OPTIMUM DESIGN OF A HELICOPTER ROTOR FOR LOW VIBRATION USING AEROELASTIC ANALYSIS AND RESPONSE SURFACE METHODS

An aeroelastic analysis based on finite elements in space and time is used to model the helicopter rotor in forward flight. The rotor blade is represented as an elastic cantilever beam undergoing flap and lag bending, elastic torsion and axial deformations. The objective of the improved design is to reduce vibratory loads at the rotor hub that are the main source of helicopter vibration. Constraints are imposed on aeroelastic stability, and move limits are imposed on the blade elastic stiffness design variables. Using the aeroelastic analysis, response surface approximations are constructed for the objective function (vibratory hub loads). It is found that second order polynomial response surfaces constructed using the central composite design of the theory of design of experiments adequately represents the aeroelastic model in the vicinity of the baseline design. Optimization results show a reduction in the objective function of about 30 per cent. A key accomplishment of this paper is the decoupling of the analysis problem and the optimization problems using response surface methods, which should encourage the use of optimization methods by the helicopter industry.     

基于特征值方法的旋翼尾迹稳定性分析

给出一个旋翼尾迹线性化稳定性分析的方法.在该方法中,尾迹涡线被离散为直线涡段,尾迹的扰动归结为涡元端点的扰动,考虑了桨尖涡的自诱导和涡线的互诱导以及桨尖涡与桨叶的实际干扰.使用该方法,分别以UH-1H和AH-1G模型旋翼为例,对悬停和前飞状态的旋翼尾迹的稳定性进行计算和分析.结果表明:旋翼尾迹运动存在大于0的特征值,是内在不稳定的,且最大发散率随波数变化呈现出一定规律性;前飞与悬停状态不同,其最大发散率减小,不稳定性减弱.     

Parametrically excited helicopter ground resonance dynamics with high blade asymmetries

The present work is aimed at verifying the influence of high asymmetries in the variation of in-plane lead-lag stiffness of one blade on the ground resonance phenomenon in helicopters. The periodical equations of motions are analyzed by using Floquet's Theory (FM) and the boundaries of instabilities predicted. The stability chart obtained as a function of asymmetry parameters and rotor speed reveals a complex evolution of critical zones and the existence of bifurcation points at low rotor speed values. Additionally, it is known that when treated as parametric excitations; periodic terms may cause parametric resonances in dynamic systems, some of which can become unstable. Therefore, the helicopter is later considered as a parametrically excited system and the equations are treated analytically by applying the Method of Multiple Scales (MMS). A stability analysis is used to verify the existence of unstable parametric resonances with first and second-order sets of equations. The results are compared and validated with those obtained by Floquet's Theory. Moreover, an explanation is given for the presence of unstable motion at low rotor speeds due to parametric instabilities of the second order.     

Helicopter noise in hover: Computational modelling and experimental validation

The aeroacoustic characteristics of a helicopter rotor are calculated by a new method, to assess its applicability in assessing rotor performance in hovering. Direct solution of the Euler equations in a noninertial coordinate system is used to calculate the near-field flow around the spinning rotor. The far-field noise field is calculated by the Ffowcs Williams–Hawkings (FW–H) method using permeable control surfaces that include the blade. For a multiblade rotor, the signal obtained is duplicated and shifted in phase for each successive blade. By that means, the spectral characteristics of the far-field noise may be obtained. To determine the integral aerodynamic characteristics of the rotor, software is written to calculate the thrust and torque characteristics from the near-field flow solution. The results of numerical simulation are compared with experimental acoustic and aerodynamic data for a large-scale model of a helicopter main rotor in an open test facility. Two- and four-blade configurations of the rotor are considered, in different hover conditions. The proposed method satisfactorily predicts the aerodynamic characteristics of the blades in such conditions and gives good estimates for the first harmonics of the noise. That permits the practical use of the proposed method, not only for hovering but also for forward flight.     

Interpolation of aerodynamic damping of lifting rotors in forward flight from measured response variance

With the necessity of experimentally measuring the aerodynamic derivatives of lifting rotors in forward flight, it becomes desirable for reasons of parametric resonance to measure the variance data of the response to white noise inputs, and then to interpolate with the numerically computed variance values for different assumed values of the damping parameter or the inertia number. Accordingly, methods are developed to compute directly the blade response variance matrix up to high rotor advance ratios for a series of inertia numbers. Numerical results refer to a rigid blade flapping model with an elastically restrained flapping hinge at the rotor centre. Different combinations of the advance ratio, tip-loss factor, flapping restraint parameter and inertia number are included. The white noise excitation is treated without and with input modulation, the latter case including azimuthwise blade input variation. The mean square response study of the model shows that up to an advance ratio of the order of 0·3 and at conventional values of the inertia number the perturbation approach is satisfactory. The numerical scheme, in addition to solving directly the response variance equations which are similar in structure to the original blade dynamic equations, makes use of the fact that the variable part of system parameter functions are independent of the inertia number. Compared to earlier studies the methods therefore offer significant saving in machine time. Numerical results pertaining to previously proposed experimental and analytical models show excellent agreement.     

Soil influence on unbalance response and stability of a simple rotor-foundation system

The equations of motion are set up for a simple rotor (Jeffcott or Laval rotor) on a rigid foundation mass resting on an elastic half space (soil). The unbalance response and the stability limit against self-excited vibrations caused by the internal damping of the rotating shaft are calculated. The numerical results presented as response diagrams and stability graphs show that the damping effect of the soil on the system, due to radiation of energy, may have a very positive influence on the smooth running of the rotors.     

On the definition of the natural frequency of oscillations in nonlinear large rotation problems

Computational multibody system algorithms allow for performing eigenvalue analysis at different time points during the simulation to study the system stability. The nonlinear equations of motion are linearized at these time points, and the resulting linear equations are used to determine the eigenvalues and eigenvectors of the system. In the case of linear systems, the system eigenvalues remain the same under a constant coordinate transformation; and zero eigenvalues are always associated with rigid body modes, while nonzero eigenvalues are associated with non-rigid body motion. These results, however, cannot in general be applied to nonlinear multibody systems as demonstrated in this paper. Different sets of large rotation parameters lead to different forms of the nonlinear and linearized equations of motion, making it necessary to have a correct interpretation of the obtained eigenvalue solution. As shown in this investigation, the frequencies associated with different sets of orientation parameters can differ significantly, and rigid body motion can be associated with non-zero oscillation frequencies, depending on the coordinates used. In order to demonstrate this fact, the multibody system motion equations associated with the system degrees of freedom are presented and linearized. The resulting linear equations are used to define an eigevalue problem using the state space representation in order to account for general damping that characterizes multibody system applications. In order to demonstrate the significant differences between the eigenvalue solutions associated with two different sets of orientation parameters, a simple rotating disk example is considered in this study. The equations of motion of this simple example are formulated using Euler angles, Euler parameters and Rodriguez parameters. The results presented in this study demonstrate that the frequencies obtained using computational multibody system algorithms should not in general be interpreted as the system natural frequencies, but as the frequencies of the oscillations of the coordinates used to describe the motion of the system.     

A COUPLED FLUID-STRUCTURE ANALYSIS FOR 3-D INVISCID FLUTTER OF IV STANDARD CONFIGURATION

A three-dimensional non-linear time-marching method and numerical analysis for aeroelastic behaviour of an oscillating blade row is presented. The approach is based on the solution of the coupled fluid-structure problem in which the aerodynamic and structural equations are integrated simultaneously in time. In this formulation of a coupled problem, the interblade phase angle at which a stability (or instability) would occur is a part of the solution. The ideal gas flow through multiple interblade passage (with periodicity on the whole annulus) is described by the unsteady Euler equations in the form of conservative laws, which are integrated by use of the explicit monotonic second order accurate Godunov-Kolgan volume scheme and a moving hybrid H-H (or H-O) grid. The structure analysis uses the modal approach and 3-D finite element model of the blade. The blade motion is assumed to be a linear combination of modes shapes with the modal coefficients depending on time. The influence of the natural frequencies on the aerodynamic coefficient and aeroelastic coupled oscillations for the Fourth Standard Configuration is shown. The stability (instability) areas for the modes are obtained. It has been shown that interaction between modes plays an important role in the aeroelastic blade response. This interaction has an essentially non-linear character and leads to blade limit cycle oscillations.     

Three-dimensional modelling and dynamic analysis of an automatic ball balancer in an optical disk drive

Dynamic behaviours and stability of an automatic ball balancer (ABB) in an optical disk drive are analyzed based on the proposed three-dimensional dynamic model. For dynamic analysis, the feeding deck with the ball balancer and a spindle motor is modelled as a rigid body with six degrees of freedom. The nonlinear equations of motion are derived using Lagrange's equation in order to describe the translational and rotational motions of the system. From the derived nonlinear equations, the linearized equations of motion in the neighbourhood of a balanced equilibrium position are obtained by the perturbation method. These equations are coupled, linear, differential equations with time-dependent periodic coefficients, from which the stability of the system is analyzed by using the Floquet theory. Finally, the time responses are computed to verify the results of the stability analysis, and to investigate the balancing performance of the ABB.     

DYNAMIC RESPONSE AND STABILITY ANALYSIS OF AN AUTOMATIC BALL BALANCER FOR A FLEXIBLE ROTOR

Dynamic stability and time responses are studied for an automatic ball balancer of a rotor with a flexible shaft. The Stodola-Green rotor model, of which the shaft is flexible, is selected for analysis. This rotor model is able to include the influence of rigid-body rotations due to the shaft flexibility on dynamic responses. Applying Lagrange's equation to the rotor with the ball balancer, the non-linear equations of motion are derived. Based on the linearized equations, the stability of the ball balancer around the balanced equilibrium position is analyzed. On the other hand, the time responses computed from the non-linear equations are investigated. This study shows that the automatic ball balancer can achieve the balancing of a rotor with a flexible shaft if the system parameters of the balancer satisfy the stability conditions for the balanced equilibrium position.     

Scale effects in the bending vibrations of helicopter rotor blades

Scale effects and dynamic similarity in the bending vibrations of helicopter rotor blades are examined by expressing the first three modes of bending vibration of a uniform, conventional rotor blade by a series of Legendre polynomials as suggested by Wilde and others. The natural frequency ratios for these three modes are then determined as functions of a dynamic similarity parameter over the entire range of designs and operating conditions from very flexible, rapidly rotating blades to highly rigid, slowly turning conditions.     

Effect of phase angle on multibladed rotor flutter

A theoretical technique for predicting the flutter characteristics of a helicopter rotor is presented. The effect of phase angle on flutter speed of a two-bladed rotor in hovering and axial flight is determined. For this purpose, a uniform and untwisted rotor blade with coupled flapwise bending and torsional degrees of freedom is considered. The transmission matrix method is used to obtain the natural vibration characteristics of the system. An unsteady aerodynamic theory is used to obtain the aerodynamic loading in compressible flow.     

A MATHEMATICAL MODEL FOR WIND TURBINE BLADES

A mathematical model for an elastic wind turbine blade mounted on a rigid test stand is derived and compared with experimental results. The linear equations of motion describe small rotations of the test stand, blade lateral deflections and rotation of the chord. Warping, extension and tilt of the cross-sections are slaved to the dependent minimal co-ordinates in order to reduce the number of state variables. Using the principle of virtual work, a procedure is employed which combines the volume discretization of general “solid”, or shell-type finite elements (FE), with the approach of global form functions (stretching over the whole blade length). The equations of motion are solved as an eigenvalue problem and the results are compared with an experimental modal analysis of a 19 m long blade. The computed eigenfrequencies fit well, but the mathematical model underestimates the pitch motion of the blade chord. Parameter studies show the effect of warping. Despite the few degrees of freedom and uncertainties in the model parameters, the mathematical model approximates the measured blade dynamics well.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

大展弦比气弹机翼柔性非线性变形的气动力效应分析与风洞试验

从飞行器刚弹耦合动力学模型出发,引入柔性机翼准定常假设,建立大柔性飞行器非线性静气动弹性气动力方程,利用非线性迭代求解思路模拟了柔性飞行器的静气动弹性响应行为,开展了大展弦比飞机静气动弹性风洞试验验证,采用气动力有限基本解与机翼的耦合计算,发现了大柔性飞机大变形状态下载荷及结构变形形式随风速的变化规律.传统基于小变形假设的线性分析方法和刚体分析由于无法考虑气动面随结构变形的曲面气动力因素和结构变形后的非线性刚度特性,均与风洞试验存在一定的误差.对于大展弦比柔性飞机的非线性静气动弹性分析十分必要.      

Circular whirling and stability due to unbalanced magnetic pull and eccentric force

This rapid communication is concerned with the circular whirling and stability of a model rotor in a synchronous generator under no load, subjected to unbalanced magnetic pull and mass eccentric force. The analysis is focused on the synchronous whirling of the rotor. Based on the existing analytical expression for unbalanced magnetic pull with any pole-pair number, the nonautonomous system of differential equations of motion with parametrically exciting force is transformed to an autonomous one by introducing a rotating coordinate frame. The circular whirlings of the model rotor are thus converted into equilibrium solutions to the autonomous system, which can be obtained by solving a system of polynomial equations with two unknowns only. Furthermore, stability of these equilibrium solutions is determined by applying the linearized stability criterion. An example is used to illustrate the proposed analytical method.     

直升机交互式桨叶振动调整仿真研究

为解决目前机务维修培训中存在的物理样机难以满足需求的问题,以实际机务维修工作中旋翼振动调整方法为参考,提出在虚拟维修训练系统的基础上实现受训人员对虚拟样机旋翼桨叶进行振动调整仿真操作,以期达到训练目的。根据非沉浸式虚拟维修训练系统的特点设计交互仿真操作逻辑,确定仿真操作步骤,分析座舱振动可视化表示方法,专用虚拟样机的构建方法以及虚拟仪器的实现方式和数据组织结构形式,并将该仿真实验以模块化形式添加到虚拟维修训练系统中,满足拆装条件下对虚拟样机的交互操作需求。最后以欧直松鼠直升机虚拟样机旋翼振动调整方法为例验证该操作仿真逻辑的可用性。     

基于多信道无线传输的旋翼载荷测试技术研究

直升机旋翼系统载荷和强度飞行试验是对真实大气环境中旋翼系统应力载荷谱的研究,它提供的真实数据是理论计算所不能提供的。因此旋翼系统载荷试飞是直升机设计定型试飞中极其重要的项目。针对直升机旋翼系统载荷测试技术需求,采用模块化、冗余度和高集成的设计理念,通过多信道无线传输设计等技术,将采集的动态载荷数据调制、发射与解调,实现了多通道、高带宽和精同步的旋翼系统载荷数据采集与监控。该技术对于直升机旋翼系统载荷试飞中遇到的类似问题具有一定的借鉴意义。     

Frequency-domain method for discrete frequency noise prediction of rotors in arbitrary steady motion

A novel frequency-domain formulation for the prediction of the tonal noise emitted by rotors in arbitrary steady motion is presented. It is derived from Farassat's ‘Formulation 1A’, that is a time-domain boundary integral representation for the solution of the Ffowcs-Williams and Hawkings equation, and represents noise as harmonic response to body kinematics and aerodynamic loads via frequency-response-function matrices. The proposed frequency-domain solver is applicable to rotor configurations for which sound pressure levels of discrete tones are much higher than those of broadband noise. The numerical investigation concerns the analysis of noise produced by an advancing helicopter rotor in blade–vortex interaction conditions, as well as the examination of pressure disturbances radiated by the interaction of a marine propeller with a non-uniform inflow.     

旋转运动声源的频率波动修正波束形成方法

针对前飞状态的旋翼气动噪声信号频率存在周期性波动,且频域波束形成方法只能应用于稳态声源的问题,提出一种频率波动声源的波束形成方法。该方法利用已知的声源频率变化规律进行频率修正,在时域将频率波动信号等效为单频信号,基于该单频信号进行波束形成声源定位,实现了旋转运动的频率波动声源准确定位。数值仿真结果表明,提出的方法能够在频率波动幅值为127 Hz的情况下准确呈现出声源分布情况。在旋翼模型的风洞试验中,利用提出的频域波束形成方法其声成像结果中声源最大能量位置均在旋转轨迹上,而未进行频率修正的波束形成方法结果无法准确呈现出声源的位置。该方法扩展了频域波束形成方法中的单频声源假设,实现了旋转运动声源在频率波动状态下的波束形成,适用于前飞状态下旋翼气动噪声源的声源定位。