Vibrations and Waves in Elastic Bars of Rectangular Cross Section

1960 ◽  
Vol 27 (1) ◽  
pp. 152-158 ◽  
Author(s):  
R. D. Mindlin ◽  
E. A. Fox
Keyword(s):  
Exact Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Modes Of Vibration

An exact solution of the equations of elasticity is found for a family of modes of vibration, or waves, in an infinite bar of rectangular cross section for certain ratios of width to depth. The solution is composed of coupled dilatational and equivoluminal waves and the four faces of the bar are free of traction.

Stress Analysis of a Helical Coil

1963 ◽  
Vol 30 (1) ◽  
pp. 122-126 ◽  
Author(s):  
Irving Stein
Keyword(s):  
Exact Solution ◽  
Stress Analysis ◽  
Cross Section ◽  
Small Angle ◽  
Rectangular Cross Section ◽  
Helical Coil ◽  
Torsional Stress ◽  
Stress Problem

The solution for the torsional-stress problem in a small-angle helical coil of rectangular cross section is derived. The exact solution is then approximated for validity at small coil indices. This work complements the solution of Gohner,2 which is valid at larger coil indices.

Torsion of Curved Beams of Rectangular Cross Section

1952 ◽  
Vol 19 (1) ◽  
pp. 49-53
Author(s):  
H. L. Langhaar
Keyword(s):  
Exact Solution ◽  
Cross Section ◽  
Approximation Method ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Curved Beams ◽  
Previously Treated

Abstract Recently, W. Freiberger obtained an exact solution of the problem of uniform torsion of a segment of a ring of circular cross section. This paper presents a solution of the problem for the rectangular cross section. O. Göhner previously treated this case by an approximation method.

Modal Curves of Pre-Twisted Beams of Rectangular Cross-Section

1969 ◽  
Vol 11 (1) ◽  
pp. 1-13 ◽  
Author(s):  
B. Dawson ◽  
W. Carnegie
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Ritz Method ◽  
Transformation Method ◽  
Mode Shapes ◽  
Modes Of Vibration ◽  
Vibrational Characteristics ◽  
Theoretical Results ◽  
Good Agreement

An important aspect of the theoretical study of the vibrational characteristics of turbine and compressor blading is the prediction of the modal curves from which the stresses along the length of the blading can be determined. The accurate prediction of the modal curves allowing for such factors as pre-twist, camber, size of cross-section, centrifugal tensile effects, aerodynamic effects, etc., is still not possible. However, a better understanding of the effects of some of these parameters can be obtained by a study of the modal curves of relatively simple idealized models. In this work the theoretical mode shapes of vibration of pre-twisted rectangular cross-section beams for various width to depth ratios and pre-twist angle in the range 0-90° are examined. The theoretical results are obtained by the transformation method given by Carnegie, Dawson and Thomas (1)† and the accuracy of these results is verified by comparison with results obtained by Dawson (2) using the Ritz method. The theoretical results are compared to modal curves determined experimentally and good agreement is shown between them. A physical explanation of the effects of the pre-twist angle upon the modal curves is given for the first three modes of vibration.

scholarly journals Estimation of Reduced Stiffness under the Influence of Crack in a Beam

2019 ◽  
Vol 9 (1) ◽  
pp. 711-714
Keyword(s):  
Natural Frequency ◽  
Cantilever Beam ◽  
Cross Section ◽  
Loading Condition ◽  
Dynamic Behaviour ◽  
Rectangular Cross Section ◽  
Fatigue Loading ◽  
Stiffness Reduction ◽  
Modes Of Vibration ◽  
The Relationship

Machine components with cantilever boundary conditions are most prominently used in mechanical engineering applications. When such components are subjected to fatigue loading, crack may get initiated and failure may occur. In order to prevent catastrophic failure and safe guard these components one has to condition monitor the dynamic behavior under fatigue loading condition. Vibration based condition monitoring is one of the most effective method to assess the fatigue failure of the component. In this paper, a cantilever beam is analyzed for its dynamic behaviour under the influence of crack. The cantilever beam considered for the analysis is of rectangular cross section and is uniform throughout its length. The characteristic equation was derived for the Euler-Bernoulli cantilever beam to obtain the relationship between the location of the crack and stiffness of the beam because the stiffness of the beam influences the natural frequency. It is found that the stiffness of the cantilever beam is varying for varying locations of the crack in different modes of vibration. It is clearly understood from the analysis that the vibration response of the cracked cantilever beam in its modes of vibration is affected by the corresponding stiffness reduction based upon the location of the crack. So, it can be inferred that the natural frequency of the cracked cantilever beam may have different values for different locations of the crack.

On the Transverse Vibration of Beams of Rectangular Cross-Section

1986 ◽  
Vol 53 (1) ◽  
pp. 39-44 ◽  
Author(s):  
J. R. Hutchinson ◽  
S. D. Zillmer
Keyword(s):  
Exact Solution ◽  
Cross Section ◽  
Plane Stress ◽  
Timoshenko Beam ◽  
Transverse Vibration ◽  
Rectangular Cross Section ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Shear Coefficient ◽  
Stress Solution

An exact solution for the natural frequencies of transverse vibration of free beams with rectangular cross-section is used as a basis of comparison for the Timoshenko beam theory and a plane stress approximation which is developed herein. The comparisons clearly show the range of applicability of the approximate solutions as well as their accuracy. The choice of a best shear coefficient for use in the Timoshenko beam theory is considered by evaluation of the shear coefficient that would make the Timoshenko beam theory match the exact solution and the plane stress solution. The plane stress solution is shown to provide excellent accuracy within its range of applicability.

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