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Fourier Series and Periodic Functions in Elasticity Theory

Explore the application of Fourier series and periodic functions in solving two-dimensional elasticity problems, with examples and explanations of displacements, orthogonal relationships, and properties of odd and even functions in engineering spaces.

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Fourier Series and Periodic Functions in Elasticity Theory

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  1. MER200: Theory of Elasticity Lecture 10 TWO DIMENSIONAL PROBLEMS Displacements Fourier Series MER200: Theory of Elasticity

  2. Example 1 • Consider the thin plate subjected to the uniform load shown. Determine the displacements in the beam. MER200: Theory of Elasticity

  3. Fourier Series SolutionsBackground • Periodic Functions • f(x) is defined for all x • f(x+p)=f(x) for all x • Examples of periodic functions • sin(x) and cos(x), period 2¶ • sin(2¶x/p) and cos(2¶x/p), period p MER200: Theory of Elasticity

  4. Periodic Functions Con’t • Periodic Functions have Many Periods • Periodic Conditions Holds for + and – Changes in Arguments MER200: Theory of Elasticity

  5. Periodic Functions in Engineering Space • Most can be represented in terms of Simple Functions • If f is periodic with period 2¶, then f can be represented in the form of an infinite series MER200: Theory of Elasticity

  6. Given a Function with Period 2¶ • What are the values of • ao • an • bn • Does this series actually represent f(x) MER200: Theory of Elasticity

  7. Orthogonal Relationships MER200: Theory of Elasticity

  8. Fourier Series with Period 2¶ MER200: Theory of Elasticity

  9. Fourier Series for an Arbitrary Period of 2a MER200: Theory of Elasticity

  10. Properties of Odd and Even Functions • Defination • Odd Function: g(-x)=-g(x) • Even Function: h(-x)=h(x) • Important Features • Integral of an odd function over a symmetric interval • Integral of an even function over a symmetric interval MER200: Theory of Elasticity

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