Pretwist Angle Research Articles

Linear Vibration of the Rotary Plate Under Combined Excitations in Subsonic Airflow

Due to strong nonlinear, unsteady characteristics and the fluid–structure interaction effect, vibration analysis of blades under the excitation of the airflow is still one of the technical difficulties. In this paper, the accurate subsonic aerodynamic force is obtained through numerical simulation, and the aerodynamic coupling model of the rotary blade is established. The distribution of the aerodynamic force of the compressor blade under the unsteady airflow is focused on. The blade is modeled as presetting a presetting pre-twisted rotary cantilever plate. Dynamic frequencies of the plate, calculated by Chebyshev–Ritz method, are compared with frequencies calculated using the finite element method (FEM). Effects of different parameters on natural frequencies of the rotary plate are discussed. Based on von-Karman nonlinear geometric relation and the first-order shear deformation theory, nonlinear dynamic equations of the pre-twisted rotary plate under the combination of the centrifugal force and the aerodynamic are derived by utilizing Hamilton’s principle. Second-order ordinary differential equations are derived by applying the Galerkin method. Analytical solution of the dynamic deformation of the plate is presented and is compared with that produced by FEM. Results indicate the accuracy of the explicit presentation of the aerodynamic of the low-pressure compressor blade. Effects of the rotary speed, the thickness, the pre-twisted angle and the presetting angle on vibration characteristics of the warping blade are studied. Mode shape shift and frequency loci veering are discussed.

A nonlinear dynamical model for rotating composite thin-walled beams subjected to hygrothermal effects

A rotating composite thin-walled beam structure usually works in environment with strong hygrothermal characteristics. Temperature variation and moisture absorption can influence physical properties of composite, and furthermore change mechanical behavior of the structure. Hygrothermal influences depend on composite, layup scheme, geometry and nonlinearity of the beam. However, a comprehensive analytical model including these factors is lacking. This work aims to establish such a model for rotating composite thin-walled beams subjected to hygrothermal effects so that relevant problems can be solved analytically, which can aid numerical simulation and experimental test. Many important factors including hygrothermal effects, nonlinearity, mode coupling, arbitrary layup schemes, rotation, external forces, transverse shear, non-uniformity of cross section, constrained torsion, warping, setting angle and pretwist angle are considered. Governing equations of the coupled vibration are established by using the generalized Hamiltonian principle. Reduction verification, simplification principles and reduced models are discussed. The solution of the model is illustrated and some examples are given. Finally, the influences of temperature, moisture absorption and ply angle on hygrothermal internal forces of constrained thin-walled beams with CUS and CAS configurations are analyzed.

Dynamic modeling and coupling characteristics of rotating inclined beams with twisted-shape sections

In the existing literature, most studies investigated the free vibrations of a rotating pre-twisted cantilever beam; however, few considered the effect of the elasticsupport boundary and the quantification of modal coupling degree among different vibration directions. In addition, Coriolis, spin softening, and centrifugal stiffening effects are not fully included in the derived equations of motion of a rotating beam in most literature, especially the centrifugal stiffening effect in torsional direction. Considering these deficiencies, this study established a coupled flapwise–chordwise–axial–torsional dynamic model of a rotating double-tapered, pre-twisted, and inclined Timoshenko beam with elastic supports based on the semi-analytic method. Then, the proposed model was verified with experiments and ANSYS models using Beam188 and Shell181 elements. Finally, the effects of setting and pretwisted angles on the degree of coupling among flapwise, chordwise, and torsional directions were quantified via modal strain energy ratios. Results showed that 1) the appearance of torsional vibration originates from the combined effect of flapwise–torsional and chordwise–torsional couplings dependent on the Coriolis effect, and that 2) the flapwise–chordwise coupling caused by the pure pre-twisted angle is stronger than that caused by the pure setting angle.

Aeroelastic analysis of swept pre-twisted wings

In this paper, aeroelastic modeling of aircraft wings with variations in sweep angle, taper ratio, and variable pre-twist angle along the span is considered. The wing structure is modeled as a classical beam with torsion and bending flexibility. The governing equations are derived based on Hamilton’s principle. Moreover, Peters’ finite state aerodynamic model which is modified to take into account the effects of the wing finite-span, the wing sweep angle, and the wing pre-twist angle, is used to simulate the aerodynamic loads on the wing. The coupled partially differential equations are discretized to a set of ordinary differential equations using Galerkin’s approach. By solving these equations the aeroelastic instability conditions are derived. The results are compared with some experimental and analytical results of previous published papers and good agreement is attained. Effects of the wing sweep angle, taper ratio, bending to torsional rigidity, and pre-twist angle on the flutter boundary in several cases are studied. Results show that these geometrical and physical parameters have considerable effects on the wing flutter boundary.

Nonlinear vibration analysis of composite blade with variable rotating speed using Chebyshev polynomials

A composite cantilever plate model with variable high speed rotating blade subject to transverse aerodynamic force and centrifugal force is established. Based on the third-order shear deformation theory, von Karman large deformation theory and Hamilton's principle, the nonlinear partial differential governing equations of motion are established for the rotating blade with variable rotating speed. The two-degree-of-freedom nonlinear ordinary differential equations of motion are obtained by using Galerkin method with Chebyshev polynomials for the rotating blade. The influences of different structural parameters on the natural frequencies of the blade with variation of the rotating speed are investigated. The method of multiple scales is applied to obtain the averaged equations with the primary parametric resonance-1/2 subharmonic resonance and the relationship of 1:2 internal resonance. Numerical simulations are performed to portray the frequency-response curves and the complex nonlinear dynamic behaviors of the rotating blade by discussing the effect of the aerodynamic force. The effects of the varying rotating speed, the centrifugal force, the pre-twist angle and the pre-setting angle are taken into account on the nonlinear dynamics of the rotating blade. It is observed from the frequency-response curves that the rotating blade exhibits the hardening nonlinear behaviors as well as the jumping phenomena. The bifurcation diagrams, the phase portraits and the waveforms are utilized to illustrate the complex nonlinear dynamic behaviors of the rotating blade, such as the periodic, the quasi-periodic and the chaotic motions.

Stability analysis of an exponentially tapered, pre-twisted asymmetric sandwich beam on a variable Pasternak foundation with viscoelastic supports under temperature gradient

The purpose of this work is to study the stability of an exponentially tapered, pre-twisted and asymmetric sandwich beam on a variable Pasternak foundation, propped at ends. Viscoelastic translational and rotational springs have been employed to include the resistance offered by pinned–pinned end supports. The system is subjected to pulsating axial loads, and the elastic layers are subjected to a temperature gradient due to steady heat flow. A set of equations of motion was obtained by using Hamilton’s principle, and instability regions were plotted using formulae developed by Saito and Otomi. The effects of pre-twist angle, temperature gradient, taper parameter, shear modulus of the core, core loss factor, stiffness of the Pasternak foundation and rotational spring stiffness on static stability and regions of parametric instability were studied. Increase in the value of pre-twist angle was observed to be detrimental to both dynamic and static stability of the system. It was found that the dynamic stability of the system also degraded with an increase in the taper parameter.

The well-posedness and regularity of a rotating blades equation

<p style='text-indent:20px;'>In this paper, a rotating blades equation is considered. The arbitrary pre-twisted angle, arbitrary pre-setting angle and arbitrary rotating speed are taken into account when establishing the rotating blades model. The nonlinear PDEs of motion and two types of boundary conditions are derived by the extended Hamilton principle and the first-order piston theory. The well-posedness of weak solution (global in time) for the rotating blades equation with Clamped-Clamped (C-C) boundary conditions can be proved by compactness method and energy method. Strong energy estimates are derived under additional assumptions on the initial data. In addition, the existence and regularity of weak solutions (global in time) for the rotating blades equation with Clamped-Free (C-F) boundary conditions are proved as well.</p>

Modal analysis of rotating pre-twisted viscoelastic sandwich beams

This paper presents an analysis of the vibrational behavior of a rotating viscoelastic sandwich pre-twisted beams with a setting angle and with various viscoelastic stiffness laws. The governing equations of motion are derived using the Lagrange formulation and the assumed modes method. The obtained nonlinear eigenvalue problems are solved by using an iterative nonlinear eigensolver leading to complex eigensolutions composed of damped frequencies and loss factors with high accuracy. Further, the effects of the rotating speed, pre-twist angle, thickness ratio of core layer on the dynamic characteristics are investigated with taking into account the dependence of Young modulus with respect to frequency. Different numerical tests on rotating pre-twisted beams are performed for both isotropic and sandwich materials with a constant then a variable core modulus and the obtained results coincide very well with those provided in literature.

Vibration characteristics of rotating pretwisted composite tapered blade with graphene coating layers

A new dynamic model of the rotating tapered cantilever cylindrical panel with the graphene coating layers is developed to investigate the vibration characteristics of the rotating pretwisted tapered blade. It is assumed that the graphene platelets (GPLs) are randomly oriented and uniformly dispersed in the top layer and the bottom layer of the rotating pretwisted composite tapered blade. The modified Halpin-Tsai model is used to estimate the effective Young's modulus. The rule of the mixture is used to calculate the effective Poisson's ratio and mass density. Based on the Green strain tensor, an accurate strain-displacement relationship is acquired. The effects of the centrifugal force and Coriolis force are considered in the formulation. The Chebyshev-Ritz method is utilized to obtain the natural frequencies and mode shapes of the rotating pretwisted composite tapered blade with the graphene coating layers. The accuracy of the proposed model is validated through several comparison studies with the results of the present literatures and ANSYS. The free vibration characteristics are analyzed by considering different material and geometry parameters of the rotating pretwisted composite tapered cantilever cylindrical panel with the graphene coating layers, such as the graphene platelet (GPL) geometry, GPL weight fraction, taper ratio, length-to-radius ratio, pretwist angle, presetting angle and rotating speed. The frequency veering and the mode shape shift phenomena are found in the rotating pretwisted tapered cantilever cylindrical panel with the graphene coating layers.

A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades

A comprehensive understanding of the vibration characteristics of rotating blades is critical to avoid resonance fracture. This paper presents the vibration analysis and lay-up optimization of rotating pre-twisted laminated functionally graded CNTs reinforced composite (FG-CNTRC) shallow conical shells. A numerical shallow conical shell model is proposed, and the displacement fields of the shell model are described via the first-order shear deformation theory. Considering the rotating speed, the governing equation of the rotating pre-twisted laminated conical shell is derived, and then solved using the meshless kp-Ritz method. We carry out the comparison studies to demonstrate the accuracy and efficiency of the present model in dealing with the vibrations of this kind of structures. Detailed parametric studies are then performed and demonstrate that the CNT configuration, pre-twisted angle, and stacking sequence play essential roles in vibration behaviors of rotating FG-CNTRC blade. With the nondimensional frequency parameter being the optimization objective function, a genetic algorithm is employed to search for the optimal layering sequence of the rotating shallow conical shells. The optimized results demonstrate that the GA serves as a useful tool for sequence optimization of rotating blade in terms of improving the vibration characteristics.

The Influences of Stagger and Pretwist Angles of Blades on Coupling Vibration in Shaft-Disk-Blade Systems

Abstract The influences of stagger angle (α) and pretwist angle (βL) of blades on the coupling vibration among shaft bending and blade bending in a shaft-disk-blade (SDB) system are investigated using a Lagrangian approach in combination with the assumed modes method (AMM). The disk is rigid, and the flexible shaft is supported with two rigid bearings. It is shown that α and βL have variable effects on the coupling vibration because their influences can be increased, reduced, or even completely eliminated for different values of disk location (λ), blade thickness ratio (δ), and blade aspect ratio (γ). To study the coupling vibration in an SDB system, consideration of λ, δ, and γ are very important because those can alter the coupling magnitude, the coupling pattern as well as the predominant modes. Nevertheless, previous researches rarely take into account these parameters. Moreover, in the present work, to investigate the natural frequencies and critical speeds versus λ, δ, and γ, new diagrams are introduced. Also, the relation between the in-plane and out-of-plane motions of the blades with the coupling vibration is precisely analyzed.

Modal Based Sensitivity Analysis of Crack in Turbine Blades Using Validated Finite Element Models

Turbines in power industry use pretwisted blades of varying lengths, which are mounted on the rotating shafts with one end fitted in the fir tree root of the rotor. The fatigue of these blades may cause crack that starts at the stress concentration regions and may propagate for catastrophic failure. This paper addresses the vibration behavior of the pretwisted blades under the influence of crack. The effect of crack on the natural frequencies of pretwisted blade is investigated by conducting parametric studies using finite element analysis software. The results obtained are used to evaluate the influence of crack on turbine blade after validating with experimental results. It has been observed that the crack in a blade had a considerable effect on the natural frequencies that change with varying pretwist angles in all the modes of vibration. In addition, the crack sensitivity for natural frequencies of long and short blades has been compared and the fatigue behavior of these blades has been analysed.

Using PSO Algorithm to Compensate Power Loss Due to the Aeroelastic Effect of the Wind Turbine Blade

Power loss due to the aeroelastic effect of the blade is becoming an important problem of large-scale blade design. Prior work has already employed the pretwisting method to deal with this problem and obtained some good results at reference wind speed. The aim of this study was to compensate for the power loss for all of the wind speeds by using the pretwisting method. Therefore, we developed an aeroelastic coupling optimization model, which takes the pretwist angles along the blade as free variables, the maximum AEP (annual energy production) as the optimal object, and the smooth of the twist distribution as one of the constraint conditions. In this optimization model, a PSO (particle swarm optimization) algorithm is used and combined with the BEM-3DFEM (blade element momentum—three-dimensional finite element method) model. Then, the optimization model was compared with an iteration method, which was recently developed by another study and can well compensate the power loss at reference wind speed. By a design test, we found that the power loss can be reduced by pretwisting the origin blade, whether using the optimization model or the iteration method. Moreover, the optimization model has better ability than the iteration method to compensate the power loss with lower thrust coefficient while keeping the twist distribution smooth.

Modified couple stress-based free vibration behavior of pre-twisted tapered BFGM rotating micro beam considering spin-softening and Coriolis effects

An improved mathematical model of pre-twisted tapered rotating micro beams made of bidirectional functionally graded material (BFGM) is presented to study its free vibration behavior. The effects of spin-softening and Coriolis acceleration are incorporated, and modified couple stress theory is employed to address the size effect. The mathematical formulation is based on first-order shear deformation theory and is developed in a global non-inertial frame incorporating appropriate transformations between the global inertial frame and the local non-inertial frame. Two different but interrelated steps are employed, where the first step determines the centrifugally deformed configuration using minimum potential energy principle, and the second step determines the free vibration behavior through tangent stiffness of the deformed rotating beam using Hamilton's principle. The direct use of tangent stiffness considers the centrifugal stiffening effect through von Kármán non-linearity and bypasses the need of strain energy functional for the vibrating beam. The governing equations are transformed to an eigenvalue problem through state-space approach and solved following Ritz method. The effects of spin-softening, Coriolis acceleration, and pre-twist angle are shown and discussed. The effects of different parameters such as size-dependent parameter, aspect ratio, material gradation indices, operating temperature, FGM constituent, taperness parameters, and slender parameter are discussed.

Aeroelastic Stability Analysis of Tailored Pretwisted Wings

In this paper, the aeroelastic stability of a tailored aircraft wing with different pretwist distributions is investigated. The structure of the wing is modeled using the geometrically exact fully intrinsic beam theory of Hodges, whereas the aerodynamic loads are simulated by an incompressible unsteady aerodynamic model. The governing nonlinear partial differential equations are discretized using a time-space scheme, and the stability of the system is sought by evaluating the eigenvalues of the linearized system. The wings have linear or quadratic pretwist distributions, and the effect of various twist angles on isotropic and tailored wings is investigated. In this study, the effect of pretwist is considered on both the aerodynamic and structural models. Moreover, the effect of wing structure taper ratio in conjunction with the pretwist is investigated. The preliminary results obtained for a wing modeled as a clamped-free beam are compared with those reported in the literature and excellent agreement is observed. It is concluded that the pretwist angle leads to mode coupling and also has a significant effect on the flutter speed of the wing. By pretwisting the wing, the flutter speed of the wing with respect to the clean wing increases until a specific twist value and then decreases. Moreover, adding the pretwist to the wing decreases the flutter frequency. Finally, results highlighting the effect of bend-twist elastic coupling and wing taper ratio in combination with the pretwist angle on the aeroelastic stability of the wing are provided.

Coupled flapwise-chordwise-axial-torsional dynamic responses of rotating pre-twisted and inclined cantilever beams subject to the base excitation

A rotating pre-twisted and inclined cantilever beam model (RPICBM) with the flapwise-chordwise-axial-torsional coupling is established with the Hamilton principle and the finite element (FE) method. The effectiveness of the model is verified via comparisons with the literatures and the FE models in ANSYS. The effects of the setting and pre-twisted angles on the dynamic responses of the RPICBM are analyzed. The results show that: (i) the increase in the setting or pre-twisted angle results in the increases in the first-order flapwise and torsional frequencies while the decrease in the first-order chordwise frequency under rotating conditions; (ii) a positive/negative setting angle leads to a positive/negative constant component, while a positive/negative pre-twisted angle leads to a negative/positive constant component; (iii) when the rotation speed is non-zero, the pre-twisted angle or non-zero setting angle will result in the coupled flapwise-chordwise-axial-torsional vibration of the RPICBM under axial base excitation.

Dynamic Characteristics of a Rotating Tapered Cantilevered Timoshenko Beam with Preset and Pre-Twist Angles

This paper is concerned with the free vibration of a rotating tapered Timoshenko beam with preset and pre-twist angles. The power series method is used to obtain the frequencies and complex modes of the structure. The rotating velocity related terms are re-classified into three types, namely, static centrifugal terms, dynamic centrifugal terms and the gyroscopic terms. This reclassification provides clearer descriptions of the varying frequencies with respect to the rotating velocity. The gyroscopic coupling among different directions are discussed. The overall contour of the complex modal vibrations is recorded and investigated by time series snapshots of neutral line motions and tip end cross-section motions.

Optimal Control of Pretwisted Rotating Thin-Walled Beams via Piezoelectrically Induced Couplings

Problems related to mathematical modeling and optimal active control of pretwisted adaptive blade are considered. The blade is modeled as a rotating thin-walled composite beam embedded with anisotropic piezo-composite layers accounting for nonclassical effects, such as transverse shear and warping inhibitions. The linear-quadratic-regulator feedback control strategy is adopted to study the tailoring of piezo-actuators on vibration suppression. Control authority of piezoelectrically induced transverse shear and bending coupling is highlighted. Tailoring studies using the present model reveal that piezoelectrically induced transverse shear plays an important role on control effectiveness. In addition, the relations between the control authority and the elastic couplings, piezoelectrically induced actuation couplings, pretwist angle, and size and position of piezo-actuators are investigated.

Numerical investigation for influence of pre-twist on stress behavioral characteristics of curved blade

To assist designers in developing blade design based on conventional and composite materials, an overview of material characteristics and stress analysis methods was taken for consideration. The significant effect of loading on the pre-twisted curved blade are evaluated in terms of its deflection and corresponding stresses. A more simplified approach was not satisfactory and furthermore to establish the feasibility of composite blade detailed stress analysis has been carried out for various pre-twist angles of composite and mild steel blade using a thick shell element. The suitability of the finite element method in predicting the stresses in externally loaded laminates is examined from the viewpoint of satisfaction of all boundary conditions and results. From the results it can be shown that the three-dimensional finite element analysis provides a useful insight for predicting deflection and stresses. The influence of ply angle for the distribution of inter laminar stresses along y and z-axis are determined.

Free Vibration of Composite Thin-Walled Beams with Chordwise Asymmetric Closed Cross-Sections

In this paper, some analytical results via extended Galerkin method on free vibration characteristics of an anisotropic composite beam, which is modeled as a nonuniform thin-walled structure with a chordwise asymmetric closed cross-section and corrected the warping functions, are newly presented. For this study, nonclassical parameters such as warping restraint, transverse shear flexibility, and structural couplings induced by two special configurations, such as circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS), are incorporated. And also, design parameters of the beam associated with preset angles, pretwist angles, taper ratios, and section ratios are additionally investigated. The results of this study could play an important role in more efficient designs of composite thin-walled beams.