# Studies on dynamic behaviour of a beam with symmetrically varying thickness

Abstract: A detailed study has been conducted on the effect of different types of variation in profile and thickness on the amplitude and the dynamic bending stress of a beam under a point harmonic load. The end conditions considered are clamped or simply supported at both the ends. The response has been calculated for the first three modes of vibration. In each case the results obtained for different types of thickness variation are compared with those obtained for a uniform thickness beam. It is observed that a considerable reduction in amplitude and bending stress can be achieved by proper selection of thickness variation.

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Topics: Beam (structure) (59%)

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Abstract: This paper analyzes lateral-torsional dynamic instability of elastic beams with rectangular cross section having constant thickness, with the depth symmetric with respect to the midpoint and either uniform or tapered linearly in each half Free vibrations are also investigated The ends of the beam are prevented from twisting and are pinned with respect to bending in the weak direction In the strong direction, one end of the beam is fixed and the other is subjected to transverse harmonic motion This problem was motivated by “butterfly-shaped links” proposed for use in seismic mitigation Uniform torsion is assumed Frequencies of free vibration are computed, critical excitation frequencies for lateral-torsional instability are determined, and critical combinations of excitation amplitude and frequency are obtained The effects of geometric parameters of double-tapered beams on instability are presented Lateral-torsional instability may occur for very small amplitudes of the moving end of the beam, as is typical for such problems involving parametric resonance

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Abstract: The differential equation is developed from the Bernoulli‐Euler equation for the free vibrations of a double‐tapered cantilever beam. The beam tapers linearly in the horizontal and in the vertical planes simultaneously. From a computer solution of this equation, a table has been developed from which the fundamental frequency, second, third, fourth, and fifth harmonic can easily be obtained for various taper ratios. Charts are plotted for selected taper ratios in the vertical plane to show the effect of taper ratios on frequency.

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84 citations

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Abstract: Explicit expressions for the element mass and stiffness matrices of a linearly tapered beam finite element including shear deformation and rotary inertia are given in this paper. The element cross section rotation is assumed to be a sum of the slope of the transverse displacement and the shear. The cross-sectional dimensions vary linearly along the length of the element. Two series of eigenvalue solutions have been carried out. One was performed with various taper ratios in order to provide a comparison with classical cases. The other was used to determine the aspect ratio effect on the eigenvalues of thin-walled tapered cantilever beam structures. The approach presented is straightforward, and allows comparisons to be made with other explicit solutions without recourse to numerical examples.

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55 citations

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Abstract: Free vibrations of nonuniform cantilever beams with an end support have been investigated, using the equations of Bernoulli‐Euler. Two configurations of interest are treated: (a) constant width and linearly variable thickness, and (b) constant thickness and linearly variable width. Charts have been plotted for each case from which the fundamental frequency, the second harmonic, and the third harmonic can be easily determined for various taper ratios. The Tables from which these charts were plotted are also included.

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53 citations

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Abstract: Non-uniform beams with cross-section varying in a continuous or non-continuous manner along their lengths are used in many structural applications in an effort to achieve an optimum distribution of strength and weight. The authors have investigated the problem of calculating the natural frequencies for beams with inertia, area and mass varying in a general manner. The procedure proposed allows the effect of each of these variables to be evaluated separately and leads to the concept of effective inertia, effective area and effective mass. It also enables one to determine the natural frequencies of a beam with varying section properties using a simple formula similar to the one used in practice for computing the natural frequencies for a uniform beam. It is also shown that the effect of shear deformation upon the natural frequencies of a beam with classical boundary conditions can be estimated within the limits of practical accuracy by using a simple formula similar to the one used for simply supported beams. An important feature of the results presented here is the simplicity and relative ease with which these expressions can be applied to beams with various types of boundary conditions. The authors believe that a practicing engineer will find these results easy to apply in computing the natural frequencies of non-uniform beams within practical limits.

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32 citations

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Abstract: Transverse vibrations of a beam with uniform inertia and uniform area have been studied extensively by scientists and engineers and closed form solutions leading to simple formulae are readily available for estimating the natural frequencies of a beam when subjected to various types of boundary conditions. However, no such simple formulae exists for beams whose inertia, cross-sectional area and mass vary along the length. In practice, the use of beams with varying sectional properties, such as inertia, area and mass is common, and to estimate the natural frequencies of these beams the engineer has to resort to more expensive methods such as the finite element method. In this paper, a procedure based on the Galerkin method is employed to estimate the first few natural frequencies of a beam with varying sectional properties and including the effects due to rotary inertia. In a particular instance, the numerical values are computed and are verified by using finite element analysis. Also, a formula based on effective inertia and effective area is proposed from which the natural frequencies can be estimated with reasonable accuracy. This formula is simple and is similar to that used for a beam with uniform section properties; also it enables one to evaluate the rotary inertia effect and the shear effect.

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26 citations