Dispersion of Torsional Waves in Uniform Elastic Rods

1974 ◽  
Vol 41 (4) ◽  
pp. 1041-1046 ◽  
Author(s):  
O. L. Engstro¨m
Keyword(s):  
Finite Element Method ◽  
Cross Sections ◽  
Equations Of Motion ◽  
Order Theory ◽  
Warping Function ◽  
Elastic Rods ◽  
The Finite Element Method ◽  
Torsional Waves ◽  
Approximate Equations ◽  
Low Order

Approximate equations of motion are derived by use of Hamilton’s variational principle. The warping function, which is part of the solution, depends on wavelength. Numerical results on dispersion for rectangular cross sections have been obtained by the finite-element method. A comparison with the experimentally verified Barr theory is given. The paper is a contribution to the low-order approximate theories of torsional waves. It shows how good the Saint Venant warping function assumption in the low-order theory is at long relative wavelengths and it provides a modification of the theory for use at shorter wavelengths.

The Accepting of Pipe Bends With Ovality and Thinning Using Finite Element Method

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
AR. Veerappan ◽  
S. Shanmugam ◽  
S. Soundrapandian
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Internal Pressure ◽  
Cross Sections ◽  
Pressure Ratio ◽  
Empirical Relationship ◽  
The Finite Element Method ◽  
Element Method

Thinning and ovality are commonly observed irregularities in pipe bends, which induce higher stress than perfectly circular cross sections. In this work, the stresses introduced in pipe bends with different ovalities and thinning for a particular internal pressure are calculated using the finite element method. The constant allowable pressure ratio for different ovalities and thinning is presented at different bend radii. The allowable pressure ratio increases, attains a maximum, and then decreases as the values of ovality and thinning are increased. An empirical relationship to determine the allowable pressure in terms of bend ratio, pipe ratio, percent thinning, and percent ovality is presented. The pipe ratio has a strong effect on the allowable pressure.

Finite Element Models for Flexible Cosserat Solids

2020 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Minghe Shan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Displacement Vector ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Discrete Equations ◽  
Computation Efficiency ◽  
Elasticity Problems ◽  
Element Method

Abstract The application of the finite element method to the modeling of Cosserat solids is investigated in detail. In two- and three-dimensional elasticity problems, the nodal unknowns are the components of the displacement vector, which form a linear field. In contrast, when dealing with Cosserat solids, the nodal unknowns form the special Euclidean group SE(3), a nonlinear manifold. This observation has numerous implications on the implementation of the finite element method and raises numerous questions: (1) What is the most suitable representation of this nonlinear manifold? (2) How is it interpolated over one element? (3) How is the associated strain field interpolated? (4) What is the most efficient way to obtain the discrete equations of motion? All these questions are, of course intertwined. This paper shows that reliable schemes are available for the interpolation of the motion and curvature fields. The interpolated fields depend on relative nodal motions only, and hence, are both objective and tensorial. Because these schemes depend on relative nodal motions only, only local parameterization is required, thereby avoiding the occurrence of singularities. For Cosserat solids, it is preferable to perform the discretization operation first, followed by the variation operation. This approach leads to considerable computation efficiency and simplicity.

In-Plane Free Vibration of Curved Beams Using Finite Element Method

2007 ◽  
Author(s):  
F. Yang ◽  
R. Sedaghati ◽  
E. Esmailzadeh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Central Line ◽  
Circular Ring ◽  
Curved Beam ◽  
The Finite Element Method ◽  
Curved Beams ◽  
Element Method

Curved beam-type structures have many applications in engineering area. Due to the initial curvature of the central line, it is complicated to develop and solve the equations of motion by taking into account the extensibility of the curve axis and the influences of the shear deformation and the rotary inertia. In this study the finite element method is utilized to study the curved beam with arbitrary geometry. The curved beam is modeled using the Timoshenko beam theory and the circular ring model. The governing equation of motion is derived using the Extended-Hamilton principle and numerically solved by the finite element method. A parametric sensitive study for the natural frequencies has been performed and compared with those reported in the literature in order to demonstrate the accuracy of the analysis.

Penalty Formulations for the Dynamic Analysis of Elastic Mechanisms

1989 ◽  
Vol 111 (3) ◽  
pp. 321-327 ◽  
Author(s):  
E. Bayo ◽  
M. A. Serna
Keyword(s):  
Finite Element Method ◽  
Dynamic Analysis ◽  
Simulation Analysis ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Numerical Algorithms ◽  
Body Motion ◽  
The Finite Element Method ◽  
Floating Frame ◽  
Flexible Mechanisms

A series of penalty methods are presented for the dynamic analysis of flexible mechanisms. The proposed methods formulate the equations of motion with respect to a floating frame that follows the rigid body motion of the links. The constraint conditions are not appended to the Lagrange’s equations in the form of algebraic or differential constraints, but inserted in them by means of a penalty formulation, and therefore the number of equations of the system does not increase. Furthermore, the discretization of the equations using the finite element method leads to a system of ordinary differential equations that can be solved using standard numerical algorithms. The proposed methods are valid for three dimensional analysis and can be very easily implemented in existing codes. Furthermore, they can be used to model any type of constraint conditions, either holonomic or nonholonomic, and with any degree of redundancy. A series of mechanisms composed of elastic members are analyzed. The results demonstrate the capabilities of the proposed methods for simulation analysis.

Use of the Finite Element Method in the Analysis of Impact-Induced Longitudinal Waves in Constrained Elastic Systems

1995 ◽  
Vol 117 (2A) ◽  
pp. 336-342 ◽  
Author(s):  
W. H. Gau ◽  
A. A. Shabana
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Virtual Work ◽  
Fourier Method ◽  
Jump Discontinuity ◽  
Elastic Rods ◽  
The Finite Element Method ◽  
Longitudinal Waves ◽  
The Impact ◽  
Element Method

In rotating elastic rods, dispersions occurs as the result of the finite rotations. By using Fourier method, it can be shown that the impact-induced longitudinal waves no longer travel with the same phase velocities. Furthermore, the speeds of the wave propagation are independent of the impact conditions including the value of the coefficient of restitution. In this investigation the use of the finite element method in the analysis of impact-induced longitudinal waves in rotating elastic rods is examined. The equations of motion are developed using the principle of virtual work in dynamics. Jump discontinuity in the system velocity vector as result of impact is predicted using the generalized impulse momentum equations. The solution obtained using the finite element method is compared with the solution obtained using Fourier method. Numerical results show that there is a good agreement between the solution obtained by using Fourier method and the finite element solution in the analysis of wave motion. However, discrepancies between the two solutions in the analysis of the velocity waves are observed and discussed in this paper.

Planar Motion of a Flexible Beam With a Tip Mass Driven by Two Kinematic Rotational Degrees of Freedom

1998 ◽  
Vol 120 (1) ◽  
pp. 206-213
Author(s):  
D. C. Winfield ◽  
B. C. Soriano
Keyword(s):  
Finite Element Method ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Planar Motion ◽  
Flexible Beam ◽  
The Finite Element Method ◽  
Rotational Degrees Of Freedom ◽  
Tip Mass

The objective was to model planar motion of a flexible beam with a tip mass that is driven by two kinematic rotational degrees of freedom which are (1) at the center of the hub and (2) at the point the beam is attached to the hub. The equations of motion were derived using Lagrange’s equations and were solved using the finite element method. The results for the natural frequencies of the beam especially at high tip masses and high rotational velocities of the hub were calculated and compared to results obtained using the Raleigh-Ritz method. The dynamic response of the beam due to a specified hub rotation was calculated for two cases.

Finite Element Analysis of Vibration of Toroidal Field Coils Coupled With Laplace Transform

1982 ◽  
Vol 49 (3) ◽  
pp. 594-600 ◽  
Author(s):  
K. Miya ◽  
M. Uesaka ◽  
F. C. Moon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Laplace Transform ◽  
Equations Of Motion ◽  
Toroidal Field ◽  
Inverse Laplace Transform ◽  
The Finite Element Method ◽  
Element Analysis ◽  
Original Domain ◽  
Element Method

A numerical analysis of a vibration of toroidal field coils in a magnetic fusion reactor is shown here on the basis of the finite element method coupled with Laplace transform. Lagrangian consisting of kinetic, elastic strain, and magnetic energies was utilized to deduce equations of motion of the coils. The equations were solved numerically by applying the Laplace transform to a formulation with respect to time and the finite element method to one with respect to space. The Fast Fourier Transform algorithm was utilized for a calculation of the inverse Laplace transform to obtain a nodal vector of the coil’s displacement in the original domain. Numerical results reasonably explain a dependency of the coil current on a frequency of the coil.

scholarly journals OPTIMIZATION OF ELASTIC BRIDGE TRUSSES / TAMPRIŲ TILTO SANTVARŲ OPTIMIZAVIMAS

2013 ◽  
Vol 5 (5) ◽  
pp. 506-512
Author(s):  
Ignas Rimkus ◽  
Šarūnas Kisevičius ◽  
Stanislovas Kalanta
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cross Section ◽  
Cross Sections ◽  
Iterative Process ◽  
Branch And Bound Method ◽  
Rolled Steel ◽  
The Finite Element Method ◽  
Section Area ◽  
Steel Sections

The article analyzes the problems of optimizing elastic bridgetrusses, which is a tool for seeking the establishment of theminimum volume (mass) of construction and optimization of thecross-section area and height as well as the structure of the truss.It has been formulated as a nonlinear discrete mathematical programmingproblem. The upper band of the truss works not onlyfor compression but also for bending. The cross-sections of theelements are designed from rolled steel sections. Mathematicalmodels are prepared by using the finite element method and complyingwith requirements for the strength, stiffness and stabilityof the structure. The formulated problems are solved referringto an iterative process and applying the mathematical softwarepackage “MATLAB” along with routine “fmincon”. The ratio ofbuckling is corrected in every case of iteration. Requirementsfor cross-section assortment (discretion) are fulfilled employingthe branch and bound method. Santrauka Darbe nagrinėjami tamprių tilto santvarų optimizavimo uždaviniai, kuriais siekiama nustatyti minimalų konstrukcijos tūrį (masę), optimizuojant strypų skerspjūvius, santvaros aukštį bei tinklelio struktūrą. Jie formuluojami kaip netiesiniai diskrečiojo matematinio programavimo uždaviniai. Santvaros viršutinės juostos elementai ne tik gniuždomieji elementai, bet ir lenkiamieji. Strypų skerspjūviai projektuojami iš plieninių valcuotųjų profiliuočių. Uždavinių matematiniai modeliai sudaromi taikant baigtinių elementų metodą ir atsižvelgiant į konstrukcijos stiprumo, standumo bei pastovumo reikalavimus. Suformuluoti uždaviniai sprendžiamai iteraciniu būdu, naudojant matematinį kompiuterinį paketą MATLAB ir jo paprogramį fmincon. Kiekvienoje iteracijoje koreguojami gniuždomųjų elementų klupumo koeficientai. Skerspjūvių sortimento (diskretiškumo) reikalavimai užtikrinami taikant šakų ir rėžių metodą.

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