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MER200: Theory of Elasticity Lecture 10

MER200: Theory of Elasticity Lecture 10. TWO DIMENSIONAL PROBLEMS Displacements Fourier Series. Example 1. Consider the thin plate subjected to the uniform load shown. Determine the displacements in the beam. Fourier Series Solutions Background. Periodic Functions

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MER200: Theory of Elasticity Lecture 10

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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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