Nonlinear Self-excited Oscillations Research Articles

A high-fidelity model for nonlinear self-excited oscillations in rotary drilling systems

This paper presents an integrated model to analyze self-excited axial and torsional vibrations in rotary drilling systems using realistic PDC bits. The model combines a multi-degree-of-freedom (MDOF) drillstring representation with a detailed PDC bit–rock interaction model, accounting for cutter layouts and rock properties. The bit–rock interaction is described by rate-independent laws for reactive force and torque, encompassing cutting and frictional contact at the cutter face and wear flat. The regenerative rock cutting introduces state-dependent delays (up to 100) due to complex cutter layout of the bit, while the unilateral nature of frictional contact is formulated as a nonlinear boundary condition. The state-dependent delays are transformed into constant angular offsets prescribed by the bit design by introducing a bit trajectory function whose evolution is governed by a partial differential equation (PDE), which is coupled with the drillstring dynamics described by ordinary differential equations (ODEs). The Galerkin method with Chebyshev polynomials transforms the set of coupled PDE–ODEs into a system of ODEs. This approach bypasses the need to search for multiple delays at each time step, allowing implementation of this integrated rock-bit-drillstring model. Simulation results replicate field observed torque-on-bit (TOB) effects including velocity-weakening and hysteresis due to torsional stick–slips, attributing them to cutting depth variation rather than downhole friction. Preliminary findings regarding the influence of bit design on torsional stick–slips align with field drilling practices. The results of parametric study correspond to field test observations, enabling analysis of bit design, drillstring configuration, and drilling parameters on self-excited torsional stick–slips.

Analytical study of self-synchronization in two unbalanced rotors based on energetic conditions

When self-synchronization occurs in a system in which several nonlinear self-excited oscillators with different natural frequencies are coupled, the frequencies of all of the oscillators are entrained due to their interaction. Although many researchers have examined the characteristics of this phenomenon, the mechanism has not yet been completely clarified. This paper treats a system comprising two co-rotating unbalanced rotors fixed to a rigid body and derives the two energetic conditions satisfied by energy dissipated in damping and energy transferred in the system from the energy integration of the equations of motion. Using these two conditions and the approximation of constant angular frequency, methods for obtaining an approximate solution and for deriving the existence limit of this solution are proposed. The two methods clarify the parameter effects on the phenomenon efficiently and are expected to contribute to the clarification of the mechanism and the application to vibration machines.

Measurement of added mass for an object oscillating in viscous fluids using nonlinear self-excited oscillations

In this study, a method of measuring the added mass for an object oscillating in a viscous fluid using nonlinear self-excited oscillations was developed. The added mass produces an additional inertial effect on the vibrating object. In previous methods, the added mass is obtained experimentally from the response frequency and amplitude at the peak in the frequency response curve under a harmonically forced excitation. However, such methods cannot be utilized in highly viscous environments because the peak becomes ambiguous as a result of damping. Moreover, in very high-viscosity environments, the resonance peak ceases to exist in the frequency response curves. To solve this problem, self-excited oscillations were induced in the system using linear velocity feedback. Because linear velocity feedback can be used to eliminate the viscous damping effect when the linear feedback gain is set near the Hopf bifurcation point related to self-excited oscillations, the object becomes self-excited at a frequency equal to the natural frequency of the object oscillating in a vacuum, i.e., the undamped natural frequency. What distinguishes the proposed method from the abovementioned previous methods is that the value of the response is not needed; however, a reduction in the oscillation amplitude is required to obtain accurate measurement results. In the proposed method, to avoid an increase in the response amplitude under the applied linear velocity feedback, nonlinear feedback is also applied to produce a limit cycle similar to that produced in a van der Pol oscillator. A prototype of the measurement system was constructed based on the proposed method. A comparison of the experimentally and theoretically obtained added mass values confirmed the validity of the proposed method for added mass sensing.

Constructing explicit homoclinic solution of oscillators: An improvement for perturbation procedure based on nonlinear time transformation

An improved procedure for perturbation method is presented for constructing homoclinic solutions of strongly nonlinear self-excited oscillators. Compared with current perturbation methods based on nonlinear time transformations, the preference of the present method is that the explicit solutions, in respect to the original time variable, can be derived. In the paper, the equivalence and unified perturbation procedure with nonlinear time transformations, by which implicit solutions can be derived at nonlinear time scales, are firstly presented. Then an explicit generating homoclinic solution for power-law strongly nonlinear oscillator is derived with proposed hyperbolic function balance procedure. An approximation scheme is presented to improve the perturbation procedure and the explicit expression for nonlinear time transformation can be achieved. Applications and comparisons with other methods are performed to assess the advantage of the present method.

Stochastic bifurcations in a prototypical thermoacoustic system.

We study the influence of noise in a prototypical thermoacoustic system, which represents a nonlinear self-excited bistable oscillator. We analyze the time series of unsteady pressure obtained from a horizontal Rijke tube and a mathematical model to identify the effect of noise. We report the occurrence of stochastic bifurcations in a thermoacoustic system by tracking the changes in the stationary amplitude distribution. We observe a complete suppression of a bistable zone in the presence of high intensity noise. We find that the complete suppression of the bistable zone corresponds to the nonexistence of phenomenological (P) bifurcations. This is a study in thermoacoustics to identify the parameter regimes pertinent to P bifurcation using the stationary amplitude distribution obtained by solving the Fokker-Planck equation.

Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

A generalized hyperbolic perturbation method for heteroclinic solutions is presented for strongly nonlinear self-excited oscillators in the more general form of . The advantage of this work is that heteroclinic solutions for more complicated and strong nonlinearities can be analytically derived, and the previous hyperbolic perturbation solutions for Duffing type oscillator can be just regarded as a special case of the present method. The applications to cases with quadratic-cubic nonlinearities and with quintic-septic nonlinearities are presented. Comparisons with other methods are performed to assess the effectiveness of the present method.

Nonlinear self-excited thermoacoustic oscillations of a ducted premixed flame: bifurcations and routes to chaos

AbstractThermoacoustic systems can oscillate self-excitedly, and often non-periodically, owing to coupling between unsteady heat release and acoustic waves. We study a slot-stabilized two-dimensional premixed flame in a duct via numerical simulations of a $G$-equation flame coupled with duct acoustics. We examine the bifurcations and routes to chaos for three control parameters: (i) the flame position in the duct, (ii) the length of the duct and (iii) the mean flow velocity. We observe period-1, period-2, quasi-periodic and chaotic oscillations. For certain parameter ranges, more than one stable state exists, so mode switching is possible. At intermediate times, the system is attracted to and repelled from unstable states, which are also identified. Two routes to chaos are established for this system: the period-doubling route and the Ruelle–Takens–Newhouse route. These are corroborated by analyses of the power spectra of the acoustic velocity. Instantaneous flame images reveal that the wrinkles on the flame surface and pinch-off of flame pockets are regular for periodic oscillations, while they are irregular and have multiple time and length scales for quasi-periodic and aperiodic oscillations. This study complements recent experiments by providing a reduced-order model of a system with approximately 5000 degrees of freedom that captures much of the elaborate nonlinear behaviour of ducted premixed flames observed in the laboratory.

Nonlinear self-excited thermoacoustic oscillations: intermittency and flame blowout

AbstractNonlinear self-excited thermoacoustic oscillations appear in systems involving confined combustion in the form of coupled acoustic pressure oscillations and unsteady heat release rate. In this paper, we investigate the nonlinear transition undergone by thermoacoustic oscillations to flame blowout via intermittency, in response to variation in the location of the combustion source with respect to the acoustic field of the confinement. A ducted laminar premixed conical flame, stabilized on a circular jet exit with a fully developed exit velocity profile, was investigated. Transition to limit cycle oscillations from a non-oscillatory state was observed to occur via a subcritical Hopf bifurcation. Limit cycle oscillations underwent a further bifurcation to quasi-periodic oscillations characterized by the repeated formation of elongated necks in the flame that pinch off as pockets of unburned fuel–air mixture. The quasi-periodic state loses stability, resulting in an intermittent state identified as type II through recurrence analysis of phase space trajectories reconstructed from the acoustic pressure time trace. In this state, the flame undergoes repeated lift-off and reattachment. Instantaneous flame images suggest that the intermittent flame behaviour is influenced by jet flow dynamics.

Self-Synchronized Phenomena Generated in Coupled Nonlinear Self-Excited Oscillators with Stick-Slip Motion

Self-Synchronized Phenomena Generated in Coupled Nonlinear Self-Excited Oscillators with Stick-Slip Motion

Nonlinear Self-Excited Combustion Oscillations of a Premixed Laminar Flame

Self-excited combustion oscillations are caused by a coupling between acoustic waves and unsteady heat release. A premixed laminar flame in a Rijke tube, anchored to a metal gauze, is considered in this work. The flame response to flow disturbances is investigated by developing a nonlinear kinematic model based on the classical-equation, with the assumption of a time-invariant laminar flame speed. Unsteady heat release from the flame is assumed to be caused by its surface variations, which results from the fluctuations of the oncoming flow velocity. The flame is acoustically compact, and its presence causes the mean temperature undergoing a jump, whose effect on the dynamics of the thermo-acoustic system is discussed. Coupling the flame model with a Galerkin series expansion of the acoustic waves present enables the time evolution of the flow disturbances to be calculated. It was found that the model can predict the mode shape and the frequencies of the excited combustion oscillations very well. Moreover, the fundamental mode is found to be the easiest one to be triggered among all acoustic modes. To gain insight about the mode selection and triggering, further numerical investigation is conducted by linearizing the flame model and recasting into the popular formulation.

The electrical asymmetry effect in capacitively coupled radio-frequency discharges

We present an analytical model to describe capacitively coupled radio-frequency (CCRF) discharges and the electrical asymmetry effect (EAE) based on the non-linearity of the boundary sheaths. The model describes various discharge types, e.g. single and multi-frequency as well as geometrically symmetric and asymmetric discharges. It yields simple analytical expressions for important plasma parameters such as the dc self-bias, the uncompensated charge in both sheaths, the discharge current and the power dissipated to electrons. Based on the model results the EAE is understood. This effect allows control of the symmetry of CCRF discharges driven by multiple consecutive harmonics of a fundamental frequency electrically by tuning the individual phase shifts between the driving frequencies. This novel class of capacitive radio-frequency (RF) discharges has various advantages: (i) A variable dc self-bias can be generated as a function of the phase shifts between the driving frequencies. In this way, the symmetry of the sheaths in geometrically symmetric discharges can be broken and controlled for the first time. (ii) Almost ideal separate control of ion energy and flux at the electrodes can be realized in contrast to classical dual-frequency discharges driven by two substantially different frequencies. (iii) Non-linear self-excited plasma series resonance oscillations of the RF current can be switched on and off electrically even in geometrically symmetric discharges. Here, the basics of the EAE are introduced and its main applications are discussed based on experimental, simulation, and modeling results.

Self-Synchronized Phenomena Generated in Coupled Self-Excited Oscillators with Stick-Slip Motion

A simple nonlinear system generating the self-synchronization is developed in order to clarify the occurrence mechanism of the phenomena. This system consists of two nonlinear self-excited oscillators with Coulomb friction and the stick-slip motion is generated. These two oscillators are directly coupled in series by a coil-spring and a dashpot. The self-synchronized phenomena generated in this system are investigated analytically and experimentally. The validity of the analytical model and the computational method based on the shooting method are verified by comparing the computational results and the experimental results. It is clarified that two types of the self-synchronized solutions exist in this system and the vibration patterns of the self-synchronized solutions are closely related to the undamped free vibration characteristics of the model without Coulomb friction. In addition, the differences between the occurrence mechanisms of the self-synchronized solutions are analytically confirmed by examining the energy transmission between two oscillators through the coupling element. It is also proved that the unstable regions caused by the internal resonance exist in the solution branch in which two oscillators vibrate nearly in phase.

305 自己同期現象における非線形自励振動子間の結合形態に関する検討

The purpose of this research is to investigate influences of coupling elements placed between nonlinear self-excited oscillators on the self-synchronization phenomena. We newly developed a simple nonlinear model that is modified from the dynamical model studied in the previous report. The new model is composed of two nonlinear self-excited oscillators subjected to Coulomb friction and a mass block. They are connected by coil-springs and dashpots. In the present model, the stick-slip vibration caused by the friction forces often generates. The self-synchronized solutions and the stability are computationally analyzed by approving the improved shooting method. As a result, we confirmed the verification of the present model by comparing numerical results and experimental results. The characteristics of the self-synchronized solutions are investigated, and the frequencies and vibration patterns closely correspond to the undamped natural frequencies and characteristic modes, respectively. In addition, we also examined that the existence regions of the self-synchronized solutions vary depending on the system parameter of the mass block. Then it was proven that appropriately setting the parameter allows the regions to extend widely.

221 摩擦力が作用する自励振動子群に生じる自己同期現象 : 振動子間のエネルギーの授受に関する検討(自励・鳴き振動,OS-1 振動基礎,総合テーマ「伝統を,未来へ!」)

We studied that the influence of a coupling element, which connects some nonlinear self-excited oscillators, on the self-synchronized phenomena. In this report, the model composed of two oscillators and a coupling element is investigated by applying the improved shooting method. As a result, in comparison with the previous model where two oscillators are directly coupled by a coil-spring and a dashpot, the self-synchronized phenomena widely occur against the considerable individual variability according to the parameter of the coupling element. In addition, the relation between the natural frequencies of the whole system and that of coupling element is also investigated in order to consider the reason why the phenomenon widely occurs.

Application of He's Parameter-expansion Method to a Nonlinear Self-excited Oscillator System

Article Application of He's Parameter-expansion Method to a Nonlinear Self-excited Oscillator System was published on January 1, 2009 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 10, issue 1).

Open-loop control of fully nonlinear self-excited oscillations

Experiments conducted using a low-density axisymmetric jet reveal that the self-excited nature can be altered by modulating the external pressure field using an acoustic driver. A region exists in the forcing frequency-amplitude space where the self-sustaining frequency is entirely absent and solely replaced by the forcing frequency and harmonics thereof. Inside this region, the centerline streamwise velocity disturbance level can be significantly increased or decreased. Visual evidence suggests these alterations coincide with dramatic changes in jet spreading and demonstrates that forcing can provide an effective means of controlling mixing in self-excited jets. Results support theoretical predictions made by Pier [B. Pier, Proc. R. Soc. London, Ser. A 459, 1105 (2003)].

AEROELASTIC OSCILLATIONS OF A SEESAW-TYPE OSCILLATOR UNDER STRONG WIND CONDITIONS

This paper is concerned with one-degree-of-freedom aeroelastic oscillations of a seesaw- type structure in a steady wind flow. Here it is assumed that strong wind conditions induce nonlinear aeroelastic stiffness forces that are of the same order of magnitude as the structural stiffness forces. As a model equation for the aeroelastic behaviour of the seesaw-type structure, a strongly nonlinear self-excited oscillator is obtained. The bifurcation and the stability of limit cycles for this equation are studied using a special perturbation method. Both the case with linear structural stiffness and the case with nonlinear structural stiffness are studied. For both cases is assumed a general cubic approximation to describe the aerodynamic coefficient. Conditions for the existence, the stability, and the bifurcation of limit cycles are given.

Nonlinear Response of Vortex Breakdown over a Pitching Delta Wing

A e ow visualization study was carried out to investigate the response of breakdown location to small amplitude pitching oscillations of a delta wing. At moderate angles of attack, left and right breakdown locations are inphase and locked to the pitching frequency. At large angles of attack, low-frequency antisymmetric modes and the symmetric mode at the pitching frequency are observed simultaneously. The existence of multiple frequencies in the spectra and other features suggest that vortex breakdown behaves like a nonlinear self-excited oscillator.

Nonlinear self-excited oscillations of a ducted flame

Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

Bifurcations of strongly non-linear self-excited oscillations

Perturbation of a single-degree-of-freedom conservative oscillator leads to the emergence and vanishing of periodic solutions and to various types of self-excited oscillations. Using techniques from dynamical systems theory, in particular a certain Poincaré map, we establish the presence of Hopf bifurcations, various types of homoclinic bifurcations and saddle-node bifurcations of the associated Poincaré map. The corresponding bifurcation sets in parameter space are computed explicitly by perturbation methods. The theory is applied to the generalized van der Pol and the generalized Rayleigh oscillator, and to the case of a non-linear spring attached to a conveyor belt.