Flexural-torsional Vibration Research Articles

Flexural-torsional vibration and buckling of thin-walled bi-directional functionally graded beams

The paper introduces an analysis of flexural-torsional vibration and buckling of thin-walled bi-directional FG beams. Several cross sections have been conducted such as mono-symmetric I-shaped and channel sections in which material properties are assumed to vary across blade thickness and along axial direction. Governing equations and finite element model are developed. The model is capable of capturing all complex eigenvalue problems, also provides a highly accurate prediction of vibrational shapes and buckling capacity. Since the material changes, the obtained results reflect the relative relation between the behaviors of the beam and the transformation of internal properties, for example, Young's modulus, density, etc. Effects of gradient parameters on the vibrational frequencies or limit loads of a thin-walled bi-directional FG beam under various configurations and boundary conditions have also been parametrically studied.

Non-linear dynamic instability analysis of thin-walled stiffener beam subjected to uniform harmonic in-plane loading

Present analysis deals with nonlinear flexural-torsional vibration and dynamic instability of thin-walled stiffener beam with open section subjected to harmonic in-plane loading. The static and dynamic components of the applied harmonic in-plane loading are assumed to vary uniformly. A set of nonlinear partial differential equations (PDEs) describing the vibration of system is derived. Using Galerkin's method, these partial differential equations are reduced into coupled Mathieu equations. The steady state response of the system is determined by solving the condition for a non-trivial solution. The principal regions of parametric resonance are determined using the method suggested by Bolotin. The numerical results are presented to investigate the effect of aspect ratios, boundary conditions and static load factor on the frequency-amplitude responses and instability regions.

Non-linear dynamic instability analysis of mono-symmetric thin walled columns with various boundary conditions

This paper presents the nonlinear flexural-torsional vibration and instability analysis of thin-walled column with different mono-symmetric open sections subjected to periodic axial load. Non-linear partial differential equations (PDEs) governing the dynamic instability behavior of the considered sections are derived. Galerkin's method is employed to reduce the coupled partial differential equations to a set of coupled Mathieu equations. The amplitudes of steady state vibrations are determined from the non-trivial solution of Mathieu equations in terms of excitation frequency. Following Bolotin's method, principal instability regions are determined for different mono-symmetric section columns having various boundary conditions. Numerical results for the natural frequencies, instability regions and frequency-amplitude responses for thin-walled columns are presented. Influence of eccentricity, shape of the cross-section and boundary conditions on frequency-amplitude responses and instability regions are investigated.

Flexural-torsional vibration of a tapered C-section beam

Previous studies have shown that numerical models of tapered thin-walled C-section beams based on a stepped or piecewise prismatic beam approximation are inaccurate regardless of the number of elements assumed in the discretization. Andrade recently addressed this problem by extending Vlasov beam theory to a tapered geometry resulting in new terms that vanish for the uniform beam. (See One-Dimensional Models for the Spatial Behaviour of Tapered Thin-Walled Bars with Open Cross-Sections: Static, Dynamic and Buckling Analyses, PhD Thesis, University of Coimbra, Portugal, 2012, https://estudogeral.sib.uc.pt) In this paper, we model the coupled bending-twisting vibration of a cantilevered tapered thin-walled C-section using a Galerkin approximation of Andrade's beam equations resulting in an 8-degree-of-freedom beam element. Experimental natural frequencies and mode shapes for 3 prismatic and 2 tapered channel beams are compared to model predictions. In addition, comparisons are made to detailed shell finite element models and exact solutions for the uniform beams to confirm the validity of the approach. Comparisons to the incorrect stepped model are also presented.

Nonlinear Control of Flexural–Torsional Vibrations of a Rotating Thin-Walled Composite Beam

In this paper, the effectiveness of a saturation control strategy in suppressing vibration of a rotating composite thin walled beam is studied. The mathematical model of the flexible beam takes into account a shear deformation effect, a warping function, a centrifugal force and the Coriolis acceleration. To extend the generality of the proposed formulation an inertia of the hub is also considered. Adaptive capability of the beam is achieved through the implementation of the saturation control algorithm. Within the performed tests, the discussed control strategy is applied for different magnitudes of flexural–torsional vibration modes resulting from different orientations of beam laminate-reinforcing fiber’s. The obtained results prove the applied nonlinear control to be the effective method for beam vibration suppression in near-by resonance zones for all studied cases. Parametric studies considered different rotating speeds of the system. It is shown that the vibration of the beam can be suppressed to similar levels independently of the transportation motion rotating speed. However, significant differences in the width of vibration suppression zones are observed.

Dynamic stability of band saw machines

The eigenfrequencies of the flexural–torsional vibrations of band saw blades and the limits of the instability regions are determined as a function of the tool parameters and the operating conditions. Analytical equations describing the dynamic stability regions of the plane form of the band saws are obtained.

On Finding the Right-Hand Sides of Equations of Flexural-Torsional Vibrations of a Bar

On Finding the Right-Hand Sides of Equations of Flexural-Torsional Vibrations of a Bar

Coupled flexural–torsional vibration of spinning smart beams with asymmetric cross sections

In this paper, the electrical–structural modeling and control electronics of spinning smart beams with distributed piezoelectric sensor and actuator layers are considered. The governing equations of beams with asymmetric cross sections have been formulated incorporating the warping effect and the influence of spinning. For the purpose of verification of the model established, numerical results are compared with those given in relevant literature. Effects of spinning on natural frequencies and mode shapes are investigated for the boom in solar array of Hubble Space Telescope. In order to control the vibration of the spinning beam whose natural characteristics are changed with respect to spinning velocity, the positive position feedback (PPF) compensator is applied. Numerical results showed that the PPF approach is effective in vibration suppression of spinning smart beams.

Modelling of Nonlinear Torsional-Flexural Vibrations of Drill Strings

Nonlinear flexural-torsional vibrations of shallow drill-strings are investigated. The drill-string is represented as an elastic rod rotating at constant angular velocity under the action of the longitudinal compressive force. The nonlinear model of flexural and torsional vibrations of drill-string is constructed on the basis of the theory of finite deformations of V.V. Novozhilov and its second system of simplifications. The numerical analysis of the model is carried out in the environment of symbolic mathematical computations - Wolfram Mathematica. The dominance of flexural vibrations of the drill-string over torsional vibrations is established. The influence of drill-string parameters on its oscillatory processes is investigated. It will allow to build up the modes of the drill-string movements to improve the quality of drilling of wells.

Dynamic analysis of bridge girders submitted to an eccentric moving load

The cross-section warping due to the passage of high-speed trains can be a relevant issue to consider in the dynamic analysis of bridges due to (i) the usual layout of railway systems, resulting in eccentric moving loads; and (ii) the use of cross-sections prone to warping deformations. A thin-walled beam formulation for the dynamic analysis of bridges including the cross section warping is presented in this paper. Towards a numerical implementation of the beam formulation, a finite element with seven degrees of freedom is proposed. In order to easily consider the compatibility between elements, and since the coupling between flexural and torsional effects occurs in non-symmetric cross-sections due to dynamic effects, a single axis is considered for the element. The coupled flexural-torsional free vibration of thin-walled beams is analysed through the presented beam model, comparing the results with analytical solutions presented in the literature. The dynamic analysis due to an eccentric moving load, which results in a coupled flexuraltorsional vibration, is considered in the literature by analytical solutions, being therefore of a limited applicability in practice engineering. In this paper, the dynamic response due to an eccentric moving load is obtained from the proposed finite element beam model that includes warping by a modal analysis.

A closed-form study on the free vibration of a grid joined by a mass-spring system

This paper deals with the coupling flexural-torsional vibration analysis of a grid formed by two members. A mass-spring system is attached to the grid at the intersecting joint. The members of the grid are assumed to resist torsion as well as bending and shear. Moreover, the mechanical and geometrical properties of each member are different. In order to analyze the problem, a closed-form solution is obtained. In doing so, the governing differential equations of the system along with the pertinent boundary and compatibility conditions of the system are introduced. Then, the frequency parameters of the mechanical system under study are derived and given for the first five modes of vibration. Furthermore, some diagrams of the mode shapes of the grid are plotted. A finite element program is also written to verify the results. The results obtained from these two methods are in good agreement and the accuracy of our results is confirmed.

On the Flexural-Torsional Vibration and Stability of Beams Subjected to Axial Load and End Moment

The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MATLAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.

Flexural-torsional vibration analysis of axially loaded thin-walled beam

The present paper considers the flexure-torsion coupled vibrations of axially loaded thin-walled beams with arbitrary open cross section, by means of an exact solution. The effects of axial force, warping stiffness and rotary inertia are included in the present formulations. In the case of simply supported thin-walled beam, a closed-form solution for the coupled natural frequencies of free harmonic vibrations was derived by using a general solution of the governing differential equations of motion based on Vlasov theory. The method is illustrated by its application to two test examples, to demonstrate the effects of bending-torsion coupling and axial force on the dynamic behavior of thin-walled beams. Compared with those available in the relevant literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.

A beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded

A coupled torsional-bending finite element with shear deformations and rotatory inertia for vibration of nonsymmetric thin walled beams axially loaded is developed. The equations of motion are based on Vlasov’s theory of thin-walled beams, which are modified to include an axial load. The formulation is also applicable to solid beams. The Hermite cubic polynomials are adopted as shape functions. Mass, elastic stiffness and geometrical stiffness matrices of unsymmetrical cross-section beams are presented. In order to verify the accuracy of this theory and the corresponding beam element developed, a numerical study is presented and compared with the literature and experimental tests.

Active control of coupled flexural-torsional vibration in a flexible rotor–bearing system using electromagnetic actuator

Rotor–shaft systems are subject to non-uniform spin speed during start-up, coast-down or any non-stationary situation changing the spin speed suddenly, e.g., load fluctuation or sudden mass-loss like loss of a blade or a part thereof. For a flexurally and torsionally compliant rotor-shaft, the dynamics under non-uniform spin-speed shows inertial coupling among transverse and torsional coordinates through mass-unbalance and gyroscopic effect. This results into coupled transverse-torsional vibration, where torsional response consists of significant harmonic components at bisynchronous spin frequency, torsional natural frequency of the shaft, and at combination frequencies corresponding to sum and difference of spin and transverse natural frequencies and twice the transverse natural frequency of the rotor-shaft. As a result of the coupling, transverse rotor motion also influences the torsional motion. The Method of Multiple Scales (MMS) is used in this work to carry out an analysis of a simplified system to get an idea about the dominant frequencies of excitation. Results of numerical simulation are presented next to show the effectiveness and influence of actively controlling the transverse rotor motion on its torsional motion, at the dominant frequencies, with the help of non-contact electromagnetic force from an actuator. Transverse vibration control is also observed to control the torsional oscillations due to coupled nature of the problem. The Stability Limit Speed (SLS) of the system is also increased as a result of application of the active control action. Constant axial torque is observed to diminish the influence of coupling, and protect the system against torsional instability, but control action is a must to stabilize the transverse vibration of the system above SLS.

The Theoretical Prediction and Experimental Verification of Modal Parameters for Carbon Fiber Reinforced Composites with Sandwich Structure

A innovative structure of sandwich composite rectangular plate with dimensions of 350 ×83.5×9.5mm was designed, it was made of unidirectional prepreg of carbon fibre and woven fabric prepreg of carbon fiber as face materials and paulownia as core material by hand lay-up performing and press molding technology for investigating the dynamic performance such as natural frequency and modal shapes. Based on testing the in-plane and out-plane mechanical properties of composite samples reinforced by unidirectional carbon fibre and carbon fiber woven fabric, a ANSYS FEA dynamic modeling was developed. According to the impulse response modal test method, a modal test system was established. The natural frequency test results showed that the minimum natural frequency of sandwich composite rectangular plate is about 616.45Hz which is higher about 27.5% than that of aluminum rectangular plate reinforced by carbon. The modal experiment indicated that the 1st modal shape, 2nd modal shape, 3rd modal shape and 4th modal shape of the sandwich composite rectangular plate were torsional vibration, flexural vibration shape, torsional flexural vibration and double-flexural vibration separately. It was found the calculating precision of FEA dynamic predication was very high, the dynamic predicating results by FEA could provide fundamental data to the optimal design high speed reciprocating sandwich composite rectangular parts.

The Effect of Reinforcement Structure on the Modal Parameters for Sandwich Structure Composite Plate

In order to investigate the effect of different reinforcement structure on the dynamic characteristics of sandwich structure composite plates used for manufacturing the high speed reciprocating motion composite components, four kinds of paulownia wood sandwich composite test specimens with dimensions of 350×83.5×9.5mm was designed and made by hand lay-up performing and press molding technology. The woven and 2D braiding fabric prepreg were both selected as top face and inner face materials , respectively, and the carbon fiber woven fabric prepreg was chosen as inner part materials. According to the impulse response modal test method, a modal test system was established. It was found that this kind of sandwich structure composite plate has bigger natural frequency value, it’s minimum natural frequency was about 609.77Hz that could meet the requirement for high speed reciprocating motion parts. The dynamic test results shown that the natural frequency of F2BAF-IUC-CPW sample is higher t about 11.17% at least, selecting 2D integral braiding pipe fabric as top face and inner face reinforced materials could effectively improve the dynamic properties of sandwich composite rectangular plates. The modal experiments indicated that the modal shapes of sandwich composite plate specimen with four kind reinforcement structures were identical, it’s 1st modal shape, 2nd modal shape and 3rd modal shape presented torsional vibration shape, flexural vibration shape and torsional flexural vibration shape, separately, the modal shapes of sandwich composite plate specimen were not obviously affected by reinforcement structure.

Nonlinear Effects in Elastic Flexural-Torsional Vibrations of Beams of Arbitrary Cross Section

In this paper a boundary element method is developed for the nonlinear flexural-torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderate large deflections and rotations under general boundary conditions, taking into account the effects of rotary and torsional warping inertia. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the nonlinear effects to the response of the beam.

Analysis of Flexural and Torsional Vibration for Turbogenerator Shafts on Power Impact

Based on a Jeffcott rotor system with rigid support, the math models of flexural and torsional vibration were built, and their vibration characteristics were concluded. A simulation experiment in this paper was designed about the flexural and torsional vibration of shafts on a simulated turbine generator during network impacts. By analysis, flexural and torsional vibration characteristics were obtained, and the results showed the shafts flexural vibration and torsional vibration were interacted with each other. It was provided that torsional vibration could inhibit the complex shafts exercise. A new idea was suggested that the characteristics of torsion vibration should be considered as the symptom of faults diagnosis

Vibration analysis of pultruded FRP frames with semi-rigid connections

Vibration frequencies and mode shapes of pultruded FRP plane and space frames with semi-rigid joints are analyzed using two-node locking- free Hermitian finite elements. A nonlinear beam displacement field, accounting for shear deformation effects due to non-uniform bending and torsion, is adopted. The equation of motion is obtained via Hamilton’s principle, and the second-order work of initial stresses is included. Flexibility of three-dimensional joints is simulated by modifying the element stiffness matrix without introducing additional degrees of freedom. The first natural frequencies of portal frames, two-bay single-storey and single-bay two-storey frames are reported as a function of in-plane rotational joint stiffness, and a mode change between flexural and flexural–torsional vibrations is sometimes observed. Finally, a pultruded space frame is analyzed, which shows the effects of semi-rigid connections, column orientation and shear deformation on the natural frequencies.