numerical integration of the dynamic equations of the system in the time domain. Some authors have proposed simplified methods allowing a direct calculation of stationary regimes, but limited to a reduced modal description of the system dynamics. In this presentation, an original method is proposed to determine approximate limit cycles from the wheel/rail contact mobilities expressed in the frequency domain. The contact condensation allows to be both more general and more functional to describe the dynamics of structures, in particular the one of the rail. Assuming a mono-harmonic vibratory cycle, the corresponding amplitude is determined from a power balance and the squeal level at the considered pulsation is obtained from analytical radiation factors.

Bats expand their vocal range by recruiting different laryngeal structures for echolocation and social communication.

Echolocating bats produce very diverse vocal signals for echolocation and social communication that span an impressive frequency range of 1 to 120 kHz or 7 octaves. This tremendous vocal range is unparalleled in mammalian sound production and thought to be produced by specialized laryngeal vocal membranes on top of vocal folds. However, their function in vocal production remains untested. By filming vocal membranes in excised bat larynges (Myotis daubentonii) in vitro with ultra-high-speed video (up to 250,000 fps) and using deep learning networks to extract their motion, we provide the first direct observations that vocal membranes exhibit flow-induced self-sustained vibrations to produce 10 to 95 kHz echolocation and social communication calls in bats. The vocal membranes achieve the highest fundamental frequencies (fo's) of any mammal, but their vocal range is with 3 to 4 octaves comparable to most mammals. We evaluate the currently outstanding hypotheses for vocal membrane function and propose that most laryngeal adaptations in echolocating bats result from selection for producing high-frequency, rapid echolocation calls to catch fast-moving prey. Furthermore, we show that bats extend their lower vocal range by recruiting their ventricular folds-as in death metal growls-that vibrate at distinctly lower frequencies of 1 to 5 kHz for producing agonistic social calls. The different selection pressures for echolocation and social communication facilitated the evolution of separate laryngeal structures that together vastly expanded the vocal range in bats.

Drill-string with cutting dynamics: A mathematical assessment of two models

In this paper, a new model to study drill-string dynamics is developed. Some of the novelties of the paper are: (1) the use of a new bit-rock interaction relation that does not restrict backward rotation of the bit nor bit-bounce; (2) the use of an advection equation to avoid dealing with a system of delay-differential equations in the simulation of the cutting process; (3) the extension of the advection approach treated in previous works to allow rotation in both directions; (4) the combination of the previous items with a distributed approach for the torsional dynamics; (5) the inclusion of the 2-DOF model as a limiting case of the continuous model. For this purpose, a strategy considering an extra parameter α is used. All these aspects lead to a coupled non-linear formulation that can exhibit self-sustained vibrations associated with the regenerative cutting process. The new model is compared, from a numerical point of view, with an established 2-DOF approach. Five scenarios considering a 1200 m column with different friction parameters and top drive conditions are simulated. Dissimilar predictions are observed, showing a more complex response for the continuum model where higher frequencies are excited as well. These results indicate that the more sophisticated model could capture other aspects of the dynamics that are neglected in the 2-DOF approach.

Nonlinear supersonic flutter of sandwich truncated conical shell with flexible honeycomb core manufactured by fused deposition modeling

The self-sustained vibrations of sandwich conical shell with a flexible honeycomb core manufactured by fused deposition modeling are studied. The interaction of the sandwich thin-walled structure with supersonic gas flow is analyzed. The geometrical nonlinearity of the shell structure is taken into account to predict the self-sustained vibrations. Vibrations of the structure layer are described by three displacements of the middle surface of each layer and two rotations of the normal to the middle surface. The higher-order shear deformation theory is used to describe the strain–displacement relationships. The self-sustained vibrations of the sandwich conical shell are described by a system of nonlinear ordinary differential equations with respect to the generalized coordinates. The assumed mode method is used to derive the equations.The bifurcations of the nonlinear self-sustained vibrations are analyzed numerically using a continuation technique. Quasi-periodic and chaotic self-sustained vibrations are numerically studied for the shell subjected to different boundary conditions. Numerical results show that the amplitudes of quasi-periodic and chaotic vibrations are significantly larger than the amplitudes of periodic vibrations for this shell.

Self-sustained vibrations in the system of a bi-harmonically driven pendulum

In this paper, we investigate self-sustained oscillations in a weakly damped pendulum system under the action of two harmonic forces with close frequencies. With no limitations on the amplitude of oscillations, we construct an asymptotic procedure; separating the dynamics of the system into fast, slow, and super-slow timescales, we show that the movements of the pendulum on the fast and slow scales are decoupled. The system of equations for the amplitude of the envelope and the phase of the state function in the leading approximation on a slow scale become autonomous when the super-slow time parameter depends rather weakly on slow time. Using this technique, it is possible to analyse relaxation oscillations in a wide frequency range along with their dependence on the friction coefficient. The results obtained are in good agreement with the numerical solution of the original system.

Aeroelastic mode decomposition framework and mode selection mechanism in fluid–membrane interaction

In this study, we present a global Fourier mode decomposition framework for unsteady fluid–structure interaction. We apply the framework to isolate and extract the aeroelastic modes arising from a coupled three-dimensional fluid–membrane system. We investigate the frequency synchronization between the vortex shedding and the structural vibration via mode decomposition analysis. We explore the role of flexibility in the aeroelastic mode selection and perform a systematic comparison of flow features among a rigid flat wing, a rigid cambered wing and a flexible membrane. The camber effect can enlarge the pressure suction area on the membrane surface and suppress the turbulent intensity, compared to the rigid flat wing counterpart. With the aid of our mode decomposition technique, we find that the dominant structural mode exhibits a chordwise second and spanwise first mode at different angles of attack. The structural natural frequency corresponding to this mode is estimated using an approximate analytical formula. By examining the dominant frequency of the coupled system, we show that the dominant membrane vibration mode is selected via the frequency lock-in between the dominant vortex shedding frequency and the structural natural frequency. From the fluid modes and the mode energy spectra at α=20∘ and 25°, the aeroelastic modes corresponding to the non-integer frequency components lower than the dominant frequency are observed, which are associated with the bluff body vortex shedding instability. The non-periodic aeroelastic behaviors observed at higher angles of attack are related to the interaction between aeroelastic modes caused by the frequency lock-in and the bluff-body-like vortex shedding. Using the mode decomposition analysis, we suggest a feedback cycle for flexible membrane wings undergoing synchronized self-sustained vibration. This feedback cycle reveals that the dominant aeroelastic modes are selected through the mode and frequency synchronization during fluid–membrane interaction to exhibit similar modal shapes in the membrane vibration and the pressure pulsation.

Flow-excited membrane instability at moderate Reynolds numbers

In this paper, we study the fluid–structure interaction of a three-dimensional (3-D) flexible membrane immersed in an unsteady separated flow at moderate Reynolds numbers. We employ a body-conforming variational fluid–structure interaction solver based on the recently developed partitioned iterative scheme for the coupling of turbulent fluid flow with nonlinear structural dynamics. Of particular interest is to understand the flow-excited instability of a 3-D flexible membrane as a function of the non-dimensional mass ratio ( $m^{*}$ ), Reynolds number ( $Re$ ) and aeroelastic number ( $Ae$ ). For a wide range of parameters, we examine two distinct stability regimes of the fluid–membrane interaction: deformed steady state (DSS) and dynamic balance state (DBS). We propose stability phase diagrams to demarcate the DSS and DBS regimes for the parameter space of mass ratio versus Reynolds number ( $m^{*}$ - $Re$ ) and mass ratio versus aeroelastic number ( $m^{*}$ - $Ae$ ). With the aid of the global Fourier mode decomposition technique, the distinct dominant vibrational modes are identified from the intertwined membrane responses in the parameter space of $m^{*}$ - $Re$ and $m^{*}$ - $Ae$ . Compared to the deformed steady membrane, the flow-excited vibration produces relatively longer attached leading-edge vortices which improve the aerodynamic performance when the coupled system is near the flow-excited instability boundary. The optimal aerodynamic performance is achieved for lighter membranes with higher $Re$ and larger flexibility. Based on the global aeroelastic mode analysis, we observe a frequency lock-in phenomenon between the vortex-shedding frequency and the membrane vibration frequency causing self-sustained vibrations in the dynamic balance state. To characterize the origin of the frequency lock-in, we propose an approximate analytical formula for the nonlinear natural frequency by considering the added mass effect and employing a large deflection theory for a simply supported rectangular membrane. Through our systematic high-fidelity numerical investigation, we find that the onset of the membrane vibration and the mode transition has a direct dependence on the frequency lock-in between the natural frequency of the tensioned membrane and the vortex-shedding frequency or its harmonics. These findings on the fluid-elastic instability of membranes have implications for the design and development of control strategies for membrane wing-based unmanned systems and drones.

A Review of Computational Methods and Reduced Order Models for Flutter Prediction in Turbomachinery

Aeroelastic phenomena in turbomachinery are one of the most challenging problems to model using computational fluid dynamics (CFD) due to their inherent nonlinear nature, the difficulties in simulating fluid–structure interactions and the considerable computational requirements. Nonetheless, accurate modelling of self-sustained flow-induced vibrations, known as flutter, has proved to be crucial in assessing stability boundaries and extending the operative life of turbomachinery. Flutter avoidance and control is becoming more relevant in compressors and fans due to a well-established trend towards lightweight and thinner designs that enhance aerodynamic efficiency. In this paper, an overview of computational techniques adopted over the years is first presented. The principal methods for flutter modelling are then reviewed; a classification is made to distinguish between classical methods, where the fluid flow does not interact with the structure, and coupled methods, where this interaction is modelled. The most used coupling algorithms along with their benefits and drawbacks are then described. Finally, an insight is presented on model order reduction techniques applied to structure and aerodynamic calculations in turbomachinery flutter simulations, with the aim of reducing computational cost and permitting treatment of complex phenomena in a reasonable time.

Transient Flutter Stability Analysis of Structural Mistuned Blisk by Energy Growth Method

Abstract Blisks suffer from flutter, a self-sustained vibration caused by aerodynamic coupled forces. This instability could cause serious damage to the blades and the machine. Flutter stability is usually analyzed based on the eigenvalue method in the aspect of the linear structural dynamic system, which transforms a dynamics stability analysis into a point of equilibrium in an infinite time scale. However, in reality, most of the blisk vibrations arise on a finite time horizon. The transient vibration amplification may cause serious damage. This paper proposes a transient flutter stability analysis method in a finite time for structural mistuned blisk based on the energy growth method. First, two common blisk models coupled aerodynamic force with different complexity are built and are all expressed in the state space representation. A novel energy growth method is then employed to analyze the transient stability and to find the maximum energy growth of the models. The optimal initial condition which leads to the maximum energy growth is obtained. A new flutter stability criterion is developed to consider the transient stability based on the energy growth method and the infinite time stability based on the eigenvalue method. The new transient stability method is verified by two numerical studies. It is found that the structural mistuned blisk model which is traditionally predicted stable still has a transient instability in a finite time due to the non-normal property of the dynamic state matrix.

Marine Propulsion Shafting: A Study of Whirling Vibrations

Whirling vibration is an important part of the calculations of the design of a marine shaft. In fact, all classification societies require a propulsion shafting whirling vibration calculation giving the range of critical speeds, i.e., free whirling vibration calculation. However, whirling vibration is a source of fatigue failure of the bracket and aft stern tube bearings, destruction of high-speed shafts with universal joints, noise, and hull vibrations. There are numerous uncertainties in the calculation of whirling vibration, namely, in the shafting system modeling and in the determination of excitement and damping forces. Moreover, whirling vibration calculation mathematics is much more complex than torsional or axial calculations. The marine propulsion shaft can be studied as a selfsustained vibration system, which can be modeled using the Van der Pol equation. In this document, a new way to solve the Van der pol equation is presented. The proposed method, based on a variational approach without local minima extra to the solution, converges for whatever initial point and parameter in the Van der Pol equation.

Fluid-structure coupled analysis of tandem 2D elastic panels

The aeroelastic characteristics of tandem positioned two-dimensional elastic panels have been numerically investigated. High-fidelity Computational Fluid Dynamics and Computational Structural Dynamics (CFD/CSD) methods have been used to simulate the aeroelastic interactions between the panels. Moreover, a CFD-based reduced order model is established through system identification method for auxiliary analyses. The results indicate strong fluid-mediated interactions between the adjacent panels. At low supersonic speeds, affected by the self-sustained vibration of upstream panel, the downstream panel exhibits the characteristics of forced vibration state for some parameter ranges where the flutter should occur. Beyond these parameter ranges, the vibration frequency of downstream panel gradually departs from the wake flow frequency and coincides with the flutter frequency of its isolated state, exhibiting the characteristics of limit cycle flutter. At high subsonic speeds, the aeroelastic response of downstream panel has a stabilizing effect on the upstream panel, and increases the dynamic stability of the entire tandem panels system. The aeroelastic behavior of the adjacent panels may enlighten a new perspective on the flutter control of the shell structures.

Suppressing Frequency Fluctuations of Self-Sustained Vibrations in Underdamped Nonlinear Resonators

We consider frequency fluctuations in self-sustained oscillators based on nonlinear underdamped resonators. An important type of such resonators are nano- and micro-electro-mechanical systems. Various noise sources are considered, with the emphasis on the fundamentally unavoidable noise that comes along with dissipation from the coupling to a thermal reservoir. The formulation in terms of the action-angle variables of the resonator allows us to study a deeply nonlinear regime. In this regime the vibration frequency as a function of the action can have an extremum. We show that frequency fluctuations can be strongly reduced by choosing the operation point at this extremum. We suggest a practical implementation of a nanoresonator that has the appropriate property and show explicit results for the corresponding model.

Performances of the double modal synthesis for the prediction of the transient self-sustained vibration and squeal noise

This paper presents an efficient reduction method for predicting the nonlinear transient and steady state squeal events in mechanical systems subjected to friction-induced vibration and noise. This proposed reduction technique is based on the Double Modal Synthesis (DMS) method that involves the use of a classical Craig & Bampton modal reduction on each substructure considering the interface surfaces associated to a condensation at the frictional interface based on complex modes.In this paper, the performances of some reduced bases based on the DMS strategy are investigated in the case of a finite element model of a simplified disc/pads system. The originality of the present work is to propose a comprehensive study on the convergence of the DMS method in order to predict not only the stability or the limit cycles of a simplified brake system but also the transient nonlinear self-excited vibrations, as well as the squeal noise. A special attention is brought to the convergence of the DMS method and more precisely the number of interfaces modes required to provide satisfactory results in regard to various criterion used to characterize the squeal.

A novel two-degree-of-freedom model of nonlinear self-excited force for coupled flutter instability of bridge decks

This paper aims to propose a novel model for predicting the nonlinear heave-torsion coupled flutter instability of flat bridge decks. The nonlinear oscillatory system during coupled post-flutter instabilities was modeled as a weak perturbation of the classical linear flutter theory. A novel nonlinear self-excited force model was then proposed by introducing extra nonlinear terms in classical linear model to consider the effects of nonlinear aerodynamic damping, amplitude-dependent vibration mode and the coupling of aerostatic deformation. An efficient algorithm for parameters identification and solving of the nonlinear coupled oscillator was also developed. The effectiveness of the proposed analytical framework was validated through an elastically-supported sectional model test in estimating the self-sustained vibrations during post-critical states of a typical closed-box bridge deck.