Natural Frequencies of In-Plane Vibration of Arcs

1983 ◽  
Vol 50 (2) ◽  
pp. 449-452 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Plane Vibration

The natural frequencies of in-plane vibration are presented for uniform arcs with circular cross section under all combinations of boundary conditions.

Natural Frequencies of Out-of-Plane Vibration of Arcs

1982 ◽  
Vol 49 (4) ◽  
pp. 910-913 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Plane Vibration ◽  
Out Of Plane

The natural frequencies of out-of-plane vibration based on the Timoshenko beam theory are calculated numerically for uniform arcs of circular cross section under all combination of boundary conditions, and the results are presented in some figures.

scholarly journals Analysis of Timoshenko beam with Koch snowflake cross-section and variable properties in different boundary conditions using finite element method

2021 ◽  
Vol 13 (11) ◽  
pp. 168781402110609
Author(s):  
Hossein Talebi Rostami ◽  
Maryam Fallah Najafabadi ◽  
Davood Domiri Ganji
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Variable Properties ◽  
The Third ◽  
Koch Snowflake

This study analyzed a Timoshenko beam with Koch snowflake cross-section in different boundary conditions and for variable properties. The equation of motion was solved by the finite element method and verified by Solidworks simulation in a way that the maximum error was about 2.9% for natural frequencies. Displacement and natural frequency for each case presented and compared to other cases. Significant research achievements illustrate that if we change the Koch snowflake cross-section of the beam from the first iteration to the second, the area and moment of inertia will increase, and we have a 5.2% rise in the first natural frequency. Similarly, by changing the cross-section from the second iteration to the third, a 10.2% growth is observed. Also, the hollow cross-section is considered, which can enlarge the natural frequency by about 26.37% compared to a solid one. Moreover, all the clamped-clamped, hinged-hinged, clamped-free, and free-free boundary conditions have the highest natural frequency for the Timoshenko beam with the third iteration of the Koch snowflake cross-section in solid mode. Finally, examining important physical parameters demonstrates that variable density from a minimum value to the standard value along the beam increases the natural frequencies, while variable elastic modulus decreases it.

Calculation of the Natural Frequencies of Transverse Vibration of Complex Beams Using the Differential-Matrix Equations

2011 ◽  
Vol 243-249 ◽  
pp. 284-289
Author(s):  
Yu Zhang
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Cross Section ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Composite Beams ◽  
Matrix Equations ◽  
Generalized Differential ◽  
Set Up ◽  
Differential Matrix

The generalized differential-matrix equations of transverse vibration of the beams were set up and they were solved by means of Cauchy sequence iterative method. Then according to the boundary conditions at two ends of the beams the natural frequencies of the transverse vibration of the different beams including the complex beams of non-uniform section and composite beams under different boundary conditions were figured out. The form of the differential-matrix is simple. The calculation of the sequence iterations can be accomplished by simple computer program. Using the method in this paper, the amount of work of calculation is reduced greatly and the results are accurate compared with the approximate method in which a beam of non-uniform section is replaced by many small segments of equal cross-section.

Finite amplitude convection cells

1967 ◽  
Vol 30 (3) ◽  
pp. 577-600 ◽  
Author(s):  
J. L. Robinson
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Cross Section ◽  
Circular Cross Section ◽  
Maximum Heat ◽  
Mean Field Equations

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.

Energy Transfer From High-to Low-Frequency Modes in a Flexible Structure via Modulation

1994 ◽  
Vol 116 (2) ◽  
pp. 203-207 ◽  
Author(s):  
S. A. Nayfeh ◽  
A. H. Nayfeh
Keyword(s):  
Cross Section ◽  
High Frequency ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Low Frequency ◽  
Cantilever Beams ◽  
Axially Symmetric ◽  
Internal Resonances ◽  
Planar Motions ◽  
High Frequency Modes

An experimental study of the response of axially-symmetric (i.e., circular cross-section) cantilever beams to planar external excitations is presented. Because of the axial symmetry, one-to-one internal resonances occur at each natural frequency. These resonances cause the planar motions to lose stability and nonplanar (whirling) motions are observed. Under certain conditions, periodically-and chaotically-modulated motions may occur. In addition, when the beam is excited near one of its high natural frequencies, large first-mode responses accompanied by slow modulations of the amplitudes and phases of high-frequency modes are observed. This interaction between high-and low-frequency modes may be extremely dangerous because the amplitudes of the responses of the low-frequency modes can be very large compared with those of the directly excited high-frequency modes.

scholarly journals Elliptical Crack Identification in a Nonrotating Shaft

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
B. Muñoz-Abella ◽  
L. Rubio ◽  
P. Rubio ◽  
L. Montero
Keyword(s):  
Cross Section ◽  
Direct Problem ◽  
Natural Frequencies ◽  
Fatigue Cracks ◽  
Circular Cross Section ◽  
Crack Identification ◽  
Bernoulli Beam ◽  
Bending Vibration ◽  
Euler Bernoulli Beam ◽  
Cracked Shaft

It is known that fatigue cracks are one of the most important problems of the mechanical components, since their propagation can cause severe loss, both personal and economic. So, it is essential to know deeply the behavior of the cracked element to have tools that allow predicting the breakage before it happens. The shafts are elements that are specially affected by the described problem, because they are subjected to alternative compression and tension stresses. This work presents, firstly, an analytical expression that allows determining the first four natural frequencies of bending vibration of a nonrotating cracked shaft, assumed as an Euler–Bernoulli beam, with circular cross section under pinned-pinned conditions, taking into account the elliptical shape of the crack. Second, once the direct problem is known, the inverse problem is approached. Genetic Algorithm technique has been used to estimate the crack parameters assuming known the natural frequencies of the cracked shaft.

On the Proper Boundary Conditions for a Beam

1992 ◽  
Vol 59 (4) ◽  
pp. 915-922 ◽  
Author(s):  
H. Fan ◽  
G. E. O. Widera
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Cross Section ◽  
Boundary Data ◽  
Circular Cross Section ◽  
Numerical Calculations ◽  
Analysis Technique ◽  
Outer Expansion ◽  
Beam Theories

Employing the asymptotic expansion approach, the boundary conditions of a beam are reconsidered in the present paper. Gregory and Wan’s (1984) decay analysis technique is extended here to formulate the boundary conditions for the outer expansion. Among the various prescribed boundary data, most of the attention is focused on the displacement case because engineering beam theories employ incorrect conditions for these data. Numerical calculations are carried out for the displacement prescribed beam having a circular cross-section.

Lateral Vibrations of a Damped Laminated Hollow Circular Cross-Section Beam

1974 ◽  
Vol 96 (3) ◽  
pp. 845-852
Author(s):  
R. A. Ditaranto
Keyword(s):  
Cross Section ◽  
Viscoelastic Material ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Material Loss ◽  
Simply Supported ◽  
Elastic Beams ◽  
Mass Moment ◽  
Elastic Layers ◽  
Loss Factors

The free lateral bending vibrations of an “infinitely” long or simply-supported thin-walled circular cross-section beams having elastic-viscoelastic-elastic layers are investigated to determine the natural frequencies and associated composite loss factors. The analysis considers the inner and outer beams to behave as elastic beams in which the mass and mass-moment of inertia are both considered along with the interaction of the two elastic beams through the viscoelastic material. The results indicate that there are two natural frequencies. The lower one associated with the two elastic beams moving together so that little damping is obtained in this mode of vibration; the higher mode in which the two elastic beams vibrate in opposite directions so that there is an amount of damping comparable to the material loss factor of the viscoelastic material. A simplified model analysis is performed which is used to corroborate the trends obtained in the computer solutions of the more rigorous analysis. A series of curves are obtained for equal thickness elastic layers which can be used to obtain natural frequencies and composite loss-factors for a realistic range of geometrical and physical properties of a laminated circular cross-section simply supported beam.

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