Free torsional vibration of cracked nanobeams incorporating surface energy effects

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton's principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.     

Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated.     

Free vibration of axially loaded thin-walled composite Timoshenko beams

Based on shear-deformable beam theory, free vibration of thin-walled composite Timoshenko beams with arbitrary layups under a constant axial force is presented. This model accounts for all the structural coupling coming from material anisotropy. Governing equations for flexural-torsional-shearing coupled vibrations are derived from Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibrations. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for thin-walled composite beams to investigate the effects of shear deformation, axial force, fiber angle, modulus ratio on the natural frequencies, corresponding vibration mode shapes and load–frequency interaction curves.     

Free vibration analysis of rotating composite Timoshenko beams with bending-torsion couplings

Meccanica - The free vibration of rotating bending-torsional composite Timoshenko beams (CTBs) with arbitrary boundary conditions is analyzed. The composite material coupled rigidity, Coriolis...     

Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model

The bending responses of functionally graded(FG) nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.     

In-Plane vibration of Timoshenko arcs with variable cross-section

Summary This paper studies in-plane vibrations of Timoshenko arcs with variable cross-section by the transfer matrix approach. For this purpose, the equations governing the in-plane vibration of the arcs are written in a coupled set of first-order differential equations by use of the transfer matrix. Once the transfer matrix has been determined by numerical integration of the equations, the natural frequencies (the eigenvalues) and the mode shapes are calculated in terms of the elements of the matrix for a given set of boundary conditions. This method is applied to arcs with linearly, parabolically and exponentially varying cross-section, and the effects of the varying cross-section and slenderness on the free vibrations of the arcs are studied.

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Übersicht Diese Abhandlung untersucht die ebenen Schwingungen von Timoshenkoskreisbogenträgern mit veränderlichem Querschnitt mit Hilfe einer Transfermatrix-Methode. Zu diesem Zweck werden die linearen Differentialgleichungen, die die ebenen Schwingungen der Kreisbogenträger beherrschen, durch Anwendung einer Transfermatrix umgeschrieben. Sobald die Transfermatrix durch numerische Integration der Gleichungen bestimmt ist, lassen sich die Eigenwerte und Schwingungsformen aus den Matrixelementen für beliebige Randbedingungen berechnen. Diese Methode wird auf die Analyse von Kreisbogenträgern mit linear, parabolisch und exponentiell veränderlichen Querschnitt angewandt. Die Einwirkungen des veränderlichen Querschnittes und der Schlankheit der Kreisbogentr äger auf die freien Schwingungen werden diskutiert.

     

Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions

A nonlocal Euler beam model with second-order gradient of stress taken into consideration is used to study the thermal vibration of nanobeams with elastic boundary.An analytical solution is proposed to investigate the free vibration of nonlocal Euler beams subjected to axial thermal stress.The effects of the nonlocal parameter,thermal stress and stiffness of boundary constraint on the vibration behaviors of nanobeams are revealed.The results show that natural frequencies including the thermal stress are lower than those without the thermal stress when temperature rises.The boundary-constrained springs have significant effects on the vibration of nanobeams.In addition,numerical simulations also indicate the importance of small-scale effect on the vibration of nanobeams.     

On the vibration of laminated nonhomogeneous orthotropic shells

In this paper, an analytical procedure is given to study the free vibration of the laminated homogeneous and non-homogeneous orthotropic conical shells with freely supported edges. The basic relations, the modified Donnell type motion and compatibility equations have been derived for laminated orthotropic truncated conical shells with variable Young’s moduli and densities in the thickness direction of the layers. By applying the Galerkin method, to the basic equations, the expressions for the dimensionless frequency parameter of the laminated homogeneous and non-homogeneous orthotropic truncated conical shells are obtained. The appropriate formulas for the single-layer and laminated complete conical and cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the non-homogeneity, the number and ordering of layers and the variations of the conical shell characteristics on the dimensionless frequency parameter are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam

In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.     

Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order

The present paper investigates the free vibration characteristics of Timoshenko beams whose cross-sectional profile and material properties vary along the beam axis with any arbitrary functions. Free vibration analysis of these beams is carried out through solving the governing differential equations of motion. Since the application of differential transformation method (DTM) does not necessarily converge to satisfactory results, an element-based differential transformation method, namely differential transformation element method (DTEM), is introduced which significantly enhances the accuracy of the results. Furthermore, differential quadrature element of the lowest order (DQEL) is introduced which is based on differential quadrature element method (DQEM). DQEL formulates the problem on the basis of the interpolation of the first differential of the functions; therefore, in contrast with DQEM higher differentials of functions are not employed in DQEL. The competency of DQEL and DTEM in free vibration analysis is verified through several numerical examples. The effects of taper ratio and material non-homogeneity on natural frequencies are investigated.     

Coupling of bending and torsional vibration of a cracked Timoshenko shaft

Summary A transverse surface crack is known to add to the shaft a local flexibility due to the stress-strain singularity in the vicinity of the crack tip. This flexibility can be represented by way of a 6 × 6 matrix describing the local flexibility in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling of motion along the directions which are indicated by the off-diagonal terms. Not all motions are coupled, however. To study the coupling of torsion and shear, a 3 × 3 flexibility matrix is used which includes the appropriate terms. Due to the shear terms of the Timoshenko beam equation of the shaft, bending vibration is finally coupled to torsional vibration. This effect is the subject of this investigation, which is of particular importance in turbomachinery operation. The equations of motion of a Timoshenko beam shaft with three degrees of freedom are derived. The free vibration of the shaft and the influence of the crack on the vibrational behaviour of the shaft is studied. The relation of the eigenvalues of the system, to the crack depth and the slenderness ratio of the shaft is derived. Moreover forced vibration analysis of the cracked shaft is performed. The significant influence of the bending vibration on the torsional vibration spectrum, and vice-versa, is demonstrated. It is believed that this effect can be very useful for rotor crack identification in service, which is of importance to turbomachinery.

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Kopplung zwischen Biege- und Torsionsschwingungen einer Welle Tom Timoshenko-Balkentyp mit Riß

Übersicht Bekanntlich verringert ein von der Oberfläche in den Querschnitt reichender Riß infolge der Spannungs- und Verzerrungssingularität an der Rißspitze örtlich die Steifigkeit einer Welle. Dies kann mit Hilfe einer 6 × 6-Nachgiebigkeitsmatrix für ein kurzes Wellenstück, das den Riß enthält, beschrieben werden. Die Matrix enthält Elemente außerhalb der Diagonalen, wodurch eine Kopplung der Bewegungen in die Richtungen erfolgt, welche das Element anzeigt. Nicht alle Bewegungen sind dabei verknüpft. Zur Untersuchung der Kopplung von Torsion und Querschub wird eine 3 × 3-Nachgiebigkeitsmatrix benutzt, die die betreffenden Elemente enthält. Infolge der Querkraft-Terme in der Timoshenko-Balkengleichung werden letztendlich Biegef- und Torsionsschwingungen gekoppelt. Dieser Effekt ist Gegenstand der Untersuchung. Die Bewegungsgleichungen eines Timoshenko-Balkens mit 3 Freiheitsgraden werden hergeleitet. Die freien Schwingungen der Welle und der Einfluß des Risses auf das Schwingungsverhalten werden untersucht. Die Beziehungen zwischen Eigenformen, Rißtiefe und Schlankheitsgrad der Welle werden hergeleitet. Darüber hinaus werden erzwungene Schwingungen der angerissenen Welle untersucht. Der deutliche Einfluß der Biegeschwingung auf das Spektrum der Torsionsschwingung und umgekehrt wird aufgezeigt. Dieser Effekt ist bei Turbomaschinen bedeutsam, da er für die Identifizierung von Rotorrissen beim Betrieb nützlich sein müßte.

     

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.     

Nonlinear vibration analysis of viscoelastic plates based on a refined Timoshenko theory

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 120–131, May 2006.     

Influence of a finite strain on vibration of a bounded Timoshenko beam

The goal of this work is to study the eigenmodes of shearable beams with initial finite strain. A three dimensional model is developed on the base of Cosserat continuum mechanics. The characteristics of waves propagation superimposed upon finite pre-stress are obtained using the (rigorous) calculation of the Hamiltonian action. The results are applied on vibration of beam supporting a finite longitudinal strain. Nonlinear effect according to the pre-stress is obtained for various boundary conditions and through a nondimensional formalism.     

Free vibration of multi-degree-of-freedom non-linear systems

The motions of n unequal masses connected by (n + 1) non-linear springs and anchored to rigid end walls were computed applying a perturbation method. A general solution was obtained exhibiting pseudonormal modes similar to the linear normal modes. Additional pseudo-normal modes are obtained for special cases. A separate solution is formulated for the cases where the above mentioned general solution breaks down.     

Dynamics and magneto-rheological control of vibration of cantilever Timoshenko beam under earthquake loads

The problem of minimizing the dynamics response of a damped cantilever Timoshenko beam subjected to earthquake excitation is investigated in this paper. The ground acceleration is expressed in terms of a Fourier series that is modulated by an enveloping function. The method of lines and modal approach are developed for analyzing the eigenvalues and the flexural vibrations. A magneto rheological damper is proposed to reduce the vibration of the structure. The device is localized at a specific point of the beam. A modal shape which characterizes the vibration of the uncontrolled and controlled system is obtained. The condition of stability of the controlled system is derived using the Routh–Hurwitz criterion.     

Free vibrations of an isotropic nonhomogeneous infinite plate of linearly varying thickness

Summary The free transverse vibrations of an isotropic nonhomogeneous infinite plate of variable thickness have been studied on the basis of classical plate theory. The governing differential equation of motion has been solved by Frobenius method by expressing the transverse displacement as an infinite series. The frequencies corresponding to the first two modes of vibration are computed for different values of thickness variation constant, nonhomogeneity parameter, and different combinations of boundary conditions.

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Sommario Viene considerato il problema delle vibrazioni flessionali libere di una lastra infinita, isotropa, non omogenea, di spessore variabile. L'equazione del moto è stata risolta col metodo di Frobenius, esprimendo lo spostamento mediante sviluppo in serie. Sono state calcolate le prime due au to frequenze per varie combinazioni delle condizioni ai limiti e per differenti valori dei parametri caratteristici della variazione di spessore e della disomogeneità.