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ResonanceIntermediateStarter track

Concept module

Damping / Resonance

Explore how damping removes energy, how driving frequency changes amplitude, and why resonance becomes dramatic near the natural frequency.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 10.0 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s10.0 s

Damping / Resonance

Compare the decay of a damped oscillator with the response near resonance.

Transient mode

Graphs

Switch graph views without breaking the live stage and time link.

Damped motion

Shows the motion fading when damping is stronger.

time (s): 0 to 10relative displacement (a.u.): -2.2 to 2.2

Transient displacementEnvelope

Hover or scrub to link the graph back to the stage.time (s) / relative displacement (a.u.)

Controls

Adjust the physical parameters and watch the motion respond.

Damping ratio

β

0.12

Controls how quickly energy is lost from the motion.

Natural frequency

ω_{0}

2 rad/s

Sets the system's preferred oscillation rate.

Driving frequency

ω_{d}

1.85 rad/s

Controls how quickly the external force pushes the system.

Drive amplitude

F_{0}

1 a.u.

Controls how strong the driver is.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

Resonance responseSwitch between a damped time-history view and a frequency-response view.

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 1

Notice that damping removes energy, so the motion envelope shrinks even when the oscillator is still cycling.

Try this

Stay in the damped-motion view and compare the first swings with the later ones instead of looking only at one instant.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

β

Damping ratio

0.12

Controls how fast energy is removed, which shortens the transient and lowers the resonance peak.

Graph: Damped motionGraph: Amplitude responseOverlay: Response envelope

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Treat the prompt as a live cue about the current representation. Some cues belong to the time trace, while others belong to the response curve.

ObservationPrompt 1 of 1

Graph: Damped motion

Notice that damping removes energy, so the motion envelope shrinks even when the oscillator is still cycling.

Try this

Stay in the damped-motion view and compare the first swings with the later ones instead of looking only at one instant.

Why it matters

This is the difference between a repeating oscillation and one that is gradually losing energy.

Control: Damping ratioGraph: Damped motionOverlay: Response envelope

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Driving force

Shows the external driver feeding energy into the system.

What to notice

The arrow follows the chosen drive, not the natural motion.

Why it matters

It separates the source of motion from the response.

Control: Driving frequencyControl: Drive amplitudeGraph: Damped motionGraph: Amplitude responseEquation

ω_{d}

Equation

F_{0}

Challenge mode

Use the same response curve and inspect-time tools to tune a driven oscillator instead of passively watching it.

0/2 solved

ConditionStretch

1 of 5 checks

Late small motion

From Free swing, make the transient decay quickly enough that a late inspected sample shows only a very small displacement.

Inspect timeGraph-linkedGuided start2 hints

Suggested start

Raise damping, then pause later in the motion to inspect the remaining amplitude.

Matched

Keep the damped-motion graph active.

Damped motion

Pending

Pause into inspect mode.

live

Pending

Inspect a late sample between

4

and

6 s

0 s

Pending

Raise damping ratio into the

0.4

0.6

band.

0.12

Pending

Make the remaining displacement amplitude smaller than

0.15

1 a.u.

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

The system is in transient decay. The driving frequency is 0.93 times the natural frequency. At t = 0 s the relative displacement is 1 a.u., and the predicted steady-state response amplitude is 1.37 a.u..

Equation detailsDeeper interpretation, notes, and worked variable context.

Driven damped oscillator

The restoring force, damping force, and external driver act on the same system.

m \overset{x}{¨} + b \overset{x}{˙} + k x = F_{0} sin (ω_{d} t)

The damping term removes energy.

The driving term adds energy at a chosen frequency.

β

Damping ratio 0.12

ω_{d}

Driving frequency 1.85 rad/s

F_{0}

Drive amplitude 1 a.u.

Natural frequency

The frequency the system prefers when no driver is forcing it.

ω_{0} = \frac{k}{m}

This is the reference frequency for resonance.

ω_{0}

Natural frequency 2 rad/s

Response amplitude

A simplified amplitude curve that peaks near the natural frequency and depends on damping.

A (ω_{d}) \approx \frac{F _{0} / m}{( ω _{0}^{2} - ω _{d}^{2} ) ^{2} + ( 2 β ω _{d} ) ^{2}}

Larger damping lowers the peak and broadens the response.

β

Damping ratio 0.12

ω_{0}

Natural frequency 2 rad/s

ω_{d}

Driving frequency 1.85 rad/s

F_{0}

Drive amplitude 1 a.u.

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 3 of 30 / 3 complete

Oscillations and Energy

Earlier steps still set up Damping / Resonance.

1. Simple Harmonic Motion2. Oscillation Energy3. Damping / Resonance

Previous step: Oscillation Energy.

Start from track beginning

Short explanation

What the system is doing

Damping and resonance are the two forces that make oscillators feel realistic. Damping drains motion away, while a driver can keep feeding energy into the system and build a larger response.

This module keeps both stories visible. In one mode you watch the motion fade. In the other you sweep driving frequency and see the response rise and fall around resonance.

Key ideas

01Higher damping makes the transient motion die out more quickly.

02The strongest response happens when the driving frequency is close to the natural frequency.

03A little damping softens the resonance peak instead of removing it entirely.

Live worked example

Solve the exact state on screen.

Use the current drive settings and, when relevant, the current inspected time. The math follows the same mode and parameters as the live oscillator.

Live valuesLive at t = 0.00 s

At $t = 0 s$ , what displacement does the oscillator have in the current mode?

t

Time

0 s

β

Damping ratio

0.12

ω_{0}

Natural frequency

2 rad/s

ω_{d}

Driving frequency

1.85 rad/s

1. Identify the current-mode relation

In transient decay mode, use

x (t) = F_{0} e^{- β ω_{0} t} cos (ω_{d} t + ϕ)

2. Substitute the current values

x (0) = 1 e^{- 0.12 \cdot 2 \cdot 0} cos (1.85 \cdot 0 + 0)

3. Compute the displacement

That gives

x = 1

in the current mode.

Current displacement

x (0) = 1

The transient view includes the decaying envelope, so the same time relation gives smaller displacements as damping removes energy.

Common misconception

Resonance always means an unlimited amplitude spike.

Real systems lose energy, so damping keeps the amplitude finite.

The exact shape of the peak depends on how strongly the system is damped.

Mini challenge

If you increase damping but keep the drive frequency unchanged, what happens to the resonance peak?

Prediction prompt

Predict whether the peak gets taller, wider, or both.

Check your reasoning

The peak gets shorter and broader.

Damping removes energy more quickly, so the system cannot build as much amplitude. The response also becomes less selective about the exact driving frequency.

Quick test

Compare cases

Question 1 of 4

Pick the explanation that best matches the motion, the transient, or the response curve. The point is to reason from the physics, not just spot a definition.

Two response curves have the same natural frequency and drive amplitude. One peak is lower and broader. What most likely changed?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation alternates between two views of the same system: a time-based damped motion and a frequency-response view that sweeps the driver across different frequencies.

The controls make the physical meaning explicit so the user can tell whether they are changing damping, the natural frequency, or the strength and rate of the external driver.

Graph summary

The transient graph shows how quickly the motion settles after energy is removed by damping.

The response graph shows how close the driver is to resonance and how damping changes the shape of the peak.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Useful comparisonMechanics

Damping / Resonance

See each variable before you move it.

Driving force

Late small motion

Oscillations and Energy

What the system is doing

Solve the exact state on screen.

At t=0s, what displacement does the oscillator have in the current mode?

Two response curves have the same natural frequency and drive amplitude. One peak is lower and broader. What most likely changed?

Keep moving through the concept map.

Projectile Motion

Simple Harmonic Motion

Oscillation Energy

At $t = 0 s$ , what displacement does the oscillator have in the current mode?