Vibration of Beams and Plate Strips With Three-Dimensional Flexibility

1989 ◽  
Vol 56 (1) ◽  
pp. 228-231 ◽  
Author(s):  
Manuel Stein
Keyword(s):  
Cross Sections ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Beam Theory ◽  
Vibration Modes ◽  
Natural Vibrations ◽  
Accurate Representation ◽  
Simply Supported

Studies making use of higher vibration modes and frequencies have indicated a need for a more accurate beam theory. Equations of motion are developed here that give a more accurate representation of the dynamic behaivor of a beam than conventional beam theory. Results are obtained using these equations for the natural vibrations of simply-supported aluminum beams of rectangular cross-sections. These results are compared to results from conventional beam theory, and they are examined to identify where various effects are important.

Dynamics of a Plate in Large Overall Motion

1989 ◽  
Vol 56 (4) ◽  
pp. 887-892 ◽  
Author(s):  
A. K. Banerjee ◽  
T. R. Kane
Keyword(s):  
Rigid Body ◽  
Reference Frame ◽  
Elastic Plate ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Present Theory ◽  
Thin Elastic Plate ◽  
Vibration Modes ◽  
Simply Supported ◽  
Rigid Body Motions

Equations of motion are formulated for a thin elastic plate that is executing small motions relative to a reference frame undergoing large rigid body motions (three-dimensional rotation and translation) in a Newtonian reference frame. As an illustrative example, a spin-up maneuver for a simply-supported rectangular plate is examined, and the vibration modes of such a plate are used to show that the present theory captures the phenomenon of dynamic stiffening.

scholarly journals Analysis of simply supported wood beams at ambient and high temperatures

2018 ◽  
Vol 7 (2.23) ◽  
pp. 180 ◽  
Author(s):  
Elza M M Fonseca ◽  
Pedro J V Gouveia
Keyword(s):  
High Temperatures ◽  
Cross Sections ◽  
Beam Theory ◽  
Sufficient Accuracy ◽  
Load Bearing Capacity ◽  
Concentrated Load ◽  
Standard Methods ◽  
Simply Supported ◽  
Wood Beams ◽  
Safe Load

The main objective of this work is to present a methodology for safety analysis of simply supported wood beams at ambient and high temperatures with a concentrated load at mid-span. Sixteen different beam configurations will be studied. All calculations were conducted according the Eurocode 5, part 1-1 and part 1-2. During this study will be analyzed the safe load bearing capacity according standards and compared with the elastic and plastic load from beam theory. The beam theory can provide sufficient accuracy up to the point of instability. The standard methods are generally conservative and they are suitable to be used for design purposes with safety. The studied beam cross sections will be in glued laminated wood, as yellow birch, with characteristics equals to a Glulam GL28H. 

scholarly journals Non uniform warping for beams

2008 ◽  
pp. 933-944
Author(s):  
Rached El Fatmi
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Beam Theory ◽  
Elastic Behavior ◽  
Cantilever Beams ◽  
Shear Forces ◽  
One Dimensional ◽  
The One ◽  
Displacement Model ◽  
Shear Bending

A non-uniform warping beam theory including the effects of torsion and shear forces is presented. Based on a displacement model using three warping parameters associated to the three St Venant warping functions corresponding to torsion and shear forces, this theory is free from the classical assumptions on the warpings or on the shears, and valid for any kind of homogeneous elastic and isotropic cross-section. This general theory is applied to analyze, for a representative set of cross-sections, the elastic behavior of cantilever beams subjected to torsion or shear-bending. Numerical results are given for the one-dimensional structural behavior and the three-dimensional stresses distributions; for the stresses in the critical region of the built-in section, comparisons with three-dimensional finite elements computations are presented. The study clearly shows when the effect of the restrained warping is localized or not.

Natural Vibrations and Stability of Elliptical Cylindrical Shells Containing Fluid

2016 ◽  
Vol 16 (10) ◽  
pp. 1550076 ◽  
Author(s):  
Sergey A. Bochkarev ◽  
Sergey V. Lekomtsev ◽  
Valery P. Matveenko
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Cylindrical Shells ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Natural Vibrations ◽  
Flowing Fluid ◽  
Thin Walled ◽  
Element Algorithm ◽  
Principle Of Virtual Displacements

The paper deals with a three-dimensional problem on natural vibrations and stability of thin-walled cylindrical shells with arbitrary cross sections, containing a quiescent or flowing ideal compressible fluid. The motion of compressible non-viscous fluid is described by a wave equation, which together with the impermeability condition and corresponding boundary conditions is transformed using the Bubnov–Galerkin method. A mathematical formulation of the problem of thin-walled structure dynamics has been developed based on the variational principle of virtual displacements. Simulation of shells with arbitrary cross sections is performed under the assumption that a curvilinear surface is approximated to sufficient accuracy by a set of plane rectangular elements. The strains are calculated using the relations of the theory of thin shells based on the Kirchhoff–Love hypothesis. The developed finite element algorithm has been employed to investigate the influence of the fluid level, the ratio of the ellipse semi-axes and types of boundary conditions on the eigenfrequencies, vibration modes and the boundary of hydroelastic stability of thin-walled circular and elliptical cylindrical shells interacting with a quiescent or flowing fluid.

scholarly journals Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Ying Hao ◽  
Wei He ◽  
Yanke Shi
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Design Principle ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Coupled System ◽  
Variable Coefficients ◽  
Curve Beam ◽  
Design Factor

The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.

Reaction Rate Constants from Classical Trajectories of Atom-Diatomic Molecule Collisions

2008 ◽  
Vol 63 (3-4) ◽  
pp. 159-169
Author(s):  
Hamzeh M. Abdel-Halim ◽  
Sawsan M. Jaafreh
Keyword(s):  
Cross Sections ◽  
Rate Constants ◽  
Potential Energy Surfaces ◽  
Diatomic Molecules ◽  
Equations Of Motion ◽  
Strong Dependence ◽  
Three Dimensional ◽  
Reaction Rates ◽  
Classical Trajectory ◽  
Reaction Rate Constants

Classical trajectory calculations for various atom-diatomic molecules were preformed using the three-dimensional Monte Carlo method. The reaction probabilities, cross-sections and rate constants of several systems were calculated. Equations of motion, which predict the positions and momenta of the colliding particles after each step, have been integrated numerically by the Runge-Kutta-Gill and Adams-Moulton methods. Morse potential energy surfaces were used to describe the interaction between the atom and each atom in the diatomic molecules. The results were compared with experimental ones and with other theoretical values. Good agreement was obtained between calculated rate constants and those obtained experimentally. Also, reasonable agreement was observed with theoretical rate constants obtained by other investigators using different calculation methods. The effects of the temperature, the nature of the colliding particles and the thermochemistry were studied. The results showed a strong dependence of the reaction rates on these factors.

scholarly journals Analysis of Excitation and Dead Vibration Modes of Quartz Resonators

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zi-Gui Huang ◽  
Zheng-Yu Chen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Resonance Frequency ◽  
Plane Strain ◽  
Excitation Frequency ◽  
Three Dimensional ◽  
Vibration Modes ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Natural Vibrations

This study uses the finite element method (FEM) to analyze the excitation and dead vibration modes of two-dimensional quartz plates. We first simplify three-dimensional quartz plates with plane strain simplification and then compare the modes of the simplified three-dimensional plates to those of two-dimensional plates. We then analyze quartz vibrating elements of AT-cut plates and SC-cut plates. To understand the regularity of the resonance frequency of plates that are excitable by voltage loading, we compare the natural vibrations of quartz plates with the excitation frequency generated after the plates are excited by voltage loading.

Analytical, Numerical, and Experimental Investigation of the Flow Structure in a Flooded Ball Bearing

1996 ◽  
Vol 118 (4) ◽  
pp. 865-871 ◽  
Author(s):  
Daniel H. Fruman ◽  
Ibtissem Benmansour ◽  
Che´rif Nouar ◽  
Thierry Bidot ◽  
Jean-Marc Vanel
Keyword(s):  
Analytical Model ◽  
Cross Sections ◽  
Large Scale ◽  
Ball Bearing ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Magnitude Estimate ◽  
Scale Model ◽  
Turbulent Regime ◽  
Dimensional Flow

Using an order of magnitude estimate of the leading terms in the equations of motion, the three-dimensional flow in a flooded ball bearing is reduced to the investigation of two-dimensional flow problems in a series of bearing cross sections. Combining, through appropriate compatibility conditions, the individual analytical solutions for the spaces confined between the cage and the inner wall of the rings, the halls and the rings and the balls and the cage’s holes, a very simple analytical model is derived. It allows the computation, in the laminar regime, of the flow rate, the pressure drop, and the velocity profile in different cross sections of the confined spaces. The results of the analytical model are confirmed by those obtained using a CFD code and extended to the turbulent regime. The analytical and numerical results are compared to those obtained from flow visualizations and velocity measurements conducted in a specially designed large scale model of a ball bearing. The agreement is very satisfactory.

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

2015 ◽  
Vol 8 (1) ◽  
pp. 82-103
Author(s):  
Palaniyandi Ponnusamy
Keyword(s):  
Classical Theory ◽  
Cross Sections ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Coupled System ◽  
Theory Of Elasticity ◽  
Inviscid Fluid ◽  
Displacement Potential ◽  
Modes Of Vibration

AbstractIn this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.

Vibration Modes and Natural Frequency Veering in Three-Dimensional, Cyclically Symmetric Centrifugal Pendulum Vibration Absorber Systems

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Chengzhi Shi ◽  
Robert G. Parker
Keyword(s):  
Degrees Of Freedom ◽  
Vibration Mode ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Vibration Absorber ◽  
Vibration Modes ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Mode Structure ◽  
Centrifugal Pendulum Vibration Absorber

This paper investigates the vibration mode structure of three-dimensional, cyclically symmetric centrifugal pendulum vibration absorber (CPVA) systems. The rotor in the system has two translational, one rotational, and two tilting degrees of freedom. The equations of motion for the three-dimensional model, including the rotor tilting, are derived to study the modes analytically and numerically. Only three mode types exist: rotational, translational-tilting, and absorber modes. The rotational and absorber modes have identical properties to those of in-plane models. Only the translational-tilting modes contain rotor tilting. The veering/crossing behavior between the eigenvalue loci is derived analytically.

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