Pre-twisted Beam Research Articles

Nonlinear free vibrations of a structurally tailored anisotropic pre-twisted thin-walled beam subjected to large deformations

It is essential to understand the nonlinear free vibration behaviour of a thin-walled composite structure as they find their applications in harsher environments prone to large deformations. Further, these structures made of fibrous materials, have directional properties for different fibre orientations resulting in a diversified and a complex nonlinear behaviour. In this regard, present work provides a comprehensive mathematical formulation on nonlinear vibration of pre-twisted thin-walled anisotropic box beam. The mathematical model is derived as coupled (flap-lag-extension-torsion) model considering shear deformation and green's strain tensor for large displacements in order to study the influence of nonlinear couplings on the dynamic behaviour. The derived energy equations are presented depending on degree of nonlinearity. These nonlinear equations those derived are nondimensionalized and solved with in the frame work of energy method using classical Ritz approximation. An iterative method is used to solve the nonlinear equations pertaining displacement terms. This nonlinear behaviour is evaluated for two important types of structural tailoring techniques available for thin-walled composite beams namely Circumferentially Asymmetric Stiffness (CAS) and Circumferentially Uniform Stiffness (CUS). It is found that CAS layup experiences significantly severe nonlinear effects compared to CUS due to the presence of 3rd degree nonlinear coupling terms. The variation of nonlinear frequency ratios over different ply angles showing the influence of nonlinearity on fibres orientation is presented for the first time for the two ply lay ups. The influence of degree of nonlinearity on the accuracy of the nonlinear frequencies is evaluated and presented. Moreover, the impact of nonlinearity on the pre-twisted beam is presented for different pre-twist angles.

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.

Static deflection of pre-twisted beam subjected to transverse load

In the current research, the static deflection of the pre-twisted beam has been calculated experimentally and numerically using finite element approach (ANSYS workbench 2020R1) and under the effects of different twist angles, different lengths, four boundary conditions of edges support of the beam. The beam was manufactured from PETG filament material by a 3D printer at room temperature. The obtained results showed reasonable acceptance between the experimental and numerical pre-twisted beam deflection. The numerical maximum deflection of the pre-twisted beam decreased with the increase of twisted angle of the beam for clamped-simply and simply-simply edges support. While it was irregularly various for clamped-clamped edge support with a minimum value of the maximum deflection occurred at the twist angle 135o. The maximum deflection of the pre-twisted beam is disappeared gradually at the twist angles greater than 270o for claimed-free edge support. In addition, the maximum deflection of the pre-twisted beam increased due to the beam length increment.

CALCULATING THE NATURAL FREQUENCY OF PRE-TWISTED BEAM

Beams in two configurations; uniform and non-uniform, are common structural components utilized for several engineering applications. Thus, the studies dealing with their behavior under dynamic and vibrations have been increased. In this research, the transverse vibration phenomena of pre-twisted beams were experimentally and theoretically studied by investigating the effect of twisting angles on the first three transverse natural frequencies. In the present experimental part, the pre-twisted beams are manufactured using a 3D printer, and the fundamental frequencies of manufactured pre-twisted beams are measured by a suitable rig. In the theoretical work, the finite element method is used to simulate the transverse vibration behavior of the pre-twisted beams. The accuracy of the simulation process is checked by comparison of the first natural frequencies calculated by the finite element method (using ANSYS Workbench Software) with those measured experimentally. The results show that there is an excellent agreement between the experimental results and finite element outputs. For the clamped-free pre-twisted beam, there is no critical twisting angle. The critical twisting angle is equal to the mode number for simply – simply and clamped simple pre-twisted beam. While there is more than one value of critical twisting angle for the clamped-clamped pre-twisted beam.

Nonlinear dynamic characteristics of horizontal-axis wind turbine blades including pre-twist

This paper focuses on the dynamic modeling and vibration analysis of horizontal axis wind turbine (HAWT) blades with emphasis on the effects of the pre-twist, rotational rigid body motion and coupling among the different blade's degrees of freedom. The blade is considered as a rotating pre-twisted beam mounted on a rigid hub with a constant angle relative to the plane of rotation, so-called precone angle. Kinetic energy of the blade based on its elastic as well as rigid motion together with its strain energy and generalized forces resulting from gravitational, centrifugal and aerodynamic loadings are obtained and used to derive new coupled dynamic equations by means of Hamilton's principle. The weak formulation is constructed and the Rayleigh-Ritz method (RRM) is adopted to extract a reduced order model (ROM) for the blade's vibration analysis. The developed nonlinear ROM is linearized around the steady-state solution and the blade's dynamic characteristics are obtained. The reference wind turbine blade called NREL 5-MW HAWT blade is selected as the case-study and effects of pre-twist, centrifugal force, angular velocity-depended terms, nonlinearities and transverse static loads on its dynamic characteristics are investigated. Results are shown to be in good agreement with available data in the literature.

Parametric Stability Investigation of a Spring-Attached and Viscoelastic-Supported Pre-twisted Sandwich Beam

Parametric Stability Investigation of a Spring-Attached and Viscoelastic-Supported Pre-twisted Sandwich Beam

Analysis of parametric instability of a spring-attached pre-twisted beam with viscoelastic end support

This study investigated the parametric instability of a single elastic beam with spring attachment on the top and viscoelastic springs as end supports. The beam considered is pre-twisted with a pin connection at both ends that supports the beam. The analytical solution of the problem is expressed in the matrix form achieved from the implementation of Hamilton’s principle and General Galerkin’s method, from which both static and dynamic stability of the beam can be investigated. The results of various influential dimensionless parameters such as stiffness, mass, length, position of the spring attachment, and stiffness of the viscoelastic springs on both the stabilities are studied. This analysis concluded that the spring attachment on the system leads to substantial contribution in improving the stability. The viscoelastic springs also contribute in upsurging the beam’s stability. Three different profiles of the beam have been considered, and for each profile, three different types of springs have been examined. The results revealed that the beam with parabolic profile and stiffness of the spring attachment with parabolic variation is most effective towards strength-to-weight ratio.

Analysis of vibration in rotating pretwisted functionally graded graphene platelets reinforced nanocomposite laminated blades with an attached point mass

This work presents an investigation on the free vibration behavior of rotating pre-twisted functionally graded graphene platelets reinforced composite (FG-GPLRC) laminated blades/beams with an attached point mass. The considered beams are constituted of [Formula: see text] layers which are bonded perfectly and made of a mixture of isotropic polymer matrix and graphene platelets (GPLs). The weight fraction of GPLs changes in a layer-wise manner. The effective material properties of FG-GPLRC layers are computed by using the modified Halpin-Tsai model together with rule of mixture. The free vibration eigenvalue equations are developed based on the Reddy’s third-order shear deformation theory (TSDT) using the Chebyshev–Ritz method under different boundary conditions. After validating the approach, the influences of the GPLs distribution pattern, GPLs weight fraction, angular velocity, the variation of the angle of twist along the beam axis, the ratio of attached mass to the beam mass, boundary conditions, position of attached mass, and geometry on the vibration behavior are investigated. The findings demonstrate that the natural frequencies of the rotating pre-twisted FG-GPLRC laminated beams significantly increases by adding a very small amount of GPLs into polymer matrix. It is shown that placing more GPLs near the top and bottom surfaces of the pre-twisted beam is an effective way to strengthen the pre-twisted beam stiffness and increase the natural frequencies.

Nonlinear dynamics of the rotating beam with time-varying speed under aerodynamic loads

In this research, the bifurcation analysis is inspected for a rotatory pre-twisted beam which is subjected to the time varying angular velocity and aerodynamic forces. Aerodynamic loads are obtained by the Piston and Whitehead theories. It is assumed that a time-dependent periodic component is superposed on the angular velocity. The extended Hamilton's principle is utilized to derive the dynamic equations of motion in flapwise and chordwise directions. Then a combination of the Galerkin and multiple scales methods is established to obtain the modulation equations describing amplitude and phases of the interacting modes. Eigen value analysis is performed on the linear part of the equations and the possibility of 1:1 internal resonance condition is examined. The evolution of amplitudes is obtained versus the amplitude and frequency of the periodic angular velocity function. Primary and subharmonic resonance conditions are tuned between the angular velocity frequency and natural frequencies of the beam. Numerical simulations indicate that the nonlinear phenomena such as jump, saturation, and double jumping appear in nonlinear vibration of rotating beam with varying speed.

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.

A twist piezoelectric beam for multi-directional energy harvesting

This paper develops a twist beam for multi-directional energy harvesting using piezoelectric transduction. The working principle of the multi-directional ability is explained by the theory of a pre-twisted beam. A simulation study is performed for concept verification and revealing the potential advantages of the proposed twist-beam vibration energy harvester (VEH). A physical prototype is fabricated and an experimental study is conducted to evaluate the actual performance of the proposed twist-beam VEH for validation. The results show that for the excitation comes from any direction in the plane in parallel with the beam cross-section, the proposed twist-beam VEH can produce a substantial amount of power. The optimal resistance is independent of the excitation direction. Moreover, it is found that the relationships between the output voltage amplitude and the excitation direction are different for the cases around the first and second resonances of the twist-beam VEH. The underlying physics behind this phenomenon is that the dominant motion of the two resonant modes are contrarily different. In addition, the experimental study shows that the first two natural frequencies of the prototyped twist-beam VEH are 22.37 and 41.47 Hz, respectively, which are much closer to each other as compared with those of a conventional plain beam. This feature benefits multi-modal energy harvesting within a targeted frequency range. The prospects regarding further optimization of the proposed twist-beam VEH are discussed.

A total Lagrangian Timoshenko beam formulation for geometrically nonlinear isogeometric analysis of spatial beam structures

This paper concerns a novel isogeometric Timoshenko beam formulation for a geometrically nonlinear analysis of spatial beams using the total Lagrangian description. Constitutive laws for hyperelastic materials, whose behavior varies with their deformations, are widely defined by using strain energy density functions that are written in terms of the Green–Lagrange strain tensor. Many finite element beam formulations for geometrically nonlinear analyses of spatial beams are developed using the Green–Lagrange strain tensor and its energy conjugate, the second Piola–Kirchhoff stress tensor. Unfortunately, there virtually exist no isogeometric Timoshenko beam formulations for this type of analysis that are derived by using this energy conjugate pair. To allow the possibility of considering hyperelastic materials, the present isogeometric beam formulation is developed in the total Lagrangian description using the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The proposed formulation is capable of simulating beam structures that are subjected to large displacements and rotations, without any restriction in magnitude. Three-dimensional beam configurations are reduced into one-dimensional structures using the beam axis and director vectors of the cross sections. The cross-sectional rotation along the beam axis is represented by an orthogonal tensor, which is parameterized by a vector-like parameter. Updating the cross-sectional rotations is performed purely by natural exponentiation and superposition of relevant rotational quantities. To show the accuracy and efficiency of the proposed beam formulation, some benchmark and well-established numerical examples with various types of beam, i.e., straight, curved, pre-twisted beams, are analyzed. The obtained results are compared with those results in the literature, obtained from both analytical and numerical methods.

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.

Isogeometric-based cross-sectional analysis of pre-twisted composite beams

Abstract This paper is intended to investigate the static behavior of pre-twisted composite beams using an isogeometric-based cross-sectional analysis. The three-dimensional pre-twisted beam problem is decomposed into a two-dimensional cross-sectional analysis and a one-dimensional beam model. The cross-sectional analysis obtains stiffness constants by considering three-dimensional warping deformation effects. The influence of pre-twist ratio is investigated on the stiffness constants and one-dimensional deformations of isotropic beams, composite strips and box-beams. Inconsistencies in results of pre-twisted composite beam theories in the literature are addressed and discussed. The present method eliminates the costly use of three-dimensional finite element analysis in the initial design steps.

Free vibration analysis of a rotating pre-twisted beam subjected to tendon-induced axial loading

The main objective of this paper is to investigate free vibration of a rotating pre-twisted beam with bending-bending-torsion coupling that is axially loaded by a tendon. The tendon is connected to the tip of the beam, passes through its body, and is fixed and loaded at the axis of rotation. Due to their connection, the motion of the beam influences the motion of the tendon and vice versa. The equations of motion of the beam-tendon system are introduced and solved numerically by a combination of a boundary value problem solver and differential quadrature method to obtain the natural frequencies and mode shapes of the system. The numerical implementation is firstly validated against literature for a non-rotating beam-tendon system and a rotating pre-twisted blade. Then, the effect of the tendon on the free vibration of the system is studied for a wide range of loading cases and rotation speeds. From the computed modal characteristics, it is found that the presence of a tendon leads to the frequency shift of beam's natural frequencies as well as frequency loci veering between the beam-dominated and tendon-dominated modes. Both of these effects strongly depend on the beam-tendon rotation speed.

Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

Geometrically Exact, Fully Intrinsic Analysis of Pre-Twisted Beams Under Distributed Follower Forces

This paper studied the dynamic behavior of pre-twisted beams subjected to applied, distributed follower forces. The analysis was based on geometrically exact, fully intrinsic nonlinear composite beam theory, which is capable of analyzing the dynamic behavior of a general, nonuniform, initially curved and twisted, anisotropic beam undergoing large deformation. The equations of motion are independent of displacement and rotation variables, and singularities caused by finite rotation are avoided. The system of nonlinear equations was linearized about a trim solution state, and the linear system governing dynamic stability was solved numerically. The Hopf bifurcation point and behavior of the eigenvalues both pre- and post-instability were determined, and nonlinear limit-cycle oscillations were analyzed in the time domain. The results obtained show that the nonconservative instability problem of the beam is highly affected by pre-twist, as well as by the magnitude and orientation of the applied, distributed follower forces.

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

Abstract A Chebyshev collocation method is presented for the free vibration analysis of geometrically exact nonlinear beams with fully intrinsic formulation. The intrinsic formulation of the governing equations of the beam contains neither displacement nor rotation variables. The proposed collocation discretization technique is based on the Chebyshev points as the collocation points and the orthogonal Chebyshev polynomials as the trial functions. This method is successfully applied to the eigenvalue analysis of the linearized intrinsic governing equations of a nonlinear beam. A number of test cases have been considered for either straight or pretwisted beams and the obtained results are compared to the analytical, numerical as well as experimental results. In order to show the applicability of current approach for real-life engineering problems, a composite wind turbine rotor blade with non-uniform distribution of properties is also considered. In all test cases a very good concordance has been observed. The proposed method bypasses the integrations common in finite element based methods and difficulties associated with finite rotations interpolation and while exhibiting a very good accuracy compared to the finite element results, it is computationally more efficient and simpler to implement in a computer programming code.

전달행렬법을 이용하여 비틀림 각과 회전관성을 고려한 보의 진동해석

전달행렬법을 이용하여 비틀림 각과 회전관성을 고려한 보의 진동해석

The finite element model research of the pre-twisted thin-walled beam

Based on the traditional mechanical model of thin-walled straight beam, the paper makes analysis and research on the pre-twisted thin-walled beam finite element numerical model. Firstly, based on the geometric deformation differential relationship, the Saint-Venant warping strain of pre-twisted thin-walled beam is deduced. According to the traditional thin-walled straight beam finite element mechanical model, the finite element stiffness matrix considering the Saint-Venant warping deformations is established. At the same time, the paper establishes the element stiffness matrix of the pre-twisted thin-walled beam based on the classic Vlasov Theory. Finally, by calculating the pre-twisted beam with elliptical section and I cross section and contrasting three-dimensional solid finite element using ANSYS, the comparison analysis results show that pre-twisted thin-walled beam element stiffness matrix has good accuracy.