Centrifugal Stiffening Effect Research Articles

Control of edgewise vibration of a pre-twisted rotating beam with shunted piezoelectric tuned mass damper

The edgewise vibration control of a pre-twisted beam, rotating in the vertical plane, is considered in this study. A piezoelectric tuned mass damper (PTMD) comprising a spring–mass–damper system and a piezoelectric transducer with its associated shunt circuit is considered the control device. A finite element formulation of the pre-twisted rotating beam with the attached PTMD is developed while considering the effect of centrifugal stiffening due to rotation and the effect of gravity force. The natural frequencies of the beam (without PTMD) obtained using the present formulation are corroborated by comparing them with the results derived from Ansys. A single PTMD attached at the tip end of the beam can effectively suppress the response of the beam subjected to a distributed load in the edgewise direction. The best values of resistance and inductance of the shunt circuit are obtained through a parametric study on the peak and root mean square value of the beam tip response. A qualitative difference in the uncontrolled as well as in the controlled responses can be noticed while considering the effect of the gravity load component in the edgewise direction. Further, it has been observed that the conventional Tuned Mass Damper (TMD) provides better control than the PTMD system for ideally tuned conditions, however, the control performance of TMD degrades rapidly as compared to the PTMD in the presence of detuning. This demonstrates the effectiveness of PTMD in a practical implementation where achieving the perfectly tuned condition is seldom realized. The control performance of PTMD marginally degrades when the beam is subjected to multi-directional excitation. Hence, the proposed control system has the potential for application in rotating beam-type structures, such as wind turbine blades.

Rubbing characteristics of a rotating blade with cracks

A novel compound fault modeling (NCFM) method is proposed to provide a qualitative numerical characterization of the cracked blade with the rubbing fault. Firstly, the proposed NCFM method incorporated the effects of centrifugal stiffening and spinning softening on the rotating blade under aerodynamic force. Secondly, multiple factors including variable speed, crack depth, and multi-failures modes (i.e., crack fault, rubbing fault) are considered to establish the effect matrixes (i.e., the additional effect matrix introduced by rubbing fault and the nonlinear crack fault) of different failures. Beside, a modified rubbing force model of the cracked blade is established. Moreover, an in-depth comparative analysis of the obtained results with the variable contact friction coefficients, casing stiffness, rotational speed, and crack stiffness underlines the relevance of compound fault. On the one hand, the continuous increase of rotational speed or crack depth will intensifies rubbing, including a decrease in rubbing interval and an increase in rubbing displacement amplitude. On the other hand, crack fusion leads to earlier development of rubbing, with smaller rotational speeds corresponding to less severe rubbing.

Dynamic Characteristics of a Rotating Blade with a Dovetail Fixture

Considering rotation-induced centrifugal stiffening, spin softening, and Coriolis effects, the reduced dynamic model of a rotating blade with a dovetail fixture is established in the ANSYS environment via the fixed-interface method for higher computational efficiency and lower memory consumption. Then some parameters such as rotating speed, friction factor, and stator blade number affecting the nonlinear vibration responses of the system under the combined actions of aerodynamic force, centrifugal force, and gravity are elaborately discussed. The results show that: (1) the contact-induced nonlinearity between the tenon and the mortise mainly results in the frequency multiplications of the aerodynamic excitation frequency; (2) a larger friction factor results in a lower magnitude of contact pressure and a higher resonance frequency, while a larger stator blade number results in a lower magnitude of the uniform and continuous contact pressure distribution; (3) the excitation of the resonant mode caused by the aerodynamic force is primarily characterized by the first-order bending mode of the system.

Three-dimensional vibration analysis of rotating pre-twisted variable thickness blades composed of spanwise graded functional materials

In this study, the isogeometric method is used to obtain a three-dimensional (3-D) solution for the vibration of rotating pre-twisted blades composed of spanwise functionally graded materials (SFGM). The material properties are assumed to change continuously in the spanwise direction according to a mixture rule. Nonuniform rational B–splines (NURBS) are used to describe the unknown displacement and geometry of the rotating blades, whose thickness linearly tapers along the spanwise direction. A typical isogeometric element for a rotating pre-twisted variable thickness SFGM blade is introduced to prevent transformations among multiple coordinate systems. This model considers centrifugal stiffening, centrifugal softening, and Coriolis force effects. Several cases are studied, including rotating rectangular plates, partial cylindrical shells and conical shells with different material properties and geometric parameters. The predicted vibration behaviors are verified by comparisons with results from the literature and commercial finite element software. The comparisons show that the current model has good accuracy. Finally, the influence of material properties, rotating speeds, installation angles, pre-twist angles, and variable thicknesses on the vibrational characteristics of rotating pre-twisted SFGM blades are examined.

Semi-analytical modeling and experimental verification of a flexible varying section disk–blades system with elastic supports

A semi-analytical model of a flexible varying section disk–blades (FVSDBs) system with elastic supports is proposed, where the spin softening, centrifugal stiffening, and Coriolis force effects of the system are considered. Timoshenko beams with a stagger angle are used to simulate blades, and a varying section wheel based on plate theory is used to construct the disk. Hamilton’s principle and Galerkin method are utilized to derive the differential governing equations of the FVSDBs system. The proposed semi-analytical model accounts for the lateral vibration of the disk, and the longitudinal and lateral vibration of the blades. Besides, the validity of the proposed model has been verified by comparing the natural frequencies, mode shapes, and vibration responses obtained from the proposed model, finite element model (FEM), and experiment. The results indicate that the proposed semi-analytical model has high precision and computational efficiency, which can provide an alternative scheme for predicting the vibration characteristics of varying section disk–blades and overcoming the high computational cost of finite element models.

Statičko ponašanje funkcionalno stepenovanih rotirajućih konzolnih greda korišćenjem B-splajn tehnike kolokacije

The present work reports the static behaviour of functionally graded rotating beam based on Timoshenko beam theory, which includes the effect of shear deformation. The principle of virtual displacement is applied to derive a governing equation for the functionally graded (FG) rotating beam, considering the centrifugal stiffening effect. The B-spline collocation technique is employed to solve the differential equation and material properties are function of power law distribution. The effect of material power index and rotational speed on the static characteristics of functionally graded rotating cantilever beams has been investigated for two different slenderness ratios. The obtained results demonstrate the beam deflection and the beam normal and shear stresses for the power gradient and rotational speed values. The results help to conclude that material gradation can be utilized to improve the functioning of the rotating structures such as helicopter rotor, aircraft propellers, windmill blades etc.

Similarity design method for dynamic characteristics of rotating composite blades in thermal environment

A similarity method for rotating composite blades in thermal environment is developed for the problems of difficulty, time consumption, high cost, and risk in the dynamic test under a high-temperature environment. Within this study, a set of scaling laws are derived for thin-walled composite plates based on their dynamic models and the applicability of the scaling laws is demonstrated in the similitude design of composite blades. The present dynamic model takes into account thermal stress and centrifugal stiffening effect. Analytical calculations show that the scaled models at relatively low temperatures can well reproduce the dynamic characteristics of their corresponding prototypes at high temperatures when taking into account the scaling of temperature change and rotational speed.

Effects of centrifugal stiffening and spin softening on nonlinear modal characteristics of cyclic blades with impact–friction coupling

This paper aims to interpret the coupling modal properties of cyclic blades under impact–friction interactions and their evolution mechanism versus operating points. Therefore, a coupling analytical model of cyclic blades is developed based on a Lagrange method and the assumed mode method (AMM), after considering centrifugal stiffening, spin softening, stagger angle, and twist angle. Then a mixed modal analysis method (MMAM) for this analytical model is extended by combining the extended periodic motion concept (EPMC) with AMM. Wherein a classic alternating frequency/time method (AFT) and the continuation method are employed to overcome the numerical divergence problem. Then damped nonlinear normal modes (dNNMs), including eigenfrequencies, modal damping ratios, and mode shapes, of the coupling system with shroud joints are finally computed and discussed under different excitation levels and contact conditions through a modal synthesis algorithm. After that, the influence laws of centrifugal stiffening and spin softening on the dNNMs are explored to reveal its evolution mechanism versus operation speeds. Finally, the Campbell diagrams of dNNMs are successfully obtained to discuss the effects of the impact–friction coupling on critical speeds (CSs) of the shrouded blades system.

Study of Centrifugal Stiffening on the Free Vibrations and Dynamic Response of Offshore Wind Turbine Blades

Due to their large and increasing size and the corrosive nature of salt water and high wind speeds, offshore wind turbines are required to be more robust, more rugged and more reliable than their onshore counterparts. The dynamic characteristics of the blade and its response to applied forces may be influenced dramatically by rotor rotational speed, which may even threaten the stability of the wind turbine. An accurate and computationally efficient structural dynamics model is essential for offshore wind turbines. A comprehensive model that takes the centrifugal stiffening effect into consideration could make rapid and accurate decisions with live data sensed from the structure. Moreover, this can enhance both the performance and reliability of wind turbines. When a rotating blade deflects in its plane of rotation or perpendicular to it, the centrifugal force exerts an inertia force that increases the natural frequencies and changes the mode shapes, leading to changes in the dynamic response of the blade. However, in the previous literature, studies of centrifugal stiffening are rarely found. This study investigates the influence of centrifugal stiffening on the free vibrations and dynamic response of offshore wind turbine blades. The National Renewable Energy Laboratory (NREL) 5 MW blade benchmark was considered to study the effect of angular speed in the flap-wise and edge-wise directions. The results demonstrate that the angular speed directly affects the modal features, which directly impacts the dynamic response. The results also show that the angular velocity effect in the flap-wise direction is more significant than its effect in the edge-wise direction.

A novel time-variant prediction model for megawatt flexible wind turbines and its application in NTM and ECD conditions

Dynamic response analysis for large-scale horizontal axis wind turbine(HAWT) plays a pivotal role in the design, stability, performance and safety. To overcome the problem of difficult measurement caused by many influence factors in wild wind field, accurately calculate the unsteady load, modal characteristics and dynamic response for megawatt flexible wind turbine components under various complex wind conditions, a novel time-variant prediction model is proposed in the manuscript. It perfectly combines the TUDK (Technical University of Denmark) dynamic inflow aerodynamic model, the hybrid beam theory considering shear deformation and centrifugal stiffening effect and Lanczos decoupling technology widely used in finite element software. Meanwhile, the rotor dynamic inflow effect, structural-damping and aerodynamic-damping of blade and tower, the structural coupling between blade and tower are considered in the model. Furthermore, the time-variant dynamic responses of blade tip and tower top under normal turbulence model (NTM) and extreme coherent gust with direction change (ECD) are further analyzed. The research results under ECD condition are compared with the experimental data from National Renewable Energy Laboratory (NREL) to validate the model accuracy.

Multi-mode vibration attenuation of mistuned bladed disks by frictional tuned mass dampers array

An effective method is presented for multi-mode vibration attenuation of mistuned bladed disk (blisk) by employing ring-like array structure composed of multiple frictional tuned mass dampers (FTMDs). The multi-mode vibration attenuation is realized by arranging multiple series of FTMDs. A novel design scheme of the FTMD array is presented. 3D finite element method is employed for quantitative modelling of the blisk-FTMD array system, and the rotordynamic effects of Coriolis forces, spin softening and centrifugal stiffening are taken into account to reach better tuning in natural frequencies in the rotating conditions. The free-interface component mode synthesis method is used to reduce the order of the model. An efficient technique, which combines the harmonic balance method, the analytical formulation of Jacobian matrix and the parameterized Newton-Raphson method, is employed to solve the nonlinear equations. Finally, a blisk, with its first 4ND and 5ND modes being targeted, is employed as an example to illustrate the effectiveness of the presented method. Numerical results show that the method can obviously mitigate the multi-mode vibration of blisk with wide levels of mistuning and is robust against the small tuning deviations of the oscillators.

Dynamically synergistic regulation mechanism for rotation energy harvesting

To realize the sustainable energy supply of wireless sensor networks, a novel dynamically synergistic regulation mechanism is proposed for rotation energy harvesting under the guidance of the dynamic regulation evolution. The connotation of synergyincludes: (1) Nonlinear magnetic force and variable stiffness (centrifugal stiffening effect and penalty stiffness of adaptive-anastomotic barricades) synergistically regulate the dynamic behavior of the system; (2) The adaptive-anastomotic barricadesalso convertmechanical energy into electrical energy when regulating the dynamic behavior of the system, i.e., the triboelectric nanogeneratorsynergistically generateselectricity with piezoelectric energy harvesterin different states (collision and vibration) of the system, respectively.Based on the energy method of Hamiltonian principle, the electromechanical coupling dynamic model of the system is developed. The parameters analysis is performed for the discussion of dynamic characteristics and electrical performance of the system in a wide rotation speed range. The prototype is fabricated and the experiments under different conditions are carried out. The experimental data are in good agreement with the simulation results, which verifies the effectiveness of the theoretical model. The experimental results show that the proposed harvester can harvest energy effectively in a wide speed range of 0–1000 r/min. The peak-peak voltage and the average power for PEH and TENG can reach 122.1 V, 1600 μW; 774.3 V, 205.1 μW, respectively. The proposed dynamically synergistic regulation mechanism provides a new perspective to improve the comprehensive performance of the rotation energy harvester, which exhibits a promising application prospect in the self-powered IoT applications.

Analytical and numerical study on centrifugal stiffening effect for large rotating wind turbine blade based on NREL 5 MW and WindPACT 1.5 MW models

To pursue high efficiency for wind turbine and harness more energy from wind, the wind power generation system has been designed larger and larger. With length scale increasing, the natural frequency of blade will be lower and changing dramatically with rotational speed, which will threaten the stability of the blade and turbine. However, in the previous research, the parametric, analytical, and quantitative study of centrifugal stiffening effect is rare to see. As a result, the current research has comprehensively studied the centrifugal stiffening effect on the structural characteristics of large wind turbine blade. Specifically, the equivalent rotating wedge beam model has been proposed for composite wind turbine blade. Based on the proposed wedge beam model and Rayleigh-Ritz method, the natural frequency variation curve has been derived and verified by ANSYS 3D simulation results. The parameters, including blade length, stiffness-mass ratio, and aspect ratio, affecting the natural frequency curve have been uncovered and analysed. It was found that the centrifugal stiffening effect has a great impact on the fundamental frequency of blade, e.g. 10% for NREL 5 MW, while it has less impact on other modal frequencies. The variation curve of fundamental natural frequency can be approximated by parabola. The product of the parabola curvature and fundamental frequency is found to be approximately constant for both NREL 5 MW and WindPACT 1.5 MW turbine blades. This research will provide guidelines for the resonance avoidance design of large wind turbine blade in the future.

Enhanced 3D solid finite element formulation for rotor dynamics simulation

In this paper, an enhanced solid finite element formulation for rotor dynamics analysis is presented. The element kinematics is first defined with a Total Lagrangian formulation in order to account for possible centrifugal stiffening effect and its associated pre-stress deformation. The novelty of the three-dimensional element proposed in this study is then to correctly consider the effect of inertia due to small rotational deformation occurring around the pre-stressed equilibrium position. The rotational motion is characterized by interpolation of the translational degrees of freedom of the element. The accuracy and convergence of the proposed finite element are tested on several numerical examples consisting of a rotating rigid disk and a rotating Timoshenko beam with different slenderness ratios. All results agree well with analytical solutions excerpted from literature and demonstrate the usefulness and applicability of the proposed finite element formulation. • Initial stress effect from centrifugal force is included. • A total-Lagrangian formulation is used. • The change of direction of the spin velocity induced by deformation is accounted for. • Results show good accuracy and convergence rate at high rotating speed.

Nonlinear solutions for the steady state oscillations of a clamped–free rotating beam

The rotating beam problem has been extensively used as a benchmark for testing nonlinear finite element implementations. The remarkable characteristic of this benchmark is the coupling between axial a transverse deformations due to the centrifugal forces. The rotating beam dynamics exhibits a centrifugal stiffening effect that can only be captured either by including a kinematic coupling between axial and transverse deformation or by using a nonlinear description of the elastic forces. This paper presents simplified models of the rotating beam that capture the centrifugal stiffening effect due to the inclusion of an axial to transverse kinematic coupling of the beam centreline. The simplified models allow a discussion on their accuracy and a meaningful analysis of the nonlinear oscillations present in the steady state rotation. The results of the simplified models are compared to finite element solutions obtained by using the absolute nodal coordinate formulation and a commercial FEM code.

Parametric study of flapwise and edgewise vibration of horizontal axis wind turbine blades

In this paper, flapwise and edgewise vibrations of a horizontal axis wind turbine (HAWT) blade are studied. Rayleigh-Ritz method is used in which; orthogonal mode functions of the Euler-Bernoulli beam having fixed-free boundary are introduced into the Lagrange function and then the dynamic equations are derived. Effect of gravity, pitch angle, centrifugal stiffening, and rotary inertia are considered. Nondimensional equations are obtained by defining nondimensional parameters like; natural frequency, blade rotation, slenderness ratio, and simple pendulum frequency. The stiffness term of the natural frequency has two speed dependent elements and it is shown that, for small pitch angles, flapwise natural frequencies of the blade are increased by the increasing blade speed while the edgewise natural frequencies of the blade are decreased with the increasing blade speed. Pitch angle values ranging from 0° to 15° has negligible effect on the nondimensional natural frequencies of the blade up to the nondimensional blade speed of 4. Since the natural frequencies are the function of the blade speed, rotor critical speeds should be calculated with Campbell diagrams. Vibrational response of the blade tip to the gravity is dominant and much greater than that of the wind speed in the edgewise and flapwise vibration.

Combining magnet-induced nonlinearity and centrifugal softening effect to realize high-efficiency energy harvesting in ultralow-frequency rotation

Few studies have successfully achieved high-efficiency energy harvesting in ultralow-frequency rotational motion where the rotational frequency is less than 2 Hz (120 rpm). In order to solve this challenge, this paper proposes a novel energy harvester with the centrifugal softening effect (CSE) and the magnet-induced nonlinearity. In a rotational coordinate system, the corresponding theoretical model is derived to reveal the mechanism of the CSE and to analyze the influence of the nonlinear magnetic force on the output power of the proposed harvester. Furthermore, the mechanism of the centrifugal stiffening effect (C_stiffen_E) is also theoretically analyzed. The analysis results indicate that based on the C_stiffen_E, the self-tuning harvester with a wider bandwidth can be obtained, and the CSE is better than the C_stiffen_E in enhancing the ultralow-frequency rotational energy harvesting. Meanwhile, the derived model is experimentally validated via comparisons among different piezoelectric energy harvesters (PEHs) including the linear inverse PEH, the nonlinear inverse PEH and the nonlinear forward PEH. The experimental results show that the nonlinear inverse PEH combining the magnet-induced nonlinearity and the CSE is capable of realizing high-efficiency energy harvesting in the rotational speed range of 60-120 rpm. The corresponding RMS output voltage is larger than 4 V, which is promising to provide the continuous power source for a wireless sensor. In addition, the parametric studies are conducted to provide theoretical guidance for the further optimization design of the proposed PEH.

Vibration Analyses of an Axial Turbine Wheel With Intentional Mistuning

Abstract The last-stage bladed disk of a steam turbine is analyzed with respect to both flutter susceptibility and limitation of forced response. Due to the lack of variable stator vanes, unfavorable flow conditions may occur, which can lead to flow separation in some circumstances. Consequently, there is the risk of flutter in principle, particularly at nominal speed under part load conditions. For this reason, intentional mistuning is employed by the manufacturer with the objective to prevent any self-excited vibrations. A first step in this direction is done by choosing alternate mistuning, which keeps the manufactural efforts in limits since only two different blade designs are allowed. In this sense, two different series of blades have been made. However, it is well known that small deviations from the design intention are unavoidable due to the manufacturing procedure, which could be proved by bonk tests carried out earlier. The influence of these additional but unwanted deviations is considered in numerical simulations. Moreover, the strong dependence of blade frequencies on the speed is taken into account since centrifugal stiffening effects significantly attenuate the blade-to-blade frequency difference in this particular case. Focusing on the first flap mode, it could be shown that a mitigation of flutter susceptibility is achieved by prescribing alternate mistuning, which indeed evokes an increase of originally small aerodynamic damping ratios. Nevertheless, the occurrence of negative damping ratios could not be completely precluded at part load conditions. That is why optimization studies are conducted based on genetic algorithms with the objective function of maximizing the lowest aerodynamic damping ratios. Again, only two different blade designs are admitted. Finally, mistuning patterns could be identified causing a tremendous increase of aerodynamic damping ratios. The robustness of the solutions found could be proved by superimposing additional random mistuning.

Modal analysis of a rotating pre-twisted beam axially loaded by an internally guided tendon

A guided tendon concept in the context of rotating helicopter blades is proposed to modify their dynamic characteristics, so that the rotorcraft can operate under a wide range of working conditions. In this paper, a rotating pre-twisted beam with bending-bending-torsion coupling axially loaded by a tendon is used to model the proposed system. An internally guided tendon is attached to the beam at several spanwise locations to avoid the resonant vibrations due to insufficient separation between the rotor harmonics and natural frequencies of the beam. A set of orthogonal polynomials that satisfy the geometric boundary conditions is used to represent the deflections of the beam and tendon. The equations of motion are derived by means of the Lagrange's equation of the second kind combined with the appropriate energy expressions. In vacuo modal analysis is then carried out after the modal properties of the rotating pre-twisted beam without the tendon are validated against the reference. The effect of the attachments and rotation on the free vibration of the beam-tendon system is studied. It is found that the attachments and rotation can improve the structural stability of the system significantly, so as to allow a significantly larger axial force to be applied. The natural frequencies of all studied modes increase with rotation due to the centrifugal stiffening effect. The tendon-guiding attachments cause an increase in the natural frequencies of the tendon-dominated modes because of their reduced effective length but decrease the frequency reduction rate of the beam-dominated modes with the applied axial force. The beam part of the mode shapes of the beam-dominated modes is not significantly sensitive to the varying axial force, rotor speed or number of attachments expect when the studied modes enter the modal veering regions with other neighbouring modes.

Analytical and experimental investigation of the centrifugal softening and stiffening effects in rotational energy harvesting

Depending on its mounting orientation, the rotational piezoelectric harvester can be subjected to the centrifugal softening or stiffening effects, which may pose a positive or negative influence on the energy harvesting performance. This paper firstly presents the comprehensive comparison of the centrifugal softening and stiffening harvesters with a unified dimensionless analytical model. The analytical expressions of the optimum short-circuit and open-circuit rotational frequencies, the optimum resistances and the optimum centrifugal coefficients are derived to reveal the influence mechanism of the centrifugal effects. The proposed model is validated experimentally and used for case studies to compare the centrifugal softening and stiffening harvesters. Results illustrate that the centrifugal softening harvester has a superior energy harvesting performance at low rotational frequencies while the centrifugal stiffening harvester is better at high rotational frequencies. Furthermore, the peak power of the centrifugal softening harvester is higher while the frequency bandwidth of the centrifugal stiffening harvester is wider. In addition, it is found that the centrifugal stiffening harvester is more sensitive to the centrifugal coefficient. Based on the centrifugal stiffening effect, the self-tuning harvester can be achieved with a wide frequency bandwidth. Meanwhile, this requires that the centrifugal coefficient is precisely determined, and the fundamental natural frequency of the harvester without the centrifugal force should be low enough when operating at low rotational frequencies. Compared with the centrifugal stiffening effect, the centrifugal softening effect may be more suitable to be used in MEMS/NEMS applications since it can reduce the fundamental resonant frequency of the harvester during the rotation.