An Analytical Approach to Vibration Analysis of Beams with Variable Properties (original) (raw)
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Transverse Vibration Analysis of Euler-Bernoulli Beams Using Analytical Approximate Techniques
Advances in Vibration Analysis Research, 2011
The vibration problems of uniform and nonuniform Euler-Bernoulli beams have been solved analytically or approximately [1-5] for various end conditions. In order to calculate fundamental natural frequencies and related mode shapes, well known variational techniques such as Rayleigh_Ritz and Galerkin methods have been applied in the past. Besides these techniques, some discretized numerical methods were also applied to beam vibration analysis successfully. Recently, by the emergence of new and innovative semi analytical approximation methods, research on this subject has gained momentum. Among these studies, Liu and Gurram [6] used He's Variational Iteration Method to analyze the free vibration of an Euler-Bernoulli beam under various supporting conditions. Similarly, Lai et al [7] used Adomian Decomposition Method (ADM) as an innovative eigenvalue solver for free vibration of Euler-Bernoulli beam again under various supporting conditions. By doing some mathematical elaborations on the method, the authors obtained i th natural frequencies and modes shapes one at a time. Hsu et al. [8] again used Modified Adomian Decomposition Method to solve free vibration of non-uniform Euler-Bernoulli beams with general elastically end conditions. Ozgumus and Kaya [9] used a new analytical approximation method namely Differential Transforms Method to analyze flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam. Hsu et al. [10] also used Modified Adomian Decomposition Method, a new analytical approximation method, to solve eigenvalue problem for free vibration of uniform Timoshenko beams. Ho and Chen [11] studied the problem of free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using Differential Transform Method. Another researcher, Register [12] found a general expression for the modal frequencies of a beam with symmetric spring boundary conditions. In addition, Wang [13] studied the dynamic analysis of generally supported beam. Yieh [14] determined the natural frequencies and natural www.intechopen.com
Finite Element Method for Vibration Analysis of Timoshenko Beams
2019 9th International Conference on Recent Advances in Space Technologies (RAST), 2019
In industry and lots of engineering applications, rotating components, turbines, helicopter blades, rotors belong to large usage area. Design, material properties and dynamic properties of these structures or components are so significant with respect to efficiency. Frequencies and mode shapes are used to identify the dynamic properties of structures. In this study, a theoretical investigation in free vibration of a functionally graded beam (FGB) is presented with using Finite Element Model. It is assumed that material properties vary along the beam thickness according to power law distributions. Timoshenko beam theory is studied and the FGB are modeled according to this theorem. Free vibration analysis of flap wise bending is studied at symmetrical functionally graded beam. The governing equations of motion and boundary conditions are derived on the basis of Hamilton principle. Analytical solutions of the natural frequencies are obtained with finite element method which the properties of FGB distribution shape functions are used for exponential FG beams with clamped-free end supports. MATLAB code is developed to analyze the free vibration of the functionally graded rotating Timoshenko beam. In the process, finite element formulation (FE) is used and the calculated results are validated with the ones in open literature.
Mathematical Models in Engineering, 2018
In this article, the free vibrations of Euler-Bernoulli and Timoshenko beams with arbitrary varying cross-section are investigated analytically using the perturbation technique. The governing equations are linear differential equations with variable coefficients and the Wentzel, Kramers, Brillouin approximation is adopted for solving these eigenvalue equations and determining the natural frequencies and mode shapes. This method relates the solution of equations with the solving of some successive algebraic equations. A parametric study is performed and the effects of different profiles and different combinations of boundary conditions on the natural frequencies are investigated. To confirm the reliability of the present method, the analytical results are checked with those obtained from the finite elements method and other literatures which are found to be in a good agreement. The calculations show that the presented procedure is very effective to find the modal characteristics of the varying cross-sections beams.
Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams
Meccanica, 2014
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.
Linear free vibration analysis of tapered Timoshenko beams using coupled displacement field method
2016
Every structure which is having some mass and elasticity is said to vibrate. Natural frequency is the one of the most important parameter associated with engineering vibration. In nature every structure has its own natural frequency. Whenever the natural frequency of the structure coincides with the frequency of external applied load, excessive deflections will occur and the structure will be failed. To avoid such condition one must be aware of the operating frequencies of the materials or structures under various conditions like simply supported, clamped and cantilever boundary conditions. There are many methods to evaluate the natural frequency of the structures. in this method the authors developed a method called “coupled displacement field method” which reduces computational efforts compared with the other methods and which is successfully applied for the Hinged-Hinged boundary condition of a tapered (rectangular cross section) Timoshenko beam and calculated the fundamental fre...
An analytical-numerical approach to vibration analysis of periodic Timoshenko beams
Composite Structures, 2018
The subject of this article is analysis of transverse vibrations of beams which geometric and material properties vary periodically along the longitudinal axis. The aim is to present averaged models that take into account the shear deformation and geometric non-linearity, and to analysand transverse vibrations of such beams in moderately large deflection range. As the theoretical foundations, we use Timoshenko beam theory with von Kármán-type non-linearity. This results in obtaining new differential equations with constant coefficients, some of which explicitly depend on the beam inhomogeneity period size. Then, a reasonably simplified model is proposed to describe the vibrations of the considered beams in the low frequency range. The differential equations are transformed into a system of algebraic equations according to the Galerkin method. The response of the beam to transverse harmonic load is investigated by means of a pseudo arc-length continuation scheme. Non-linear coupling between vibration modes and the possibility of superharmonic resonance occurrence are taken into account. As an example of application, few special cases of beam geometry and boundary conditions are examined and compared. The results have the potential application to structural vibration control.
IOP Conference Series: Materials Science and Engineering, Volume 576, 2019 International Conference on Advances in Materials, Mechanical and Manufacturing (AMMM 2019) 22–24 March 2019, Beijing, China, 2019
Rotating beams are extensively used in different mechanical and aeronautical installations. In this paper, a systematic approach is presented in order to solve the eigenvalues problem through the Timoshenko beam theory. The equations of motion are deduced by using the Hamiltonian approach. These equations are then solved by the differential transform method (DTM). The obtained numerical results using DTM are compared with the exact solution. Natural frequencies are determined, and the effects of the rotational speed and axial force on the natural frequencies are investigated. Results show high accuracy and efficiency of the differential transform method. 1. Introduction Rotating structures can be found in turning machinery systems such as motors, engines, and turbines. The bending vibrations analysis of the beams aroused considerable interest for the engineers. The natural frequencies and mode shapes of such systems are indispensable in the design of structures. Zu and Han [1] analytically solved the free flexural vibration of a spinning Timoshenko beam with classical boundary conditions. Zhang [2] studied the free vibration of axially loaded shear beam column and obtained a very simple frequency equation to utilize for axially loaded beam as well as to obtain the buckling loads by setting the natural frequencies to disappear. Farchaly and Shebl [3] determined two sets of exact general frequency and mode shape equations to study the vibration and stability of a Timoshenko beam that carries an end masses of finite length. Lee [4] enclosed the constant axial force and found that it had a considerable effect on the magnitude of the dynamic response. Ouyang [5] established a dynamic model for a rotating Timoshenko beam subjected to three force components acting on the surface. The deflection of the beam examined and found it had proportional increases with respect to the deflection and the frequency components when the axial force component is included. The effect of such parameters such as moving velocity, the skew force angle, and the rotating speed on the system dynamic response is investigated by utilizing the global assumes mode method by considering boundary conditions [6]. The dynamic green function is used to introduce the free vibration of elastically supported Timoshenko beam on a partly Winkler foundation [7]. The finite element method is used the investigate the behaviour of the natural frequencies and to determine the influence of the rotating speed profile on the vibration of the cantilevered beam based on the dynamic modelling method by using the stretch deformation [8].
Journal of Sound and Vibration, 2000
Two sets of governing equations for transverse vibration of non-uniform Timoshenko beam subjected to both axial and tangential loads have been presented. In the "rst set, the axial and tangential loads were taken perpendicular to the shearing force, i.e., normal to the cross-section inclined at an angle , while in the second set, the axial force is assumed to be tangential to the axis of the beam-column. For each case, there exist a pair of di!erential equations coupled in terms of the #exural displacement and the angle of rotation due to bending. The two coupled second order governing di!erential equations were combined into one fourth order ordinary di!erential equation with variable coe$cients. The parameters of the frequency equation were determined for di!erent boundary conditions. The exact fundamental solutions could be found by expressing the coe$cients of the reduced di!erential equation in a polynomial form before applying the Frobenius method. Several illustrative examples of uniform and non-uniform beams with various boundary conditions such as clamped supported, elastically supported, and free end mass and pinned end mass, have been presented. The stability analysis, for the variation of the natural frequencies of the uniform and non-uniform beams with the axial force, has also been investigated. Moreover, the present work illustrates the frequency behavior of the beam under a tangential load.
A finite beam element for vibration analysis of rotating tapered timoshenko beams
Journal of Sound and Vibration, 1992
A finite beam element for vibration analysis of a rotating tapered beam including shear deformations and rotary inertia is derived in this paper. The finite element has four degrees of freedom, and accounts for linear tapering in two planes. This formulation permits any combination of taper ratios as well as unequal element lengths. Explicit expressions for the finite element mass and stiffness matrices are derived using the consistent mass approach, while accounting for the centrifugal stiffness effects. The generalized eigenvalue problem is defined and numerical solutions are generated for a wide range of rotational speed and taper ratios. Results obtained include the first ten free vibrational modes for both fixed and hinged end conditions. Comparisons are made whenever possible with exact solutions and with numerical results available in the literature. The results display high accuracy when compared with other numerical results.
An Exact Solution for the Free Vibration Analysis of Timoshenko Beams
This work presents a new approach to find the exact solutions for the free vibration analysis of a beam based on the Timoshenko type with different boundary conditions. The solutions are obtained by the method of Lagrange multipliers in which the free vibration problem is posed as a constrained variational problem. The Legendre orthogonal polynomials are used as the beam eigenfunctions. Natural frequencies and mode shapes of various Timoshenko beams are presented to demonstrate the efficiency of the methodology.