Non-uniform Timoshenko Beam Research Articles

Vibration analysis of timoshenko beams with non-homogeneity and varying cross-section

In this paper, an analytic solution for free and forced vibrations of stepped Timoshenko beams is presented and used for the approximate analysis of generally non-uniform Timoshenko beams. In the case of free vibrations, the frequency equation is expressed in terms of some initial parameters at one end of the beam; while in the case of forced vibrations, the solution may be obtained by solving a set of algebraic equations with only two unknowns. Several examples are presented to illustrate the validity and accuracy of the analysis.

Vibrations of elastically restrained non-uniform timoshenko beams

The free vibration of an elastically restrained symmetric non-uniform Timoshenko beam resting on a non-uniform elastic foundation and subjected to an axial load is studied. The two coupled governing characteristic differential equations are reduced into two separate fourth order ordinary differential equations with variable coefficients in the angle of rotation due to bending and the flexural displacement. The frequency equation is expressed in terms of the four normalized fundamental solutions of the associated differential equation. A simple and efficient algorithm is developed to find the approximate fundamental solutions of the governing characteristic differential equation. The relation between problems with elastically restrained boundary conditions and those with tip-mass boundary conditions is explored. Finally, several limiting cases are examined and examples are given to illustrate the validity and accuracy of the analysis. It is noted that the proposed analysis can also be applied to stepped beam problems.

Dynamic stiffness analysis of non-uniform timoshenko beams

A dynamic stiffness method is introduced to analyze non-uniform Timoshenko beams with various boundary conditions. A dynamic stiffness matrix is formed by using frequency dependent shape functions which are exact solutions of the governing differential equations. This eliminates spatial discretization error and is capable of predicting many natural modes with use of a small number of degrees of freedom. The method is now widely used in various dynamic problems [1-4]. In this study it is not necessary to reduce the two coupled governing differential equations into one equation, as has been done by many authors [5-8]. The natural frequencies of the non-uniform beams can be found by equating to zero the determinant of the dynamic stiffness matrix of the system. Several cases are studied and the results are compared with those obtained by the finite element method.

Bending Vibrations of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

Without considering the Coriolis force, the governing differential equations for the pure bending vibrations of a rotating nonuniform Timoshenko beam are derived. The two coupled differential equations are reduced into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The explicit relation between the flexural displacement and the angle of rotation due to bending is established. The frequency equations of the beam with a general elastically restrained root are derived and expressed in terms of the four normalized fundamental solutions of the associated governing differential equations. Consequently, if the geometric and material properties of the beam are in polynomial forms, then the exact solution for the problem can be obtained. Finally, the limiting cases are examined. The influence of the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and taper ratio on the natural frequencies, and the phenomenon of divergence instability (tension buckling) are investigated.

Static analysis of nonuniform timoshenko beams

With the assumption that the bending rigidity of a beam is second-order differentiable with respect to the coordinate variable, the exact static deflection of a nonuniform Timoshenko beam with typical kinds of boundary conditions are given in closed form and expressed in terms of the four fundamental solutions of the governing differential equation. Finally, the limiting cases are studied and the results are shown to be consistent with those in the existing literature.

Exact Vibration Solutions for Nonuniform Timoshenko Beams with Attachments

The exact solution for the free vibration of a symmetric nonuniform Timoshenko beam with tip mass at one end and elastically restrained at the other end of the beam is derived. The two coupled governing characteristic differential equations are reduced into one complete fourth-order ordinary differential equation with variable coefficients in the angle of rotation due to bending. The frequency equation is derived in terms of the four normalized fundamental solutions of the differential equation. It can be shown that, if the coefficients of the reduced differential equation can be expressed in polynomial form, the exact fundamental solutions can be found by the method of Frobenius. Finally, several limiting cases are studied and the results are compared with those in the existing literature. A (x) E(x) G(x) I(x) J(x)

Design of Beams Subjected to Random Loads

ABSTRACT Nonuniform Timoshenko beams subjected to a given stationary random excitation are considered. The general equations relating the spectral density function of the response to the cross spectral density of the load are derived. The optimal shape of the beam is defined as the shape which, for given constant volume of the beam, minimizes the maximum root-mean-square value of the bending stresses in the beam. The shape of the beam is described by a limited number of orthogonal design functions, and their optimal combination is found by sequential linear programming with move limits. From numerical results it is seen that slight modifications of the beam shape give a considerable reduction of maximum r.m.s. stress for most loading cases.

Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.

Propagation of Discontinuous Waves In Nonuniform Timoshenko Beams

Propagation of Discontinuous Waves In Nonuniform Timoshenko Beams