Detection and description of non-linear phenomena in experimental modal analysis via linearity plots

Ground vibration tests (GVTs) on aircraft prototypes are mainly performed to experimentally identify the structural dynamic behaviour in terms of a modal model. This assumes a linear dynamic behaviour of the structure. However, in the practice of ground vibration testing it is often observed that structures do not behave in a perfectly linear manner. Non-linearities can be determined, for example, by free play in junctions, hydraulic systems in control surfaces, or friction. This paper compiles measured, typical, non-linear phenomena from various GVTs on large aircraft. The standard procedure in GVTs nowadays is the application of the Harmonic Balance method which linearizes the dynamic behaviour on the level of excitation. The procedure requires a harmonic excitation of the structure which is usually performed during phase resonance testing. The non-linear behaviour is investigated in terms of linearity plots in which the resonance frequency of a mode is plotted as a function of the excitation level. The experimental data is then compatible with all post-processing procedures for the measured results, e.g. updating of the finite element model or flutter calculations. This paper shows measured linearity plots for some typical non-linear phenomena. In the second part of the paper analytical linearity plots for different non-linear stiffness and damping models are considered in order to investigate whether the type of non-linearity can be identified from measured linearity plots. The analytical linearity plots are discussed with respect to their application limits. The analytical linearity plots are used to interpret the experimental linearity plots stemming from various GVTs on different aircraft prototypes. Finally, the observability of non-linear stiffness and non-linear damping characteristics via linearity plots is assessed.     

Self-tuning PID controller to three-axis stabilization of a satellite with unknown parameters

This paper addresses the three-axis stabilization of a satellite system in the presence of the gravity gradient and orbital eccentricity. Multivariable non-linear dynamics of the satellite system are converted into three well-known non-linear canonical independent models with unknown parameters. The new model is efficient and practical for designers to implement and analyze different control methodologies on satellite systems. A self-tuning PID controller is designed on the basis of the new proposed model to produce control signals for three reaction wheels in three axes. An adaptive algorithm is applied to tune and update gains of the PID controller and stability of the closed-loop system is guaranteed by using Lyapunov approach. Numerical simulations are performed to demonstrate feasibility and effectiveness of the self-tuning PID controller and a comparison with a fixed gain PD controller and a variable-structure controller is made.     

基于反演滑模的机动飞行器横向回路控制设计

针对再入机动飞行器模型的参数不确定性以及外界干扰对飞行器控制性能的影响,基于反演控制和滑模控制理论,结合飞行器的动态特性要求,设计了一种基于标称模型的再入机动飞行器横向回路姿态控制方案,并基于Lyapunov方法,给出了整个系统的稳定性证明。控制系统阶跃响应仿真结果表明:系统响应无超调,调节时间为0.6 s,稳态误差为1%,优于指标要求的超调量15%,调节时间1 s,稳态误差5%,证明所提方法对模型参数大范围摄动具有强鲁棒性,且在较大程度上提高了系统的动态性能,最终达到姿态指令的快速高精度跟踪效果。     

Semi-implicit two-level predictor–corrector methods for non-linearly coupled,hydrostatic, barotropic/baroclinic flows

The unsteady shallow-water equations for barotropic/baroclinic (free-surface/density-stratified) flows with non-linear coupling of density transport and momentum are solved using a family of two-time-level, semi-implicit predictor–corrector methods (PC2). The PC2 methods are a general family that includes the popular TRIM method for hydrostatic flows. PC2 is characterised by four ‘θ’ parameters controlling the time ‘n’ and ‘n + 1’ weighting of (1) free surface gradient, (2) predictor step, (3) baroclinic gradient and (4) density temporal interpolation. Stability of the non-linear coupling between momentum and density transport for PC2 is examined in the inviscid limit. Central difference and quadratic (QUICK) spatial interpolation for density are compared. Second-order temporal accuracy for both barotropic and baroclinic flows is simultaneously obtained with appropriate θ parameters, which has previously been shown to be impractical for TRIM. The 2nd-order PC2 method has near-neutral non-linear stability (slightly positive amplification factor) where linear theory predicts exactly neutral stability. QUICK is shown to be preferable to central difference spatial discretisation to reduce the amplification factor. Adjusting the baroclinic weighting or adding small artificial viscosities can stabilise the model for non-linear internal wave simulations.     

基于径向基点插值法的旋转Mindlin板高次刚柔耦合动力学模型

将无网格径向基点插值法(radial point interpolation method,RPIM)用于中心刚体?旋转柔性板的动力学分析.基于浮动坐标系方法和一阶剪切变形理论即Mindlin板理论,考虑剪切变形的影响,并计入板面内变形的非线性耦合变形项,采用径向基点插值法描述板的变形场,保留动能中有关非线性耦合变形项...     

Identification of tire forces using Dual Unscented Kalman Filter algorithm

Nowadays, application of active control systems in vehicles has been developed in order to increase safety and steerability. In these systems, using an appropriate dynamic model can be very effective in increasing the accuracy of simulations and analysis. Tire-road forces are crucial in vehicle dynamics and control since they are the only forces that a vehicle experiences from the ground and have maximum uncertainty on vehicle dynamic model. In order to simulate the non-linear regimes of vehicle motion, the ‘Pacejka’ tire model is being utilized. In this paper, a dynamic model with Dual Unscented Kalman Filter algorithm has been utilized to identify the lateral forces, side slip angle, and normal forces of tires. In order to solve the non-linear least squares problem, these parameters were given as input to the hybrid Levenberg–Marquardt and quasi Newton algorithm to find the Pacejka tire model coefficients in the offline mode. Four degrees of freedom vehicle model combined with Pacejka tire model are used for simulation in various maneuvers. Results show appropriate compatibility with CarSim software.     

Finite time integral sliding mode control of hypersonic vehicles

This study investigates the tracking control problem for the longitudinal model of an airbreathing hypersonic vehicle (AHV) with external disturbances. By introducing finite time integral sliding mode manifolds, a novel finite time control method is designed for the longitudinal model of an AHV. This control method makes the velocity and altitude track the reference signals in finite time. Meanwhile, considering the large chattering phenomenon caused by high switching gains, an improved sliding mode control method based on nonlinear disturbance observer is proposed to reduce chattering. Through disturbance estimation for feedforward compensation, the improved sliding mode controller may take a smaller value for the switching gain without sacrificing disturbance rejection performance. Simulation results are provided to confirm the effectiveness of the proposed approach.     

Damage localization using transient non-linear elastic wave spectroscopy on composite structures

To reduce the costs related to maintenance of aircraft structures, there is the need to develop new robust, accurate and reliable damage detection methods. A possible answer to this problem is offered by newly developed non-linear acoustic/ultrasonic techniques, which monitor the non-linear elastic wave propagation behaviour introduced by damage, to detect its presence and location.In this paper, a new transient non-linear elastic wave spectroscopy (TNEWS) is presented for the detection and localization of a scattered zone (damage) in a composite plate. The TNEWS analyses the uncorrelations between two structural dynamic responses generated by two different pulse excitation amplitudes by using a time-frequency coherence function. A numerical validation of the proposed method is presented. Damage was introduced and modelled using a multi-scale material constitutive model (Preisach-Mayergoyz space).The developed technique identified in a clear manner the faulted zone, showing its robustness to locate and characterize non-linear sources in composite materials     

Non-linear dynamic of rotor–stator system with non-linear bearing clearanceDynamique non-linéaire d'un ensemble rotor–stator comportant un roulement non-linéaire avec jeu

The study deals with a rotor–stator contact inducing vibration in rotating machinery. A numerical rotor–stator system, including a non-linear bearing with Hertz contact and clearance is considered. To determine the non-linear responses of this system, non-linear dynamic equations can be integrated numerically. However, this procedure is both time consuming and costly to perform. The aim of this Note is to apply the Alternate Frequency/Time Method and the ‘path following continuation’ in order to obtain the non-linear responses to this problem. Then the orbits of rotor and stator responses at various speeds are investigated. To cite this article: J.-J. Sinou, F. Thouverez, C. R. Mecanique 332 (2004).     

Finite deformation analysis of linearly viscoelastic simple structures2

In order to determine the effect of finite deformations on the stability and non-linear time-deflection behaviour of linearly viscoelastic uniaxially stressed structures, a series of simple rigid-bar-spring dashpot models were analysed ‘exactly’. The material representation was also kept as simple as possible using the standard three-element solid model.Results obtained indicate that the relaxation behaviour of such a structure depends only on its material properties. The creep response is influenced not only by the load level but most significantly by the instantaneous non-linear elastic characteristics of the structure. For structures exhibiting instantaneous elastic local instability a ‘critical time’ may be defined beyond which equilibrium is impossible. The definition for ‘safe-load-limit’ or viscoelastic critical force usually used in linear stability analyses of viscoelastic columns is generalized.     

Center manifold and multivariable approximants applied to non-linear stability analysis

This paper presents a research devoted to the study of instability phenomena in non-linear model with a constant brake friction coefficient. This paper outlines the stability analysis and a procedure to reduce and simplify the non-linear system, in order to obtain limit cycle amplitudes. The center manifold approach, the multivariable approximants theory, and the alternate frequency/time domain (AFT) method are applied. Brake vibrations, and more specifically heavy trucks grabbing are concerned. The modelling introduces sprag-slip mechanism based on dynamic coupling due to buttressing. The non-linearity is expressed as a polynomial with quadratic and cubic terms. This model does not require the use of brake negative coefficient, in order to predict the instability phenomena. Finally, the center manifold approach, the multivariable approximants, and the AFT method are used in order to obtain equations for the limit cycle amplitudes. These methods allow the reduction of the number of equations of the original system in order to obtain a simplified system, without loosing the dynamics of the original system, as well as the contributions of non-linear terms. The goal is the validation of this procedure for a complex non-linear model by comparing results obtained by solving the full system and by using these methods. The brake friction coefficient is used as an unfolding parameter of the fundamental Hopf bifurcation point.     

Non-linear dynamics of asymmetric structures under 2:2:1 resonance

The non-linear dynamic behavior of a novel model of a single-story asymmetric structure under earthquake and harmonic excitations and near two-to-one internal resonance is investigated. The non-linearities of the proposed model, ignored in conventional linear models, are caused by non-linear inertial coupling between translational and torsional degrees of freedom defined in the directions of a non-inertial rotational system of reference, attached to the center of mass of the floor. The multiple scales method is used to achieve approximately linear solutions for the originally non-linear equations near a two-to-one ratio of external and internal resonant conditions. The suitability of the proposed model is justified by the similarity between the simulated response of the non-linear model and the experimental results. The numerical results of time history and frequency domain analyses illustrate the difference between the non-linear and linear models. Energy transfer from a lower natural frequency excited mode to a higher one due to non-linear interaction in the novel model is shown. The effects of amplitude, frequency detuning parameters, uncoupled lateral and torsional frequencies, and damping ratio on the responses are inspected and some non-linear phenomena such as hysteresis, jumping, hardening, and softening are observed.     

平尾翼型后加载对飞机配平的影响研究

利用MGAERO 软件对搭配不同尾翼模型的某典型布局客机翼身组合体流场进行了计算,研究了平尾翼型后加载对全机配平的影响. 分析表明,采用后加载设计的平尾翼型在配平时升力和阻力损失较小,配合其他手段可以提高全机升阻比,进而改善全机配平后的气动性能. 这一结论从流场计算的角度揭示了客机平尾翼型采用后加载设计这一发展趋势的重要原因.     

A semi-collocated ocean model based on the SOMS approach

In application to the Gulf of Mexico (GOM), a new DieCAST ocean model, which uses a modified Arakawa ‘a’ grid, and the SOMS model, which uses an Arakawa ‘c’ grid, give remarkably similar results. The new model avoids ‘null space’ problems of the standard ‘a’ grid approach by first using fourth-order interpolations to a ‘c’ grid advection velocity, then applying incompressibility to the result. Accuracy is further improved by using fourth-order pressure gradient approximations at ‘a’ grid locations. Incompressibility with general geometry is satisfied efficiently by a fast-converging iteration with a regular gteometry elliptic solver. Results are compared with satellite-measured r.m.s. sea surface elevation anomaly and detailed front structures, climatological mean thermocline and empirical orthogonal functions and other observations.     

Dynamics modeling and stability of robotic systems with discrete-time force control

This paper investigates the dynamic behavior of robotic echanical systems with discrete-time force control. Force control is associated with the constrained motion of a mechanical system. A novel approach is presented to analyze the stability and performance based on the separation of constrained and admissible motions. This results in a model representing the dynamics of the constrained motion of the system. The analysis connects the complex nonlinear model of a mechanical system to a set of abstract delayed oscillators. These oscillator models make it possible to perform a detailed closed-form mathematical analysis of the stability behavior. A planar two-degree-of-freedom (DoF) mechanism is presented as an example to illustrate the material. Results are illustrated by stability charts in the parameter space of mechanical parameters, control gains and the sampling rate.     

FINITE ELEMENT SIMULATION OF GUST LOADING

In the structural design of civil aircraft the critical loads are often those encountered in a gust or atmospheric turbulence. The traditional ‘indicial’ solution is restricted to a simple plate. In this paper a finite element formulation is proposed for an aerofoil or arbitrary shape entering a uniform sharp-edged or sinusoidal gust. The thin rotational gust front and wake in an irrotational flow field are successfully modelled by a novel superposition technique. The finite element solutions are compared with the Kussner function and results by other numerical methods. The agreement is good.     

基于RBF插值技术的CFD/CSD非线性耦合分析方法研究

基于非结构混合网格的N-S方程求解器和结构柔度影响系数法,发展了一种考虑气动、结构非线性的基于RBF插值技术CFD/CSD耦合分析方法,适用于解决现代大展弦比飞机的非线性静气动弹性问题。该方法采用时间相关法（即求解非定常方程组,用长时间的渐近解趋于定常状态）求解静气弹分析时的定常流动。考虑大展弦比飞机结构变形问题为大变形小应力问题,在利用柔度系数法求解结构方程时,假设每次求解结构方程时应力与应变为线性关系,整体静气弹分析过程为非线性关系,因此每次求解结构方程时要更新柔度影响系数矩阵。在非定常N-S方程每求解一个时间步耦合一次结构有限元分析,由于结构有限元分析的时间相对于气动分析时间是很短的,所以这种方法实际上近似使用了一次求解非定常气动力的时间完成了整个静气动弹性分析的过程。对于气动网格与结构有限元网格不一致性,本文采用径向基函数（RBF）插值方法中的TPS方法进行结构弹性变形和气动载荷插值,采用虚功原理完成气动载荷数据交换。为了节省气弹分析时间,采用动网格方法对气动网格进行更新,本文基于RBF插值方法发展一种适用于混合网格（四面体、三棱柱、金字塔和六面体）变形的动网格方法,可以保证附面层网格的质量与分布从而准确模拟其流动。利用该方法对M6机翼、DLR-F6翼身组合体和某大型客机机翼进行了静气动弹性特性分析,结果验证了本文开发的非线性CFD/CSD耦合分析方法的可行性、精确性和高效性。     

Separated Flow and Buffeting Control

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and the flow separations on the upper wing surfaces of civil aircraft induce flow instabilities, ‘buffet’ and then structural vibrations, ‘buffeting’. Buffeting can greatly affect aerodynamic behavior. The buffeting phenomenon appears when the aircraft's Machnumber or angle of attack increases. This phenomenon limits the aircraft's flight envelope. The objectives of this study are to cancel out or decrease the aerodynamic instabilities (unsteady separation, movement of the shock position) due to this type of flow by using control systems. The following actuators can be used: ‘Vortex Generators’ situated upstream of the shock position, a ‘Bump’ located at the shock position, and a new moving part designed by ONERA, situated on the trailing edge of the wing, the ‘Trailing Edge Deflector’ or TED. It looks like an adjustable ‘Divergent Trailing Edge’. It is an active actuator and can take different deflections or be driven by dynamic movements up to 250 Hz. Tests were performed in transonic 2D flow with models well equipped with unsteady pressure transducers. For high lift coefficients, a selected static position of the ‘Trailing Edge Deflector’ increases the wing's aerodynamic performances and delays the onset of buffet. Furthermore, in 2D flow buffet conditions, the ‘Trailing Edge Deflector’, driven by a closed-loop active control using the measurements of the unsteady wall static pressures, can greatly reduce buffet. The aerodynamic performances are not improved to the same extent by the bump actuator. From our experience, there is no effect on buffet or separated flow because of the incorrect positioning of the bump. All that can be observed is a local improvement on the intensity of the shock wave when the bump is very precisely situated at the shock position. Vortex generators have a great impact on the separated flow. The separated flow instabilities are greatly reduced and buffet is totally controlled even for strong instabilities. The aerodynamic performances of the airfoil are also greatly improved.     

An investigation into the significance of the non-linear terms in the equations of motion for a cantilevered beam

Non-linear beam models can provide accurate and efficient results and are therefore used frequently in the structural dynamic analysis of large-aspect-ratio cantilevered structures. Using a beam model is especially beneficial in aeroelastic applications, which are computationally expensive.This paper investigates the significance of each of the non-linear terms in a third-order non-linear beam model and shows the importance of retaining terms up to the third order. The approach presented in this paper provides an effective method to simplify the non-linear equations of motion by including the most significant non-linear terms. The resulting equations preserve the fidelity of the original equations and are easier to manipulate.     

Application of cellular automata modeling to seismic elastodynamics

This article details application of a physics-based cellular automata (CA) computational approach to model seismic events in an idealized linear-elastic medium. Application of rectangular-celled CA to the seismic problem is shown to yield discrete equations equivalent to the centered-difference finite difference (FD) approach. However, it is emphasized that the discrete equations are arrived at from the ‘bottom up’ using local rules vice ‘top-down’ discretization of global partial differential equations. A further distinction between the two methods concerns the location of stresses and its impact on boundary conditions: the CA approach assigns stresses to the cell faces while the FD approach assigns stress collocated with displacement components at a single node. These differences may provide important perspective on modeling arbitrary geometry with a finite difference-like approach based on cell assembly, similar to finite element analysis. Implementation of the CA paradigm using autonomous, local cells fits naturally with object-oriented programming practices and lends itself readily to distributed computing. Results are provided for an example ground-shock simulation in which a differentiated Gaussian pulse acts on the surface of a linear-elastic half-space. The CA perspective suggests a simple treatment for the free-surface boundary condition. Comparison of the computed pressure, shear, and surface waves to those computed using a staggered-grid finite difference approach demonstrates very good agreement. In addition, the simulation results suggest that the CA approach may exhibit less ‘ringing’ as waves pass, and more symmetry in left-ward and right-ward moving waves. Future directions exploiting attractive attributes of the CA approach are suggested, to include large-scale simulation, multi-resolution analysis, and coupled-field modeling.