18.4. Fourier Series

Try these first

1. 1.

   Define a periodic function.

   Hint

   Think about how periodic functions relate to time.

2. 2.

   What is a Fourier series?

   Hint

   Consider the 'Fourier' in context of periodic signals.

3. 3.

   What is the primary use of Fourier Series in engineering applications?

   • To approximate functions
   • To analyze periodic functions
   • To solve ODEs only

   Hint

   Think about the periodic nature of many engineering problems.

4. 4.

   True or False: Fourier Series can only represent functions that are continuous and differentiable.

   • True
   • False

   Hint

   Review the nature of periodic functions.

5. 5.

   Given the function f(x) = x^2 on the interval [-L, L], find its Fourier series representation.

   Hint

   Focus on the integrals and observe the symmetry of the function on the interval.

6. 6.

   Consider a beam with fixed ends; how would you formulate the Fourier sine series for its vibration mode shape?

   Hint

   Remember boundary conditions drive the selection of the series.