Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass

Wang, C.Y.; Wang, C. M.

doi:10.1115/1.4005835

Cited by 35 publications

(23 citation statements)

References 11 publications

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“…Therefore, in this example, we investigated a cantilever beam with a tip mass. This system was also investigated in Wang and Wang (2012), and the same results were reported. As reported in Wang and Wang (2012), as the value of b decreases, the value of V for the first mode decreases, but values of V for the higher modes increase.…”

Section: Numerical Resultssupporting

confidence: 73%

“…These cases are thoroughly investigated in Wang and Wang (2012) and Li, Kang, and Wu (2013). Therefore, herein, we just investigate the case 4λ 2 > b 2 in detail which occurs for most cases.…”

Section: Substituting Equation (3) Into Equation (2) Results Inmentioning

confidence: 97%

“…As a first example, a non-uniform cantilever beam without the mass-spring system is analyzed. The results of this beam are available in Ece, Aydogdu, and Taskin (2007) and Wang and Wang (2012). To have such a system, the values of k t and k r approach infinity and values of g and k are set equal to zero.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The height of the cross section is assumed to be constant. Such geometry was previously taken into account in Suppiger and Taleb (1956), Ece, Aydogdu, and Taskin (2007), De Rosa et al (2010) and Wang and Wang (2012). Therefore, assuming h.x/ D h 0 and b.x/ D b 0 e ax L , in which h 0 and b 0 are the height and width of the beam at x D 0, gives the subsequent relations for the cross-sectional area, A, and moment of inertia, I :…”

Section: Problem Statementmentioning

confidence: 99%

“…Various numerical examples were also investigated. Wang and Wang (2012) analyzed the problem of free vibration of an exponentially cantilever beam with a tip mass. They presented the exact characteristic equation for generating the complete vibration frequencies of the system.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Hozhabrossadati

2015

The IES Journal Part A: Civil & Structural Engineering

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This article focuses on the free vibration analysis of a non-uniform cantilever beam with an attached mass-spring system at the free end. One end of the beam is elastically restrained against rotation and translation. The height of the beam is assumed constant but the width of the beam exponentially varies. The governing differential equations of the beam, which is a partial differential equation with variable coefficients, and that of the mass-spring system, which is an ordinary differential equation, are found. The exact solution of the problem is then obtained using the pertinent boundary conditions. The eigenvalues and eigenfunctions of the problem which are frequencies and mode shapes of the system are derived for various properties of the system such as stiffness of springs and attached mass. Some limiting cases with available results in the literature are analyzed. The comparison of the proposed results and the reference results shows the accuracy of the solution.

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Section: Numerical Resultssupporting

confidence: 73%

Section: Substituting Equation (3) Into Equation (2) Results Inmentioning

confidence: 97%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Problem Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Hozhabrossadati

2015

The IES Journal Part A: Civil & Structural Engineering

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Vibrations of flexible pylons with time‐dependent mass attachments under ground induced motions

Manolis

Dadoulis

2021

Earthq Engng Struct Dyn

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This work examines how mass attachments that vary with time influence the motion of flexible pylons undergoing ground-induced vibrations. Specifically, an analytical solution is derived for a time-dependent, lumped mass attachment placed at the top of a cantilevered pylon undergoing time-harmonic longitudinal vibrations. At first, the solution to the governing equation of motion entails the recovery of the eigenvalue problem in the presence of a fixed mass. This is then followed by a generalization of this solution to allow for mass variability with time. It is noted that modal analysis no longer yields an uncoupled system of equations for the generalized coordinates, this necessitating a second step modal analysis, which is complicated by the fact that the system matrices are now time dependent. Results are presented in terms of time histories and frequency plots for the pylon displacement amplitude at the top, resulting from harmonic base motion of unit amplitude. The aim is to identify the frequency range over which the presence of a time-varying mass is either beneficial or detrimental in minimizing the pylon's kinematic response. Since the rate of change of the mass can be either positive or negative, the special cases of constant mass and of no mass can be recovered from the solution. Finally, this methodology can be generalized to include transverse and rotational vibrations of flexible pylons.

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Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

Saitta

2023

Earthq Engng Struct Dyn

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The modal analysis and seismic response of a vibrating cantilever, with or without tip mass and rotary inertia, are investigated in this study using a shear deformable beam model and including the effect of vertical load. Based on existing approaches, an original method is proposed that does not use fourth‐order uncoupled equations to determine modal deflection and rotation. In fact, the approach presented herein transforms the second‐order coupled system into a first‐order system which can be solved more easily using matrix algebra and Laplace transform. Furthermore, the proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self‐adjoint, and the solution in the case of forced response using modal superposition. In addition, even if the solution presented herein is applicable only to the cantilever with a tip mass and rotary inertia, the scope is general, and the approach can be applied to shear deformable beams with other boundary conditions. Finally, the seismic response by modal superposition is shown, and some examples are proposed and discussed for the case of uniform or continuously varying cross‐sectional properties.

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Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass

Cited by 35 publications

References 11 publications

Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Vibrations of flexible pylons with time‐dependent mass attachments under ground induced motions

Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

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