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Related Concept Videos

Deflection of a Beam01:19

Deflection of a Beam

383
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...
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Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

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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...
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Beams with Unsymmetric Loadings01:17

Beams with Unsymmetric Loadings

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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...
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Shear on the Horizontal Face of a Beam Element01:16

Shear on the Horizontal Face of a Beam Element

293
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...
293
Singularity Functions for Bending Moment01:18

Singularity Functions for Bending Moment

279
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...
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Elastic Curve from the Load Distribution01:16

Elastic Curve from the Load Distribution

262
The structural behavior of beams under distributed loads is critical for engineering analysis, which focuses on predicting how beams bend and react under such conditions. Different types of beams (e.g., cantilever, supported, or overhanging) behave differently under distributed load conditions.
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Updated: Sep 17, 2025

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Published on: August 12, 2013

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Direct generation of helical Mathieu-Gauss beams using phase-only profile.

Zheng Xu1, Yubo Shi1, Xuemei Ren1

  • 1Institute of Acoustics, Tongji University, Shanghai 200092, China.

The Journal of the Acoustical Society of America
|July 3, 2025
PubMed
Summary
This summary is machine-generated.

Researchers developed a simplified method to create diffraction-free helical Mathieu-Gauss (HMG) beams. This technique simplifies calculations and maintains beam quality for acoustic applications.

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Area of Science:

  • Acoustic physics
  • Beam optics

Background:

  • Diffraction-free beams offer advantages in particle manipulation, imaging, and communication.
  • Helical Mathieu-Gauss (HMG) beams are a type of diffraction-free beam with unique properties.

Purpose of the Study:

  • To demonstrate a simplified phase-only method for generating HMG beams.
  • To provide a computationally efficient alternative for producing structured acoustic beams.

Main Methods:

  • Employed a single-step phase modulation during diffraction.
  • Developed an approximate analytical expression for phase distribution, avoiding complex Mathieu function calculations.
  • Utilized numerical simulations to validate beam properties.

Main Results:

  • Successfully generated HMG beams with propagation invariance.
  • Simulated beams closely matched theoretical HMG beam characteristics.
  • The simplified method showed no compromise in beam quality compared to existing techniques.

Conclusions:

  • The proposed simplified phase-only method is a practical approach for generating HMG beams.
  • This method offers mathematical simplification without sacrificing beam quality for acoustic applications.
  • The technique is suitable for applications requiring structured acoustic beam fields.