Transverse Vibrations of Beams, Exact Versus Approximate Solutions

1981 ◽  
Vol 48 (4) ◽  
pp. 923-928 ◽  
Author(s):  
J. R. Hutchinson
Keyword(s):  
Exact Solution ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Approximate Solutions ◽  
Approximate Methods ◽  
Transverse Vibrations ◽  
Elasticity Equations ◽  
Beam Approximation ◽  
Free Beam

An exact solution for the natural frequencies of vibration of a finite length free-free beam with a circular cross section is found and compared to approximate solutions. This exact solution is a series solution of the general linear elasticity equations which converges to correct natural frequencies. Correctness of the frequencies is established by comparison to previous experiments. Comparison of the exact to approximate solutions is made with the Pochhammer-Chree approximation, the Timoshenko beam approximation and the Pickett approximation. The comparisons clearly show the range of applicability of the approximate methods as well as their accuracy. The correct shear coefficient for use in the Timoshenko beam approximation is investigated and conclusions which differ with, yet at the same time complement, those of previous researchers are reached.

Vibrations of Thick Free Circular Plates, Exact Versus Approximate Solutions

1984 ◽  
Vol 51 (3) ◽  
pp. 581-585 ◽  
Author(s):  
J. R. Hutchinson
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Approximate Solutions ◽  
Mindlin Plate ◽  
Circular Plates ◽  
Shear Coefficient ◽  
Elasticity Equations ◽  
Mindlin Plate Theory ◽  
Plate Solution

An exact solution for the natural frequencies of a thick free circular plate is compared to approximate solutions. The exact solution is a series solution of the general linear elasticity equations that converges to the correct natural frequencies. The approximate solutions to which this exact solution is compared are the Mindlin plate theory and a modification of a solution method proposed by Pickett. 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 Mindlin plate theory is considered by evaluating the shear coefficient that would make the exact and modified Pickett method match the Mindlin plate solution.

Equilibrium Temperatures in a Boundary-Layer Flow Over a Flat Plate of Absorbing-Emitting Gas

1968 ◽  
Vol 90 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Yehuda Taitel ◽  
J. P. Hartnett
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Flat Plate ◽  
Interaction Mechanism ◽  
Approximate Solutions ◽  
Equilibrium Temperature ◽  
Constant Heat Flux ◽  
Adiabatic Wall ◽  
Approximate Methods ◽  
Layer Flow

The effect of radiation on the equilibrium temperature for a flow of emitting-absorbing gas over a flat plate is studied. Three methods of solution are formulated: An approximate solution for a thin boundary layer, a similarity solution for the limiting case when the boundary layer is optically thick, and an exact solution. Emphasis is put on the study of the recovery or adiabatic wall case, where conduction to the wall is balanced by the net radiation away from the wall. Results are reported for the limiting cases of a black plate and completely reflective plate and for a unit Prandtl number. The exact solution reflects very favorably on the use of the approximate methods and points out clearly the conditions for which the approximate solutions are applicable. Results are also reported for the equilibrium wall temperature for the case of constant heat flux and for the recovery factor in the case of blowing and suction; both for optically thin boundary layers. Special attention is put on the interaction mechanism and the role of the emitting-absorbing coefficient on this process. It is shown that, for small absorption coefficient, high wall emissivity, and Mach number, the results approach the case where the gas is transparent.

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.

Vibration Analysis of Functionally Graded Timoshenko Beams

2018 ◽  
Vol 18 (01) ◽  
pp. 1850007 ◽  
Author(s):  
Wei-Ren Chen ◽  
Heng Chang
Keyword(s):  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Longitudinal Vibration ◽  
Numerical Data ◽  
Functionally Graded ◽  
Approximate Methods ◽  
Vibration Behavior ◽  
Fg Beam

The vibration behavior of a functionally graded Timoshenko beam is investigated by applying the transformed-section method. The material properties of a functionally graded (FG) beam are assumed to vary across the thickness according to a simple power law. The cross section of FG beam with two constituents is first transformed into an equivalent cross section of the material on the top. Then, the lateral and longitudinal vibration equations of a homogeneous Timoshenko beam are separately applied to the beam with the transformed section. The bending natural frequencies of FG beam are evaluated using the Chebyshev collocation method, and the longitudinal natural frequencies are also obtained from the known closed-form solutions. Some of the analytical results are compared with the existing numerical data to validate the present model accuracy. Good agreement has been observed between the analytical and numerical data. The effects of aspect ratio, volume fraction, and boundary conditions on the free-vibration behavior of FG beam are discussed. The present analytical solutions provide an insight to the effects of various parameters on the vibration behavior of the beam. They also serve as benchmarks for testing the vibration results obtained by other analytical or approximate methods.

Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System

1993 ◽  
Vol 115 (2) ◽  
pp. 202-209 ◽  
Author(s):  
Ming Une Jen ◽  
E. B. Magrab
Keyword(s):  
Exact Solution ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Attached Mass ◽  
Approximate Methods ◽  
Mass System ◽  
The Laplace Transform ◽  
Rigid Mass ◽  
Two Degree Of Freedom

An exact solution for the natural frequencies and mode shapes for a beam elastically constrained at its ends and to which a rigid mass is elastically mounted is obtained. The attached mass can both translate and rotate. The general solution is obtained using the Laplace transform with respect to the spatial variable and yields the exact solutions to several previously published simpler configurations that were obtained using approximate methods. Numerous numerical results are presented for the natural frequency coefficients that extend previously reported results and that show the transition between various limiting cases. In addition, values are presented for the lowest two natural frequency coefficients for a beam that is clamped at both ends and is carrying a two dof spring-mass system. Representative mode shapes at selected values of the system’s parameters are also given.

Explicit Modal Analysis of an Axially Loaded Timoshenko Beam With Bending-Torsion Coupling

1999 ◽  
Vol 67 (2) ◽  
pp. 307-313 ◽  
Author(s):  
J. R. Banerjee
Keyword(s):  
Finite Element ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Computationally Efficient ◽  
Approximate Methods ◽  
Symbolic Computing ◽  
End Conditions ◽  
Axially Loaded ◽  
Analytical Expressions

Exact analytical expressions for the natural frequencies and mode shapes of a uniform bending-torsion coupled Timoshenko beam are presented. The beam is taken to be axially loaded, and for which cantilever end conditions apply. A symbolic computing package (REDUCE) has been used, which simplifies the analysis greatly. Results are presented for an illustrative example, which confirms the accuracy of the method, and provides a convenient benchmark for the validation of the finite element or other alternative approximate methods. The approach offers the prospect of aeroelastic development, and is computationally efficient, thus holding out the promise of eventual optimization. [S0021-8936(00)01002-3]

Solution of heat and mass transfer problems with non-linear coefficients

2019 ◽  
Vol 5 (4) ◽  
pp. 10-20
Author(s):  
Boris G. Aksenov ◽  
Yuri E. Karyakin ◽  
Svetlana V. Karyakina
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Numerical Methods ◽  
Heat And Mass Transfer ◽  
Unknown Function ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Maximum Deviation ◽  
Approximate Methods ◽  
Nonmonotonic Dependence

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

Radiant Heat Transfer From Isothermal Dispersions With Isotropic Scattering

1967 ◽  
Vol 89 (4) ◽  
pp. 300-308 ◽  
Author(s):  
R. H. Edwards ◽  
R. P. Bobco
Keyword(s):  
Heat Transfer ◽  
Approximate Solutions ◽  
Radiant Heat ◽  
Second Order ◽  
Isotropic Scattering ◽  
Radiant Heat Transfer ◽  
Approximate Methods ◽  
First Order ◽  
Order Solution ◽  
Radiant Transfer

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

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