Damped oscillations | PhysicsRank

A door closer should shut a door without letting it slam or bounce back and forth. Too little resistance allows oscillation; too much can make the return sluggish. That ordinary design problem introduces a useful distinction between energy storage, energy loss and external driving.

This chapter connects force equations, free-decay measurements, phase and power. You will distinguish the natural frequency, the damped frequency and the frequency where a chosen observable has its maximum.

What you will learn to do

Build a linear damping model from a force diagram and an energy balance.
Use two initial conditions to obtain a correct free response.
Distinguish underdamped, critical and overdamped recovery.
Infer decay rate from repeated peaks without confusing amplitude and energy.
Explain steady-state amplitude and phase from competing forces.
Identify displacement, velocity and power maxima separately.
Interpret half-power bandwidth and the limits of Q approximations.

Useful before this chapterOscillations: restoring forces, phase and energy →Rates and accumulation: calculus for physics →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

In an ideal spring oscillator, where is energy stored as the mass moves through equilibrium? What new destination appears when drag is present?

Compare your reasoning

At equilibrium the ideal oscillator’s energy is kinetic. With dissipative drag, some mechanical energy is transferred to the surroundings, so the amplitude can shrink. The spring still provides the restoring effect.

Revisit the explanation →

01 / Damping, driving and the meaning of resonance

Begin with a force that removes energy

A spring oscillator is useful because it separates two jobs. The spring stores energy and returns it; a resisting interaction transfers some of that energy to the surroundings. The mass may cross equilibrium many times while its motion shrinks. Calling this motion “slower SHM” misses both the changing amplitude and the physical destination of the energy.

We measure displacement x from a stable equilibrium and assume a linear spring, a fixed mass and a resistive force −bv. The coefficient b is positive and has units kg/s. This is a model for approximately velocity-proportional resistance over the speeds considered. Dry sliding friction, turbulent drag and an actively controlled force have different laws and need different equations.

Draw the forces before writing the equation: the spring contributes −kx, the resistance contributes −b dx/dt, and an optional external driver contributes F(t). A constant force such as gravity can be absorbed into the equilibrium position if it does not change the stiffness. Every term below has units of force.

m\ddot x+b\dot x+kx=F(t),\qquad \omega_0=\sqrt{k/m},\quad \gamma=\frac{b}{2m}

Dots mean time derivatives. The undamped natural angular frequency is ω₀; γ is an amplitude decay rate, measured in s⁻¹. Convert an angular oscillation frequency to hertz by dividing by 2π. The decay rate γ is not an oscillation frequency.

WORKED EXAMPLE

Find the energy destination before solving the motion

An unforced oscillator has m = 1 kg, k = 25 N/m and b = 2 kg/s. At an instant its speed is 0.40 m/s. How quickly is mechanical energy being lost?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: Begin with a force that removes energy
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
m\ddot x+b\dot x+kx=F(t),\qquad \omega_0=\sqrt{k/m},\quad \gamma=\frac{b}{2m}
Meaning: Dots mean time derivatives. The undamped natural angular frequency is ω₀; γ is an amplitude decay rate, measured in s⁻¹. Convert an angular oscillation frequency to hertz by dividing by 2π. The decay rate γ is not an oscillation frequency.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Damping, driving and the meaning of resonance

Distinguish decay rate, oscillation rate and initial conditions

Set F(t) = 0. Try a motion proportional to exp(st): differentiation multiplies it by s, so the differential equation becomes a quadratic for s. The roots decide whether the motion oscillates. For γ < ω₀ the roots have a negative real part and an imaginary part; decay and oscillation occur together.

The underdamped solution can be written as an exponentially shrinking sine/cosine combination. Its angular oscillation rate is ωd, slightly below ω₀. The multiplying exponential changes amplitude, so this is not strict simple harmonic motion with constant amplitude.

Two initial conditions are still needed. “Released from rest at x₀” means both x(0) = x₀ and v(0) = 0. The tempting guess x₀ exp(−γt) cos(ωdt) has initial velocity −γx₀ and therefore does not describe that release.

s^2+2\gamma s+\omega_0^2=0,\quad \omega_d=\sqrt{\omega_0^2-\gamma^2},\quad x=e^{-\gamma t}(A\cos\omega_dt+B\sin\omega_dt)

A and B are set by the actual starting state. The oscillation period is 2π/ωd; 1/γ is the amplitude-envelope e-folding time, not the period.

WORKED EXAMPLE

Make the release condition survive substitution

The oscillator above is released from rest at x₀ = 0.10 m. Write its underdamped motion and test its initial acceleration.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: Distinguish decay rate, oscillation rate and initial conditions
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
s^2+2\gamma s+\omega_0^2=0,\quad \omega_d=\sqrt{\omega_0^2-\gamma^2},\quad x=e^{-\gamma t}(A\cos\omega_dt+B\sin\omega_dt)
Meaning: A and B are set by the actual starting state. The oscillation period is 2π/ωd; 1/γ is the amplitude-envelope e-folding time, not the period.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Damping, driving and the meaning of resonance

A stronger resistance can remove the oscillation

The discriminant γ² − ω₀² divides the free response into three regimes. Under damping produces repeated crossings; critical damping gives a repeated real root; over damping gives two distinct negative real roots. The latter two have no sustained sinusoidal crossings, although an initial velocity can still carry the mass across equilibrium once.

Critical damping is often described as the fastest return without oscillation. Read that statement with its conditions: it compares passive linear systems with the same m and k, typically released from displacement with zero initial velocity. A claim about a settling tolerance or a different starting velocity needs its own comparison.

Increasing b far beyond critical does not force the body to settle instantly. One root approaches −k/b, giving a very slow relaxation. Strong resistance can hold the mass away from equilibrium for longer.

Free linear response with fixed m and k
Regime	Condition	Form of the response
Underdamped	b² < 4mk	Decaying sinusoidal motion; ωd is real
Critical	b² = 4mk	(A + Bt) exp(−ω₀t)
Overdamped	b² > 4mk	Sum of two decaying real exponentials

WORKED EXAMPLE

Classify a system without confusing b with γ

A 0.50 kg mass has k = 8 N/m. Compare b = 2, 4 and 8 kg/s.

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STOP & THINK

An overdamped mass is released from rest away from equilibrium. Which claim is justified by the linear model?

Its displacement decays without repeated oscillations
Its acceleration is zero at all times
Its energy cannot decrease

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: A stronger resistance can remove the oscillation
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / Damping, driving and the meaning of resonance

Infer damping from comparable points on successive cycles

Take two maxima on the same side of equilibrium, separated by n damped periods. Their phases match, so the common sinusoidal factor cancels from their ratio. This is why repeated peak measurements are more useful than two arbitrary displacements.

The logarithmic decrement per cycle is ln(Aj/Aj+1) = γTd. A plot of the logarithm of same-side peak magnitude against time is a straight line of slope −γ for this model. Curvature or a nearly fixed loss of amplitude per cycle suggests the simple velocity-proportional damping model may be inadequate.

At a turning point all mechanical energy is spring energy. Therefore same-side peak energies have exactly the square of the displacement-peak ratio. Between peaks, instantaneous mechanical energy is not generally one pure exponential; its derivative remains exactly −bv². Cycle-averaged exponential energy decay is a weak-damping approximation.

WORKED EXAMPLE

Separate an amplitude lifetime from an energy lifetime

Same-side displacement peaks fall from 4.0 cm to 1.0 cm over three cycles lasting 6.0 s. Find γ and the ratio of the corresponding peak energies.

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STOP & THINK

Comparable amplitude peaks fall by a factor of e over 5.0 s. What is γ in s⁻¹?

Answer in s⁻¹.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: Infer damping from comparable points on successive cycles
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Damping, driving and the meaning of resonance

A sustained response needs an energy supply and a phase

Now drive the mass with F₀ cos(ωt). The full motion contains a free transient fixed by the starting conditions and a forced response at the driver frequency. With b > 0 the transient dies away; the equations below describe the settled response, not the instant the driver is switched on.

Use x = X cos(ωt − φ). The acceleration opposes displacement; the damping force opposes velocity, which is a quarter-cycle out of step with displacement. Matching the in-phase and quadrature parts produces both an amplitude and a phase. Adding the magnitudes kX, mω²X and bωX as ordinary positive numbers loses that phase information.

At very low frequency the spring follows the changing force almost statically, X ≈ F₀/k and φ ≈ 0. At high frequency inertia dominates, X ≈ F₀/(mω²) and displacement is almost opposite to the force. Near ω₀ the stiffness and inertia contributions cancel, leaving the resistance to limit the response.

X=\frac{F_0}{\sqrt{(k-m\omega^2)^2+b^2\omega^2}},\qquad \phi=\operatorname{atan2}(b\omega,k-m\omega^2)

Choose the phase in the interval 0 to π for positive damping and positive frequency. A one-argument inverse tangent can incorrectly put the phase in the wrong quadrant.

WORKED EXAMPLE

Explain why the motion can be large while the spring is stiff

Use m = 1 kg, k = 25 N/m, b = 2 kg/s and F₀ = 3 N at ω = 5 rad/s. Find the settled displacement amplitude, velocity amplitude and mean input power.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: A sustained response needs an energy supply and a phase
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
X=\frac{F_0}{\sqrt{(k-m\omega^2)^2+b^2\omega^2}},\qquad \phi=\operatorname{atan2}(b\omega,k-m\omega^2)
Meaning: Choose the phase in the interval 0 to π for positive damping and positive frequency. A one-argument inverse tangent can incorrectly put the phase in the wrong quadrant.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Damping, driving and the meaning of resonance

Ask which measured quantity has the peak

“The resonant frequency” is incomplete until a response quantity and driving condition are named. For a fixed-amplitude sinusoidal force and constant b, the displacement maximum occurs below ω₀ if damping is small enough. The velocity-amplitude and mean-power maxima occur at ω₀. They are different measurements of the same oscillator.

Differentiate the squared displacement denominator with respect to ω. Its nonzero minimum gives ωr² = ω₀² − 2γ². If γ ≥ ω₀/√2, there is no nonzero-frequency displacement peak: displacement decreases from its static value. The mass may still be underdamped in free motion for part of this range.

For velocity, multiply displacement amplitude by ω before locating the maximum. For mean power, use bω²X²/2. These operations change the frequency dependence, so a maximum of X is not automatically a maximum of its time derivative or of dissipated energy.

\omega_{X\max}=\sqrt{\omega_0^2-2\gamma^2}\quad(\gamma<\omega_0/\sqrt2),\qquad \omega_{v\max}=\omega_{P\max}=\omega_0

These results hold for a linear oscillator with frequency-independent m, k, b and a fixed force amplitude. A driver that maintains displacement or changes force with frequency is a different experiment.

INTERACTIVE LAB

Compare displacement, velocity and absorbed power

Keep the damping ratio fixed and sweep the driving frequency. Find the displacement peak, then compare it with the frequency where velocity and absorbed power peak. Increase damping and explain which peak disappears.

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WORKED EXAMPLE

An underdamped system need not have a displacement resonance peak

Two systems have the same ω₀ = 10 rad/s but γ = 2 s⁻¹ and γ = 8 s⁻¹. Compare their free motion and driven displacement peaks.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: Ask which measured quantity has the peak
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
\omega_{X\max}=\sqrt{\omega_0^2-2\gamma^2}\quad(\gamma<\omega_0/\sqrt2),\qquad \omega_{v\max}=\omega_{P\max}=\omega_0
Meaning: These results hold for a linear oscillator with frequency-independent m, k, b and a fixed force amplitude. A driver that maintains displacement or changes force with frequency is a different experiment.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

07 / Damping, driving and the meaning of resonance

Use a response curve to estimate loss, with the right definition

At ω₀, the average absorbed power is F₀²/(2b). At the two half-power angular frequencies, the reactive force mismatch has magnitude bω. Solving those two conditions gives an exact angular-frequency width Δω = b/m = 2γ for this model.

Define Q = ω₀/Δω = mω₀/b. A larger Q gives a narrower power peak. “Half-power” means power has fallen to one half; a velocity-amplitude plot reaches that level at 1/√2 of its maximum, not at half its maximum. Displacement half-height is another width and must not be silently substituted.

For weak damping Q is also approximately 2π times stored energy divided by energy dissipated in one cycle. The approximation assumes energy changes little during the cycle. Do not use that weak-loss argument to infer an exact large-loss energy ratio.

WORKED EXAMPLE

Infer damping from a measured power peak

A fixed-force oscillator with mass 0.20 kg has a power peak at 8.0 rad/s and half-power frequencies separated by 2.0 rad/s. Find b, γ and Q.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Damping, driving and the meaning of resonance
Focus: Use a response curve to estimate loss, with the right definition
Learning goals:
- Build a linear damping model from a force diagram and an energy balance.
- Use two initial conditions to obtain a correct free response.
- Distinguish underdamped, critical and overdamped recovery.
- Infer decay rate from repeated peaks without confusing amplitude and energy.
- Explain steady-state amplitude and phase from competing forces.
- Identify displacement, velocity and power maxima separately.
- Interpret half-power bandwidth and the limits of Q approximations.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Resistance removes energy at the rate bv²; it does not merely reduce the frequency.
An initial displacement and an initial velocity determine the free response together.
The same oscillator has different response maxima when different quantities are measured.
State the damping law, drive condition and approximation before using a resonance formula.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

For m = 2 kg and k = 18 N/m, what is the critical b in kg/s?

Answer in kg/s.

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SELF CHECK 2

A settled sinusoidal speed has amplitude 0.80 m/s and b = 0.50 kg/s. What is mean dissipated power?

Answer in W.

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SELF CHECK 3

A displacement resonance maximum is observed below ω₀. Where is the velocity-amplitude maximum for the same constant-force linear model?

At the same lower frequency necessarily
At ω₀
At zero frequency

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SELF CHECK 4

Which energy statement is exact for unforced linear viscous damping?

Instantaneous energy always equals E₀ exp(−2γt)
dE/dt = −bv²
The speed stays constant while amplitude decays

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

Infer damping from peak energiesAdvanced · Free sample · Hints and guided reasoning Which frequency maximizes displacement?Advanced · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

Comparable displacement peaks halve. What happens to their spring energies, and why must the compared phases match?

Compare your reasoning

Peak spring energies become one quarter because energy is proportional to displacement squared. Matching phases isolates the decay envelope instead of confusing it with the oscillating part of the motion.

Revisit the explanation →

A few days later: change the situation

Can one driven oscillator have different frequencies for maximum displacement and maximum absorbed power?

Compare your reasoning

Yes, with fixed force amplitude and positive constant viscous damping. Power and velocity peak at the natural frequency, while a nonzero displacement peak is shifted lower and can disappear at sufficiently large damping. Name the observable first.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑