scholarly journals Vibrations of the Euler–Bernoulli Beam Under a Moving Force based on Various Versions of Gradient Nonlocal Elasticity Theory: Application in Nanomechanics

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Śniady Paweł ◽  
Katarzyna Misiurek ◽  
Olga Szyłko-Bigus ◽  
Idzikowski Rafał
Keyword(s):  
Constant Velocity ◽  
Stress Gradient ◽  
Nonlocal Elasticity Theory ◽  
Theory Of Elasticity ◽  
Gradient Theory ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Moving Force ◽  
Theory Application ◽  
Euler Bernoulli Beam

AbstractTwo models of vibrations of the Euler–Bernoulli beam under a moving force, based on two different versions of the nonlocal gradient theory of elasticity, namely, the Eringen model, in which the strain is a function of stress gradient, and the nonlocal model, in which the stress is a function of strains gradient, were studied and compared. A dynamic response of a finite, simply supported beam under a moving force was evaluated. The force is moving along the beam with a constant velocity. Particular solutions in the form of an infinite series and some solutions in a closed form as well as the numerical results were 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.

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.

Study on Criterion of Vibration Stability for Tether of Submerged Floating Tunnel

2012 ◽  
Vol 446-449 ◽  
pp. 2071-2074
Author(s):  
Xiao Jun Zhou ◽  
Yin Gang Qin ◽  
Gang Luo
Keyword(s):  
Lyapunov Function ◽  
Galerkin Method ◽  
Parametric Excitation ◽  
Stability Criterion ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Vibration Stability ◽  
Submerged Floating Tunnel ◽  
Set Up ◽  
Euler Bernoulli Beam

The submerged floating tunnel moored by tethers is simplified as a simply supported Euler-Bernoulli beam, and its vibration caused by vehicles passing through the tubular is also reduced as parametric excitation, moreover, the vibration governing equation of tether is still propounded by means of Galerkin method, simultaneously, the dynamic stability criterion of tether is also set up by means of Lyapunov function in this paper.

Mechanical Characters of Six Engineering Timoshenko Beams

2014 ◽  
Vol 668-669 ◽  
pp. 201-204
Author(s):  
Hong Liang Tian
Keyword(s):  
Timoshenko Beam ◽  
Analytic Solutions ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Normality Assumption ◽  
Length Direction ◽  
The Right ◽  
Euler Bernoulli Beam ◽  
Mechanical Characters

Timoshenko beam is an extension of Euler-Bernoulli beam to interpret the transverse shear impact. The more refined Timoshenko beam relaxes the normality assumption of plane section that remains plane and normal to the deformed centerline. The manuscript presents some exact concise analytic solutions on deflection and stress resultants of NET single-span Timoshenko beam with general distributed force and 6 kinds of standard boundary conditions, adopting its counterpart Euler-Bernoulli beam solutions. Engineering example shows that scale impact would not unveil itself for micro structure with micrometer μm-order length, yet will be prominent for nanostructure with nanometer nm-order length. When simply supported CNTs is undergone to a concentrative force at the median and complete bend moment, scale action is observed along the ensemble CNTs, while it unfurls itself the most at the position of the concentrated strength. When a clamped-free CNTs is exposed to a centralized force at the mesial and distributed force, there is no scale impact about the deflection at all positions on the left border of the concentrated strength position, while such operation inspires at once at all positions on the right margin of the concentrated strength position. When a clamped-clamped CNTs is lain under a concentrative strength at the middle, the deflection of NET Euler-Bernoulli CNTs reflects scale effect completely. Notable differences between the deflection of Euler-Bernoulli CNTs and that of Timoshenko CNTs are reflected at large ratio of diameter versus length. The deflection of NET clamped-free and simply supported Timoshenko beam doesn’t introduce surplus scale process in terms of its counterpart, NET Euler-Bernoulli beam. However, the deflection of NET clamped-clamped Timoshenko beam does involve additional scale impact solely including the method when the concentrated strength position is at the midway in the beam-length direction.

Dynamics of Eccentric Shafts Under Linearly Varying Tension Rotating in a Viscous Medium

1974 ◽  
Vol 96 (4) ◽  
pp. 1285-1290
Author(s):  
V. Prodonoff ◽  
C. D. Michalopoulos
Keyword(s):  
Steady State ◽  
Beam Theory ◽  
Viscous Medium ◽  
Bernoulli Beam ◽  
Axial Coordinate ◽  
Simply Supported ◽  
Deterministic Function ◽  
Mass Eccentricity ◽  
Bending Stresses ◽  
Euler Bernoulli Beam

Using Euler-Bernoulli beam theory an investigation is made of the dynamic behavior of an eccentric vertical circular shaft rotating in viscous medium. The shaft is subjected to linearly-varying tension and has distributed mass and elasticity. The mass eccentricity is assumed to be a deterministic function of the axial coordinate. The solution is obtained by modal analysis. An example is considered wherein the shaft is simply supported at the top and vertically guided at the bottom. Steady-state deflections and bending stresses are computed for a particular eccentricity function over a range of speeds of rotation which includes a resonant frequency.

scholarly journals Composite patch reinforcement of a cracked simply-supported beam traversed by moving mass

2020 ◽  
Vol 14 (1) ◽  
pp. 6403-6415
Author(s):  
M. S. Aldlemy ◽  
S. A. K. Al-jumaili ◽  
R. A. M. Al-Mamoori ◽  
T. Ya ◽  
Reza Alebrahim
Keyword(s):  
Finite Element Method ◽  
Edge Crack ◽  
Time History ◽  
Beam Theory ◽  
Beam Deflection ◽  
Moving Mass ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Composite Patch ◽  
Euler Bernoulli Beam

In this study dynamic analysis of a metallic beam under travelling mass was investigated. A beam with an edge crack was considered to be reinforced using composite patch. Euler-Bernoulli beam theory was applied to simulate the time-history behavior of the beam under dynamic loading. Crack in the beam was modeled using a rotational spring. Dimension of the composite patch, crack length, stress intensity factor at crack tip and beam deflection are some parameters which were studied in details. Results were validated against those which were found through Finite Element Method.

Transverse Vibration of a Two-Span Beam Under Action of a Moving Constant Force

1950 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
R. S. Ayre ◽  
George Ford ◽  
L. S. Jacobsen
Keyword(s):  
Mechanical Model ◽  
Constant Velocity ◽  
Natural Mode ◽  
Theoretical Studies ◽  
Constant Force ◽  
Bending Stress ◽  
Transient Vibration ◽  
Simply Supported ◽  
Moving Force ◽  
Theory And Experiment

Abstract The problem relates to the transient vibration of a symmetrical, continuous, simply supported two-span beam which is traversed by a constant force moving with constant velocity. The beam is of slender proportions, flexure alone being considered. Damping is zero, and there is no mass associated with the moving force. Exact theoretical solutions for bending stress have been derived in general form. They consist of three infinite series, each related to one of three time eras as follows: (a) Where force is crossing first span; (b) is crossing second span; (c) has left the beam. Each term of a series is related to a natural mode of vibration. Quantitative theoretical studies show the variation in individual terms of the series, and also in summations of the first five terms, as the traversing velocity is varied. A mechanical model with electrical recording of stress was employed to obtain a more complete quantitative solution than was feasible analytically. The agreement between theory and experiment was reasonably good. Large magnifications of stress (of the order of 2.5) were found in the neighborhood of resonance with the fundamental mode.

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.

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