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9.4 Application of Fourier Series

\[ mx'' + cx' + kx = F(t) \]
  • m: mass
  • c: damping constant
  • k: spring constant

Solution Structure

\[ x(t) = x_c(t) + x_p(t) \]
  • \( x_c(t) \): Complementary (homogeneous part)
  • \( x_p(t) \): particular (due to \( F(t) \))

Case: Sinusoidal Forcing Function

If \( F(t) = F_0 \sin(\omega t) \) and \( \omega \neq \sqrt{\frac{k}{m}} \), then:

\[ x_p = A \cos(\omega t) + B \sin(\omega t) \]

Case: General Periodic Forcing Function

If \( F(t) \) is periodic but not necessarily sine/cosine:

→ expand into sine and/or cosine series
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Example Problem

For example, \( c=0, m=1, k=5 \)

\[ F(t) = \begin{cases} 1 & 0 < t < 3 \\ -1 & 3 < t < 6 \end{cases} \quad \text{period } 6 \]
Coordinate graph of a square wave function F(t) with period 6, oscillating between 1 and -1.

Origin Symmetry

\( F(t) \) is a sine series \( (a_n = 0, b_n = \frac{2}{L} \int_0^L F(t) \sin(\frac{n \pi t}{L}) dt) \)

Fourier Expansion

\[ F(t) = \sum_{n=1}^{\infty} \frac{2}{n\pi} [1 - (-1)^n] \sin\left(\frac{n\pi t}{3}\right) \] \[ = \frac{4}{\pi} \sin\left(\frac{\pi t}{3}\right) + \frac{4}{3\pi} \sin\left(\frac{3\pi t}{3}\right) + \frac{4}{5\pi} \sin\left(\frac{5\pi t}{3}\right) + \dots \]

Differential Equation Application

\[ x'' + 5x = F(t) \]
Each term results in its own particular solution.
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Particular and General Solutions for Differential Equations

In the form of \( A_n \cos\left(\frac{n\pi t}{3}\right) + B_n \sin\left(\frac{n\pi t}{3}\right) \), then the entire particular solution is:

\[ x_p = \sum_{n=1}^{\infty} A_n \cos\left(\frac{n\pi t}{3}\right) + B_n \sin\left(\frac{n\pi t}{3}\right) \]

It must satisfy:

\[ x'' + 5x = F(t) = \sum_{n=1}^{\infty} \frac{2}{n\pi} [1 - (-1)^n] \sin\left(\frac{n\pi t}{3}\right) \]

Sub \( x_p \) into the eq. above:

. . .

\( A_n = 0 \) for all \( n \)

\[ B_n = \frac{18 [1 - (-1)^n]}{n\pi (45 - n^2\pi^2)} \]

General Solution

\[ x(t) = \underbrace{C_1 \cos(\sqrt{5}t) + C_2 \sin(\sqrt{5}t)}_{x_c} + \underbrace{\sum_{n=1}^{\infty} \frac{18 [1 - (-1)^n]}{n\pi (45 - n^2\pi^2)} \sin\left(\frac{n\pi t}{3}\right)}_{x_p \text{ (steady periodic solution)}} \]
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\( 45 - n^2\pi^2 \neq 0 \) for \( n = 1, 2, 3, \dots \)

but if \( n = 2 \), it is close to zero because the input frequency \( \frac{n\pi}{3} = \frac{2\pi}{3} \approx 2.094 \) is close to the natural freq. \( \sqrt{\frac{k}{m}} = \sqrt{5} \approx 2.236 \).

(near resonance)

Can be a problem, but we are lucky here because

\[ B_n = \frac{18 [1 - (-1)^n]}{n\pi (45 - n^2\pi^2)} \text{ is } 0 \text{ if } n = 2 \quad ([1 - (-1)^n] = 0) \]

if we had

\[ x'' + 5x = \begin{cases} 1 & 0 < t < 10 \\ -1 & 10 < t < 20 \end{cases} \quad \text{period } 20 \]
. . .
\[ x_p = \sum_{n=1}^{\infty} \frac{200 [1 - (-1)^n]}{n\pi (500 - n^2\pi^2)} \sin\left(\frac{n\pi t}{10}\right) \]

\( 500 - n^2\pi^2 \) is close to \( 0 \) if \( n = 7 \)

\( [1 - (-1)^n] \) if \( n = 7 \) is NOT zero \( \rightarrow \) near resonance is visible

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Mass-Spring Response to \( F(t) = \pm 1 \)

Differential Equation: \( x'' + 5x = F(t) \)

Plot of amplitude vs time showing square wave forcing and the resulting natural, forced, and total responses.

Legend

  • Forcing \( F(t) \) (Square Wave)
  • Homogeneous \( x_c(t) \) (Natural Response)
  • Steady Periodic \( x_p(t) \) (Forced Response)
  • General Solution \( x(t) \) (Total)

The graph illustrates the interaction between the natural frequency of the system and the external square wave forcing. The total solution \( x(t) \) is the sum of the homogeneous solution (natural ringing) and the particular solution (steady periodic response).

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Mass-Spring Response with \( T = 20 \): Visualizing Near-Resonance

Dominant \( n = 7 \) harmonic due to near-resonance \( (\omega_7 \approx \omega_0) \)

Amplitude vs time plot showing high-frequency oscillations amplified by near-resonance with the 7th harmonic.

Legend

  • Forcing \( F(t) \) (Input)
  • Homogeneous \( x_c(t) \) (Natural Ringing)
  • Steady Periodic \( x_p(t) \) (Forced Response)
  • General Solution \( x(t) \) (Total Motion)

This visualization highlights the phenomenon of near-resonance. When a specific harmonic of the forcing function (in this case, the 7th harmonic) aligns closely with the system's natural frequency, that component's amplitude is significantly amplified in the steady periodic response.

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Resonance in Differential Equations

now look at

\[ x'' + 9x = \begin{cases} 1 & 0 < t < \pi \\ -1 & \pi < t < 2\pi \end{cases} \quad \text{period } 2\pi \]
\( \vdots \)
\[ x_p = \sum_{n=1}^{\infty} \frac{2[1-(-1)^n]}{n(9-n^2)} \sin(nt) \]

\( n=3 \) makes \( 9-n^2 = 0 \) exactly

part of \( F(t) \) hits resonance freq. exactly (pure resonance)

the series solution \( x_p \) is not valid when \( n=3 \)

so, we need to handle that term separately

\[ x_p = At \cos(3t) + Bt \sin(3t) \]
\( \vdots \)
\[ A = -\frac{2}{3} \quad B = 0 \]
\[ x_p = \underbrace{-\frac{2}{3} t \cos(3t)}_{n=3 \text{ only}} + \sum_{\substack{n=1 \\ n \neq 3}}^{\infty} \frac{2[1-(-1)^n]}{n(9-n^2)} \sin(nt) \]
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Pure Resonance vs. Damped Response: \( x'' + cx' + 9x = F(t) \)

Graph comparing undamped pure resonance (blue) with linear growth and damped steady state (orange) over time.
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Damped Forced Oscillations

If \( c \neq 0 \), we consider the differential equation for a damped forced system:

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

Where the forcing function is given by a Fourier sine series:

\[ F(t) = \sum_{n=1}^{\infty} F_n \sin\left(\frac{n\pi t}{L}\right) \]

The particular solution \( x_p \) is assumed to be of the form:

\[ x_p = \sum_{n=1}^{\infty} A_n \cos\left(\frac{n\pi t}{L}\right) + B_n \sin\left(\frac{n\pi t}{L}\right) \]
\(\vdots\)
\[ x_p = \sum_{n=1}^{\infty} \frac{F_n}{\sqrt{(k - m\omega_n^2)^2 + (c\omega_n)^2}} \sin\left(\frac{n\pi t}{L} - \tan^{-1}\left(\frac{c\omega_n}{k - m\omega_n^2}\right)\right) \]

Definitions:

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

Note: if \( c \neq 0 \), the denominator \( \neq 0 \).

Observations:

  • The term \( \tan^{-1}\left(\frac{c\omega_n}{k - m\omega_n^2}\right) \) represents the phase shift.
  • The particular solution lags behind the complementary solution.
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Damped Response at Resonance

System Parameters: \( c = 0.5 \)

\[ x'' + 0.5x' + 9x = F(t) \]
A plot of Amplitude vs Time (t) showing four curves: a dashed forcing function, a decaying green transient, a yellow steady-state sine wave, and a thick blue total solution curve that converges to the yellow curve.

Transient Behavior

The transient "ringing" \( (x_c) \) decays over time due to damping in the system.

Steady-State Convergence

The total motion \( (x) \) eventually converges to the steady periodic solution \( (x_p) \).

Graph Legend

  • Forcing \( F(t) \)
  • Homogeneous \( x_c(t) \) (Transient)
  • Steady Periodic \( x_p(t) \) (Steady-State)
  • General Solution \( x(t) \) (Total)