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4.7
Math 3331 Differential Equations4.7 Forced Harmonic Motion
Jiwen He
Department of Mathematics, University of Houston
[email protected]/∼jiwenhe/math3331
Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 1 / 14
4.7 Undamped Case Damped Case
4.7 Forced Harmonic Motion
Periodically Forced Harmonic Motion
Forced Undamped Harmonic Motion: Beats
Forced Undamped Harmonic Motion: Resonance
Forced Damped Harmonic Motion
Amplitude and Phase
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4.7 Undamped Case Damped Case
Periodically Forced Harmonic Motion
Sinusoidal forcing: F (t) = A cosωt
where A is the amplitude and ω isthe driving frequency.
General solution:x(t) = xh(t) + xp(t)
where
xp(t): steady state part(persistent oscillation)
xh(t): transient part (d > 0 )(xh(t)→ 0 for t →∞)
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4.7 Undamped Case Damped Case
Forced Undamped Harmonic Motion: Beats (ω 6= ω0)
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4.7 Undamped Case Damped Case
Beats: ω 6= ω0
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4.7 Undamped Case Damped Case
Beats in Forced, Undamped, Harmonic Motion
In acoustics, a beat is an
interference between two sounds of
slightly different frequencies,
perceived as periodic variations in
volume whose rate is the difference
between the two frequencies.
x(t) = coswt − cosw0t = 2 sin δt sin ω̄t
where the mean frequency ω̄ and the half difference δ are definedby
ω̄ = (ω0 + ω)/2, δ = (ω0 − ω)/2.
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4.7 Undamped Case Damped Case
Forced Undamped Harmonic Motion: Resonance (ω = ω0)
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4.7 Undamped Case Damped Case
Resonance in Forced, Undamped, Harmonic Motion
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4.7 Undamped Case Damped Case
Forced Damped Harmonic Motion
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4.7 Undamped Case Damped Case
Particular Solution of (7)
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4.7 Undamped Case Damped Case
Amplitude and Phase
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4.7 Undamped Case Damped Case
Solution of (6)
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4.7 Undamped Case Damped Case
Qualitative Forms
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4.7 Undamped Case Damped Case
Example 4.7.18
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4.7 Undamped Case Damped Case
Example 4.7.18 (cont.)
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