A Note on the Use of Approximate Solutions for the Bending Vibrations of Simply Supported Cracked Beams

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
L. Rubio ◽  
J. Fernández-Sáez
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Direct Method ◽  
Approximate Solutions ◽  
Cracked Beam ◽  
Bending Vibrations ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
A Direct Method ◽  
Cracked Beams

The main goal of this note is to discuss the applicability of approximate closed-form solutions to evaluate the natural frequencies for bending vibrations of simply supported Euler–Bernoulli cracked beams. From the well-known model, which considers the cracked beam as two beams connected by a rotational spring, different approximate solutions are revisited and compared with those found by a direct method, which has been chosen as reference.

Apparently First Closed-Form Solution for Frequencies of Deterministically and/or Stochastically Inhomogeneous Simply Supported Beams

2000 ◽  
Vol 68 (2) ◽  
pp. 176-185 ◽  
Author(s):  
S. Candan ◽  
I. Elishakoff
Keyword(s):  
Closed Form ◽  
Infinite Number ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Benchmark Solutions

An infinite number of closed-form solutions is reported for a deterministically or stochastically nonhomogeneous beam, for both natural frequencies and reliabilities, for specialized cases. These solutions may prove useful as benchmark solutions. Numerical examples are evaluated.

scholarly journals New Conditions for Obtaining the Exact Solutions of the General Riccati Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lazhar Bougoffa
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Riccati Equation ◽  
Direct Method ◽  
Equivalent Equation ◽  
A Direct Method

We propose a direct method for solving the general Riccati equationy′=f(x)+g(x)y+h(x)y2. We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functionsf(x),g(x), andh(x)of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method.

Relaxation of Thick-Walled Cylinders and Spheres

1982 ◽  
Vol 49 (3) ◽  
pp. 487-491 ◽  
Author(s):  
N. S. Ottosen
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Approximate Solutions ◽  
Nonlinear Creep ◽  
Closed Form Solutions ◽  
Cylinders And Spheres

Using the nonlinear creep law proposed by Soderberg, closed-form solutions are derived for the relaxation of incompressible thick-walled spheres and cylinders in plane strain. These solutions involve series expressions which, however, converge very quickly. By simply ignoring these series expressions, extremely simple approximate solutions are obtained. Despite their simplicity these approximations possess an accuracy that is superior to approximations currently in use. Finally, several physical aspects related to the relaxation of cylinders and spheres are discussed.

A Direct Method for Analysis of Branched Torsional Systems

1974 ◽  
Vol 96 (3) ◽  
pp. 1006-1009 ◽  
Author(s):  
N. Shaikh
Keyword(s):  
Natural Frequencies ◽  
Direct Method ◽  
Torsional Vibrations ◽  
Transfer Matrices ◽  
Usual Procedure ◽  
A Direct Method ◽  
Number Of Branches

A general and direct method for the analysis of branched systems is presented, in which transfer matrices are used in Holzer-type solutions. The method is shown here for torsional vibrations, but should be applicable to other branched systems. As presented here, this method is much different than the usual procedure found in the literature. Unlike other methods, no matrix inversions (or equivalent operations) are required to account for branches at a junction. A single determinant giving natural frequencies is arrived at irrespective of the number of branches and junctions. Thus the method is straightforward, compact, and economical for computer solutions.

Natural Frequencies and Critical Buckling Loads of Marine Risers

1988 ◽  
Vol 110 (1) ◽  
pp. 2-8 ◽  
Author(s):  
Y. C. Kim
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Simple Procedure ◽  
Variable Cross Section ◽  
Mode Shapes ◽  
Variable Cross ◽  
Bending Rigidity ◽  
Buckling Loads ◽  
Closed Form Solutions ◽  
Marine Risers

Natural frequencies, mode shapes and critical buckling loads of marine risers simply supported at both ends are given in closed form by using the WKB method. These solutions allow variable cross section, bending rigidity, tension and mass distribution along the riser length. Furthermore, a simple procedure to predict natural frequencies for other boundary conditions is described. Some special forms of these closed-form solutions are compared with existing solutions in the literature.

Free Vibration of the Triple-Walled Carbon Nanotubes

2009 ◽  
Author(s):  
Demetris Pentaras ◽  
Isaac Elishakoff
Keyword(s):  
Carbon Nanotubes ◽  
Natural Frequencies ◽  
Polynomial Equation ◽  
Galerkin Methods ◽  
Frequency Spectra ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Cubic Polynomial ◽  
Walled Carbon Nanotubes ◽  
Frequency Squares

Natural frequencies of the triple-walled carbon nanotubes (TWCNTs) are determined both exactly and approximately. For the case of TWCNT which is simply supported at it ends closed-form solutions are obtained. It is shown that there are three series of natural frequencies corresponding to the cubic polynomial equation for natural frequency squares. For the TWCNT that has other boundary conditions approximate Bubnov-Galerkin and Petrov-Galerkin methods are applied. Simple polynomial coordinate functions are utilized. Each of these methods yields three natural frequencies corresponding to the lower and of each frequency spectra.

Dynamic Response of a Timoshenko Beam to a Moving Force

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
Paweł Śniady
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Constant Velocity ◽  
Free Vibrations ◽  
Dynamical Response ◽  
Transverse Displacement ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Moving Force

We consider the dynamical response of a finite, simply supported Timoshenko beam loaded by a force moving with a constant velocity. The classical solution for the transverse displacement and the rotation of the cross section of a Timoshenko beam has a form of a sum of two infinite series, one of which represents the force vibrations (aperiodic vibrations) and the other one free vibrations of the beam. We show that one of the series, which represents aperiodic (force) vibrations of the beam, can be presented in a closed form. The closed form solutions take different forms depending if the velocity of the moving force is smaller or larger than the velocities of certain shear and bar velocities.

An Extended Separation-of-Variable Method for Free Vibration of Rectangular Mindlin Plates

2021 ◽  
pp. 2150154
Author(s):  
Gen Li ◽  
Yufeng Xing ◽  
Zekun Wang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Closed Form ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Arbitrary Boundary ◽  
Unknown Variable ◽  
Closed Form Solutions ◽  
Solution Methods ◽  
Separation Of Variable

For rectangular thick plates with non-Levy boundary conditions, it is important to explore analytical free vibration solutions because the classical inverse and semi-inverse exact solution methods are not applicable to this category of problems. This work is to develop an extended separation-of-variable (SOV) method to find closed-form analytical solutions for the free vibration of rectangular Mindlin plates with arbitrary homogeneous boundary conditions. In the extended SOV method, characteristic differential equations and boundary conditions in two directions are obtained by employing the Rayleigh principle and the assumption that the mode functions are in the SOV form, and two transcendental eigenvalue equations are achieved through boundary conditions. But these two eigenvalue equations cannot be solved simultaneously since there are two equations and only the natural frequency is the unknown variable. Considering this, the second assumption in this method is that the natural frequencies corresponding to two-direction mode functions are independent of each other in the mathematical sense, thus there are two unknowns in two transcendental eigenvalue equations, and the closed-form solutions for plates with arbitrary boundary conditions can be obtained non-iteratively. From the physical sense, the natural frequencies pertaining to different direction mode functions should be the same, and this conclusion is validated analytically and numerically. The present natural frequencies and mode shapes agree well with those obtained by other analytical and numerical methods. Especially, for the plates with at least two opposite sides simply supported, the present solutions are exact.

On the Prediction of the Strain Rate Intensity Factor in Metal Forming Processes

2008 ◽  
Author(s):  
Elena Lyamina
Keyword(s):  
Intensity Factor ◽  
Strain Rate ◽  
Closed Form ◽  
Upper Bound ◽  
Approximate Solutions ◽  
Closed Form Solutions ◽  
Perfectly Plastic ◽  
Strain Rate Intensity Factor ◽  
Rate Intensity ◽  
Strain Rate Intensity

The strain rate intensity factor is the coefficient of the principal singular term in a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. Such singular behaviour occurs in the case of several rigid plastic models (rigid perfectly plastic solids, the double-shearing model, the double slip and rotation model, some of viscoplastic models). Since it is only possible to introduce the strain rate intensity factor for singular velocity fields, it is obvious that standard finite element codes cannot be used to calculate it. The currently available distributions of the strain rate intensity factor have been found from closed form solutions or with the use of simple approximate solutions (for instance upper bound solutions). Closed form solutions are available for boundary value problems with simple geometry (flow through infinite rough channels, compression of infinite layers between rough plates and so on) and, therefore, are mostly of academic interest. Simple approximate solutions can predict general tendencies in the distribution of the strain rate intensity factor but cannot predict its distribution with a sufficient accuracy for industrial applications. For, the strain rate intensity factor reflects a very local effect inherent in the velocity field whereas simple approximate methods, such as the upper bound method, estimate global parameters, such as the limit load. The purpose of the present research is to propose a special numerical technique for calculating the strain rate intensity factor in the case of plane strain deformation of rigid perfectly plastic materials and to verify it by means of comparison with an analytical solution. The technique is based on the method of Riemann.

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