Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end

doi:10.1016/0022-460X(76)90559-9

Journal of Sound and Vibration

Volume 48, Issue 4, 22 October 1976, Pages 565-568

https://doi.org/10.1016/0022-460X(76)90559-9 Get rights and content

Abstract

The purpose of the present study is to deal with the free vibration of a beam hinged at one end by a rotational spring and subjected to the restraining action of a translational spring at the other end. The eigenfrequencies for the fundamental mode are presented for different values of the parameters K_RL/EI and K_TL³/EI, where KR and K_T are the stiffness constants of the springs.

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Vibration frequencies for a uniform beam with central mass and elastic supports
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Dynamics of Euler-Bernoulli beams with unknown viscoelastic boundary conditions under a moving load
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Citation Excerpt :
Once the beam's inherent properties are specified, its natural frequencies will vary with the stiffness and damping coefficients of the viscoelastic boundaries. Previous works [40-48] illustrated that, under elastic or viscoelastic support conditions, when the dimensionless stiffness coefficients increase, the natural frequencies of the beam will converge from free condition to fixed condition. Therefore, it's reasonable to consider the viscoelastic boundary as fixed to the ground at the supporting direction when these dimensionless stiffness coefficients (Eq. (40)) are multiplied by a large number; 1000 is used in this work.
A new methodology is presented in this work to identify the viscoelastic boundary conditions and dynamical response of Euler-Bernoulli beams under a moving load. A new expression for the transmissibility function of the beam output responses with unknown viscoelastic boundary conditions with Voigt and more generalized four-element models is derived. The proposed identification method uses modal shapes with a pattern search optimization scheme. Finite element simulation was used to demonstrate the validity of the proposed method. The acceleration response from different locations on the beam was utilized in the identification of the boundary conditions considering six optimization cases. The proposed method provided solutions of the boundaries that can satisfy the requirements of the natural frequencies and damping ratios simultaneously. The effects of the number of selected measurement locations, participant modes, and measurement noise on the accuracy of the resulting boundary parameters were investigated. The results showed that the use of three complex modes and eight measurement points provided an accurate estimation of the modal parameters and reduced the relative error effectively in the resulting eight unknown boundary coefficients, under noise-free and corrupted acceleration signals with 5% noise conditions.
Analytic responses of slender beams supported by rotationally restrained hinges during support motions
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Because of the great number of publications, some references are cited for a brief survey. Free vibrations of uniform Euler-Bernoulli beam with elastically restrained support or supports are treated by Chun [2], Lee [3], Grant [4], Hibbeler [5], Maurizi, et al. [6], Goel [7], Laura et al. [8], Rao et al. [9], and Digilov et al. [10]. Non-uniform beams with elastically restrained ends have also been extensively investigated by Verniere de Irassar et al. [11], Laura et al. [12], Alvarez et al. [13], Grossi et al. [14], and Hozhabrossadati [15].
This paper presents an analytic solution procedure of the rotationally restrained hinged-hinged beam subjected to transverse motions at supports based on EBT (Euler-Bernoulli beam theory). The EBT solutions are compared with the solutions based on TBT (Timoshenko beam theory) for a wide range of the rotational restraint parameter ( $k L / E I$ ) of slender beams whose slenderness ratio is greater than 100. The comparison shows the followings. The internal loads such as bending moment and shearing force of an extremely thin beam obtained by EBT show a good agreement with those obtained by TBT. But the discrepancy between two solutions of internal loads tends to increase as the slenderness ratio decreases. A careful examination shows that the discrepancy of the internal loads originates from their dynamic components whereas their static components show a little difference between EBT and TBT. This result suggests that TBT should be employed even for slender beams to consider the rotational effect and the shear deformation effect on dynamic components of the internal loads. The influence of the parameter on boundary conditions is examined by manipulating the spring stiffness from zero to a sufficiently large value.
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Similarly, the next two modes exhibit similar accuracy. The result thus obtained agrees with the work of Maurizi, Rossi and Reyes [30] for a degenerate case, as elaborated in references [21,22]. Fig. 11 summarizes the sensing and actuating procedure.
An efficient analytical method for vibration analysis of a Euler–Bernoulli beam with Spring Loading at the Tip has been developed as a baseline for treating flexible beam attached to central-body space structure, followed by the development of MATLAB© finite element method computational routine. Extension of this work is carried out for the generic problem of Active Vibration Suppression of a cantilevered Euler–Bernoulli beam with piezoelectric sensor and actuator attached as appropriate along the beam. Such generic example can be further extended for tackling light-weight structures in space applications, such as antennas, robot’s arms and solar panels. For comparative study, three generic configurations of the combined beam and piezoelectric elements are solved. The equation of motion of the beam is expressed using Hamilton’s principle, and the baseline problem is solved using Galerkin based finite element method. The robustness of the approach is assessed.
Fundamental frequencies of a nano beam used for atomic force microscopy (AFM) in tapping mode
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Letter to the editorVibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end

Abstract

Mechanical Vibrations

Engineering Vibrations

The Mechanics of Vibration

Tables of characteristic functions representing normal modes of vibration of a beam

The University of Texas, Austin, Publication No. 4913 (Engineering Research Series No. 44)

Vibration Analysis Tables

Tables of eigenmodes for vibrating uniform one-span beams

Chalmers University of Technology, Division of Solid Mechanics, Gothenburg, Publication No. 23

Vibration frequencies for a uniform beam with central mass and elastic supports

Journal of Applied Mechanics

Vibration frequencies for a uniform beam with central mass and elastic supports

Trans. ASME

Letter to the editor
Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end