4.7

Math 3331 Differential Equations4.7 Forced Harmonic Motion

Jiwen He

Department of Mathematics, University of Houston

jiwenhe@math.uh.edumath.uh.edu/∼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.

Jiwen He, University of Houston Math 3331 Differential Equations Summer, 2014 6 / 14

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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