PROPAGATION AND RESONANCE OF LONGITUDINAL WAVES IN PRISMATIC RODS

1936 ◽  
Vol 14a (2) ◽  
pp. 48-55
Author(s):  
R. Ruedy
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Equations Of Motion ◽  
Poisson's Ratio ◽  
Longitudinal Waves ◽  
Fundamental Frequencies ◽  
Cubic Equation ◽  
Resonance Frequencies ◽  
Fourth Series ◽  
Shearing Motion

For vibrations involving shearing and rotation, and for those involving both distortion and dilatation, the equations of motion combined with the boundary conditions yield in the simplest case a cubic equation for the resonance frequencies; its solution depends on Poisson's ratio and on the resonance frequencies fx, fy, fz, which the rod possesses when in pure shearing motion in the direction of its three axes. Three series of resonance frequencies are obtained when fy and fz are constant and the frequencies of the overtones are inserted for fx. A fourth series of resonance frequencies begins above the highest of the fundamental frequencies fx, fy, fz.

PROPAGATION AND RESONANCE OF COMPOSITE WAVES IN PRISMATIC RODS

1936 ◽  
Vol 14a (3) ◽  
pp. 66-70
Author(s):  
R. Ruedy
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Experimental Results ◽  
Cubic Equation ◽  
Resonance Frequencies ◽  
Good Agreement

By taking into account the three main terms of the equation of motion of the prismatic rod, there is obtained for the frequency a cubic equation which is in good agreement with the experimental results when the thickness of the rod is not negligible compared with its length but does not exceed about one-fifth of the length. It corresponds to the equation obtained for a system with three degrees of freedom.For a composite vibration consisting of a wave of dilatation and a wave of distortion in the direction of the smallest dimension of the rod, and waves of dilatation in the two other directions, the equations of motion combined with some of the boundary conditions yield another cubic equation for the resonance frequencies.

scholarly journals Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods

Vibration ◽  
2019 ◽  
Vol 2 (1) ◽  
pp. 157-173 ◽  
Author(s):  
Guadalupe Leon ◽  
Hung-Liang Chen
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Elastic Constants ◽  
Frequency Ratio ◽  
Poisson’S Ratio ◽  
Direct Determination ◽  
Poisson's Ratio ◽  
Torsional Mode ◽  
Bending Mode ◽  
Resonance Frequencies

In this paper, the exact solution of the Timoshenko circular beam vibration frequency equation under free-free boundary conditions was determined with an accurate shear shape factor. The exact solution was compared with a 3-D finite element calculation using the ABAQUS program, and the difference between the exact solution and the 3-D finite element method (FEM) was within 0.15% for both the transverse and torsional modes. Furthermore, relationships between the resonance frequencies and Poisson’s ratio were proposed that can directly determine the elastic constants. The frequency ratio between the 1st bending mode and the 1st torsional mode, or the frequency ratio between the 1st bending mode and the 2nd bending mode for any rod with a length-to-diameter ratio, L/D ≥ 2 can be directly estimated. The proposed equations were used to verify the elastic constants of a steel rod with less than 0.36% error percentage. The transverse and torsional frequencies of concrete, aluminum, and steel rods were tested. Results show that using the equations proposed in this study, the Young’s modulus and Poisson’s ratio of a rod can be determined from the measured frequency ratio quickly and efficiently.

Evaluation of Effective Elastic Mechanical Properties of Graphene Sheets

2012 ◽  
Author(s):  
Khalid I. Alzebdeh
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Graphene Sheet ◽  
Poisson’S Ratio ◽  
Elastic Moduli ◽  
Young's Modulus ◽  
Unit Cell ◽  
Poisson's Ratio ◽  
Graphene Sheets ◽  
Numerical Predictions

The mechanical behaviour of a single-layer nanostructured graphene sheet is investigated using an atomistic-based continuum model. This is achieved by equating the stored energy in a representative unit cell for a graphene sheet at atomistic scale to the strain energy of an equivalent continuum medium under prescribed boundary conditions. Proper displacement-controlled (essential) boundary conditions which generate a uniform strain field in the unit cell model are applied to calculate one elastic modulus at a time. Three atomistic finite element models are adopted with an assumption that force interactions among carbon atoms can be modeled by either spring-like or beam elements. Thus, elastic moduli for graphene structure are determined based on the proposed modeling approach. Then, effective Young’s modulus and Poisson’s ratio are extracted from the set of calculated elastic moduli. Results of Young’s modulus obtained by employing the different atomistic models show a good agreement with the published theoretical and numerical predictions. However, Poisson’s ratio exhibits sensitivity to the considered atomistic model. This observation is supported by a significant variation in estimates as can be found in the literature. Furthermore, isotropic behaviour of in-plane graphene sheets was validated based on current modeling.

scholarly journals Nonlinear analysis of buckling and postbuckling for axially compressed functionally graded cylindrical panels with the Poisson's ratio varying smoothly along the thickness

2012 ◽  
Vol 34 (1) ◽  
pp. 27-44 ◽  
Author(s):  
Dao Van Dung ◽  
Le Kha Hoa
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Compressive Load ◽  
Cylindrical Panel ◽  
Functionally Graded ◽  
Poisson's Ratio ◽  
Postbuckling Behavior ◽  
Nonlinear Buckling ◽  
Dimensional Parameter ◽  
Classical Shell Theory

In this paper an approximate analytical solution to analyze the nonlinear buckling and postbuckling behavior of imperfect functionally graded panels with the Poisson's ratio also varying smoothly along the thickness is investigated. Based on the classical shell theory and von Karman's assumption of kinematic nonlinearity and applying Galerkin procedure, the equations for finding critical loads and load-deflection curves of cylindrical panel subjected to axial compressive load with two types boundary conditions, are given. Especially, the stiffness coefficients are analyzed in explicit form. Numerical results show various effects of the inhomogeneous parameter, dimensional parameter, boundary conditions on nonlinear stability of panel. An accuracy of present theoretical results is verified by the previous well-known results.

scholarly journals Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension

1952 ◽  
Vol 19 (4) ◽  
pp. 526-528
Author(s):  
M. L. Williams
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Poisson's Ratio ◽  
Vertex Angle ◽  
Stress Singularities ◽  
Stress Concentrations ◽  
Various Boundary Conditions ◽  
Free Case

Abstract As an analog to the bending case published in an earlier paper, the stress singularities in plates subjected to extension in their plane are discussed. Three sets of boundary conditions on the radial edges are investigated: free-free, clamped-clamped, and clamped-free. Providing the vertex angle is less than 180 degrees, it is found that unbounded stresses occur at the vertex only in the case of the mixed boundary condition with the strength of the singularity being somewhat stronger than for the similar bending case. For vertex angles between 180 and 360 degrees, all the cases considered may have stress singularities. In amplification of some work of Southwell, it is shown that there are certain analogies between the characteristic equations governing the stresses in extension and bending, respectively, if ν, Poisson’s ratio, is replaced by −ν. Finally, the free-free extensional plate behaves locally at the origin exactly the same as a clamped-clamped plate in bending, independent of Poisson’s ratio. In conclusion, it is noted that the free-free case analysis may be applied to stress concentrations in V-shaped notches.

Classical Solution and Edge Effect in the Problem of Stability of an Axially Compressed Cylindrical Shell

2020 ◽  
Vol 22 (1) ◽  
pp. 48-54
Author(s):  
Guyk A. Manuylov ◽  
Sergey B. Kosytsyn ◽  
Maksim M. Begichev
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Critical Load ◽  
Classical Solution ◽  
Edge Effect ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
Critical Stresses ◽  
Membrane Deformations ◽  
Initial Arrangement

The classical solution for critical stresses in the problem of stability of a circular longitudinally compressed cylindrical shell consists of two terms, reflecting the ability of the shell to resist buckling due to bending and membrane deformations. However, with usual boundary conditions the classical solution appears only with the absence of the Poisson expansion of a shell. With a non-zero Poisson's ratio, an axisymmetric edge effect presents. It reduces the critical load and causes the initial arrangement of its own forms to change as the load increases.

Fundamental Frequencies of U-Tubes in Tube Bundles

1985 ◽  
Vol 107 (2) ◽  
pp. 207-209
Author(s):  
P. M. Moretti
Keyword(s):  
Poisson’S Ratio ◽  
Estimation Procedure ◽  
Moment Of Inertia ◽  
Poisson's Ratio ◽  
Dimensionless Number ◽  
Bend Radius ◽  
Cross Sectional ◽  
Strong Function ◽  
Fundamental Frequencies ◽  
Axial Moment

The natural frequencies of U-tubes on multiple supports have been studied as a complement to the author’s previous work on the vibration of straight heat-exchanger tubes (reference [1]). A rapid estimation procedure for fundamental frequencies of tubes on symmetrical support spacings has been developed by expressing the frequency in the form f1≥12π•a1•1Ls2•EIμ where the square root contains the tube properties of Young’s modulus, cross-sectional second moment, and linear density; Ls is a characteristic length of support spacing; and a1 is a dimensionless number which is a strong function of the support geometry (as expressed by the ratio of bend radius to span lengths) and weak function of Poisson’s ratio and of tube axial moment of inertia. a1 has been plotted as a function of the ratio of the bend radius to the straight-span length, for usual values of Poisson’s ratio and small axial moment of inertia. The underlying assumptions for the use of such plots are examined and the theoretical basis for the statement of a lower bound is given, in order to show where the use of this method is applicable.

The Center of Shear Again

1943 ◽  
Vol 10 (2) ◽  
pp. A62-A64
Author(s):  
W. R. Osgood
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
Unique Point ◽  
Usual Definition ◽  
Different Boundary Conditions ◽  
Definition Of

Abstract There is disagreement in the literature as to the location of the center of shear. Timoshenko, for example, states that the position of this point depends upon Poisson’s ratio, whereas Trefftz says that it does not. Both Timoshenko’s and Trefftz’ solutions are compatible with the usual definition of the center of shear. The disagreement may be attributed to the assumptions of different boundary conditions. In this paper the boundary conditions are examined, and a definition of the center of shear is proposed that leads to a unique point for any cross section.

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