Vibration Of Non-uniform Beams Research Articles

STUDY ON VIBRATION RESPONSE OF A NON-UNIFORM BEAM WITH NONLINEAR BOUNDARY CONDITION

Forced vibration of non-uniform beam with nonlinear boundary condition is studied in this paper by proposing an iterative model combining Adomian Decomposition Method and modal analysis. An exponentially tapered beam with a hardening nonlinearity spring boundary is simulated as a case study. The model accuracy is proved by comparing iteration results and analysis solutions with linear and weakly nonlinear boundary conditions. Sin-weep nonlinear frequency spectrum is then obtained by the proposed model. The influence of boundary nonlinearity on the vibration response of non-uniform beam is analyzed. And the effect of different excitation amplitudes on nonlinearity in the vibration response is studied. The mathematical model and numerical solutions proposed in this paper can be used to solve and analysis broad vibration problems on general non-uniform beams with different nonlinear boundary conditionsunder various excitations.

Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Natural frequencies and free vibration are important characteristics of beams with non-uniform cross section. Hence, the solution for free vibrations of non-uniform beams is presented using a Laguerre collocation method. The elastically restrained beam model is based on the Euler–Bernoulli theory. Also, the non-uniform beam is rested on a non-uniform foundation (Winkler type). The Laguerre collocation method is introduced for solving the differential equation. This approach reduces the governing differential equation to a system of algebraic equations, and finally, the problem is greatly simplified. Properties of Laguerre polynomials and the operational matrix of derivation are first presented. Eventually, the proposed method is applied for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.

Modelling a spade rudder as a hollow two-way tapered Kirchhoff’s plate : free dry and wet vibration study with numerical verification

A semi-analytical approach to free dry and wet vibration of a trapezoidal, 2-way tapered, pivoted hollow spade rudder is presented. The rudder is modeled as a hollow Kirchhoff’s plate, with a NACA0018 profile chord section. The rudder pivot is modeled as a combination of a translational spring and a rotational spring. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, to establish the non-uniform beam modeshapes, which act as admissible functions to the Galerkin’s method for plate vibration. Eigenvalue analysis generates the plate natural frequencies and the plate modeshapes. Alternately, uniform beam modeshapes themselves are used as admissible functions into the Galerkin’s method. Frequencies are analyzed for various pivot positions, taper ratios, and NACA sections. For the wet vibration, constant strength source distribution technique is used to generate the added mass of a 2D aerofoil. Also, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, aspect ratios.

Transverse vibration of nonuniform Euler-Bernoulli beams on bounded Time scales

In this article, we consider Euler-Bernoulli equation of transverse vibrations of nonuniform beams on bounded time scales $\mathbb{T}.$ We will give a description of all maximal dissipative, maximal accretive, self adjoint and other extensions of such operators.

Simulation of the vibrations of a non-uniform beam loaded with both a transversely and axially eccentric tip mass

The main purpose of this work is to employ the Adomian modified decomposition method (AMDM) to calculate free transverse vibrations of non-uniform cantilever beams carrying a transversely and axially eccentric tip mass. The effects of the variable axial force are taken into account here, and Hamilton’s principle and Timoshenko beam theory are used to obtain a single governing non-linear partial differential equation of the system as well as the appropriate boundary conditions. Two product non-linearities result from the analysis and the respective Cauchy products are computed using Adomian polynomials. The use of AMDM to make calculations for such a cantilever beam/tip mass arrangement has not, to the authors’ knowledge, been used before. The obtained analytical results are compared with numerical calculations reported in the literature and good agreement is observed. The qualitative and quantitative knowledge gained from this research is expected to enable the study of the effects of an eccentric tip mass and beam non-uniformity on the vibration of beams for improved dynamic performance.

A New Fourier series solution for free vibration of non-uniform beams, resting on variable elastic foundation

In this research, the combination of Fourier sine series and Fourier cosine series is employed to develop an analytical method for free vibration analysis of an Euler-Bernoulli beam of varying cross- section, fully or partially supported by a variable elastic foundation. The foundation stiffness and cross section of the beam are considered as arbitrary functions in the beam length direction. The idea of the proposed method is to superpose Fourier sine and Fourier cosine series to satisfy general elastically end constraints and therefore no auxiliary functions are required to supplement the Fourier series. This method provides a simple, accurate and flexible solution for various beam problems and is also able to be extended to other cases whose governing differential equations are nonlinear. Moreover, this method is applicable for plate problems with different boundary conditions if two-dimensional Fourier sine and cosine series are taken as displacement function.Numerical examples are carried out illustrating the accuracy and efficiency of the presented approach.

Vibration of Nonuniform Beams Under Moving Point Loads: An Approximate Analytical Solution in Time Domain

The forced-free responses of nonuniform beams under moving point loads are analyzed in this paper. Simple approximate analytical formulae for the forced responses of undamped nonuniform beams, derived using the fundamental mode by the Rayleigh–Ritz (R–R) method, are presented. The responses of both simply supported and clamped–clamped beams are analyzed. The responses are also determined by the finite element method (FEM) in which nonuniform elements are used for fast convergence. It is found that the present method yields results that are very close to those obtained by the FEM. As this method does not require time integration, it is faster and computationally more efficient. Though the single-mode analysis of forced vibration of uniform beams under moving loads has been done by several researchers, its application to nonuniform beams has not been reported.

Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)

The natural frequencies of non-uniform beams resting on two layer elastic foundations are numerically obtained using the Generalized Differential Quadrature (GDQ) method. The Differential Quadrature (DQ) method is a numerical approach effective for solving partial differential equations. A new combination of GDQM and Newton’s method is introduced to obtain the approximate solution of the governing differential equation. The GDQ procedure was used to convert the partial differential equations of non-uniform beam vibration problems into a discrete eigenvalues problem. We consider a homogeneous isotropic beam with various end conditions. The beam density, the beam inertia, the beam length, the linear (k₁) and nonlinear (k₂) Winkler (normal) parameters and the linear (k₃) Pasternak (shear) foundation parameter are considered as parameters. The results for various types of boundary conditions were compared with the results obtained by exact solution in case of uniform beam supported on elastic support.

Non-uniform beam vibration using Differential Transform Method

The paper focuses on the vibration characteristics of non-uniform Euler- Bernoulli beam using Differential Transform Method (DTM). DTM is a numerical method to solve differential equations where the governing equations are reduced into a set of polynomials. Non-uniformity is considered corresponding to linear variation in breadth and height of the beam. The effect of taper ratio on the fundamental frequency of tapered beams is also analysed. The method has proved to be accurate, simple and effective for eigenvalue analysis. For the two cases of non-uniform beam analysed, the frequency computed by the method of differential transform is found to be comparable with the previously available results.

Analytical modeling of large amplitude free vibration of non-uniform beams carrying a both transversely and axially eccentric tip mass

The objective of this paper is to study large amplitude flexural–extensional free vibration of non-uniform cantilever beams carrying a both transversely and axially eccentric tip mass. The effects of variable axial force is also taken into account. Hamilton׳s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. A numerical finite difference scheme is proposed to find the natural frequencies and mode shapes of the system which is validated specifically for a beam with linearly varying cross section. Using a single mode approximation in conjunction with the Lagrange method, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. These temporal coupled equations are then solved analytically using the multiple time scales perturbation technique. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. The qualitative and quantitative knowledge resulting from this research is expected to enable the study of the effects of eccentric tip mass and non-uniformity on the large amplitude flexural–extensional vibration of beams for improved dynamic performance.

Nonlinear vibrations of non-uniform beams by the MTS asymptotic expansion method

The frequency response curves of a non-uniform beam undergoing nonlinear oscillations are deter- mined analytically by the multiple time scale method, which provides approximate, but accurate results. The axial inertia in neglected, and so the equations of motion are statically condensed on the transversal displace- ment only. The nonlinearity due to the stretching of the axis of the beam is considered. The effects of variable cross-section, of variable material properties and of the distributed axial loading are taken into account in the formulation. They have been illustrated by means of two examples and are also compared with existing results. The main result of this work is that the effects of any type of non-uniformity can be detected by simple formulas.

Transverse vibrations of arbitrary non-uniform beams

Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Vibration of non-uniform beams with different boundary conditions subjected to a moving mass is investigated. The beam is modeled using Euler–Bernoulli beam theory. Applying the method of eigenfunction expansion, equation of motion has been transformed into a number of coupled linear time-varying ordinary differential equations. In non-uniform beams, the exact vibration functions do not exist and in order to solve these equations using eigenfunction expansion method, an adequate set of functions must be selected as the assumed vibration modes. A set of polynomial functions called as beam characteristic polynomials, which is constructed by considering beam boundary conditions, have been used along with the vibration functions of the equivalent uniform beam with similar boundary conditions, as the assumed vibration functions. Orthogonal polynomials which are generated by utilizing a Gram–Schmidt process are also used, and results of their application show no advantage over the set of simple non-orthogonal polynomials. In the numerical examples, both natural frequencies and forced vibration of three different non-uniform beams with different shapes and boundary conditions are scrutinized.

Frequency Analysis of an Arbitrarily Supported, Exponentially Tapered Beam

In most introductory vibration textbooks, the vibration of non-uniform beams is not covered. In this paper, the free vibration of an arbitrarily supported Euler–Bernoulli beam of constant thickness and exponentially varying width is investigated. The ‘assumed modes‘ method is utilized to obtain the approximate natural frequencies of the tapered beam, where the basis functions used in the expansion correspond to the eigenfunctions of the uniform beam, including its rigid-body modes. Expressing the eigenfunctions in terms of exponentials, the generalized masses and stiffnesses for the non-uniform beam can all be obtained in closed form, which substantially simplifies the analysis. The proposed scheme is simple to code, efficient to apply and can be easily modified to accommodate a tapered beam with various boundary conditions. Additionally, with slight modifications, the proposed method can also be used to analyze the free vibration of a combined system consisting of an exponentially varying beam carrying lumped attachments. Numerical experiments validate the present approach, and show that the approximate natural frequencies converge very quickly to the exact solution.

Approximate Solution for Longitudinal Vibration of Non-Uniform Beams by Differential Transform Method (DTM)

In this paper, differential transform method is applied for longitudinal vibration analysis of beams with non-uniform cross section. Natural frequencies and corresponding normalized mode shapes are calculated for different cases in cross section and boundary conditions. Comparison of results with the previous solutions proves the accuracy and versatility of the presented paper.     2 0 sin A xA a bx  . Li (4) combined the transfer matrix method and closed form solution of one step non-uniform rod to obtain a single frequency equation for a multi-step non-uniform rod. In another papers, he derived an exact solution for free longitudinal vibrations of one-step non-uniform rods with classical and non-classical boundary conditions (5) and an analytical solution for determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar with several boundary conditions (6). Li, et al. (7) found closed form solution for the longitudinal vibration of non-uniform structures with lumped masses and spring supports whose cross-section varies as   1 n

Exact frequency equations of free vibration of exponentially functionally graded beams

Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler–Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited for pseudo-frequencies lower than the critical frequency. For some gradient index with small change, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating nonhomogeneous beam structures. Obtained results also apply to free vibration of nonuniform beams with constant thickness and exponentially decaying width.

Transverse Vibration of Non-Uniform Euler-Bernoulli Beam, Using Differential Transform Method (DTM)

Analysis of transverse vibration of beams is presented in this paper. Unfortunately, complexities which appear in solving differential equation of transverse vibration of non-uniform beams, limit analytical solution to some special cases, so that the numerical method is presented. DTM is a numerical method for solving linear and some non-linear, ordinary and partial differential equations. In this paper, this technique has been applied for solving differential equation of transverse vibration of conical Euler-Bernoulli beam. Natural circular frequencies and mode shapes have been calculated. Comparing results with the cases which exact solution have been presented, shows that DTM is a strong method especially for solving quasi-linear differential equations.

Research on Numerical Computation of Natural Frequency of Transverse Vibration for Simply Supported Beam of Non-Uniform

Recursive and inexplicit differential equation of the second order with variable coefficients is derived from the fourth order linear homogeneous differential equation with variable coefficients of transverse vibration of non-uniform beam, which is about deflection and bending moment according to boundary conditions and order reduction. By finite difference method, numerical computation and accuracy are studied for natural frequency of transverse vibration for simply supported beam of non-uniform. Theoretical analysis and orthogonal computation examples show that numerical computation algorithm is very simple, and accuracy of computation depends on variety rate of gradually changed cross section in vertical direction and numbers of computation step, which is independent of width and length of beam; numerical accuracy of computation is estimable for given length or numbers of computation step; and reasonable length or numbers of computation step is determinable for given accuracy demand.

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh–Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys ©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.

Linear and nonlinear vibration of non-uniform beams on two-parameter foundations using p-elements

A hierarchical finite element is presented for the geometrically nonlinear free and forced vibration of a non-uniform Timoshenko beam resting on a two-parameter foundation. Legendre orthogonal polynomials are used as enriching shape functions to avoid the shear-locking problem. With the enriching degrees of freedom, the accuracy of the computed results and the computational efficiency are greatly improved. The arc-length iterative method is used to solve the nonlinear eigenvalue equation. The computed results of linear and nonlinear vibration analyses show that the convergence of the proposed element is very fast with respect to the number of Legendre orthogonal polynomials used. Since the elastic foundation and the axial load applied at both ends of the beam affect the ratios of linear frequencies associated with the internal resonance, they influence the nonlinear vibration characteristics of the beam. The axial tensile stress of the beam in nonlinear vibration is investigated in this paper, and attention should be paid to the geometrically nonlinear vibration resulting in considerably large axial tensile stress in the beam.