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Related Concept Videos

Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

546
The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
The M/EI...
546
Singularity Functions for Bending Moment01:18

Singularity Functions for Bending Moment

698
Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a...
698
Shear on the Horizontal Face of a Beam Element01:16

Shear on the Horizontal Face of a Beam Element

668
To understand shear on the flat side of a prismatic beam element, consider the vertical and horizontal shearing forces, and the normal forces, acting on the element. The element's upper (U) and lower (L) sections, which are divided by the beam's neutral axis, are examined. The equilibrium of these forces is determined by applying the equilibrium equation, which helps identify the horizontal shearing force. This force is directly related to the bending moments and the cross-section's...
668
Beams with Unsymmetric Loadings01:17

Beams with Unsymmetric Loadings

531
Analyzing a supported beam under unsymmetrical loadings is essential in structural engineering to understand how beams respond to varied force distributions. This analysis involves calculating the deflection and identifying points where the slope of the beam is zero, which are crucial for ensuring structural stability and functionality.
The first moment-area theorem determines the slope at any point on the beam. This theorem indicates that the change in slope between two points on a beam...
531
Deflection of a Beam01:19

Deflection of a Beam

962
Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
Singularity functions, described in an earlier lesson, are powerful mathematical tools that represent discontinuities within a function commonly encountered in structural loading...
962
Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

641
In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...
641

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The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Generalized Bessel beams with two indices.

Marco Ornigotti, Andrea Aiello

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    |November 1, 2014
    PubMed
    Summary
    This summary is machine-generated.

    Researchers engineered Bessel beams to create novel solutions for the scalar Helmholtz equation. These generalized beams exhibit unique propagation characteristics and radial structures similar to Laguerre-Gaussian beams.

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    Area of Science:

    • * Physics
    • * Optics
    • * Mathematical Physics

    Background:

    • * The scalar Helmholtz equation is fundamental in wave propagation phenomena.
    • * Bessel beams are known for their non-diffracting properties.
    • * Laguerre-Gaussian beams possess a specific radial structure relevant in optical applications.

    Purpose of the Study:

    • * To derive new exact solutions for the scalar Helmholtz equation.
    • * To explore generalized Bessel beams with engineered angular spectra.
    • * To investigate the propagation characteristics of these novel optical beams.

    Main Methods:

    • * Engineering the angular spectrum of a Bessel beam.
    • * Analyzing solutions within the paraxial approximation.
    • * Comparing the radial structure to Laguerre-Gaussian beams.

    Main Results:

    • * A new class of exact solutions to the scalar Helmholtz equation was obtained.
    • * Generalized Bessel beams with specific radial structures were identified.
    • * Peculiar propagation properties of these novel beams were revealed.

    Conclusions:

    • * The study presents novel exact solutions for wave propagation problems.
    • * Engineered Bessel beams offer new possibilities in optical beam shaping.
    • * These findings contribute to the understanding of generalized beam propagation.