On the Resolution of Existing Discontinuities in the Dynamic Responses of an Euler-Bernoulli Beam Subjected to the Moving Mass

2006 ◽  
Author(s):  
M. H. Kargarnovin ◽  
K. Saeedi
Keyword(s):  
Contact Point ◽  
Bending Moment ◽  
Dynamic Responses ◽  
Distributed Parameter ◽  
Moving Mass ◽  
Bernoulli Beam ◽  
Concentrated Mass ◽  
Uniform Cross Section ◽  
Support Conditions ◽  
Euler Bernoulli Beam

The dynamic response of a one-dimensional distributed parameter system subjected to a moving mass with constant speed is investigated. An Euler-Bernoulli beam with the uniform cross-section and finite length with specified boundary support conditions is assumed. In this paper, rather a new method based on the time dependent series expansion for calculating the bending moment and the shear force due to motion of the mass is suggested. Governing differential equations of the motion are derived and solved. The accuracy of the numerical results primarily is verified and further the rapid convergence of this new technique was illustrated over other existing methods. Finally, it is shown that a considerable improvement is obtained in capturing the incurred discontinuities at the contact point of traveling concentrated mass.

Dynamic Stability and Response of Inclined Beams Under Moving Mass and Follower Force

2020 ◽  
pp. 2043004
Author(s):  
D. S. Yang ◽  
C. M. Wang ◽  
J. D. Yau
Keyword(s):  
Dynamic Stability ◽  
Moving Load ◽  
Dynamic Responses ◽  
Follower Force ◽  
Load Force ◽  
Moving Mass ◽  
Concentrated Mass ◽  
Spatial Part ◽  
Method Of Solution ◽  
Fourier Sine Series

This paper is concerned with the dynamic stability and response of an inclined Euler–Bernoulli beam under a moving mass and a moving follower force. The extended Hamilton’s principle is used to derive the governing equation of motion and the boundary conditions for this general moving load/force problem. Considering a simply supported beam, one can solve the problem analytically by approximating the spatial part of the deflection with a Fourier sine series. Based on the formulation and method of solution, sample dynamic responses are determined for a beam that is inclined at 30[Formula: see text] with respect to the horizontal. It is shown that the dynamic response of the beam under a moving mass is rather different from an equivalent moving follower force. Also investigated herein are the dynamic stability of inclined beams under moving load/follower force which are described by four key variables, viz. the speed of the moving mass/follower force, concentrated mass to the beam distributed mass, vibration frequency and the magnitude of the moving mass/follower force. The critical axial load and the critical follower force are different when they are located at different positions in the beam; except for the special case when they are at the end of the beam.

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.

scholarly journals Macaulay's Method for a Timoshenko Beam

2007 ◽  
Vol 35 (4) ◽  
pp. 285-292 ◽  
Author(s):  
N. G. Stephen
Keyword(s):  
Timoshenko Beam ◽  
Bending Moment ◽  
Shearing Force ◽  
Bernoulli Beam ◽  
Deflection Curve ◽  
Axial Coordinate ◽  
Deflection Analysis ◽  
Euler Bernoulli Beam

The Macaulay bracket notation is familiar to many engineers for the deflection analysis of a Euler–Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

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.

Nonlinear Dynamics of High-Dimensional Models of a Rotating Euler-Bernoulli Beam Under the Gravity Load

2014 ◽  
Author(s):  
J. L. Huang ◽  
W. D. Zhu
Keyword(s):  
Slope Angle ◽  
Period Doubling ◽  
Dynamic Responses ◽  
Phase Portraits ◽  
High Dimensional ◽  
Bernoulli Beam ◽  
Gravity Load ◽  
Dimensional Models ◽  
Euler Bernoulli Beam ◽  
Ihb Method

Nonlinear dynamic responses of an Euler-Bernoulli beam attached to a rotating rigid hub with a constant angular velocity under the gravity load are investigated. The slope angle of the centroid line of the beam is used to describe its motion, and the nonlinear integro-partial differential equation that governs the motion of the rotating hub-beam system is derived using Hamilton’s principle. Spatially discretized governing equations are derived using Lagrange’s equations based on discretized expressions of kinetic and potential energies of the system, yielding a set of second-order nonlinear ordinary differential equations with combined parametric and forced harmonic excitations due to the gravity load. The incremental harmonic balance (IHB) method is used to solve for periodic responses of high-dimensional models of the system and period-doubling bifurcation. The multivariable Floquet theory along with the precise Hsu’s method is used to investigate the stability of the periodic responses. Phase portraits and bifurcation points obtained from the IHB method agree very well with those from numerical integration.

On the Curvature of an Euler–Bernoulli Beam

2003 ◽  
Vol 31 (2) ◽  
pp. 132-142 ◽  
Author(s):  
Osman Kopmaz ◽  
Ömer Gündoğdu
Keyword(s):  
Nonlinear Theory ◽  
Quantitative Assessment ◽  
Bending Moment ◽  
Point Of View ◽  
Bernoulli Beam ◽  
Cantilevered Beam ◽  
Single Moment ◽  
The Difference ◽  
Euler Bernoulli Beam ◽  
The Relationship

This paper deals with different approaches to describing the relationship between the bending moment and curvature of a Euler—Bernoulli beam undergoing a large deformation, from a tutorial point of view. First, the concepts of the mathematical and physical curvature are presented in detail. Then, in the case of a cantilevered beam subjected to a single moment at its free end, the difference between the linear theory and the nonlinear theory based on both the mathematical curvature and the physical curvature is shown. It is emphasized that a careless use of the nonlinear mathematical curvature and moment relationship given in most standard textbooks may lead to erroneous results. Furthermore, a numerical example is given for the reader to make a quantitative assessment.

scholarly journals Dynamic Response of a Beam Subjected to Moving Load and Moving Mass Supported by Pasternak Foundation

2012 ◽  
Vol 19 (2) ◽  
pp. 205-220 ◽  
Author(s):  
Rajib Ul Alam Uzzal ◽  
Rama B. Bhat ◽  
Waiz Ahmed
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamic Response ◽  
Moving Load ◽  
Bending Moment ◽  
Dynamic Responses ◽  
Moving Mass ◽  
Pasternak Foundation ◽  
Partial Differential ◽  
Foundation Parameters

This paper presents the dynamic response of an Euler- Bernoulli beam supported on two-parameter Pasternak foundation subjected to moving load as well as moving mass. Modal analysis along with Fourier transform technique is employed to find the analytical solution of the governing partial differential equation. Shape functions are assumed to convert the partial differential equation into a series of ordinary differential equations. The dynamic responses of the beam in terms of normalized deflection and bending moment have been investigated for different velocity ratios under moving load and moving mass conditions. The effect of moving load velocity on dynamic deflection and bending moment responses of the beam have been investigated. The effect of foundation parameters such as, stiffness and shear modulus on dynamic deflection and bending moment responses have also been investigated for both moving load and moving mass at constant speeds. Numerical results obtained from the study are presented and discussed.

Sliding mode boundary control for vibration suppression in a pinned-pinned Euler–Bernoulli beam with disturbances

2016 ◽  
Vol 24 (6) ◽  
pp. 1109-1122 ◽  
Author(s):  
Dimitri Karagiannis ◽  
Verica Radisavljevic–Gajic
Keyword(s):  
Sliding Mode Control ◽  
Boundary Conditions ◽  
Sliding Mode ◽  
Bending Moment ◽  
Control Technique ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Mode Control ◽  
Control Functions ◽  
Euler Bernoulli Beam

This work addresses the control of a pinned-pinned beam represented by the fourth order partial differential equation commonly known as the Euler–Bernoulli beam model. The system under consideration has pinned boundary conditions on one end (displacement and bending moment fixed at zero) and controlled boundary conditions on the other end (displacement and bending moment are prescribed by control functions). There are also unknown bounded disturbances included on the controlled boundary. A backstepping control technique which introduces arbitrary damping into the system is discussed, and a method for applying this control in the presence of unknown disturbances is developed using sliding mode control theory. Sliding mode controllers are developed in a way that does not create a chattering effect, which is a common issue with sliding mode control. Simulation results are presented to show how the system dampens out vibrations at an arbitrarily determined rate and how the control functions respond to unmodeled disturbances.

Dynamic Responses of Two Beams Connected by a Spring-Mass Device

2012 ◽  
Vol 29 (1) ◽  
pp. 143-155 ◽  
Author(s):  
H.- P. Lin ◽  
D. Yang
Keyword(s):  
Natural Frequencies ◽  
Beam Theory ◽  
Free Vibrations ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Entire System ◽  
Bernoulli Beam ◽  
Continuous Beams ◽  
Experimental Validations ◽  
Euler Bernoulli Beam

AbstractThis paper deals with the transverse free vibrations of a system in which two beams are coupled with a spring-mass device. The dynamics of this system are coupled through the motion of the mass. The entire system is modeled as two two-span beams and each span of the continuous beams is assumed to obey the Euler-Bernoulli beam theory. Considering the compatibility requirements across each spring con-nection position, the eigensolutions (natural frequencies and mode shapes) of this system can be obtained for different boundary conditions. Some numerical results and experimental validations are presented to demonstrate the method proposed in this article.

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