Cantilever Blade Research Articles

Reduced-Order Models of Blisks With Small Geometric Mistuning

A technique for generating reduced-order models (ROMs) of bladed disks with small geometric mistuning is proposed. Discrepancies in structural properties (mistuning) from blade to blade can cause a significant increase in the maximum vibratory stress. The effects of mistuning have been studied over the past few decades. Many researchers have studied the dynamic behavior of mistuned bladed disks by using ROMs. Many of these techniques rely on the fact that the modes of a mistuned system can be approximated by a linear combination of modes of the corresponding tuned system. In addition, the tuned system modes have been modeled in component mode mistuning by using modal participation factors of cantilevered blade modes. Such techniques assume that mistuning can be well modeled as variations in blade-alone frequencies. However, since geometric deformations contain stiffness and mass variations, mistuning can no longer be captured by cantilevered blade modes alone. To address this, several studies have focused on large and small geometric mistuning. These studies exploited the difference between tuned (with perturbed geometry) and nominal tuned mode shapes. In this work, we extend on that approach and devote particular attention to the development of ROMs of bladed disks with small geometric mistuning. The methodology requires only sector-level calculations and therefore can be applied to highly refined, realistic models of industrial size.

Aerodynamic Performance of a Vibrating Piezoelectric Blade Under Varied Operational and Confinement States

This paper investigates the aerodynamic performance of a piezoelectrically actuated cantilever blade operating in the first mode of vibration. Bulk air-moving capability and local velocity field features have been experimentally measured for different operational conditions and blade confinement states. A testing facility has been developed to determine the pressure-flow rate characteristics of a vibrating blade and simultaneously conduct particle image velocimetry. Under high system resistance, similar to that found in densely packed electronics, flow reversal occurs at the blade outlet region analogous to stall. This is accompanied by increased local unsteadiness in the airflow being emitted. Local turbulence intensity increases by 50% between maximum flow rate and maximum efficiency operating points. The introduction of confinement around a vibrating blade reduces bulk air-moving capability significantly, with pressure rise and volumetric flow rate measured as low as 20% of what can be achieved for an unconfined blade. However, there is an accompanying change in spatial velocity distribution, which focuses the flow normal to the outlet. This has potentially beneficial uses for targeted electronics cooling applications. The findings identify the relationship between local and bulk airflow characteristics that ultimately provides information about the sensitivity of this type of air-moving device in practical installations.

A Study of an Effective Heat-Dissipating Piezoelectric Fan for High Heat Density Devices

Heat dissipation per unit volume has grown rapidly, as the size of modern electronic devices has continued to decrease. The air flow induced by an oscillating cantilever blade enhances the heat transfer performance of high heat density devices. The heat transfer improvement mainly depends on the velocity magnitude and distribution of air streams induced by the vibrating blade. Accordingly, this study numerically and experimentally examines the time-varying flow characteristics of a vibrating cantilever for five blade types. The blades are rectangular or trapezoidal with various widths and actuated at various frequencies. The fluid domain is numerically discretized using a dynamic meshing scheme to model the three-dimensional time-varying vibrating blade. The experiment utilizes nine hot-wire velocity meters to measure the average velocities. The flow structure with streamlines and velocity contours of the induced air flow are determined at various section planes. The results show that a major maximum-velocity region appears around the blade tip and that four minor local-maximum-velocity regions appear at the four corners. In addition, the width and width ratio of the blade significantly affects the velocity distribution of the flow induced by the vibrating cantilever blade.

Dynamic characteristics and enhanced performance of a novel wind turbine rotor analyzed via experiments and numerical simulation

The enormous demand for large wind turbine rotors has led to a need to develop high-performance and reliable wind turbine rotors. The flexibility of the huge blade was a challenge in creating a balanced design with regard to dynamic behavior, mass, and power output. To enhance the wind turbine rotor, a newly designed wind turbine system with a supporting rod and damper was proposed and investigated. A scaled blade was experimentally tested, with the results indicating an increase in both frequency and damping of the system. Through the use of a self-coded numerical model, the correlations between the design constraints and the dynamic behavior, tip displacement, and additional mass of the rotor were demonstrated. This showed that the novel rotor has some preferable characteristics in both static and dynamic aspects. In particular, this blade is stiffer and has a smaller tip displacement compared with a traditional cantilevered blade. These characteristics enabled the effective application of the novel rotor to a 5-MW wind turbine to achieve a 15.16% power output increase based on the blade element momentum theory with Prandtl correction, as well as 5.1% mass savings. Copyright © 2016 John Wiley & Sons, Ltd.

Experimental evaluation of a cruciform piezoelectric energy harvester

This paper describes the development and experimental evaluation of a particular type of piezoelectric energy harvester, composed of four aluminum cantilever blades to which piezoelectric patches are bonded, in such way that electric energy is generated when the blades undergo bending vibrations. Concentrated masses, whose values can be varied, are attached to the tips of the blades. Due to the geometric shape of the harvester, in which the four blades are oriented forming right angles, the harvester is named cruciform. As opposed to the large majority of previous works on the subject, in which harvesters are excited at their bases by prescribed acceleration, herein the harvester is connected to a vibrating structure excited by an imbalance force. Hence, the amount of harvested energy depends upon the dynamic interaction between the harvester and the host structure. Laboratory experiments were carried-out on a prototype connected to a tridimensional truss. The experimental setup includes a force generator consisting of an imbalanced disc driven by an electrical motor whose rotation is controlled electronically, a voltage rectifier circuit, and a battery charged with the harvested energy. After characterization of the dynamic behavior of the harvester and the host structure, both numerically and experimentally, the results of experiments are presented and discussed in terms of the voltage output of the piezoelectric transducers as function of the excitation frequency and the values of the tip masses. Also, the capacity of the harvester to charge a Lithium battery is evaluated.

Damage localization in a residential-sized wind turbine blade by use of the SDDLV method

The stochastic dynamic damage location vector (SDDLV) method has previously proved to facilitate effective damage localization in truss- and plate-like structures. The method is based on interrogating damage-induced changes in transfer function matrices in cases where these matrices cannot be derived explicitly due to unknown input. Instead, vectors from the kernel of the transfer function matrix change are utilized; vectors which are derived on the basis of the system and state-to-output mapping matrices from output-only state-space realizations. The idea is then to convert the kernel vectors associated with the lowest singular values into static pseudo-loads and apply these alternately to an undamaged reference model with known stiffness matrix. By doing so, the stresses in the potentially damaged elements will, theoretically, approach zero. The present paper demonstrates an application of the SDDLV method for localization of structural damages in a cantilevered residential-sized wind turbine blade. The blade was excited by an unmeasured multi-impulse load and the resulting dynamic response was captured through accelerometers mounted along the blade. The static pseudo-loads were applied to a finite element (FE) blade model, which was tuned against the modal parameters of the actual blade. In the experiments, an undamaged blade configuration was analysed along with different damage scenarios, hereby testing the applicability of the SDDLV method.

Statistical evaluation of characteristic SDDLV-induced stress resultants to discriminate between undamaged and damaged elements

The stochastic dynamic damage location vector (SDDLV) method utilizes the vectors from the kernel of a damaged-induced transfer function matrix change to localize damages in a structure. The kernel vectors associated with the lowest singular values are converted into static pseudo-loads and applied alternately to an undamaged reference model with known stiffness matrix, hereby, theoretically, yielding characteristic stress resultants approaching zero in the damaged elements. At present, the discrimination between potentially damaged elements and undamaged ones is typically conducted on the basis of modified characteristic stress resultants, which are compared to a pre-defined tolerance value, without any thorough statistical evaluation. In the present paper, it is tested whether three widely-used statistical pattern-recognition-based damage-detection methods can provide an effective statistical evaluation of the characteristic stress resultants, hence facilitating general discrimination between damaged and undamaged elements. The three detection methods in question enable outlier analysis on the basis of, respectively, Euclidian distance, Hotelling's T2 statistics, and Mahalanobis distance. The study of the applicability of these methods is based on experimentally obtained accelerations of a cantilevered residential-sized wind turbine blade subjected to an unmeasured multi-impulse load. The characteristic stress resultants are derived by applying the static pseudo-loads to a representative finite element (FE) model of the actual blade.

Experimental investigation on forced response of mistuned bladed disc rotor in an aero-engine

An experimental investigation on forced response of a first stage compressor mistuned bladed disc rotor is performed in an operational aero-engine. It is desired to evaluate the maximum blade amplitude response of the bladed disc for the existing blade mistuning pattern depending on blade mode, exciting engine order, and aerodynamic influences. Finite element analysis on a single cantilever blade is performed and the possible resonances are estimated. Bench tests on a stand-alone blade are conducted to verify the finite element predictions. Experiments are conducted on a full-scale core engine using blade tip timing system. Blade tip timing system utilizes noncontacting sensors installed on the engine casing which provides vibration responses of all the blades in a rotor. The vibration responses obtained from blade tip timing system is validated using simultaneously measured strains from a few blade-mounted strain gages. A good correlation is observed between the predicted and measured blade responses. The effect of blade mistuning on the amplitude resonant response of the bladed disc is explored experimentally. It is observed that the maximum amplitude response of the blades due to forced resonant excitation is far below the fatigue limits for the prevailing blade mistuning pattern. The present exercise of quantifying the blade resonant responses through efficient implementation of the blade tip timing system is crucial to understand and thus avoid high-cycle fatigue, which is one of the most serious problems in aero-engine development.

Full-field dynamic strain prediction on a wind turbine using displacements of optical targets measured by stereophotogrammetry

Health monitoring of rotating structures (e.g. wind turbines and helicopter blades) has historically been a challenge due to sensing and data transmission problems. Unfortunately mechanical failure in many structures initiates at components on or inside the structure where there is no sensor located to predict the failure. In this paper, a wind turbine was mounted with a semi-built-in configuration and was excited using a mechanical shaker. A series of optical targets was distributed along the blades and the fixture and the displacement of those targets during excitation was measured using a pair of high speed cameras. Measured displacements with three dimensional point tracking were transformed to all finite element degrees of freedom using a modal expansion algorithm. The expanded displacements were applied to the finite element model to predict the full-field dynamic strain on the surface of the structure as well as within the interior points. To validate the methodology of dynamic strain prediction, the predicted strain was compared to measured strain by using six mounted strain-gages. To verify if a simpler model of the turbine can be used for the expansion, the expansion process was performed both by using the modes of the entire turbine and modes of a single cantilever blade. The results indicate that the expansion approach can accurately predict the strain throughout the turbine blades from displacements measured by using stereophotogrammetry.

Fiber Bragg Grating-Based System for 2-D Analysis of Vibrational Modes of a Steel Propeller Blade

This paper reports results obtained using fiber Bragg grating (FBG)-based sensors to investigate the displacement mode shapes of a cantilevered steel propeller blade, using FBG arrays for vibration monitoring for the first time. The experimental data obtained are cross compared with those from a finite element analysis of the same blade, undertaken using proprietary software. In the experimental configuration used, a network of gratings, forming a series of sensor arrays, was mounted on the blade under study to monitor its bending modes, while a further set was mounted perpendicular to this array to monitor torsional modes. To obtain the shape of the strain modes generated in the blade at specific frequencies, the dynamic response of the FBG arrays, as a function of time, was captured and then processed using Fourier transform algorithms to show the natural frequencies of the blade. As a result, the displacement modes shapes for the bending, torsional, and coupled modes of the first nine natural frequencies of the plate were obtained. The experimental data show very good agreement with theoretical analysis. This paper demonstrates the potential of using the lightweight, minimally invasive sensing technique described for the analysis of propeller blades and, thus, illustrating an effective method to overcome the deleterious effects of propellers seen in some commercial propeller designs.

광섬유 브레그격자 센서를 이용한 풍력발전기 날개의 모드 해석

소형 풍력발전기 날개의 동적 거동에 대한 실험적 연구를 수행하였다. 날개를 따라 배열된 광 브레그 격자 (Fiber Bragg Grating) 센서를 이용하여 날개 표면에서의 변형률(strain)을 측정하였다. 충격 햄머 실험 (Impact Hammer Test)을 통하여, 1차 및 고차 모드의 공진주파수를 측정하였다. 광섬유 센서를 이용한 실험결과를 스트레인 게이지를 이용한 실험 결과와 비교한 결과 모드 주파수는 매우 유사하였다. 하지만, 광섬유 센서의 경우 스트레인 게이지에서 감지하지 못하는 모드를 감지할 수 있었다. 또한, 실험으로부터 얻은 변형률 모드를 이용하여 근사적으로 날개의 변위 모드를 추정하였다.

Natural frequencies of a pre-twisted blade in a centrifugal force field

In this paper, starting with the thin shell theory, the governing partial differential equation of motion for the transverse deflection of a rotating pre-twisted plate is derived. Strain–displacement relationships include the effect of warping of the cross-section due to twist–bend coupling effect introduced as a result of pre-twist in the plate of non-circular (rectangular) cross-section. Then the equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil blade by considering it as a plate of an equivalent rectangular cross-section subjected to a quasi-static load due to centrifugal force field. The analytical derivation considers both the stress-stiffening as well as stress-softening effects of the centrifugal forces on the spinning airfoil. The partial differential equation governing the flexural motion of the plate is transformed into a matrix-eigenvalue form using a Rayleigh–Ritz technique. The plate deformations are represented by a set of ‘admissible’ sinusoidal trial functions, which fully satisfy all the clamped-end constrains as well as the free-edge boundary conditions. The results of the analytical model exhibit an excellent agreement with the previously published test data both for thin and thick plate geometries and even in highly twisted configurations. The results of the eigenvalue solution are presented in a non-dimensional form for plates of varying aspect ratios and different amounts of pre-twist in the plate. The numerical results are directly applicable in determining the static and running frequencies of typical blades used in turbo-machinery.

Nonlinear Vibration Characteristics of a Flexible Blade with Friction Damping due to Tip-Rub

An approximate approach is proposed in this paper for analyzing the two-dimensional friction contact problem so as to compute the dynamic response of a structure constrained by friction interfaces due to tip-rub. The dynamical equation of motion for a rotational cantilever blade in a centrifugal force field is established. Flow-induced distributed periodic forces and the internal material damping in the blade are accounted for in the governing equation of motion. The Galerkin method is employed to obtain a three-degree-of-freedom oscillator with friction damping due to tip-rub. The combined motion of impact and friction due to tip-rub produced a piecewise linear vibration which is actually nonlinear. Thus, a complete vibration cycle is divided into successive intervals. The system possesses linear vibration characteristic during each of these intervals, which can be determined using analytical solution forms. Numerical simulation shows that the parameters such as gap of the tip and the rotational speed of the blades have significant effects on the dynamical responses of the system. Finally, the nonlinear vibration characteristics of the blade are investigated in terms of the Poincare graph, and the frequency spectrum of the responses and the amplitude-frequency curves.

Nonlinear Vibration Characteristics of a Flexible Blade with Friction Damping due to Tip-Rub

An approximate approach is proposed in this paper for analyzing the two-dimensional friction contact problem so as to compute the dynamic response of a structure constrained by friction interfaces due to tip-rub. The dynamical equation of motion for a rotational cantilever blade in a centrifugal fo rce field is established. Flow-induced distributed periodi c forces and the internal material damping in the blade are accounted for in the governing equation of motion. The Galerkin method is employed to obtain a three-degree-of-freedom oscillator with frict ion damping due to tip-rub. The combined motion of impact and friction due to tip-rub produced a piecewise linear vibration which is actually nonlinear. Thus, a complete vibration cycle is divided into successive intervals. The system possesses linear vibrati on characteristic during each of these intervals, which can be determined using analytical solution forms. Numerical simulation shows that the parameters such as gap of the tip and the rotational speed of the blades have significant effects on the dynamical respons es of the system. Finally, the nonlinear vibration characte ristics of the blade are investigated in terms of the Poincare graph, and the frequency spectrum of the responses and the amplitude-frequency curves.

Extended-time-scale creep measurement on Maraging cantilever blade springs

Two controlled temperature facilities were built to induce an accelerated creep rate in a Maraging steel GAS spring and to measure the material's creep over an artificially extended period of time. The data acquisition of the first experiment lasted for almost a year, but then the blades were allowed to creep for six more years before measuring the permanent deformation integrated over time. The data from this first experiment was polluted by a defect in the data acquisition software, but yielded overall creep limits and an evaluation of the Arrhenius acceleration of creep speed with temperature (1.28±0.13 °C −1). The duration of the second experiment was only 1 year but more free of systematic errors. The effective test period of this second experiment (normalized with the Arrhenius acceleration measured in the first experiment) extends in billions of years showing no sign of anomalous creep. The result of both experiments also produced a simple procedure capable of eliminating all practical effects of creep from the Advanced LIGO seismic isolation and suspensions. Measurements of creep under various stress levels, and of the thermal variations of Young's modulus (2.023 (±0.013)×10 −4 °C −1) are reported as well.

Research on the Chatter Suppression During Machining Thin-Walled Complex Blades

This paper presents a new method for chatter suppression during finishing thin-walled blades. Traditionally auxiliary support is used to increase the stiffness of the structure, however, the shrink or dilate effect of the packing material, such as wax and rosin, is difficult to control, so the machining precision could not be assured. Based on the principle of stiffness optimization principle, the rigidity of the cantilever blade tip and leading/trailing edge region are improved with the non-uniform allowances distribution. The milling experiments have showed that the proposed strategy could be successfully used to realize the suppression of chatter vibration during flexible parts machining.

Monolithic geometric anti-spring blades

In this article we investigate the principle and properties of a vertical passive seismic noise attenuator conceived for ground based gravitational wave interferometers. This mechanical attenuator based on a particular geometry of cantilever blades called monolithic geometric anti springs (MGAS) permits the design of mechanical harmonic oscillators with very low resonant frequency (below 10 mHz). Here we address the theoretical description of the mechanical device, focusing on the most important quantities for the low-frequency regime, on the distribution of internal stresses, and on the thermal stability. In order to obtain physical insight of the attenuator peculiarities, we devise some simplified models, rather than use the brute force of finite element analysis. Those models have been used to optimize the design of a seismic attenuation system prototype for LIGO advanced configurations and for the next generation of the TAMA interferometer.

Dynamic characteristics of a flexible bladed-rotor with Coulomb damping due to tip-rub

Starting with the basic dynamical equations for a rotating radial cantilever blade in a centrifugal force field, a system of equations are derived for a fully-bladed flexible rotor (shaft and disk) supported by a set of bearings at multiple locations. The dynamical equations include the effect of the rotary inertia and gyroscopic moments as a result of both shaft bending as well as staggered blades flexing in-and-out of the plane of the disk. The governing equations also account for internal material damping in the shaft and the external damping in the bearing. In addition to the unbalance load at the disk location, the shaft may be subjected to a torque and axial forces. In the analytical derivation, the blade tips are considered to be rubbing against the outer case introducing Coulomb damping in the system. Transient response of the rotor with the blades deforming due to rub during the acceleration and deceleration through the resonance is discussed. Numerical results are presented for this highly non-linear impact dynamics problem of hard rub with Coulomb friction. The effect of blade tip rub forces transmitted to the shaft are analyzed in terms of the dynamic stability of the rotor.

Fundamental vibration of rotating cantilever blades with pre-twist

A non-linear strain–displacement relationship of a pre-twisted conical shell on the general shell theory is utilized, and a method for vibration of a rotating cantilever conical shell with pre-twist is developed by the principle of virtual work and the Rayleigh–Ritz method. Firstly, deformation and stress resultants caused by rotation are analyzed. Secondly, an equilibrium of energy for vibration of a pre-twisted conical shell having the conditions achieved in the first process is given and then an eigenfrequency equation of a rotating cantilever conical shell with pre-twisted is formulated. The effects of parameters such as an angular velocity, a radius of a hub, a setting angle, a twist angle, a subtended angle and a tapered ratio of cross-section on the fundamental vibration are investigated.

지그재그 삼각형 유한요소를 이용한 점탄성물질이 심어진 비틀린 복합재료판의 진동해석

고차 지그재그이론을 이용하여 면내회전 자유도를 갖는 3절점 삼각형요소를 개발하여 점탄성물질이 심어진 비틀린 복합재료 판의 진동을 해석하였다. 평면상에서 점탄성물질과 가장자리에서 박리현상을 방지하기 위하여 사용된 경계물질사이의 경계면상에서 전단력 적합조건이 사용되었다. 본 연구에서 개발된 지그재그 삼각형요소를 이용하여 점탄성물질이 심어진 비틀린 외팔보 복합재료 판의 고유진동수와 감쇠계수를 계산하고 실험치 및 판과 고체요소를 혼합한 MSC/NASTRAN결과치와 비교하였다. A three node triangular element with drilling rotations incorporating improved higher-order zig-zag theory(HZZT) is developed to analyze the vibration of pretwisted composite plates with embedded damping layer. Shear force matching conditions are enforced along the interfaces between the embedded damping patch and the border patch by matching the shear forces of the embedded damping patch to the shear forces of the adjacent border patch. The natural frequencies and modal loss factors are calculated for cantilevered pretwisted composite blade with damping core with the present triangular element, and compared to experiments and MSC/NASTRAN using a layered combination of plate and solid elements.