1.11 Applications of Fourier Series and Integrals 123
Figure 13 Graphs of approximation using sampling: Eq. (4) withN=100 and
=4 and 10.
EXERCISES
- Use the method of Part A to find a particular solution of
d^2 u
dt^2+ 0. 4 du
dt+ 1. 04 u=r(t),wherer(t)is periodic with period 4πandr(t)= t
4 π, 0 <t< 4 π.- In the solution of Exercise 1, calculate the magnitude of the coefficients of
the Fourier series ofu(t)(periodic part). - A simply supported beam of lengthLhas a point loadwin the middle and
axial tensionT. (See Exercises in Section 0.3.) Its displacementu(x)satisfies
the boundary value problem
d^2 u
dx^2 −T
EIu=w
EIh(x),^0 <x<L,
u( 0 )= 0 , u(L)= 0 ,whereh(x)is the “triangle function”h(x)={
2 x/L, 0 <x<L/2,
2 (L−x)/L, L/ 2 <x<L.Use the method of Part B to findu(x)as a sine series.- The inhomogeneity in the differential equation in Exercise 3 has a dis-
continuous derivative. Find another way to solve the differential equation.
Hint: Bothu(x)andu′(x)must be continuous for 0<x<L.