Freeplay Nonlinearity Research Articles

Reliability of Hypersonic Airfoil with Freeplay and Stochasticity via Nonlinear Energy Sink

The reliability of a pitch-plunge hypersonic airfoil in random fluctuating flow with both cubic and freeplay nonlinearity is examined. The Hopf bifurcation and dynamic responses of the hypersonic airfoil are performed. To analyze the reliability, the effects of stochasticity on the dynamic behaviors of the hypersonic airfoil model are discussed in detail. Several unwanted phenomena that result in the failure of the airfoil structure are induced by random fluctuations. Subsequently, the reliability of the airfoil model is defined and analyzed according to the first passage failure criteria. The effects of different parameters on the reliability are investigated. Furthermore, a nonlinear energy sink is introduced to suppress the vibration of the airfoil and enhance the reliability. Two-dimensional reliability regions of the airfoil model are given to provide the safety parameter region. The results show that the reliability of the airfoil model is significantly improved with the nonlinear energy sink. This work will provide new insights into the safety design of hypersonic aircraft.

Model Order Reduction and Nonlinear Dynamic Analysis of the Folding Fin with Freeplay Nonlinearity

Model Order Reduction and Nonlinear Dynamic Analysis of the Folding Fin with Freeplay Nonlinearity

Numerical Analysis of Free Play-Induced Aeroelastic Phenomena: A Numerical Approach With Adaptive Step Size Control

This study presents a detailed numerical analysis of nonlinear aeroelastic behavior in a two degree of freedom (DOF) model, focusing on plunge and pitch motions and employing the continuation method (CM) with an adaptive step size control algorithm. The research incorporates free-play nonlinearity at the plunge hinge, a common structural nonlinearity in aeronautics that can induce detrimental limit cycle oscillations (LCOs) during flight. By examining three scenarios—linear response, unhindered plunge motion, and nonlinear stiffness behavior—the study assesses the effects of free play on flutter and LCO phenomena, including discontinuity-induced bifurcations like grazing bifurcation. Additionally, the study explores parameter variation for nonlinear flutter analysis, revealing the dynamics of grazing bifurcation and its impact on LCO behavior. The research also demonstrates the method’s superior accuracy in flutter speed estimation and mode-switching identification, despite higher computational demands. The findings underscore the diminishing influence of nonlinear free-play behavior on LCO amplitude, providing insights with significant implications for aeroelastic design and aircraft safety.

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 concern. 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 comprehensive Limit Cycle Oscillation (LCO) solution route is proposed for complex aircraft with multiple freeplays, which can consider both accuracy and efficiency. For the first time, this study reveals the unique LCO characteristics exhibited by twin-tail boom UAVs with rudder freeplays and provides 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 continuation 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 differing 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 understanding of the aeroelastic behavior of twin-tail boom UAVs with rudder freeplays.

Aeroelasticity Analysis of Folding Wing with Structural Nonlinearity Using Modified DLM

The structural nonlinear aeroelastic analysis of folding wings is conducted by integrating the modified Doublet Lattice Method in this paper. Firstly, a modified Doublet Lattice Method is investigated to relax the aspect ratio limitation of aerodynamic elements, and its correctness is verified through numerical examples. Secondly, the rational function approximation method is discussed, and the modified Doublet Lattice Method is approximated using the minimum state method to obtain the unsteady aerodynamic forces in the time domain. Finally, the aeroelastic characteristics of a folding wing with free-play and friction nonlinearity are studied by integrating time-domain unsteady aerodynamics with the nonlinear structural dynamics model. The results show that the modified Doublet Lattice Method, which relaxes the aspect ratio limitation of aerodynamic grid elements, is feasible and effective; Friction can suppress the influence of free-play nonlinearity on aeroelastic response.

Aeroelastic investigation on an all-movable horizontal tail with free-play nonlinearity

Free-play-induced nonlinear dynamic behavior has been one of the most important topics of aeroelastic research in recent decades. In this paper, the describing function (DF) method is developed to investigate the complex dynamic response of a popular all-movable horizontal tail with free-play. Piecewise expressions for the time history and phase portrait of limit cycle oscillation (LCO) are derived by the developed DF method, which is conducive to understand the mechanism of free-play-induced LCO. Another advantage of the developed DF method is the ability to predict the high-order harmonics, which cannot be realized by the classic DF method. A three-dimensional (3D) all-movable horizontal tail model with torsional free-play was designed and manufactured to implement wind tunnel tests via various initial parameters. A good agreement was found between the numerical and experimental results, which can demonstrate the effectiveness of the proposed method. The influence of the initial parameters of the all-movable horizontal tail on the LCO characteristics is analyzed by both numerical calculations and wind tunnel tests. The method and results in this paper can provide a significant reference for the design of all-movable horizontal tail versus the free-play-induced LCO.

Nonlinear dynamics, bifurcations, and multi-stability in a vibro-impact system with geometric and multi-segmented freeplay nonlinearities

Nonlinear dynamics, bifurcations, and multi-stability in a vibro-impact system with geometric and multi-segmented freeplay nonlinearities

Limit Cycle Prediction for a Multi-Degree-of-Freedom System with a Freeplay Non-Linearity

Limit Cycle Prediction for a Multi-Degree-of-Freedom System with a Freeplay Non-Linearity

Evaluating route to impact convergence of the harmonic balance method for piecewise-smooth systems

In this work, we investigate the applicability of the harmonic balance method (HBM) to predict periodic solutions of a single degree-of-freedom forced Duffing oscillator with freeplay nonlinearity. By studying the route to impact, which refers to a parametric study as the contact stiffness increases from soft to hard, the convergence behavior of the HBM can be understood in terms of the strength of the non-smooth forcing term. HBM results are compared to time-integration results to facilitate an evaluation of the accuracy of nonlinear periodic responses. An additional contribution of this study is to perform convergence and stability analysis specifically for isolas generated by the non-smooth nonlinearity. Residual error analysis is used to determine the approximate number of harmonics required to get results accurate to a given error tolerance. Hill’s method and Floquet theory are employed to compute the stability of periodic solutions and identify the types of bifurcations in the system.

Nonlinear Dynamics of an All-Movable Rudder System with Freeplay Nonlinearity

Nonlinear Dynamics of an All-Movable Rudder System with Freeplay Nonlinearity

Nonlinear Aeroelastic Characteristics of a Supersonic All-Movable Fin with Single or Multiple Freeplay Nonlinearities

Establishing the aeroelastic characteristic of all-movable fins with freeplay nonlinearities is one of the most common problems in the design of supersonic flight vehicles. In this context, this study provided novel points of view on the nonlinear aeroelastic characteristics of an all-movable fin with freeplay nonlinearities in its root. The unsteady aerodynamic model that was employed uses the second-order piston theory considering thickness effects. For a system with multiple freeplay nonlinearities, a discrete scanning method based on the describing function method was established to solve the limit-cycle oscillations (LCOs) and avoid the loss of solutions. Combining this with the time-domain integration method, the influences of the support stiffness at the root of the fin and the freeplay size ratio of the bending and torsional degrees of freedom on the dynamical response of the system were analyzed. The results demonstrate that systems with a single freeplay nonlinearity exhibit two completely different types of LCO, while systems with multiple freeplay nonlinearities exhibit complex dynamical behaviors such as LCO and quasi-periodic and chaotic motions. The path of a quasi-periodic torus breaking into chaos was observed. Furthermore, a harmonic initial condition for the time-domain integration is proposed; this can be used for a quick check of the frequency-domain calculation results.

Nonlinear system identification framework of folding fins with freeplay using backbone curves

Due to wear and manufacturing tolerance, the freeplay is unavoidable in the hinges of folding fins, which exerts significant effects on the aerodynamic characteristics. This paper proposes a backbone-curve-based framework for the dynamical identification of folding fins containing the freeplay nonlinearity. With no need to measure the input force signal and the response signals of nonlinear related Degrees of Freedom (DOFs), the proposed method is more direct and elegant than most existing nonlinear identification approaches, and it contains three steps: Firstly, the underlying linear model of the folding fin structure is obtained through the modal test on its linear sub-parts, and then, the harmonic approximation solves the analytical expressions of the backbone curves of measurable DOFs. Secondly, response data measured from the sine-sweep test are used to extract the fitting points of backbone curves for these DOFs. Finally, the curve fitting approach is applied to identify the freeplay parameters. A series of numerical experiments verify the effectiveness of the proposed method. A real-life folding fin structure is also employed to illustrate how the method can be applied. These examples demonstrate that the identification framework can give an accurate dynamic model of the folding fin structure.

Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems

Characterisation and quantification of nonlinearities in the engineering structures include selecting and fitting a good mathematical model to a set of experimental vibration data with significant nonlinear features. These tasks involve solving an optimisation problem where it is difficult to choose a priori the best optimisation technique. This paper presents a systematic comparison of ten optimisation methods used to select the best nonlinear model and estimate its parameters through nonlinear system identification. The model selection framework fits the structure’s equation of motions using time-domain dynamic response data and takes into account couplings due to the presence of the nonlinearities. Three benchmark problems are used to evaluate the performance of two families of optimisation methods: (i) deterministic local searches and (ii) global optimisation metaheuristics. Furthermore, hybrid local–global optimisation methods are examined. All benchmark problems include a free play nonlinearity commonly found in mechanical structures. Multiple performance criteria are considered based on computational efficiency and robustness, that is, finding the best nonlinear model. Results show that hybrid methods, that is, the multi-start strategy with local gradient-based Levenberg–Marquardt method and the particle swarm with Levenberg–Marquardt method, lead to a successful selection of nonlinear models and an accurate estimation of their parameters within acceptable computational times.

Fixed-interval smoothing of an aeroelastic airfoil model with cubic or free-play nonlinearity in incompressible flow

Fixed-interval smoothing of an aeroelastic airfoil model with cubic or free-play nonlinearity in incompressible flow

Revisit of the variable stiffness method for aeroelastic computations with/without thermal effects based on computational fluid dynamics

In the present study, a variable stiffness method (VSM) is revisited for aeroelastic computations with/without thermal effects. Unlike the traditional CFD/CSM coupling method (TM), which predicts aeroelastic responses by varying freestream conditions (e.g. freestream density, velocity), the freestream conditions in VSM can be fixed. The VSM is first verified theoretically and adopted to predict nonlinear aeroelastic responses. For aeroelastic computations, the method is applied to predict flutter onset of Isogai wing section and limit cycle oscillation (LCO) of Goland+ wing at transonic conditions. The structural free-play nonlinearity is included in the Isogai wing section aeroelastic system to further demonstrate the method. The aeroelastic computation with thermal effects is considered as the flutter onset prediction of a simply supported panel in supersonic flow. It is shown that the VSM method can replicate the nonlinear aeroelastic responses predicted by its traditional CFD/CSM coupling method counterpart under the same flow similarity parameters (e.g. Mach number, Reynolds number), whereby the freestream conditions need to be adjusted. Limitations of the VSM are also pointed out and discussed. To further assess the VSM, an ARMA model is constructed under the framework of the VSM and demonstrated by flutter onset prediction of the Isogai wing section at transonic regime.

Routes to Synchronization in a pitch–plunge aeroelastic system with coupled structural and aerodynamic nonlinearities

The present study aims to investigate the synchronization characteristics of a pitch–plunge aeroelastic system subjected to coupled structural free-play and dynamic stall-induced aerodynamic nonlinearities. The semi-empirical Leishman–Beddoes (LB) aerodynamic model is used to capture the dynamic stall phenomena at large angles-of-attack (AoA). A bifurcation analysis, considering the flow-speed as the control parameter, reveals that the free-play nonlinearity is primarily responsible for the dynamical transition in the presence of the attached flow. In this case, a pitch and plunge frequency coalescence takes place through an aperiodic to periodic signature transition. Furthermore, by tracking the flow specific variables estimated using the LB model, it is observed that the onset of the dynamic stall plays a pivotal role in the bifurcation scenario at high flow speed regimes, marking the presence of stall flutter. The framework of synchronization is used to discern the response dynamics at low speed and high speed flow regime by estimating the pitch and plunge modal frequencies and their corresponding phase components. The routes to synchronization are identified to be through phase-locking (frequency coalescence) and through suppression of the plunge mode by the dominance of pitching frequency in the regimes before and after the onset of dynamic stall, respectively. Finally, it is shown that the different synchronization routes can possibly be a useful tool to identify the presence of stall flutter oscillations in a coupled aeroelastic system.

Insights on the continuous representations of piecewise-smooth nonlinear systems: limits of applicability and effectiveness

Dynamical systems subject to intermittent contact are often modeled with piecewise-smooth contact forces. However, the discontinuous nature of the contact can cause inaccuracies in numerical results or failure in numerical solvers. Representing the piecewise contact force with a continuous and smooth function can mitigate these problems, but not all continuous representations may be appropriate for this use. In this work, five representations used by previous researchers (polynomial, rational polynomial, hyperbolic tangent, arctangent, and logarithm-arctangent functions) are studied to determine which ones most accurately capture nonlinear behaviors including super- and subharmonic resonances, multiple solutions, and chaos. The test case is a single-DOF forced Duffing oscillator with freeplay nonlinearity, solved using direct time integration. This work intends to expand on past studies by determining the limits of applicability for each representation and what numerical problems may occur.

Fuzzy control of nonlinear aeroelastic system based on neural network identification

This article presents a fuzzy control method for the limit cycle oscillation (LCO) suppression of nonlinear aeroelastic systems based on the neural network identification algorithm. A prototypical 2D wing section with a single control surface at the trailing edge of the main wing, which contains a symmetrical free play nonlinearity in the pitch degree of freedom, is modeled to illustrate the proposed method. A neural network is used to identify the fuzzy control rules from the existing LCO suppression input and output data. A new fuzzy control rate of the nonlinear aeroelastic system is obtained by adjusting the parameters of the fuzzy control surface. Numerical simulations are conducted to verify the effectiveness of the proposed method.

Adaptive fuzzy control of nonlinear aeroelastic system with measurement noise

This paper presents a limit cycle oscillation (LCO) suppression method of nonlinear aeroelastic system based on adaptive neuro-fuzzy control. A prototypical 2D wing section with a single control surface at the trailing edge of the main wing, which contains a symmetrical freeplay nonlinearity in the pitch degree of freedom, is modelled by SIMULINK (Matlab 2016R) to illustrate the proposed method. Proportional integral differential (PID) controller is used to suppression the LCO of nonlinear aeroelastic system. The control law of the PID controller is identified by neural network. A new fuzzy control law of the nonlinear aeroelastic system is obtained by adjusting the parameters of the fuzzy control system. A nonlinear aeroelastic system with measurement noise in the measurement feedback loop is conducted to verify the effectiveness of the proposed method.

Stability analysis of whirl flutter in rotor-nacelle systems with freeplay nonlinearity

Tiltrotor aircraft are growing in prevalence due to the usefulness of their unique flight envelope. However, aeroelastic stability—particularly whirl flutter stability—is a major design influence that demands accurate prediction. Several nonlinearities that may be present in tiltrotor systems, such as freeplay, are often neglected for simplicity, either in the modelling or the stability analysis. However, the effects of such nonlinearities can be significant, sometimes even invalidating the stability predictions from linear analysis methods. Freeplay is a nonlinearity that may arise in tiltrotor nacelle rotation actuators due to the tension–compression loading cycles they undergo. This paper investigates the effect of a freeplay structural nonlinearity in the nacelle pitch degree of freedom. Two rotor-nacelle models of contrasting complexity are studied: one represents classical whirl flutter (propellers) and the other captures the main effects of tiltrotor aeroelasticity (proprotors). The manifestation of the freeplay in the systems’ dynamical behaviour is mapped out using Continuation and Bifurcation Methods, and consequently the change in the stability boundary is quantified. Furthermore, the effects on freeplay behaviour of (a) model complexity and (b) deadband edge sharpness are studied. Ultimately, the freeplay nonlinearity is shown to have a complex effect on the dynamics of both systems, even creating the possibility of whirl flutter in parameter ranges that linear analysis methods predict to be stable. While the size of this additional whirl flutter region is finite and bounded for the basic model, it is unbounded for the higher complexity model.