Natural frequencies for out-of-plane vibrations of curved beams on elastic foundations

The governing differential equations for the out-of-plane, free vibration of circular curved beams resting on elastic foundations are derived and solved numerically. The formulation takes into consideration the effects of rotary inertia and transverse shear deformation. The lowest three natural frequencies are calculated for beams with hinged–hinged, hinged-clamped, and clamped–clamped end constraints. The effects of various system parameters as well as rotary inertia and shear deformation on the natural frequencies are investigated.

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Natural frequencies for out-of-plane vibrations of continuous curved beams

Journal of Sound and Vibration ◽

10.1016/0022-460x(80)90397-1 ◽

1980 ◽

Vol 68 (3) ◽

pp. 427-436 ◽

Cited By ~ 25

Author(s):

T.M. Wang ◽

R.H. Nettleton ◽

B. Keita

Keyword(s):

Natural Frequencies ◽

Curved Beams ◽

Out Of Plane

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Out-of-Plane Free Vibration and Forced Harmonic Response of a Curved Beam

Shock and Vibration ◽

10.1155/2020/8891585 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

Hancheng Mao ◽

Guangbin Yu ◽

Wei Liu ◽

Tiantian Xu

Keyword(s):

Free Vibration ◽

Natural Frequencies ◽

Amplitude Ratio ◽

Harmonic Wave ◽

Wave Number ◽

Curved Beam ◽

Curved Beams ◽

Propagation Behavior ◽

Out Of Plane ◽

Governing Differential Equation

Based on the governing differential equation of out-of-plane curved beam, the wave propagation behavior, free vibration, and transmission properties are presented theoretically in this paper. Firstly, harmonic wave solutions are given to investigate the dispersion relation between frequency and wave number, cut-off frequency, displacement, amplitude ratio, and phase diagram. The frequency spectrum results are obtained to verify the work by Kang and Lee. Furthermore, natural frequencies of the single and composite curved beam are calculated through solving the characteristic equation in the case of free-free, clamped-clamped, and free-clamped boundaries. Finally, the transfer matrices of the out-of-plane curved beam are derived by combining the continuity between the different interfaces. The transmissibility curves of the single and composite curved beam are compared to find the vibration attention band. This work will be valuable to extend the study of the out-of-plane vibration of curved beams.

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Natural frequencies for out-of-plane vibrations of continuous curved beams considering shear and rotary inertia

International Journal of Solids and Structures ◽

10.1016/0020-7683(84)90037-4 ◽

1984 ◽

Vol 20 (3) ◽

pp. 257-265 ◽

Cited By ~ 16

Author(s):

Wang Tung-Ming ◽

Andrea J. Laskey ◽

Mohamed F. Ahmad

Keyword(s):

Natural Frequencies ◽

Rotary Inertia ◽

Curved Beams ◽

Out Of Plane

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Out-of-Plane Vibration of Curved Uniform and Tapered Beams with Additional Mass

Mathematical Problems in Engineering ◽

10.1155/2017/8178703 ◽

2017 ◽

Vol 2017 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Hamdi Alper Özyiğit ◽

Mehmet Yetmez ◽

Utku Uzun

Keyword(s):

Finite Element ◽

Free Vibration ◽

Boundary Conditions ◽

Natural Frequencies ◽

Variable Cross ◽

Additional Mass ◽

Inertia Effects ◽

Tapered Beam ◽

Out Of Plane ◽

Good Agreement

As there is a gap in literature about out-of-plane vibrations of curved and variable cross-sectioned beams, the aim of this study is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. Out-of-plane free vibration of curved uniform and tapered beams with additional mass is also investigated. Finite element method is used for all analyses. Curvature type is assumed to be circular. For the different boundary conditions, natural frequencies of both symmetrical and unsymmetrical tapered beams are given together with that of uniform tapered beam. Bending, torsional, and rotary inertia effects are considered with respect to no-shear effect. Variations of natural frequencies with additional mass and the mass location are examined. Results are given in tabular form. It is concluded that (i) for the uniform tapered beam there is a good agreement between the results of this study and that of literature and (ii) for the symmetrical curved tapered beam there is also a good agreement between the results of this study and that of a finite element model by using MSC.Marc. Results of out-of-plane free vibration of symmetrically tapered beams for specified boundary conditions are addressed.

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Stress concentration factors for the torsion of curved beams of arbitrary cross-section

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/0954406jmes303 ◽

2006 ◽

Vol 220 (12) ◽

pp. 1709-1726 ◽

Cited By ~ 1

Author(s):

R E Cornwell

Keyword(s):

Finite Element ◽

Stress Concentration ◽

Cross Section ◽

Cross Sections ◽

Shear Stresses ◽

Curved Beams ◽

Finite Element Implementation ◽

Out Of Plane ◽

Concentration Factors ◽

Stress Concentration Factors

There are numerous situations in machine component design in which curved beams with cross-sections of arbitrary geometry are loaded in the plane of curvature, i.e. in flexure. However, there is little guidance in the technical literature concerning how the shear stresses resulting from out-of-plane loading of these same components are effected by the component's curvature. The current literature on out-of-plane loading of curved members relates almost exclusively to the circular and rectangular cross-sections used in springs. This article extends the range of applicability of stress concentration factors for curved beams with circular and rectangular cross-sections and greatly expands the types of cross-sections for which stress concentration factors are available. Wahl's stress concentration factor for circular cross-sections, usually assumed only valid for spring indices above 3.0, is shown to be applicable for spring indices as low as 1.2. The theory applicable to the torsion of curved beams and its finite-element implementation are outlined. Results developed using the finite-element implementation agree with previously available data for circular and rectangular cross-sections while providing stress concentration factors for a wider variety of cross-section geometries and spring indices.

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Stochastic Modal Models of Slender Uncertain Curved Beams Preloaded Through Clamping

Volume 1: 24th Conference on Mechanical Vibration and Noise, Parts A and B ◽

10.1115/detc2012-71169 ◽

2012 ◽

Author(s):

Javier Avalos ◽

Lanae A. Richter ◽

X. Q. Wang ◽

Raghavendra Murthy ◽

Marc P. Mignolet

Keyword(s):

Stochastic Model ◽

Stiffness Matrix ◽

Natural Frequencies ◽

Simulated Data ◽

Mode Shapes ◽

Curved Beams ◽

Shape Information ◽

Final Shape ◽

Modal Model ◽

Clamped Beams

This paper addresses the stochastic modeling of the stiffness matrix of slender uncertain curved beams that are forced fit into a clamped-clamped fixture designed for straight beams. Because of the misfit with the clamps, the final shape of the clamped-clamped beams is not straight and they are subjected to an axial preload. Both of these features are uncertain given the uncertainty on the initial, undeformed shape of the beams and affect significantly the stiffness matrix associated with small motions around the clamped-clamped configuration. A modal model using linear modes of the straight clamped-clamped beam with a randomized stiffness matrix is employed to characterize the linear dynamic behavior of the uncertain beams. This stiffness matrix is modeled using a mixed nonparametric-parametric stochastic model in which the nonparametric (maximum entropy) component is used to model the uncertainty in final shape while the preload is explicitly, parametrically included in the stiffness matrix representation. Finally, a maximum likelihood framework is proposed for the identification of the parameters associated with the uncertainty level and the mean model, or part thereof, using either natural frequencies only or natural frequencies and mode shape information of the beams around their final clamped-clamped state. To validate these concepts, a simulated, computational experiment was conducted within Nastran to produce a population of natural frequencies and mode shapes of uncertain slender curved beams after clamping. The application of the above concepts to this simulated data led to a very good to excellent matching of the probability density functions of the natural frequencies and the modal components, even though this information was not used in the identification process. These results strongly suggest the applicability of the proposed stochastic model.

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Vibrations of an Intermediately Supported U-Bend Tube

Journal of Engineering for Industry ◽

10.1115/1.3438545 ◽

1975 ◽

Vol 97 (1) ◽

pp. 23-32 ◽

Cited By ~ 8

Author(s):

L. S. S. Lee

Keyword(s):

Natural Frequency ◽

Natural Frequencies ◽

Experimental Tests ◽

Bending Stiffness ◽

Stiffness Ratio ◽

Plane Vibration ◽

Straight Beam ◽

Out Of Plane ◽

Straight Segments ◽

Good Agreement

Vibrations of an intermediately supported U-bend tube fall into two independent classes as an incomplete ring of single span does, namely, the in-plane vibration and the coupled twist-bending out-of-plane vibration. Natural frequencies may be expressed in terms of a coefficient p which depends on the stiffness ratio k, the ratio of lengths of spans, and the supporting conditions. The effect of the torsional flexibility of a curved bar acts to release the bending stiffness of a straight beam and hence decrease the natural frequency. Some conclusions for an incomplete ring of single span may not be equally well applicable to the U-tube case due to the effects of intermediate supports and the presence of the supporting straight segments. Results of the analytical predictions and the experimental tests of an intermediately supported U-tube are in good agreement.

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