The Fundamental Flexural Vibration of a Cantilever Beam of Rectangular Cross Section with Uniform Taper

Rao, J. S.

doi:10.1017/s000192590000336x

Cited by 21 publications

(4 citation statements)

References 7 publications

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“…Rao [1] used the Galerkin method, Housner and Keightley [2] applied the Myklestad procedure, Rao and Carnegie [3] used the "nite di!erences approach, Martin [4] adopted a perturbation technique and Mabie and Rogers [5] solved the di!erential equations using Bessel functions to "nd the natural frequencies of vibration of tapered cantilever beams.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Vibration Analysis of Rotating Timoshenko Beams

Rao

Gupta

2001

Journal of Sound and Vibration

128

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Section: Introductionmentioning

confidence: 99%

Finite Element Vibration Analysis of Rotating Timoshenko Beams

Rao

Gupta

2001

Journal of Sound and Vibration

128

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“…Similar problems were often solved by the Ritz–Galerkin method in the second half of the 20th century. Significant achievements in its applications were made by Rao [ 11 ], who solved the equations of motion of cantilever bars with a rectangular tapered cross-section. Computational methods were applied too.…”

Section: Introductionmentioning

confidence: 99%

“…The functions 𝑦 1 (𝑥) and 𝑦 2 (𝑥), which are solutions of these equations, can be represented as linear combinations of Bessel functions (11) and (12): Since the natural frequency is of interest, the problem can be reduced to solving an ordinary differential Equation ( 6):…”

Section: Theoretical Introduction To the Approachmentioning

confidence: 99%

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Solution for Determining Modulus of Elasticity of Natural Materials Using Vibrations of Non-Uniform Circular Cross-Section Cantilevers

Podgórski

Kawecki

2023

Materials

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The article presents an original method for determining the modulus of elasticity of natural materials. A studied solution was based on vibrations of non-uniform circular cross-section cantilevers solved using Bessel functions. The derived equations, together with experimental tests, allowed for calculating the material’s properties. Assessments were based on the measurement of the free-end oscillations in time using the Digital Image Correlation (DIC) method. They were induced manually and positioned at the end of a cantilever and monitored in time using a fast Vision Research Phantom v12.1 Camera with 1000 fps. GOM Correlate software tools were then used to find increments of deflection on a free end in every frame. It provided us with the ability to make diagrams containing a displacement–time relation. To find natural vibration frequencies, fast Fourier transform (FFT) analyses were conducted. The correctness of the proposed method was compared with a three-point bending test performed on a Zwick/Roell Z2.5 testing machine. The presented solution generates trustworthy results and can provide a method to confirm the elastic properties of natural materials obtained in various experimental tests.

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Exact Bernoulli–Euler dynamic stiffness matrix for a range of tapered beams

Banerjee

Williams

1985

Numerical Meth Engineering

148

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Bernoulli-Euler theory and Bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial, torsional and flexural vibrations of any beam which is tapered such that A varies as J'"and GJ and I both vary as y(" ' ), where A, GJ and I have their usual meanings; y = (cx:L) + 1; c is a constant such that c > -1; I, is the length of the beam; and x is the distance from onc end of the beam. Numerical checks give better than seven-figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements. A procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped. This enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members. and S =k'8 at x = L) can be substituted in equations (F9) and (Flu) to give i.e. where Equations (F13) and (F17) give F, = K,U, i.e. f(Symmetric K 8 . 8 where K, = D,B,Therefore, one way of finding K, is to use a computer, at each required frequency, to calculate the elements of B, and D, (from equations (F12), (F13), (F16) and (F17)) and then to invert B, and pre-multiply by D,. For large structures the computer is likely to spend much less time performing this arithmetic for all the tapered members in a frame than it spends on the other operations which it has to perform on the frame at the same frequency (see, for example. Reference 31, where the much larger arithmetic effort needed to incorporate stayed columns as members of a space truss proved to be quite a small proportion of the total solution time for the truss). Therefore, this simple procedure would be adequate for many purposes. However, computer time can be saved by using equation (F21), and the expressions for the elements of B, and D, given in equations (F12) and (F16), to obtain algebraic expressions for the elements of K,. Such expressions are particularly useful when only some of the stiffnesses are needed, and are given below. They are not necessarily in the shortest possible form, but arc surprisingly concise, and represent a lot of work by the first author which is too extensive and detailed to be presented. The correctness of the expressions was confirmed by numerical checks which arc discussed in the next section.

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The Fundamental Flexural Vibration of a Cantilever Beam of Rectangular Cross Section with Uniform Taper

Cited by 21 publications

References 7 publications

Finite Element Vibration Analysis of Rotating Timoshenko Beams

Finite Element Vibration Analysis of Rotating Timoshenko Beams

Solution for Determining Modulus of Elasticity of Natural Materials Using Vibrations of Non-Uniform Circular Cross-Section Cantilevers

Exact Bernoulli–Euler dynamic stiffness matrix for a range of tapered beams

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