Vibration Of Thin-walled Beams Research Articles

An Analysis of the Effect of a Change in the Support Point Location on the Vibration of Thin-Walled Beams

This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.

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.

Flexural-Torsional Vibration of Thin-Walled Beams Subjected to Combined Initial Axial Load and End Bending Moment: Application to the Design of Saw Tooth Blades

The present paper analyzes the vibration issue of thin-walled beams under combined initial axial load and end moment in two cases with different boundary conditions, specifically the simply supported-end and the laterally fixed-end boundary conditions. The analytical expressions for the first natural frequencies of thin-walled beams were derived by two methods that are a method based on the existence of the roots theorem of differential equation systems and the Rayleigh method. In particular, the stability boundary of a beam can be determined directly from its first natural frequency expression. The analytical results are in good agreement with those from the finite element analysis software ANSYS Mechanical APDL. The research results obtained here are useful for those creating tooth blade designs of innovative frame saw machines.

Analytical and finite element solutions of free and forced vibration of unrestrained and braced thin-walled beams

In this article, vibration of thin-walled beams with arbitrary open cross-section shape is investigated. Based on the beam element model accounting for warping and flexural–torsional coupling, analytical solutions for different boundary conditions are derived for higher free vibration modes in bending, torsion and flexural–torsional coupled modes. In the model, the effects of rotational inertial kinematic terms are considered. The finite element approach of the model is also investigated. Three-dimensional beams with seven degrees of freedom per node are adopted in the mesh process. Free vibration and forced vibration analyses are possible. In forced vibration, the behaviour of the beams is studied in the frequency domain using the steady-state method (modal analysis). Damping is considered using the Rayleigh model. The model is validated by comparing the results to benchmark solutions found in the literature and to other recent numerical and experimental results. Additional finite element simulations are performed by means of commercial softwares (Abaqus and Adina). In slender unrestrained beams, the vibration behaviour is predominated by torsion and lateral bending modes. In design, recourse to braces is a good compromise. This solution is discussed, and improvement of the vibration behaviour in the presence of intermediate braces is confirmed. Application of higher vibration modes in building and bridge design is outlined. The effects of the number and distribution of the intermediate braces to improve structural stability against vibration behaviour is outlined.

Free Vibration Analysis of Cross-Ply Laminated Thin-Walled Beams with Open Cross Sections: Exact Solution

AbstractThe objective of the present paper is to analyze the free vibrations of thin-walled beams with arbitrary open cross section, made of cross-ply laminates with midplane symmetry, by means of an exact solution. The theory of thin-walled composite beams is based on assumptions consistent with Vlasov’s beam theory and classical lamination theory. The governing differential equations for coupled bending-torsional vibrations were obtained using the principle of virtual displacements. To simplify the coupled system of differential equations, an ideal center of gravity and shear center were introduced. In the case of a simply supported thin-walled beam, the closed-form solution for the natural frequencies of free harmonic vibrations was derived. The frequency equation, given in determinantal form, is expanded in an explicit analytical form. To demonstrate the validity of this method, the natural frequencies of nonsymmetric thin-walled beams having coupled deformation modes are evaluated and compared with r...

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.

Nonlinear Torsional Vibrations of Thin-Walled Beams of Open Section

An attempt has been made to derive and solve the governing differential equation of large amplitude torsional vibrations of simply supported doubly symmetric thin-walled beams of open section. Graphs indicating the influence of large amplitudes on nonlinear period of torsional vibrations for various nondimensional beam constants are presented.

Non-linear and parametric vibration of thin-walled beams of equal angle-section

The linear coupling of the various types of motion of thin-walled beams of equal angle-section was discussed in reference [1]. Here their interaction due to non-linear effects is studied. In addition to the usual displacements in the axial direction, a second-order effect is introduced. The mutual interaction of two types of motion at a time is studied. This is done by retaining quadratic non-linearities while ignoring cubic non-linearities in the derived differential equations of motion, so obtaining pairs of (quadratically) coupled non-linear partial differential equations in the relevant dependent variables. Application of Galerkin's approximation gives a set of ordinary differential equations with periodic coefficients. The regions of instability are evaluated, and are expressed in non-dimensional terms. Experimental tests on a series of cantilever beams show that torsional and cross-sectional modes are parametrically excited by support motion in either the axial or the transverse direction. When the forcing frequency approaches one of the natural frequencies (longitudinal or transverse), it is shown that the parametric instability region becomes substantially wider. The effect on the instability region due to changes in the basic dimensions of the beam is also discussed.

Linear vibration of thin-walled beams of equal angle-section

The linear coupling of the various possible types of motion of thin-walled beams of anglesection with equal arms is studied theoretically and experimentally. In the theory proposed the displacements are considered as comprising lateral, transverse and torsional motion of the beam as a whole along with deformation of its cross-section. By application of Hamilton's principle the differential equations of motion of the beam are derived. It is shown that in the main the various types of motion are coupled in pairs (e.g., transverse bending couples with symmetric distortion); a frequency-wavelength or dispersion relation is thus obtained for the coupled motion in the form of a simple quadratic equation in non-dimensional terms, suitable for comparison with experimental results. A series of beams were made, one dimension at a time being varied so that the effect of such variations on the natural frequencies could be studied. The experimental results show reasonable agreement with the calculated curves. Within the range of parameters investigated the torsional and lateral motions are very weakly coupled, whereas the coupling between the transverse vibration and the cross-sectional distortion is much more marked and is clearly shown by the experiments.

Flexural-torsional vibrations of thin-walled beams

Description of a method of calculation based on trasfer matrices for determining the principal modes of vibrations of T beams taking into account torsional and flexional coupling.

Cross-Section Deformation Effects in the Vibration of Thin-Walled Beams of Arc Section

Differential equations describing the coupling of ordinary bending motion with cross-sectional distortion are obtained for thin-walled beams of circular-arc cross-section using Hamilton's principle. In deriving the theory the cross-sectional deformation is assumed to take the form of the characteristic functions of a curved beam of the shape of the section. The variation with wavelength of the frequency spectra which result from the coupling is obtained. Experimental results showing the effects of the variation of the parameters of the beam geometry on the natural frequencies are in reasonable agreement with the theory.

Influence of Cross-Section Distortion on the Bending Vibration of Thin-Walled Beams of H Section

Approximate differential equations describing the bending vibration of beams of thin-walled H section, in which the distortion of the cross-section in its own plane is taken into account, are derived from Hamilton's principle using an assumed form for the cross-section deformation. Only the simplest of the cross-sectional deformation configurations which will couple with ordinary bending is considered. The variation with wavelength of the two spectra of frequencies which result from this coupling of the bending and cross-sectional motions is shown for several section geometries. Theoretical curves show reasonable agreement with experimental results from free beams.

THEORY OF VIBRATIONS OF THIN-VALLED BEAMS

In this paper, the problem of vibrations of thin-walled beams is investigated. The fundamental assumption is the same as that in a preceding paper~[1],i.e. the non-deformability of contour lines of transverse cross-sections in it's own plane. Equations of vibrations show that the vibrations of thin-walled beams generally take place as a combination of bending-torsional and longitudinal vibrations. Corresponding to a certain type of bending-torsional vibrations, instead of only one frequency as in the well-known Vlasof's (B. э. Bдасов) theory, there are an infinite number of frequencies, corresponding to different types of longitudinal vibrations. The natural frequencies as calculated by Vlasof's theory correspond to the lowest frequencies in our theory, but the former are generally higher than the latter as should be expected.