MA 30300: Differential Equations

9.4. Fourier Series Method (Exercises)

Goal: Use Fourier series to solve mass-spring systems with external forces and analyze resonance.

1. Mass-Spring Systems

Find the steady periodic solution \( x_{sp}(t) \) of each differential equation. Plot enough Fourier terms to visualize the solution.

Problem 1

\[ x'' + 5x = F(t) \]

where \(F(t)\) is periodic with period \(2\pi\) and

\[ F(t) = \begin{cases} 3, & 0 < t < \pi, \\ -3, & \pi < t < 2\pi. \end{cases} \]
Graph of the steady periodic solution.
Graph of the steady periodic solution. Check out this desmos graph.

Problem 2

\[ x'' + 10x = F(t) \]

where \(F(t)\) is an even function with period \(4\) and

\[ F(t) = \begin{cases} 3, & 0 < t < 1, \\ -3, & 1 < t < 2. \end{cases} \]

Problem 3

\[ x'' + 3x = F(t) \]

where \(F(t)\) is an odd function with period \(2\pi\) and

\[ F(t) = 2t, \quad 0 < t < \pi. \]

Problem 4

\[ x'' + 4x = F(t) \]

where \(F(t)\) is an even function with period \(4\) and

\[ F(t) = 2t, \quad 0 < t < 2. \]

Problem 5

\[ x'' + 10x = F(t) \]

where \(F(t)\) is an odd function with period \(2\) and

\[ F(t) = t - t^2, \quad 0 < t < 1. \]

Problem 6

\[ x'' + 2x = F(t) \]

where \(F(t)\) is an even function with period \(2\pi\) and

\[ F(t) = \sin t, \quad 0 < t < \pi. \]

2. Pure Resonance

In each of the problems 1-6, the mass \(m\) and Hooke’s constant \(k\) for a mass–spring system are given. Determine whether or not pure resonance will occur under the influence of the given external periodic force \(F(t)\).

Problem 1

\[ m x'' + kx = F(t), \quad m=1,\; k=9 \]

where \(F(t)\) is an odd function with period \(2\pi\) and

\[ F(t) = 1, \quad 0 < t < \pi. \]

Problem 2

\[ m x'' + kx = F(t), \quad m=2,\; k=10 \]

where \(F(t)\) is an odd function with period \(2\) and

\[ F(t) = 1, \quad 0 < t < 1. \]

Problem 3

\[ m x'' + kx = F(t), \quad m=3,\; k=12 \]

where \(F(t)\) is an odd function with period \(2\pi\) and

\[ F(t) = 3, \quad 0 < t < \pi. \]

Problem 4

\[ m x'' + kx = F(t), \quad m=1,\; k=4\pi^2 \]

where \(F(t)\) is an odd function with period \(2\) and

\[ F(t) = 2t, \quad 0 < t < 1. \]

Problem 5

\[ m x'' + kx = F(t), \quad m=3,\; k=48 \]

where \(F(t)\) is an even function with period \(2\pi\) and

\[ F(t) = t, \quad 0 < t < \pi. \]

Problem 6

\[ m x'' + kx = F(t), \quad m=2,\; k=50 \]

where \(F(t)\) is an odd function with period \(2\pi\) and

\[ F(t) = \pi t - t^2, \quad 0 < t < \pi. \]

Problem 7

A particular mass-spring system is modeled by:

\[ x'' + 10x = F(t) \]

where \( F(t) \) is an odd periodic force with period 4, whose Fourier series representation is:

\[ F(t) = \frac{20}{\pi} \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin\frac{n \pi t}{2} \]

Tasks:

  1. Find a steady-periodic solution \( x(t) \).
  2. Check whether pure resonance occurs.
  3. If no pure resonance occurs, identify the dominant term in the response.

3: Damped Mass-Spring Systems (\(c > 0\))

\[ mx'' + cx' + kx = F(t) \]

Steady Periodic Solution \( x_{sp}(t) \)

Denote
\[ \omega_n = \frac{n\pi}{L} \]

1. Even Force

\[ F(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(\omega_n t) \]
\[ x_{sp}(t) = \frac{a_0}{2k} + \sum_{n=1}^{\infty} C_n \cos(\omega_n t - \alpha_n) \]
where
\[ C_n = \dfrac{a_n}{\sqrt{(k - m\omega_n^2)^2 + (c\omega_n)^2}}. \]

2. Odd Force

\[ F(t) = \sum b_n \sin(\omega_n t) \]
\[ x_{sp}(t) = \sum_{n=1}^{\infty} C_n \sin(\omega_n t - \alpha_n) \]
where
\[ C_n = \dfrac{b_n}{\sqrt{(k - m\omega_n^2)^2 + (c\omega_n)^2}}. \]
In both cases, \( \alpha_n \) represents the phase lag of the response relative to the driving force
\[ \tan(\alpha_n) = \left( \frac{c\omega_n}{k - m\omega_n^2} \right), \quad 0 \le \alpha_n \le \pi \]

In each of the following problems, the values of \(m\), \(c\), and \(k\) for a damped mass–spring system are given. Find the steady periodic motion \(x_{sp}(t)\) of the mass under the influence of the given external force \(F(t)\). Compute the coefficients and phase angles for the first three nonzero terms in the series for \(x_{sp}(t)\).

Problem 1

\[ m x'' + c x' + kx = F(t) \]

\(m=1,\; c=0.1,\; k=4\), where \(F(t)\) is periodic with period \(2\pi\) and

\[ F(t) = \begin{cases} 3, & 0 < t < \pi, \\ -3, & \pi < t < 2\pi. \end{cases} \]

Problem 2

\[ m x'' + c x' + kx = F(t) \]

\(m=2,\; c=0.1,\; k=18\), where \(F(t)\) is an odd function with period \(2\pi\) and

\[ F(t) = 2t, \quad 0 < t < \pi. \]

Problem 3

\[ m x'' + c x' + kx = F(t) \]

\(m=3,\; c=1,\; k=30\), where \(F(t)\) is an odd function with period \(2\) and

\[ F(t) = t - t^2, \quad 0 < t < 1. \]

Problem 4

\[ m x'' + c x' + kx = F(t) \]

\(m=1,\; c=0.01,\; k=4\), where \(F(t)\) is an even function with period \(4\) and

\[ F(t) = 2t, \quad 0 < t < 2. \]