Flexural-torsional Vibration Research Articles

The vibration response of five beam theories under eccentric moving harmonic loads

The shear rotary inertia caused by the shear deformation affects the natural frequencies and the vibration characteristics of bridges. To guarantee the safety of bridges, it is necessary to investigate the shear rotary inertia as a crucial content. Unlike previous studies, the motivation of this paper is to improve the Timoshenko beam model and further explore the influence of the shear rotary inertia on flexural–torsional natural frequencies and flexural–torsional vibration properties of bridges. Based on the Timoshenko beam theory, a modified dynamic equation of the Timoshenko beam is established. Its natural frequencies are obtained, and its analytical solutions under moving harmonic loads are derived using the integral transform method. In order to further understand the shear rotary inertia, the other four beams (namely, the Euler–Bernoulli beam, the shear beam, the Rayleigh beam, and the Timoshenko beam) are introduced to make comparisons with the modified Timoshenko beam. Subsequently, in the parameter analysis, the natural frequencies and vibration responses of five different beams with various lengths are analyzed numerically. It is discovered that the shear deformation and the shear rotary inertia have great effect on the high-mode vertical and bridge flexural–torsional frequencies, and it can improve the accuracy of calculating the natural frequencies of beams by about 6.00%. Furthermore, the shear deformation effect on vertical dynamic responses under harmonic moving loads intensifies with the beam spans increasing. The shear beam theory and the Timoshenko beam theory are appropriate for the vibration analyses of short beams, and the Rayleigh beam theory and the Timoshenko beam theory are suitable for the vibration analyses of long beams, while the modified Timoshenko beam theory, being more generally, is suitable for both beams.

Comparing the in-plane shear moduli of cardboard measured by flexural vibration, torsional vibration, static torsion, off-axis vibration, and off-axis tension tests

Abstract Flexural vibration (FV), torsional vibration (TV), static torsion (ST), longitudinal vibration (OAV), and tension (OAT) of 45° off-axis tests of flat pieces of cardboards were analyzed to determine the in-plane shear modulus (IPSM) using samples of various widths. The analyses were performed with simulations using the finite element method (FEM) for models with and without lamination and physical experiments using cardboard samples. For the FV, TV, and ST tests, the IPSM was calculated using binary search (BS) and linear regression (LR). The FEM results suggest that the IPSM values obtained from the TV and ST test simulations were affected by lamination. In contrast, the effect of lamination was not clear in the FV, OAV, and OAT test simulations. The coefficient of variation obtained from the FV experiments was often larger than that obtained from other tests. Despite this variation, the FV test is more attractive than the other tests because the experiments could be performed easily, and the effect of lamination could be reduced.

Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.

A new theoretical method for torsional-flexural vibrations response of thin-walled beams with closed cross-sections under uniform distributed moving load

This article presents a new theoretical method for the torsional-flexural vibration response of a thin-walled beam with closed cross sections under a uniformly distributed moving load. Based on Vlasov’s theory and restrained torsion theory, a comprehensive equation of motion for torsional flexural vibration is established by considering the effects of additional torque caused by the change of the shear center and the center of mass and the warpage coefficient. Using the Fourier transform, Laplace transform, and their inverse transformations, the torsional-flexural vibration response expression of thin-walled beams with closed cross-sections under a uniformly distributed moving load was derived. The results calculated by the analytical solutions in this study were compared with those from finite element method, demonstrating the reliability and superiority of the analytical solutions. Compared with the method that ignores the warpage coefficient, the lateral displacement and torsion angle of the beam calculated by the proposed theoretical method decreased by more than 12.00%. The effect of the cross-sectional properties on the natural frequency of the torsional-flexural vibration of thin-walled beams is analyzed, and the results show that the moment of inertia and mass per unit length have a considerable influence on the torsional-flexural frequency, which increases with the moment of inertia and decreases with an increase in the mass per unit length. The influences of the load magnitude, load velocity, and load eccentricity on the dynamic response were further investigated. According to the results, the lateral displacement of the beam increased significantly with increases in the load eccentricity and load magnitude. In practical engineering, the conclusions of this study can be used to limit the torsional-flexural vibration response of simple maglev support bridges to ensure the safety of vehicles and passengers.

Flexural–torsional vibration of thin-walled beams with open cross sections considering the additional torsional moment

Flexural–torsional vibration of thin-walled beams with open cross sections considering the additional torsional moment

Flexural–torsional vibration reduction of an eccentric phononic crystal pipe conveying fluid

Flexural vibration and torsional vibration pervasively exist in slender structures, which often lead to noise emission and fatigue failure of structures. In order to effectively suppress coupled flexural–torsional vibrations of defective devices transporting fluid, this paper introduces an eccentric fluid-conveying pipe axially made of periodically varying materials, and the flexural–torsional bandgap (BG) characteristics of such phononic crystal pipe system are investigated. With consideration of warping effect, coupled flexural–torsional motion equations are established. The complex band structure of sub-structure and frequency response function and attenuation configuration of topological structure are attained by applying the spectral element method. In comparison with literatures only yielding total merging BG of flexural and torsional vibrations, this paper presents a novel result of completely separate flexural BG and torsional BG, which enables individual manipulation of the two elastic waves. Meanwhile, broadband frequency BGs are harvested in the proposed pipe structure, especially for torsional vibration, demonstrating an excellent vibration isolation property. Complicated regulation mechanism of eccentric, warping effects and some critical parameters is explored. This study is expected to provide a new approach for flexural–torsional vibration reduction of engineering fluid-transporting devices.

Coupled flexural and torsional vibration attenuation with locally resonant metamaterials

A metamaterial beam with coupled bending and torsional resonators for broadband flexural–torsional vibration suppression is studied. The main objective is to design and analyse the metamaterial beam for simultaneous attenuation of flexural and torsional vibration waves using FEM and the Bloch-Floquet theorem. The bandgaps of the single unit cell with an infinite periodic structure assumption are studied using Bloch-Floquet reduction with a commercial multi-physics program. Initially, the beam with a bending resonator is examined, and a high-frequency broadband vibration frequency bandgap above 10 kHz is observed. With the addition of a torsional resonator, the existing bending bandgaps are widened, and further new bandgaps are formed. Transmissibility analysis of a metamaterial beam with the above unit cells is performed using the finite element method. The FRFs show similar bandgaps observed in the dispersion curves, thereby validating unit cell analysis. It is observed that coupled bending and torsional resonators create the opportunity to widen and introduce new bandgaps compared to uncoupled resonators. These metamaterial beams with bending and coupled resonators can aid in vibration isolation and frequency filtering of structures subjected to bending and torsional impact loads.

Attenuation bands for flexural–torsional vibrations of locally resonant Vlasov beams

In this paper, a study on flexural–torsional vibrations of finite beams with a large number of resonators periodically attached along the length is presented. Emphasis is given in determining bandgaps, defined as frequency ranges free of resonances. The structural model is based on a modified Vlasov theory that incorporates the resonators effect by means of sectional inertial properties depending on the excitation frequency. Analyzing these last properties, the mechanism of weak and strong bandgap formation is enlightened. Analytical formulas for the edge frequencies of bandgaps are obtained. Also, an exact analytical solution for the free and forced vibration of simply supported beams is presented. This approach allows exploring, from another point of view, the location of attenuation bands.

Direct FE numerical simulation for dynamic instability of frame structures

Dynamic instability analysis of structures has been a critical issue in the application of engineering and mechanical designs. Can we directly simulate the dynamic stability of structures like buckling analysis in finite element software? To address this question, in this study, we explore the dynamic instability of frame structures, discretized into finite element structures, through developing a direct FE numerical simulation procedure. Benefiting from the advances of eigenvalue analysis of partial differential equations and finite element technology, a direct numerical simulation procedure is developed by using Floquet theorem, harmonic balance methods and finite element method. Dynamic instability characteristics of frame structures, concerning fundamental vibration, multi-mode coupling vibration and flexural-torsional vibration deformation due to their spatial characteristics, are thoroughly investigated. The results reveal that, the developed procedure can achieve dynamic stability analysis of damped and non-damped structures. Frame's flexural-torsional vibration deformation can significantly enlarge the size and instability intensity of unstable regions, threatening dynamic stability of frame structures. Moreover, damping ratio has a slight impact on instability mitigation of the frame induced by flexural-torsional vibration. Unstable regions of mode-coupling vibration of frame have a hidden attribute. No existing approximate formula can be available to estimate instability regions of frame's flexural-torsional vibration and mode-coupling vibration. The dynamic instability index presented here can characterize unstable regions intuitively. The proposed insights can provide design guidelines for frame structures, the developed direct FE simulation procedure may be insightful for extending to explore instability mechanisms of generalized structures with complex forms owing to parametric resonance.

Coupled Flexural–Torsional Forced Vibration Analysis of a Piezoelectrically Actuated Double-Cantilever Structure

Abstract In this paper, the forced coupled flexural–torsional vibration of a piezoelectrically actuated double-cantilever structure is investigated. The double-cantilever structure is composed of two uniform and identical Euler–Bernoulli cantilever beams connected by a rigid tip connection at their free ends. There is also a piezoelectric layer attached on the top surface of each cantilever beam. The characteristic equation for the coupled flexural–torsional vibrations of the structure is derived and solved to determine the natural frequencies. The time response to the forced vibrations of the structure is studied using the Galerkin approximation method. The effects of dimensional parameters, including the length of the cantilever beams and the length of the tip connection, and the piezoelectric input voltage on the coupled flexural–torsional natural frequencies and amplitude of vibrations of the structure are investigated analytically and experimentally. The results show that the coupled flexural–torsional fundamental frequency of the piezoelectrically actuated double-cantilever structure decreases as either the length of the cantilever beams or the tip connection is increased. Moreover, the amplitude of the coupled flexural–torsional vibrations of the structure is proportional to the piezoelectric input voltage; however, the slope of the curves depends on dimensional parameters. For a given voltage, the effect of either of the aforementioned dimensional parameters on the amplitude of vibrations depends on the other dimensional parameter such that there is a turning point in all the curves, whose location depends on the configuration of the structure.

Dynamic Finite Element Modelling and Vibration Analysis of Prestressed Layered Bending–Torsion Coupled Beams

Free vibration analysis of prestressed, homogenous, Fiber-Metal Laminated (FML) and composite beams subjected to axial force and end moment is revisited. Finite Element Method (FEM) and frequency-dependent Dynamic Finite Element (DFE) models are developed and presented. The frequency results are compared with those obtained from the conventional FEM (ANSYS, Canonsburg, PA, USA) as well as the Homogenization Method (HM). Unlike the FEM, the application of the DFE formulation leads to a nonlinear eigenvalue problem, which is solved to determine the system’s natural frequencies and modes. The governing differential equations of coupled flexural–torsional vibrations, resulting from the end moment, are developed using Euler–Bernoulli bending and St. Venant torsion beam theories and assuming linear harmonic motion and linearly elastic materials. Illustrative examples of prestressed layered, FML, and unidirectional composite beam configurations, exhibiting geometric bending-torsion coupling, are studied. The presented DFE and FEM results show excellent agreement with the homogenization method and ANSYS modeling results, with the DFE’s rates of convergence surpassing all. An investigation is also carried out to examine the effects of various combined axial loads and end moments on the stiffness and fundamental frequencies of the structure. An illustrative example, demonstrating the application of the presented methods to the buckling analysis of layered beams is also presented.

Multiagent Control of Airplane Wing Stability with “Feathers” under the Flexural Torsional Flutter

This paper proposes a novel method for the unbounded oscillation prevention of an aircraft wing under the flexural torsional flutter, an innovative multiagent attitude to control an aircraft wing with a surface consisting of managed rotating “feathers” (agents). Theoretical evaluation of the method demonstrates its high aptitude to avoid an aircraft wing’s flexural-torsional vibrations via expansion of the model’s ability to dampen the wing oscillations. It potentially allows increasing an aircraft’s speed without misgiving of the flutter. A new way to control an aircraft wing based on the Speed-Gradient methodology is suggested to increase the maximal possible flight speed without a flutter occurrence. Provided experiments demonstrate the theoretical advantage of the multiagent approach to the “feathers” rotation control.

Flexural–Torsional Free Vibration Analysis of a Double-Cantilever Structure

Abstract This paper aims to investigate the free coupled flexural–torsional vibrations of a double-cantilever structure. The structure consists of two identical Euler–Bernoulli cantilever beams with a piezoelectric layer on top. The beams are connected by a rigid tip connection at their free ends. The double-cantilever structure in this study vibrates in two distinct modes: flexural mode or coupled flexural–torsional mode. The flexural mode refers to the in-phase flexural vibrations of the two cantilever beams resulting in translation of the tip connection, while the coupled flexural–torsional mode refers to the coupled flexural–torsional vibrations of the cantilever beams resulting in rotation of the tip connection. The latter is the main interest of this research. The governing equations of motion and boundary conditions are developed using Hamilton’s principle. Two uncoupled equations are realized for each beam: one corresponding to the flexural vibrations and the other one corresponding to the torsional vibrations. The characteristic equations for both the flexural and the coupled flexural–torsional vibration modes are derived and solved to find the natural frequencies corresponding to each mode of vibration. The orthogonality condition among the mode shapes is derived and utilized to determine the modal coefficients corresponding to each mode of vibration. Moreover, the analytical and experimental investigations show that the coupled flexural–torsional fundamental frequency of the structure is dependent on dimensional parameters including the length of the cantilever beams and the length of the tip connection.

Free vibration performance of curved composite box-girders with corrugated steel webs

The curved composite box girder with corrugated steel webs (CBG with CSWs) shows a more complex performance compared to a straight one due to the coupled flexural-torsional effect caused by its curvature. However, the relevant studies on it were limited. In this paper, considering the shear lag effect of the flanges, the shear deformation of the steel webs, and the coupled flexural-torsional effect, the natural frequencies of the curved CBGs with CSWs were derived using the energy variation method and Hamilton principle. Free vibration experiments were conducted on three simply-supported curved CBGs with CSWs. Numerical analysis was performed using ANSYS software. According to the parametric analysis, the torsional kinetic energy and the additional effect of the restrained torsion have little effect on the natural frequencies in the flexural-torsional vibration, while they significantly affect the flexural mode component. The natural frequencies decrease with the increase of λ (the ratio of calculation span to the radius of the curved axis). When λ ≤ 0.5, the reduction is smaller than 5%, but larger than 10% when λ > 0.7. The span presents an insignificant influence on the fundamental frequency if λ remains constant. Based on the analytical and experimental researches, a simple-to-use equation was proposed to calculate the fundamental frequency of the curved CBGs with CSWs.

Investigation of the dynamic stability of bending-torsional deformations of the pipeline

Nonlinear mathematical models are proposed that describe the dynamics of a pipeline with a fluid flowing in it: a) the model of bending-torsional vibrations with two degrees of freedom; b) the model describing flexural-torsional vibrations taking into account the nonlinearity of the bending moment and centrifugal force; c) the model that takes into account joint longitudinal, bending (transverse) and torsional vibrations. All proposed models are described by nonlinear partial differential equations for unknown strain functions. To describe the dynamics of a pipeline, the nonlinear theory of a rigid deformable body is used, which takes into account the transverse, tangential and longitudinal deformations of the pipeline. The dynamic stability of bending-torsional and longitudinal-flexural-torsional vibrations of the pipeline is investigated. The definitions of the stability of a deformable body adopted in this work correspond to the Lyapunov concept of stability of dynamical systems. The problem of studying dynamic stability, namely, stability according to initial data, is formulated as follows: at what values of the parameters characterizing the gas-body system, small deviations of the body from the equilibrium position at the initial moment of time will correspond to small deviations and at any moment of time. For the proposed models, positive definite functionals of the Lyapunov type are constructed, on the basis of which the dynamic stability of the pipeline is investigated. Sufficient stability conditions are obtained that impose restrictions on the parameters of a mechanical system.

Nonclassical vibrations of a monoclinic composite strip

A mathematical model of damped flexural-torsional vibrations of monoclinic composite strip of constant length rectangular cross section is proposed. The model is based on the refined bending theory Timoshenko beams, the theory of generalized Voigt — Lekhnitskii torsion and the elastic-viscoelastic correspondence principle in the linear theory of viscoelasticity. A two-stage method for solving a coupled system of differential equations is developed. First, using the Laplace transform in spatial variable, real natural frequencies and natural forms are found. To determine the complex natural frequencies of the strip in found real values are used as their initial values of natural frequencies, and then the complex frequencies are calculated by the method iterations of the third order. An assessment of the reliability of the mathematical model and method of numerical solution, performed by comparing calculated and experimental values of natural frequencies and loss factors is given. The results of a numerical study of the effect angles of orientation of reinforcing fibers and lengths by the values of natural frequencies and loss factors for free-free and cantilever monoclinic stripes are discussed. It is shown that for the free-free strip the region of mutual transformation eigenmodes of coupled vibration modes arise for quasi-bending and quasi-twisting vibrations of either even or odd tones. In the console strip of the region of mutual transformation of eigenforms of coupled modes vibrations occur for both even and odd tones.

Prediction of stability against subsonic flutter for axial turbine machine compressor blade assemblies

The developed method for prediction of their dynamic stability limit is described for the first flexural and torsional modes of vibrations against subsonic flutter. It involves the use of the database of the critical reduced frequencies of the blade vibrations as a dynamic stability limit within the wide range of variations of the following characteristics: phase angle of flexural, torsional and flexural-torsional vibrations of the adjacent blades; attack angle; reduced vibration frequency; geometric parameters for the airfoil and cascade; law of motion for blade airfoils. The algorithm realized as the numerical software assumes the determination of the critical reduced frequencies in the most loaded section of the full-scale blade assembly via the analysis of the distribution of the relative flow rate, attack angle, amplitudes of vibrations over the blade height and the subsequent comparison with the functional defining its critical value for different attack angles. The results of practical implementation of the developed procedure are given for the assessment of the stability of the blade assemblies of axial compressors in some modern aircraft gas-turbine engine at the first flexural mode of vibrations. It is implied that the proposed method allows one to select the optimal reduced frequency of vibrations of the blade assembly for the specified geometry of its peripheral sections and attack angle of the mainstream flow upon the condition of subsonic flutter at the engine design state.

Numerical and Experimental Analyses of Free and Forced Vibration of Thin-Walled Beams

The flexural–torsional vibration behavior of unrestrained and braced thin-walled beams is investigated by experimental and finite elements approaches. In the experimental part, tests in free and forced vibrations of thin walled beams with arbitrary sections are analyzed. By the help of an instrumental hammer test and a shaker machine, the natural frequencies and the response spectra of the beams are extracted in the range 1–400[Formula: see text]Hz. Beam displacements are measured by some accelerometer transducers. The behavior is also investigated by the finite element method. In mesh process, 3D beams are adopted and an additional DOF is affected to the warping. The model is implemented in a home-made model. The numerical and experimental results are compared to numerical simulations of a commercial code. Test results and numerical simulations of the present model agree well. The model seems to be accurate especially in the presence of higher vibration and coupled modes.

Instability Regions in Flexural-Torsional Vibrations of Plates

The paper is devoted to study of the parametric resonance of torsional vibrations caused by large-amplitude flexural vibrations of cantilevered plates. In this study we considered the first two resonant modes of flexural vibrations. The analytical study of the original problem which is reduced to Hill equation, is based on the WKB approximation. According to the obtained results, we built a map of instability gaps in the plane of the two process control parameters. The conclusions are made about the possibility of experimentally detection the studied effect in real systems.

On Determining Initial Conditions of Equations Flexural-Torsional Vibrations of a Bar

The problem of finding the initial conditions in the boundary-value problem for the system of flexural-torsional vibrations of a bar with additional conditions on the straight line is reduced to an optimal control problem and studied by the methods of optimal control theory. The gradient of the functional is calculated and using the gradient expression a necessary and sufficient optimality condition are proved.