scholarly journals An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework

2017 ◽  
Vol 24 (2) ◽  
pp. 361-376 ◽  
Author(s):  
Cagan Diyaroglu ◽  
Erkan Oterkus ◽  
Selda Oterkus
Keyword(s):  
Three Dimensional ◽  
Equation Of Motion ◽  
Computational Time ◽  
Benchmark Problems ◽  
Geometrical Shape ◽  
Bernoulli Beam ◽  
Lagrange Equations ◽  
The World ◽  
Euler Bernoulli Beam ◽  
Model Structures

Every object in the world has a three-dimensional geometrical shape and it is usually possible to model structures in a three-dimensional fashion, although this approach can be computationally expensive. In order to reduce computational time, the three-dimensional geometry can be simplified as a beam, plate or shell type of structure depending on the geometry and loading. This simplification should also be accurately reflected in the formulation that is used for the analysis. In this study, such an approach is presented by developing an Euler–Bernoulli beam formulation within ordinary state-based peridynamic framework. The equation of motion is obtained by utilizing Euler–Lagrange equations. The accuracy of the formulation is validated by considering various benchmark problems subjected to different loading and displacement/rotation boundary conditions.

Dynamic Responses of a Cracked Gas-Bearing Spindle

2004 ◽  
Author(s):  
Bo-Wun Huang ◽  
Jao-Hwa Kuang
Keyword(s):  
Rotation Speed ◽  
Crack Depth ◽  
Applied Pressure ◽  
Equation Of Motion ◽  
Dynamic Responses ◽  
Bernoulli Beam ◽  
Spindle System ◽  
Depth Rotation ◽  
Euler Bernoulli Beam ◽  
Gas Bearing

The dynamic response of a cracked gas-bearing spindle system is studied in this work. A round Euler-Bernoulli beam is used to approximate the spindle system. The stiffness effect of the gas bearing spindle is considered as massless springs and the Hamilton principle is employed to derive the equation of motion for the spindle system. The effects of crack depth, rotation speed and air applied pressure on the dynamic characteristics of a rotating gas-bearing spindle system are studied.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

Active vibration control of a nonlinear three-dimensional Euler–Bernoulli beam

2016 ◽  
Vol 23 (19) ◽  
pp. 3196-3215 ◽  
Author(s):  
Wei He ◽  
Chuan Yang ◽  
Juxing Zhu ◽  
Jin-Kun Liu ◽  
Xiuyu He
Keyword(s):  
Differential Equations ◽  
Boundary Control ◽  
Coupling Effect ◽  
Three Dimensional ◽  
Uniform Boundedness ◽  
Transverse Displacement ◽  
Axial Deformation ◽  
Bernoulli Beam ◽  
Parameter Uncertainties ◽  
Euler Bernoulli Beam

In this paper, boundary control is designed to suppress the vibration of a nonlinear three-dimensional Euler–Bernoulli beam. Considering the coupling effect between the axial deformation and the transverse displacement, the dynamics of the beam are modeled as a distributed parameter system described by three partial differential equations (PDEs) and 12 ordinary differential equations (ODEs). Firstly, model-based boundary control is designed based on a mathematical model of the system. Subsequently, adaptive control is proposed when there are parameter uncertainties in the model. The uniform boundedness and uniform ultimate boundedness are proved under the proposed control laws. Finally, numerical simulations illustrate the effectiveness of the results.

Damping of Vibrations of Layered Elastic-Viscoelastic Beams

1993 ◽  
Vol 46 (11S) ◽  
pp. S305-S311 ◽  
Author(s):  
Richard B. Hetnarski ◽  
Ray A. West ◽  
Joseph S. Torok
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Bernoulli Beam ◽  
Viscoelastic Effects ◽  
Viscoelastic Layers ◽  
Euler Bernoulli Beam ◽  
Damping Of Vibrations ◽  
Fourth Order Differential Equation

A five-layer cantilever beam consisting of an elastic core, two symmetric viscoelastic layers, and two elastic constraining layers is considered. The viscoelastic effects are incorporated in the Euler-Bernoulli beam theory. If the contraction and extension of the constraining layers is neglecterd a fourth order differential equation of motion is received. Inclusion of contraction and extension of the constraining layers results in a more accurate sixth order differential equation. Appropriate boundary conditions are derived. Laplace transforms are used extensively. Both the analytical solution and the numerical results are presented.

scholarly journals The Vibration of a Beam under a Traversing Load

2021 ◽  
Author(s):  
Kan-Chen Jane Wu
Keyword(s):  
Boundary Conditions ◽  
Constant Velocity ◽  
Equation Of Motion ◽  
Linear Strain ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Runge Kutta Method ◽  
Extended Hamilton’S Principle ◽  
Gyroscopic Forces ◽  
Euler Bernoulli Beam

The objective of this study is to investigate the response of an Euler-Bernoulli beam under a force or mass traversing with constant velocity. Simply-supported and clamped-clamped boundary conditions are considered. The linear strain-displacement scenario is applied to both boundary conditions, while the von Kármán nonlinear scenario is applied only to the former boundary condition. The governing equation of motion is derived via the extended Hamilton's principle. Simulations are performed with the fourth-order Runge-Kutta method via Matlab software. The equation of motion is first validated and then used to investigate the effects of the beam second moment of area, the magnitude of the traversing velocity, and centrifugal and gyroscopic forces.

Noether symmetries and exact solutions of an Euler–Bernoulli beam model

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640011 ◽  
Author(s):  
Aeeman Fatima ◽  
Fazal M. Mahomed ◽  
Chaudry Masood Khalique
Keyword(s):  
Exact Solutions ◽  
Order Equation ◽  
Beam Equation ◽  
Beam Model ◽  
Noether Symmetries ◽  
Bernoulli Beam ◽  
Lagrange Equations ◽  
New Exact Solutions ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

In this paper, a Noether symmetry analysis is carried out for an Euler–Bernoulli beam equation via the standard Lagrangian of its reduced scalar second-order equation which arises from the standard Lagrangian of the fourth-order beam equation via its Noether integrals. The Noether symmetries corresponding to the reduced equation is shown to be the inherited Noether symmetries of the standard Lagrangian of the beam equation. The corresponding Noether integrals of the reduced Euler–Lagrange equations are deduced which remarkably allows for three families of new exact solutions of the static beam equation. These are shown to contain all the previous solutions obtained from the standard Lie analysis and more.

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