Non-linear 2-DOF model and centre manifold theory to study limit cycle oscillations caused by drum-brake judder

This paper presents the research on the laws of systematic-parameter dependent variation in the vibration amplitude of drum-brake limit cycle oscillations (LCO). We established a two-degree non-linear dynamic model to describe the low-frequency vibration of the drum brake, applied the centre manifold theory to simplify the system, and obtained the LCO amplitude by calculating the normal form of the simplified system at the Hopf bifurcation point. It is indicated that when the friction coefficient is smaller than the friction coefficient at the bifurcation point, the amplitude decreases; whereas with a friction coefficient larger than the friction coefficient of bifurcation point, LCO occurs. The results suggest that it is applicable to suppress the LCO amplitude by changing systematic parameters, and thus improve the safety and ride comfort when applying brake. These findings can be applied to guiding the design of drum brakes.

Limit cycle oscillation suppression of 2-DOF airfoil using nonlinear energy sink

This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.

Reversed rotation of limit cycle oscillation and dynamics of low-intermediate-high confinement transition

The dynamics of the confinement transition from L mode to H mode（LH） is investigated in detail theoretically via the extended three-wave coupling model describing the interaction of turbulence and zonal flow（ZF） for the first time.Thereinto, turbulence is divided into a positive-frequency（PF） wave and a negative-frequency（NF） one, and the gradient of pressure is added as the auxiliary energy for the system. The LH confinement transition is observed for a sufficiently high input energy. Moreover, it is found that the rotation direction of the limit cycle oscillation（LCO） of PF wave and pressure gradient is reversed during the transition. The mechanism is illustrated by exploring the wave phases. The results presented here provide a new insight into the analysis of the LH transition, which is helpful for the experiments on the fusion devices.

Generation of Optimal Voltage Reference for Limit Cycle Oscillation in Digital Control-Based Switching Power Supply

Digital control of a general-purpose switching power supply is one of the key technologies to perform the high reliability and the intelligent function demanded for the next generation. The contribution of this paper is the development of a digital control-based switching power supply. In the developed system, the generation method of the optimal voltage reference to eliminate the limit cycle oscillation of output voltage due to the AD/DA resolution is proposed. In the proposed method, the variation of the input power source voltage can be also compensated. The effectiveness of the proposed optimal reference generation method is experimentally verified.

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.

Calculations of the bounds on limit cycle oscillations in nonlinear aeroelastic systems based on equivalent linearization

The nonlinear aeroelastic system of an airfoil with an external store was investigated,with emphasis on the bounds of limit cycle oscillations(LCOs).Based on the equivalent linearization,an approach was proposed to calculate the bounds on LCOs over the full flight envelope.The bounds are determined directly without solving LCOs one by one as the flow speed varies.The presented approach can provide us with the maximal LCO amplitudes and the lower threshold for flow speed beyond which LCOs may arise.Numerical examples show that the obtained bounds are in nice agreement with numerical simulation results.The speed threshold can be predicted to a relative error less than 0.1%,and the maximal LCO amplitude to about 3%.The influences of the system parameters on the speed threshold for speed were investigated efficiently by the proposed approach.

Dynamic analysis of two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flow

The nonlinear aeroelastic response of a two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flows is investigated. The second-order piston theory is used to analyze a double-wedge airfoil. Then, the fold bifurcation and the amplitude jump phenomenon are detected by the averaging method and the multi-variable Floquet theory. The analyticall results are further verified by numerical simulations. Finally, the influence of the freeplay parameters on the aeroelastic response is analyzed in detail.

Multiple Limit Cycles in an Immune System

The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied by Huang has at least three limit cycles in the system. The conditions for the multiple limit cycles are useful in analyzing the nonlinear oscillation in immune response.

LCO flutter of cantilevered woven glass/epoxy laminate in subsonic flow

The paper presents a cantilevered composite wing, aeroelastic characteristics of idealized as a composite flat plate laminate. The composite laminate was made from woven glass fibers with epoxy matrix. The elastic and dynamic properties of the laminate were determined experimentally for aeroelastic calculations. Aeroelastic wind tunnel testing of the laminate was performed and the result showed that flutter, a dynamic instability occurred. The cantilevered laminate also displayed limit cycle amplitude, post-flutter oscillation. The experimental flutter velocity and frequency were verified by our computational analysis.

Experimental and numerical flutter analysis of a folding fin with multiple asymmetric free-plays

Experimental folding fin models with an adjustable free-play are tested in a wind tunnel.The fin structure is modeled using the free-interface component mode synthesis method,and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness.A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow.Surface spline interpolation is improved through projection and partition.The aeroelastic characteristics of folding fins with different free-play magnitudes,initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision.The results show good consistency among the presented method,the wind tunnel test and the harmonic balance method.There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude.If either the free-play magnitude or the initial condition is extreme(tiny or vast),the system nonlinearity degenerates to linearity.Generally,the flutter prevention design of a linear model can be applied to a nonlinear model,such as moving the elastic-axis position aftward.The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.

Nonlinear aeroelastic analysis of a twin-tail boom UAV with rudder freeplay nonlinearities

The issue of nonlinear structural freeplays in aircraft has always been a significant con-cern.This study investigates the aeroelastic characteristics of a twin-tail boom Unmanned Aerial Vehicle(UAV)with simultaneous freeplay nonlinearity in its left and right rudders.A comprehen-sive Limit Cycle Oscillation(LCO)solution route is proposed for complex aircraft with multiple freeplays,which can consider both accuracy and effciency.For the first time,this study reveals the unique LCO characteristics exhibited by twin-tail boom UAVs with rudder freeplays and pro-vides simulations and explanations of interesting phenomena observed during actual flight.The governing equations are established using the free-interface component mode synthesis method,and the LCOs of the system are mainly solved through the improved time-domain numerical con-tinuation method and frequency-domain numerical continuation method.Furthermore,the study investigates the influence of the left and right rudder freeplay size ratio on the LCO characteristics.The results demonstrate that the twin-tail boom UAV exhibits two stable LCO types:close and dif-fering left and right rudder amplitudes.The proposed method successfully describes the complete LCO behaviors of the system.Overall,this study makes significant contributions to our understand-ing of the aeroelastic behavior of twin-tail boom UAVs with rudder freeplays.

A nonlinear POD reduced order model for limit cycle oscillation prediction

As the amplitude of the unsteady flow oscillation is large or large changes occur in the mean background flow such as limit cycle oscillation,the traditional proper orthogonal decomposition reduced order model based on linearized time or frequency domain small disturbance solvers can not capture the main nonlinear features.A new nonlinear reduced order model based on the dynamically nonlinear flow equation was investigated.The nonlinear second order snapshot equation in the time domain for proper orthogonal decomposition basis construction was obtained from the Taylor series expansion of the flow solver.The NLR 7301 airfoil configuration and Goland+ wing/store aeroelastic model were used to validate the capability and efficiency of the new nonlinear reduced order model.The simulation results indicate that the proposed new reduced order model can capture the limit cycle oscillation of aeroelastic system very well,while the traditional proper orthogonal decomposition reduced order model will lose effectiveness.

Theoretical assessment of the Ca^2＋ oscillations in the afferent arteriole smooth muscle cell of the rat kidney

The afferent arteriole （AA） of rat kidney exhibits the myogenic response, in which the vessel constricts in response to an elevation in blood pressure and dilates in response to a pressure reduction. Additionally, the AA exhibits spontaneous oscillations in vascular tone at physiological luminal pressures. These time-periodic oscillations stem from the dynamic exchange of Ca^2＋ between the cytosol and the sarcoplasmie reticulum, cou- pled to the stimulation of Ca^2＋-activated potassium and chloride channels, and to the modulation of voltage-gated L-type Ca^2＋ channels. The effects of physiological factors, including blood pressure and vasoactive substances, on AA vasomotion remain to be well characterized. In this paper, we analyze a mathematical model of Ca^2＋ signaling in an AA smooth muscle cell. The model represents detailed transmembrane ionic trans- port, intracellular Ca^2＋ dynamics as well as kinetics of nitric oxide （NO） and superoxide （O2^-） formation, diffusion and reaction. NO is an important factor in the maintenance of blood pressure and O2^- has been shown to contribute significantly to the functional alternations of blood vessels in hypertension. We perform a bifurcation analysis of the model equations to assess the effect of luminal pressure, NO and O2^- on the behaviors of limit cycle oscillations.

Flutter analysis of an airfoil with nonlinear damping using equivalent linearization

The equivalent linearization method （ELM） is modified to investigate the nonlinear flut- ter system of an airfoil with a cubic damping. After obtaining the linearization quantity of the cubic nonlinearity by the ELM, an equivalent system can be deduced and then investigated by linear flut- ter analysis methods. Different from the routine procedures of the ELM, the frequency rather than the amplitude of limit cycle oscillation （LCO） is chosen as an active increment to produce bifurca- tion charts. Numerical examples show that this modification makes the ELM much more efficient. Meanwhile, the LCOs obtained by the ELM are in good agreement with numerical solutions. The nonlinear damping can delay the occurrence of secondary bifurcation. On the other hand, it has marginal influence on bifurcation characteristics or LCOs.

Bifurcation in a 3-DOF Airfoil with Cubic Structural Nonlinearity

Limit cycle oscillations （LCOs） as well as nonlinear aeroelastic analysis of a 3-DOF aeroelastic airfoil motion with cubic restoring moments in the pitch degree of freedom are investigated. Aeroelastic equations of an airfoil with control surface in an incompressible potential flow are presented in the time domain. The harmonic balance （HB） method is utilized to calculate the LCO frequency and amplitude for the airfoil. Also the semi-analytical method has revealed the presence of stable and unstable limit cycles, along with stability reversal in the neighborhood of a Hopf bifurcation. The system response is determined by nu- merically integrating the governing equations using a standard Runge-Kutta algorithm and the obtained results are compared with the HB method. Also the results by the third order HB （HB3） method for control surface are consistent with the other numerical solution. Finally, by combining the numerical and the HB methods, types of bifiarcation, be it supercritical, subcritical, or diver- gent flutter area are identified.

Active Control Law Design for Flutter/LCO Suppression Based on Reduced Order Model Method

Active stability augmentation system is an attractive and promising technology to suppress flutter and limit cycle oscillation (LCO). In order to design a good active control law, the control plant model with low order and high accuracy must be provided, which is one of the most important key points. The traditional model is based on low fidelity aerodynamics model such as panel method, which is unsuitable for transonic flight regime. The physics-based high fidelity tools, reduced order model (ROM) and CFD/CSD coupled aeroservoelastic solver are used to design the active control law. The Volterra/ROM is applied to constructing the low order state space model for the nonlinear unsteady aerodynamics and static output feedback method is used to active control law design. The detail of the new method is demonstrated by the Goland+ wing/store system. The simulation results show that the effectiveness of the designed active augmentation system, which can suppress the flutter and LCO successfully.

Aeroelastic analysis and flutter control of wings and panels:A review

Flutter is a self‐excited vibration under the interaction of the inertial force,aero-dynamic force,and elastic force of the structure.After the flutter occurs,the aircraft structures will exhibit limit cycle oscillation,which will cause catastrophic accidents or fatigue damage to the structures.Therefore,it is of great theoretical and practical significance to study the aeroelastic characteristics and flutter control for improving the aeroelastic stability of aircraft structures.This paper reviews the recent advances in aeroelastic analysis and flutter control of wings and panel structures.The me-chanism of aeroelastic flutter of wings and panels is presented.The research methods of aeroelastic flutter for different structures developed in recent years are briefly summarized.Various control strategies including the linear and nonlinear control algorithms as well as the active flutter control results of wings and panels are presented.Finally,the paper ends with conclusions,which highlight challenges of the development in aeroelastic analysis and flutter control,and provide a brief outlook on the future investigations.This study aims to present a comprehensive under-standing of aeroelastic analysis and flutter control.It can also provide guidance on the design of new wings and panel structures for improving their aeroelastic stability.