Unit 7 · Topic 10

Damping, driving, and resonance

Dissipation and driving separate natural, damped, and driving frequencies.

Unit 7Damping and resonanceOpen navigation

Concept 01

Loss of mechanical amplitude

Essential The minimum you should retain

m x''+b x'+kx=0 models a dissipative force proportional to velocity.

UnderstandInterpret and connect

With ω₀=sqrt(k/m) and γ=b/(2m), the underdamped regime requires γ<ω₀.

DeepenFormulation and conditions

Then ω_d=sqrt(ω₀²−γ²) and the envelope is A₀e^(−γt).

ExploreConnections for further study

Mechanical energy transfers to other forms; it does not vanish.

Mathematical relation

Damped oscillator

ω0=km,γ=b2m,ωd=ω₀2−γ2 \omega_0=\sqrt{k/m},\ \gamma=b/(2m),\ \omega_d=\sqrt{\omega_0^2-\gamma^2},\ x=A_0e^{-\gamma t}\cos(\omega_dt+\phi)
Represents

Natural frequency, decay rate, and damped frequency.

Physical interpretation

The A₀e^(−γt) envelope decays without destroying total energy.

DeepenVariables, conditions, and checks

Variables

γ
decay constant; usual unit: s⁻¹
ω_d
damped angular frequency; usual unit: rad/s

Conditions of application

  • Underdamped regime γ<ω₀.

Dimensional check

s⁻¹ and rad/s.

Concept 02

Oscillating or nonoscillating return

Essential The minimum you should retain

Underdamped, critical, and overdamped describe qualitatively different responses.

UnderstandInterpret and connect

Only the underdamped case has real ω_d and decaying oscillations.

DeepenFormulation and conditions

Critical damping separates oscillatory from nonoscillatory return.

ExploreConnections for further study

Complete algebra for every regime lies outside the core.

Concept 03

An external frequency

Essential The minimum you should retain

m x''+b x'+kx=F₀cos(Ωt) uses Ω for the driving frequency.

UnderstandInterpret and connect

After transients, A(Ω)=F₀/sqrt[(k−mΩ²)²+(bΩ)²].

DeepenFormulation and conditions

With b>0 the denominator keeps amplitude finite.

ExploreConnections for further study

Ω must not be confused with damped frequency ω_d.

Worked example

Steady amplitude of a driven oscillator

m=1.0 kg, k=100 N/m, b=4.0 kg/s, F₀=10 N, and Ω=10 rad/s.

Given
  • m=1.0 kg
  • k=100 N/m
  • b=4.0 kg/s
  • F₀=10 N
  • Ω=10 rad/s
Target

Find A(Ω).

  1. Model

    A=F₀/sqrt[(k−mΩ²)²+(bΩ)²].

  2. Reactive term

    k−mΩ²=0.

  3. Dissipative term

    bΩ=40.

  4. Result

    A=10/40=0.25 m.

Conclusion

The steady amplitude is finite: 0.25 m.

Mathematical relation

Steady driven response

mx''+bx'+kx=F0cos(Ωt),A(Ω)=F0k−mΩ²2+bΩ2 mx''+bx'+kx=F_0\cos(\Omega t),\quad A(\Omega)=F_0/\sqrt{(k-m\Omega^2)^2+(b\Omega)^2}
Represents

Steady amplitude of a damped linear oscillator.

Physical interpretation

The peak is finite and its maximum need not occur exactly at ω₀.

DeepenVariables, conditions, and checks

Variables

Ω
driving angular frequency; usual unit: rad/s
F₀
external-force amplitude; usual unit: N

Conditions of application

  • b>0 for the parameterized family; steady linear response.

Dimensional check

m.

Finite A(Ω) for b>0

Normalized curves compare the time envelope and response for two damping values.finite peaks
  • envelope e^(−γt)
  • small b
  • larger b

The free response decays and the driven peak remains finite; more damping lowers and broadens it.

Concept 04

A finite, broad peak

Essential The minimum you should retain

Resonance is a large response near the system's natural-response region.

UnderstandInterpret and connect

Increasing b lowers and broadens the peak.

DeepenFormulation and conditions

With damping the amplitude maximum need not occur exactly at ω₀.

ExploreConnections for further study

Designing for or against resonance requires amplitude, phase, and dissipation.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Saying damping destroys energy.

It reduces mechanical energy and transfers it to other forms.

Claiming resonance always means infinite amplitude.

For b>0, steady amplitude is finite and the peak depends on b.