Closed form solutions for the dynamic response of Euler–Bernoulli beams with step changes in cross section

2006 ◽  
Vol 295 (1-2) ◽  
pp. 214-225 ◽  
Author(s):  
Michael A. Koplow ◽  
Abhijit Bhattacharyya ◽  
Brian P. Mann
Keyword(s):  
Dynamic Response ◽  
Closed Form ◽  
Cross Section ◽  
Closed Form Solutions

Elastic–plastic deformation and residual stresses in helical springs

2019 ◽  
Vol 16 (3) ◽  
pp. 448-475
Author(s):  
Vladimir Kobelev
Keyword(s):  
Residual Stresses ◽  
Closed Form ◽  
Cross Section ◽  
Circular Cross Section ◽  
Content Type ◽  
Closed Form Solutions ◽  
Helical Springs ◽  
First Case ◽  
Setting Process ◽  
The Wire

Purpose The purpose of this paper is to develop the method for the calculation of residual stress and enduring deformation of helical springs. Design/methodology/approach For helical compression or tension springs, a spring wire is twisted. In the first case, the torsion of the straight bar with the circular cross-section is investigated, and, for derivations, the StVenant’s hypothesis is presumed. Analogously, for the torsion helical springs, the wire is in the state of flexure. In the second case, the bending of the straight bar with the rectangular cross-section is studied and the method is based on Bernoulli’s hypothesis. Findings For both cases (compression/tension of torsion helical spring), the closed-form solutions are based on the hyperbolic and on the Ramberg–Osgood material laws. Research limitations/implications The method is based on the deformational formulation of plasticity theory and common kinematic hypotheses. Practical implications The advantage of the discovered closed-form solutions is their applicability for the calculation of spring length or spring twist angle loss and residual stresses on the wire after the pre-setting process without the necessity of complicated finite-element solutions. Social implications The formulas are intended for practical evaluation of necessary parameters for optimal pre-setting processes of compression and torsion helical springs. Originality/value Because of the discovery of closed-form solutions and analytical formulas for the pre-setting process, the numerical analysis is not necessary. The analytical solution facilitates the proper evaluation of the plastic flow in torsion, compression and bending springs and improves the manufacturing of industrial components.

A Conical Beam Finite Element for Rotor Dynamics Analysis

1985 ◽  
Vol 107 (4) ◽  
pp. 421-430 ◽  
Author(s):  
L. M. Greenhill ◽  
W. B. Bickford ◽  
H. D. Nelson
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Cross Section ◽  
Rotor Dynamics ◽  
General Purpose ◽  
Closed Form Expression ◽  
Dynamics Analysis ◽  
Closed Form Solutions ◽  
Analysis Computer ◽  
Rotor Dynamic

The development of finite element formulations for use in rotor dynamics analysis has been the subject of many recent publications. These works have included the effects of rotatory inertia, gyroscopic moments, axial load, internal damping, and shear deformation. However, for most closed-form solutions, the element geometry has been limited to a cylindrical cross-section. This paper extends these previous works by developing a closed-form expression including all of the above effects in a linearly tapered conical cross-section element. Results are also given comparing the formulation to previously published examples, to stepped cylinder representations of conical geometry, and to a general purpose finite element elasticity solution. The elimination of numerical integration in the generation of the element matrices, and the ability of the element to represent both conical and cylindrical geometries, make this formulation particularly suited for use in rotor dynamic analysis computer programs.

Torsion of Composite Elastic Bars of Arbitrary Cross Section

1968 ◽  
Vol 90 (3) ◽  
pp. 435-440 ◽  
Author(s):  
E. M. Sparrow ◽  
H. S. Yu
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Elastic Properties ◽  
Cross Section ◽  
Excellent Agreement ◽  
Solution Method ◽  
Circular Cross Section ◽  
High Accuracy ◽  
Arbitrary Cross Section ◽  
Closed Form Solutions

A method of analysis is presented for determining closed-form solutions for torsion of inhomogeneous prismatic bars of arbitrary cross section, the inhomogeneity stemming from the layering of materials of different elastic properties. It is demonstrated that the solution method is very easy to apply and provides results of high accuracy. As an application, solutions are obtained for the torsion of a bar of circular cross section consisting of two materials separated by a plane interface. The results are compared with those of various limiting cases and excellent agreement is found to exist. Among the limiting cases, an exact solution was derived by Green’s functions for the problem in which the interface between the materials coincides with a diameter of the circular cross section.

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.

Analytical modal analysis of thin-film flat lenses

2005 ◽  
Vol 219 (1) ◽  
pp. 55-59 ◽  
Author(s):  
H R Hamidzadeh ◽  
L Moxey
Keyword(s):  
Thin Film ◽  
Analytical Solution ◽  
Dynamic Response ◽  
Modal Analysis ◽  
Closed Form ◽  
Material Properties ◽  
Mathieu Equation ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Closed Form Solutions

The free vibrations of circular and elliptical thin-film lens are investigated. In particular, linear closed-form solutions for free vibrations of these structures were achieved and modal analysis was performed. The vibration response of the thin-film membranes were mathematically modelled using the Mathieu equation. Numerical results for various nodal diameters were computed. For the limited case, when an elliptical lens becomes circular, an excellent comparison was established with the available analytical solution. Experimental analyses were conducted to determine the effects of various parameters, such as material properties, membrane pre-strain rate, and the geometry, on natural frequency and mode shapes of these structures. The comparison verified the adequacy of linear solutions to predict the dynamic response of thin-film lenses.

Analytical Modal Analysis of Thin-Film Flat Lenses

2003 ◽  
Author(s):  
L. Moxey ◽  
H. Hamidzadeh
Keyword(s):  
Thin Film ◽  
Analytical Solution ◽  
Dynamic Response ◽  
Modal Analysis ◽  
Closed Form ◽  
Material Properties ◽  
Mathieu Equation ◽  
Free Vibrations ◽  
Vibration Response ◽  
Closed Form Solutions

Dynamics of circular and elliptical thin-film lens are investigated. In particular, linear closed-form solutions for free vibrations of these structures were achieved and modal analysis was performed. The vibration response of the thin film membranes were mathematically modeled using Mathieu equation. Numerical results for various nodal diameters were computed. For the limited case when an elliptical lens becomes circular, an excellent comparison was established with the available analytical solution. Experimental analyses were conducted to determine the effects of various parameters such as material properties, membrane pre strain and the geometry on the dynamic response of these structures. The comparison verified the adequacy of linear solutions to predict the dynamic response of thin film lenses.

On Dynamics of Bar of Rectangular Cross Section

2000 ◽  
Vol 68 (4) ◽  
pp. 662-666 ◽  
Author(s):  
N. B. Rassoulova
Keyword(s):  
Fourier Transform ◽  
Closed Form ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Original Method ◽  
The Body ◽  
Closed Form Solutions ◽  
Nonstationary Waves ◽  
Parameters Inversion ◽  
Laplace And Fourier Transform

The propagation of nonstationary waves in semi-infinite elastic rectangular bars is studied. It is assumed that two opposite lateral surfaces of the body are free of forces, while the two others are subjects to cross conditions. By introducing three new potential functions, the author succeeded in getting closed-form solutions in Laplace and Fourier transform parameters. Inversion of the transform solutions, carried out by an original method of inversion, is suggested herein.

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