Mathematics (Civil Engineering -1) | 20. Rectangular Membrane, Use of Double Fourier Series by Abraham | Learn Smarter
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20. Rectangular Membrane, Use of Double Fourier Series

The chapter delves into the behavior of rectangular membranes under various conditions, focusing on the mathematical modeling using the two-dimensional wave equation and double Fourier series. It emphasizes the formulation and solution of vibration problems, particularly under the constraints of fixed boundaries. Key insights include the method of separation of variables and the determination of vibration modes, highlighting practical applications in civil engineering.

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Sections

  • 20

    Rectangular Membrane, Use Of Double Fourier Series

    This section introduces the mathematical modeling of vibrating rectangular membranes using double Fourier series to analyze their behavior under various conditions.

    Learning Practice

  • 20.1

    The Two-Dimensional Wave Equation

    This section discusses the two-dimensional wave equation governing the transverse vibration of a rectangular membrane, focusing on its mathematical formulation and boundary conditions.

    Learning Practice

  • 20.2

    Solution By Separation Of Variables

    This section discusses the method of separation of variables to solve the two-dimensional wave equation governing the behavior of rectangular membranes.

    Learning Practice

  • 20.3

    Solving The Spatial Equations

    This section covers solving spatial equations in the context of vibrating rectangular membranes using double Fourier series.

    Learning Practice

  • 20.4

    Solving The Time Equation

    The section discusses how to solve the time-dependent part of the two-dimensional wave equation for a rectangular membrane using harmonic functions.

    Learning Practice

  • 20.5

    General Solution

    The general solution for the vibration of a rectangular membrane is presented as a double Fourier series expansion, integrating the contributions from various modes of vibration.

    Learning Practice

  • 20.6

    Determining Coefficients A And B

    This section discusses how to determine the coefficients A and B in the double Fourier series solution of the vibrating rectangular membrane using initial conditions.

    Learning Practice

  • 20.7

    Modes Of Vibration

    This section covers the distinct modes of vibration for rectangular membranes, highlighting their significance in civil engineering applications.

    Learning Practice

  • 20.8

    Applications In Civil Engineering

    This section discusses the application of double Fourier series methods in analyzing vibrations in civil engineering structures.

    Learning Practice

  • 20.9

    Orthogonality Of Sine Functions

    The orthogonality of sine functions is a crucial property for deriving Fourier coefficients in the context of rectangular membranes.

    Learning Practice

  • 20.10

    Eigenvalues And Eigenfunctions

    This section introduces eigenvalues and eigenfunctions in the context of solving boundary value problems using separation of variables for vibrating membranes.

    Learning Practice

  • 20.11

    Nodal Lines And Mode Shapes

    Learning Practice

  • 20.12

    Forced Vibrations And Damping (Overview)

    This section addresses the phenomena of forced vibrations and damping in membranes, highlighting their significance in real-world applications such as civil engineering.

    Learning Practice

  • 20.13

    Computational Considerations

    This section outlines the computational aspects of analyzing vibrations in rectangular membranes, emphasizing the use of truncated double Fourier series for practical engineering applications.

    Learning Practice

  • 20.14

    Practical Problems In Civil Engineering Using Double Fourier Series

    This section discusses real-life applications of the double Fourier series in civil engineering, highlighting various scenarios where this mathematical approach is utilized.

    Learning Practice

References

Chapter_20_Recta.pdf

Class Notes

Memorization

What we have learnt

  • Rectangular membranes oscil...
  • The solution involves separ...
  • Distinct modes of vibration...

Final Test

Revision Tests